1 Introduction

Rotor-bearing systems are widely used in the transmission process of mechanical movement and momentum. The performances of rotor-bearing systems need to meet different requirements in different application scenarios. Thereby, the performance design has always been the dominant issue in the stage of system development design. The randomness and time variability of kinematic precision can be expressed by a reliability index [1]. The precision reliability can be earmarked as an important constraint index in the research on tolerance allocation [2], and it is also an important basis for formulating maintenance strategy [3]. Consequently, it is necessary to establish a kinematic precision reliability evaluation method with more accurate performance for rotor-bearing systems to serve the precision design.

An effective and correct precision expression model plays an important role in the research of kinematic precision reliability. In recent years, much attention has been paid to the aspect of the precision modeling theories for rotor-bearing systems. Tang et al. [4] introduced the stream of variation theory into geometric error modeling for mechanical systems, regarded each moving component as a station, and calculated the accumulated geometric errors according to the transmission sequence from the upstream station to the downstream station. Yang and Ding [5] represented the error propagation process from one component to the other different component with a differential motion matrix, and built the integrated precision model of the whole kinematics chain by homogeneous transformation matrices. Bozca [6] calculated the transmission precision of the gear system based on a state-space model. Liu et al. [7] regarded the error accumulation between components as the screw motion of rigid bodies, and established the precision model of systems by screw theory and Jacobian function. Yang et al. [8] established a six-degree-of-freedom geometric motion errors model of the precision rotary stage by adopting the rigid body hypothesis and the spatial transformation matrix. Wang et al. [9] proposed a transmission error model of the ball screw by converting manufacturing errors and installation errors to the errors in normal and axial direction. However, the assembly precision caused by the geometric errors of components and the transmission precision caused by the relative motion relationship of kinematic pairs are not distinguished in the current precision modeling methods, which cannot reflect the motion essence of mechanical systems. Moreover, there are various error sources for the complex system with many components which lead to the cumbersome calculation process and inaccurate results. The main cause of this problem is that the error sources are not reasonably classified in the traditional structured decomposition method. From the perspective of motion modularization, meta-action theory [10] produces a reasonable balance in the subordinate relationship between components and systems. And some applications in reliability analysis have been carried out [11]; the advantages in failure mode and effects analysis have also been proved [12, 13]. Mu et al. [14] introduced the application of meta-action theory in the remaining useful life prediction on account of its advantage in the reflection of the interaction among the components. Chen et al. [15] applied meta-action theory for the reliability optimization design of mechanical systems. Yang et al. [16] presented a performance degradation prediction approach by combining meta-action theory with the digital twin technology. Xiao et al. [17] further explored the motion modularization strategy of the meta-action unit, which can reflect the characteristics of the mechanical systems that the function determined by the motion. Therefore, meta-action theory also lay a good foundation for the precision modeling.

The random geometric errors of components caused by the uncertainty of the manufacturing process are considered as the initial error sources affecting the precision reliability of rotor-bearing systems, and some meaningful research work on the analysis of such errors has been proposed. Huang et al. [18] considered the original manufacturing errors of gears as the random variables affecting the transmission error of the gear mechanism, and presented a kinematic accuracy reliability analysis method using the saddle-point approximation technique. Cai et al. [19] established a geometric/kinematic error model for mechanical systems using homogenous transformation matrices, regarded the geometric error values as random quantities within the tolerance range, and developed an accuracy reliability analysis model based on the high-order moment standardization technique. Wang et al. [20] estimated the probability distribution of the precision state function for mechanical systems with random geometric errors based on the statistical fourth moment method, and employed the Edgeworth series to analyze the dynamic precision reliability. Zhang et al. [21] conducted the importance sampling method to analyze the accuracy reliability of mechanical systems after establishing a comprehensive error model affected by the random geometric errors. Yang et al. [22] analyzed the error propagation between parts based on small displacement torsor and multi-tolerance coupling, presented the assembly precision model of mechanical rotating systems in accordance with the Jacobian-torsor method, and calculated the assembly precision reliability according to the probability distribution of the assembly error index. Ma et al. [23] performed the effects of the stochastic rotor/stator gap caused by manufacturing on vibrational behaviors and reliability of the rotor systems. In addition, there are some time-variant error sources such as the wears on gears and bearings during the operation of the rotor-bearing systems. Wang et al. [24] calculated the wear depths of the parts by the Archard wear model, considered the part geometric parameters in the wear formula as input random variables, and constructed the Kriging models to predict the time-variant reliability of mechanical systems. Chen et al. [25] decomposed the time-varying dynamic transmission error of the gear mechanism into many random variables according to the meshing period, thus realizing the transformation of a time-dependent reliability solution into a time-independent reliability solution, and estimated the reliability based on the saddle-point approximation method. Jiang et al. [26] calculated the cumulative wear of gears in the gear transmission system and the meshing force and stress distribution under the influence of gear wear, and predicted the system dynamic reliability according to the gear strength degradation rule. Furthermore, stochastic models [27] can also be used to describe the process of performance degradation, such as the gamma process model [28] and the Wiener process model [29], and the precision reliability evaluation using the PHI2 method [30], support vector machine [31], response surface method [32], and the Kriging method [33] have been reported in literature. Unfortunately, to the best of our current knowledge, there are few reports in literature which have combined the manufacturing error sources with the time-variant wear sources to evaluate the effects on the kinematic precision reliability of rotor-bearing systems, and that leads to many important error items being ignored and the inaccuracy model.

Motivated by the above-mentioned limitations, a novel kinematic precision reliability modeling method is proposed to improve the accuracy of reliability assessment for rotor-bearing systems in this paper. Specifically, based on the meta-action structural decomposition method, the precision analysis model containing each source related to the output precision of rotor-bearing systems is established. When the distribution of random errors and the expression of time-variant wears are determined, the kinematic precision reliability can be calculated by the time-variant stochastic process discretization method. The remainder of this manuscript is organized as follows: Sect. 2 provides the time-varying wear models for the probably worn components in rotor-bearing systems. In Sect. 3, the output precision model based on the meta-action structural decomposition method is established. Section 4 presents the precision reliability evaluation model considering multi-source wear degradations and random errors. The applicative case is reported in Sect. 5 to validate the proposed method. The conclusions are finally drawn in the final section.

2 Time-variant wear models

The precision time variation and failure of rotor-bearing systems are primarily caused by the wearing of components, especially the wear occurring on gears and bearings. Therefore, these two types of wear are analyzed specifically and respectively as follows.

2.1 Tooth thickness wear

Tooth thickness wear is the gear material continuous reduction process under stress. The wear quantity of the tooth profile surface at a certain time can be calculated by the Archard model. Let V represent the wear volume of the tooth surface and s the relative sliding distance; then

$$\frac{V}{s} = k\frac{W}{H}$$
(1)

where k is the wear coefficient, W the normal pressure on the contact surface, and H the hardness of the material.

By discretizing the contact area on the tooth surface into several contact points, the infinitesimal wear quantity of the arbitrary contact point at a certain time can be obtained and calculated as

$$\delta h = \int_{0}^{s} {\frac{kP}{H}ds}$$
(2)

where P represents the contact stress at the contact point.

The wear coefficient k is a parameter comprehensively affected by factors such as working condition, surface roughness, lubrication characteristics, and material characteristics. According to the Janakiraman regression formula [34], it can be calculated as

$$k = \frac{{3.981 \times 10^{29} }}{{E_{e} }}L^{1.219} G^{ - 7.377} S^{1.589}$$
(3)

And \(L = \frac{W\prime }{{E_{e} R_{e} }}\), \(G = \alpha_{0} E_{e}\), \(S = \frac{{\sqrt {R_{\alpha 1}^{2} + R_{\alpha 2}^{2} } }}{{R_{e} }}\).

Where equivalent elastic modulus \(E_{e} = {1 \mathord{\left/ {\vphantom {1 {\left( {\frac{{1 - \upsilon_{\left( 1 \right)}^{2} }}{{E_{\left( 1 \right)} }} + \frac{{1 - \upsilon_{\left( 2 \right)}^{2} }}{{E_{\left( 2 \right)} }}} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\frac{{1 - \upsilon_{\left( 1 \right)}^{2} }}{{E_{\left( 1 \right)} }} + \frac{{1 - \upsilon_{\left( 2 \right)}^{2} }}{{E_{\left( 2 \right)} }}} \right)}}\) and \(\upsilon_{\left( 1 \right)}\), \(\upsilon_{\left( 2 \right)}\) and \(E_{\left( 1 \right)}\), \(E_{\left( 2 \right)}\) are Poisson ratio and elastic modulus of the driving and driven wheel’s material, respectively. \(W\prime\) is the load per unit length. Equivalent radius of curvature \(R_{e} = {1 \mathord{\left/ {\vphantom {1 {\left( {\frac{1}{{r_{b\left( 1 \right)} }} + \frac{1}{{r_{b\left( 2 \right)} }}} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\frac{1}{{r_{b\left( 1 \right)} }} + \frac{1}{{r_{b\left( 2 \right)} }}} \right)}}\), \(r_{b\left( 1 \right)}\), and \(r_{b\left( 2 \right)}\) are the base circle radius of the driving and driven wheel, respectively. \(\alpha_{0}\) is the pressure-viscosity coefficient of lubricating oil (assuming it is a constant not affected by pressure during the wearing process). \(R_{\alpha 1}\) and \(R_{\alpha 2}\) are the roughness of these two contact surfaces, respectively.

According to Hertz contact theory, the contact stress P can be solved by taking the tooth surface meshing problem of the driving and driven gear equivalent to the mutual extrusion problem of two cylinders. When these two tooth surfaces are in line contact under stress, a rectangular contact area is formed due to elastic deformation. The contact stress distribution of a certain cross section in the contact area at a certain time is shown in Fig. 1.where Pmax represents the maximum Hertz contact stress. Half width of Hertz contact area \(a = \sqrt {\frac{{4W^{\prime}R_{e} }}{{\pi E_{e} }}}\) and the variation range of the relative sliding distance s is [0, a].

Fig. 1
figure 1

Hertz contact stress distribution

Full size image

As demonstrated in Fig. 1, the maximum contact stress occurs in the center of the contact area and has a decaying trend to both sides. Contact stress at the arbitrary contact point is calculated as

$$P\left( x \right) = \frac{{2P_{{{\text{max}}}} }}{{\pi a^{2} }}\sqrt {\left( {a^{2} - x^{2} } \right)}$$
(4)

where the maximum contact stress \(P_{{{\text{max}}}} = {{T_{\left( 1 \right)} } \mathord{\left/ {\vphantom {{T_{\left( 1 \right)} } {B_{\left( 1 \right)} r_{b\left( 1 \right)} }}} \right. \kern-\nulldelimiterspace} {B_{\left( 1 \right)} r_{b\left( 1 \right)} }}\). \(T_{\left( 1 \right)}\) is the load transmitted by the gear pair; \(B_{\left( 1 \right)}\) is the tooth width of the driving gear. Therefore, the calculation equation of the single infinitesimal wear quantity can be rewritten as

$$\delta h = \int_{0}^{a} {\frac{{2kT_{\left( 1 \right)} }}{{\pi a^{2} B_{\left( 1 \right)} r_{b\left( 1 \right)} H}}\sqrt {\left( {a^{2} - x^{2} } \right)} dx}$$
(5)

When the driving wheel rotates once, the contact point of the tooth surface completes single wear accumulation, while the driven wheel completes the corresponding number of times of wear accumulations due to the relationship of the transmission ratio. Therefore, the wear quantity of the driving gear tooth thickness can be calculated as

$$\Delta e_{g\left( 1 \right)} \left( t \right) = \delta h_{g\left( 1 \right)} \cdot n_{g\left( 1 \right)} \cdot t$$
(6)

The wear quantity of the driven gear tooth thickness can be calculated as

$$\Delta e_{g\left( 2 \right)} \left( t \right) = \delta h_{g\left( 2 \right)} \cdot n_{g\left( 2 \right)} \cdot t$$
(7)

2.2 Bearing wear

Rolling bearings are generally composed of rolling elements, bearing outer ring, locking inner ring, and bearing cage. The roller and the raceway move relative to each other in a point-contact manner. A single roller not only rotates around its own axis, but also rolls around the bearing axis on the raceway, and under the force of the bearing cage, the roller also slides on the raceway to a certain extent. Therefore, the internal friction of the rolling system is a compound friction including rolling and sliding friction.

To facilitate the analysis and calculation, it is assumed that the rolling elements are uniformly worn, that is, the wear loss of each roller at a certain time is the same. The rollers move independently, and there is no radial clearance between the rollers and the inner and outer rings. Similar to the calculation of tooth thickness wear, the continuous wear process of bearing is firstly discretized into several infinitesimal wearing moments. Assuming that the wear loss of each point on the contact line at each moment is the same, according to the Archard wear model, the infinitesimal wear loss of the contact point at a certain moment can be calculated as

$$\delta h = \int_{0}^{t} {kPv_{s} dt}$$
(8)

where k represents the material wear coefficient, and the calculation equation is the same as Eq. (3). P represents the contact stress at the contact point. vs represents the relative sliding speed. dt is the infinitesimal contact time.

Normally, the rollers contact with the outer raceway due to the centrifugal force Fc. The force analysis of the roller is shown in Fig. 2. Considering that the rotation speed ω is small, the influence of centrifugal force can be ignored.where Ni represents the normal pressure exerted by the inner raceway on the rollers. No represents the support force of the outer raceway to the roller. \(F_{k}^{1}\) and \(F_{k}^{2}\) represent the force of the bearing cage. fi and fo represent the friction between the inner and outer raceways and the rollers, respectively.

Fig. 2
figure 2

Force analysis of the cylindrical roller in the load-bearing area

Full size image

According to Hertz contact stress distribution in Fig. 1, the half-width a of the contact area is discretized into m units with a width of \(\Delta a = {a \mathord{\left/ {\vphantom {a m}} \right. \kern-\nulldelimiterspace} m}\). Taking the contact between the inner raceway and the roller as an example, the calculation equation of the half width of the contact area \(a^{{\left( {i,g} \right)}}\) is

$$a^{{\left( {i,g} \right)}} = \sqrt {\frac{{4W^{{\left( {i,g} \right)}} }}{\pi }\left( {\frac{{1 - \upsilon_{i}^{2} }}{{E_{i} }} + \frac{{1 - \upsilon_{g}^{2} }}{{E_{g} }}} \right)\left( {\frac{{r_{i} r_{g} }}{{r_{i} + r_{g} }}} \right)}$$
(9)

where \(W^{{\left( {i,g} \right)}}\) is the contact load. \(\upsilon_{i}\), \(\upsilon_{g}\) and \(E_{i}\), \(E_{g}\) are Poisson ratio and elastic modulus of the inner ring and roller material, respectively. \(r_{i}\) and \(r_{g}\) are the contact curvature radius of the inner raceway and the roller, respectively.

Considering that the effective meshing width is much smaller than the meshing length, when the contact area is discretized into units, the slight difference in the equivalent curvature radius between contact units is ignored. Thus, the normal contact stress of the k-th unit within the half width of each contact area can be obtained as

$$P_{k}^{{\left( {i,g} \right)}} = P_{\max } \times \sqrt {1 - \left( {\frac{{\Delta a^{{\left( {i,g} \right)}} \times k - {{\Delta a^{{\left( {i,g} \right)}} } \mathord{\left/ {\vphantom {{\Delta a^{{\left( {i,g} \right)}} } 2}} \right. \kern-\nulldelimiterspace} 2}}}{{a^{{\left( {i,g} \right)}} }}} \right)}$$
(10)

Let Ri be the diameter of the inner raceway, Ro the diameter of the contact point of the outer raceway, Rg the diameter of the rollers, Ror the diameter of the bearing rollers’ center circle, \(\vartheta = {{R_{g} } \mathord{\left/ {\vphantom {{R_{g} } {R_{or} }}} \right. \kern-\nulldelimiterspace} {R_{or} }}\), and n the rotational speed of the inner ring, and let the angular velocity of the inner ring and the worm gear be kept the same. Then, the revolution angular velocity of rollers is calculated as

$$\omega_{g}^{R} = \frac{1}{2}\omega \left( {1 - \vartheta } \right)$$
(11)

The rotation angular velocity of rollers is calculated as

$$\omega_{g}^{r} = \frac{{R_{or} }}{{2R_{g} }}\omega \left( {1 + \vartheta } \right)\left( {1 - \vartheta } \right)$$
(12)

Since the inner ring rotates and the outer ring remains stationary during the motion of the rolling bearing, there are two pairs of friction pairs in the bearing, so the wear value of the bearing includes three parts: the outer ring, the inner ring, and the rollers. The wear values of the inner and outer rings are determined by the single wear value and the number of rolling contact at a contact point in one revolution cycle. The bearing including z rollers, when the inner ring of the bearing makes one rotation on its axis,

$$N_{i} = \frac{{\omega - \omega_{g}^{R} }}{\omega } \times z = \frac{z}{2}\left( {1 + \vartheta } \right)$$
(13)

The number of rollers in contact with the outer raceway is

$$N_{o} = \frac{{\omega_{g}^{R} }}{\omega } \times z = \frac{z}{2}\left( {1 - \vartheta } \right)$$
(14)

Therefore, the calculation equation of the wear quantity of the contact point on the inner raceway is obtained as

$$\delta h_{i} = a^{{\left( {i,g} \right)}} \cdot z \cdot \left( {1 + \vartheta } \right) \cdot n \cdot t \cdot \sum\limits_{k}^{m} {\left[ {k_{k}^{{\left( {i,g} \right)}} \left( {P_{k}^{{\left( {i,g} \right)}} } \right) \cdot P_{k}^{{\left( {i,g} \right)}} } \right]}$$
(15)

The calculation equation of the wear quantity of the contact point on the outer raceway is obtained as

$$\delta h_{o} = a^{{\left( {o,g} \right)}} \cdot z \cdot \left( {1 - \vartheta } \right) \cdot n \cdot t \cdot \sum\limits_{k}^{m} {\left[ {k_{k}^{{\left( {o,g} \right)}} \left( {P_{k}^{{\left( {o,g} \right)}} } \right) \cdot P_{k}^{{\left( {o,g} \right)}} } \right]}$$
(16)

For the roller, its revolution speed is \(n_{g}^{R} = \frac{1}{2}n\left( {1 - \vartheta } \right)\). Since the roller is in contact with both the inner and outer raceways during one rotation, the calculation equation of the wear quantity of the contact point on the roller can be obtained as

$$\begin{aligned} \delta h_{g} & { = a^{{\left( {i,g} \right)}} \cdot \frac{{n_{g}^{r} }}{{n_{g}^{R} }} \cdot n \cdot t \cdot \sum\limits_{k}^{m} {\left[ {k_{k}^{{\left( {i,g} \right)}} \left( {P_{k}^{{\left( {i,g} \right)}} } \right) \cdot P_{k}^{{\left( {i,g} \right)}} } \right]} + a^{{\left( {o,g} \right)}} \cdot \frac{{n_{g}^{r} }}{{n_{g}^{R} }} \cdot n \cdot t \cdot \sum\limits_{k}^{m} {\left[ {k_{k}^{{\left( {o,g} \right)}} \left( {P_{k}^{{\left( {o,g} \right)}} } \right) \cdot P_{k}^{{\left( {o,g} \right)}} } \right]} } \\ & { = \frac{{R_{or} }}{{R_{g} }} \cdot \left( {1 + \vartheta } \right) \cdot n \cdot t \cdot \left\{ {a^{{\left( {i,g} \right)}} \cdot \sum\limits_{k}^{m} {\left[ {k_{k}^{{\left( {i,g} \right)}} \left( {P_{k}^{{\left( {i,g} \right)}} } \right) \cdot P_{k}^{{\left( {i,g} \right)}} } \right]} + a^{{\left( {o,g} \right)}} \cdot \sum\limits_{k}^{m} {\left[ {k_{k}^{{\left( {o,g} \right)}} \left( {P_{k}^{{\left( {o,g} \right)}} } \right) \cdot P_{k}^{{\left( {o,g} \right)}} } \right]} } \right\}} \end{aligned}$$
(17)

Finally, the total wear volume of the bearing \(\Delta e_{r} \left( t \right)\) is calculated as

$$\begin{aligned} \Delta e_{r} \left( t \right) & = {\delta h_{i} + \delta h_{o} + \delta h_{g} = a^{{\left( {i,g} \right)}} \cdot \left( {z + \frac{{R_{or} }}{{R_{g} }}} \right) \cdot \left( {1 + \vartheta } \right) \cdot n \cdot t \cdot \sum\limits_{k}^{m} {\left[ {k_{k}^{{\left( {i,g} \right)}} \left( {P_{k}^{{\left( {i,g} \right)}} } \right) \cdot P_{k}^{{\left( {i,g} \right)}} } \right]} } \\ & { + a^{{\left( {o,g} \right)}} \cdot \left[ {z \cdot \left( {1 - \vartheta } \right) + \frac{{R_{or} }}{{R_{g} }} \cdot \left( {1 + \vartheta } \right)} \right] \cdot n \cdot t \cdot \sum\limits_{k}^{m} {\left[ {k_{k}^{{\left( {o,g} \right)}} \left( {P_{k}^{{\left( {o,g} \right)}} } \right) \cdot P_{k}^{{\left( {o,g} \right)}} } \right]} } \end{aligned}$$
(18)

3 Output kinematic precision model based on meta-action structural decomposition

The performance of rotor-bearing systems is often presented as multiple indicators, and the essential factors that form and affect these indicators are the geometric characteristics of the components. Traditional precision modeling often leads to imprecise model expression due to the large number of components, and cannot consider all the error terms that affect the motion. The meta-action unit (MAU) is a structural unit composed of the moving part and other related components that ensure its correct action. Meta-action theory finds a reasonable balance between the dependencies of components and systems. Most of the components that compose the meta-action unit are assembled in a fixed connection, and its movement is mainly accomplished by plane or cylindrical contact. To solve the relationship between indicators and geometric error items, the accumulation geometric error of intra-MAU and transmission kinematic precision of inter-MAU are calculated separately.

3.1 Meta-action basic concepts

The meta-action unit refers to the most basic form of motion in mechanical products which can accomplish motion and power transmission. The most common forms of motion in mechanical systems are the rotary motion around the fixed axis and the linear motion along the mating surface. Therefore, these two motions are defined as rotation and translation MAU, respectively. Rotor-bearing systems are composed of rotation MAUs.

M represents the transmission torque, θ the output angular displacement, ω the angular velocity, and α the angular acceleration, as shown in Fig. 3.

Fig. 3
figure 3

Rotation meta-action unit

Full size image

MAU is the smallest granularity to accomplish an action, and the most basic structural unit that transmits motion and power in mechanical products. As is shown in Fig. 4, MAU consists of a power input part, middle parts, fasteners, support parts, and a power output part. The power input part provides the power source or receives the output power of the upper MAU, such as gears and screws. The role of the middle parts is to transmit motion and ensure the correct position of the power input and output parts, such as connecting keys, rotating shafts, locating pins, bearings, and bushings. Fasteners play the role of connection, tightening, anti-loosening, and sealing, such as screws, springs, sealing rings, and end caps. The boxes and bases are generally taken as the support parts. The power output part is used as the end movement or power output.

Fig. 4
figure 4

Structural conceptual model of the meta-action unit

Full size image

3.2 Spatial pose of the power output part in MAU

The essence of MAU is to ensure that the power output part completes the designed motion. In order to achieve the expected motion trajectory, the position and orientation errors of the power output part in the X, Y, Z, α, β, and γ directions in the space coordinate system need to be within the specified error range, and the geometric errors of the power input and output parts. The spatial error of the power output part is the accumulation result of machining errors, assembly errors, and wear of the components in MAU. During the assembly process of MAU, the actual pose of the power output part is offset relative to the ideal pose in the X, Y, Z, α, β, and γ directions in the space coordinate system, as shown in Fig. 5.

Fig. 5
figure 5

Space pose formation process of the power output part

Full size image

The screw theory is used to calculate the accumulation error between the parts in MAU. When the arbitrary point in the space coordinate system rotates around a unit vector \({\varvec{\omega}}\) by an angle \(\theta\), its rotation matrix can be calculated according to Rodrigues’ rotation equation as

$$e^{\theta\overset\frown\omega}=\mathbf E+\overset{\boldsymbol\frown}{\mathbf\omega}\;\sin\;\theta+\overset{\boldsymbol\frown}{\mathbf\omega}^2(1-\cos\theta)$$
(19)

where the antisymmetric matrix \(\overset{\boldsymbol\frown}{\mathbf\omega}{ = }\left( {\begin{array}{*{20}c} 0 & { - \omega_{z} } & {\omega_{y} } \\ {\omega_{z} } & 0 & { - \omega_{x} } \\ { - \omega_{y} } & {\omega_{x} } & 0 \\ \end{array} } \right)\).

The motion of the arbitrary rigid body in the space coordinate system can be regarded as a spiral motion around a certain vector axis. The screw-exponential product form of its coordinate change is expressed as

$${\mathbf{g}} = \left( {\begin{array}{*{20}c} {e^{{\theta \overset{\boldsymbol\frown}{\mathbf\omega}}} } & {({\mathbf{E}} - e^{{\theta \overset{\boldsymbol\frown}{\mathbf\omega}}} )({\varvec{\omega}} \times {\varvec{\nu}}) + \theta \varvec{\omega \omega }^{T} {\varvec{\nu}}} \\ 0 & 1 \\ \end{array} } \right)$$
(20)

where \({\varvec{\nu}}\) represents the displacement vector moving along the vector axis.

The pose change of the power output part in the space coordinate system can be decomposed into the spiral motion around the X, Y, and Z directions, and further decomposed into the movement \(\delta_{i}\)(i = x, y, z) and rotation \(\varepsilon_{j}\)(j = x, y, z) around the X, Y, and Z axes. Considering that the error amount is infinitesimal relative to the geometric size of the part, let \(\sin \varepsilon_{i} = \varepsilon_{i}\), \(\cos \varepsilon_{i} = 1\) and ignore the high-order infinitesimal. Then, according to Eq. (20), the pose change screw expression of the power output part is obtained as

$${\mathbf{g}}^{o} = \prod\limits_{{\begin{array}{*{20}c} {i = x,y,z} \\ {i = x,y,z} \\ \end{array} }} {{\mathbf{g}}_{{\delta_{i} }} {\mathbf{g}}_{{\varepsilon_{j} }} } = \left( {\begin{array}{*{20}c} 1 & { - \varepsilon_{z} } & {\varepsilon_{y} } & {\delta_{x} } \\ {\varepsilon_{z} } & 1 & { - \varepsilon_{x} } & {\delta_{y} } \\ { - \varepsilon_{y} } & {\varepsilon_{x} } & 1 & {\delta_{z} } \\ 0 & 0 & 0 & 1 \\ \end{array} } \right)$$
(21)

The basic fit forms in rotor-bearing systems include plane fit and cylindrical fit, which are shown in Fig. 6.

Fig. 6
figure 6

Two fit forms in rotor-bearing systems

Full size image

As shown in Fig. 6a, the plane fit constrains three degrees of freedom in the X, β, and γ directions, so errors in these three directions are also generated. Assuming that all parts in MAU are rigid bodies, and their contact is a rigid fit, the spatial errors of plane fit during the assembly process are represented by homogeneous coordinate transformation as

$${\mathbf{g}}_{p}^{o} = \left( {\begin{array}{*{20}c} 1 & { - \varepsilon_{z} } & {\varepsilon_{y} } & {\delta_{x} } \\ {\varepsilon_{z} } & 1 & 0 & 0 \\ { - \varepsilon_{y} } & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} } \right)$$
(22)

where \(\delta_{x} \approx \frac{E}{2}\), \(\varepsilon_{y} \approx \frac{E}{D}\), and \(\varepsilon_{z} \approx \frac{E}{D}\). D represents the diameter of the rotating shaft and E represents the error value.

As is shown in Fig. 6b, the plane fit constrains four degrees of freedom in the Y, Z, β, and γ directions, so errors in these four directions are also generated. The spatial errors of cylindrical fit during the assembly process are represented by homogeneous coordinate transformation as

$${\mathbf{g}}_{c}^{o} = \left( {\begin{array}{*{20}c} 1 & { - \varepsilon_{z} } & {\varepsilon_{y} } & 0 \\ {\varepsilon_{z} } & 1 & 0 & {\delta_{y} } \\ { - \varepsilon_{y} } & 0 & 1 & {\delta_{z} } \\ 0 & 0 & 0 & 1 \\ \end{array} } \right)$$
(23)

where \(\delta_{y} \approx \frac{E}{2}\), \(\delta_{z} \approx \frac{E}{2}\), \(\varepsilon_{y} \approx \frac{E}{L}\), and \(\varepsilon_{z} \approx \frac{E}{L}\). L represent the length of the rotating shaft and E represents the error value.

The spatial pose of the power output part is determined by the geometric errors \({\mathbf{g}}_{i}^{o}\)(i = 1, 2, …, n). The error expression model of the power output part after assembly is

$${\mathbf{g}}_{m} = \prod\limits_{i}^{n} {{\mathbf{g}}_{i}^{o} }$$
(24)

3.3 Output precision of MAU chain

According to the principle of freedom degree, the function of a MAU chain is generally accomplished by series structures. The MAU chain transmits power from the input end to the output end in the motion transmission direction (Fig. 7). Few rotor systems with redundant structures or dual driving structures are beyond the scope of this paper.

Fig. 7
figure 7

The structural model of the MAU chain

Full size image

There are several motion precision indicators, such as indexing precision, repositioning precision, and radial runout, which are used to characterize the performance of the motion transmitted by the MAU chain. These indicators are obtained through the accumulation of the error components in each direction, which are caused by the spatial pose of the power output part and the geometric characteristics of the power input and output part. In this paper, the indexing precision reliability is selected as the performance evaluation index. Indexing precision is determined by the tooth profile error, tooth thickness error, and eccentricity errors of the power input and output parts. And the eccentricity errors include the assembly error component of the power output part in the direction perpendicular to the rotation axis.

The tooth profile error refers to the normal distance between the two tooth profile lines with the smallest distance within the tooth width effective range on a given section (Fig. 8). The tooth profile error is approximately equivalent to a harmonic function of the tooth-to-tooth and total tangential composite errors.

Fig. 8
figure 8

One-tooth meshing error of the worm wheel

Full size image

Let the gear meshing frequency be ωm and the gear rotation frequency be ωs; then, the tooth profile error is calculated as

$$\Delta e_{s} = 0.5\Delta f_{{i{\text{c}}}}^{^{\prime}} \sin (2\pi \omega_{m} t) + 0.5\Delta F_{ic}^{^{\prime}} \sin (2\pi \omega_{s} t)$$
(25)

Let the initial tooth thickness error of the power input part in MAU be \(\Delta e_{ig}^{I}\), the tooth thickness wear be \(\Delta e_{g}^{I} \left( t \right)\), the tooth profile deviation be \(\Delta e_{s}^{I}\), and the eccentricity error caused by the accumulated assembly error be \(\Delta e_{ip}^{I}\). Angular velocity is \(\omega_{m}^{I}\), pressure angle is α. And initial phase angle \(\beta^{I}\) are random variables that obey uniform distribution in the interval [0, 2π] .

Let the initial tooth thickness error of the power output part in MAU be \(\Delta e_{ig}^{O}\), the tooth thickness wear be \(\Delta e_{g}^{O} \left( t \right)\), the tooth profile deviation be \(\Delta e_{s}^{O}\), the eccentricity error caused by the accumulated assembly error be \(\Delta e_{ip}^{O}\), and the eccentricity error caused by wear be \(\Delta e_{p}^{O} \left( t \right)\). Angular velocity is \(\omega_{m}^{O}\), pressure angle is α. And initial phase angle \(\beta^{O}\) are random variables that obey uniform distribution in the interval [0, 2π].

Then the indexing precision of MAU is obtained as

$$\begin{aligned}\Delta e_{i} \left( t \right){ = } & {\Delta e_{ig}^{I} + \Delta e_{g}^{I} \left( t \right) + \Delta e_{s}^{I} + \Delta e_{ip}^{I} \sin \left( {2\pi \omega_{m}^{I} t + \beta^{I} + \alpha } \right) + \Delta e_{ig}^{O} + \Delta e_{g}^{O} \left( t \right) + } \\ & {\Delta e_{s}^{O} + \left( {\Delta e_{ip}^{O} + \Delta e_{p}^{O} \left( t \right)} \right)\sin \left( {2\pi \omega_{m}^{O} t + \beta^{O} + \alpha } \right)} \end{aligned}$$
(26)

Assuming that the rotor-bearing system is composed of n MAUs, then the indexing precision of the rotor-bearing system can be obtained as

$$\Delta e\left( t \right) = \sum\limits_{i}^{n} {\Delta e_{i} \left( t \right)}$$
(27)

4 The construction of kinematic precision reliability evaluation model

Assuming that the allowable indexing accuracy of rotor-bearing systems is \(\Delta e^{ * }\), the precision reliability function of rotor-bearing systems is built as

$$g\left( {{\varvec{X}}\left( t \right),{\varvec{Y}},t} \right) = \sum\limits_{i}^{n} {\Delta e_{i} \left( t \right)} - \Delta e^{ * }$$
(28)

This function includes both the initial random error variables and the time-variant wear quantity. Stochastic process discretization is adopted to solve the time-variant reliability question. Firstly, the time-variant process in a certain time interval is discretized into many micro time units. When the discretization degree reaches a certain level and the influence of time variables on the output value is ignored, the error in a micro time unit approximately obeys a Gaussian distribution. In this way, the time-variant reliability problem is transformed into the random variable problem in traditional reliability research.

Assuming that the running time of a rotor-bearing system is [0, T], the time interval is discretized into m equal micro time units with step length \(\Delta t = \frac{T}{m}\). Then, the random process vector \({\varvec{X}}\left( t \right) = \left( {X_{1} \left( t \right),X_{2} \left( t \right),...,X_{n} \left( t \right)} \right)\) with n time-variant wear parameters is discretized as

$${\varvec{X}}_{i} = \left[ {{\varvec{X}}_{1} \left( {t_{i} } \right),{\varvec{X}}_{2} \left( {t_{i} } \right),...,{\varvec{X}}_{n} \left( {t_{i} } \right)} \right]^{T} ,i = 1,2,...,m$$
(29)

Thus, the kinematic precision reliability of rotor-bearing systems is expressed as [35]

$$P_{s} \left( T \right) = P\left\{ {\bigcap\limits_{i = 1}^{m} {\left[ {g\left( {{\varvec{X}}_{i} ,{\varvec{Y}},t_{i} } \right) > 0,t_{i} = \left( {i - \frac{1}{2}} \right)\Delta t,\Delta t = \frac{T}{m}} \right]} } \right\}$$
(30)

where \(g\left( {{\varvec{X}}_{i} ,{\varvec{Y}},t_{i} } \right)\) is the random performance function discretized from the time-variant performance function \(g\left( {{\varvec{X}}\left( t \right),{\varvec{Y}},t} \right)\), and Y is an l-dimensional random vector independent of \({\varvec{X}}\left( t \right)\).

Considering the correlation between the discrete random vectors, the parameters are converted into independent standard normal distributions before solving. Based on Nataf transform [36], the random vectors \({\varvec{X}}_{i}\) and \({\varvec{Y}}\) in Eq. (30) are converted into standard normal distribution vectors \({\varvec{U}}_{i}\) and \({\varvec{V}}\) as follows:

$$U_{i,k} = \Phi^{ - 1} \left( {F_{{X_{i,k} }} \left( {X_{i,k} } \right)} \right),i = 1,2,...,m;k = 1,2,...,n$$
(31)
$$V_{p} = \Phi^{ - 1} \left( {F_{{Y_{p} }} \left( {Y_{p} } \right)} \right),p = 1,2,...,l$$
(32)

where \(\Phi^{ - 1} ( \cdot )\) is the cumulative distribution inverse function of the standard normal distribution, and \(F_{{X_{i,k} }} ( \cdot )\) and \(F_{{Y_{p} }} ( \cdot )\) are the cumulative distribution functions of random vectors \({\varvec{X}}_{i}\) and \({\varvec{Y}}\), respectively.

Let \(g^{ * } \left( {{\varvec{U}}_{i} ,{\varvec{V}},t_{i} } \right)\) represent the transformed performance function of the random vectors \({\varvec{U}}_{i}\) and \({\varvec{V}}\); then, the most probable point of the function is solved as

$$\left\{ {\begin{array}{*{20}l} {\mathrm{min} \left\| {\left( {{\varvec{U}}_{i}^{T} ,{\varvec{V}}^{T} } \right)^{T} } \right\|} \hfill \\ {s.t.g^{ * } \left( {{\varvec{U}}_{i} ,{\varvec{V}},t_{i} } \right) = 0} \hfill \\ \end{array} } \right.$$
(33)

Thus, the calculation equation of \(\beta_{i}\), the i-th index value in the reliability index vector \({\varvec{\beta}}\) corresponding to \(g^{ * } \left( {{\varvec{U}}_{i} ,{\varvec{V}},t_{i} } \right)\), is obtained as

$$\beta_{i} = \left\| {\left( {\overline{U}_{i} ,\overline{V} } \right)} \right\|$$
(34)

where (\(\overline{U}_{i}\),\(\overline{V}\)) is the most probable point of the i-th performance function \(g^{ * } \left( {{\varvec{U}}_{i} ,{\varvec{V}},t_{i} } \right)\).

Using the first-order Taylor series to expand the functional function \(g^{ * } \left( {{\varvec{U}}_{i} ,{\varvec{V}},t_{i} } \right)\) at the most probable point, the reliability calculation equation of the approximate performance function is obtained as

$$P_{s} \left( T \right) = P\left\{ {\bigcap\limits_{i = 1}^{m} {\beta_{i} + {\varvec{\alpha}}_{U,i} {\varvec{U}}_{i}^{T} + {\varvec{\alpha}}_{V,i} {\varvec{V}}^{T} > 0} } \right\}$$
(35)

where \({\varvec{\alpha}}_{U,i} \user2{ = }\frac{{\overline{{\varvec{U}}}_{i} }}{{\beta_{i} }}\) and \({\varvec{\alpha}}_{V,i} \user2{ = }\frac{{\overline{{\varvec{V}}} }}{{\beta_{i} }}\) are the gradient vectors of the random vector \({\varvec{U}}_{i}\) and \({\varvec{V}}\), respectively.

Let \(L_{i} = \beta_{i} + {\varvec{\alpha}}_{U,i} {\varvec{U}}_{i}^{T} + {\varvec{\alpha}}_{V,i} {\varvec{V}}^{T}\). Since the transformed random vectors \({\varvec{U}}_{i}\) and \({\varvec{V}}\) are independent standard normal distribution vectors, \(L_{i}\) obeys a normal distribution with the expected value \(\mu_{i} = \beta_{i}\) and the standard deviation \(\sigma_{i} = 1\). Considering that the time-variant wear quantity function is equivalent to a Gaussian process, the correlation coefficient between random variables is obtained as

$$\rho_{i,j} = \frac{{Cov\left( {L_{i} ,L_{j} } \right)}}{{\sigma_{{L_{i} }} \sigma_{{L_{j} }} }} = Cov\left( {L_{i} ,L_{j} } \right) = \sum\limits_{k = 1}^{m} {\alpha_{U,i,k} \alpha_{U,j,k} C_{k} \left( {t_{i} ,t_{j} } \right) + {\varvec{\alpha}}_{V,i}^{T} {\varvec{\alpha}}_{V,j} }$$
(36)

where \(C_{k} ( \cdot )\) represents the autocorrelation function of time-variant parameter \(X_{k} \left( t \right)\).

Let the correlation coefficient matrix \({\varvec{\rho}}{ = }\left[ {\rho_{i,j} } \right]\). Thus, using the first-order second-moment method to solve Eq. (30), the precision reliability of the rotor-bearing system is obtained as

$$P_{s} \left( T \right) = \Phi_{m} \left( {{\varvec{\beta}},{\varvec{\rho}}} \right)$$
(37)

5 Case investigations

In this section, a numerical control rotary table is taken as an example to elaborate the proposed method in detail, as shown in Fig. 9. The rotary table is composed of the gear MAU, the worm shaft MAU, and the worktable MAU. The gear MAU provides power input, the worm shaft MAU transmits motion and power, and the worktable MAU realizes the function of indexing and positioning.

Fig. 9
figure 9

The numerical control rotary table

Full size image

As shown in Fig. 10, the wear of the worktable MAU mainly occurs from the worm wheel and bearing. The structural parameters of the worm wheel are listed in Table 1; the structural parameters of the bearing are listed in Table 2.

Fig. 10
figure 10

The structure of the worktable MAU

Full size image
Table 1 Structural parameters of the worm wheel
Full size table
Table 2 Structural parameters of the bearing
Full size table

According to Eqs. (3)–(6), the time-variant wear (μm/h) of the worm wheel is calculated as

$$\Delta e_{g}^{\left( w \right)I} \left( t \right) = 0.034707t$$

According to Eqs. (9)–(18), the time-variant wear (μm/h) of the bearing is calculated as

$$\Delta e_{p}^{\left( w \right)O} \left( t \right) = 0.0005063t$$

Generally, the initial random errors of the components are subject to normal distribution within the tolerance range. The distribution information of the random errors affecting spatial pose errors of the power output part during the worktable MAU assembly process are listed in Table 3.

Table 3 Distribution information of each random error/μm
Full size table

The values of each random error are obtained by Monte Carlo simulation method [37]. Substituting these values into Eqs. (22–24), the spatial pose errors of the power input part and the power output part are calculated. The fitting joint distribution of error components in each degree of freedom can be obtained with enough sampling times. The combined distribution of the eccentric error of the worm wheel and worktable is as follows:

$$\Delta e_{ip}^{\left( w \right)I} \sim N\left( {4.25,2.25^{2} } \right),\Delta e_{ip}^{\left( w \right)O} \sim N\left( {28.25,6.80^{2} } \right)$$

The geometric characteristic error distribution of the worm wheel is listed in Table 4:

Table 4 The geometric characteristic error distribution of the worm wheel
Full size table

According to the model in Sect. 3, the indexing precision of the worktable MAU is calculated as

$$\begin{aligned}\Delta e^{\left( w \right)} \left( t \right)&{ = } {\Delta e_{ig}^{\left( w \right)I} + 0.5\Delta f_{{i{\text{c}}}}^{^{\prime}\left( w \right)I} \sin (2\pi \omega^{\left( w \right)} t) + 0.5\Delta F_{ic}^{^{\prime}\left( w \right)I} \sin (2\pi \omega^{\left( w \right)} t) + } \\ & {0.034707t + \Delta e_{ip}^{\left( w \right)I} \sin \left( {2\pi \omega^{\left( w \right)} t + \beta^{\left( w \right)I} + \alpha^{\left( w \right)} } \right) + } \\ & {\left( {\Delta e_{ip}^{\left( w \right)O} + 0.0005063t} \right)\sin \left( {2\pi \omega^{\left( w \right)} t + \beta^{\left( w \right)O} + \alpha } \right)} \end{aligned}$$

Similarly, the indexing precision of the worm shaft MAU is calculated as

$$\begin{aligned}\Delta e^{\left( s \right)} \left( t \right)&{ = } {\Delta e_{ig}^{\left( s \right)I} + 2.9264 \times 10^{ - 5} t + 0.5\Delta f_{{i{\text{c}}}}^{^{\prime}\left( s \right)I} \sin (2\pi \omega^{\left( s \right)} t) + 0.5\Delta F_{ic}^{^{\prime}\left( s \right)I} \sin (2\pi \omega^{\left( s \right)} t) + } \\ & {\Delta e_{ip}^{\left( s \right)I} \sin \left( {2\pi \omega^{\left( s \right)} t + \beta^{\left( s \right)I} + \alpha^{\left( s \right)I} } \right) + \Delta e_{ig}^{\left( s \right)O} + \Delta e_{is}^{\left( s \right)O} + } \\ & {\left( {\Delta e_{ip}^{\left( s \right)O} + {0}{\text{.0167}}t} \right)\sin \left( {2\pi \omega^{\left( s \right)} t + \beta^{\left( s \right)O} + \alpha^{\left( s \right)O} } \right)} \end{aligned}$$

The probability distributions of random variables in worm shaft MAU are listed in Table 5.

Table 5 The probability distributions of random variables in worm shaft MAU
Full size table

The indexing precision of the gear MAU is calculated as

$$\begin{aligned}\Delta e^{\left( g \right)} \left( t \right)&{ = } {\Delta e_{ig}^{\left( g \right)O} + 5.8528 \times 10^{ - 5} t + 0.5\Delta f_{{i{\text{c}}}}^{^{\prime}\left( g \right)O} \sin (2\pi \omega^{\left( g \right)} t) + 0.5\Delta F_{ic}^{^{\prime}\left( g \right)O} \sin (2\pi \omega^{\left( g \right)} t) + } \\ & { + \Delta e_{ip}^{\left( g \right)O} \sin \left( {2\pi \omega^{\left( g \right)} t + \beta^{\left( g \right)O} + \alpha^{\left( g \right)} } \right)} \end{aligned}$$

The probability distributions of random variables in gear MAU are listed in Table 6.

Table 6 The probability distributions of random variables in gear MAU
Full size table

The indexing precision of the rotary table is calculated as

$$\Delta e\left( t \right) = \Delta e^{\left( w \right)} \left( t \right) + \Delta e^{\left( s \right)} \left( t \right) + \Delta e^{\left( g \right)} \left( t \right)$$

Considering that the output precision of the servo motor is generally fixed, the allowable indexing precision value after subtracting the servo motor precision is \(\Delta e^{ * } = 80{\mu m}\). The numerical analysis is carried out according to the method in Sect. 4. According to the simulation results shown in Fig. 11, when the NC rotary table has been operating for 930 h, its kinematic precision reliability begins to decrease. The confidence interval is generally 95%; when the operation time reaches 1255 h, the kinematic precision reliability is reduced to 0.95. One thousand two hundred fifty-five hours is considered as the precision life limit of the rotary table, and maintenance is required. In comparison with the expected 1200-h precision life, it can be determined that the precision design of the NC turntable is reasonable and the proposed evaluation model is effective. As demonstrated in the figure, the curve is gentle at the initial stage of kinematic precision reliability decline, and the precision reliability decreases significantly when reaching the precision limit. This is also identical with the actual operation situation. Consequently, it shows that this method can accurately predict the precision life of the rotary table in the design stage.

Fig. 11
figure 11

Kinematic precision reliability of the rotary table with multi-source wears and random errors

Full size image

Further, we investigate the influence of different wear resistances of component materials on the kinematic precision life of the rotary table. From the calculation results, it indicates that the main wears locate on the worm wheel of the NC rotary table. Therefore, three different grades of copper are selected for simulation. Based on the results shown in Fig. 12, when ZCuSn10P1 is selected as the worm wheel material, the precision life of the rotary table is 806 h. When ZCuAl10Fe3Mn2 is used, the precision life increases by 449 h. When the worm wheel is made of the ZCuAl9Fe4Ni4Mn2, the precision life is further improved to 1682 h. Combined with the conversion cost of each material and the maintenance cost of the rotary table, the material selection in the design stage can be realized.

Fig. 12
figure 12

Effects of the material type on the kinematic precision reliability

Full size image

Different tolerance ranges of components also have an impact on the kinematic precision reliability. In order to reveal the influence, the assembly errors of the worm wheel are taken as an example; different distribution ranges of the worm wheel eccentric error are selected for simulation. According to the results shown in Fig. 13, when \(\Delta e_{ip}^{\left( w \right)O} \sim N\left( {19.78,\,6.80^{2} } \right)\), the manufacturing error of the components is smaller, and the precision life of the rotary table increases to 1300 h. When \(\Delta e_{ip}^{\left( w \right)O} \sim N\left( {36.73,\,6.80^{2} } \right)\), this indicates that the manufacturing quality is relatively poor and leads to a reduction in precision life to 1170 h. Thus, the tolerance optimization design can be realized according to the precision demand and cost.

Fig. 13
figure 13

Effects of the tolerance range on the kinematic precision reliability

Full size image

6 Conclusions

In this paper, a practical and effective approach for kinematic precision reliability evaluation of rotor-bearing systems is proposed innovatively. This method has advantages in the quantity of error terms and the accuracy of assessment results, and can be provided with great application potential in precision design and maintenance strategy for rotor-bearing systems. The study contains the following:

  1. 1.

    A comprehensive kinematic precision model describing the mapping relationship from component geometric errors to the system output precision is proposed based on the meta-action theory and screw theory, and it can overcome the limitations of the traditional methods in the number of error sources.

  2. 2.

    Both the initial errors conforming to specified random distributions and the time-dependent wears are contained in the precision reliability function of rotor-bearing systems; a kinematic precision reliability evaluation model with multiple error sources for rotor-bearing systems is provided by introducing the stochastic process discretization method.

  3. 3.

    An example for a numerical control rotary table is investigated to demonstrate the effectiveness of this approach. The simulation results obtained by changing the parameters of error sources show that the proposed method is practical and helpful to increase the precision reliability of rotor-bearing systems in the design phase.

However, this method only calculates the wear errors among the time-variant error sources; some random errors affected by changeable environmental factors during the operation of rotor-bearing systems also need to be seriously considered, such as thermal errors, deformation errors caused by residual stress release, and runout errors caused by fastener loosening. In addition, different lubrication states will affect the wear rate, and the fatigue wear is also an important constituent part in the wear of gears. The influence of the above-mentioned factors on the kinematic precision reliability of the systems will be comprehensively taken into consideration in our future investigations.