*3.3.1 Reliability mathematical modeling based on MU*

As the smallest action unit in enabling a CNC machine tool's function, MU's reliability affects the whole system [20]. Therefore, the technology incorporated in the MU should be studied and the reliability mathematical model of the MU should be established first.

**Figure 5.** *The assembly model of a MU.*

MU's reliability means the ability of the MU to remain functional. It can be also characterized by the degree of reliability. The reliability degree of the MU means the probability that the MU will perform its required function under given conditions for a stated time interval [21], namely, the probability that MU output characteristic parameters are within acceptable ranges in specified time periods. This is shown in Eq. (1).

$$R = P[Y\_{\min} \le Y(t) \le Y\_{\max}] \tag{1}$$

where, *Y t*ð Þ means the output quality characteristic parameters (such as precision, accuracy life, performance stability, etc.), ½ � *Y*min*; Y*max defines the ranges of MU's output quality characteristic parameters under design requirements.

Taking the motion precision of MU, for example, and assuming that motion error values of MU follow the normal distribution, then the reliability of the MU can be described as below:

$$R = P(e\_{\min} \le E \le e\_{\max}) = P(E \le e\_{\max}) - P(E \le e\_{\min}) = \Phi\left(\frac{e\_{\max} - \mu}{\sigma}\right) - \Phi\left(\frac{e\_{\min} - \mu}{\sigma}\right)$$

In a practical situation, the CNC machine tool needs to accomplish multiple different missions by different MUs, so the system's mission reliability is actually a dynamic combination of each MU's reliability, shown in **Figure 6**.

The calculation of the machine system's mission reliability is shown in Eq. (2).

$$R^W = \sum\_{i=1}^n a\_i R^{A\_i} \tag{2}$$

problem. To simplify the design, the reliability design technology by using FMA has

*Mission reliability model of MU. (a) Single meta-action unit and (b) machine system based on meta-action unit.*

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

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

*3.3.2.1 Design process planning for machine center driven by FMA structural*

*Design process planning driven by FMA structural decomposition.*

The design process of the machine center is optimized by using the FMA structural decomposition methodology [22]. There is a large amount of information coupling among each design unit; the basic planning of the design process is obtained based on the consideration of each unit's coupling, as shown in **Figure 7**, which can speed-up the design and the development of machining centers. Firstly, the machine center is decomposed into sub-function design units, motion design units, and meta-action design units by FMA structural

Secondly, the initial design sequence (IDS) of the function layer is obtained by considering the coupling among the design units of the sub-function layer. Next, the IDS of the meta-action layer is determined in the light of a sub-function, by taking its motion layer as a transition layer. The last step is to design the mechanical

structures following an ascending order, i.e., from bottom to top (from the

been studied.

**Figure 7.**

**Figure 6.**

decomposition.

**55**

*decomposition*

meta-action layer to the entire machine).

where, *R<sup>W</sup>* means the reliability of the *w*th mission, *α<sup>i</sup>* means the weight that the *i*th MU relative to the *w*th mission, *RAi* means the reliability of the *i*th MU.

#### *3.3.2 Reliability design for CNC machine tool based on MU*

Reliability design is a basic guarantee of the CNC machine tools' reliability. As everyone knows, design of the machine tool is a difficult system engineering *Reliability Technology Based on Meta-Action for CNC Machine Tool DOI: http://dx.doi.org/10.5772/intechopen.85163*

**Figure 6.** *Mission reliability model of MU. (a) Single meta-action unit and (b) machine system based on meta-action unit.*

**Figure 7.** *Design process planning driven by FMA structural decomposition.*

problem. To simplify the design, the reliability design technology by using FMA has been studied.
