Research and Development

**Chapter 4**

**Abstract**

**1. Introduction**

**67**

Accuracy

*Peter Demeč and Tomáš Stejskal*

in the development of new machine designs.

numerical experiment, static stiffness measurement

**Keywords:** machine tool, virtual prototyping, virtual machining,

Reducing the development time of modern machines is still a topical issue. Manufacturing experience is not sufficient to predict the characteristics of the machines being developed. Many solutions have to be tested and subsequently modified. This process requires experience but mainly time and cost. Computational technology in the form of virtual prototyping brings considerable acceleration of developing process. In addition to static parameters, the dynamic parameters of a virtual machine tool are now well verified. At the end of the development, a real machine tool is verified, which already requires considerably less modifications, as it was produced based on virtual testing of the virtual machine. Of course it is not possible to verify all the properties, but the system of virtual prototyping is constantly improving and expanding. This trend is confirmed by world trade fairs for machine tools (EMO). More and more sophisticated products supporting virtual prototyping are emerging on the market. Prototyping on the principle of virtual machining is a progressive direction. This type of analysis better simulates real production conditions and thus tests the designed machine using implemented mathematical models. In this chapter, the practical application of virtual machining for virtual prototyping of machine tools will be explained, supplemented by the measurement of static stiffness of the machine tool. Typical sources of working inaccuracy are

Analytical and Experimental

In this chapter, one of the ways of modelling the working accuracy of machine tools is elaborated. The method of constructing a numerical model based on transmission matrices is applicable in the field of virtual prototyping of modern machine tools. The results of numerical simulation of working precision are closely related to the machine stiffness in the machine workspace. A new way of measuring static stiffness can support numerical simulation results. Measurement of static stiffness is based on the measurement of the positioning accuracy of machine tools. The difference of the method lies in the load of the measured axis during the measurement. In this way, the stiffness of the movable machine partly relative to the base can be determined. The method for measuring static stiffness can advantageously be used

Research of Machine Tool

#### **Chapter 4**

## Analytical and Experimental Research of Machine Tool Accuracy

*Peter Demeč and Tomáš Stejskal*

#### **Abstract**

In this chapter, one of the ways of modelling the working accuracy of machine tools is elaborated. The method of constructing a numerical model based on transmission matrices is applicable in the field of virtual prototyping of modern machine tools. The results of numerical simulation of working precision are closely related to the machine stiffness in the machine workspace. A new way of measuring static stiffness can support numerical simulation results. Measurement of static stiffness is based on the measurement of the positioning accuracy of machine tools. The difference of the method lies in the load of the measured axis during the measurement. In this way, the stiffness of the movable machine partly relative to the base can be determined. The method for measuring static stiffness can advantageously be used in the development of new machine designs.

**Keywords:** machine tool, virtual prototyping, virtual machining, numerical experiment, static stiffness measurement

#### **1. Introduction**

Reducing the development time of modern machines is still a topical issue. Manufacturing experience is not sufficient to predict the characteristics of the machines being developed. Many solutions have to be tested and subsequently modified. This process requires experience but mainly time and cost. Computational technology in the form of virtual prototyping brings considerable acceleration of developing process. In addition to static parameters, the dynamic parameters of a virtual machine tool are now well verified. At the end of the development, a real machine tool is verified, which already requires considerably less modifications, as it was produced based on virtual testing of the virtual machine. Of course it is not possible to verify all the properties, but the system of virtual prototyping is constantly improving and expanding. This trend is confirmed by world trade fairs for machine tools (EMO). More and more sophisticated products supporting virtual prototyping are emerging on the market. Prototyping on the principle of virtual machining is a progressive direction. This type of analysis better simulates real production conditions and thus tests the designed machine using implemented mathematical models.

In this chapter, the practical application of virtual machining for virtual prototyping of machine tools will be explained, supplemented by the measurement of static stiffness of the machine tool. Typical sources of working inaccuracy are

kinematic errors, thermo-mechanical errors, static and dynamic loads [1], a motion control method and control software [2]. In addition, the working inaccuracy is also determined by the appropriate choice of technological conditions and the required power performance.

movement. In this way, the so-called ideal machined surface can be modelled. The surface is created as a trace of the ideal toolpath given by mathematical models. To derive the mathematical model of the ideal tool trajectory, we will accept the

1.The relative movements of the machine's executive members will be examined as relative movements of the Cartesian coordinate systems linked with the workpiece, the individual executive members of the machine and its stationary nodes (e.g., the bed). All moving and stationary machine nodes involved in generating the resulting relative movement of the tool and workpiece will be

2.The workpiece will have index 0, and its coordinate system *S*<sup>0</sup> *(O*0, *X*0, *Y*0, *Z*0*)*

3.For each of the model bodies, we assign the index i ∈ h 1; ni, where index 1 will have the model body immediately next to the workpiece and index n will have the model body that is the tool carrier. For other model bodies, we assign the

4.The individual model bodies *T*<sup>i</sup> will have the coordinate system *S*<sup>i</sup> *(O*i, *X*i, *Y*i, *Z*i*)*, whose axes are in the starting position (i.e., to the beginning of the machining), parallel to the corresponding coordinate axes of the workpiece

5.Model bodies can either stationary or can perform translational, respectively, rotational movement. One model body can perform only one of these

movements and only in direction, respectively, around one coordinate axis of

6.The movements of all model bodies will follow the workpiece coordinate system.

Under these rules it is possible to derive relatively simple formulas for general

In monograph [11] the fundamental mathematical relationships for the model-

Y *i*

**<sup>R</sup>** *<sup>j</sup>*, *<sup>j</sup>*�<sup>1</sup>ð Þ*<sup>t</sup>* � � ! ð Þ f g **<sup>T</sup>***<sup>i</sup>*þ1, *<sup>i</sup>*ð Þ*<sup>t</sup>* <sup>þ</sup> f g **<sup>K</sup>***<sup>i</sup>*þ1, *<sup>i</sup>* !

(1)

*j*¼1

f g **r***<sup>n</sup>* is the tool contact point position vector in the tool carrier coordinate system

application defining the position of any point *A* � [*x*i, *y*i, *z*i] in the coordinate system *S*i*(O*i, *X*i, *Y*i, *Z*i*)*, if you know the definition of his position *A* � [*x*i+1, *y*i+1, *z*i+1] in the coordinate system *S*i+1*(O*i+1, *X*i+1, *Y*i+1, *Z*i+1*)* and if the coordinate system given the previous coordinate system *S*i*(O*i, *X*i, *Y*i, *Z*i*)* performs any of the possible

ling of working accuracy of machine tools are derived with a serial kinematic structure of general shape. The mathematical model of the ideal tool trajectory is in

f g **<sup>r</sup>***<sup>n</sup>* <sup>þ</sup> <sup>X</sup>*<sup>n</sup>*�<sup>1</sup>

*i*¼1

where f g **r**0ð Þ*t* is the tool contact point position vector in the workpiece

increasing indexes in the direction from the workpiece to the tool.

following presumptions:

called common how to model bodies.

*(X*i||*X*0, *Y*i||*Y*0, *Z*i||*Z*0*)*.

the previous model body.

movements listed in the fifth rule.

the form of a matrix equation:

½ � **R***<sup>i</sup>*, *<sup>i</sup>*�<sup>1</sup>ð Þ*t* !

coordinate system (model body *T*0),

þ f g **T**10ð Þ*t* þ f g **K**<sup>10</sup>

*i*¼1

f g **<sup>r</sup>**0ð Þ*<sup>t</sup>* <sup>¼</sup> <sup>Y</sup>*<sup>n</sup>*

(model body *T*n),

**69**

will be considered as stationary in the space.

*Analytical and Experimental Research of Machine Tool Accuracy*

*DOI: http://dx.doi.org/10.5772/intechopen.91345*

The issue of dynamic software adaptation of Tool Center Point (TCP) is addressed [3]. In many works [4–8], the main influence on the TCP position has to do with the temperature factor. Stiffness experiments have shown that besides temperature, the clearance and the manifestation of contact stiffness between bearings and ball screw also play an important role in precision.

The stiffness of the machine tool has a significant influence on the accuracy of its work. Machine tools with a serial cinematic structure are working on the principle that the resulting relative movement of the tool and the workpiece is required to change the shape and dimensions of the workpiece and the results of the superposition of the individual machine nodes. Several mathematical and experimental methods are used to determine this stiffness. Consequently, many works deal with the calculation of stiffness by the finite element method [9, 10].

The classical static stiffness measurement is performed on a stationary machine most often at the tool clamping point relative to the workpiece clamping point. The load jig is gradually loaded and the working area subsequently uploaded in the direction of the machine axes. The deflection gauge measures the deformation at a given point in the load direction. Based on the measured values, the stiffness at the place location is determined. At the same time, deflections are also measured outside the load location. This determines the contribution of deflections to the stiffness change from individual machine components. Such measurement has great importance in prototype testing. Measurement of stiffness contributions reveals machine design imperfections. Some parts need to be redesigned to increase their stiffness. This achieves an acceptable machine stiffness. Normally stiffness measurements are rarely performed in in-process inspection.

The actual position of the moving part of the machine tool in relation to its base, even under static load, is influenced by three basic nonlinearities. These are friction, backlash and compliance [7].

The aim of this work is to design experimentally verified mathematical models that will significantly streamline, shorten and improve the quality of the work of the constituents. The aim of this chapter was to verify experimentally static stiffness using an interferometer, which opens new possibilities of evaluation. Interferometer measurement is used to evaluate positioning accuracy under the not loaded machine. The proposed stiffness measurement methodology with the use of an interferometer opens up new effects of the individual system pulses on the overall accuracy of the machine work. However, there are many nonlinear elements in this field, so the construction of a reliable mathematical model is considerably limited.

#### **2. Theoretical fundamentals**

The relative movement of the tool is determined by contributions from all parts of the kinematic chain. This kinematic chain is generally made up of serially connected machine components from the cutting tool to the workpiece holder. The relative movement between the tool and the workpiece is given by the precision of the kinematic components, the force deformations of the components and the random values that result from the nonlinear elements. The simplest solution for virtual motion is based on the kinematic motion of rigid components. The executive elements are characterised by nodes that can perform rotational or translational

#### *Analytical and Experimental Research of Machine Tool Accuracy DOI: http://dx.doi.org/10.5772/intechopen.91345*

kinematic errors, thermo-mechanical errors, static and dynamic loads [1], a motion control method and control software [2]. In addition, the working inaccuracy is also determined by the appropriate choice of technological conditions and the required

The stiffness of the machine tool has a significant influence on the accuracy of its work. Machine tools with a serial cinematic structure are working on the principle that the resulting relative movement of the tool and the workpiece is required to change the shape and dimensions of the workpiece and the results of the superposition of the individual machine nodes. Several mathematical and experimental methods are used to determine this stiffness. Consequently, many works deal with

The classical static stiffness measurement is performed on a stationary machine most often at the tool clamping point relative to the workpiece clamping point. The load jig is gradually loaded and the working area subsequently uploaded in the direction of the machine axes. The deflection gauge measures the deformation at a given point in the load direction. Based on the measured values, the stiffness at the place location is determined. At the same time, deflections are also measured outside the load location. This determines the contribution of deflections to the stiffness change from individual machine components. Such measurement has great importance in prototype testing. Measurement of stiffness contributions reveals machine design imperfections. Some parts need to be redesigned to increase their stiffness. This achieves an acceptable machine stiffness. Normally stiffness mea-

The actual position of the moving part of the machine tool in relation to its base, even under static load, is influenced by three basic nonlinearities. These are friction,

The aim of this work is to design experimentally verified mathematical models that will significantly streamline, shorten and improve the quality of the work of the constituents. The aim of this chapter was to verify experimentally static stiffness using an interferometer, which opens new possibilities of evaluation. Interferometer measurement is used to evaluate positioning accuracy under the not loaded machine. The proposed stiffness measurement methodology with the use of an interferometer opens up new effects of the individual system pulses on the overall accuracy of the machine work. However, there are many nonlinear elements in this field, so the construction of a reliable mathematical model is

The relative movement of the tool is determined by contributions from all parts

of the kinematic chain. This kinematic chain is generally made up of serially connected machine components from the cutting tool to the workpiece holder. The relative movement between the tool and the workpiece is given by the precision of the kinematic components, the force deformations of the components and the random values that result from the nonlinear elements. The simplest solution for virtual motion is based on the kinematic motion of rigid components. The executive elements are characterised by nodes that can perform rotational or translational

The issue of dynamic software adaptation of Tool Center Point (TCP) is addressed [3]. In many works [4–8], the main influence on the TCP position has to do with the temperature factor. Stiffness experiments have shown that besides temperature, the clearance and the manifestation of contact stiffness between

bearings and ball screw also play an important role in precision.

the calculation of stiffness by the finite element method [9, 10].

surements are rarely performed in in-process inspection.

backlash and compliance [7].

considerably limited.

**68**

**2. Theoretical fundamentals**

power performance.

*Machine Tools - Design, Research, Application*

movement. In this way, the so-called ideal machined surface can be modelled. The surface is created as a trace of the ideal toolpath given by mathematical models.

To derive the mathematical model of the ideal tool trajectory, we will accept the following presumptions:


Under these rules it is possible to derive relatively simple formulas for general application defining the position of any point *A* � [*x*i, *y*i, *z*i] in the coordinate system *S*i*(O*i, *X*i, *Y*i, *Z*i*)*, if you know the definition of his position *A* � [*x*i+1, *y*i+1, *z*i+1] in the coordinate system *S*i+1*(O*i+1, *X*i+1, *Y*i+1, *Z*i+1*)* and if the coordinate system given the previous coordinate system *S*i*(O*i, *X*i, *Y*i, *Z*i*)* performs any of the possible movements listed in the fifth rule.

In monograph [11] the fundamental mathematical relationships for the modelling of working accuracy of machine tools are derived with a serial kinematic structure of general shape. The mathematical model of the ideal tool trajectory is in the form of a matrix equation:

$$\begin{aligned} \{\mathbf{r}\_0(t)\} &= \left(\prod\_{i=1}^n [\mathbf{R}\_{i,i-1}(t)]\right) \{\mathbf{r}\_n\} + \sum\_{i=1}^{n-1} \left( \left(\prod\_{j=1}^i [\mathbf{R}\_{j,j-1}(t)]\right) (\{\mathbf{T}\_{i+1,i}(t)\} + \{\mathbf{K}\_{i+1,i}\}) \right) \\ &+ \{\mathbf{T}\_{10}(t)\} + \{\mathbf{K}\_{10}\} \end{aligned} \tag{1}$$

where f g **r**0ð Þ*t* is the tool contact point position vector in the workpiece coordinate system (model body *T*0),

f g **r***<sup>n</sup>* is the tool contact point position vector in the tool carrier coordinate system (model body *T*n),

[**R** *<sup>j</sup>*, *<sup>j</sup>*�1ð Þ*t* ] is the rotational motion transformation matrix of the model body *T*<sup>i</sup> relative to the *T*i-1 model body,

f g **T**<sup>i</sup>þ1, *<sup>i</sup>*ð Þ*t* is the linear motion transformation vector of the model body *Ti+1* relative to the *T*<sup>i</sup> model body,

f g **K***<sup>i</sup>*þ1, *<sup>i</sup>* is the starting position vector of the model body *T*i+1 coordinate system in the *T*<sup>i</sup> model body coordinate system,

*t* is the time.

The final machining inaccuracy is determined by the sum of part deformation due to production forces and position inaccuracy of all model bodies (machine nodes) from tool to workpiece in a coordinate system of workpiece in time *t* that in mathematical language could be written in a form [12].

$$\{\mathbf{A}(t)\} = \{\mathbf{A}\_0(t)\} \; + \sum\_{i=1}^{n} \left( \left( \prod\_{j=1}^{i} \left[ \mathbf{R}\_{j,j-1}(t) \right] \right) \left( \{\mathbf{\dot{s}}\_i(t)\} \; + \; \left[ \mathbf{e}\_i(t) \right] \{\mathbf{r}\_i(t)\} \right) \right), \tag{2}$$

where the vector of deformations of the machined part is

$$\{\Delta\_0(t)\} = \{\mathbf{\dot{6}}\_0(t)\} \; \; \; \; \; \left[\mathbf{e}\_0(t)\right] \{\mathbf{r}\_0(t)\} \tag{3}$$

and the vectors

$$\{\Delta\_i(t)\} = \begin{Bmatrix} \mathfrak{d}\_i(t) \end{Bmatrix} + \begin{bmatrix} \mathfrak{e}\_i(t) \end{bmatrix} \begin{Bmatrix} \mathfrak{r}\_i(t) \end{Bmatrix} \tag{4}$$

represent the final position inaccuracies of active tool's point position caused by position inaccuracies of individual model bodies *T*<sup>i</sup> expressed in the coordinate systems of these model bodies. The second part of Eq. (2) therefore represents a summary position inaccuracy of the active tool's point that is caused by position inaccuracies of all model bodies *T*<sup>i</sup> and is reflected in the workpiece coordinate system.

The vector of linear inaccuracies of model body *T*<sup>i</sup> in Eq. (2) defined relationship

$$\{\delta\_i(t)\} = \begin{Bmatrix} \delta\_{\text{xi}}(t) \ \delta\_{\text{yi}}(t) \ \delta\_{\text{zi}}(t) \end{Bmatrix}^T,\tag{5}$$

where *δxi(t), δyi(t)* and *δzi(t)* are linear inaccuracies in the direction of corresponding coordinate axes.

The matrix of angular inaccuracies of model body Ti is defined by the relationship

$$\begin{aligned} \begin{bmatrix} \mathbf{e}\_i(t) \end{bmatrix} = \begin{bmatrix} \mathbf{0} & -\boldsymbol{\nu}\_i(t) & \boldsymbol{\nu}\_i(t) \\ \boldsymbol{\nu}\_i(t) & \mathbf{0} & -\boldsymbol{\phi}\_i(t) \\ -\boldsymbol{\nu}\_i(t) & \boldsymbol{\phi}\_i(t) & \mathbf{0} \end{bmatrix}, \end{aligned} \tag{6}$$

where *φi(t), υi(t)* and *ψi(t)* are angular inaccuracies (rotations about the axes *X*i, *Y*i, *Z*i).

The position vector of active tool's point in coordinate system of model body *T*<sup>i</sup> – vector *ri(t)* in Eqs. (2) and (4) is defined by the relationship

$$\begin{aligned} \{\mathbf{r}\_i(t)\} &= \left(\prod\_{j=i}^{n-1} [\mathbf{R}\_{j+1,j}(t)]\right) \{\mathbf{r}\_n\} \\ &+ \sum\_{j=i}^{n-2} \left( \left(\prod\_{k=i}^j [\mathbf{R}\_{k+1,k}(t)]\right) \left(\{\mathbf{T}\_{j+2,j+1}(t)\} + \{\mathbf{K}\_{j+2,j+1}\}\right) \right) + \{\mathbf{T}\_{i+1,i}(t)\} \\ &+ \{\mathbf{K}\_{i+1,i}\} \end{aligned}$$

(7)

Virtual machining with thinking of acting forces calculating the inaccuracies of the position of all model bodies in the corresponding *t* ∈ h *O; T*i is mathematically defined by Eqs. (2)–(6). The result is a mathematical model of the real machined

Practical implementation of the virtual machining on the virtual machine tool model is represented by the example of a virtual model of a machining centre with a horizontal spindle axis for machining non-rotating components, the design of which stems from the Box-in-Box concept and is complemented by a rotary table with the relvant construction dimensions (**Table 1**). This can be considered in a simplified

**Dimension** *a bB4 B5 h h3 h4 h5* Value (mm) 120 235 300 300 72 20 20 62 Dimension *H3 H4 H5 L2 L3 L4 L5 p* Value (mm) 134 200 700 300 750 760 380 66

**f**ðÞ¼ *t* **r**0ðÞ þ *t* **Δ**ð Þ*t :* (8)

surface in the form of vector function

*DOI: http://dx.doi.org/10.5772/intechopen.91345*

*Analytical and Experimental Research of Machine Tool Accuracy*

*Design dimensions of the horizontal machining centre.*

**3. Practical example**

**Table 1.**

**Figure 1.**

**71**

*Virtual model of the machining Centre.*

*Analytical and Experimental Research of Machine Tool Accuracy DOI: http://dx.doi.org/10.5772/intechopen.91345*

Virtual machining with thinking of acting forces calculating the inaccuracies of the position of all model bodies in the corresponding *t* ∈ h *O; T*i is mathematically defined by Eqs. (2)–(6). The result is a mathematical model of the real machined surface in the form of vector function

$$\mathbf{f}(t) = \mathbf{r}\_0(t) \, | \, + \, \Delta(t). \tag{8}$$

#### **3. Practical example**

Practical implementation of the virtual machining on the virtual machine tool model is represented by the example of a virtual model of a machining centre with a horizontal spindle axis for machining non-rotating components, the design of which stems from the Box-in-Box concept and is complemented by a rotary table with the relvant construction dimensions (**Table 1**). This can be considered in a simplified


**Table 1.**

[**R** *<sup>j</sup>*, *<sup>j</sup>*�1ð Þ*t* ] is the rotational motion transformation matrix of the model body *T*<sup>i</sup>

f g **K***<sup>i</sup>*þ1, *<sup>i</sup>* is the starting position vector of the model body *T*i+1 coordinate system

The final machining inaccuracy is determined by the sum of part deformation due to production forces and position inaccuracy of all model bodies (machine nodes) from tool to workpiece in a coordinate system of workpiece in time *t* that in

represent the final position inaccuracies of active tool's point position caused by

The vector of linear inaccuracies of model body *T*<sup>i</sup> in Eq. (2) defined relationship

The matrix of angular inaccuracies of model body Ti is defined by the relationship

where *φi(t), υi(t)* and *ψi(t)* are angular inaccuracies (rotations about the axes

The position vector of active tool's point in coordinate system of model body

� � � � !

0 �*ψi*ð Þ*t υi*ð Þ*t ψi*ð Þ*t* 0 �*ϕi*ð Þ*t* �*υi*ð Þ*t ϕi*ð Þ*t* 0

**<sup>T</sup>** *<sup>j</sup>*þ2, *<sup>j</sup>*þ<sup>1</sup>ð Þ*<sup>t</sup>* � � <sup>þ</sup> **<sup>K</sup>***<sup>j</sup>*þ2, *<sup>j</sup>*þ<sup>1</sup>

3 7

f g **<sup>δ</sup>***i*ð Þ*<sup>t</sup>* <sup>¼</sup> *<sup>δ</sup>xi*ð Þ*<sup>t</sup> <sup>δ</sup>yi*ð Þ*<sup>t</sup> <sup>δ</sup>zi*ð Þ*<sup>t</sup>* � �*<sup>T</sup>*

where *δxi(t), δyi(t)* and *δzi(t)* are linear inaccuracies in the direction of

position inaccuracies of individual model bodies *T*<sup>i</sup> expressed in the coordinate systems of these model bodies. The second part of Eq. (2) therefore represents a summary position inaccuracy of the active tool's point that is caused by position inaccuracies of all model bodies *T*<sup>i</sup> and is reflected in the workpiece coordinate

ð Þ f g **δ***i*ð Þ*t* þ ½ � **ε***i*ð Þ*t* f g **r***i*ð Þ*t*

, (2)

, (5)

<sup>5</sup>, (6)

þ f g **T***<sup>i</sup>*þ1, *<sup>i</sup>*ð Þ*t*

(7)

!

f g **Δ**0ð Þ*t* ¼ f g **δ**0ð Þ*t* þ ½ � **ε**0ð Þ*t* f g **r**0ð Þ*t* (3)

f g **Δ***i*ð Þ*t* ¼ f g **δ***i*ð Þ*t* þ ½ � **ε***i*ð Þ*t* f g **r***i*ð Þ*t* (4)

f g **T**<sup>i</sup>þ1, *<sup>i</sup>*ð Þ*t* is the linear motion transformation vector of the model body *Ti+1*

relative to the *T*i-1 model body,

relative to the *T*<sup>i</sup> model body,

f g **<sup>Δ</sup>**ð Þ*<sup>t</sup>* <sup>¼</sup> f g **<sup>Δ</sup>**0ð Þ*<sup>t</sup>* <sup>þ</sup> <sup>X</sup>*<sup>n</sup>*

corresponding coordinate axes.

½ �¼ **ε***i*ð Þ*t*

2 6 4

*T*<sup>i</sup> – vector *ri(t)* in Eqs. (2) and (4) is defined by the relationship

f g **r***<sup>n</sup>*

½ � **R***<sup>k</sup>*þ1, *<sup>k</sup>*ð Þ*t* !

*t* is the time.

and the vectors

system.

*X*i, *Y*i, *Z*i).

**70**

f g **<sup>r</sup>***i*ð Þ*<sup>t</sup>* <sup>¼</sup> <sup>Y</sup>*<sup>n</sup>*�<sup>1</sup>

*j*¼*i*

þX*<sup>n</sup>*�<sup>2</sup> *j*¼*i*

þ f g **K***<sup>i</sup>*þ1, *<sup>i</sup>*

**<sup>R</sup>** *<sup>j</sup>*þ1, *<sup>j</sup>*ð Þ*<sup>t</sup>* � � !

Y *j*

*k*¼*i*

in the *T*<sup>i</sup> model body coordinate system,

*Machine Tools - Design, Research, Application*

mathematical language could be written in a form [12].

Y *i*

**<sup>R</sup>** *<sup>j</sup>*, *<sup>j</sup>*�1ð Þ*<sup>t</sup>* � � !

*j*¼1

where the vector of deformations of the machined part is

*i*¼1

*Design dimensions of the horizontal machining centre.*

**Figure 1.** *Virtual model of the machining Centre.*

version as a simple "indexing" table (allowing positioning of the workpiece for example by 90°) or as a variant with a controlled axis B (allowing machining when the workpiece is rotated about the vertical axis continuously).

model bodies (gradually from the workpiece towards the tool): *T*1, rotary table; *T*2, table; *T*3, bed (longitudinal part); *T*4, bed (transverse part); *T*5, stand; *T*6, headstock; and *T*7, spindle. The illustration also shows the basic layout of the machine's coordinate system. A detailed dimensional computational model of the machining

*Analytical and Experimental Research of Machine Tool Accuracy*

*Detailed computational model of the machining centre with rotary table (ground plan).*

*Detailed computational model of the machining centre with rotary table (side view).*

centre is shown in **Figures 3**–**5**.

*DOI: http://dx.doi.org/10.5772/intechopen.91345*

**Figure 4.**

**Figure 5.**

**73**

The proposed machine (see **Figure 1**) has a work area defined by the maximum workpiece dimensions *Lomax* = 600 mm, *Bomax* = 500 mm a *Homax* = 600 mm. The simplified machine model is illustrated in **Figure 2** and consists of the following

**Figure 2.** *Simplified computational model of the machining Centre with rotary table.*

**Figure 3.** *Detailed computational model of the machining centre with rotary table (front view).*

*Analytical and Experimental Research of Machine Tool Accuracy DOI: http://dx.doi.org/10.5772/intechopen.91345*

version as a simple "indexing" table (allowing positioning of the workpiece for example by 90°) or as a variant with a controlled axis B (allowing machining when

workpiece dimensions *Lomax* = 600 mm, *Bomax* = 500 mm a *Homax* = 600 mm. The simplified machine model is illustrated in **Figure 2** and consists of the following

The proposed machine (see **Figure 1**) has a work area defined by the maximum

the workpiece is rotated about the vertical axis continuously).

*Machine Tools - Design, Research, Application*

*Simplified computational model of the machining Centre with rotary table.*

*Detailed computational model of the machining centre with rotary table (front view).*

**Figure 2.**

**Figure 3.**

**72**

model bodies (gradually from the workpiece towards the tool): *T*1, rotary table; *T*2, table; *T*3, bed (longitudinal part); *T*4, bed (transverse part); *T*5, stand; *T*6, headstock; and *T*7, spindle. The illustration also shows the basic layout of the machine's coordinate system. A detailed dimensional computational model of the machining centre is shown in **Figures 3**–**5**.

**Figure 4.** *Detailed computational model of the machining centre with rotary table (ground plan).*

**Figure 5.** *Detailed computational model of the machining centre with rotary table (side view).*

The stand: [**R**54(*t*)], {**T**54(*t*)}, {**K**54}.

*DOI: http://dx.doi.org/10.5772/intechopen.91345*

workpiece and the table are identified.

adjusted to *nv* = 2700 min�<sup>1</sup>

**4. Virtual machining result**

resistance) according to [13]

milling:

**75**

The bed—transverse part: [**R**43(*t*)], {**T**43(*t*)}, {**K**43}. The bed—longitudinal part: [**R**32(*t*)], {**T**32(*t*)}, {**K**32}. The longitudinal table: [**R**21(*t*)], {**T**21(*t*)}, {**K**21}. The rotary table: [**R**10(*t*)], {**T**10(*t*)}, {**K**10}.

*Analytical and Experimental Research of Machine Tool Accuracy*

rate on the milling tooth), cutting speed vc = 170 m�min�<sup>1</sup>

corresponding numerical simulations are performed.

*Fc* <sup>¼</sup> <sup>682</sup>*h*<sup>0</sup>*:*<sup>86</sup>

from the top right corner of the workpiece (see **Figures 6** and **7**).

The virtual workpiece was designed as a quad shape body with dimensions *Lo* = 450 mm, *Bo* = *Bomax* = 500 mm and *Ho* = *Homax* = 600 mm. The position of the workpiece on the table is symmetrical, and the vertical planes of symmetry of the

piece, we cut the front of the workpiece over the entire height machining starting

The machining of the respective planar faces of the workpiece by the front cylindrical milling cutter begins at the coordinate *y0(0)* = 590 mm and ends at *y0*(*T*) = *k* = 110 mm, resulting from the construction of the headstock and spindle. Thus, the headstock travel at time *T* is 480 mm, with the total machining process lasting at selected feed rates, and the number of milling teeth and spindle speed *T* = 480/16.2 ≈ 29.63 s. If we divide the total machining time into, for example, 10 equal sections, Δ*t* = 2.963 s; the headstock path is simultaneously divided into 10 equal sections of 48 mm in length. In these positions of the headstock, the

The power ratios for machining (see **Figure 8**) were simulated on the basis of the structural equation for the tangential component of the cutting force (cutting

> 0*:*72 *<sup>Y</sup> <sup>d</sup>*�0*:*<sup>86</sup>

> > 9 >=

> > >;

*<sup>f</sup> bf zf f*

*FX* ¼ ð Þ 0*:*60 ÷ 0*:*90 *Fc FY* ¼ ð Þ 0*:*45 ÷ 0*:*70 *Fc FZ* ¼ ð Þ 0*:*50 ÷ 0*:*55 *Fc*

Due to the proportions of the design of the individual model bodies of the machine, only the deformations of the stand were considered in the other calculations. For numerical simulations, relationships (10) were used for the most

where the appropriate dimensions are set in millimetres (mm). On the basis of Eq. (9), according to [13], the components of the cutting force (cutting resistance) in the directions of the individual coordinate axes are valid for asymmetrical down-

In virtual machining, the simultaneous machining of two vertical planar faces of the workpiece by the front and cylindrical peripheral surfaces of the tool—the front cylindrical cutter—was considered. It is considered an unsymmetrical downmilling. The tool will be the front cylindrical milling cutter R215.59–06, the diameter *df* = 20 mm and the number of teeth *zf* = 4. Milling depth *hf* = 10 mm, milling width *bf* is equal to half the diameter of the cutter *df*, machining with motion of the headstock (direction *Y*) is from the top down, feed rate *fY* = 0.09 mm (feed

, and spindle speed is

*<sup>f</sup>* ð Þ N (9)

, (10)

. When looking from the machine stand to the work-

**Figure 6.** *Detailed computational model of the situation at the beginning of virtual machining (front view).*

#### **Figure 7.**

*Detailed computational model of the situation at the beginning of virtual machining (side view).*

Based on the detailed computational models of each model body of the virtual machine, the corresponding transformation matrices and vectors were defined as follows:

The spindle: [**R**76(*t*)], {**T**76(*t*)}, {**K**76}. The headstock: [**R**65(*t*)], {**T**65(*t*)}, {**K**65}. *Analytical and Experimental Research of Machine Tool Accuracy DOI: http://dx.doi.org/10.5772/intechopen.91345*

The stand: [**R**54(*t*)], {**T**54(*t*)}, {**K**54}. The bed—transverse part: [**R**43(*t*)], {**T**43(*t*)}, {**K**43}. The bed—longitudinal part: [**R**32(*t*)], {**T**32(*t*)}, {**K**32}. The longitudinal table: [**R**21(*t*)], {**T**21(*t*)}, {**K**21}. The rotary table: [**R**10(*t*)], {**T**10(*t*)}, {**K**10}.

The virtual workpiece was designed as a quad shape body with dimensions *Lo* = 450 mm, *Bo* = *Bomax* = 500 mm and *Ho* = *Homax* = 600 mm. The position of the workpiece on the table is symmetrical, and the vertical planes of symmetry of the workpiece and the table are identified.

In virtual machining, the simultaneous machining of two vertical planar faces of the workpiece by the front and cylindrical peripheral surfaces of the tool—the front cylindrical cutter—was considered. It is considered an unsymmetrical downmilling. The tool will be the front cylindrical milling cutter R215.59–06, the diameter *df* = 20 mm and the number of teeth *zf* = 4. Milling depth *hf* = 10 mm, milling width *bf* is equal to half the diameter of the cutter *df*, machining with motion of the headstock (direction *Y*) is from the top down, feed rate *fY* = 0.09 mm (feed rate on the milling tooth), cutting speed vc = 170 m�min�<sup>1</sup> , and spindle speed is adjusted to *nv* = 2700 min�<sup>1</sup> . When looking from the machine stand to the workpiece, we cut the front of the workpiece over the entire height machining starting from the top right corner of the workpiece (see **Figures 6** and **7**).

#### **4. Virtual machining result**

The machining of the respective planar faces of the workpiece by the front cylindrical milling cutter begins at the coordinate *y0(0)* = 590 mm and ends at *y0*(*T*) = *k* = 110 mm, resulting from the construction of the headstock and spindle. Thus, the headstock travel at time *T* is 480 mm, with the total machining process lasting at selected feed rates, and the number of milling teeth and spindle speed *T* = 480/16.2 ≈ 29.63 s. If we divide the total machining time into, for example, 10 equal sections, Δ*t* = 2.963 s; the headstock path is simultaneously divided into 10 equal sections of 48 mm in length. In these positions of the headstock, the corresponding numerical simulations are performed.

The power ratios for machining (see **Figure 8**) were simulated on the basis of the structural equation for the tangential component of the cutting force (cutting resistance) according to [13]

$$F\_c = 682h\_f^{0.86} b\_f z\_f f\_Y^{0.72} d\_f^{-0.86} \text{ (N)}\tag{9}$$

where the appropriate dimensions are set in millimetres (mm). On the basis of Eq. (9), according to [13], the components of the cutting force (cutting resistance) in the directions of the individual coordinate axes are valid for asymmetrical downmilling:

$$\begin{aligned} F\_X &= (\mathbf{0.60} \div \mathbf{0.90}) F\_c \\ F\_Y &= (\mathbf{0.45} \div \mathbf{0.70}) F\_c \\ F\_Z &= (\mathbf{0.50} \div \mathbf{0.55}) F\_c \end{aligned} \tag{10}$$

Due to the proportions of the design of the individual model bodies of the machine, only the deformations of the stand were considered in the other calculations. For numerical simulations, relationships (10) were used for the most

Based on the detailed computational models of each model body of the virtual machine, the corresponding transformation matrices and vectors were defined as

*Detailed computational model of the situation at the beginning of virtual machining (side view).*

*Detailed computational model of the situation at the beginning of virtual machining (front view).*

follows:

**74**

**Figure 7.**

**Figure 6.**

*Machine Tools - Design, Research, Application*

The spindle: [**R**76(*t*)], {**T**76(*t*)}, {**K**76}. The headstock: [**R**65(*t*)], {**T**65(*t*)}, {**K**65}. unfavourable values; therefore, the respective components of the cutting forces (resistances) are for down-milling:

$$\begin{aligned} F\_X &= 0.90 \, F\_c = 3386.0563 \text{ N} \approx 3386 \text{ N} \\ F\_Y &= 0.70 \, F\_c = 2633.5993 \text{ N} \approx 2634 \text{ N} \end{aligned} \tag{11}$$
 
$$\begin{aligned} F\_Z &= 0.55 F\_c = 2069.2566 \text{ N} \approx 2070 \text{ N} \end{aligned} \tag{11}$$

**Figure 8.** *Layout of cutting forces and resistances for virtual machining.*

Some results of numerical experiments are shown in **Figures 9**–**11**; numerical

 0 590 0.254 215 215.00025 0.155 235 235.00016 2963 542 0.197 215 215.00020 0.120 235 235.00012 5926 494 0.149 215 215.00015 0.091 235 235.00009 8889 446 0.110 215 215.00011 0.067 235 235.00007 11.852 398 0.078 215 215.00008 0.048 235 235.00005 14.815 350 0.053 215 215.00005 0.032 235 235.00003 17.778 302 0.034 215 215.00003 0.021 235 235.00002 20.741 254 0.020 215 215.00002 0.012 235 235.00001 23.704 206 0.011 215 215.00001 0.007 235 235.00001 26.667 158 0.005 215 215.00000 0.003 235 235.00000 29.63 110 0.002 215 215.00000 0.001 235 235.00000

*The real coordinates of the machined surface in Z direction by down-milling.*

*Analytical and Experimental Research of Machine Tool Accuracy*

*DOI: http://dx.doi.org/10.5772/intechopen.91345*

**Down-milling** *t y0i (t) Δx0i(t) x0i,id (t) x0i,sk (t) Δz0i(t) z0i,id (t) z0i,sk (t)* **(s) (mm) (μm) (mm) (mm) (μm) (mm) (mm)**

Inspiration to modify the commonly used extended static stiffness measurement method resulted from the significantly different experimentally measured static stiffness values of the new loading method compared to the standard stiffness measurement method. Static stiffness measurements in this experiment were performed under the axis load immediately after reaching the desired position. From the known force and deflection, it is possible to determine the stiffness of the table relative to the base at a given programmed linear unit position. Thus, the stiffness measurement results more faithfully reflect the actual ratios that occur during machining. This note is important in that relatively slow changes, such as a

**5. Experimental measurement of milling machine stiffness during**

values are given in **Table 2**.

*Virtual machining results with down-milling.*

**Table 2.**

**77**

**Position of the headstock i**

**Figure 11.**

**X-axis positioning**

**Figure 9.**

*Graphical representation of workpiece inaccuracies.*

**Figure 10.** *The real coordinates of the machined surface in X direction by down-milling.*

#### *Analytical and Experimental Research of Machine Tool Accuracy DOI: http://dx.doi.org/10.5772/intechopen.91345*

**Figure 11.** *The real coordinates of the machined surface in Z direction by down-milling.*


**Table 2.**

unfavourable values; therefore, the respective components of the cutting forces

*FX* ¼ 0*:*90*Fc* ¼ 3386*:*0563 N ≈ 3386 N *FY* ¼ 0*:*70*Fc* ¼ 2633*:*5993 N ≈ 2634 N *FZ* ¼ 0*:*55*Fc* ¼ 2069*:*2566 N ≈ 2070 N

9 >=

>;

, (11)

(resistances) are for down-milling:

*Machine Tools - Design, Research, Application*

**Figure 9.**

**Figure 8.**

**Figure 10.**

**76**

*Graphical representation of workpiece inaccuracies.*

*Layout of cutting forces and resistances for virtual machining.*

*The real coordinates of the machined surface in X direction by down-milling.*

*Virtual machining results with down-milling.*

Some results of numerical experiments are shown in **Figures 9**–**11**; numerical values are given in **Table 2**.

#### **5. Experimental measurement of milling machine stiffness during X-axis positioning**

Inspiration to modify the commonly used extended static stiffness measurement method resulted from the significantly different experimentally measured static stiffness values of the new loading method compared to the standard stiffness measurement method. Static stiffness measurements in this experiment were performed under the axis load immediately after reaching the desired position. From the known force and deflection, it is possible to determine the stiffness of the table relative to the base at a given programmed linear unit position. Thus, the stiffness measurement results more faithfully reflect the actual ratios that occur during machining. This note is important in that relatively slow changes, such as a

change in the thickness of the oil layer in the contact surfaces, may occur after the movement has stopped, which may affect the static stiffness.

From this, stiffness was determined in each position in both directions of the table starting with the given position (**Figure 13**). At each position, the average

As shown, the stiffness measured by this method is approximately the same in either direction. The highest stiffness was detected around position 8. This fact is consistent with the machine design since the lowest matrix and motor distance is at this position (**Figure 12**). The screw in this position is least involved in the stiffness

The static loading of the table was performed to verify the continuous loading method. The working procedure consisted of moving the table to the measured position. In the given position, it was gradually loaded and relieved by the *Fr* force. In each position, the loading and unloading cycle was repeated twice. The reason was that only at the second cycle the backlash from the load was determined in the given direction. The stiffness of the system has been determined from the charted slope (**Figure 14**). This method of stiffness measurement can practically be consid-

offset from the slope can be calculated.

*DOI: http://dx.doi.org/10.5772/intechopen.91345*

*5.1.2 Second experiment: static loading of table positions*

*Analytical and Experimental Research of Machine Tool Accuracy*

*Measured stiffness of the shift in the a and r directions, respectively.*

*Stiffness measurement in position 8 (second loading). The slope shows stiffness (169 N/μm).*

deterioration.

ered a classic one.

**Figure 13.**

**Figure 14.**

**79**

An important observation from the measurements is that the measured static stiffness of the table greatly depended on the previous operation of the table and the way it was loaded. The classical stiffness measurement showed up to three times higher stiffness values than the modified method using identical tools and conditions. The reasons for such a considerable difference in stiffness are given by the structural arrangement and the effect of nonlinearities on the contact surfaces of the cross-table components. During the measurement, all machine parts were stationary. This modified view of the static stiffness can be used to change the design philosophy of the machine tool. This is applicable in engineering practice, especially in the field of machine tool design, which will ensure higher machining accuracy under comparable conditions. In the experiments performed, the Renishaw XL80 laser interferometer was used to measure deformation and displacement. Measuring accuracy of the manufacturer guarantees better than 0.5 μm/m.

#### **5.1 Arrangement of experiments**

For experimental static stiffness measurement, the Kondia B 640 CNC Vertical Milling Machine was used as a production machine with three fully controlled axes [14].

#### *5.1.1 First experiment: gradual loading of table positions*

During gradual loading of table positions, the work procedure consisted of alternating cycles with and without the load. The reason for such a procedure was to estimate the thermal expansion of the table against the base during one cycle. One cycle consisted of the table's gradual positioning of nine positions in the *X* axis (**Figure 12**). Using the laser interferometer, an exact position was measured.

**Figure 12.**

*Measurement sequencing in direction of the X axis on the cross-milling table.*

*Analytical and Experimental Research of Machine Tool Accuracy DOI: http://dx.doi.org/10.5772/intechopen.91345*

From this, stiffness was determined in each position in both directions of the table starting with the given position (**Figure 13**). At each position, the average offset from the slope can be calculated.

As shown, the stiffness measured by this method is approximately the same in either direction. The highest stiffness was detected around position 8. This fact is consistent with the machine design since the lowest matrix and motor distance is at this position (**Figure 12**). The screw in this position is least involved in the stiffness deterioration.

#### *5.1.2 Second experiment: static loading of table positions*

The static loading of the table was performed to verify the continuous loading method. The working procedure consisted of moving the table to the measured position. In the given position, it was gradually loaded and relieved by the *Fr* force. In each position, the loading and unloading cycle was repeated twice. The reason was that only at the second cycle the backlash from the load was determined in the given direction. The stiffness of the system has been determined from the charted slope (**Figure 14**). This method of stiffness measurement can practically be considered a classic one.

**Figure 13.**

change in the thickness of the oil layer in the contact surfaces, may occur after the

An important observation from the measurements is that the measured static stiffness of the table greatly depended on the previous operation of the table and the way it was loaded. The classical stiffness measurement showed up to three times higher stiffness values than the modified method using identical tools and conditions. The reasons for such a considerable difference in stiffness are given by the structural arrangement and the effect of nonlinearities on the contact surfaces of the cross-table components. During the measurement, all machine parts were stationary. This modified view of the static stiffness can be used to change the design philosophy of the machine tool. This is applicable in engineering practice, especially in the field of machine tool design, which will ensure higher machining accuracy under comparable conditions. In the experiments performed, the Renishaw XL80 laser interferometer was used to measure deformation and displacement. Measuring

movement has stopped, which may affect the static stiffness.

*Machine Tools - Design, Research, Application*

accuracy of the manufacturer guarantees better than 0.5 μm/m.

*5.1.1 First experiment: gradual loading of table positions*

*Measurement sequencing in direction of the X axis on the cross-milling table.*

For experimental static stiffness measurement, the Kondia B 640 CNC Vertical Milling Machine was used as a production machine with three fully

During gradual loading of table positions, the work procedure consisted of alternating cycles with and without the load. The reason for such a procedure was to estimate the thermal expansion of the table against the base during one cycle. One cycle consisted of the table's gradual positioning of nine positions in the *X* axis (**Figure 12**). Using the laser interferometer, an exact position was

**5.1 Arrangement of experiments**

controlled axes [14].

measured.

**Figure 12.**

**78**

*Measured stiffness of the shift in the a and r directions, respectively.*

**Figure 14.** *Stiffness measurement in position 8 (second loading). The slope shows stiffness (169 N/μm).*

structure. All three sources cause nonlinearity between the load and the

procedure requires feedback that is provided by sensors, for example, temperature, vibration sensors and force and deflection sensors. The last two sensors can be used to the stiffness measure of the measured point at a given

interval. The system works very much like the weather forecast.

not be detectable from a classic stiffness measurement.

the previous load in the individual parts of the machine.

*Analytical and Experimental Research of Machine Tool Accuracy*

*DOI: http://dx.doi.org/10.5772/intechopen.91345*

As follows from the experiments, the classic machine stiffness measurement does not detect design deficiencies in terms of drive quality and the machine's alignment. The measured stiffness values are in the order of three times the values measured when the machine is shifted to the measured position. Yet, the method of shifting the movable parts under load clearly reflects the real state during machin-

The stiffness analysis shows that if the load direction is not changed, the stiffness will be higher. This is related to the definition of the backlash and the direction of

The purpose of the construction of the mathematical model is based on the idea of improving the design of the machine in terms of work accuracy of production. There are two possible procedures. The first is to minimise the adverse effect of nonlinearities in the construction nodes. This can be achieved, for example, by selecting preloaded connections, reducing the number of kinematic elements to a minimum and the like. The second procedure is control of nonlinear states. This

Theoretically, there is no analytical method for calculating nonlinearities. For the first time, French mathematician Henri Poincaré demonstrated it in the task of solving the movement of the three heavenly bodies (three body problem) at the end of the nineteenth century. However, the awareness of this fact was very gradual. It was not until the development of computer technology and chaos theory in the 1970s of the last century that nonlinear problems began to be solved by numerical simulation. Such a nonlinear model gives a solution, but the trajectory of the output functions is uncertain. It moves with some probability that the result is in some

The stiffness measurement by laser interferometer opens up new possibilities of evaluation, not exactly feasible under classic measurement with dial micrometre indicators. The results obtained by measurements point to the fact that the static stiffness depends both on the previous method of loading and the direction of the start of the measured position. The proposed shift stiffness measurement methodology allows for a better assessment of the machine's working ability. It also allows for the detection of faults caused by, for example, incorrect assembly of machine components. The method also points to structural design deficiencies that would

The comparison of the theoretical inaccuracy resulting from the design of the machine components to some extent is also related to the experimental results of the measurement of the static stiffness at different axis positions. Experiments show that the stiffness change is also related to the location of the nut and bolt (**Figure 13**, position 8). This is the influence of design accuracy, which was theoretically described in the introductory chapters. In addition, the previous working action and the amount of load, which is also a design matter, also affect the stiffness and hence the accuracy. However, there are many nonlinear elements in this field, so the construction of a reliable mathematical model is considerably limited. When designing the design of good machines, it is advisable to keep these effects in mind.

position change.

ing better.

time and space.

**7. Conclusion**

**81**

**Figure 15.**

*Size comparison of (a) the static stiffness, (b) the shift stiffness in direction* a *and (c) the shift stiffness in direction* r*.*

The course of static stiffness in individual positions is shown in **Figure 15**. The resulting course was compared with the results of the first experiment. As the graph clearly shows, the static stiffness is considerably different from the shift stiffness.

#### **6. Discussion**

As the experiments show, the actual position of the table depends on the previous method of loading. Immediate static load can only be given a partial credit for the assumed position. This fact places special demands on how to compose a mathematical model. In common mathematical models, the input values translate into a clear result. This is not the case. For that reason, we did not proceed with creating the mathematical model. Only a schematic model has been created (**Figure 16**) [14].

The basis of the model is the source of resistance to movement. These are friction, deformation of the contact surfaces and elastic deformation of the machine

**Figure 16.** *Machine positioning model.*

*Analytical and Experimental Research of Machine Tool Accuracy DOI: http://dx.doi.org/10.5772/intechopen.91345*

structure. All three sources cause nonlinearity between the load and the position change.

As follows from the experiments, the classic machine stiffness measurement does not detect design deficiencies in terms of drive quality and the machine's alignment. The measured stiffness values are in the order of three times the values measured when the machine is shifted to the measured position. Yet, the method of shifting the movable parts under load clearly reflects the real state during machining better.

The stiffness analysis shows that if the load direction is not changed, the stiffness will be higher. This is related to the definition of the backlash and the direction of the previous load in the individual parts of the machine.

The purpose of the construction of the mathematical model is based on the idea of improving the design of the machine in terms of work accuracy of production. There are two possible procedures. The first is to minimise the adverse effect of nonlinearities in the construction nodes. This can be achieved, for example, by selecting preloaded connections, reducing the number of kinematic elements to a minimum and the like. The second procedure is control of nonlinear states. This procedure requires feedback that is provided by sensors, for example, temperature, vibration sensors and force and deflection sensors. The last two sensors can be used to the stiffness measure of the measured point at a given time and space.

Theoretically, there is no analytical method for calculating nonlinearities. For the first time, French mathematician Henri Poincaré demonstrated it in the task of solving the movement of the three heavenly bodies (three body problem) at the end of the nineteenth century. However, the awareness of this fact was very gradual. It was not until the development of computer technology and chaos theory in the 1970s of the last century that nonlinear problems began to be solved by numerical simulation. Such a nonlinear model gives a solution, but the trajectory of the output functions is uncertain. It moves with some probability that the result is in some interval. The system works very much like the weather forecast.

#### **7. Conclusion**

The course of static stiffness in individual positions is shown in **Figure 15**. The resulting course was compared with the results of the first experiment. As the graph clearly shows, the static stiffness is considerably different from the shift stiffness.

*Size comparison of (a) the static stiffness, (b) the shift stiffness in direction* a *and (c) the shift stiffness in*

As the experiments show, the actual position of the table depends on the previous method of loading. Immediate static load can only be given a partial credit for the assumed position. This fact places special demands on how to

compose a mathematical model. In common mathematical models, the input values translate into a clear result. This is not the case. For that reason, we did not proceed with creating the mathematical model. Only a schematic model has been created

The basis of the model is the source of resistance to movement. These are friction, deformation of the contact surfaces and elastic deformation of the machine

**6. Discussion**

**Figure 15.**

*Machine Tools - Design, Research, Application*

*direction* r*.*

(**Figure 16**) [14].

**Figure 16.**

**80**

*Machine positioning model.*

The stiffness measurement by laser interferometer opens up new possibilities of evaluation, not exactly feasible under classic measurement with dial micrometre indicators. The results obtained by measurements point to the fact that the static stiffness depends both on the previous method of loading and the direction of the start of the measured position. The proposed shift stiffness measurement methodology allows for a better assessment of the machine's working ability. It also allows for the detection of faults caused by, for example, incorrect assembly of machine components. The method also points to structural design deficiencies that would not be detectable from a classic stiffness measurement.

The comparison of the theoretical inaccuracy resulting from the design of the machine components to some extent is also related to the experimental results of the measurement of the static stiffness at different axis positions. Experiments show that the stiffness change is also related to the location of the nut and bolt (**Figure 13**, position 8). This is the influence of design accuracy, which was theoretically described in the introductory chapters. In addition, the previous working action and the amount of load, which is also a design matter, also affect the stiffness and hence the accuracy. However, there are many nonlinear elements in this field, so the construction of a reliable mathematical model is considerably limited. When designing the design of good machines, it is advisable to keep these effects in mind.

#### **Acknowledgements**

This work was supported by the Slovak Research and Development Agency under the Contract no. APVV-18-0413: Modular architecture of structural elements of production machinery.

**References**

323-336

2010

771-791

[1] Chen XB, Geddam A, Yuan ZJ. Accuracy improvement of three-axis CNC machining centers by quasi-static

*DOI: http://dx.doi.org/10.5772/intechopen.91345*

*Analytical and Experimental Research of Machine Tool Accuracy*

Machine Tools and Manufacture. 2001;

[10] Altintas Y et al. Virtual machine tool. CIRP Annals - Manufacturing Technology. 2005;**54**(2):115-138

[11] Demeč P, Svetlík J. Virtual Prototyping of Machine Tools. 1st ed. RAM-Verlag: Lüdenscheid; 2017. p. 158.

[12] Demeč P, Svetlík J. Virtual machining and its experimental verification. Acta Mechanica Slovaca.

[13] Lipták O et al. Production Technology - Machining. 1st ed. Bratislava: ALFA; 1979. (in Slovak)

[14] Stejskal T, Svetlík J, Dovica M, Demeč P, Kráľ J. Measurement of static stiffness after motion on a three-axis CNC milling table. Applied Sciences. 2018;**8**(1):15. DOI: 10.3390/app8010015

ISBN 978-3-942303-61-3

2009;**13**(4):68-73

**41.8**:1149-1163

error compensation. Journal of Manufacturing Systems. 1997;**16**(5):

[2] Lin M-T, Shih-Kai W. Modeling and improvement of dynamic contour errors for five-axis machine tools under synchronous measuring paths. International Journal of Machine Tools and Manufacture. 2013;**72**:58-72

[3] Mayr J et al. Comparing different cooling concepts for ball screw systems. Proceedings of ASPE Annual Meeting;

[4] Mayr J et al. Thermal issues in machine tools. CIRP Annals -

Operations. Vol. 2; 2009

Technology. 1998;**75**(1):45-53

**56**(1):521-524

**83**

Manufacturing Technology. 2012;**61.2**:

[5] Brecher Chr, Wissmann A. Modelling of thermal behaviour of a milling machine due to spindle load. 12th CIRP Conference on Modelling of Machining

[6] Wang Y. et al. compensation for the thermal error of a multi-axis machining center. Journal of Materials Processing

[7] Donmez MA, Hahn M-H, Soons JA. A novel cooling system to reduce thermally-induced errors of machine tools. CIRP Annals - Manufacturing Technology. 2007;

[8] Gebhardt M et al. High precision grey-box model for compensation of thermal errors on five-axis machines.

[9] Huang DT-Y, Lee J-J. On obtaining

CIRP Annals - Manufacturing Technology. 2014;**63.1**:509-512

machine tool stiffness by CAE techniques. International Journal of

#### **Author details**

Peter Demeč and Tomáš Stejskal\* Department of Manufacturing Machinery, Faculty of Mechanical Engineering, Technical University of Košice, Košice, Slovakia

\*Address all correspondence to: tomas.stejskal@tuke.sk

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Analytical and Experimental Research of Machine Tool Accuracy DOI: http://dx.doi.org/10.5772/intechopen.91345*

#### **References**

**Acknowledgements**

*Machine Tools - Design, Research, Application*

of production machinery.

**Author details**

**82**

Peter Demeč and Tomáš Stejskal\*

Technical University of Košice, Košice, Slovakia

provided the original work is properly cited.

\*Address all correspondence to: tomas.stejskal@tuke.sk

Department of Manufacturing Machinery, Faculty of Mechanical Engineering,

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

This work was supported by the Slovak Research and Development Agency under the Contract no. APVV-18-0413: Modular architecture of structural elements [1] Chen XB, Geddam A, Yuan ZJ. Accuracy improvement of three-axis CNC machining centers by quasi-static error compensation. Journal of Manufacturing Systems. 1997;**16**(5): 323-336

[2] Lin M-T, Shih-Kai W. Modeling and improvement of dynamic contour errors for five-axis machine tools under synchronous measuring paths. International Journal of Machine Tools and Manufacture. 2013;**72**:58-72

[3] Mayr J et al. Comparing different cooling concepts for ball screw systems. Proceedings of ASPE Annual Meeting; 2010

[4] Mayr J et al. Thermal issues in machine tools. CIRP Annals - Manufacturing Technology. 2012;**61.2**: 771-791

[5] Brecher Chr, Wissmann A. Modelling of thermal behaviour of a milling machine due to spindle load. 12th CIRP Conference on Modelling of Machining Operations. Vol. 2; 2009

[6] Wang Y. et al. compensation for the thermal error of a multi-axis machining center. Journal of Materials Processing Technology. 1998;**75**(1):45-53

[7] Donmez MA, Hahn M-H, Soons JA. A novel cooling system to reduce thermally-induced errors of machine tools. CIRP Annals - Manufacturing Technology. 2007; **56**(1):521-524

[8] Gebhardt M et al. High precision grey-box model for compensation of thermal errors on five-axis machines. CIRP Annals - Manufacturing Technology. 2014;**63.1**:509-512

[9] Huang DT-Y, Lee J-J. On obtaining machine tool stiffness by CAE techniques. International Journal of

Machine Tools and Manufacture. 2001; **41.8**:1149-1163

[10] Altintas Y et al. Virtual machine tool. CIRP Annals - Manufacturing Technology. 2005;**54**(2):115-138

[11] Demeč P, Svetlík J. Virtual Prototyping of Machine Tools. 1st ed. RAM-Verlag: Lüdenscheid; 2017. p. 158. ISBN 978-3-942303-61-3

[12] Demeč P, Svetlík J. Virtual machining and its experimental verification. Acta Mechanica Slovaca. 2009;**13**(4):68-73

[13] Lipták O et al. Production Technology - Machining. 1st ed. Bratislava: ALFA; 1979. (in Slovak)

[14] Stejskal T, Svetlík J, Dovica M, Demeč P, Kráľ J. Measurement of static stiffness after motion on a three-axis CNC milling table. Applied Sciences. 2018;**8**(1):15. DOI: 10.3390/app8010015

**Chapter 5**

**Abstract**

**1. Introduction**

**85**

Geometric Accuracy, Volumetric

Accuracy and Compensation of

*Jiri Marek, Michal Holub,Tomas Marek and Petr Blecha*

The production of geometrically and dimensionally defined workpieces is what the user expects from a machine tool. Deviations from these prescribed dimensions and geometry are due to machine inaccuracies. Therefore, it was necessary to develop tests and tests on the properties and parameters of machine tools that can detect these. Every new machine tool undergoes these tests.How to perform and evaluate these tests is determined and recommended primarily by standards and regulations. When testing the properties of machines, it is not only about knowing and knowing how to measure machines, but also how I can analyze and apply the obtained results. Is it necessary to do a mechanical intervention of the machine or is

**Keywords:** geometric accuracy, volumetric accuracy, compensation, machine tools

The production of geometrically and dimensionally defined workpieces is what the user expects from a machine tool. Deviations from these prescribed dimensions and geometry are due to machine inaccuracies. Therefore, it was necessary to develop trials and tests of machine tool properties and parameters that can detect these errors. Every new machine tool, a newly developed machine, or a machine

Testing of machine tools is an important part of the product life cycle-machine tool. Tests of machine tools can be divided into three groups. The first group of tests is associated with a contractual obligation between the seller and the buyer of the machine. They are, therefore, a part of the contract. Acceptance tests usually take place in two steps—first, directly at the machine manufacturer and then, after the machine is assembled, at the customer. These tests aim to verify the declared properties of the machine. The prototype tests serve to verify the properties of newly designed and manufactured machines. Prototype tests extend the acceptance tests with a series of measurements to provide important information, especially to machine designers. The proposed and expected properties of the new product are examined and the unknown properties, which cannot be expected when the product is being developed, are revealed. Statistical acceptance (process competence

CNC Machine Tools

it enough to compensate the software?

overhauled is subjected to these tests [1].

#### **Chapter 5**

## Geometric Accuracy, Volumetric Accuracy and Compensation of CNC Machine Tools

*Jiri Marek, Michal Holub,Tomas Marek and Petr Blecha*

#### **Abstract**

The production of geometrically and dimensionally defined workpieces is what the user expects from a machine tool. Deviations from these prescribed dimensions and geometry are due to machine inaccuracies. Therefore, it was necessary to develop tests and tests on the properties and parameters of machine tools that can detect these. Every new machine tool undergoes these tests.How to perform and evaluate these tests is determined and recommended primarily by standards and regulations. When testing the properties of machines, it is not only about knowing and knowing how to measure machines, but also how I can analyze and apply the obtained results. Is it necessary to do a mechanical intervention of the machine or is it enough to compensate the software?

**Keywords:** geometric accuracy, volumetric accuracy, compensation, machine tools

#### **1. Introduction**

The production of geometrically and dimensionally defined workpieces is what the user expects from a machine tool. Deviations from these prescribed dimensions and geometry are due to machine inaccuracies. Therefore, it was necessary to develop trials and tests of machine tool properties and parameters that can detect these errors. Every new machine tool, a newly developed machine, or a machine overhauled is subjected to these tests [1].

Testing of machine tools is an important part of the product life cycle-machine tool. Tests of machine tools can be divided into three groups. The first group of tests is associated with a contractual obligation between the seller and the buyer of the machine. They are, therefore, a part of the contract. Acceptance tests usually take place in two steps—first, directly at the machine manufacturer and then, after the machine is assembled, at the customer. These tests aim to verify the declared properties of the machine. The prototype tests serve to verify the properties of newly designed and manufactured machines. Prototype tests extend the acceptance tests with a series of measurements to provide important information, especially to machine designers. The proposed and expected properties of the new product are examined and the unknown properties, which cannot be expected when the product is being developed, are revealed. Statistical acceptance (process competence

test) is used for exacting customers, where it is necessary to maintain the quality of the workpiece in the long term [2].

immediate effect on the machine installation site and can adversely affect the machine operation. Coldness or sudden temperature changes are equally unfavorable. In cases where this does not impede the operation of the machine (e.g., thermal protection failure, functionality of motion mechanisms) and the temperature changes (sudden temperature difference) are not too high, the machine can be operated satisfactorily. This state can be compared to a temperature steady state (tempered state). Therefore, manufacturers usually report the temperature range at which their machine operates. Rather, a sudden change in the temperature field is

*Geometric Accuracy, Volumetric Accuracy and Compensation of CNC Machine Tools*

In addition to these external effects, several factors, referred to collectively as production accuracy (production uncertainty), affect the operation and, in particular, its machining accuracy. When machining a workpiece over time, its dimensions vary within or outside the given and permitted limits. Workpiece dimensional variations are caused by three main factors affecting the machine tool and the

Every CNC machine tool is exposed to temperature effects, both even and uneven, during its operation and also in its sleep mode. Due to this temperature effect, temperature deformations arise which lead to a change in the position of the workpiece relative to the tool and thus to inaccuracies. This will be striking if we are focused on the stability of the machined dimension in case of a smaller series of workpieces, respecting the shape and position errors defined on the machined parts. The causes of heating up the individual parts of the machine tool can be found either in the machine itself (passive resistors in the motion axes or the cutting process itself) or outside it. The thermal stability of machine tools today is one of the most important factors for maintaining the specified tolerances on the work-

Almost all the mechanical work that is done in the cutting process turns into heat. In addition, losses occur in the machine motion groups. Heat is dissipated

from the place of origin (cutting process or in drives, guides) by [5]:

Heat dissipates from the cutting process by:

detrimental [4].

piece [5].

• conduction;

• convection;

• radiation.

• workpiece;

• chip;

• tool;

**87**

• ambient.

manufacturing process [4]:

• temperature influence;

*DOI: http://dx.doi.org/10.5772/intechopen.92085*

• static rigidity of the machine-tool-workpiece system;

• dynamic compliance of machine-tool-workpiece system.

How to perform and evaluate these tests is determined and recommended primarily by standards and regulations. When testing the properties of machines, it is not only about knowing and being capable of how to measure machines (what kind of equipment to use, what method and procedure), but also how to analyze and apply the results in future. Is it necessary to do a mechanical intervention into the machine or is it sufficient to compensate the machine software? [1].

The inspector should be able to answer these and other questions related to machine tool diagnostics. Machine diagnostics is not only a knowledge of the measurement method, but also a set of knowledge that the inspector must know. The first is the knowledge of the measuring equipment itself and its management, monitoring its properties, accuracy, and ensuring a regular calibration (if necessary). Next, it is the knowledge of working with these devices (procedures) and what standards and regulations apply to the measured quantity, the machine, and the device itself. However, it is also important to know the measured machine, without which we cannot adequately perform diagnostics and propose suitable measures to improve the accuracy of the machine [1].

The publication [3] describes the effects of an improperly selected method of measuring the volumetric accuracy of a machine tool. Various methods of placing the temperature sensors on the machine were carried out. These are then reflected in the size of individual machine errors, but also in the resulting volumetric accuracy in the range of 8–12%. This is an example of a different approach to measuring of volumetric accuracy, which is, in this case, affected by the human factor.

#### **2. Effects influencing CNC machine tool operation**

The machine tool must be seen as a technical system, which must always be considered in a comprehensive way, with all the impacting effects. In operation, the CNC machine tool is influenced by a number of effects. By this, we understand the effect not only of the ambient where it is installed, but also the influence of the operator on the machine itself and its impacts on the ambient. These influences affect the properties that all machine tool users call for, namely run stability, repeated machining accuracy, and trouble-free operation. We must assess machine tools in a comprehensive, hierarchical, and structured way. The deviations in the dimensions of the machined component provide the user with direct information on the accuracy of the parts from which the machine is assembled, on the care devoted to the assembly and, last but not least, on its construction. The workshop environment where the machine is installed affects the machine tool by [4]:


On the other hand, the machine can have the same effects on the environment. The machine can cause vibrations (not common), exhaust gases from the supply of coolant and cutting fluid to the cutting site and can also cause ambient warming. By impurities we do not mean coarse dirt and excessive dust, but the standard ambient of normal workshop operation. Heat flow and radiation from the ambient have an

#### *Geometric Accuracy, Volumetric Accuracy and Compensation of CNC Machine Tools DOI: http://dx.doi.org/10.5772/intechopen.92085*

immediate effect on the machine installation site and can adversely affect the machine operation. Coldness or sudden temperature changes are equally unfavorable. In cases where this does not impede the operation of the machine (e.g., thermal protection failure, functionality of motion mechanisms) and the temperature changes (sudden temperature difference) are not too high, the machine can be operated satisfactorily. This state can be compared to a temperature steady state (tempered state). Therefore, manufacturers usually report the temperature range at which their machine operates. Rather, a sudden change in the temperature field is detrimental [4].

In addition to these external effects, several factors, referred to collectively as production accuracy (production uncertainty), affect the operation and, in particular, its machining accuracy. When machining a workpiece over time, its dimensions vary within or outside the given and permitted limits. Workpiece dimensional variations are caused by three main factors affecting the machine tool and the manufacturing process [4]:

• temperature influence;

test) is used for exacting customers, where it is necessary to maintain the quality of

How to perform and evaluate these tests is determined and recommended primarily by standards and regulations. When testing the properties of machines, it is not only about knowing and being capable of how to measure machines (what kind of equipment to use, what method and procedure), but also how to analyze and apply the results in future. Is it necessary to do a mechanical intervention into the

The inspector should be able to answer these and other questions related to machine tool diagnostics. Machine diagnostics is not only a knowledge of the measurement method, but also a set of knowledge that the inspector must know. The first is the knowledge of the measuring equipment itself and its management, monitoring its properties, accuracy, and ensuring a regular calibration (if necessary). Next, it is the knowledge of working with these devices (procedures) and what standards and regulations apply to the measured quantity, the machine, and the device itself. However, it is also important to know the measured machine, without which we cannot adequately perform diagnostics and propose suitable

The publication [3] describes the effects of an improperly selected method of measuring the volumetric accuracy of a machine tool. Various methods of placing the temperature sensors on the machine were carried out. These are then reflected in the size of individual machine errors, but also in the resulting volumetric accuracy in the range of 8–12%. This is an example of a different approach to measuring

of volumetric accuracy, which is, in this case, affected by the human factor.

The machine tool must be seen as a technical system, which must always be considered in a comprehensive way, with all the impacting effects. In operation, the CNC machine tool is influenced by a number of effects. By this, we understand the effect not only of the ambient where it is installed, but also the influence of the operator on the machine itself and its impacts on the ambient. These influences affect the properties that all machine tool users call for, namely run stability, repeated machining accuracy, and trouble-free operation. We must assess machine tools in a comprehensive, hierarchical, and structured way. The deviations in the dimensions of the machined component provide the user with direct information on the accuracy of the parts from which the machine is assembled, on the care devoted to the assembly and, last but not least, on its construction. The workshop environment where the machine is installed affects the machine tool by [4]:

On the other hand, the machine can have the same effects on the environment. The machine can cause vibrations (not common), exhaust gases from the supply of coolant and cutting fluid to the cutting site and can also cause ambient warming. By impurities we do not mean coarse dirt and excessive dust, but the standard ambient of normal workshop operation. Heat flow and radiation from the ambient have an

machine or is it sufficient to compensate the machine software? [1].

measures to improve the accuracy of the machine [1].

**2. Effects influencing CNC machine tool operation**

• vibrations;

• impurities;

• heat.

**86**

the workpiece in the long term [2].

*Machine Tools - Design, Research, Application*


Every CNC machine tool is exposed to temperature effects, both even and uneven, during its operation and also in its sleep mode. Due to this temperature effect, temperature deformations arise which lead to a change in the position of the workpiece relative to the tool and thus to inaccuracies. This will be striking if we are focused on the stability of the machined dimension in case of a smaller series of workpieces, respecting the shape and position errors defined on the machined parts. The causes of heating up the individual parts of the machine tool can be found either in the machine itself (passive resistors in the motion axes or the cutting process itself) or outside it. The thermal stability of machine tools today is one of the most important factors for maintaining the specified tolerances on the workpiece [5].

Almost all the mechanical work that is done in the cutting process turns into heat. In addition, losses occur in the machine motion groups. Heat is dissipated from the place of origin (cutting process or in drives, guides) by [5]:


Heat dissipates from the cutting process by:


It follows that almost all the heat is stored in the machine tool and must be dissipated or stabilized. Uneven heating up of machine tool parts can occur, which can lead to thermal expansion and deformation. This results in fluctuations of workpiece dimensions and tolerance variations in shape and position. All temperature effects cause a temperature increase during machine tool operation, which then stabilizes at a certain value—the so-called steady temperature, which is different for each machine. Therefore, some manufacturers insist on this condition and then recommend machining. However, they must ensure that there is no sudden change in temperature. The harm caused to the machining process may not be the temperature itself, but rather harms of temperature changes during machining. For this reason, in addition to efficient cooling, some manufacturers also heat their machines [5].

forced vibration is by fixing the machine on a flexible foundation or by using a vibration absorber. On the other hand, self-excited vibrations limit the machining quality. The self-excited vibration of the machine arises without an external power supply (excitation source), since this is due to the interaction between the workpiece and the tool. If there is an excess of energy obtained, i.e., if this energy is greater than the energy consumed, self-excited vibrations occur. This is manifested as a chatter of the machine; this is caused by a number of mechanisms. Self-excited vibrations occur during roughing and finishing operations. This does not mean that if less chip is removed, self-excited vibrations are avoided. For example, self-excited vibrations may occur when removing a chip of small depth on a vertical lathe

*Geometric Accuracy, Volumetric Accuracy and Compensation of CNC Machine Tools*

(0.3 mm) with a large load of the ram on the tool tip (1500 mm) [5].

absorbers that can replace the system) [5].

*DOI: http://dx.doi.org/10.5772/intechopen.92085*

**3. Types of accuracy of CNC machine tools**

Each of these accuracies has its own justification [6].

**volumetric accuracy, and thermal expansion**.

or under finishing conditions of machining [6].

that may affect the accuracy of the machine operation [7].

**3.1 Geometric accuracy**

**89**

Self-excited vibrations occur suddenly; stable conditions of cutting process can also suddenly change to unstable ones. Stable conditions become unstable when a certain value of chip depth, which is called a limit chip depth, is exceeded. The basics of the self-excited vibration theory were developed in the 1950s at VÚOSO Praha, founded by Tlustý, Poláček, and others. The theory was based on equality of energy in the feedback system. Energy is generated by the cutting process, which is the source of excitation, and consumed by vibrations (inertial mass, springs and

Under the term accuracy of machine tools, you can imagine several partial features of the machine. Accuracy will be taken differently from the perspective of the designer and from the perspective of the metrologist. From the metrological point of view, accuracy describes how close the measurement result is to the true value of the quantity. In the field of machine tools, we can talk about several types

These basic three types of accuracy of CNC machine tools are complemented by other types of accuracy, namely **positioning accuracy, interpolation accuracy,**

Geometric accuracy describes the geometric structure of a machine tool from which the properties of functional parts affecting its working accuracy can be evaluated. It also describes the production quality of the machine and its assembly in an unloaded state. The tests are carried out on machines working under no load

Geometric accuracy of axes, their measurement and evaluation are given by the standard ČSN ISO 230-1. This section applies only to accuracy tests. It does not deal with the functional tests of the machine (vibrations, jerky movements of parts, etc.) or the determination of characteristic parameters (revolutions, feeds), as these tests are to be performed prior to the accuracy tests. Geometric tests consist of verifying the dimensions, shapes, and positions of components and their relative alignment. They include all operations that affect a part of the machine, such as planeness, alignment, intersection of axes, parallelism, squareness of straight lines or planar surfaces. They relate only to dimensions, shapes, positions, and relative motions

of accuracy, while the determination of accuracy is only qualitative (small, medium, and high). These are **geometric, working, and production accuracies**.

This state is called a thermally stabilized machine tool. The cold machine tool heats up slowly, because we cannot achieve smooth operation and even workload of the machine tool at the beginning of machining. This is because machining must often be interrupted and this causes cooling. Therefore, at first, the machine is thermally stabilized by heating to the operating temperature and then by controlling and maintaining its temperature. Our aim is that, in spite of the thermally stabilized state of the machine, the changes in temperature and its manifestations of thermal deformation could affect as little as possible the position of the tool relative to the workpiece and thus the machining accuracy by [5]:


Undesirable and harmful side effects of time-varying loading can be vibrations, and thus also the accompanying phenomenon of these vibrations—noise of the machine or its parts. Vibrations deteriorate the working conditions of the working process, deteriorate the quality of machined surface, and reduce the tool edge life. The vibrations that occur in machine tools are called forced and self-excited vibration. The source of forced vibration in machine tools is the periodic force.

Forced vibrations are dangerous for the machine construction itself if their frequencies or higher harmonic frequencies of this force, e.g., from the cutting process, are equal to the eigen frequencies of the machine-tool-workpiece system.

If the source of the forced vibration is caused by the cutting process, the suppression of subsequent vibrations can be accomplished by selecting the cutting conditions. However, it should be borne in mind that, for example, the eigen frequencies of the workpiece can sometimes vary considerably depending on the depth of the chip being removed.

Similarly, the eigen frequency of the machine or the eigen frequency of tool clamping in the spindle may not be suitable. Another way how to suppress the

*Geometric Accuracy, Volumetric Accuracy and Compensation of CNC Machine Tools DOI: http://dx.doi.org/10.5772/intechopen.92085*

forced vibration is by fixing the machine on a flexible foundation or by using a vibration absorber. On the other hand, self-excited vibrations limit the machining quality. The self-excited vibration of the machine arises without an external power supply (excitation source), since this is due to the interaction between the workpiece and the tool. If there is an excess of energy obtained, i.e., if this energy is greater than the energy consumed, self-excited vibrations occur. This is manifested as a chatter of the machine; this is caused by a number of mechanisms. Self-excited vibrations occur during roughing and finishing operations. This does not mean that if less chip is removed, self-excited vibrations are avoided. For example, self-excited vibrations may occur when removing a chip of small depth on a vertical lathe (0.3 mm) with a large load of the ram on the tool tip (1500 mm) [5].

Self-excited vibrations occur suddenly; stable conditions of cutting process can also suddenly change to unstable ones. Stable conditions become unstable when a certain value of chip depth, which is called a limit chip depth, is exceeded. The basics of the self-excited vibration theory were developed in the 1950s at VÚOSO Praha, founded by Tlustý, Poláček, and others. The theory was based on equality of energy in the feedback system. Energy is generated by the cutting process, which is the source of excitation, and consumed by vibrations (inertial mass, springs and absorbers that can replace the system) [5].

#### **3. Types of accuracy of CNC machine tools**

Under the term accuracy of machine tools, you can imagine several partial features of the machine. Accuracy will be taken differently from the perspective of the designer and from the perspective of the metrologist. From the metrological point of view, accuracy describes how close the measurement result is to the true value of the quantity. In the field of machine tools, we can talk about several types of accuracy, while the determination of accuracy is only qualitative (small, medium, and high). These are **geometric, working, and production accuracies**. Each of these accuracies has its own justification [6].

These basic three types of accuracy of CNC machine tools are complemented by other types of accuracy, namely **positioning accuracy, interpolation accuracy, volumetric accuracy, and thermal expansion**.

#### **3.1 Geometric accuracy**

Geometric accuracy describes the geometric structure of a machine tool from which the properties of functional parts affecting its working accuracy can be evaluated. It also describes the production quality of the machine and its assembly in an unloaded state. The tests are carried out on machines working under no load or under finishing conditions of machining [6].

Geometric accuracy of axes, their measurement and evaluation are given by the standard ČSN ISO 230-1. This section applies only to accuracy tests. It does not deal with the functional tests of the machine (vibrations, jerky movements of parts, etc.) or the determination of characteristic parameters (revolutions, feeds), as these tests are to be performed prior to the accuracy tests. Geometric tests consist of verifying the dimensions, shapes, and positions of components and their relative alignment. They include all operations that affect a part of the machine, such as planeness, alignment, intersection of axes, parallelism, squareness of straight lines or planar surfaces. They relate only to dimensions, shapes, positions, and relative motions that may affect the accuracy of the machine operation [7].

It follows that almost all the heat is stored in the machine tool and must be dissipated or stabilized. Uneven heating up of machine tool parts can occur, which can lead to thermal expansion and deformation. This results in fluctuations of workpiece dimensions and tolerance variations in shape and position. All temperature effects cause a temperature increase during machine tool operation, which then stabilizes at a certain value—the so-called steady temperature, which is different for each machine. Therefore, some manufacturers insist on this condition and then recommend machining. However, they must ensure that there is no sudden change in temperature. The harm caused to the machining process may not be the temperature itself, but rather harms of temperature changes during machining. For this reason, in addition to efficient cooling, some manufacturers also heat their machines [5]. This state is called a thermally stabilized machine tool. The cold machine tool heats up slowly, because we cannot achieve smooth operation and even workload of the machine tool at the beginning of machining. This is because machining must often be interrupted and this causes cooling. Therefore, at first, the machine is thermally stabilized by heating to the operating temperature and then by controlling and maintaining its temperature. Our aim is that, in spite of the thermally stabilized state of the machine, the changes in temperature and its manifestations of thermal deformation could affect as little as possible the position of the tool relative

• increasing the efficiency of all nodes and elements, thus minimizing losses that

• dissipating the heat by cooling, chip removal, or by dimensioning the surfaces

• checking the air flow and its temperature, or shielding the external thermal

Undesirable and harmful side effects of time-varying loading can be vibrations, and thus also the accompanying phenomenon of these vibrations—noise of the machine or its parts. Vibrations deteriorate the working conditions of the working process, deteriorate the quality of machined surface, and reduce the tool edge life. The vibrations that occur in machine tools are called forced and self-excited vibra-

tion. The source of forced vibration in machine tools is the periodic force.

Forced vibrations are dangerous for the machine construction itself if their frequencies or higher harmonic frequencies of this force, e.g., from the cutting process, are equal to the eigen frequencies of the machine-tool-workpiece system. If the source of the forced vibration is caused by the cutting process, the suppression of subsequent vibrations can be accomplished by selecting the cutting conditions. However, it should be borne in mind that, for example, the eigen frequencies of the workpiece can sometimes vary considerably depending on the

Similarly, the eigen frequency of the machine or the eigen frequency of tool clamping in the spindle may not be suitable. Another way how to suppress the

• placing heat sources efficiently so that they do not affect the design of the

to the workpiece and thus the machining accuracy by [5]:

• selecting a thermo-symmetrical machine design;

change into heat;

for efficient heat dissipation;

*Machine Tools - Design, Research, Application*

• compensating the machine;

depth of the chip being removed.

**88**

machine;

radiation.

According to the standard, there are six geometric errors in linear (according to ČSN ISO 230 - 1) and rotary (according to ČSN ISO 230 - 7) axes, namely three translational errors—positioning error, horizontal and vertical straightness error and three angular errors. A typical three-axis CNC machine tool contains 21 geometric errors—3 3 translation errors, 3 3 angular errors. To these errors, the errors of the relative squareness of the linear axes are added. All of these errors can adversely affect the overall positioning accuracy of the machine and thus also the accuracy of the machined parts. Errors usually occur when the actual position differs from the position displayed on the machine control unit. Errors increase with dynamic effects arising from the interpolation of axes [4].

As early as in 1932, German professor Georg Schlesinger published a book "Inspection Test on Machine Tools," which became the basis for a unified system for assessing the accuracy of machine tools. In this book, he introduced guidelines for the use of devices and equipment for machine tool inspections. Measurement procedures and tolerances for permitted deviations are also given. The name of prof. Schlesinger is used to informally call the geometric accuracy tests of

*Geometric Accuracy, Volumetric Accuracy and Compensation of CNC Machine Tools*

*DOI: http://dx.doi.org/10.5772/intechopen.92085*

The devices and aids most commonly used to measure geometric errors in machine tools are, for example, granite rulers and cubes, dial gauges, digital inclinometers, autocollimators or laser interferometers, which are increasingly used for measurement. The principle of light interference as a measuring tool dates back to 1880, when Albert Michelson developed interferometry. The Michelson interferometer consists of a light source of one wavelength (monochromatic light), a silver-coated mirror and two other mirrors. Although modern

interferometers are more sophisticated and measure with accuracy of the order of 1 ppm and higher, they still use the basic principles of the Michelson

The straightness measurement shows deflection (bent component) or misalignment in the machine guides. This may be due to wear, an accident that may have damaged them, or poor machine foundations that cause the axis or the entire

Squareness is measured by comparing the straightness of two nominally orthogonal axes. Measurements can be carried out using different fixtures and devices with different arrangements. Measuring prisms, mandrels, or granite cubes may be included among fixtures while dial gauges and lasers among

Planeness measurement is performed to check the planeness of CMM tables and machine tools, plate fields and surfaces. It determines whether there are any significant peaks or valleys and quantifies them. If these errors are significant, corrective operations are required. A certain number of measuring lines are required to mea-

This parameter describes the accuracy and repeatability of positioning in linear and rotary numerically controlled axes. "Determination of accuracy and repeatability of positioning in numerically controlled axes" is described in the standard ISO 230-2/6 (ISO 230-2 Test code for machine tools—Determination of accuracy and repeatability of positioning numerically controlled axes; ISO 230-6 Test code for machine tools—Determination of positioning accuracy on body and face diagonals),

Positioning accuracy is the most common form of measurement made with a laser interferometer (**Figure 2**). The laser system measures linear positioning accuracy and repeatability by comparing the position displayed on the machine with the

A more advanced device for measurement of positioning accuracy of the machine is the Laser Tracker, which allows for immediate evaluation of the x, y, and z deviations. The geometric accuracy of the machine and the accuracy of positioning can be evaluated simultaneously (**Figure 3**) for an already assembled and activated machine. For this reason, the aforementioned accuracies are usually

but very often the directive VDI/DGQ 3441is also used [6].

actual position measured by the laser system.

considered simultaneously [9].

**91**

machine tools.

interferometer [4].

machine to drop.

devices [4].

sure the planeness of the surface.

**3.2 Positioning accuracy**

In the case of three-axis kinematics, we can find 21 error parameters, 18 translational errors and 3 parameters of squareness of individual machine axes. These errors, including spindle errors, are shown for the three-axis vertical milling machine in **Figure 1**. The kinematic chain of the three-axis machine tool presented below corresponds to W (Workpiece) -X-Y-Z-T (Tool) [8].

The error description for one linear X-axis and one rotary C-axis is given in **Table 1**.

#### **Figure 1.**

*Scheme of deviations of three-axis kinematics at the machine MCV 754 QUICK, KOVOSVIT-MAS [8].*


#### **Table 1.**

*Error description for one linear axis.*

#### *Geometric Accuracy, Volumetric Accuracy and Compensation of CNC Machine Tools DOI: http://dx.doi.org/10.5772/intechopen.92085*

As early as in 1932, German professor Georg Schlesinger published a book "Inspection Test on Machine Tools," which became the basis for a unified system for assessing the accuracy of machine tools. In this book, he introduced guidelines for the use of devices and equipment for machine tool inspections. Measurement procedures and tolerances for permitted deviations are also given. The name of prof. Schlesinger is used to informally call the geometric accuracy tests of machine tools.

The devices and aids most commonly used to measure geometric errors in machine tools are, for example, granite rulers and cubes, dial gauges, digital inclinometers, autocollimators or laser interferometers, which are increasingly used for measurement. The principle of light interference as a measuring tool dates back to 1880, when Albert Michelson developed interferometry. The Michelson interferometer consists of a light source of one wavelength (monochromatic light), a silver-coated mirror and two other mirrors. Although modern interferometers are more sophisticated and measure with accuracy of the order of 1 ppm and higher, they still use the basic principles of the Michelson interferometer [4].

The straightness measurement shows deflection (bent component) or misalignment in the machine guides. This may be due to wear, an accident that may have damaged them, or poor machine foundations that cause the axis or the entire machine to drop.

Squareness is measured by comparing the straightness of two nominally orthogonal axes. Measurements can be carried out using different fixtures and devices with different arrangements. Measuring prisms, mandrels, or granite cubes may be included among fixtures while dial gauges and lasers among devices [4].

Planeness measurement is performed to check the planeness of CMM tables and machine tools, plate fields and surfaces. It determines whether there are any significant peaks or valleys and quantifies them. If these errors are significant, corrective operations are required. A certain number of measuring lines are required to measure the planeness of the surface.

#### **3.2 Positioning accuracy**

According to the standard, there are six geometric errors in linear (according to ČSN ISO 230 - 1) and rotary (according to ČSN ISO 230 - 7) axes, namely three translational errors—positioning error, horizontal and vertical straightness error and three angular errors. A typical three-axis CNC machine tool contains 21 geometric errors—3 3 translation errors, 3 3 angular errors. To these errors, the errors of the relative squareness of the linear axes are added. All of these errors can adversely affect the overall positioning accuracy of the machine and thus also the accuracy of the machined parts. Errors usually occur when the actual position differs from the position displayed on the machine control unit. Errors increase

In the case of three-axis kinematics, we can find 21 error parameters, 18 translational errors and 3 parameters of squareness of individual machine axes. These errors, including spindle errors, are shown for the three-axis vertical milling machine in **Figure 1**. The kinematic chain of the three-axis machine tool presented

The error description for one linear X-axis and one rotary C-axis is given in

*Scheme of deviations of three-axis kinematics at the machine MCV 754 QUICK, KOVOSVIT-MAS [8].*

EAX – angular roll error EAC - tilt error motion around the X of the C axis EBX - angular pitch error EBC - tilt error motion around the Y of the C axis

EXX – positioning error EXC – radial motion in X direction EYX – straightness error in Y direction EYC - radial motion in Y direction EZX - straightness error in Z direction EZC - axial motion of C axis

ECX - angular yaw error ECC - angular positioning error

**Linear axis X Rotary axis C**

with dynamic effects arising from the interpolation of axes [4].

*Machine Tools - Design, Research, Application*

below corresponds to W (Workpiece) -X-Y-Z-T (Tool) [8].

**Table 1**.

**Figure 1.**

**Table 1.**

**90**

*Error description for one linear axis.*

This parameter describes the accuracy and repeatability of positioning in linear and rotary numerically controlled axes. "Determination of accuracy and repeatability of positioning in numerically controlled axes" is described in the standard ISO 230-2/6 (ISO 230-2 Test code for machine tools—Determination of accuracy and repeatability of positioning numerically controlled axes; ISO 230-6 Test code for machine tools—Determination of positioning accuracy on body and face diagonals), but very often the directive VDI/DGQ 3441is also used [6].

Positioning accuracy is the most common form of measurement made with a laser interferometer (**Figure 2**). The laser system measures linear positioning accuracy and repeatability by comparing the position displayed on the machine with the actual position measured by the laser system.

A more advanced device for measurement of positioning accuracy of the machine is the Laser Tracker, which allows for immediate evaluation of the x, y, and z deviations. The geometric accuracy of the machine and the accuracy of positioning can be evaluated simultaneously (**Figure 3**) for an already assembled and activated machine. For this reason, the aforementioned accuracies are usually considered simultaneously [9].

under ideal machining conditions if the diameter and feed are the same for both

*Geometric Accuracy, Volumetric Accuracy and Compensation of CNC Machine Tools*

Advanced and highly progressive methods include the assessment of volumetric accuracy and its subsequent compensation. The purpose of these advanced compensations is to minimize the tool center point (TCP) deviation at any point in the machine measured workspace. TCP volumetric deviation is defined as the sum of

Volumetric accuracy of machine tools is represented by a vector map of error deviations in the workspace. In the standard ISO 230-1, the concept of volumetric accuracy for a three-axis center is defined as the maximum range of relative deviations between the actual and ideal position in the X, Y, Z directions and the maximum range of deviations orientation for directions of A, B, C axes for motions in X, Y, Z axes in the specified volume, where the deviations are the relative deviations between the tool and the workpiece on the machine tool for specified alignment of

The LaserTRACER measuring device (**Figure 4**) is mainly used for measuring of volumetric accuracy and subsequent volumetric compensation. The principle of the LaserTRACER measurement is based on measurement of beam lengths (HeNe laser wavelengths, 632.8 nm) and calculation of the measured point in the workspace by

With this method, it is necessary to measure gradually from multiple locations on the machine (it is recommended to measure from at least four LaserTRACER positions). The method is presented as an analogy to the GPS system [10].

This is a property of a machine tool that expresses the quality and productivity of a potential workpiece production. Working accuracy is expressed by the production of a test workpiece or a series of test workpieces. The working accuracy of the

machine is affected by the accuracy of the relative tool path [6].

machining and interpolation testing [1, 7].

*DOI: http://dx.doi.org/10.5772/intechopen.92085*

partial deviations in the individual axes [6].

the primary and secondary axes [1, 10].

the method of sequential multilateration.

**3.5 Working accuracy**

*Principle of measurement with LaserTRACER [etalon].*

**Figure 4.**

**93**

**3.4 Volumetric accuracy**

**Figure 2.** *Setting of measuring system for measurement of positioning accuracy [Renishaw].*

**Figure 3.**

*Synergy when evaluating geometric and positioning accuracy using a laser tracker [9].*

#### **3.3 Interpolation accuracy**

Theoretically, if the CNC machines were perfectly accurate, then the circular path of the machine would exactly match the programmed circular path. In practice, however, any of the errors (measuring error, straightness, clearance, reverse error, etc.) will cause the radius of the circle to deviate from the programmed circle. If we are able to accurately measure the actual circular path and compare it with the programmed (nominal) path, we would get a scale of the machine tool accuracy. Measurement and evaluation of circular interpolation accuracy are the subject of, for example, the standard ČSN ISO 230-4. The aim of the tests is to provide a method for estimating the properties of contour forming of numerically controlled machine tools. These errors are affected by the geometric errors and dynamic behavior of the machine at the feed used. Results are visible on machined parts

*Geometric Accuracy, Volumetric Accuracy and Compensation of CNC Machine Tools DOI: http://dx.doi.org/10.5772/intechopen.92085*

under ideal machining conditions if the diameter and feed are the same for both machining and interpolation testing [1, 7].

#### **3.4 Volumetric accuracy**

Advanced and highly progressive methods include the assessment of volumetric accuracy and its subsequent compensation. The purpose of these advanced compensations is to minimize the tool center point (TCP) deviation at any point in the machine measured workspace. TCP volumetric deviation is defined as the sum of partial deviations in the individual axes [6].

Volumetric accuracy of machine tools is represented by a vector map of error deviations in the workspace. In the standard ISO 230-1, the concept of volumetric accuracy for a three-axis center is defined as the maximum range of relative deviations between the actual and ideal position in the X, Y, Z directions and the maximum range of deviations orientation for directions of A, B, C axes for motions in X, Y, Z axes in the specified volume, where the deviations are the relative deviations between the tool and the workpiece on the machine tool for specified alignment of the primary and secondary axes [1, 10].

The LaserTRACER measuring device (**Figure 4**) is mainly used for measuring of volumetric accuracy and subsequent volumetric compensation. The principle of the LaserTRACER measurement is based on measurement of beam lengths (HeNe laser wavelengths, 632.8 nm) and calculation of the measured point in the workspace by the method of sequential multilateration.

With this method, it is necessary to measure gradually from multiple locations on the machine (it is recommended to measure from at least four LaserTRACER positions). The method is presented as an analogy to the GPS system [10].

**Figure 4.** *Principle of measurement with LaserTRACER [etalon].*

#### **3.5 Working accuracy**

This is a property of a machine tool that expresses the quality and productivity of a potential workpiece production. Working accuracy is expressed by the production of a test workpiece or a series of test workpieces. The working accuracy of the machine is affected by the accuracy of the relative tool path [6].

**3.3 Interpolation accuracy**

**Figure 2.**

**Figure 3.**

**92**

Theoretically, if the CNC machines were perfectly accurate, then the circular path of the machine would exactly match the programmed circular path. In practice, however, any of the errors (measuring error, straightness, clearance, reverse error, etc.) will cause the radius of the circle to deviate from the programmed circle. If we are able to accurately measure the actual circular path and compare it with the programmed (nominal) path, we would get a scale of the machine tool accuracy. Measurement and evaluation of circular interpolation accuracy are the subject of, for example, the standard ČSN ISO 230-4. The aim of the tests is to provide a method for estimating the properties of contour forming of numerically controlled machine tools. These errors are affected by the geometric errors and dynamic behavior of the machine at the feed used. Results are visible on machined parts

*Setting of measuring system for measurement of positioning accuracy [Renishaw].*

*Machine Tools - Design, Research, Application*

*Synergy when evaluating geometric and positioning accuracy using a laser tracker [9].*

The three main influences that affect the machine tool and the production process and cause workpiece dimensional variations can be more closely assigned

*Relationships between individual accuracies of a CNC machine tool throughout its life cycle.*

*Geometric Accuracy, Volumetric Accuracy and Compensation of CNC Machine Tools*

The above-mentioned partial accuracies of the machine tool can be divided into individual parts of the life cycle (**Figure 6**). Production accuracy can, therefore, be monitored at the phase of customer's machine use and is influenced by both the working accuracy of the machine and long-term stability of geometric accuracy.

One of the possibilities of compensating the error of linear and rotary axis is to use the so-called interpolation compensations, which include the compensation of leadscrew errors and measuring system errors [12]. In the SIEMENS control system, errors are referred to as LEC and MSEC (*LEC*-Leadscrew Error Compensation and *MSEC*-Measuring System Error Compensation). Compensation values are entered into the system in the form of tables, which are either manually entered or subprograms can be generated using various software solutions, which automatically load and write the table into the machine control system. Here, it should be taken into account that this automatic table loading can only work for given versions of the machine control system with the appropriate service pack. The MSEC compensation is also referred to as ENC\_COMP in the machine control system and, through this parameter, the compensation is gradually set and activated. The abbreviations

**4. Example of basic compensation: positioning accuracy**

depend on the type of machine control system.

to [4]:

**95**

**Figure 6.**

• production technology 15%,

*DOI: http://dx.doi.org/10.5772/intechopen.92085*

• measurement 15%,

• machined part 5%,

• ambient conditions 20%,

• machine operator 20%.

• working accuracy of the machine 25%,

**Figure 5.**

*Overview of the error budget in a machine tool and the factors affecting it [11].*


An overall summary of factors affecting the accuracy of the machine tool is shown in **Figure 5**. The resulting error in the Cartesian coordinate system is shown by Eq. (1) as a spatial error between the programmed and the actual TCP position [6].

Test workpieces to be tested for working accuracy are given, for example, by ISO 10791–7. Here, a test workpiece for three-axis machining is designed. Furthermore, test workpieces are aimed at continuous five-axis machining. An example is the test workpiece defined by the directive VDI NCG 5211-1.

#### **3.6 Production accuracy**

Production accuracy describes the production process accuracy evaluated on the workpiece. Production accuracy is influenced by geometrical accuracy, positioning accuracy, working accuracy, and also by the errors of machine operator (incorrectly adjusted tool, poorly clamped workpiece) and by changes of ambient conditions. Variations in the dimensions of the test workpieces during the production process provide direct information on production accuracy [6].

Production accuracy is usually monitored by SPC (statistical process control). This method has already been overcome in some production processes with 100% product control. Due to the spectrum of workpieces of medium-sized and large CNC machine tools, the SPC method can still be considered valid [6].

*Geometric Accuracy, Volumetric Accuracy and Compensation of CNC Machine Tools DOI: http://dx.doi.org/10.5772/intechopen.92085*


#### **Figure 6.**

• geometric accuracy of the machine;

*Machine Tools - Design, Research, Application*

*Overview of the error budget in a machine tool and the factors affecting it [11].*

• selection of cutting conditions, etc.

workpiece weight, etc.);

**Figure 5.**

**3.6 Production accuracy**

**94**

• tool positioning accuracy relative to the workpiece (positioning accuracy);

• resistance of the machine to elastic deformations (caused by cutting forces,

An overall summary of factors affecting the accuracy of the machine tool is shown

Production accuracy describes the production process accuracy evaluated on the workpiece. Production accuracy is influenced by geometrical accuracy, positioning accuracy, working accuracy, and also by the errors of machine operator (incorrectly adjusted tool, poorly clamped workpiece) and by changes of ambient conditions. Variations in the dimensions of the test workpieces during the production process

Production accuracy is usually monitored by SPC (statistical process control). This method has already been overcome in some production processes with 100% product control. Due to the spectrum of workpieces of medium-sized and large

CNC machine tools, the SPC method can still be considered valid [6].

• resistance of the machine to thermal expansion ("thermal stability");

in **Figure 5**. The resulting error in the Cartesian coordinate system is shown by Eq. (1) as a spatial error between the programmed and the actual TCP position [6]. Test workpieces to be tested for working accuracy are given, for example, by ISO 10791–7. Here, a test workpiece for three-axis machining is designed. Furthermore, test workpieces are aimed at continuous five-axis machining. An example is

the test workpiece defined by the directive VDI NCG 5211-1.

provide direct information on production accuracy [6].

*Relationships between individual accuracies of a CNC machine tool throughout its life cycle.*

The three main influences that affect the machine tool and the production process and cause workpiece dimensional variations can be more closely assigned to [4]:


The above-mentioned partial accuracies of the machine tool can be divided into individual parts of the life cycle (**Figure 6**). Production accuracy can, therefore, be monitored at the phase of customer's machine use and is influenced by both the working accuracy of the machine and long-term stability of geometric accuracy.

#### **4. Example of basic compensation: positioning accuracy**

One of the possibilities of compensating the error of linear and rotary axis is to use the so-called interpolation compensations, which include the compensation of leadscrew errors and measuring system errors [12]. In the SIEMENS control system, errors are referred to as LEC and MSEC (*LEC*-Leadscrew Error Compensation and *MSEC*-Measuring System Error Compensation). Compensation values are entered into the system in the form of tables, which are either manually entered or subprograms can be generated using various software solutions, which automatically load and write the table into the machine control system. Here, it should be taken into account that this automatic table loading can only work for given versions of the machine control system with the appropriate service pack. The MSEC compensation is also referred to as ENC\_COMP in the machine control system and, through this parameter, the compensation is gradually set and activated. The abbreviations depend on the type of machine control system.

Only unidirectional compensations can be made by ENC\_COMP compensation. In the event that a clearance error is found from the test, it is possible to use the Backlash compensation in combination with ENC\_COMP.

#### **4.1 Backlash**

During the transfer of force between the movable part of the machine and its drive—e.g., a ball screw and its mounting—there are clearances (gaps) at different load directions. Conversely, a complete clearance-free mechanical adjustment will dramatically increase machine wear and heat generation. Mechanical clearances cause deviations in the reverse path of axes or spindles with indirect measuring systems. This means that if the direction changes, the axis will travel depending on the gap size. These clearances are compensated by the function listed below as Backlash.

Backlash can be entered into the control system in several ways. The first option is to use the machine parameter and enter the value as a constant for the selected axis.

The second option is to use the SAG compensations and the CEC table, which will be described in the next step and eliminate the clearance error by bidirectional compensation. The advantage of the first solution is to specify only one constant. In the case of non-linear behavior, it is preferable to enter the clearance in the form of a CEC table.

when the motion axes are not properly aligned at the correct angle (e.g., vertical). As the deviation from the zero position increases, the positioning errors also increase. Both types of errors can occur as a result of shifting the weights of individual machine parts, replaceable heads, workpiece diversity, and machine compliance. Measured correction values are calculated based on the relevant standards or own algorithms and are stored in the machine control system in the form of

*Geometric Accuracy, Volumetric Accuracy and Compensation of CNC Machine Tools*

During machine operation and motion of axes, the corresponding value is interpolated between the values of the "interpolation points" table. For each motion in a continuous path, there is always both the base axis and the compensation axis. If the perpendicular y-axis is not in the continuous path of the x-axis and the y-axis, this inaccuracy is compensated by the x-axis in the continuous path. **Figure 7** shows the

principle of compensation on an example of a horizontal machine tool. The

straightness error of EYZ is largely due to the machine compliance, while, through the ram travel, the sag occurs which is caused by the load of the assembly spindle-

To use the SAG compensation for sagging compensations, the table for the

\$AN\_CEC[0,0]=0 ; first compensation value (interpolation point 0); for Z:

\$AN\_CEC[0,1]=0.01 ; second compensation value (interpolation point 1); for Z:

\$AN\_CEC[0,2]=0.012 ; third compensation value (interpolation point 2); for Z:

This compensation provides a wide range of options for elimination of geometric errors. Here, an example will be given to compensate a sag, e.g., caused by changing the load of the replaceable heads, where there may be significant differences in their weights. If the machine is without a replaceable head, the sag is shown in **Figure 7**. If a milling head with a certain weight is used, the travel will be more loaded;

a compensation table during commissioning.

therefore, a greater deformation will occur.

Siemens control system will be as follows:

0μm

+10μm

+12μm

ram-slide-accessory.

**Figure 7.**

*Error EYZ of horizontal machine tool.*

*DOI: http://dx.doi.org/10.5772/intechopen.92085*

%\_N\_NC\_CEC\_INI CHANDATA(1) ;

**97**

To use the MSEC compensation, the table for the Siemens control system will be as follows:


#### **5. Example of basic compensation: sag compensation**

In the previous paragraph, compensation in one MSEC axis was described [12]. In a large number of cases, MSEC compensation is insufficient and it is advisable to introduce corrections of two dependent axes. The sag compensation is performed when the weight of the individual machine elements leads to the positioning displacement and inclination of the moving parts, as this causes the related machine parts—including guide systems—to bend. The compensation error of angle is used

*Geometric Accuracy, Volumetric Accuracy and Compensation of CNC Machine Tools DOI: http://dx.doi.org/10.5772/intechopen.92085*

Only unidirectional compensations can be made by ENC\_COMP compensation. In the event that a clearance error is found from the test, it is possible to use the

During the transfer of force between the movable part of the machine and its drive—e.g., a ball screw and its mounting—there are clearances (gaps) at different load directions. Conversely, a complete clearance-free mechanical adjustment will dramatically increase machine wear and heat generation. Mechanical clearances cause deviations in the reverse path of axes or spindles with indirect measuring systems. This means that if the direction changes, the axis will travel depending on the gap size. These clearances are compensated by the function listed below as

Backlash can be entered into the control system in several ways. The first option is to use the machine parameter and enter the value as a constant for the

The second option is to use the SAG compensations and the CEC table, which will be described in the next step and eliminate the clearance error by bidirectional compensation. The advantage of the first solution is to specify only one constant. In the case of non-linear behavior, it is preferable to enter the clearance in the form of

To use the MSEC compensation, the table for the Siemens control system will be

\$AA\_ENC\_COMP[0,0,X1]=0.003 ; first compensation value (interpolation point 0):+3μm \$AA\_ENC\_COMP[0,1,X1]=0.01 ; second compensation value (interpolation

\$AA\_ENC\_COMP[0,2,X1]=0.012 ; third compensation value (interpolation

\$AA\_ENC\_COMP\_MIN[0,X1]=-200.0 ; Start of compensation -200.0 mm \$AA\_ENC\_COMP\_MAX[0,X1]=600.0 ; End of compensation +600.0 mm \$AA\_ENC\_COMP\_IS\_MODULO[0,X1]=0 ; Compensation without modulo

M17 function

**5. Example of basic compensation: sag compensation**

\$AA\_ENC\_COMP[0,800,X1]=-0.0 ; last compensation value (interpolation point 800): 0μm \$AA\_ENC\_COMP\_STEP[0,X1]=1.0 ; Distance between two compensation values 1.0 mm

In the previous paragraph, compensation in one MSEC axis was described [12]. In a large number of cases, MSEC compensation is insufficient and it is advisable to introduce corrections of two dependent axes. The sag compensation is performed when the weight of the individual machine elements leads to the positioning displacement and inclination of the moving parts, as this causes the related machine parts—including guide systems—to bend. The compensation error of angle is used

point 1): +10μm

point 2): +12μm

Backlash compensation in combination with ENC\_COMP.

*Machine Tools - Design, Research, Application*

**4.1 Backlash**

Backlash.

selected axis.

a CEC table.

as follows:

**96**

%\_N\_AX\_EEC\_INI CHANDATA(1)

when the motion axes are not properly aligned at the correct angle (e.g., vertical). As the deviation from the zero position increases, the positioning errors also increase. Both types of errors can occur as a result of shifting the weights of individual machine parts, replaceable heads, workpiece diversity, and machine compliance. Measured correction values are calculated based on the relevant standards or own algorithms and are stored in the machine control system in the form of a compensation table during commissioning.

During machine operation and motion of axes, the corresponding value is interpolated between the values of the "interpolation points" table. For each motion in a continuous path, there is always both the base axis and the compensation axis. If the perpendicular y-axis is not in the continuous path of the x-axis and the y-axis, this inaccuracy is compensated by the x-axis in the continuous path. **Figure 7** shows the principle of compensation on an example of a horizontal machine tool. The straightness error of EYZ is largely due to the machine compliance, while, through the ram travel, the sag occurs which is caused by the load of the assembly spindleram-slide-accessory.

This compensation provides a wide range of options for elimination of geometric errors. Here, an example will be given to compensate a sag, e.g., caused by changing the load of the replaceable heads, where there may be significant differences in their weights. If the machine is without a replaceable head, the sag is shown in **Figure 7**. If a milling head with a certain weight is used, the travel will be more loaded; therefore, a greater deformation will occur.

To use the SAG compensation for sagging compensations, the table for the Siemens control system will be as follows:


\$AN\_CEC[0,100]=0 ; last compensation value (interpolation point 101); for Z: 0μm \$AN\_CEC\_INPUT\_AXIS[0]=(AX2) ; base axis Y \$AN\_CEC\_OUTPUT\_AXIS[0]=(AX3) ; compensation in Z axis \$AN\_CEC\_STEP[0]=8 ; distance between interpolation points 8.0 mm \$AN\_CEC\_MIN[0]=0 ; start of compensation Y=0 mm \$AN\_CEC\_MAX[0]=800.0 ; end of compensation Y=800 mm \$AN\_CEC\_DIRECTION[0]=0 ; table applies to both directions of Y axis motions \$AN\_CEC\_MULT\_BY\_TABLE[0]=0; \$AN\_CEC\_IS\_MODULO[0]=0 ; compensation without modulo function M17 ;

tables, where one axis is determined as the base axis and the other as compensated. An example will be given to compensate the squareness of, for example, the Y and Z axes of a horizontal machining center. From the measured values obtained, for example, from measurements with a laser interferometer, ballbar or calibration cubes and dial gauges, we obtain information on the size and orientation of squareness, which may be, for example, 22.4 μm/m. It is necessary to respect the machine coordinate system and orientation of axes when preparing the measurements. Otherwise, for the verification measurement, the resulting error value will be multiplied. For a ram travel (Z axis), this means that for a travel length of 750 mm, the measured error of 22.4 μm/m must first be converted by a ratio of 750/1000 mm. After multiplying by the measured value, we obtain the value for entering the correction into the machine control system. In this case, the value at the 750 mm

*Geometric Accuracy, Volumetric Accuracy and Compensation of CNC Machine Tools*

For the above example, the compensation table for travel of the ram axis Z will

\$AN\_CEC[0,0]=0 ; first compensation value (interpolation point 0); for Z: 0μm \$AN\_CEC[0,100]=0.0168 ; last compensation value (interpolation point 11); for Z:

16.8μm

\$AN\_CEC\_OUTPUT\_AXIS[0]=(AX2) ; compensation in Y axis

\$AN\_CEC\_MIN[0]=0.0 ; start of compensation in Z=0 mm \$AN\_CEC\_MAX[0]=750.0 ; end of compensation in Z=750 mm

\$AN\_CEC\_STEP[0]=750 ; distance between interpolation points 750.0

\$AN\_CEC\_DIRECTION[0]=0 ; table applies to for both directions of Z axis motions

\$AN\_CEC\_IS\_MODULO[0]=0 ; compensation without modulo function

**6. Example of advanced compensation: volumetric compensation**

The DMU 75 monoBlock® machine (**Figure 8**) is kinematically adapted to have three linear motions in the tool (X = 750, Y = 650, Z = 560 mm) and two rotary motions in the workpiece (swinging about the X axis and rotation around the Z axis). It is equipped with the Heidenhain TNC 640 control system. This machine

The measurement and compensation of the volumetric accuracy of the linear machine axes are shown in **Figure 9**. After compensation, the workspace was

Before verification measurement of the volumetric accuracy, the machine was measured by a DBB device to verify the successful activation of volumetric compensation. **Figure 10** shows an improvement in the accuracy of circular interpolation on the shape of roundness (especially squareness); therefore, the machine was verified by the LaserTRACER to detect an improvement in overall volumetric

After compensating the volumetric accuracy of the linear axes, the rotary axis that is the first in the kinematic chain from the workpiece to the tool, i.e., the C axis, must first be measured. This axis was measured with an example of the results in

\$AN\_CEC\_INPUT\_AXIS[0]=(AX3) ; base axis Z

\$AN\_CEC\_MULT\_BY\_TABLE[0]=0 ;

has a positioning accuracy of 8 μm per axis.

improved by approx. 60%.

accuracy [13].

**99**

**Figures 11** and **12** [13].

position will be 16.8 μm.

*DOI: http://dx.doi.org/10.5772/intechopen.92085*

%\_N\_NC\_CEC\_INI ; CHANDATA(1) ;

be as follows.

M17 ;

If we use the SAG compensations for bidirectional axis compensation, the table for the Siemens control system will be as follows. The parameters of both the base axis and the compensated axis will be the same and match the axis designation. The direction parameter will be first set to 1 and then to 1. As an example of a horizontal boring machine, for the Z axis of ram travel, it will be as follows.

```
%_N_NC_CEC_INI ;
CHANDATA(1) ;
$AN_CEC[0,0]=0 ; first compensation value (interpolation point 0); for Z: 0μm
$AN_CEC[0,1]=0.01 ; second compensation value (interpolation point 1); for Z:
                    +10μm
$AN_CEC[0,2]=0.012 ; third compensation value (interpolation point 2); for Z:
                       +12μm
$AN_CEC[0,10]=0 ; last compensation value (interpolation point 11); for Z:
                      0μm
$AN_CEC_INPUT_AXIS[0]=(AX3) ; base axis Z
$AN_CEC_OUTPUT_AXIS[0]=(AX3) ; compensation in Z axis
$AN_CEC_STEP[0]=75 ; distance between interpolation points 75.0 mm
$AN_CEC_MIN[0]=0.0 ; start of compensation in Z=0 mm
$AN_CEC_MAX[0]=750.0 ; end of compensation in Z=750 mm
$AN_CEC_DIRECTION[0]=1 ; table applies to only positive direction of Z axis
$AN_CEC_MULT_BY_TABLE[0]=0 ;
$AN_CEC_IS_MODULO[0]=0 ; compensation without modulo function
$AN_CEC[0,0]=0 ; first compensation value (interpolation point 0); for Z: 0μm
$AN_CEC[0,1]=0.01 ; second compensation value (interpolation point 1); for Z:
                      +10μm
$AN_CEC[0,2]=0.012 ; third compensation value (interpolation point 2); for Z:
                       +12μm
$AN_CEC[0,11]=0 ; last compensation value (interpolation point 11); for Z:
                    0μm
$AN_CEC_INPUT_AXIS[0]=(AX3) ; base axis Z
$AN_CEC_OUTPUT_AXIS[0]=(AX3) ; compensation in Z axis
$AN_CEC_STEP[0]=75 ; distance between interpolation points 75.0 mm
$AN_CEC_MIN[0]=0.0 ; start of compensation in Z=0 mm
$AN_CEC_MAX[0]=750.0 ; end of compensation in Z=750 mm
$AN_CEC_DIRECTION[0]=-1 ; table applies to only positive direction of Z axis
$AN_CEC_MULT_BY_TABLE[0]=0 ;
$AN_CEC_IS_MODULO[0]=0 ; compensation without modulo function
M17 ;
```
Furthermore, SAG compensations are used to compensate squareness error. The squareness compensations of the Siemens control system are entered using CEC

*Geometric Accuracy, Volumetric Accuracy and Compensation of CNC Machine Tools DOI: http://dx.doi.org/10.5772/intechopen.92085*

tables, where one axis is determined as the base axis and the other as compensated. An example will be given to compensate the squareness of, for example, the Y and Z axes of a horizontal machining center. From the measured values obtained, for example, from measurements with a laser interferometer, ballbar or calibration cubes and dial gauges, we obtain information on the size and orientation of squareness, which may be, for example, 22.4 μm/m. It is necessary to respect the machine coordinate system and orientation of axes when preparing the measurements. Otherwise, for the verification measurement, the resulting error value will be multiplied. For a ram travel (Z axis), this means that for a travel length of 750 mm, the measured error of 22.4 μm/m must first be converted by a ratio of 750/1000 mm. After multiplying by the measured value, we obtain the value for entering the correction into the machine control system. In this case, the value at the 750 mm position will be 16.8 μm.

For the above example, the compensation table for travel of the ram axis Z will be as follows.

```
%_N_NC_CEC_INI ;
CHANDATA(1) ;
$AN_CEC[0,0]=0 ; first compensation value (interpolation point 0); for Z: 0μm
$AN_CEC[0,100]=0.0168 ; last compensation value (interpolation point 11); for Z:
                        16.8μm
$AN_CEC_INPUT_AXIS[0]=(AX3) ; base axis Z
$AN_CEC_OUTPUT_AXIS[0]=(AX2) ; compensation in Y axis
$AN_CEC_STEP[0]=750 ; distance between interpolation points 750.0
$AN_CEC_MIN[0]=0.0 ; start of compensation in Z=0 mm
$AN_CEC_MAX[0]=750.0 ; end of compensation in Z=750 mm
$AN_CEC_DIRECTION[0]=0 ; table applies to for both directions of Z axis
                             motions
$AN_CEC_MULT_BY_TABLE[0]=0 ;
$AN_CEC_IS_MODULO[0]=0 ; compensation without modulo function
M17 ;
```
#### **6. Example of advanced compensation: volumetric compensation**

The DMU 75 monoBlock® machine (**Figure 8**) is kinematically adapted to have three linear motions in the tool (X = 750, Y = 650, Z = 560 mm) and two rotary motions in the workpiece (swinging about the X axis and rotation around the Z axis). It is equipped with the Heidenhain TNC 640 control system. This machine has a positioning accuracy of 8 μm per axis.

The measurement and compensation of the volumetric accuracy of the linear machine axes are shown in **Figure 9**. After compensation, the workspace was improved by approx. 60%.

Before verification measurement of the volumetric accuracy, the machine was measured by a DBB device to verify the successful activation of volumetric compensation. **Figure 10** shows an improvement in the accuracy of circular interpolation on the shape of roundness (especially squareness); therefore, the machine was verified by the LaserTRACER to detect an improvement in overall volumetric accuracy [13].

After compensating the volumetric accuracy of the linear axes, the rotary axis that is the first in the kinematic chain from the workpiece to the tool, i.e., the C axis, must first be measured. This axis was measured with an example of the results in **Figures 11** and **12** [13].

\$AN\_CEC[0,100]=0 ; last compensation value (interpolation point 101); for Z:

\$AN\_CEC\_DIRECTION[0]=0 ; table applies to both directions of Y axis motions

If we use the SAG compensations for bidirectional axis compensation, the table for the Siemens control system will be as follows. The parameters of both the base axis and the compensated axis will be the same and match the axis designation. The

\$AN\_CEC[0,0]=0 ; first compensation value (interpolation point 0); for Z: 0μm \$AN\_CEC[0,1]=0.01 ; second compensation value (interpolation point 1); for Z:

\$AN\_CEC[0,2]=0.012 ; third compensation value (interpolation point 2); for Z:

\$AN\_CEC[0,10]=0 ; last compensation value (interpolation point 11); for Z:

\$AN\_CEC\_STEP[0]=75 ; distance between interpolation points 75.0 mm

\$AN\_CEC\_DIRECTION[0]=1 ; table applies to only positive direction of Z axis

\$AN\_CEC\_IS\_MODULO[0]=0 ; compensation without modulo function \$AN\_CEC[0,0]=0 ; first compensation value (interpolation point 0); for Z: 0μm \$AN\_CEC[0,1]=0.01 ; second compensation value (interpolation point 1); for Z:

\$AN\_CEC[0,2]=0.012 ; third compensation value (interpolation point 2); for Z:

\$AN\_CEC\_DIRECTION[0]=-1 ; table applies to only positive direction of Z axis

Furthermore, SAG compensations are used to compensate squareness error. The squareness compensations of the Siemens control system are entered using CEC

\$AN\_CEC[0,11]=0 ; last compensation value (interpolation point 11); for Z:

\$AN\_CEC\_STEP[0]=75 ; distance between interpolation points 75.0 mm

\$AN\_CEC\_IS\_MODULO[0]=0 ; compensation without modulo function

\$AN\_CEC\_STEP[0]=8 ; distance between interpolation points 8.0 mm

\$AN\_CEC\_IS\_MODULO[0]=0 ; compensation without modulo function

direction parameter will be first set to 1 and then to 1. As an example of a horizontal boring machine, for the Z axis of ram travel, it will be as follows.

0μm \$AN\_CEC\_INPUT\_AXIS[0]=(AX2) ; base axis Y

+10μm

\$AN\_CEC\_MULT\_BY\_TABLE[0]=0 ;

\$AN\_CEC\_MULT\_BY\_TABLE[0]=0 ;

M17 ;

**98**

+12μm

0μm \$AN\_CEC\_INPUT\_AXIS[0]=(AX3) ; base axis Z

+10μm

+12μm

\$AN\_CEC\_OUTPUT\_AXIS[0]=(AX3) ; compensation in Z axis

\$AN\_CEC\_MIN[0]=0.0 ; start of compensation in Z=0 mm \$AN\_CEC\_MAX[0]=750.0 ; end of compensation in Z=750 mm

0μm \$AN\_CEC\_INPUT\_AXIS[0]=(AX3) ; base axis Z

\$AN\_CEC\_OUTPUT\_AXIS[0]=(AX3) ; compensation in Z axis

\$AN\_CEC\_MIN[0]=0.0 ; start of compensation in Z=0 mm \$AN\_CEC\_MAX[0]=750.0 ; end of compensation in Z=750 mm

\$AN\_CEC\_MULT\_BY\_TABLE[0]=0;

*Machine Tools - Design, Research, Application*

M17 ;

%\_N\_NC\_CEC\_INI ; CHANDATA(1) ;

\$AN\_CEC\_OUTPUT\_AXIS[0]=(AX3) ; compensation in Z axis

\$AN\_CEC\_MIN[0]=0 ; start of compensation Y=0 mm \$AN\_CEC\_MAX[0]=800.0 ; end of compensation Y=800 mm

**Figure 8.** *View of DMU 75 monoBlock ® [DMG Mori].*

**7. Conclusion**

*Error of EYA axis A [13].*

**Figure 12.**

**Figure 13.**

**101**

*Cascading of accuracies in machine tools [13].*

**Figure 11.**

*Error of EAA axis A [13].*

The aforementioned accuracies are related to one another and it cannot be assumed, for example, that the desired working accuracy can be achieved by poor

**Figure 13** shows a machine tool with linear axes. If there are rotary axes on the machine, it is necessary to check the linear axes first and then check the rotary axes. These are also checked for geometrical, positioning, and volumetric accuracy. If all

geometric accuracy. **Figure 13** shows cascading of these accuracies.

*Geometric Accuracy, Volumetric Accuracy and Compensation of CNC Machine Tools*

*DOI: http://dx.doi.org/10.5772/intechopen.92085*

**Figure 9.** *Results of volumetric accuracy measurement of linear axes before and after compensation [13].*

**Figure 10.** *Accuracy of circular interpolation in XY plane before and after volumetric compensation [13].*

*Geometric Accuracy, Volumetric Accuracy and Compensation of CNC Machine Tools DOI: http://dx.doi.org/10.5772/intechopen.92085*

**Figure 11.** *Error of EAA axis A [13].*

**Figure 12.** *Error of EYA axis A [13].*

#### **7. Conclusion**

**Figure 9.**

**Figure 10.**

**100**

**Figure 8.**

*View of DMU 75 monoBlock ® [DMG Mori].*

*Machine Tools - Design, Research, Application*

*Results of volumetric accuracy measurement of linear axes before and after compensation [13].*

*Accuracy of circular interpolation in XY plane before and after volumetric compensation [13].*

The aforementioned accuracies are related to one another and it cannot be assumed, for example, that the desired working accuracy can be achieved by poor geometric accuracy. **Figure 13** shows cascading of these accuracies.

**Figure 13** shows a machine tool with linear axes. If there are rotary axes on the machine, it is necessary to check the linear axes first and then check the rotary axes. These are also checked for geometrical, positioning, and volumetric accuracy. If all

**Figure 13.** *Cascading of accuracies in machine tools [13].*

the accuracies are within the required tolerances, the working accuracy related to the machining of the workpiece can be stepped to. Individual accuracies are described in the following section.

**References**

[1] Marek T. Měření a kontrola

Obráběcích Strojů IV. Praha: MM Publishing, s.r.o.; 2018.

[2] Marek J. Konstrukce CNC Obráběcích Strojů III. Praha: MM Publishing, s.r.o.; 2014. ISBN:

[3] Holub M, Andrs O, Kovar J, Vetiska J. Effect of position of temperature sensors on the resulting volumetric accuracy of the machine tool. Measurement [Online]. 2020;**150**: 107074. DOI: 10.1016/j.measurement.

2019.107074. ISSN: 02632241

ISBN: 978-80-87017-20-3

Grumant s.r.o.; 2017

978-3-030-03821-2

Zkušebnictví; 2014

201534. ISSN: 18031269

**103**

[4] Marek T, Marek J. Mít Sondu Nestačí. Brno: Renishaw s.r.o.; 2017.

[5] Marek J. Stavba CNC Obráběcích Strojů—Souvislosti a Fakta. Praha:

[6] Holub M. Geometric accuracy of machine tools. In: Measurement in Machining and Tribology [Online]. Materials Forming, Machining and Tribology. Cham: Springer International Publishing; 2019. pp. 89-112. DOI: 10.1007/978-3-030-03822-9. ISBN:

[7] ČSN ISO 230-1 Zásady Zkoušek Obráběcích Strojů - Část 1: Geometrická Přesnost Strojů Pracujících bez Zatížení Nebo za Dokončovacích Podmínek Obrábění. Praha: Úřad pro Technickou Normalizaci, Metrologii a Státní

[8] Holub M, Blecha P, Bradac F, Kana R. Volumetric compensation of threeaxis vertical machining centre. MM Science Journal [Online]. 2015;**2015**(03): 677-681. DOI: 10.17973/MMSJ.2015\_10\_

978-80-260-6780-1

obráběcích strojů. In: Konstrukce CNC

*DOI: http://dx.doi.org/10.5772/intechopen.92085*

*Geometric Accuracy, Volumetric Accuracy and Compensation of CNC Machine Tools*

[9] Knobloch J, Holub M, Kolouch M. Laser tracker measurement for prediction of workpiece geometric accuracy. In: Engineering Mechanics. Vol. 1. Svratka: Institute of Solid Mechanics, Mechatronics and

Biomechanics; 2014. pp. 296-300. ISBN: 978-80-214-4871-1. ISSN: 1805-8248

[10] Holub M, Kol. GTS—Testování Obráběcích Strojů. Brno: VUT Brno –

[11] Ramesh R, Mannan MA, Poo AN. Error compensation in machine tools— A review. International Journal of Machine Tools and Manufacture [Online]. 2000;**40**(9):1235-1256. DOI: 10.1016/S0890-6955(00)00009-2.

[12] Holub M. Kompenzace geometrické přesnosti CNC obráběcích strojů. In: Konstrukce CNC Obráběcích Strojů IV.

pp. 352-364. ISBN: 978-80-906310-8-3

[13] Marek T. Predikování Vybraných Vlastností Rotačních Kinematických Dvojic Obráběcích Strojů [Online]. Vysoké Učení Technické v Brně. 2019. Available from: https://www.vutbr.cz/ studenti/zav-prace/detail/122509

Praha: MM Publishing; 2018.

UVSSR; 2016

ISSN: 08906955

pp. 336-350. ISBN: 978-80-906310-8-3

### **Acknowledgements**

These results were obtained with the financial support of the Faculty of Mechanical Engineering, Brno University of Technology (Grant No. FSI-S-20-6335).

### **Author details**

Jiri Marek\*, Michal Holub, Tomas Marek and Petr Blecha Institute of Production Machines, Systems and Robotics, Brno University of Technology, Brno, Czech Republic

\*Address all correspondence to: marek@fme.vutbr.cz

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Geometric Accuracy, Volumetric Accuracy and Compensation of CNC Machine Tools DOI: http://dx.doi.org/10.5772/intechopen.92085*

#### **References**

the accuracies are within the required tolerances, the working accuracy related to the machining of the workpiece can be stepped to. Individual accuracies are

These results were obtained with the financial support of the Faculty of Mechanical

Engineering, Brno University of Technology (Grant No. FSI-S-20-6335).

described in the following section.

*Machine Tools - Design, Research, Application*

**Acknowledgements**

**Author details**

**102**

Technology, Brno, Czech Republic

provided the original work is properly cited.

Jiri Marek\*, Michal Holub, Tomas Marek and Petr Blecha

\*Address all correspondence to: marek@fme.vutbr.cz

Institute of Production Machines, Systems and Robotics, Brno University of

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

[1] Marek T. Měření a kontrola obráběcích strojů. In: Konstrukce CNC Obráběcích Strojů IV. Praha: MM Publishing, s.r.o.; 2018. pp. 336-350. ISBN: 978-80-906310-8-3

[2] Marek J. Konstrukce CNC Obráběcích Strojů III. Praha: MM Publishing, s.r.o.; 2014. ISBN: 978-80-260-6780-1

[3] Holub M, Andrs O, Kovar J, Vetiska J. Effect of position of temperature sensors on the resulting volumetric accuracy of the machine tool. Measurement [Online]. 2020;**150**: 107074. DOI: 10.1016/j.measurement. 2019.107074. ISSN: 02632241

[4] Marek T, Marek J. Mít Sondu Nestačí. Brno: Renishaw s.r.o.; 2017. ISBN: 978-80-87017-20-3

[5] Marek J. Stavba CNC Obráběcích Strojů—Souvislosti a Fakta. Praha: Grumant s.r.o.; 2017

[6] Holub M. Geometric accuracy of machine tools. In: Measurement in Machining and Tribology [Online]. Materials Forming, Machining and Tribology. Cham: Springer International Publishing; 2019. pp. 89-112. DOI: 10.1007/978-3-030-03822-9. ISBN: 978-3-030-03821-2

[7] ČSN ISO 230-1 Zásady Zkoušek Obráběcích Strojů - Část 1: Geometrická Přesnost Strojů Pracujících bez Zatížení Nebo za Dokončovacích Podmínek Obrábění. Praha: Úřad pro Technickou Normalizaci, Metrologii a Státní Zkušebnictví; 2014

[8] Holub M, Blecha P, Bradac F, Kana R. Volumetric compensation of threeaxis vertical machining centre. MM Science Journal [Online]. 2015;**2015**(03): 677-681. DOI: 10.17973/MMSJ.2015\_10\_ 201534. ISSN: 18031269

[9] Knobloch J, Holub M, Kolouch M. Laser tracker measurement for prediction of workpiece geometric accuracy. In: Engineering Mechanics. Vol. 1. Svratka: Institute of Solid Mechanics, Mechatronics and Biomechanics; 2014. pp. 296-300. ISBN: 978-80-214-4871-1. ISSN: 1805-8248

[10] Holub M, Kol. GTS—Testování Obráběcích Strojů. Brno: VUT Brno – UVSSR; 2016

[11] Ramesh R, Mannan MA, Poo AN. Error compensation in machine tools— A review. International Journal of Machine Tools and Manufacture [Online]. 2000;**40**(9):1235-1256. DOI: 10.1016/S0890-6955(00)00009-2. ISSN: 08906955

[12] Holub M. Kompenzace geometrické přesnosti CNC obráběcích strojů. In: Konstrukce CNC Obráběcích Strojů IV. Praha: MM Publishing; 2018. pp. 352-364. ISBN: 978-80-906310-8-3

[13] Marek T. Predikování Vybraných Vlastností Rotačních Kinematických Dvojic Obráběcích Strojů [Online]. Vysoké Učení Technické v Brně. 2019. Available from: https://www.vutbr.cz/ studenti/zav-prace/detail/122509

**Chapter 6**

**Abstract**

with an accuracy of micrometers.

**1. Introduction**

**105**

Actively Controlled Journal

Bearings for Machine Tools

*Renata Wagnerová and Stanislav Žiaran*

*Jiří Tůma, Jiří Šimek, Miroslav Mahdal, Jaromír Škuta,*

The advantage of journal hydrodynamic bearings is high radial load capacity and operation at high speeds. The disadvantage is the excitation of vibrations, called an oil whirl, after crossing a certain threshold of the rotational speed which depends on the radial bearing clearance, the viscosity of lubricating oil, and the rotor mass. A passive way of how to suppress vibrations consists of adjusting the shape of the bearing bushing. Vibrations can be suppressed using the system of active vibration damping with piezoactuators to move the bearing bushing in two directions. The displacement of the bearing bushing is actuated by two piezoactuators, which respond to the position of the bearing journal relative to the bearing housing. Two stacked linear piezoactuators are used to actuate the location of the bearing bushing. A pair of capacitive sensors senses the position of the journal or shaft. The system of the actively controlled journal bearings is the first functional prototype in the known up to now. It works with a cylindrical bushing which does not require special technology of manufacturing and assembly. This new bearing enables not only to damp vibrations but also serves to maintain the desired bearing journal position

**Keywords:** journal bearings, active vibration control, piezoactuators, oil whirl

called sleeve bearings or plain bearings for radial load) lie in demand for the introduction of high-speed cutting or machining as a technology for the future. Increasing the machining speed was required to get beyond the limits of the interval, where unwanted temperature increases. Some researchers define high-speed machining as machining whereby conventional cutting speeds are exceeded by a factor of 5–10. Increased machining speed has advantages. The ability to benefit the advantages of high-speed cutting in steel, cast iron, and nickel-based alloys can be obtained with spindle speeds in the range of 8k to 12k rpm. High-speed cutting of nonferrous materials such as brass, aluminum, and engineered plastics demands a significantly higher rpm capability. For these materials, we must focus on milling equipment capable of operating at high-speed spindle speeds of 25k to 50k rpm or

more. High-speed machining can also include grinding and turning.

The reasons for the interest in actively controlled journal bearings (alternatively

#### **Chapter 6**

## Actively Controlled Journal Bearings for Machine Tools

*Jiří Tůma, Jiří Šimek, Miroslav Mahdal, Jaromír Škuta, Renata Wagnerová and Stanislav Žiaran*

#### **Abstract**

The advantage of journal hydrodynamic bearings is high radial load capacity and operation at high speeds. The disadvantage is the excitation of vibrations, called an oil whirl, after crossing a certain threshold of the rotational speed which depends on the radial bearing clearance, the viscosity of lubricating oil, and the rotor mass. A passive way of how to suppress vibrations consists of adjusting the shape of the bearing bushing. Vibrations can be suppressed using the system of active vibration damping with piezoactuators to move the bearing bushing in two directions. The displacement of the bearing bushing is actuated by two piezoactuators, which respond to the position of the bearing journal relative to the bearing housing. Two stacked linear piezoactuators are used to actuate the location of the bearing bushing. A pair of capacitive sensors senses the position of the journal or shaft. The system of the actively controlled journal bearings is the first functional prototype in the known up to now. It works with a cylindrical bushing which does not require special technology of manufacturing and assembly. This new bearing enables not only to damp vibrations but also serves to maintain the desired bearing journal position with an accuracy of micrometers.

**Keywords:** journal bearings, active vibration control, piezoactuators, oil whirl

#### **1. Introduction**

The reasons for the interest in actively controlled journal bearings (alternatively called sleeve bearings or plain bearings for radial load) lie in demand for the introduction of high-speed cutting or machining as a technology for the future. Increasing the machining speed was required to get beyond the limits of the interval, where unwanted temperature increases. Some researchers define high-speed machining as machining whereby conventional cutting speeds are exceeded by a factor of 5–10. Increased machining speed has advantages. The ability to benefit the advantages of high-speed cutting in steel, cast iron, and nickel-based alloys can be obtained with spindle speeds in the range of 8k to 12k rpm. High-speed cutting of nonferrous materials such as brass, aluminum, and engineered plastics demands a significantly higher rpm capability. For these materials, we must focus on milling equipment capable of operating at high-speed spindle speeds of 25k to 50k rpm or more. High-speed machining can also include grinding and turning.

Let us now notice the machine tool spindles. Roller bearings support these spindles. Prestressed ball or tapered roller bearings are used to eliminate play. This chapter focuses on plain radial bearings, namely, hydrodynamic bearings. Plain bearings of this type require a clearance for their function, which is selected in the range of 0.1–0.3% of the journal diameter.

The value of the Sommerfeld number for the given bearing size and the rotor mass of 0.83 kg is as follows: *R* = 0.014 *N*, where *N* is the mentioned rotational

The magnitude of friction coefficient in the plain bearings was analysed in the past by the McKee brothers [2]. It has been found that bearing friction is dependent on a dimensionless bearing characteristic given by a ratio *μN=P* whose parameters are defined above. If the rotor does not rotate or rotate slowly, there is only a very thin film between the journal and the bushing. Boundary or thin-film boundary lubrication occurs with a considerably increased coefficient of friction. Many experiments show that the journal axle moves chaotically at low speeds or the journal starts to oscillate. It is uncertain at the start of run-up whether the axle of the bearing journal moves to the left or right, regardless of the direction of rotation. Only when the specified speed limit is exceeded the lubrication becomes hydrodynamic, and thick film of the lubricant is formed, and the trajectory of motion can be predicted. The limit value of the bearing characteristic for the boundary lubrication is described in [1]. Designers keep the value *<sup>μ</sup>N=P*≥1*:*<sup>7</sup> <sup>10</sup><sup>6</sup> (reyn rev/s/psi), which is about five times the value the McKee brothers have determined. The measurement in our test rig shows the limit of the boundary lubrication at about 1k rpm, which corresponds to the value of the dimensionless characteristic *μN=P* equaled to 3*:*<sup>8</sup> <sup>10</sup><sup>5</sup> (Pa.s rev/s/Pa) when using SI units for the input parameters. Our estimate for the lower limit of hydrodynamic lubrication corresponds to the recommendations in the handbook [1]. In experiments with the active vibration

An example of a gradual change of position of the bearing journal centre during an increase in speed up to 7k rpm at the constant increase rate is shown in **Figure 2**. The lubrication is of the boundary type in the range up to about 1.2k rpm and is

The reason for the oscillations is the step change of speed to about 300 rpm after switching on because it is not possible to increase the rotational speed continuously from zero. Hydrodynamic lubrication at stable motion is produced for rotational speed up to 5k rpm. Motion instability of the whirl type occurs when this speed of 5k rpm is exceeded. Fluid force makes sense to be modeled just for stable motion

control, the feedback is closed only for stable lubrication.

*Actively Controlled Journal Bearings for Machine Tools DOI: http://dx.doi.org/10.5772/intechopen.92272*

accompanied by oscillations.

**Figure 2.**

**107**

*The run-up of a journal bearing.*

speed.

Journal hydrodynamic bearings are a standard solution to support rotors. Their advantage is a possibility to carry the high radial load and to operate at high rotational speeds. The disadvantage of the journal bearings is the excitation of unwanted rotor vibrations by whirling of the journal in the bearing bushing. The bearing journal becomes unstable as the journal axis begins to perform a circular motion that is bounded only by the walls of the bearing bushing. When the speed threshold is exceeded, the axis of the bearing journal starts to circulate, causing the rotor to vibrate. These vibrations are called whirl. A passive way of how to suppress vibrations consists in adjusting the shape of the bearing bushing, such as lemon or elliptical bore of the bushing, or the use of tilting pads. Even though there are several solutions based on mentioned passive improvements, this article deals with the use of active vibration control (AVC) with piezo-actuators as a measure to prevent instability.

The disadvantage of bearings of this type is whirl instability, which can cause machine tool vibrations. The following chapter describes the possible operating range of the spindle speed.

#### **2. Operating speed range of plain bearings**

Special oil for high-speed spindle bearing of the OL-P03 type was used for testing (VG 10 grade, viscosity *μ* = 0.027 Pa.s at 20°C). Tests were carried out without preheating the lubricant at a normal temperature. The oil viscosity at ambient temperature in the laboratory corresponded to the oil viscosity at 40°C in industrial bearings. The journal bearing cross-section is shown in **Figure 1**.

The operating conditions of the hydrodynamic bearing are described by the Sommerfeld number [1]:

$$\mathbf{S} = \left(\mathbf{R}/\mathbf{c}\right)^{2} \mu \mathbf{N}/\mathbf{P},\tag{1}$$

where *N* is a rotational speed of the rotor in rev/s, *μ* is a dynamic viscosity in Pa.s, *R* is a radius of the journal, *c* is a radial clearance, *P* is a load per unit of projected bearing area (2*RL*) in N/m<sup>2</sup> , where *L* is a bearing length.

**Figure 1.** *Side-view technical drawing and a photo of bearing housing.*

*Actively Controlled Journal Bearings for Machine Tools DOI: http://dx.doi.org/10.5772/intechopen.92272*

Let us now notice the machine tool spindles. Roller bearings support these spindles. Prestressed ball or tapered roller bearings are used to eliminate play. This chapter focuses on plain radial bearings, namely, hydrodynamic bearings. Plain bearings of this type require a clearance for their function, which is selected in the

Journal hydrodynamic bearings are a standard solution to support rotors. Their

The disadvantage of bearings of this type is whirl instability, which can cause machine tool vibrations. The following chapter describes the possible operating

Special oil for high-speed spindle bearing of the OL-P03 type was used for testing (VG 10 grade, viscosity *μ* = 0.027 Pa.s at 20°C). Tests were carried out without preheating the lubricant at a normal temperature. The oil viscosity at ambient temperature in the laboratory corresponded to the oil viscosity at 40°C in

The operating conditions of the hydrodynamic bearing are described by the

2

, where *L* is a bearing length.

where *N* is a rotational speed of the rotor in rev/s, *μ* is a dynamic viscosity in Pa.s, *R* is a radius of the journal, *c* is a radial clearance, *P* is a load per unit of

*μN=P*, (1)

industrial bearings. The journal bearing cross-section is shown in **Figure 1**.

*S* ¼ ð Þ *R=c*

advantage is a possibility to carry the high radial load and to operate at high rotational speeds. The disadvantage of the journal bearings is the excitation of unwanted rotor vibrations by whirling of the journal in the bearing bushing. The bearing journal becomes unstable as the journal axis begins to perform a circular motion that is bounded only by the walls of the bearing bushing. When the speed threshold is exceeded, the axis of the bearing journal starts to circulate, causing the rotor to vibrate. These vibrations are called whirl. A passive way of how to suppress vibrations consists in adjusting the shape of the bearing bushing, such as lemon or elliptical bore of the bushing, or the use of tilting pads. Even though there are several solutions based on mentioned passive improvements, this article deals with the use of active vibration control (AVC) with piezo-actuators as a measure to

range of 0.1–0.3% of the journal diameter.

*Machine Tools - Design, Research, Application*

prevent instability.

range of the spindle speed.

Sommerfeld number [1]:

**Figure 1.**

**106**

projected bearing area (2*RL*) in N/m<sup>2</sup>

*Side-view technical drawing and a photo of bearing housing.*

**2. Operating speed range of plain bearings**

The value of the Sommerfeld number for the given bearing size and the rotor mass of 0.83 kg is as follows: *R* = 0.014 *N*, where *N* is the mentioned rotational speed.

The magnitude of friction coefficient in the plain bearings was analysed in the past by the McKee brothers [2]. It has been found that bearing friction is dependent on a dimensionless bearing characteristic given by a ratio *μN=P* whose parameters are defined above. If the rotor does not rotate or rotate slowly, there is only a very thin film between the journal and the bushing. Boundary or thin-film boundary lubrication occurs with a considerably increased coefficient of friction. Many experiments show that the journal axle moves chaotically at low speeds or the journal starts to oscillate. It is uncertain at the start of run-up whether the axle of the bearing journal moves to the left or right, regardless of the direction of rotation. Only when the specified speed limit is exceeded the lubrication becomes hydrodynamic, and thick film of the lubricant is formed, and the trajectory of motion can be predicted. The limit value of the bearing characteristic for the boundary lubrication is described in [1]. Designers keep the value *<sup>μ</sup>N=P*≥1*:*<sup>7</sup> <sup>10</sup><sup>6</sup> (reyn rev/s/psi), which is about five times the value the McKee brothers have determined. The measurement in our test rig shows the limit of the boundary lubrication at about 1k rpm, which corresponds to the value of the dimensionless characteristic *μN=P* equaled to 3*:*<sup>8</sup> <sup>10</sup><sup>5</sup> (Pa.s rev/s/Pa) when using SI units for the input parameters. Our estimate for the lower limit of hydrodynamic lubrication corresponds to the recommendations in the handbook [1]. In experiments with the active vibration control, the feedback is closed only for stable lubrication.

An example of a gradual change of position of the bearing journal centre during an increase in speed up to 7k rpm at the constant increase rate is shown in **Figure 2**. The lubrication is of the boundary type in the range up to about 1.2k rpm and is accompanied by oscillations.

The reason for the oscillations is the step change of speed to about 300 rpm after switching on because it is not possible to increase the rotational speed continuously from zero. Hydrodynamic lubrication at stable motion is produced for rotational speed up to 5k rpm. Motion instability of the whirl type occurs when this speed of 5k rpm is exceeded. Fluid force makes sense to be modeled just for stable motion

**Figure 2.** *The run-up of a journal bearing.*

and hydrodynamic lubrication. It is almost impossible to determine the initial conditions for boundary lubrication. Notice that the centre of the journal rises to the level of the centre of the bearing bore and gradually approaches this centre so that the small eccentricity gradually decreases to zero as is shown on the right panel of **Figure 2**. The data for this orbit was approximated by the five-degree polynomial in the time interval which begins at the 3rd second and ends at the 12th second. The difference between thin and thick film lubrication is also evident on the right panel of **Figure 2**, which depicts an orbit plot for the entire measurement time up to sixteenth-second.

instability but also enables to maintain the desired bearing journal position with an

The bearing journal can be considered as a rigid body rotating within the bearing housing at an angular velocity Ω. For simplicity, it is assumed that the rotation axis does not change its direction in contrast to a model [6]. Fluid forces are caused by the hydrodynamic pressure generated in the oil film, whose total mass relative to the journal and rotor is negligible. The oil pumped by the rotating journal surface produces an oil wedge that lifts up the bearing journal so that it does not touch the inner walls of the housing. The coordinate system of a cylindrical journal bearing is shown on the left side in **Figure 4**. The planar motion of the bearing journal at the *x* and *y* coordinates can be described by two motion equations arranged into a matrix

accuracy of micrometers.

*Actively Controlled Journal Bearings for Machine Tools DOI: http://dx.doi.org/10.5772/intechopen.92272*

**4. Mathematical model**

**4.1 Equation of motion**

equation:

**Figure 4.**

**109**

*M* 0 0 *M x*€

€*y* þ

*X* or *Y* and *V* is equal to *X* or *Y* as well.

values such as stiffness and damping.

*A cross-section of the hydrodynamic bearing.*

*BXX BXY BYX BYY <sup>x</sup>*\_

the gravity force *G*, as is shown in the right panel of **Figure 4**.

*y*\_ þ

where *M* is a mass of the rotor; *FX* is a force acting on the journal in the horizontal direction; *FY* is a force acting on the journal in the vertical direction; *CUV* is a stiffness coefficient; and *BUV* is a damping coefficient, where *U* is equal to

In addition to force components in the horizontal and vertical directions, the force balance will be solved in other possible directions. Force in the direction of the line of the centers is denoted as a direct force *FD*, while force which is perpendicular to the line of centers is denoted as a quadrature force *FQ* . Both these forces balance

The system is described by two motion equations, and therefore the total order of the system is four. This system may become unstable even for positive parameter

*CXX CXY CYX CYY <sup>x</sup>*

*y* 

<sup>¼</sup> *FX FY*

, (2)

The threshold of the instability of the journal movement in the bearing is given by the clearance and viscosity of the oil, which depends on the temperature.

#### **3. Instrumentations**

For developing a new design of the actively controlled bearing, a test rig was built; see **Figure 3**. This figure provides different views of the test rig. An inductive motor of 400 Hz drives the rotor, and therefore the maximum rotational speed is 23k rpm. The engine is connected to the rotor via the Huco diaphragm coupling. The bearing diameter is 30 mm, and the length-to-diameter ratio is equal to about 0.77. The span of bearing pedestals is of 200 mm. The results of the experiments presented in this article are for the radial clearance of 45 μm. Also, the journals of the other clearance are available for testing. The performance of the actively controlled bearing was tested on the test bench (Rotorkit) of the TECHLAB design [3, 4]. Additionally, it should be emphasised that research was focused at rigid rotors and the journal bushing of the cylindrical bore, where the journal motion is measured at the location closest to the bearing bushing. The research work resulted in putting into operation of the active vibration control system, which became the first functional bearing prototype known up to now [5].

The mechanical arrangement of the actively controlled bearing is shown on the right of **Figure 1**. Oil leakage from the volume between the bearing body and the loose bushing and the piezoactuator rod is sealed with rubber O-rings. As it was stated before, vibrations of the rotor is suppressed using the system for active vibration control with piezoelectric actuators enabling to move the non-rotating loose bushing. The motion of the bearing bushing is controlled by the controller, which responds to the change in position of the bearing journal related to the bearing housing. Two stacked linear piezoactuators are used to actuate the position of the bearing journal via the position of the bearing bushing. The bearing uses a cylindrical bushing which did not require unique technology of production and assembly. This new bearing enables not only to damp vibrations and to prevent

**Figure 3.** *Actively controlled journal bearings.*

instability but also enables to maintain the desired bearing journal position with an accuracy of micrometers.

### **4. Mathematical model**

#### **4.1 Equation of motion**

and hydrodynamic lubrication. It is almost impossible to determine the initial conditions for boundary lubrication. Notice that the centre of the journal rises to the level of the centre of the bearing bore and gradually approaches this centre so that the small eccentricity gradually decreases to zero as is shown on the right panel of **Figure 2**. The data for this orbit was approximated by the five-degree polynomial in the time interval which begins at the 3rd second and ends at the 12th second. The difference between thin and thick film lubrication is also evident on the right panel of **Figure 2**, which depicts an orbit plot for the entire measurement time up to

The threshold of the instability of the journal movement in the bearing is given

For developing a new design of the actively controlled bearing, a test rig was built; see **Figure 3**. This figure provides different views of the test rig. An inductive motor of 400 Hz drives the rotor, and therefore the maximum rotational speed is 23k rpm. The engine is connected to the rotor via the Huco diaphragm coupling. The bearing diameter is 30 mm, and the length-to-diameter ratio is equal to about 0.77. The span of bearing pedestals is of 200 mm. The results of the experiments presented in this article are for the radial clearance of 45 μm. Also, the journals of the other clearance are available for testing. The performance of the actively controlled bearing was tested on the test bench (Rotorkit) of the TECHLAB design [3, 4]. Additionally, it should be emphasised that research was focused at rigid rotors and the journal bushing of the cylindrical bore, where the journal motion is measured at the location closest to the bearing bushing. The research work resulted in putting into operation of the active vibration control system, which became the

The mechanical arrangement of the actively controlled bearing is shown on the right of **Figure 1**. Oil leakage from the volume between the bearing body and the loose bushing and the piezoactuator rod is sealed with rubber O-rings. As it was stated before, vibrations of the rotor is suppressed using the system for active vibration control with piezoelectric actuators enabling to move the non-rotating loose bushing. The motion of the bearing bushing is controlled by the controller, which responds to the change in position of the bearing journal related to the bearing housing. Two stacked linear piezoactuators are used to actuate the position of the bearing journal via the position of the bearing bushing. The bearing uses a cylindrical bushing which did not require unique technology of production and assembly. This new bearing enables not only to damp vibrations and to prevent

by the clearance and viscosity of the oil, which depends on the temperature.

first functional bearing prototype known up to now [5].

sixteenth-second.

*Machine Tools - Design, Research, Application*

**3. Instrumentations**

**Figure 3.**

**108**

*Actively controlled journal bearings.*

The bearing journal can be considered as a rigid body rotating within the bearing housing at an angular velocity Ω. For simplicity, it is assumed that the rotation axis does not change its direction in contrast to a model [6]. Fluid forces are caused by the hydrodynamic pressure generated in the oil film, whose total mass relative to the journal and rotor is negligible. The oil pumped by the rotating journal surface produces an oil wedge that lifts up the bearing journal so that it does not touch the inner walls of the housing. The coordinate system of a cylindrical journal bearing is shown on the left side in **Figure 4**. The planar motion of the bearing journal at the *x* and *y* coordinates can be described by two motion equations arranged into a matrix equation:

$$
\begin{bmatrix} \mathbf{M} & \mathbf{0} \\ \mathbf{0} & \mathbf{M} \end{bmatrix} \begin{bmatrix} \ddot{\boldsymbol{x}} \\ \ddot{\boldsymbol{y}} \end{bmatrix} + \begin{bmatrix} \mathbf{B}\_{\mathbf{X}\mathbf{X}} & \mathbf{B}\_{\mathbf{XY}} \\ \mathbf{B}\_{\mathbf{Y}\mathbf{X}} & \mathbf{B}\_{\mathbf{YY}} \end{bmatrix} \begin{bmatrix} \dot{\boldsymbol{x}} \\ \dot{\boldsymbol{y}} \end{bmatrix} + \begin{bmatrix} \mathbf{C}\_{\mathbf{XX}} & \mathbf{C}\_{\mathbf{XY}} \\ \mathbf{C}\_{\mathbf{Y}\mathbf{X}} & \mathbf{C}\_{\mathbf{YY}} \end{bmatrix} \begin{bmatrix} \mathbf{x} \\ \mathbf{y} \end{bmatrix} = \begin{bmatrix} \mathbf{F}\_{\mathbf{X}} \\ \mathbf{F}\_{\mathbf{Y}} \end{bmatrix},\tag{2}
$$

where *M* is a mass of the rotor; *FX* is a force acting on the journal in the horizontal direction; *FY* is a force acting on the journal in the vertical direction; *CUV* is a stiffness coefficient; and *BUV* is a damping coefficient, where *U* is equal to *X* or *Y* and *V* is equal to *X* or *Y* as well.

In addition to force components in the horizontal and vertical directions, the force balance will be solved in other possible directions. Force in the direction of the line of the centers is denoted as a direct force *FD*, while force which is perpendicular to the line of centers is denoted as a quadrature force *FQ* . Both these forces balance the gravity force *G*, as is shown in the right panel of **Figure 4**.

The system is described by two motion equations, and therefore the total order of the system is four. This system may become unstable even for positive parameter values such as stiffness and damping.

**Figure 4.** *A cross-section of the hydrodynamic bearing.*

#### **4.2 Muszynska model**

The motion equation of the rotor with the journal bearing in coordinates *x* and *y* was designed by Muszynska. The derivation is based on the design of the formula to calculate the already mentioned direct and quadrature forces. Compared to Eq. (2), the stiffness and damping matrices are designed in such a way that the oil film is replaced by a spring and a dashpot system that rotates at an angular velocity Ω, where *λ* is a dimensionless parameter, which is slightly less than 0.5. The stiffness of the spring is designated by *K*, and the damper has a damping factor *D*:

On double integrating, see [7], we get

*Actively Controlled Journal Bearings for Machine Tools DOI: http://dx.doi.org/10.5772/intechopen.92272*

*<sup>d</sup><sup>θ</sup>* <sup>¼</sup> <sup>6</sup>*μUR* <sup>ð</sup> *dh*

*<sup>d</sup><sup>θ</sup>* <sup>¼</sup> <sup>6</sup>*μUR* <sup>1</sup>

where *K*, *K*1, and *p*<sup>0</sup> are integration constants. The solution must meet the boundary condition:

*<sup>p</sup>*ð Þ¼ *<sup>θ</sup>* <sup>6</sup>*μUR c*2

> 6*μUR c*2

*θ*¼ð 2*π*

*θ*¼0

) *K*1 *c* ¼ �

The result of double integration is as follows:

*n*ð Þ 2 þ *n* cos *θ* sin *θ*

*<sup>n</sup>*ð Þ <sup>2</sup> <sup>þ</sup> <sup>2</sup> ð Þ <sup>1</sup> <sup>þ</sup> *<sup>n</sup>* cos *<sup>θ</sup>* <sup>2</sup> <sup>þ</sup> *<sup>p</sup>*<sup>0</sup> <sup>¼</sup> <sup>6</sup>*μUR*

The first integration constant was selected to meet the boundary condition *p*0ð Þ¼ 0 *p*0ð Þ 2*π* as is described by Dwivedy et al. The oil pressure distribution on the journal for *n* ¼ 0, 0*:*1, 0*:*2, ⋯, 0*:*9 is shown in **Figure 5**. It should be noticed that the

*<sup>p</sup>*ð Þ¼ *<sup>θ</sup>* <sup>6</sup>*μUR c*2

*dθ*

ð Þ <sup>1</sup> <sup>þ</sup> *<sup>n</sup>* cos *<sup>θ</sup>* <sup>2</sup> <sup>þ</sup>

1 ð Þ <sup>1</sup> <sup>þ</sup> *<sup>n</sup>* cos *<sup>θ</sup>* <sup>2</sup> <sup>þ</sup>

Ð *<sup>θ</sup>*¼2*<sup>π</sup>*

Ð *<sup>θ</sup>*¼2*<sup>π</sup>*

ð Þ <sup>1</sup> <sup>þ</sup> *<sup>n</sup>* cos *<sup>θ</sup>* <sup>2</sup> <sup>þ</sup>

ð *dθ*

*dθ* ¼ 6*μURh* þ *K*

" #

" #

*K*1 *<sup>c</sup>*ð Þ <sup>1</sup> <sup>þ</sup> *<sup>n</sup>* cos *<sup>θ</sup>* <sup>3</sup>

*p*ð Þ¼ *θ* ¼ 0 *p*ð Þ) *θ* ¼ 2*π p*ð Þ� *θ* ¼ 0 *p*ð Þ¼ *θ* ¼ 2*π* 0, (8)

*<sup>θ</sup>*¼<sup>0</sup> <sup>1</sup>*<sup>=</sup>* ð Þ <sup>1</sup> <sup>þ</sup> *<sup>n</sup>* cos *<sup>θ</sup>* <sup>2</sup> � �*d<sup>θ</sup>*

*<sup>θ</sup>*¼<sup>0</sup> <sup>1</sup>*<sup>=</sup> <sup>c</sup>*ð Þ <sup>1</sup> <sup>þ</sup> *<sup>n</sup>* cos *<sup>θ</sup>* <sup>3</sup> � �*d<sup>θ</sup>*

" #

On simplifying, we get a formula for calculating the first integration constant *K*1:

Extreme oil pressure values as a function of the attitude angle *θ* are achieved if

The first integration constant is related to the thickness of the oil film at the perimeter of the journal, where the maximum and minimum oil pressure is

*hm* <sup>¼</sup> ð Þ *<sup>h</sup> <sup>p</sup>*<sup>¼</sup> *min* <sup>¼</sup> ð Þ *<sup>h</sup> <sup>p</sup>*<sup>¼</sup> *max* ¼ �*K*<sup>1</sup> <sup>¼</sup> <sup>2</sup>*<sup>c</sup>* <sup>1</sup> � *<sup>n</sup>*<sup>2</sup> ð Þ

The attitude angle where the maximum and minimum pressure occur is given by

*K*1 *<sup>c</sup>*ð Þ <sup>1</sup> <sup>þ</sup> *<sup>n</sup>* cos *<sup>θ</sup>* <sup>3</sup>

*K*1*dθ <sup>c</sup>*ð Þ <sup>1</sup> <sup>þ</sup> *<sup>n</sup>* cos *<sup>θ</sup>* <sup>3</sup>

þ *p*0,

*dθ* ¼ 0

*<sup>n</sup>*ð Þ <sup>2</sup> <sup>þ</sup> <sup>2</sup> *:* (12)

*<sup>c</sup>*<sup>2</sup> *β θ*ð Þþ , *<sup>n</sup> <sup>p</sup>*0*:* (14)

*:*

*<sup>K</sup>*<sup>1</sup> <sup>¼</sup> <sup>2</sup>*c n*<sup>2</sup> � <sup>1</sup> � �*<sup>=</sup> <sup>n</sup>*<sup>2</sup> <sup>þ</sup> <sup>2</sup> � �*:* (10)

*K*<sup>1</sup> ¼ �*h* ¼ �*c*ð Þ 1 þ *n* cos *θ :* (11)

cos *<sup>θ</sup><sup>m</sup>* ¼ �3*n<sup>=</sup> <sup>n</sup>*<sup>2</sup> <sup>þ</sup> <sup>2</sup> � �*:* (13)

(7)

(9)

*<sup>h</sup>*<sup>3</sup> *dp*

*dp*

which gives

*dp=dθ* ¼ 0:

achieved:

**111**

$$
\begin{bmatrix} M & \mathbf{0} \\ \mathbf{0} & M \end{bmatrix} \begin{bmatrix} \ddot{\boldsymbol{x}} \\ \ddot{\boldsymbol{y}} \end{bmatrix} + \begin{bmatrix} D & \mathbf{0} \\ \mathbf{0} & D \end{bmatrix} \begin{bmatrix} \dot{\boldsymbol{x}} \\ \dot{\boldsymbol{y}} \end{bmatrix} + \begin{bmatrix} K & D\lambda\mathbf{0} \\ -D\lambda\mathbf{0} & K \end{bmatrix} \begin{bmatrix} \boldsymbol{x} \\ \boldsymbol{y} \end{bmatrix} = \begin{bmatrix} F\_{\mathcal{X}} \\ F\_{\mathcal{Y}} \end{bmatrix}.\tag{3}
$$

The derivation of the motion equation is described, for example, in the article [5].

#### **4.3 Analytical solution of the Reynolds equation**

The theory of hydrodynamic bearing is based on a differential equation derived by Osborne Reynolds. Reynolds equation is based on the following assumptions: The lubricant obeys Newton's law of viscosity and is incompressible. The inertia forces of the oil film are negligible. The viscosity *μ* of the lubricant is constant, and there is a continuous supply of lubricant. The effect of the curvature of the film concerning film thickness is neglected. It is assumed that the film is so thin that the pressure is constant across the film thickness. The shaft and bearing are rigid.

Furthermore, it is assumed that the thickness *h* of the oil film depends on the other two coordinates, namely, the coordinate z along the axis of rotation and the location on the perimeter of the journal which is described by the angle *θ* as is shown on the right side in **Figure 4**. If the radius of the bearing journal is equal to *R*, then the most general version of the Reynolds equation for calculation of the oil pressure distribution *p*ð Þ *θ*, *z* is as follows (Dwivedy) [7]:

$$\frac{1}{R^2}\frac{\partial}{\partial\theta}\left(h^3\frac{\partial p}{\partial\theta}\right) + \frac{\partial}{\partial z}\left(h^3\frac{\partial p}{\partial z}\right) = 6\mu\Omega\frac{\partial h}{\partial\theta} + 12\mu\frac{\partial h}{\partial t}.\tag{4}$$

There is no analytical solution for the Reynolds equation.

During operation, the journal axis shifts from the centre of the bearing bushing to the distance of *e*, called eccentricity, which is related to a radial clearance *c*. Variable is called an eccentricity ratio *n* ¼ *e=c*. The film thickness as a function of *θ* is defined as follows:

$$h = c(\mathbf{1} + n \cos \theta). \tag{5}$$

The oil film moves in adjacent parallel layers at different speeds, and shear stress results between them. The oil layer at the surface of the journal moves at the peripheral velocity of the journal, while the oil layers at the surface of the bearing bushing do not move (at zero velocity). The surface of the journal moves at a velocity of *U* ¼ *RΩ* in m/s. Reynolds equation will be solved for the steady state and independence of the pressure distribution on the coordinate of *z*:

$$\frac{d}{d\theta} \left[ h^3 \frac{dp}{d\theta} \right] = 6\mu U R \frac{dh}{d\theta}.\tag{6}$$

*Actively Controlled Journal Bearings for Machine Tools DOI: http://dx.doi.org/10.5772/intechopen.92272*

On double integrating, see [7], we get

$$\begin{aligned} h^3 \frac{dp}{d\theta} &= 6\mu U R \left[ \frac{dh}{d\theta} d\theta = 6\mu U R h + K \right. \\\ \frac{dp}{d\theta} &= 6\mu U R \left[ \frac{1}{\left(1 + n \cos \theta\right)^2} + \frac{K\_1}{c \left(1 + n \cos \theta\right)^3} \right] \\\ p(\theta) &= \frac{6\mu U R}{c^2} \left[ \left[ \frac{d\theta}{\left(1 + n \cos \theta\right)^2} + \frac{K\_1 d\theta}{c \left(1 + n \cos \theta\right)^3} \right] + p\_0, \end{aligned} \tag{7}$$

where *K*, *K*1, and *p*<sup>0</sup> are integration constants. The solution must meet the boundary condition:

$$p(\theta = 0) = p(\theta = 2\pi) \Rightarrow p(\theta = 0) - p(\theta = 2\pi) = 0,\tag{8}$$

which gives

**4.2 Muszynska model**

*M* 0 0 *M x*€

article [5].

€*y* þ

*Machine Tools - Design, Research, Application*

The motion equation of the rotor with the journal bearing in coordinates *x* and *y* was designed by Muszynska. The derivation is based on the design of the formula to calculate the already mentioned direct and quadrature forces. Compared to Eq. (2), the stiffness and damping matrices are designed in such a way that the oil film is replaced by a spring and a dashpot system that rotates at an angular velocity Ω, where *λ* is a dimensionless parameter, which is slightly less than 0.5. The stiffness of

> *K Dλ*Ω �*Dλ*Ω *K x*

*y*  <sup>¼</sup> *FX FY* 

*:* (3)

the spring is designated by *K*, and the damper has a damping factor *D*:

*y*\_ þ

The derivation of the motion equation is described, for example, in the

The theory of hydrodynamic bearing is based on a differential equation derived by Osborne Reynolds. Reynolds equation is based on the following assumptions: The lubricant obeys Newton's law of viscosity and is incompressible. The inertia forces of the oil film are negligible. The viscosity *μ* of the lubricant is constant, and there is a continuous supply of lubricant. The effect of the curvature of the film concerning film thickness is neglected. It is assumed that the film is so thin that the pressure is constant across the film thickness. The shaft and bearing are rigid. Furthermore, it is assumed that the thickness *h* of the oil film depends on the other two coordinates, namely, the coordinate z along the axis of rotation and the location on the perimeter of the journal which is described by the angle *θ* as is shown on the right side in **Figure 4**. If the radius of the bearing journal is equal to *R*, then the most general version of the Reynolds equation for calculation of the oil

*D* 0 0 *D x*\_

**4.3 Analytical solution of the Reynolds equation**

pressure distribution *p*ð Þ *θ*, *z* is as follows (Dwivedy) [7]:

þ *∂ ∂z*

There is no analytical solution for the Reynolds equation.

independence of the pressure distribution on the coordinate of *z*:

*d <sup>d</sup><sup>θ</sup> <sup>h</sup>*<sup>3</sup> *dp dθ* 

*<sup>h</sup>*<sup>3</sup> *<sup>∂</sup><sup>p</sup> ∂z* 

During operation, the journal axis shifts from the centre of the bearing bushing

The oil film moves in adjacent parallel layers at different speeds, and shear stress

<sup>¼</sup> <sup>6</sup>*μUR dh dθ*

to the distance of *e*, called eccentricity, which is related to a radial clearance *c*. Variable is called an eccentricity ratio *n* ¼ *e=c*. The film thickness as a function of *θ*

results between them. The oil layer at the surface of the journal moves at the peripheral velocity of the journal, while the oil layers at the surface of the bearing bushing do not move (at zero velocity). The surface of the journal moves at a velocity of *U* ¼ *RΩ* in m/s. Reynolds equation will be solved for the steady state and

<sup>¼</sup> <sup>6</sup>*μ*<sup>Ω</sup> *<sup>∂</sup><sup>h</sup>*

*<sup>∂</sup><sup>θ</sup>* <sup>þ</sup> <sup>12</sup>*<sup>μ</sup>*

*h* ¼ *c*ð Þ 1 þ *n* cos *θ :* (5)

*∂h ∂t*

*:* (6)

*:* (4)

1 *R*2 *∂ <sup>∂</sup><sup>θ</sup> <sup>h</sup>*<sup>3</sup> *<sup>∂</sup><sup>p</sup> ∂θ* 

is defined as follows:

**110**

$$\begin{split} \frac{6\mu UR}{c^2} \int\_{\theta=0}^{\theta=2\pi} \left[ \frac{1}{\left(1+n\cos\theta\right)^2} + \frac{K\_1}{c\left(1+n\cos\theta\right)^3} \right] d\theta &= 0 \\ \Rightarrow \frac{K\_1}{c} = -\frac{\int\_{\theta=0}^{\theta=2\pi} 1/\left(\left(1+n\cos\theta\right)^2\right) d\theta}{\int\_{\theta=0}^{\theta=2\pi} 1/\left(c\left(1+n\cos\theta\right)^3\right) d\theta} .\end{split} \tag{9}$$

On simplifying, we get a formula for calculating the first integration constant *K*1:

$$K\_1 = 2x(n^2 - 1)/(n^2 + 2). \tag{10}$$

Extreme oil pressure values as a function of the attitude angle *θ* are achieved if *dp=dθ* ¼ 0:

$$K\_1 = -h = -c(\mathbf{1} + n\cos\theta). \tag{11}$$

The first integration constant is related to the thickness of the oil film at the perimeter of the journal, where the maximum and minimum oil pressure is achieved:

$$h\_m = (h)\_{p=\min} = (h)\_{p=\max} = -K\_1 = \frac{2c(1-n^2)}{(n^2+2)}\,. \tag{12}$$

The attitude angle where the maximum and minimum pressure occur is given by

$$
\cos \theta\_m = -\mathfrak{B}n/(n^2 + 2). \tag{13}
$$

The result of double integration is as follows:

$$p(\theta) = \frac{6\mu U R}{c^2} \frac{n(2 + n\cos\theta)\sin\theta}{(n^2 + 2)(1 + n\cos\theta)^2} + p\_0 = \frac{6\mu U R}{c^2} \beta(\theta, n) + p\_0. \tag{14}$$

The first integration constant was selected to meet the boundary condition *p*0ð Þ¼ 0 *p*0ð Þ 2*π* as is described by Dwivedy et al. The oil pressure distribution on the journal for *n* ¼ 0, 0*:*1, 0*:*2, ⋯, 0*:*9 is shown in **Figure 5**. It should be noticed that the

**Figure 5.** *Pressure distribution along the angular coordinate.*

second integration constant has not any effect on the force excited by the oil pressure. The subatmospheric pressure creates a condition for the formation of the cavitation zones.

#### **4.4 Fluid force**

The forces acting on the journal in the centre of gravity along the bearing length of *L* can be calculated for the direction of the line of the centres and the perpendicular direction. Force in the direction of the line of centres is denoted as a direct force *FD*, while force which is perpendicular to the line of centres is denoted as a quadrature force *FQ* . Both these forces balance the gravity force *G* as is shown in **Figure 4**:

$$\begin{split} F\_D &= \int\_0^{2\pi} p\_\theta \cos \left( \pi - \theta \right) LR \, \mathrm{d}\theta = \\ &= F\_N \int\_0^{2\pi} \beta(\theta, n) \cos \left( \pi - \theta \right) \, \mathrm{d}\theta = F\_N \beta\_D(n) \\\ F\_Q &= \int\_0^{2\pi} p\_\theta \sin \left( \pi - \theta \right) LR \, \mathrm{d}\theta = \\ &= F\_N \int\_0^{2\pi} \beta(\theta, n) \sin \left( \pi - \theta \right) \, \mathrm{d}\theta = F\_N \beta\_Q(n), \end{split} \tag{15}$$

where the force *FN* can be designated as a nominal force because it corresponds to the maximum force according to the linear model (*n* ¼ 1):

$$F\_N = \left\| \mu U R^2 L / c^2 \right\| \tag{16}$$

For speeds ranging from 1.25k to 5k rpm, the force factor *FN* varies from 0.46 to 1.8 kN. However, this force is reduced by multiplying the coefficients *βD*ð Þ *n* and *βQ*ð Þ *n* , which depend on the eccentricity ratio *n* ranging from 0 to 0.33 (0*:*02*=*0*:*06). This case can only theoretically arise in an entirely flooded plain bearing with a vertical axis. The balance of forces *FD*, *FQ* , and *FQ* allows to calculate an attitude

The presence of direct force can be explained, e.g., by the cavitation or the inability to achieve high vacuum, but the mathematical model is more complicated (Ferfecki) [8]. The lubricant flows through the bearing, but in the part of the bearing journal circumference where the pressure is below the barometric pressure, the lubricant can also be sucked. The magnitude of the negative pressure for *π* < *θ* <2*π* is multiplied by a factor *γ*. Therefore the total force is given by the sum of

ð Þ … d*θ* þ *γ*

The effect of negative pressure reduction is demonstrated in the right panel of **Figure 6**. Negative pressure is limited to 1% of the magnitude of positive pressure for the angle interval of 0 <*θ* < *π*. The formulas for the calculation of the quadrature

dence on the peripheral speed *U* and therefore on the rotor angular velocity. The coefficients *β<sup>Q</sup>* ð Þ *n* and *βD*ð Þ *n* differ considerably, in the experiment, the results of which are shown in **Figure 6**. The eccentricity ratio decreases approximately from 0.3 to 0.07 in the operation at the stable bearing position and stable lubrication. The diagrams confirm the linearity of the quadrature and direct force to eccentricity ratio up to 0.6. The *β<sup>Q</sup>* ð Þ *n* and *βD*ð Þ *n* coefficients can be approximated in this range

*β<sup>Q</sup>* ð Þ *n* ≈*qcn* ¼ *qe*

*<sup>L</sup>=c*<sup>2</sup> � *<sup>d</sup>*.

*<sup>β</sup>D*ð Þ *<sup>n</sup>* <sup>≈</sup> *dcn* <sup>¼</sup> *de*, (20)

ð<sup>2</sup>*<sup>π</sup> π*

*<sup>α</sup>* <sup>¼</sup> arctan *<sup>β</sup>D*ð Þ *<sup>n</sup> <sup>=</sup>β<sup>Q</sup>* ð Þ *<sup>n</sup>* � �*:* (18)

ð Þ … d*θ*, (19)

*L=c*<sup>2</sup> and hence the depen-

*<sup>L</sup>=c*<sup>2</sup> � *<sup>q</sup>* and *<sup>d</sup>* deter-

angle *α*:

**Figure 6.**

integrals (Eq. (15)) as follows:

as a linear function:

**113**

mines the direct stiffness *CD* <sup>¼</sup> <sup>6</sup>*μUR*<sup>2</sup>

*FD FQ* � �

*Dependence of the direct and quadrature force on the eccentricity ratio.*

*Actively Controlled Journal Bearings for Machine Tools DOI: http://dx.doi.org/10.5772/intechopen.92272*

and direct forces contain the same factor *FN* <sup>¼</sup> <sup>6</sup>*μUR*<sup>2</sup>

¼ ð*π* 0

where *<sup>q</sup>* determines the quadrature stiffness *CQ* <sup>¼</sup> <sup>6</sup>*μUR*<sup>2</sup>

Note that according to Eq. (14) the pressure on the part of the journal surface is negative, which is, in fact, a relative negative pressure. Since the pressure distribution is antisymmetric with respect to *θ* ¼ *π*, without evidence, it is clear that these formulas can be applied. Only quadrature force *FQ* >0 acts on the bearing journal, and the direct force are zero *FD* ¼ 0, as is shown on the left panel in **Figure 6**.

The nominal force that multiplies the dimensionless functions *βD*ð Þ *n* and *β<sup>Q</sup>* ð Þ *n* can be calculated with the use of the dimensionless Sommerfeld number *S* and the load *P* per unit of projected bearing area as follows:

$$F\_N = \mathfrak{G}\mu U R^2 L/c^2 = \mathfrak{G}\pi \text{SG},\tag{17}$$

where *G* is a gravity force.

*Actively Controlled Journal Bearings for Machine Tools DOI: http://dx.doi.org/10.5772/intechopen.92272*

**Figure 6.**

second integration constant has not any effect on the force excited by the oil pressure. The subatmospheric pressure creates a condition for the formation of the

*FD*, while force which is perpendicular to the line of centres is denoted as a quadrature force *FQ* . Both these forces balance the gravity force *G* as is shown in

*p<sup>θ</sup> cos*ð Þ *π* � *θ LR* d*θ* ¼

*p<sup>θ</sup>* sin ð Þ *π* � *θ LR* d*θ* ¼

*FN* <sup>¼</sup> <sup>6</sup>*μUR*<sup>2</sup>

*FD* ¼

*Pressure distribution along the angular coordinate.*

*Machine Tools - Design, Research, Application*

*FQ* ¼

ð<sup>2</sup>*<sup>π</sup>* 0

> ð<sup>2</sup>*<sup>π</sup>* 0

> ð<sup>2</sup>*<sup>π</sup>* 0

to the maximum force according to the linear model (*n* ¼ 1):

¼ *FN*

¼ *FN*

load *P* per unit of projected bearing area as follows:

where *G* is a gravity force.

**112**

*FN* <sup>¼</sup> <sup>6</sup>*μUR*<sup>2</sup>

ð<sup>2</sup>*<sup>π</sup>* 0

The forces acting on the journal in the centre of gravity along the bearing length of *L* can be calculated for the direction of the line of the centres and the perpendicular direction. Force in the direction of the line of centres is denoted as a direct force

*β θ*ð Þ , *n* cosð Þ *π* � *θ* d*θ* ¼ *FNβD*ð Þ *n*

(15)

*β θ*ð Þ , *n* sin ð Þ *π* � *θ* d*θ* ¼ *FNβ<sup>Q</sup>* ð Þ *n* ,

*L=c* 2

*:* (16)

<sup>2</sup> <sup>¼</sup> <sup>6</sup>*πSG*, (17)

where the force *FN* can be designated as a nominal force because it corresponds

Note that according to Eq. (14) the pressure on the part of the journal surface is negative, which is, in fact, a relative negative pressure. Since the pressure distribution is antisymmetric with respect to *θ* ¼ *π*, without evidence, it is clear that these formulas can be applied. Only quadrature force *FQ* >0 acts on the bearing journal, and the direct force are zero *FD* ¼ 0, as is shown on the left panel in **Figure 6**. The nominal force that multiplies the dimensionless functions *βD*ð Þ *n* and *β<sup>Q</sup>* ð Þ *n* can be calculated with the use of the dimensionless Sommerfeld number *S* and the

*L=c*

cavitation zones.

**Figure 5.**

**4.4 Fluid force**

**Figure 4**:

*Dependence of the direct and quadrature force on the eccentricity ratio.*

For speeds ranging from 1.25k to 5k rpm, the force factor *FN* varies from 0.46 to 1.8 kN. However, this force is reduced by multiplying the coefficients *βD*ð Þ *n* and *βQ*ð Þ *n* , which depend on the eccentricity ratio *n* ranging from 0 to 0.33 (0*:*02*=*0*:*06). This case can only theoretically arise in an entirely flooded plain bearing with a vertical axis. The balance of forces *FD*, *FQ* , and *FQ* allows to calculate an attitude angle *α*:

$$a = \arctan\left(\beta\_D(n)/\beta\_Q(n)\right). \tag{18}$$

The presence of direct force can be explained, e.g., by the cavitation or the inability to achieve high vacuum, but the mathematical model is more complicated (Ferfecki) [8]. The lubricant flows through the bearing, but in the part of the bearing journal circumference where the pressure is below the barometric pressure, the lubricant can also be sucked. The magnitude of the negative pressure for *π* < *θ* <2*π* is multiplied by a factor *γ*. Therefore the total force is given by the sum of integrals (Eq. (15)) as follows:

$$
\begin{bmatrix} F\_D \\ F\_Q \end{bmatrix} = \int\_0^\pi (\dots) \mathbf{d}\theta + \chi \int\_\pi^{2\pi} (\dots) \mathbf{d}\theta,\tag{19}
$$

The effect of negative pressure reduction is demonstrated in the right panel of **Figure 6**. Negative pressure is limited to 1% of the magnitude of positive pressure for the angle interval of 0 <*θ* < *π*. The formulas for the calculation of the quadrature and direct forces contain the same factor *FN* <sup>¼</sup> <sup>6</sup>*μUR*<sup>2</sup> *L=c*<sup>2</sup> and hence the dependence on the peripheral speed *U* and therefore on the rotor angular velocity. The coefficients *β<sup>Q</sup>* ð Þ *n* and *βD*ð Þ *n* differ considerably, in the experiment, the results of which are shown in **Figure 6**. The eccentricity ratio decreases approximately from 0.3 to 0.07 in the operation at the stable bearing position and stable lubrication. The diagrams confirm the linearity of the quadrature and direct force to eccentricity ratio up to 0.6. The *β<sup>Q</sup>* ð Þ *n* and *βD*ð Þ *n* coefficients can be approximated in this range as a linear function:

$$\begin{aligned} \beta\_Q(n) &\approx qcn = qe \\ \beta\_D(n) &\approx dcn = de, \end{aligned} \tag{20}$$

where *<sup>q</sup>* determines the quadrature stiffness *CQ* <sup>¼</sup> <sup>6</sup>*μUR*<sup>2</sup> *<sup>L</sup>=c*<sup>2</sup> � *<sup>q</sup>* and *<sup>d</sup>* determines the direct stiffness *CD* <sup>¼</sup> <sup>6</sup>*μUR*<sup>2</sup> *<sup>L</sup>=c*<sup>2</sup> � *<sup>d</sup>*.

The stiffness in the directions of the Cartesian coordinates *x*, *y* and the attitude angle *α* which is defined in **Figure 5** can be obtained by substitution:

$$\begin{aligned} \mathbf{x}(t) &= -e \sin a \\ \mathbf{y}(t) &= +\mathbf{e} \ \cos a. \end{aligned} \tag{21}$$

the y coordinate, therefore it is possible to denote it as r ¼ x þ jy. The origin (0, 0) of the coordinate system in the complex plane is situated in the center of the

There are many ways how to model journal bearings, but this paper prefers a lumped parameter model, which is based on the concept developed by Muszynska [10]. This concept assumes that the oil film acts as a combination of the spring and damper, which rotates at an angular speed λΩ (variables will be defined hereinafter). The reason for using this concept was that it offers an effective way to understand the rotor instability problem and to create a model of a journal vibration active control by manipulating the bushing position with the use of actuators, which are a part of the closed-loop system composed of the proximity probes and the controller. If the bearing bushing is stationary (*u* ¼ 0), then the equation of motion

where M is the total rotor mass, Ω is the rotor angular velocity, K and D are specifying proportionality of stiffness and damping to the relative position of the journal center displacement vector, λ is a dimensionless parameter which is slightly less than 0.5, and FP is an oscillating disturbing force defined by ΔFP exp j ωt þ φ<sup>0</sup> ð Þ ð Þ *,* where ω is a synchronous or nonsynchronous excitation frequency and φ<sup>0</sup> is an initial phase. Excitation frequency is ω ¼ 0 for a static force and ω ¼ Ω for

Force action of the oil film on the bearing journal can be modeled by Reynolds partial differential equations. The good accuracy of the Muszynska approximate model confirms Mendes and Cavalca [9]. Eq. (2) can be rewritten in matrix form

The entries of the stiffness and damping matrices according to Muszynska model and the calculation of these matrix entries using Reynolds equation agree except for very low rotor speed. Some entries are constants, and others are linear function of the rotational speed. Even entries of the damping matrix are similar. Coordinates of the bearing journal axis for the force of gravity *F<sup>P</sup>* ¼ 0 � *jMg* is

*<sup>r</sup>*<sup>0</sup> <sup>¼</sup> *Mg <sup>D</sup>λ*<sup>Ω</sup> � *jK*

*K Dλ*Ω �*Dλ*Ω *K*

*x t*ð Þ

*y t*ð Þ <sup>¼</sup> *Fx*ð Þ*<sup>t</sup>*

ð Þ *<sup>D</sup>λ*<sup>Ω</sup> <sup>2</sup> <sup>þ</sup> *<sup>K</sup>*<sup>2</sup> (27)

*Fy*ð Þ*t :* (26)

M€r þ Dr\_ þ ð Þ K � jDλΩ r ¼ FP (25)

mentioned cylindrical bearing bore.

*Actively Controlled Journal Bearings for Machine Tools DOI: http://dx.doi.org/10.5772/intechopen.92272*

*Coordinate system in the complex plane.*

imbalance.

**Figure 7.**

*M* 0 0 *M x t* €ð Þ

given by the formula:

**115**

€*y t*ð Þ <sup>þ</sup>

at the stationary rotation angular velocity is as follows:

*D* 0 0 *D x t* \_ð Þ

*y t* \_ð Þ <sup>þ</sup>

The vector of the direct and quadrature forces depends on the coordinates *x*, *y* according to the following formula:

$$
\begin{bmatrix}
\mathbf{C}\_{Q}e\sin a + \mathbf{C}\_{D}e\cos a
\end{bmatrix} = \begin{bmatrix}
\mathbf{C}\_{D}\mathbf{x} + \mathbf{C}\_{Q}\mathbf{y} \\ 
\end{bmatrix} = \begin{bmatrix}
\mathbf{C}\_{D} & \mathbf{C}\_{Q} \\
\end{bmatrix} \begin{bmatrix}
\mathbf{x} \\ 
\mathbf{y}
\end{bmatrix}.\tag{22}
$$

The cross-coupled stiffness *Dλ*Ω according to the Muszynska model corresponds to the expression 6*μUR*<sup>2</sup> *<sup>L</sup>=c*<sup>2</sup> � *<sup>q</sup>*. The direct stiffness *<sup>K</sup>* is orderly less than the cross-coupled stiffness; however, the analytical calculation of the stiffness matrix shows the dependence on the rotational speed.

The damping matrix can be derived based on its relationship to the stiffness matrix according to the model that was designed by Muszynska:

$$
\begin{bmatrix} D & \mathbf{0} \\ \mathbf{0} & D \end{bmatrix} \begin{bmatrix} \dot{\mathbf{x}} \\ \dot{\mathbf{y}} \end{bmatrix} = \begin{bmatrix} \mathbf{C}\_{Q}/\lambda \mathbf{\Omega} & \mathbf{0} \\ \mathbf{0} & \mathbf{C}\_{Q}/\lambda \mathbf{\Omega} \end{bmatrix} \begin{bmatrix} \dot{\mathbf{x}} \\ \dot{\mathbf{y}} \end{bmatrix}. \tag{23}
$$

As is D ¼ CQ *=*λΩ, the damping coefficient D is a constant. The motion equation for the rigid rotor in the plain bearing is as follows:

$$
\begin{bmatrix} M & \mathbf{0} \\ \mathbf{0} & M \end{bmatrix} \begin{bmatrix} \ddot{\boldsymbol{x}} \\ \ddot{\boldsymbol{y}} \end{bmatrix} + \begin{bmatrix} \mathbf{C}\_{Q}/\lambda\mathbf{\Omega} & \mathbf{0} \\ \mathbf{0} & \mathbf{C}\_{Q}/\lambda\mathbf{\Omega} \end{bmatrix} \begin{bmatrix} \dot{\boldsymbol{x}} \\ \dot{\boldsymbol{y}} \end{bmatrix} + \begin{bmatrix} \mathbf{C}\_{D} & \mathbf{C}\_{Q} \\ -\mathbf{C}\_{Q} & \mathbf{C}\_{D} \end{bmatrix} \begin{bmatrix} \boldsymbol{x} \\ \boldsymbol{y} \end{bmatrix} = \begin{bmatrix} F\_{X} \\ F\_{Y} \end{bmatrix}. \tag{24}
$$

The sum of direct and quadrature forces must compensate for the gravitational force that does not depend on the speed of rotation. The suitability of this model is confirmed by Mendes [9].

If the attitude angle approaches the angle of *π=*2 in the radians, then the gravitational force is balanced only by the quadrature force. This state represents the stability limit. The experiment described in this article demonstrates the fact that hydrodynamic lubrication occurs when the eccentricity ratio *n* is decreased to below the value of 0.33, which happens after a certain speed margin has been exceeded. Unstable lubrication occurs during low revolutions when the eccentricity ratio is higher than the mentioned boundary. The problem of modelling the motion of the bearing journal at low rotational speeds raises the impossibility of determining the initial conditions.

#### **5. Linear time-invariant mathematical model**

The coordinate system in the complex plane for the bearing journal position is shown in **Figure 7**. A variable u is a control variable, and a variable r is a controlled variable. The controlled variable is a two-component coordinate of the bearing journal axis, while the control variable is a two-component coordinates of the bushing axis as is shown in **Figure 4**. Because both the variables indicate coordinates in the plane, then they can be considered as two-component vectors. The same meaning as the vector has a complex variables. The real part of this variable has the meaning of the x coordinate, while the imaginary part has the meaning of

*Actively Controlled Journal Bearings for Machine Tools DOI: http://dx.doi.org/10.5772/intechopen.92272*

**Figure 7.** *Coordinate system in the complex plane.*

The stiffness in the directions of the Cartesian coordinates *x*, *y* and the attitude

*y t*ðÞ¼þe cos *<sup>α</sup>:* (21)

<sup>¼</sup> *CD CQ* �*CQ CD x*

> *y*\_

> > *y*

*<sup>L</sup>=c*<sup>2</sup> � *<sup>q</sup>*. The direct stiffness *<sup>K</sup>* is orderly less than the

*y* 

*:* (23)

<sup>¼</sup> *FX FY* 

*:* (24)

*:* (22)

*x t*ðÞ¼�*e*sin *α*

The vector of the direct and quadrature forces depends on the coordinates *x*, *y*

The cross-coupled stiffness *Dλ*Ω according to the Muszynska model corresponds

<sup>¼</sup> *CDx* <sup>þ</sup> *CQ <sup>y</sup>* �*CQ x* þ *CDy* 

cross-coupled stiffness; however, the analytical calculation of the stiffness matrix

The damping matrix can be derived based on its relationship to the stiffness

<sup>¼</sup> *CQ <sup>=</sup>λ*<sup>Ω</sup> <sup>0</sup>

As is D ¼ CQ *=*λΩ, the damping coefficient D is a constant. The motion equation

*y*\_ þ

The sum of direct and quadrature forces must compensate for the gravitational force that does not depend on the speed of rotation. The suitability of this model is

If the attitude angle approaches the angle of *π=*2 in the radians, then the gravitational force is balanced only by the quadrature force. This state represents the stability limit. The experiment described in this article demonstrates the fact that hydrodynamic lubrication occurs when the eccentricity ratio *n* is decreased to below the value of 0.33, which happens after a certain speed margin has been exceeded. Unstable lubrication occurs during low revolutions when the eccentricity ratio is higher than the mentioned boundary. The problem of modelling the motion of the bearing journal at low rotational speeds raises the impossibility of determining the

The coordinate system in the complex plane for the bearing journal position is shown in **Figure 7**. A variable u is a control variable, and a variable r is a controlled variable. The controlled variable is a two-component coordinate of the bearing journal axis, while the control variable is a two-component coordinates of the bushing axis as is shown in **Figure 4**. Because both the variables indicate coordinates in the plane, then they can be considered as two-component vectors. The same meaning as the vector has a complex variables. The real part of this variable has the meaning of the x coordinate, while the imaginary part has the meaning of

0 *CQ =λ*Ω *x*\_

> *CD CQ* �*CQ CD x*

matrix according to the model that was designed by Muszynska:

*y*\_ 

angle *α* which is defined in **Figure 5** can be obtained by substitution:

according to the following formula:

*Machine Tools - Design, Research, Application*

�*CDe*sin *α* þ *CQ e* cos *α CQ e*sin *α* þ *CDe* cos *α* 

shows the dependence on the rotational speed.

*D* 0 0 *D x*\_

for the rigid rotor in the plain bearing is as follows:

**5. Linear time-invariant mathematical model**

*CQ =λ*Ω 0 0 *CQ =λ*Ω *x*\_

to the expression 6*μUR*<sup>2</sup>

*M* 0 0 *M x*€

initial conditions.

**114**

€*y* þ

confirmed by Mendes [9].

the y coordinate, therefore it is possible to denote it as r ¼ x þ jy. The origin (0, 0) of the coordinate system in the complex plane is situated in the center of the mentioned cylindrical bearing bore.

There are many ways how to model journal bearings, but this paper prefers a lumped parameter model, which is based on the concept developed by Muszynska [10]. This concept assumes that the oil film acts as a combination of the spring and damper, which rotates at an angular speed λΩ (variables will be defined hereinafter). The reason for using this concept was that it offers an effective way to understand the rotor instability problem and to create a model of a journal vibration active control by manipulating the bushing position with the use of actuators, which are a part of the closed-loop system composed of the proximity probes and the controller. If the bearing bushing is stationary (*u* ¼ 0), then the equation of motion at the stationary rotation angular velocity is as follows:

$$\mathbf{M}\ddot{\mathbf{r}} + \mathbf{D}\dot{\mathbf{r}} + (\mathbf{K} - \mathbf{j}\mathbf{D}\lambda\mathbf{Q})\mathbf{r} = \mathbf{F}\_\mathbf{P} \tag{25}$$

where M is the total rotor mass, Ω is the rotor angular velocity, K and D are specifying proportionality of stiffness and damping to the relative position of the journal center displacement vector, λ is a dimensionless parameter which is slightly less than 0.5, and FP is an oscillating disturbing force defined by ΔFP exp j ωt þ φ<sup>0</sup> ð Þ ð Þ *,* where ω is a synchronous or nonsynchronous excitation frequency and φ<sup>0</sup> is an initial phase. Excitation frequency is ω ¼ 0 for a static force and ω ¼ Ω for imbalance.

Force action of the oil film on the bearing journal can be modeled by Reynolds partial differential equations. The good accuracy of the Muszynska approximate model confirms Mendes and Cavalca [9]. Eq. (2) can be rewritten in matrix form

$$
\begin{bmatrix} M & \mathbf{0} \\ \mathbf{0} & M \end{bmatrix} \begin{bmatrix} \ddot{\mathbf{x}}(t) \\ \ddot{y}(t) \end{bmatrix} + \begin{bmatrix} D & \mathbf{0} \\ \mathbf{0} & D \end{bmatrix} \begin{bmatrix} \dot{\mathbf{x}}(t) \\ \dot{y}(t) \end{bmatrix} + \begin{bmatrix} K & D\lambda\mathbf{2} \\ -D\lambda\mathbf{2} & K \end{bmatrix} \begin{bmatrix} x(t) \\ y(t) \end{bmatrix} = \begin{bmatrix} F\_x(t) \\ F\_y(t) \end{bmatrix}. \tag{26}
$$

The entries of the stiffness and damping matrices according to Muszynska model and the calculation of these matrix entries using Reynolds equation agree except for very low rotor speed. Some entries are constants, and others are linear function of the rotational speed. Even entries of the damping matrix are similar. Coordinates of the bearing journal axis for the force of gravity *F<sup>P</sup>* ¼ 0 � *jMg* is given by the formula:

$$r\_0 = \text{Mg} \frac{D\lambda\Omega - jK}{\left(D\lambda\Omega\right)^2 + K^2} \tag{27}$$

where *g* is gravitational acceleration. Stiffness of the oil film for static force increases proportionally with rotational speed of the bearing journal:

$$\frac{M\text{g}}{r\_0} = D\lambda\Omega + jK.\tag{28}$$

frequency of rotation. The stead-state gain of *GS*ð Þ*s* is equal to unit *GS*ð Þ¼ *j*0 1, while the stead-state gain of *GF*ð Þ*s* depends on the rotor angular velocity as it results from the formula *GF*ð Þ¼ *j*0 1*=*ð Þ *K* � *jDλ*Ω . The reciprocal value of *GF*ð Þ *j*0 can be considered as the static stiffness of the oil film without influence of the active vibration control. Stiffness which is defined as a complex value determines the direction of journal axis displacement relative to the direction of the force. The radial force has theoretically identical direction with the journal displacement only

The transfer functions relating the set point of the closed-loop system to the displacement of the shaft *Gw*ð Þ*s* and the disturbance force to the displacement of the

The stability margin can be calculated under assumption that the open-loop frequency transfer function *G*0ðÞ¼ *s KPGS*ð Þ*s* of the control loop in **Figure 8** is equal to �1. The frequency of the steady-state vibration at the stability margin is given by *<sup>ω</sup>* <sup>¼</sup> *<sup>λ</sup>*<sup>Ω</sup> and *KP* <sup>¼</sup> *<sup>ω</sup>*<sup>2</sup>*M=<sup>K</sup>* � 1. If the feedback gain *KP* is positive, then the maximal

If the proportional controller is disconnected, i.e. *KP* = 0, then the critical angular

*Ms*<sup>2</sup> <sup>þ</sup> ð Þ <sup>1</sup> <sup>þ</sup> *KP Ds* <sup>þ</sup> ð Þ <sup>1</sup> <sup>þ</sup> *KP* ð Þ *<sup>K</sup>* � *jDλ*<sup>Ω</sup> , (34)

*Ms*<sup>2</sup> <sup>þ</sup> ð Þ <sup>1</sup> <sup>þ</sup> *KP Ds* <sup>þ</sup> ð Þ <sup>1</sup> <sup>þ</sup> *KP* ð Þ *<sup>K</sup>* � *jDλ*<sup>Ω</sup> *:* (35)

*KP* <sup>þ</sup> <sup>1</sup> <sup>p</sup> *:* (36)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*GD*ð Þ*<sup>s</sup>* <sup>¼</sup> *Ms*<sup>2</sup> <sup>þ</sup> ð Þ <sup>1</sup> <sup>þ</sup> *KP Ds* <sup>þ</sup> ð Þ <sup>1</sup> <sup>þ</sup> *KP* ð Þ *<sup>K</sup>* � *jDλ*<sup>Ω</sup> *:* (37)

*GD*ð Þ *<sup>j</sup>*<sup>0</sup> <sup>¼</sup> ð Þ <sup>1</sup> <sup>þ</sup> *KP* ð Þ *<sup>K</sup>* � *jDλ*<sup>Ω</sup> (38)

*BYX BYY* � � (39)

<sup>1</sup> <sup>þ</sup> *KPGS*ð Þ*<sup>s</sup>* <sup>¼</sup> *KP*ð Þ *Ds* <sup>þ</sup> ð Þ *<sup>K</sup>* � <sup>1</sup>*Dλ*<sup>Ω</sup>

<sup>1</sup> <sup>þ</sup> *KPGS*ð Þ*<sup>s</sup>* <sup>¼</sup> <sup>1</sup>

rotational speed Ω*MAX* for the rotor stable behaviour is as follows:

Ω*MAX* ¼ Ω*CRIT*

velocity Ω*MAX* coincides with the critical frequency Ω*CRIT* of the closed loop. Increasing of the stability margin for the rotational speed of the rotor is possible by

The reciprocal value of the transfer function (Eq. (35)) has the meaning of

If a static force is applied to a rotating journal, then the stiffness of the journal

which means that the stiffness is 1ð Þ þ *KP* times greater than the journal stiffness

Analysis of the effect of active vibration control on the stiffness of the bearing journal assumes a linear mathematical model. Practical calculation of matrix entries

*CYX CYY* � �, **<sup>B</sup>** <sup>¼</sup> *BXX BXY*

shows that the linear model does not differ substantially. The dependence of the matrix entries for the journal bearing of the test rig on the rotor rpm is given by the

at zero speed bearing journal.

*KPGS*ð Þ*s*

*Actively Controlled Journal Bearings for Machine Tools DOI: http://dx.doi.org/10.5772/intechopen.92272*

*GF*ð Þ*s*

introducing an additional feedback.

1

of stiffness **C** and damping **B** matrices

*CD*ðÞ¼ *s*

bearing is given by

without feedback.

**117**

dynamic stiffness for radial force acting at the rotor:

*CD*ð Þ¼ *<sup>j</sup>*<sup>0</sup> <sup>1</sup>

**<sup>C</sup>** <sup>¼</sup> *CXX CXY*

shaft *GD*ð Þ*s* are given by

*Gw*ðÞ¼ *s*

*GD*ðÞ¼ *s*

Movement of the bearing journal inside the bearing bushing may be unstable as is apparent from Eq. (26). This phenomenon is called a whirl. The threshold of stability in angular velocity of the rotor can be calculated by the Muszynska's formula:

$$
\Omega\_{\rm CRIT} = \sqrt{\mathcal{K}/\mathcal{M}}/\lambda. \tag{29}
$$

The equation of motion in the complex form for the rigid rotor operating in a small, localised region in the journal bearing with the movable bushing (*u* 6¼ 0) is as follows (Eq. (26)):

$$M\ddot{\mathbf{r}} + D(\dot{\mathbf{r}} - \dot{\mathbf{u}}) + (K - jD\lambda\Omega)(\mathbf{r} - \mathbf{u}) = \mathbf{F}\_P. \tag{30}$$

The Laplace transform specifies the transfer functions relating the displacement of the bushing to the displacement of the shaft *GS*ð Þ*s* , and the disturbance force to the displacement of the shaft *GF*ð Þ*s* are given by

$$
\mathfrak{r}(\mathfrak{s}) = G\_{\mathbb{S}}(\mathfrak{s})\mathfrak{u}(\mathfrak{s}) + G\_{\mathbb{P}}(\mathfrak{s})F\_{\mathbb{P}}(\mathfrak{s}), \tag{31}
$$

where the mentioned transfer functions are as follows:

$$G\_{\rm S}(s) = \frac{\Delta r(s)}{\Delta \mathfrak{u}(s)} = \frac{Ds + (K - \mathbf{1}D\lambda\Omega)}{\mathbf{M}s^2 + Ds + (K - jD\lambda\Omega)}\tag{32}$$

$$G\_F(\mathfrak{s}) = \frac{\Delta r(\mathfrak{s})}{\Delta \mathbf{F}\_P(\mathfrak{s})} = \frac{1}{M\mathfrak{s}^2 + D\mathfrak{s} + (K - jD\lambda\mathfrak{\Omega})} \tag{33}$$

The active vibration control of journal bearings uses the bushing position as the control variable *u* and the shaft position as a controlled variable *r*. The control variable is an output of a controller. The controller transforms an error signal computed as a difference of a reference (SP set point) and actual position of the journal. As is evident from the block diagram in **Figure 8**, the controller is of the proportional type with the gain of *KP*.

Substituting *s* ¼ *jω* we can obtain frequency responses of the journal bearing system to harmonic oscillation of the bearing bushing position and disturbance force at the angular frequency *ω* ¼ 2*πf* which can differ from the angular velocity Ω. Milling tool excites the strength of a frequency that is an integer multiple of the

**Figure 8.** *Closed control loop.*

*Actively Controlled Journal Bearings for Machine Tools DOI: http://dx.doi.org/10.5772/intechopen.92272*

where *g* is gravitational acceleration. Stiffness of the oil film for static force

Movement of the bearing journal inside the bearing bushing may be unstable as

is apparent from Eq. (26). This phenomenon is called a whirl. The threshold of stability in angular velocity of the rotor can be calculated by the Muszynska's

<sup>Ω</sup>*CRIT* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffi

The equation of motion in the complex form for the rigid rotor operating in a small, localised region in the journal bearing with the movable bushing (*u* 6¼ 0) is as

The Laplace transform specifies the transfer functions relating the displacement of the bushing to the displacement of the shaft *GS*ð Þ*s* , and the disturbance force to

<sup>Δ</sup>*u*ð Þ*<sup>s</sup>* <sup>¼</sup> *Ds* <sup>þ</sup> ð Þ *<sup>K</sup>* � <sup>1</sup>*Dλ*<sup>Ω</sup>

The active vibration control of journal bearings uses the bushing position as the

Substituting *s* ¼ *jω* we can obtain frequency responses of the journal bearing system to harmonic oscillation of the bearing bushing position and disturbance force at the angular frequency *ω* ¼ 2*πf* which can differ from the angular velocity Ω. Milling tool excites the strength of a frequency that is an integer multiple of the

<sup>Δ</sup>*FP*ð Þ*<sup>s</sup>* <sup>¼</sup> <sup>1</sup>

control variable *u* and the shaft position as a controlled variable *r*. The control variable is an output of a controller. The controller transforms an error signal computed as a difference of a reference (SP set point) and actual position of the journal. As is evident from the block diagram in **Figure 8**, the controller is of the

*M*€*r* þ *D*ð Þþ *r*\_ � *u*\_ ð Þ *K* � *jDλ*Ω ð Þ¼ *r* � *u FP:* (30)

*r*ðÞ¼ *s GS*ð Þ*s u*ðÞþ*s GF*ð Þ*s FP*ð Þ*s* , (31)

*Ms*<sup>2</sup> <sup>þ</sup> *Ds* <sup>þ</sup> ð Þ *<sup>K</sup>* � *jDλ*<sup>Ω</sup> (32)

*Ms*<sup>2</sup> <sup>þ</sup> *Ds* <sup>þ</sup> ð Þ *<sup>K</sup>* � *jDλ*<sup>Ω</sup> (33)

¼ *Dλ*Ω þ *jK:* (28)

*K=M* p *=λ:* (29)

increases proportionally with rotational speed of the bearing journal:

*Machine Tools - Design, Research, Application*

*Mg r*0

formula:

follows (Eq. (26)):

the displacement of the shaft *GF*ð Þ*s* are given by

*GS*ðÞ¼ *s*

*GF*ðÞ¼ *s*

proportional type with the gain of *KP*.

**Figure 8.** *Closed control loop.*

**116**

where the mentioned transfer functions are as follows:

Δ*r*ð Þ*s*

Δ*r*ð Þ*s*

frequency of rotation. The stead-state gain of *GS*ð Þ*s* is equal to unit *GS*ð Þ¼ *j*0 1, while the stead-state gain of *GF*ð Þ*s* depends on the rotor angular velocity as it results from the formula *GF*ð Þ¼ *j*0 1*=*ð Þ *K* � *jDλ*Ω . The reciprocal value of *GF*ð Þ *j*0 can be considered as the static stiffness of the oil film without influence of the active vibration control. Stiffness which is defined as a complex value determines the direction of journal axis displacement relative to the direction of the force. The radial force has theoretically identical direction with the journal displacement only at zero speed bearing journal.

The transfer functions relating the set point of the closed-loop system to the displacement of the shaft *Gw*ð Þ*s* and the disturbance force to the displacement of the shaft *GD*ð Þ*s* are given by

$$G\_w(\mathbf{s}) = \frac{K\_P G\_S(\mathbf{s})}{1 + K\_P G\_S(\mathbf{s})} = \frac{K\_P(\text{Ds} + (K - \mathbf{1}D\lambda\Omega))}{M\mathbf{s}^2 + (1 + K\_P)\mathbf{D}\mathbf{s} + (1 + K\_P)(K - jD\lambda\Omega)},\tag{34}$$

$$G\_{D}(\boldsymbol{s}) = \frac{G\_{F}(\boldsymbol{s})}{1 + K\_{P}G\_{S}(\boldsymbol{s})} = \frac{1}{M\mathfrak{s}^{2} + (1 + K\_{P})\mathrm{D}\mathfrak{s} + (1 + K\_{P})(K - jD\lambda\mathfrak{\Omega})}.\tag{35}$$

The stability margin can be calculated under assumption that the open-loop frequency transfer function *G*0ðÞ¼ *s KPGS*ð Þ*s* of the control loop in **Figure 8** is equal to �1. The frequency of the steady-state vibration at the stability margin is given by *<sup>ω</sup>* <sup>¼</sup> *<sup>λ</sup>*<sup>Ω</sup> and *KP* <sup>¼</sup> *<sup>ω</sup>*<sup>2</sup>*M=<sup>K</sup>* � 1. If the feedback gain *KP* is positive, then the maximal rotational speed Ω*MAX* for the rotor stable behaviour is as follows:

$$
\Omega\_{MAX} = \Omega\_{CRIT}\sqrt{K\_P + 1}.\tag{36}
$$

If the proportional controller is disconnected, i.e. *KP* = 0, then the critical angular velocity Ω*MAX* coincides with the critical frequency Ω*CRIT* of the closed loop. Increasing of the stability margin for the rotational speed of the rotor is possible by introducing an additional feedback.

The reciprocal value of the transfer function (Eq. (35)) has the meaning of dynamic stiffness for radial force acting at the rotor:

$$\mathbf{C}\_{D}(\mathfrak{s}) = \frac{\mathbf{1}}{\mathbf{G}\_{D}(\mathfrak{s})} = M\mathfrak{s}^{2} + (\mathbf{1} + K\_{P})D\mathfrak{s} + (\mathbf{1} + K\_{P})(K - jD\lambda\mathfrak{\omega}).\tag{37}$$

If a static force is applied to a rotating journal, then the stiffness of the journal bearing is given by

$$\mathbf{C}\_{\rm D}(\ j\mathbf{0}) = \frac{1}{\mathbf{G}\_{\rm D}(\ j\mathbf{0})} = (\mathbf{1} + K\_{\rm P})(\mathbf{K} - jD\lambda\mathbf{0}) \tag{38}$$

which means that the stiffness is 1ð Þ þ *KP* times greater than the journal stiffness without feedback.

Analysis of the effect of active vibration control on the stiffness of the bearing journal assumes a linear mathematical model. Practical calculation of matrix entries of stiffness **C** and damping **B** matrices

$$\mathbf{C} = \begin{bmatrix} \mathbf{C}\_{XX} & \mathbf{C}\_{XY} \\ \mathbf{C}\_{YX} & \mathbf{C}\_{YY} \end{bmatrix}, \mathbf{B} = \begin{bmatrix} \mathbf{B}\_{XX} & \mathbf{B}\_{XY} \\ \mathbf{B}\_{YX} & \mathbf{B}\_{YY} \end{bmatrix} \tag{39}$$

shows that the linear model does not differ substantially. The dependence of the matrix entries for the journal bearing of the test rig on the rotor rpm is given by the

magnitude depends on the voltage *V*. Displacement of the bearing bushing in the

The force produced by the piezoactuator balances the force effect of the oil film. Virtual motion of the unloaded piezoactuator is proportional to its control supply

Operation graph with all the limitations for the piezoactuator of the P-844.60 type is shown in the left panel of **Figure 11**. If the stiffness of sealing rings is taken into account and the stiffness of the support is assumed to be infinite, then the original range of motion of the bearing bushing is reduced to the size as follows:

½ � *ux MAX* <sup>¼</sup> *<sup>u</sup>*<sup>∗</sup>

The effect of the stiffness of the O-ring seal is shown in the middle panel of **Figure 11**. Displacement of the bearing bushing is reduced from 90 to 77 μm for the given parameters of the control loop. The ultimate stiffness of the support affects

> <sup>¼</sup> *KPA* 1 þ *KPA=KS*

where *a* is a multiple. Maximum displacement of the bearing bushing respecting

*x MAX* 1 þ ð Þ 1 þ *a KOR=KPA*

½ � *ux MAX* <sup>¼</sup> *<sup>u</sup>*<sup>∗</sup>

**7. Experiments with active control of journal bearings**

which relate to control the position of the bearing journal.

First experiments with an imperfect support showed displacement of the bearing bushing of about 20 microns at maximum electrical voltage to supply the piezoactuators. This happened at the beginning of the development when the support arrangement was provisionally extended due to the use of longer piezoactuators. The ideal solution is to install piezoactuators into the bearing housing.

Experiments with the active vibration control run for several years, while the hardware and software of the control system was upgraded step by step. We have improved design of the piezoactuator support, found suitable sensors for measuring the position of the bearing journal, upgraded the lubrication system, and improved the control algorithm. Properties of the active vibration system have previously been described in the paper [11], and now the main results will be described only,

The instability onset of the bearing journal motion inside the bushing arises when crossing the threshold value of rotational speed Eq. (5). This phenomenon means that the steady-state rotation of the journal is not stable and the journal axis starts to whirl at the frequency, which is 0.42–0.48 multiple of the frequency of rotational speed of the rotor. Measurements in this article were carried out on the shaft with the radial clearance of 45 μm. Rotor speed increases according to a ramp function as it is shown in the left panel of **Figure 12**. The time history of the axis coordinates of the bearing journal is shown in the other panels of **Figure 12**. The x

<sup>¼</sup> *KPA* 1 þ *a*

*<sup>x</sup>* ¼ *kV*. The resulting motion *ux* of the bearing bushing also depends on

*x MAX* 1 þ *KOR=KPA*

(40)

(41)

*:* (42)

horizontal direction is designated as *ux*.

*Actively Controlled Journal Bearings for Machine Tools DOI: http://dx.doi.org/10.5772/intechopen.92272*

the load force, as shown in the working graph of **Figure 11**.

the virtual stiffness of the piezoactuator as follows:

all influences is given by the following formula:

*<sup>K</sup>*<sup>Σ</sup> <sup>¼</sup> *KSKPA KS* þ *KPA*

voltage *u*<sup>∗</sup>

**119**

**Figure 9.** *Real stiffness and damping matrices according to the Reynolds model.*

graphs in **Figure 9**. Pertinent stiffness and damping coefficients are obtained by solving Reynolds equation providing that the journal performs small harmonic motion in neighborhood of its equilibrium position.

#### **6. Limits of the bearing bushing motion**

The range of the manipulated variable, which is the position of bearing bushing and at the same time, the controller gain, determines the way to install piezoactuators. The equivalent circuit of the mechanical branch of the control loop for the horizontal direction is shown in **Figure 10**. Parameters that indicate stiffness in the scheme in **Figure 10** are associated to the individual elements of the control loop as follows: *KPA* is for the piezoactuator, *KS* is for the support, and *KOR* is for the O-ring seal. The piezoactuator is a source of the mechanical travel *u*<sup>∗</sup> *<sup>x</sup>* whose

**Figure 10.**

*Mechanical branch of the control loop.*

**Figure 11.** *Piezoactuator operation graphs.*

*Actively Controlled Journal Bearings for Machine Tools DOI: http://dx.doi.org/10.5772/intechopen.92272*

magnitude depends on the voltage *V*. Displacement of the bearing bushing in the horizontal direction is designated as *ux*.

The force produced by the piezoactuator balances the force effect of the oil film. Virtual motion of the unloaded piezoactuator is proportional to its control supply voltage *u*<sup>∗</sup> *<sup>x</sup>* ¼ *kV*. The resulting motion *ux* of the bearing bushing also depends on the load force, as shown in the working graph of **Figure 11**.

Operation graph with all the limitations for the piezoactuator of the P-844.60 type is shown in the left panel of **Figure 11**. If the stiffness of sealing rings is taken into account and the stiffness of the support is assumed to be infinite, then the original range of motion of the bearing bushing is reduced to the size as follows:

$$[u\_{\mathbf{x}}]\_{MAX} = \frac{[u\_{\mathbf{x}}^{\*}]\_{MAX}}{1 + K\_{OR}/K\_{PA}}\tag{40}$$

The effect of the stiffness of the O-ring seal is shown in the middle panel of **Figure 11**. Displacement of the bearing bushing is reduced from 90 to 77 μm for the given parameters of the control loop. The ultimate stiffness of the support affects the virtual stiffness of the piezoactuator as follows:

$$K\_{\Sigma} = \frac{K\_{S}K\_{PA}}{K\_{S} + K\_{PA}} = \frac{K\_{PA}}{1 + K\_{PA}/K\_{S}} = \frac{K\_{PA}}{1 + a} \tag{41}$$

where *a* is a multiple. Maximum displacement of the bearing bushing respecting all influences is given by the following formula:

$$\left[u\_{\rm x}\right]\_{MAX} = \frac{\left[u\_{\rm x}^{\*}\right]\_{MAX}}{1 + (1 + a)K\_{OR}/K\_{PA}}.\tag{42}$$

First experiments with an imperfect support showed displacement of the bearing bushing of about 20 microns at maximum electrical voltage to supply the piezoactuators. This happened at the beginning of the development when the support arrangement was provisionally extended due to the use of longer piezoactuators. The ideal solution is to install piezoactuators into the bearing housing.

#### **7. Experiments with active control of journal bearings**

Experiments with the active vibration control run for several years, while the hardware and software of the control system was upgraded step by step. We have improved design of the piezoactuator support, found suitable sensors for measuring the position of the bearing journal, upgraded the lubrication system, and improved the control algorithm. Properties of the active vibration system have previously been described in the paper [11], and now the main results will be described only, which relate to control the position of the bearing journal.

The instability onset of the bearing journal motion inside the bushing arises when crossing the threshold value of rotational speed Eq. (5). This phenomenon means that the steady-state rotation of the journal is not stable and the journal axis starts to whirl at the frequency, which is 0.42–0.48 multiple of the frequency of rotational speed of the rotor. Measurements in this article were carried out on the shaft with the radial clearance of 45 μm. Rotor speed increases according to a ramp function as it is shown in the left panel of **Figure 12**. The time history of the axis coordinates of the bearing journal is shown in the other panels of **Figure 12**. The x

graphs in **Figure 9**. Pertinent stiffness and damping coefficients are obtained by solving Reynolds equation providing that the journal performs small harmonic

The range of the manipulated variable, which is the position of bearing bushing

*<sup>x</sup>* whose

piezoactuators. The equivalent circuit of the mechanical branch of the control loop for the horizontal direction is shown in **Figure 10**. Parameters that indicate stiffness in the scheme in **Figure 10** are associated to the individual elements of the control loop as follows: *KPA* is for the piezoactuator, *KS* is for the support, and *KOR* is for the

and at the same time, the controller gain, determines the way to install

O-ring seal. The piezoactuator is a source of the mechanical travel *u*<sup>∗</sup>

motion in neighborhood of its equilibrium position.

*Real stiffness and damping matrices according to the Reynolds model.*

*Machine Tools - Design, Research, Application*

**6. Limits of the bearing bushing motion**

**Figure 9.**

**Figure 10.**

**Figure 11.**

**118**

*Piezoactuator operation graphs.*

*Mechanical branch of the control loop.*

**Figure 12.** *rpm and the journal position as a function of time.*

coordinate corresponds to horizontal direction, and the y coordinate is for vertical direction. Active vibration control is switched off for the time histories in the second panel. Instability occurs at about 2k rpm. This threshold of instability depends on the viscosity of the oil and the bearing clearance. Oscillations of the bearing journal position are limited by the journal clearance within the bushing.

If the active vibration control is switched on and rates of increase of rotational speed are identical for both measurements, the instability of the bearing occurs at the rotational speed about 12k rpm as is shown in the third and fourth panel of **Figure 12** from the left. Vibrations during the instable motion of the journal are also limited by the journal clearance within the inner gap of the bearing bushing.

Dohnal [14] has solved a similar problem for magnetic bearings. Our experiments on the test bench were conducted for the following amplitudes of excitation *α* = 0, 0.1, 0.15, and 0.2. The static gain was the same as the gain of the previous experiments with the linear controller. The excitation frequency was selected 30 Hz, which is approximately equal to the frequency of vibration at the low rpm. Rotor speed increases according to a ramp function as is shown in the first left panel of **Figure 13**. The effect of the amplitude of the parametric excitation on the journal movement during rotational velocity run-up is shown in other panels of **Figure 13** [11]. The best choice of the excitation amplitude is α = 0.15, which is the position of the journal almost without oscillations. The amplitude of the residual oscillation of the journal does not exceed 8 μm. Precision ball bearings (so-called deep groove ball bearings) which are offered by SKF have a radial clearance (radial internal clearance C2) to a diameter of 30 mm in the range from 1 to 11 micrometers. The maximum rotational speed of the 206-SFFC bearing type is only 7.5k to 13k rpm.

*The journal position as a function of time for tests with active vibration control on.*

*Actively Controlled Journal Bearings for Machine Tools DOI: http://dx.doi.org/10.5772/intechopen.92272*

**Figure 13.**

**Figure 14.**

**121**

**8. Reducing mechanical power losses in actively controlled bearings**

*The electric power consumed by the frequency converter, motor, and bearings.*

Power losses in the journal bearings were estimated from the electric power which is consumed by frequency convertor and motor. Dependence of electrical power upon rotational speed of the motor was measured with and without active control as it is shown in **Figure 14**. Basic power consumption of the motor and

The transient of the journal seems to be reverse for the vertical motion (y-axis) of the bearing journal in the second and third panel of **Figure 12**. The scale for the vertical motion is reverse in these figures, meaning upside-down. The relationship between horizontal and vertical motion of the journal shows the orbit of the journal axis in the rightmost fourth panel of **Figure 12**. The shape of the orbit is approximately circular when instability occurs.

Threshold of instability is increased six times now using a proportional feedback. According to Eq. (12) this multiple corresponds to the open-loop gain, which is equal to 35. Years ago, we achieved an increase in the threshold of instability only about by 70% for a piezoactuator support with insufficient stiffness. Such an increase of the instability threshold corresponds to the open-loop gain equal to 2.

The active vibration control is not turned on at 0 rpm of the rotor but after finishing a transient process, which ends by lifting the journal to approximately the middle position in the vertical direction which takes approximately 15 seconds for the given rate of the increase of speed.

Through the experiment under specific conditions, the observed onset of instability was at 8450 rpm for control only in the x-axis direction and at 7100 rpm for control only in the y-axis direction [12]. It confirms the rule that static load delays the onset of instability at higher speeds. Control in both directions is required if the direction of the radial force may change or if the rotor has a vertical axis, i.e., the radial force is missing.

The linear proportional controller was used for active vibration control for measurements presented in **Figure 13**. Parametric excitation means that at least one parameter of the system varies periodically in time according to a sinusoidal function, as was suggested by Tondl and Dohnal [13, 14]. The gain of the proportional controller was selected as this varying parameter. The system becomes nonlinear and nonstationary. The gain of the proportional controller is given as follows:

$$K\_P = K\_{P0}(1 + a \sin\left(\alpha\_0 t\right)),\tag{43}$$

where *α* is dimensionless amplitude of excitation, *KP*<sup>0</sup> is static gain factor, and *ω*<sup>0</sup> is angular frequency of excitation.

*Actively Controlled Journal Bearings for Machine Tools DOI: http://dx.doi.org/10.5772/intechopen.92272*

#### **Figure 13.**

coordinate corresponds to horizontal direction, and the y coordinate is for vertical direction. Active vibration control is switched off for the time histories in the second panel. Instability occurs at about 2k rpm. This threshold of instability depends on the viscosity of the oil and the bearing clearance. Oscillations of the bearing journal position are limited by the journal clearance within the bushing. If the active vibration control is switched on and rates of increase of rotational speed are identical for both measurements, the instability of the bearing occurs at the rotational speed about 12k rpm as is shown in the third and fourth panel of **Figure 12** from the left. Vibrations during the instable motion of the journal are also limited by the journal clearance within the inner gap of the bearing bushing.

The transient of the journal seems to be reverse for the vertical motion (y-axis) of the bearing journal in the second and third panel of **Figure 12**. The scale for the vertical motion is reverse in these figures, meaning upside-down. The relationship between horizontal and vertical motion of the journal shows the orbit of the journal axis in the rightmost fourth panel of **Figure 12**. The shape of the orbit is approxi-

Threshold of instability is increased six times now using a proportional feedback.

Through the experiment under specific conditions, the observed onset of instability was at 8450 rpm for control only in the x-axis direction and at 7100 rpm for control only in the y-axis direction [12]. It confirms the rule that static load delays the onset of instability at higher speeds. Control in both directions is required if the direction of the radial force may change or if the rotor has a vertical axis, i.e., the

The linear proportional controller was used for active vibration control for measurements presented in **Figure 13**. Parametric excitation means that at least one parameter of the system varies periodically in time according to a sinusoidal function, as was suggested by Tondl and Dohnal [13, 14]. The gain of the proportional controller was selected as this varying parameter. The system becomes nonlinear and nonstationary. The gain of the proportional controller is given as follows:

where *α* is dimensionless amplitude of excitation, *KP*<sup>0</sup> is static gain factor, and

*KP* ¼ *KP*0ð Þ 1 þ *α* sin ð Þ *ω*0*t* , (43)

According to Eq. (12) this multiple corresponds to the open-loop gain, which is equal to 35. Years ago, we achieved an increase in the threshold of instability only about by 70% for a piezoactuator support with insufficient stiffness. Such an increase of the instability threshold corresponds to the open-loop gain equal to 2. The active vibration control is not turned on at 0 rpm of the rotor but after finishing a transient process, which ends by lifting the journal to approximately the middle position in the vertical direction which takes approximately 15 seconds for

mately circular when instability occurs.

*rpm and the journal position as a function of time.*

*Machine Tools - Design, Research, Application*

the given rate of the increase of speed.

*ω*<sup>0</sup> is angular frequency of excitation.

radial force is missing.

**120**

**Figure 12.**

*The journal position as a function of time for tests with active vibration control on.*

Dohnal [14] has solved a similar problem for magnetic bearings. Our experiments on the test bench were conducted for the following amplitudes of excitation *α* = 0, 0.1, 0.15, and 0.2. The static gain was the same as the gain of the previous experiments with the linear controller. The excitation frequency was selected 30 Hz, which is approximately equal to the frequency of vibration at the low rpm. Rotor speed increases according to a ramp function as is shown in the first left panel of **Figure 13**. The effect of the amplitude of the parametric excitation on the journal movement during rotational velocity run-up is shown in other panels of **Figure 13** [11]. The best choice of the excitation amplitude is α = 0.15, which is the position of the journal almost without oscillations. The amplitude of the residual oscillation of the journal does not exceed 8 μm. Precision ball bearings (so-called deep groove ball bearings) which are offered by SKF have a radial clearance (radial internal clearance C2) to a diameter of 30 mm in the range from 1 to 11 micrometers. The maximum rotational speed of the 206-SFFC bearing type is only 7.5k to 13k rpm.

#### **8. Reducing mechanical power losses in actively controlled bearings**

Power losses in the journal bearings were estimated from the electric power which is consumed by frequency convertor and motor. Dependence of electrical power upon rotational speed of the motor was measured with and without active control as it is shown in **Figure 14**. Basic power consumption of the motor and

frequency convertor was measured with the disconnected clutch between the motor and rotor; it means that the bearings were inoperative. The friction loss of a pair of bearings at 7k rpm is 66 W in an unstable operation, and if the active vibration control is on, then the friction loss is of only 48 W. The active vibration control reduces the friction losses of journal bearings by 27%. The bearing clearance amounts to 90 μm for the bearing journal of the diameter 30 mm. As a lubricant the hydraulic oil of the OL-P03 type (VG 10 grade, kinematic viscosity 2.5 to 4 mm<sup>2</sup> /s at 40°C) was used. All tests were undertaken at ambient temperature about 20°C. For small power loss by friction in the bearings, the actively controlled bearings can be used in systems for storing the kinetic energy as they are flywheels that spin at high speed. Longer life compared with roller bearings is another advantage of this type of bearings [11].

rig. The bearing diameter is 30 mm, and the length-to-diameter ratio is equal to about 0.77. The radial clearance of the journal is 45 μm and the very low viscosity oil is used. This combination causes instability of the oil whirl type from the rotational speed of 2k rpm. The active vibration control extends stable operating rotational speed up to 12k rpm, i.e., six times. Also the stiffness of the bearing journal increases significantly during a displacement from equilibrium position. The friction loss of a pair of bearings at 7k rpm is 66 W in an unstable operation, and if the active vibration control is switched ON, then the friction loss is of only 48 W. The active vibration control reduces the friction losses by 27%. The linear proportional controller was used for the active vibration control. The quality of control has been enhanced with the use of periodic changes of the controller gain, which is known as a parametric excitation. The effect of this way of control reduces the journal residual oscillation to the limits which does not exceed 8 μm. This amplitude is comparable with the radial clearance of the ball bearings of the deep groove type. The experiments with the time-periodic changes of the controller gain confirm the

This work was supported by the European Regional Development Fund in the Research Centre of Advanced Mechatronic Systems project, CZ.02.1.01/0.0/0.0/ 16\_019/0000867 within the Operational Programme Research, Development and Education and the project SP2020/57 Research and Development of Advanced Methods in the Area of Machines and Process Control supported by the Ministry of Education, Youth and Sports. This publication was issued thanks to supporting within the operational programme research and innovation for the project: "New generation of freight railway wagons" (project code in ITMS2014+: 313010P922) co-financed from the resources of the European Regional Development Fund.

, Jaromír Škuta<sup>1</sup>

, Renata Wagnerová<sup>1</sup>

positive effect on the vibration response.

*Actively Controlled Journal Bearings for Machine Tools DOI: http://dx.doi.org/10.5772/intechopen.92272*

**Acknowledgements**

**Author details**

and Stanislav Žiaran<sup>3</sup>

Bratislava, Slovakia

**123**

\*, Jiří Šimek<sup>2</sup>

of Ostrava, Ostrava, Czech Republic

2 TECHLAB Ltd., Prague, Praha, Czech Republic

\*Address all correspondence to: jiri.tuma@vsb.cz

provided the original work is properly cited.

, Miroslav Mahdal<sup>1</sup>

1 Department of Control Systems and Instrumentation, VSB Technical University

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

3 Mechanical Engineering Faculty, The Slovak University of Technology in

Jiří Tůma<sup>1</sup>

#### **9. Stiffness of actively controlled bearings**

The bearing bushing is suspended on a pair of piezoactuators, and the bearing journal is supported by an oil wedge. According to catalogue data, we used a linear piezoactuator, which is able to generate force of 3 kN in pressure or 700 N in tension on the track in the range of 90 μm. These parameters correspond to the piezoactuator stiffness of 33 MN/m. Stiffness of the force transducer is 2000 MN/ m, which is two orders of magnitude higher than the stiffness of piezoactuators. Stiffness of the O-ring seal is 5.5 MN/m which increases the stiffness of the piezoactuator by this value. Force is transmitted to the bearing journal through the oil film. Based on the simulations, it can be estimated that the direct stiffness (*CXX*) of the oil film in neighborhood of the central position within the bushing bore is of 185 kN/m and the quadrature stiffness (*CXY* and *CYX*) is still an order of magnitude larger, and increases proportionally with the rotational speed; see Eq. (39) and **Figure 9**. This stiffness increases by many orders of magnitude if the journal is approaching the bearing bushing wall. Stiffness of the journal support is defined first of all by stiffness of the oil film. A steady-state error in a noncontrolled bearing originates in a radial load which can be considered as a disturbance. A proportional controller which governs a system of journal bearings in the closed-loop with an open-loop gain *KP* reduces the steady-state error 1ð Þ þ *KP* times which results in increase of the oil wedge stiffness 1ð Þ þ *KP* times compared to the design without a feedback. An integration controller reduces steady-state error to zero, which corresponds to a theoretically infinite stiffness. The use of a derivative component of the controller was also analysed, but additive noise would reduce its effect [15]. Allowable forces, however, are limited by the load capacity of the piezoactuators. The pressure force is thus less than 3 kN. Notice that on the market there are piezoactuators enabling to generate forces up to 20 kN.

Stiffness of precision rolling bearings ranges from 100 to 200 kN/m, regardless of the load, while the stiffness of hydrodynamic bearings in neighborhood of central position (low load) is of the order of several kN/m. However, with active control, the stiffness can increase as much as 35 times, i.e., it can achieve values around 100 kN/m, which is comparable to that of precision ball bearing.

#### **10. Conclusion**

Experiments prove the correctness of the theoretical prediction which refers to the extending of the operating range of plain bearings when active vibration control is used. The performance of the actively controlled bearing was tested on the test

*Actively Controlled Journal Bearings for Machine Tools DOI: http://dx.doi.org/10.5772/intechopen.92272*

rig. The bearing diameter is 30 mm, and the length-to-diameter ratio is equal to about 0.77. The radial clearance of the journal is 45 μm and the very low viscosity oil is used. This combination causes instability of the oil whirl type from the rotational speed of 2k rpm. The active vibration control extends stable operating rotational speed up to 12k rpm, i.e., six times. Also the stiffness of the bearing journal increases significantly during a displacement from equilibrium position. The friction loss of a pair of bearings at 7k rpm is 66 W in an unstable operation, and if the active vibration control is switched ON, then the friction loss is of only 48 W. The active vibration control reduces the friction losses by 27%. The linear proportional controller was used for the active vibration control. The quality of control has been enhanced with the use of periodic changes of the controller gain, which is known as a parametric excitation. The effect of this way of control reduces the journal residual oscillation to the limits which does not exceed 8 μm. This amplitude is comparable with the radial clearance of the ball bearings of the deep groove type. The experiments with the time-periodic changes of the controller gain confirm the positive effect on the vibration response.

#### **Acknowledgements**

frequency convertor was measured with the disconnected clutch between the motor and rotor; it means that the bearings were inoperative. The friction loss of a pair of bearings at 7k rpm is 66 W in an unstable operation, and if the active vibration control is on, then the friction loss is of only 48 W. The active vibration control reduces the friction losses of journal bearings by 27%. The bearing clearance amounts to 90 μm for the bearing journal of the diameter 30 mm. As a lubricant the hydraulic oil of the OL-P03 type (VG 10 grade, kinematic viscosity 2.5 to 4 mm<sup>2</sup>

40°C) was used. All tests were undertaken at ambient temperature about 20°C. For small power loss by friction in the bearings, the actively controlled bearings can be used in systems for storing the kinetic energy as they are flywheels that spin at high speed. Longer life compared with roller bearings is another advantage

The bearing bushing is suspended on a pair of piezoactuators, and the bearing journal is supported by an oil wedge. According to catalogue data, we used a linear piezoactuator, which is able to generate force of 3 kN in pressure or 700 N in tension on the track in the range of 90 μm. These parameters correspond to the piezoactuator stiffness of 33 MN/m. Stiffness of the force transducer is 2000 MN/ m, which is two orders of magnitude higher than the stiffness of piezoactuators. Stiffness of the O-ring seal is 5.5 MN/m which increases the stiffness of the

piezoactuator by this value. Force is transmitted to the bearing journal through the oil film. Based on the simulations, it can be estimated that the direct stiffness (*CXX*) of the oil film in neighborhood of the central position within the bushing bore is of 185 kN/m and the quadrature stiffness (*CXY* and *CYX*) is still an order of magnitude larger, and increases proportionally with the rotational speed; see Eq. (39) and **Figure 9**. This stiffness increases by many orders of magnitude if the journal is approaching the bearing bushing wall. Stiffness of the journal support is defined first of all by stiffness of the oil film. A steady-state error in a noncontrolled bearing originates in a radial load which can be considered as a disturbance. A proportional controller which governs a system of journal bearings in the closed-loop with an open-loop gain *KP* reduces the steady-state error 1ð Þ þ *KP* times which results in increase of the oil wedge stiffness 1ð Þ þ *KP* times compared to the design without a feedback. An integration controller reduces steady-state error to zero, which corresponds to a theoretically infinite stiffness. The use of a derivative component of the controller was also analysed, but additive noise would reduce its effect [15]. Allowable forces, however, are limited by the load capacity of the piezoactuators. The

pressure force is thus less than 3 kN. Notice that on the market there are

Stiffness of precision rolling bearings ranges from 100 to 200 kN/m, regardless of the load, while the stiffness of hydrodynamic bearings in neighborhood of central position (low load) is of the order of several kN/m. However, with active control, the stiffness can increase as much as 35 times, i.e., it can achieve values around 100

Experiments prove the correctness of the theoretical prediction which refers to the extending of the operating range of plain bearings when active vibration control is used. The performance of the actively controlled bearing was tested on the test

piezoactuators enabling to generate forces up to 20 kN.

kN/m, which is comparable to that of precision ball bearing.

**10. Conclusion**

**122**

of this type of bearings [11].

*Machine Tools - Design, Research, Application*

**9. Stiffness of actively controlled bearings**

/s at

This work was supported by the European Regional Development Fund in the Research Centre of Advanced Mechatronic Systems project, CZ.02.1.01/0.0/0.0/ 16\_019/0000867 within the Operational Programme Research, Development and Education and the project SP2020/57 Research and Development of Advanced Methods in the Area of Machines and Process Control supported by the Ministry of Education, Youth and Sports. This publication was issued thanks to supporting within the operational programme research and innovation for the project: "New generation of freight railway wagons" (project code in ITMS2014+: 313010P922) co-financed from the resources of the European Regional Development Fund.

#### **Author details**

Jiří Tůma<sup>1</sup> \*, Jiří Šimek<sup>2</sup> , Miroslav Mahdal<sup>1</sup> , Jaromír Škuta<sup>1</sup> , Renata Wagnerová<sup>1</sup> and Stanislav Žiaran<sup>3</sup>

1 Department of Control Systems and Instrumentation, VSB Technical University of Ostrava, Ostrava, Czech Republic

2 TECHLAB Ltd., Prague, Praha, Czech Republic

3 Mechanical Engineering Faculty, The Slovak University of Technology in Bratislava, Slovakia

\*Address all correspondence to: jiri.tuma@vsb.cz

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### **References**

[1] Budynas RG, Nisbett JK. Shigley's Mechanical Engineering Design. 9th ed. New York: McGraw-Hill; 2011

[2] McKee SA, McKee TR. Journal bearing friction in the region of thin film lubrication. SAE Journal. 1932;**31**: 371-377

[3] Simek J, Tuma J, Skuta J, Klecka R. Unorthodox behavior of a rigid rotor supported in sliding bearings. In: Proceedings of the Colloquium Dynamics of Machines. Prague: Institute of Thermomechanics; 2010. pp. 85-90

[4] Simek J, Tuma J, Skuta J, Mahdal M. Test stand for affecting rotor behavior by active control of sliding bearings. In: Proceedings of the Colloquium Dynamics of Machines. Prague: Institute of Thermomechanics; 2014. pp. 151-156

[5] Tůma J, Šimek J, Škuta J, Los J. Active vibrations control of journal bearings with the use of piezoactuators. Mechanical Systems and Signal Processing. 2013;**36**:618-629

[6] Wagnerová R, Tůma J. Use of complex signals in modeling of journal bearings. In: 8th Vienna International Conference on Mathematical Modeling. Vienna; Austria: MATHMOD; 2015. pp. 520-525

[7] Dwivedy SK, Tiwari R. Dynamics of Machinery, A Lecture Notes. Guwahati, India: All India Council of Technical Education; 2006

[8] Ferfecki P, Zapomel J. Investigation of vibration mitigation of flexible support rigid rotors equipped with controlled elements. In: 5th International Conference on Modeling of Mechanical and Mechatronic Systems. Vol. 48. Zemplinska Sirava; Slovakia: MMaMS; 2012. pp. 135-142

[9] Mendes RU, Cavalca KJ. On the instability threshold of journal bearing supported rotors. International Journal of Rotating Machinery. 2014;**2014**: 351261

[10] Muszynska A. Whirl and whip – rotor/bearing stability problems. Journal of Sound and Vibration. 1986;**110**(3): 443-462

[11] Tůma J, Šimek J, Mahdal M, Škuta J, Wagnerová R. Actively controlled journal bearings. In: Ecker H, editor. Proceedings of the 12th SIRM Conference. Graz, Österreich; 2017. pp. 443-462

[12] Los J. Mechatronické systémy s piezoaktuátory. VSB Technical University of Ostrava. PhD. Thesis.; 2016

[13] Tondl A. Quenching of Self-Excited Vibrations. Prague: Academia; 1991

[14] Dohnal F, Markert R, Hilsdorf T. Enhancement of external damping of a flexible rotor in active magnetic bearings by time-periodic stiffness. In: Proceedings of the SIRM, Internationale Tagung Schwingungen in rotierenden Maschinen. Darmstadt, Deutschland; 2011

[15] Víteček A, Tůma J, Vítečková M. Stability of rigid rotor in journal bearing. In: Transactions of the VŠB, Mechanical Series. No. 2. Vol. LIV. Technical University of Ostrava; 2008. pp. 159-164

**125**

Section 3

Application Usage
