*Reliability Technology Based on Meta-Action for CNC Machine Tool DOI: http://dx.doi.org/10.5772/intechopen.85163*

**Table 3.** *MU-FMEA (partial).*

Taking the worm rotation meta-action (shown in **Figure 10**) and the end-toothed disc indexing table (shown in **Figure 11**) as examples, the MU-FTA and MU-FMEA are shown below [24]. Therefore, MU-FTA and MU-FMEA are shown in **Figure 12** and **Table 3**, and the specific contents of the fault tree are shown in **Table 2**.

A Worm vibration X5 Interference between bearing and shaft is too large

Manufacturing technology is important to guarantee the reliability of CNC machine tools. Assembly, the last step of manufacture, also inadvertently affects the reliability of CNC machine tools. The research on CNC machine tools' assembly reliability by using FMA can be categorized into the following two areas:

The main methodologies of assembly error modeling by using FMA are assembly error transfer link graph [25] and assembly error propagation state space model [26].

*3.3.3 Manufacturing technology of CNC machine tool by using FMA*

**Label Event definitions Label Event definitions**

*Reliability and Maintenance - An Overview of Cases*

C2 Insufficient bearing preload X8 Unclean grease

X2 Breaking liner X10 Aging of shim gaskets X3 Teeth bonding X11 Bearing preload is too large

B1 Bearing vibration X6 Fatigue failure of disc springs C1 Bad lubrication of bearings X7 Unreasonable grease injection

X1 Bad assembling of coupling X9 Loosening round nut loosening

• assembly error analysis;

**Figure 12.**

**Table 2.**

**58**

*FTA of worm's vibration.*

X4 Teeth pitting

*FTA event definition of worm vibration.*

• assembly reliability modeling.

*3.3.3.1 Assembly error modeling technology by using FMA*

1. Assembly error transfer link graph.

The assembly errors of the MU can be categorized into five aspects, namely:


The transfer and accumulation processes are shown in the assembly error transfer link graph (**Figure 18**) by using the error propagation link. The graph is a basic encapsulation unit that represents the error propagation and accumulation rules in assembly parts or between assembly parts. The function models of each part from the meta-action assembly units (MU) are presented by the circles, whereas the error propagation rules (they consist of one or several functional relations) between the function models for the parts before assembling (input) and after assembling (output) are presented by rectangles. The linkages between the function models and the rules are presented by arrows. The positive direction of the arrows is directed from the error references to the functions, shown in **Figure 13**.

For the first to the fifth error, *gij* means the *j*th function model in the part *i i*ð Þ ≥1*; j*≥ 0 , *dk* are the error models of the first to the fifth error, thus, the first to fifth error model 0ð Þ ≤*k*≤ 5 of the part *i i*ð Þ ≥1*; j*≥ 0 , *Eimk* means the *k*th error between part *i* and *m*.

There are two kinds of error propagation relation between two parts: coupling and nesting, as shown by **Figure 14**.

The complex assembly error propagation relation network (i.e., link network) is generated by the coupling and nesting of the assembly error transfer link diagram for multiple parts, shown in **Figure 15**.

At last, the link network of error propagation is transformed into the structural link matrix to predict the error propagation of MUs or the entire machine.

The link matrix is made of three aspects:


The above are presented in **Table 4**. This methodology is used to define and describe on one hand the error source among the parts, and on the other, the error source relations during the assembly process.

**Figure 15.**

**Table 4.**

**61**

**Figure 14.**

*and lk*<sup>2</sup> *.*

*Link network of the assembly error propagation (NBL).*

*lk*<sup>2</sup> *k k kk*

*Matrix of error propagation link (MBL).*

*lk*<sup>1</sup> *k k*

*lk*<sup>3</sup> *kkk*

*Link coupling and nesting of error propagation. (a) Coupling between lk*<sup>1</sup> *and lk*<sup>2</sup> *(b) nesting between lk*<sup>1</sup>

*Reliability Technology Based on Meta-Action for CNC Machine Tool*

*DOI: http://dx.doi.org/10.5772/intechopen.85163*

*A BC D*

*lk*<sup>4</sup> *k k kkk*

*lk*<sup>5</sup> *k kk k*

*gpi gpl gij gi***<sup>1</sup>** *gin gik gij gmj gi***<sup>0</sup>** *gkj gk***<sup>0</sup>** *gn***<sup>0</sup>** *gk***<sup>1</sup>** *gk***<sup>2</sup>** *gkn gr***<sup>0</sup>** *gr***<sup>1</sup>** *gr***<sup>2</sup>** *grn*

*lk*<sup>6</sup> *kkkkk*

**Figure 13.** *Link of error propagation. (a) 1st error flow; (b) 2nd error flow.*

*Reliability Technology Based on Meta-Action for CNC Machine Tool DOI: http://dx.doi.org/10.5772/intechopen.85163*

#### **Figure 14.**

• assembly torque (deformation) error; and

*Reliability and Maintenance - An Overview of Cases*

the error references to the functions, shown in **Figure 13**.

The transfer and accumulation processes are shown in the assembly error transfer link graph (**Figure 18**) by using the error propagation link. The graph is a basic encapsulation unit that represents the error propagation and accumulation rules in assembly parts or between assembly parts. The function models of each part from the meta-action assembly units (MU) are presented by the circles, whereas the error propagation rules (they consist of one or several functional relations) between the function models for the parts before assembling (input) and after assembling (output) are presented by rectangles. The linkages between the function models and the rules are presented by arrows. The positive direction of the arrows is directed from

For the first to the fifth error, *gij* means the *j*th function model in the part *i i*ð Þ ≥1*; j*≥ 0 , *dk* are the error models of the first to the fifth error, thus, the first to fifth error model 0ð Þ ≤*k*≤ 5 of the part *i i*ð Þ ≥1*; j*≥ 0 , *Eimk* means the *k*th error

There are two kinds of error propagation relation between two parts: coupling

The complex assembly error propagation relation network (i.e., link network) is generated by the coupling and nesting of the assembly error transfer link diagram

At last, the link network of error propagation is transformed into the structural

The above are presented in **Table 4**. This methodology is used to define and describe on one hand the error source among the parts, and on the other, the error

link matrix to predict the error propagation of MUs or the entire machine.

• measuring error of the parts.

between part *i* and *m*.

• linkage,

**Figure 13.**

**60**

• error sources.

and nesting, as shown by **Figure 14**.

for multiple parts, shown in **Figure 15**.

• error propagation model, and

The link matrix is made of three aspects:

source relations during the assembly process.

*Link of error propagation. (a) 1st error flow; (b) 2nd error flow.*

*Link coupling and nesting of error propagation. (a) Coupling between lk*<sup>1</sup> *and lk*<sup>2</sup> *(b) nesting between lk*<sup>1</sup> *and lk*<sup>2</sup> *.*

**Figure 15.** *Link network of the assembly error propagation (NBL).*


**Table 4.** *Matrix of error propagation link (MBL).*

The link matrix is constructed according to the two-level composite matrix architecture. In this table, the row elements represent the links, the elements in the first level column stand for the assembly parts or components, the second level column elements signify the parts contained in the components, and the center cells are identified by the error source type *k*ð Þ 0≤*k*≤ 5 . However, if there is no error propagation or if there are no effects in assembly quality and accuracy during the error propagation, the cells should be empty.

Suppose the geometric characteristics of motion assembly units are affected by single factor of the MUs, thus, by sorting the MUs that affect the *h*th geometric error of the *g*th motion unit according to the assembly sequence number, shown in

According to the assembly process, after finishing the assembly of *k*th MU, the assembly error outputs are represented by the small displacement screw *Xgh*ð Þ*k* as

where, *k* ¼ 1*,* 2*,* …*, i*,*i* is the total number of the MUs that affect the *h*th geometric error of the *g*th motion unit, *dk* is the translation component of geometric error,

The errors introduced by the dynamic uncertain factors of assembly force and measurement, etc., are considered in the actual assembly process and are shown in

> *Xgh*ð Þ¼ *k Agh*ð Þ*k Xgh*ð Þþ *k* � 1 *Bgh*ð Þ*k μgh*ð Þþ *k vgh*ð Þ*k Tgh*ð Þ¼ *k Cgh*ð Þ*k Xgh*ð Þþ *k ξgh*ð Þ*k*

where, *Agh*ð Þ*k* is the transformation matrix of the geometric error vector among characteristic co-ordinate systems, *Bgh*ð Þ*k* is the error input matrix that reflects the affection of new input geometric characteristic error on assembly units, and *μgh*ð Þ*k*

The error vector consists of the errors generated by the assembly, grinding and repairing of the MUs; and *vgh*ð Þ*k* is the assembly error introduced by the assembly force, *ξgh*ð Þ*k* is the measurement noise obeying the normal distribution with a mean value of 0. However, it is worth noting that there is no error input if this station

is the geometric error vector introduced by the assembly of the *k*th MU.

*Assembly process from meta-action assembly units to motion assembly units.*

<sup>¼</sup> *ak bk ck <sup>α</sup><sup>k</sup> <sup>β</sup><sup>k</sup> <sup>γ</sup><sup>k</sup>* ½ �*<sup>T</sup>*

(4)

*Xgh*ð Þ¼ *k*

*DOI: http://dx.doi.org/10.5772/intechopen.85163*

and *δ<sup>k</sup>* is the rotation component of geometric error.

(

*dk δk* � �

*Reliability Technology Based on Meta-Action for CNC Machine Tool*

**Figure 17**.

below:

Eq. (4).

**Figure 17.**

**63**
