**2.3. Dynamic analysis**

While static analysis of *SBS* describes spindle behaviour in a static mode, dynamic analysis describes SBS behaviour under real conditions, in real running time, and so the real operational state is better represented. It is very important to know the dynamic characteristics, especially in high-speed headstocks. It is important to ensure that the operational revolving frequencies do not fall within the resonant zone. When this happens, the vibration amplitude of the spindle is considerably increased, and the spindle's total stiffness falls to unacceptable levels.

Radial Ball Bearings with Angular Contact in Machine Tools 57

External spindle loading

Other influences

 

 

(10)

 

Spindle loading

Gyroskopical wheels effect

**DYNAMIC DEFLECTION**

Spindle material

mounting Spindle and rot. parts

**Figure 6.** Factors affected by SBS dynamic properties

**2.4. Theoretical research on bearing nodes** 

*2.4.1. Arrangements of nodal points* 

2, 3 or more bearings, see Figure 7.

Spindle mat. damping

Spindle and rotating parts dimensions

Spindle and rot.parts running inaccuracy

Bearing nodes stiffness

Spindle

Bearings damping

(rad.s-1).

in [6].

**NATURAL FREQENCIES SPECTRUM**

The application of the aforementioned equations and their modification for masses of "n" value, will create a system of homogenised algebraic equations where the results of determinant D are angular natural frequencies of the transverse vibrations of the spindle i

The procedure for determining the dynamic deflections yi, when the dimensions of the spindle, rotating parts, stiffness of bearing arrangement, and the external radial forces are taken into consideration, is very similar to the previous one. These procedures are described

> 2 2 <sup>2</sup> 11 1 12 2 1n n 22 2 21 1 22 2 2n n

*a m am a m am am a m*

*a m am a m*

It is relatively easy to transform this mathematical model into a computer readable format

Usually, radial ball bearings with angular contact arrangements in their nodal points contain

<sup>1</sup> . . . = = 0 . . . . . . . . . . . .

1 . . .

and the calculation of the dynamic characteristics can be quickly achieved.

. . . 1

2 2 <sup>2</sup> n1 1 n2 2 nn n

The most common determining dynamic characteristics of SBS are:


The SBS dynamic properties (dynamic deflection of spindle front-end, natural frequencies spectrum) [5], are affected by factors shown in Figure 6.

### **Mathematical models for determining the dynamic properties of a spindle**

Currently, the only reliable method for determining dynamic properties is to use experimental measurements. Therefore it is very useful to create reliable mathematical models for determining these dynamic properties.

In line with spindle mass reduction, mathematical models are divided into:

1/ discrete with 1º, 2º and Nº degrees of freedom,

2/ continuous.

The discrete mathematical model developed for measuring the revolving vibration of spindles with No degrees of freedom is worked out in [1], [5]. This mathematical model for calculating the dynamic properties of the spindle enables us to include in our calculation the effects of the materials, the dimensions of the rotating parts, the bearing node stiffness, and the radial forces generated by the cutting process and drive. The results calculated reflect a spectrum of natural frequencies and the dynamic deflection of the spindle under discrete masses.

The deflection of spindle yi loaded with concentrated forces at the ith point can be expressed in the form:

$$\mathbf{i} \text{ yi} = \mathbf{a}\_{\text{i1}} \mathbf{F}\_{\text{1o}} + \mathbf{a}\_{\text{i2}} \mathbf{F}\_{\text{2o}} + \dots \\ \dots + \mathbf{a}\_{\text{ik}} \mathbf{F}\_{\text{ko}} + \dots \\ \dots + \mathbf{a}\_{\text{in}} \mathbf{F}\_{\text{no}} \left(\mathbf{m}\right) \tag{8}$$

where aik (m/N) is Maxwell's affecting factor. Every mass point on the spindle produces centrifugal force

$$\mathbf{F}\_{\rm io} = \mathbf{m}\_{\rm i} \mathbf{y}\_{\rm i} o^2 \quad \text{(N)}\tag{9}$$

where mi (kg) is mass ith discrete segment.

**Figure 6.** Factors affected by SBS dynamic properties

**2.3. Dynamic analysis** 

of the spindle,

2/ continuous.

in the form:

centrifugal force

stiffness falls to unacceptable levels.


spectrum) [5], are affected by factors shown in Figure 6.

models for determining these dynamic properties.

1/ discrete with 1º, 2º and Nº degrees of freedom,

where mi (kg) is mass ith discrete segment.

The most common determining dynamic characteristics of SBS are:


**Mathematical models for determining the dynamic properties of a spindle** 

In line with spindle mass reduction, mathematical models are divided into:

frequencies and the dynamic deflection of the spindle under discrete masses.

While static analysis of *SBS* describes spindle behaviour in a static mode, dynamic analysis describes SBS behaviour under real conditions, in real running time, and so the real operational state is better represented. It is very important to know the dynamic characteristics, especially in high-speed headstocks. It is important to ensure that the operational revolving frequencies do not fall within the resonant zone. When this happens, the vibration amplitude of the spindle is considerably increased, and the spindle's total


The SBS dynamic properties (dynamic deflection of spindle front-end, natural frequencies

Currently, the only reliable method for determining dynamic properties is to use experimental measurements. Therefore it is very useful to create reliable mathematical

The discrete mathematical model developed for measuring the revolving vibration of spindles with No degrees of freedom is worked out in [1], [5]. This mathematical model for calculating the dynamic properties of the spindle enables us to include in our calculation the effects of the materials, the dimensions of the rotating parts, the bearing node stiffness, and the radial forces generated by the cutting process and drive. The results calculated reflect a spectrum of natural

The deflection of spindle yi loaded with concentrated forces at the ith point can be expressed

where aik (m/N) is Maxwell's affecting factor. Every mass point on the spindle produces

io i i F my N 

<sup>2</sup>

i1 1o i2 2o ik ko in no yi a F a F . . . a F . . . a F m (8)

(9)

The application of the aforementioned equations and their modification for masses of "n" value, will create a system of homogenised algebraic equations where the results of determinant D are angular natural frequencies of the transverse vibrations of the spindle i (rad.s-1).

The procedure for determining the dynamic deflections yi, when the dimensions of the spindle, rotating parts, stiffness of bearing arrangement, and the external radial forces are taken into consideration, is very similar to the previous one. These procedures are described in [6].

$$
\Lambda = \begin{vmatrix}
1 - a\_{11} & m\_{11} \ o^2 & -a\_{12} \ m\_{22} \ o^2 & \dots \ \dots & -a\_{1n} \ m\_{n1} \ o^2 \\
\dots & \dots & \dots & \ \dots & \dots & \dots \\
\end{vmatrix}^T = 0 \tag{10}
$$

It is relatively easy to transform this mathematical model into a computer readable format and the calculation of the dynamic characteristics can be quickly achieved.

### **2.4. Theoretical research on bearing nodes**

### *2.4.1. Arrangements of nodal points*

Usually, radial ball bearings with angular contact arrangements in their nodal points contain 2, 3 or more bearings, see Figure 7.
