Time-Varying Reliability Analysis of Mechanisms with Strength Degradation Considering Clearances and Local Stiffness Effects

Abstract

Mechanism systems are important components of dynamic mechanical systems. The time-varying reliability assessment of mechanism systems is of great significance for the safety and fault diagnosis of mechanical products. In this paper, a time-varying reliability evaluation method for mechanisms under the fatigue failure mode is proposed, which takes into account the joint effects of clearance, local structural stiffness, random load and random strength degradation. The proposed mechanism reliability models are established by nonlinear dynamic analysis considering failure factors and establishing random stress history characterization matrices. Additionally, the models are validated by the Monte Carlo simulation method. The results show that the mechanism clearance, local stiffness, and initial strength dispersion have a significant effect on the time-varying reliability of the mechanisms. Furthermore, both local stiffness and initial strength dispersion have a greater influence on the effect of clearance on the time-varying reliability of the mechanisms. The proposed reliability models can quantitatively analyze the dynamic effects of failure factors on the reliability of the mechanisms, which can be used for the stochastic remaining life evaluation and fault diagnosis of the mechanism systems.

1. Introduction

Mechanism systems are the core functional modules in mechanical products, which are mainly used for the transmission of motion and force. Through the design of mechanism structure, the specified input motion form can be transformed into reciprocating motion, rotary motion, intermittent motion and other target motion forms to complete the displacement and speed conversion. Mechanisms are also capable of changing the magnitude and direction of force to enable mechanical products to perform complex operations, such as lifting heavy objects, amplifying and reducing force, etc. Mechanism systems are widely used in aerospace machinery systems, automobiles, industrial robots and other important mechanical equipment [1,2,3]. With the increase in product requirements for high speed, high precision and heavy loads, the requirements for mechanisms in terms of accuracy, stability, durability and safety are constantly increasing. If failures occur, such as excessive clearance wear and component stiffness degradation, the mechanisms may fail with serious consequences. For example, the fault in a satellite connection mechanism or a failure of the synchronization mechanism may affect the normal separation or deployment of the satellite and prevent the satellite from working properly [4]. The normal operation of the aircraft landing gear mechanism has a very important impact on the safe operation of the aircraft and passenger safety. Therefore, it is very necessary to carry out the reliability analysis of the mechanism, identify the key mechanism failure, analyze the impact of failure, and improve the service life and reliability of mechanisms.

Since the mechanism systems need to meet the requirements of motion accuracy and force transmission during the working process, the failures to be considered in the reliability analysis of the mechanisms are often related to the elements affecting the velocity and displacement of the mechanism motion as well as the force applied to them. In general, the failure factors that affect the reliability of a mechanism include the clearances of the kinematic joints, the machining dimensional errors of the linkage lengths of the members that make up the mechanism, the stiffness of the linkages, the mounting position errors, the operating loads, and the environmental stresses. The failure modes involved in these factors include fatigue failure, wear failure, corrosion failure, and so on. It should be noted that the failure factors such as clearance of the kinematic joints, defective material of the members, etc. cause the nonlinear characteristics of the systems. Such nonlinear characteristics under the combined influence of random loads, component stiffness, manufacturing and installation errors, etc. cause difficulties in analyzing the stochastic dynamics and time-varying reliability of the systems.

In order to solve the above difficulties, a large number of researchers have carried out numerous analyses of the dynamics of mechanisms with faults [5,6,7,8]. Erkaya, Doğan and Şefkatlıoğlu developed a method for analyzing the dynamics of a partially compliant mechanism with a joint clearance [9]. The results show that the joint clearance leads to chaotic behavior in the kinematic and dynamic output of the mechanism. The flexibility effect of small pivot and the force-closure kinematic joint between the ball and the socket in the joint clearance reduce the chaotic phenomena caused by the clearance in the output of the mechanism. Xiao, Song and Zhang investigated the dynamic behavior of a slider crank mechanism with clearance faults [10]. The dynamic equations of the slider crank mechanism with clearance were established by equating the rotating link with clearance to a virtual light mass rod and utilizing the Lagrangian method. The results show that the clearance has a small effect on the displacement and velocity response of the slider crank mechanism, but has a significant effect on the acceleration response of the mechanism. Three states of motion, separation, collision and contact occur during the friction collision process. Erkaya and Uzmay analyzed the effect of joint clearance on the dynamics of a four-bar mechanism. Several properties of the joint clearance were simulated by using a neural network model and the kinematics and dynamics were studied using a continuous contact model [11]. Yaqubi, Dardel and Daniali investigated the nonlinear dynamics of a planar crank-slider mechanism considering joint clearance and linkage flexibility [12]. A control scheme that maintained continuous contact were proposed. Significant improvement in mechanism performance and vibration reduction were observed with the use of additional actuators. The effect of joint clearance on the dynamics of a slider crank mechanism was experimentally analyzed by Erkaya and Uzmay [13]. The vibration and noise characteristics of the mechanism with and without joint clearance were compared in the paper. These studies for fault-containing mechanisms provide an important basis for time-varying reliability analysis of mechanism systems.

The above research on the dynamics of fault-containing mechanisms are mainly deterministic. In fact, due to the existence of a large number of randomness in the environment, manufacturing, design, installation, etc., the reliability analysis of the mechanism system needs to be further performed based on the theory of reliability calculation and to consider the above uncertainty factors based on a more effective time-varying reliability evaluation model. The reliability models of mechanisms are mostly based on the displacement accuracy index as the reliability evaluation objective. In the solution process, the models obtain the displacement response of the mechanisms by modeling the mechanism dynamics and calculate the probability that the displacement response does not exceed the allowable displacement threshold based on the generalized stress–strength interference model. The models acquire the extreme values of the dynamic response through methods such as the response curve envelope method and statically equate the response time-varying effects. A similar solution process to this method is to compare the displacement or velocity response at each moment with the allowable threshold to obtain the reliability at each moment.

Some important research findings have been provided in the analysis of the mechanism reliability. Sleesongsom and Bureerat presented a multi-objective, reliability design optimization method for steering linkage design [14]. The method described the uncertainty through the fuzzy variables and transformed the constraints into worst-case constraints. By transforming the single-objective reliability design optimization problem into a multi-objective problem, the computational complexity can be reduced and the computational efficiency can be improved. Chen and Gao proposed a method for analyzing the dynamic accuracy reliability of a planar multi-link mechanism with clearances [15]. The dynamic model of the multi-link mechanism with clearances was established. Then, the generalized coordinate, velocity and acceleration equations of the mechanism were derived. Based on the stress–strength interference theory, a dynamic accuracy reliability model was established to evaluate the reliability of the mechanism by calculating the interference probability between stress and strength. Wan, Li and Zhao developed a method to evaluate the motion reliability of planar linkage mechanisms [16]. A parametric model of the planar linkage mechanism was established using ADAMS software, taking into account the manufacturing error and the connection clearance following the normal distribution. The Monte Carlo theory was then utilized to perform calculations in order to obtain the mechanism reliability. Wei, Chen, Ma and Xu derived the expressions for the mean and variance of the intrinsic frequency of an elastic linkage mechanism with random physical and geometrical parameters by using the generalized random factor method [17]. In addition, models for the mean, variance and coefficient of variation of the mean square values of the displacement and velocity response of the mechanism under random excitation were obtained by the algebraic synthesis method. Then, the mean and variance of the dynamic reliability of the system were obtained by means of the Poisson process theory.

In addition to the movement accuracy of the mechanism, fatigue is an important failure mode in the reliability evaluation of mechanisms. Since mechanisms always contain malfunctions and perform cyclic work, the mechanism components are subjected to the repeated action of dynamic additional stresses, which causes fatigue damage of components and systems. In practical engineering, important industrial mechanisms such as robots need to undergo rigorous dynamic stress tests and fatigue experiments before leaving their factories. Some publications have analyzed the fatigue life of mechanism components, providing a theoretical basis for the fatigue reliability analysis of mechanisms [18,19]. At present, there are fewer models for time-varying reliability analysis of mechanisms under the fatigue failure mode considering multiple failure factors of mechanisms. The main difference between the time-varying reliability models of the mechanisms based on stress analysis and the time-varying reliability models based on motion accuracy analysis is that the accuracy reliability models focus on the possible failures of the mechanisms at each position. However, it may cause large errors to use this method to analyze the reliability of the mechanism considering the influences of fatigue and dynamic stress. As shown in Figure 1, if the stress at each moment is calculated and interfered with the strength, the reliability of the system without failure at each moment is obtained. However, practical engineering requirements as well as the definition of system reliability state that reliability is the ability of a system not to fail within a specified time period. This requires the attention to the cumulative effect of damage caused by stress. In addition, organizations often have multiple failure factors that affect each other and accelerate the fatigue failure of the system. These failure factors will degrade due to different degradation mechanisms. For example, mechanism clearances will expand due to constant wear. Manufacturing defects may cause the local stiffness of a member to weaken and degrade due to repeated loads. The uncertainty of random operating loads increases the impact of multi-failure degradation factors on system reliability. Therefore, it is necessary to establish time-varying reliability models of the mechanisms with multi-failure factors under fatigue failure mode and analyze the dynamic effects of these failure factors on the time-varying reliability of the mechanism under different degradation states. Existing literature on the reliability of motion mechanisms mostly focuses on the study of reliability evaluation methods with accuracy as the performance index. There are few analyses on time-varying reliability analysis of mechanisms considering clearance, local stiffness variation and strength degradation of mechanisms.

To address the above problems, the crank slider is taken as the research object in this paper and time-varying reliability models of multi-failure mechanism under fatigue failure mode are developed considering the joint influence of clearance, local stiffness weakening, strength degradation and random load, which can be used to analyze the dynamic influence of failure factors on the time-varying reliability of the mechanism under different degradation states. The article is organized as follows: in Section 2, the dynamic response analysis of the mechanism with failure factors is carried out. In Section 3, the time-varying reliability models of the mechanisms considering the influences of multiple failure factors are established on the basis of the results in Section 2. In Section 4, the proposed method is verified by the Monte Carlo simulation method and the effects of different failure factors on the time-varying reliability of the mechanisms are analyzed through numerical examples. The conclusions are provided in Section 5.

2. Stochastic Dynamic Response Analysis of Mechanisms

2.1. Dynamics Modeling

Bearings and journals are often required to connect the moving parts of a mechanism in order to form an effective kinematic joint that allows the mechanism to accomplish the specified task. During the connection process, the existence of a clearance ensures the relative motion between the moving parts of the mechanism. However, due to manufacturing and installation errors, the clearance may obey a certain distribution and deviate from the ideal value. Moreover, as the working time of the mechanism grows, the bearings and journals are constantly rubbing against each other, causing wear between the two and increasing the clearance. The existence of the clearance increases the overall discontinuity of the mechanism system. More importantly, the existence of the clearance enhances the acceleration fluctuation of the moving parts, forming an obvious dynamic additional stress in the moving parts. This stress is the main external source of fatigue damage to the mechanism. With the repeated action of the dynamic stress, the accumulated damage of the member increases continuously, which eventually causes fatigue damage of the member and the system. Therefore, the effect of the clearance state on the time-varying reliability of the system needs to be considered. The clearance between the members of a mechanism can be simplified to the following mechanical model.

As shown in Figure 2, the revolute joint between the mechanism members is represented in the form of bearings and journals. The position vectors of a bearing and a journal are expressed as $r_1$ and $r_2$, respectively. Without considering the flexible deformation of the two, the vector differences of the two can be expressed as

$$\mathit{\epsilon }={\mathit{r}}_1-{\mathit{r}}_2$$

(1)

When the radius difference between the bearing and journal is ${\epsilon }_1$, the penetration of the two contacts can be expressed as

$${\epsilon }_2=∣\begin{array}{c}{\mathit{r}}_1-{\mathit{r}}_2\end{array}∣-{\epsilon }_1$$

(2)

The penetration of the two contacts ${\mathsf{\epsilon }}_2$ can represent the contact state of the bearing and journal. When ${\epsilon }_2⩽0$, the bearing and journal do not come into contact. The normal contact force between the bearing and journal can be obtained as follows [20]

$$F_n=\left\{\begin{array}{cc}0& {\epsilon }_2⩽0\\ K_{\epsilon }{\epsilon }_2^{\mathrm{v}}+\delta \dot{{\epsilon }_2}& {\epsilon }_2>0\end{array}\right.$$

(3)

where $K_{\epsilon }$ is the normal contact stiffness and $\delta$ is the damping coefficient. The tangential force orthogonal to $F_n$ can be expressed as

$$F_{\mathrm{t}}=\mu F_{\mathrm{n}}$$

(4)

where $\mu$ is the friction coefficient. By considering the collision and friction effects between the connecting members of the revolute joint during the modeling of the mechanism dynamics, it is possible to reflect the influences of the clearance on the force response of the mechanism.

The traditional mechanism dynamics analysis is mainly based on the important influence of the mechanism clearance on the accuracy and reliability of the system. For a mechanism in fatigue failure mode, in addition to the clearance which may have a greater influence on the dynamic stress state and structural damage of the mechanism system, the local stiffness characteristics of the system may also have a greater influence on the dynamic additional forces. In essence, the clearance causes collision and friction between the connected members, thus generating dynamic loads on the members, while the stiffness characteristics of the members in the system will transform such dynamic load characteristics into different forms of dynamic stress response. Defects such as micro-cracks and holes may exist in the material processing and manufacturing process, which may cause local stiffness changes in the components of the mechanism. From the point of view of dynamic analysis, material defects in the member only affect the stiffness in the local neighborhood, which in turn affects the dynamic response of the system. However, due to the local stiffness reduction caused by the material defects, the member may generate large dynamic stresses near the location containing the localized defects and accumulate more fatigue damage, which will continuously reduce the residual strength of the member and cause premature failure of the system.

In this paper, in order to facilitate the description of the effect of localized defects on the time-varying reliability of a mechanism system, a mechanism dynamics model is constructed using the method shown in Figure 3. Material defects tend to affect the mechanical properties of the system such as stiffness and strength to a limited extent. Therefore, in this paper, the localized flexuralization of the member is carried out and the mechanism dynamics model is constructed by changing the elastic modulus parameter of the element, based on the ADAMS software. In this paper, the localized structure in the connecting rod as shown in Figure 3 is flexibilized and the other structures are considered as rigid bodies. The flexibilized structure is modeled with the solid 185 element in the Ansys software and the generated MNF file is imported into the Adams software to complete the structural flexibility. When the mechanism contains stiff components, the components can be approximated as rigid body structures in the dynamic modeling process. It should be noted that the local material defects are mechanically simplified in order to consider the effect of the above local stiffness change. However, the reliability analysis method proposed in this paper is able to calculate the time-varying reliability of the mechanism system by combining the nonlinear factors such as the system clearance and analyzing the influence of these key factors on the time-varying reliability of the system. In practice, according to different application scenarios and defect types, local defects can be modeled in a refined way and analyzed using the reliability analysis method proposed in this paper.

Through the flexibilization treatment of the local structure, the displacements and formulated shape functions of each finite element node, the displacements and element stresses of each node inside the local structure can be obtained according to the finite element theory as follows:

$$\mathit{\delta }=\mathit{N}{\mathit{\delta }}^{\mathbf{e}}$$

(5)

$$\mathit{s}=\mathit{\tau }\mathit{B}{\mathit{\delta }}^{\mathbf{e}}$$

(6)

where $\mathit{\delta }$ is the displacement of any point in the element, ${\mathit{\delta }}^{\mathbf{e}}$ is the displacement of the node of the element, $\mathit{N}$ is the shape function, $\mathit{\tau }$ is the elasticity matrix related to the modulus of elasticity, etc., and $\mathit{B}$ is the strain matrix of the element. Then, the kinetic energy, potential energy and energy loss of each element inside the flexible structure can be calculated. In addition, considering the kinetic energy, potential energy and energy loss of the rigid body part, the Lagrangian equation of the system can be established as follows:

$$\left\{\begin{array}{lll}{\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{\mathsf{\partial }\mathit{U}}{\mathsf{\partial }\dot{\mathsf{\vartheta }}}}\right)-{\displaystyle \frac{\mathsf{\partial }\mathit{U}}{\mathsf{\partial }\mathsf{\vartheta }}}+{\displaystyle \frac{\mathsf{\partial }\mathit{\Gamma }}{\mathsf{\partial }\dot{\mathsf{\vartheta }}}}+{\left[{\displaystyle \frac{\mathsf{\partial }\mathsf{\beta }}{\mathsf{\partial }\mathsf{\vartheta }}}\right]}^T\mathit{\lambda }-\mathit{Q}=0& & \\ \mathsf{\beta }=0& & \end{array}\right.$$

(7)

where $\mathsf{\vartheta }$ is the generalized coordinate, $Q$ is the generalized force, $U$ is the Lagrange term, $\Gamma$ is the energy dissipation function, $\mathsf{\beta }$ is the constraint equation and $\lambda$ is the multiplier corresponding to the constraint equation. The system dynamics equations can be further obtained as follows [21].

$$\mathit{M}\ddot{\mathsf{\vartheta }}+\dot{\mathit{M}}\dot{\mathsf{\vartheta }}-{\displaystyle \frac{1}{2}}{\left[{\displaystyle \frac{\mathsf{\partial }\mathit{M}}{\mathsf{\partial }\mathsf{\vartheta }}}\dot{\mathsf{\vartheta }}\right]}^T\dot{\mathsf{\vartheta }}+\mathit{K}\mathsf{\vartheta }+\mathit{D}\dot{\mathsf{\vartheta }}+{\left[{\displaystyle \frac{\mathsf{\partial }\mathsf{\beta }}{\mathsf{\partial }\mathsf{\vartheta }}}\right]}^T\mathit{\lambda }=\mathit{Q}$$

(8)

where $M$ is the system mass matrix, $K$ is the system stiffness matrix and $D$ is the system damping matrix. In this paper, only the planar motion of the various components of the system is considered. Therefore, the displacements, velocities and decelerations in Equations (7) and (8) consider only the plane degrees of freedom as shown in Figure 2.

2.2. Random Stress Analysis

Stresses at specified observation points for reliability assessment can be obtained through system dynamics analysis. In the working process of the mechanism, the stochastic characteristics that exist in the environmental loads and working loads make the system dynamic stresses have significant stochastic characteristics. Since the mechanism stresses have time-varying characteristics, their randomness gives the stresses a time-varying stochastic characteristic. Hence, the random stresses need to be reasonably characterized according to the system reliability modeling requirements.

For reliability calculations, it is necessary to establish a stochastic stress characterization method. The working time of a mechanism is divided into N time periods (TP). In the ith TP $N_i$(i = 1, 2, …, N), the external load of the mechanism $L_i$ follows a certain probability distribution with the probability density function (PDF) $f\left(L_i\right)$. Theoretically, the load distribution has infinite possible values and cannot be used directly in reliability calculations. However, in practice, the load distribution can be discretized according to the numerical calculation method and the reliability calculation can be performed based on the corresponding discrete values. In order to obtain the stresses based on the above deterministic dynamics calculation method, it is necessary to discretize the continuous distribution, expressed as ${L_{i1},L}_{i2},...,L_{ij}$(j=), where $n_L$ is the total number of discretely distributed values of random loads within each TP. Then, the load matrix during the whole working time can be expressed as:

$$\mathit{L}=\left[{\mathit{L}}_1,{\mathit{L}}_2,...,{\mathit{L}}_{\mathit{N}}\right]=\left[\begin{array}{cc}\begin{array}{cc}{L}_{11}& L_{21}\\ L_{12}& L_{22}\end{array}& \begin{array}{cc}\cdots & L_{\mathrm{N}1}\\ \cdots & L_{\mathrm{N}2}\end{array}\\ \begin{array}{cc}⋮& ⋮\\ L_{1n_L}& L_{2n_L}\end{array}& \begin{array}{cc}\ddots & ⋮\\ \cdots & L_{\mathrm{N}{n}_L}\end{array}\end{array}\right]$$

(9)

In addition, the mechanism clearance may have different values at different TPs due to the manufacturing and installation errors of the mechanism and the continuous wear of the clearance during operation. Meanwhile, because of the continuous action of the load, the local stiffness of the structure may be degraded due to the presence of material defects, thus presenting different states in different TPs. It is impractical to use experimental methods to obtain the stress response under random loading. Therefore, it is necessary to obtain the stresses of the mechanism under different states of the clearance and local stiffness and use fitting techniques such as neural networks to realize the random stress characterization and time-varying reliability calculation of the mechanisms. Considering the sample values of the clearance vectors $\phi \left(N_i\right)$ consisting of each clearance value of the mechanism in the ith TP denoted by $\left[{\phi }_1\left(N_i\right),{\phi }_2\left(N_i\right),...,{\phi }_{n1}\left(N_i\right)\right]$, where $n1$ is the total number of samples for $\phi \left(N_i\right)$ and considering the sample values of the elastic modulus $E\left(N_i\right)$ of the locally flexible structure in the ith TP denoted by $\left[E_1\left(N_i\right),E_2\left(N_i\right),...,E_{n2}\left(N_i\right)\right]$, where $n2$ is the total number of samples for $E\left(N_i\right)$, then the stress matrix of the mechanism under stochastic load $L$ can be obtained from Equation (8) as follows:

$$\mathit{S}\left(\mathit{t}\right)=\left[{\mathit{S}}_1\left(N_1\right),{\mathit{S}}_2\left(N_2\right),...,{\mathit{S}}_{\mathit{N}}\left(N_N\right)\right]$$

(10)

where

$${\mathit{S}}_{\mathit{i}}\left(N_i\right)=\left[{\mathit{S}}_{\mathit{i}}\left(L_{i1}\right),{\mathit{S}}_{\mathit{i}}\left(L_{i2}\right),...,{\mathit{S}}_{\mathit{i}}\left(L_{i{n}_L}\right)\right]$$

$${\mathit{S}}_{\mathit{i}}(L_{ij})=\left[\begin{array}{cc}\begin{array}{cc}{\mathit{S}}_{\mathit{i}}\left(L_{ij},{\phi }_1\left(N_i\right), E_1\left(N_i\right)\right)& {\mathit{S}}_{\mathit{i}}\left(L_{ij},{\phi }_2\left(N_i\right), E_1\left(N_i\right)\right)\\ {\mathit{S}}_{\mathit{i}}\left(L_{ij},{\phi }_1\left(N_i\right), E_2\left(N_i\right)\right)& {\mathit{S}}_{\mathit{i}}\left(L_{ij},{\phi }_2\left(N_i\right), E_2\left(N_i\right)\right)\end{array}& \begin{array}{cc}\cdots & {\mathit{S}}_{\mathit{i}}\left(L_{ij},{\phi }_{n1}\left(N_i\right), E_1\left(N_i\right)\right)\\ \cdots & {\mathit{S}}_{\mathit{i}}\left(L_{ij},{\phi }_{n1}\left(N_i\right), E_2\left(N_i\right)\right)\end{array}\\ \begin{array}{cc}⋮& ⋮\\ {\mathit{S}}_{\mathit{i}}\left(L_{ij},{\phi }_1\left(N_i\right), E_{n2}\left(N_i\right)\right)& {\mathit{S}}_{\mathit{i}}\left(L_{ij},{\phi }_2\left(N_i\right), E_{n2}\left(N_i\right)\right)\end{array}& \begin{array}{cc}\ddots & ⋮\\ \cdots & {\mathit{S}}_{\mathit{i}}\left(L_{ij},{\phi }_{n1}\left(N_i\right), E_{n2}\left(N_i\right)\right)\end{array}\end{array}\right]$$

$$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(j=\mathrm{1,2},\dots ,n_L)$$

2.3. Random Stress Characterization

With the stress matrix, a stochastic stress characterization method for the mechanism can be obtained based on the stochastic properties of the random loads and the nonlinear dynamics model of the system. However, it should be noted that in the stress matrix, any matrix element represents the time-varying response of the stress within the ith TP rather than a static scalar. Thus, this matrix cannot be directly used for modeling time-varying reliability, nor can the stress matrix be used to express stresses directly through fitting techniques such as neural networks. In order to solve the above problems, a stochastic stress characterization method based on rainflow counting method and neural network techniques is proposed.

The stresses within the ith TP are calculated by maximum stress statistics and equivalent stress statistics based on the rainflow counting method. According to the Goodman model, the equivalent stress obtained by the rainflow counting method can be expressed as [22]

$$S={\displaystyle \frac{S}{1-\alpha S_m}}$$

(11)

where $S_a$ is the equivalent stress when considering the effect of the average stress $S_m$, which is obtained from a stress $S$ without stress averaging correction and $\alpha$ is the correction factor. The damage of the member $d\left(S_a\right)$ can be calculated from the equivalent stress and the equivalent number $n\left(S_a\right)$ of actions of the response as follows:

$$d(S_a)={\displaystyle \frac{n\left(S_a\right)}{N\left(S_a\right)}}$$

(12)

where $N\left(S_a\right)$ is the fatigue life of the member under stress $S_a$ and $n\left(S_a\right)$ is the actual number of stress application. In this paper, the Goodman model is used for reliability analysis. The proposed method is not limited to the Goodman model. In the reliability analysis, different equivalent stress formulas can be adopted and brought into the proposed models in the case of different materials. The maximum stress matrix and damage matrix required for the time-varying reliability analysis of the mechanism can be further obtained from Equation (10) as follows:

$${\mathit{S}}_{\mathit{m}\mathit{a}\mathit{x}}\left(t\right)=\left[{\mathit{S}}_{1\mathit{m}\mathit{a}\mathit{x}}\left(N_1\right),{\mathit{S}}_{2\mathit{m}\mathit{a}\mathit{x}}\left(N_2\right),...,{\mathit{S}}_{\mathit{N}\mathit{m}\mathit{a}\mathit{x}}\left(N_N\right)\right]$$

(13)

where

$${\mathit{S}}_{\mathit{i}\mathit{m}\mathit{a}\mathit{x}}\left(N_i\right)=\left[{\mathit{S}}_{\mathit{i}\mathit{m}\mathit{a}\mathit{x}}\left(L_{i1}\right),{\mathit{S}}_{\mathit{i}\mathit{m}\mathit{a}\mathit{x}}\left(L_{i2}\right),...,{\mathit{S}}_{\mathit{i}\mathit{m}\mathit{a}\mathit{x}}\left(L_{i{n}_L}\right)\right]$$

$$\begin{array}{c}{\mathit{S}}_{\mathit{i}\mathit{m}\mathit{a}\mathit{x}}(L_{ij})\hfill \\ =\left[\begin{array}{cc}\begin{array}{cc}Max\left({\mathit{S}}_{\mathit{i}}\left(L_{ij},{\phi }_1\left(N_i\right), E_1\left(N_i\right)\right)\right)& Max\left({\mathit{S}}_{\mathit{i}}\left(L_{ij},{\phi }_2\left(N_i\right), E_1\left(N_i\right)\right)\right)\\ Max\left({\mathit{S}}_{\mathit{i}}\left(L_{ij},{\phi }_1\left(N_i\right), E_2\left(N_i\right)\right)\right)& Max\left({\mathit{S}}_{\mathit{i}}\left(L_{ij},{\phi }_2\left(N_i\right), E_2\left(N_i\right)\right)\right)\end{array}& \begin{array}{cc}\cdots & Max\left({\mathit{S}}_{\mathit{i}}\left(L_{ij},{\phi }_{n1}\left(N_i\right), E_1\left(N_i\right)\right)\right)\\ \cdots & Max\left({\mathit{S}}_{\mathit{i}}\left(L_{ij},{\phi }_{n1}\left(N_i\right), E_2\left(N_i\right)\right)\right)\end{array}\\ \begin{array}{cc}⋮& ⋮\\ Max\left({\mathit{S}}_{\mathit{i}}\left(L_{ij},{\phi }_1\left(N_i\right), E_{n2}\left(N_i\right)\right)\right)& Max\left({\mathit{S}}_{\mathit{i}}\left(L_{ij},{\phi }_2\left(N_i\right), E_{n2}\left(N_i\right)\right)\right)\end{array}& \begin{array}{cc}\ddots & ⋮\\ \cdots & Max\left({\mathit{S}}_{\mathit{i}}\left(L_{ij},{\phi }_{n1}\left(N_i\right), E_{n2}\left(N_i\right)\right)\right)\end{array}\end{array}\right]\hfill \end{array}$$

$$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(j=\mathrm{1,2},\dots ,n_L)$$

$$\mathit{d}\left(t\right)=\left[{\mathit{d}}_1\left(N_1\right),{\mathit{d}}_2\left(N_2\right),...,{\mathit{d}}_{\mathit{N}}\left(N_N\right)\right]$$

(14)

where

$${\mathit{d}}_{\mathit{i}}\left(N_i\right)=\left[{\mathit{d}}_{\mathit{i}}\left(L_{i1}\right),{\mathit{d}}_{\mathit{i}}\left(L_{i2}\right),...,{\mathit{d}}_{\mathit{i}}\left(L_{i{n}_L}\right)\right]$$

$${\mathit{d}}_{\mathit{i}}(L_{ij})=\left[\begin{array}{cc}\begin{array}{cc}d\left({\mathit{S}}_{\mathit{i}}\left(L_{ij},{\phi }_1\left(N_i\right), E_1\left(N_i\right)\right)\right)& d\left({\mathit{S}}_{\mathit{i}}\left(L_{ij},{\phi }_2\left(N_i\right), E_1\left(N_i\right)\right)\right)\\ d\left({\mathit{S}}_{\mathit{i}}\left(L_{ij},{\phi }_1\left(N_i\right), E_2\left(N_i\right)\right)\right)& d\left({\mathit{S}}_{\mathit{i}}\left(L_{ij},{\phi }_2\left(N_i\right), E_2\left(N_i\right)\right)\right)\end{array}& \begin{array}{cc}\cdots & d\left({\mathit{S}}_{\mathit{i}}\left(L_{ij},{\phi }_{n1}\left(N_i\right), E_1\left(N_i\right)\right)\right)\\ \cdots & d\left({\mathit{S}}_{\mathit{i}}\left(L_{ij},{\phi }_{n1}\left(N_i\right), E_2\left(N_i\right)\right)\right)\end{array}\\ \begin{array}{cc}⋮& ⋮\\ d\left(S_i\left(L_{ij},{\phi }_1\left(N_i\right), E_{n2}\left(N_i\right)\right)\right)& d\left({\mathit{S}}_{\mathit{i}}\left(L_{ij},{\phi }_2\left(N_i\right), E_{n2}\left(N_i\right)\right)\right)\end{array}& \begin{array}{cc}\ddots & ⋮\\ \cdots & d\left({\mathit{S}}_{\mathit{i}}\left(L_{ij},{\phi }_{n1}\left(N_i\right), E_{n2}\left(N_i\right)\right)\right)\end{array}\end{array}\right]$$

$$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(j=\mathrm{1,2},\dots ,n_L)$$

It should be noted that the elements in the same position of the maximum stress matrix and the damage matrix originate from the same history and attention should be paid to the correlation between the two elements when fitting the neural network to avoid errors due to independent assumptions and fitting with different input parameters. Neural network models can capture complex nonlinear relationships between input data and output data through multi-layer structures and activation functions. Neural networks can effectively handle high-dimensional data, utilize their powerful representation to extract effective features, which can maintain good prediction performance on new data to avoid overfitting. Therefore, neural network models have very obvious advantages in data fitting. In order to complete the stochastic stress characterization based on the above stress analysis results using the neural network technique for data fitting, the fitted input parameter matrix within the ith TP is listed as follows:

$$\mathit{\beta }\left(t\right)=\left[{\mathit{\beta }}_1\left(N_1\right),{\mathit{\beta }}_2\left(N_2\right),...,{\mathit{\beta }}_{\mathit{N}}\left(N_N\right)\right]$$

(15)

where

$${\mathit{\beta }}_{\mathit{i}}\left(N_i\right)=\left[{\mathit{\beta }}_{\mathit{i}}\left(L_{i1}\right),{\mathit{\beta }}_{\mathit{i}}\left(L_{i2}\right),...,{\mathit{\beta }}_{\mathit{i}}\left(L_{i{n}_L}\right)\right]$$

$${\mathit{\beta }}_{\mathit{i}}(L_{ij})=\left[\begin{array}{cc}\begin{array}{cc}\left(L_{ij},{\phi }_1\left(N_i\right), E_1\left(N_i\right)\right)& \left(L_{ij},{\phi }_2\left(N_i\right), E_1\left(N_i\right)\right)\\ \left(L_{ij},{\phi }_1\left(N_i\right), E_2\left(N_i\right)\right)& \left(L_{ij},{\phi }_2\left(N_i\right), E_2\left(N_i\right)\right)\end{array}& \begin{array}{cc}\cdots & \left(L_{ij},{\phi }_{n1}\left(N_i\right), E_1\left(N_i\right)\right)\\ \cdots & \left(L_{ij},{\phi }_{n1}\left(N_i\right), E_2\left(N_i\right)\right)\end{array}\\ \begin{array}{cc}⋮& ⋮\\ \left(L_{ij},{\phi }_1\left(N_i\right), E_{n2}\left(N_i\right)\right)& \left(L_{ij},{\phi }_2\left(N_i\right), E_{n2}\left(N_i\right)\right)\end{array}& \begin{array}{cc}\ddots & ⋮\\ \cdots & \left(L_{ij},{\phi }_{n1}\left(N_i\right), E_{n2}\left(N_i\right)\right)\end{array}\end{array}\right]$$

$$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(j=\mathrm{1,2},\dots ,n_L)$$

Thus, data fitting as shown in Figure 4 can be carried out using $\beta \left(t\right)$ as input matrix, and $S_{imax}\left(L_{ij}\right)$ and $d\left(S_i\left(L_{ij}\right)\right)$ as output matrix based on the BP neural network model. Then, the time-varying reliability calculation regarding the maximum stress or damage PDF can be performed based on the numerical integration theory, based on the resulting maximum stress and damage stress values at the calculation point, based on Equation (16).

$${\int }_{x1}^{x2}f\left(x\right)dx\approx \sum _{k1=0}^{nk}{A}_{k1}f\left(x_{k1}\right)$$

(16)

where $f\left(x_{k1}\right)$ is the function value of the maximum stress or damage at the integration calculation point ${\mathrm{x}}_{\mathrm{k}1}$, and ${\mathrm{A}}_{\mathrm{k}1}$ is the corresponding numerical integration coefficient, which is determined by the number of calculation points and the integration approximation method.

3. Mechanism Reliability Modeling

3.1. Mechanism Reliability Calculation

The maximum stress and damage expressions within the ith TP considering the clearance, local stiffness of the member and random loading effects are obtained above. For ease of calculation, the PDF of the maximum value of stress generated by the load $L\left(N_i\right)$ with the PDF of $f\left(L\left(N_i\right)\right)$ within the ith TP is expressed as $\sigma \left(L\left(N_i\right),\phi \left(N_i\right),E\left(N_i\right)\right)$. Considering the structural damage due to random stresses, the initial strength is denoted by $r_0$ and then the equivalent damage within the ith TP can be expressed as

$${\int }_{-\infty }^{\infty }f\left(L\left(N_i\right)\right)d\left(L\left(N_i\right),\alpha ,S_m\left(N_i\right),\phi \left(N_i\right),E\left(N_i\right)\right)dL\left(N_i\right)$$

(17)

The result from Equation (17) considers the combined effect of the distribution characteristics of the load $L\left(N_i\right)$ within the $N_i$th TP on the damage. The residual strength within the n1th TP can be expressed as

$$r\left(n1\right)=r_0\left(1-{\sum }_{i=0}^{n1-1}{\int }_{-\infty }^{\infty }f\left(L\left(N_i\right)\right)d\left(L\left(N_i\right),\alpha ,S_m\left(N_i\right),\phi \left(N_i\right),E\left(N_i\right)\right)dL\left(N_i\right)\right)$$

(18)

Equation (18) describes the effects of cumulative damage due to stress application. The difference between the residual strength and the initial strength is the value of the decrease in strength due to cumulative damage. Considering the maximum stress distribution within different TPs, the reliability within N TPs is calculated by

$$\prod _{i=1}^N{\int }_{-\infty }^{r\left(i\right)}{f}_{max}\left(\sigma \left(L\left(N_i\right),\phi \left(N_i\right),E\left(N_i\right)\right)\right)\mathrm{d}\sigma \left(L\left(N_i\right),\phi \left(N_i\right),E\left(N_i\right)\right)$$

(19)

Considering the randomness and the distribution of the initial strength $r_0$ with the PDF of $f\left({\mathrm{r}}_0\right)$, the reliability of the mechanism running N TPs is

$$R\left(N\right)={\int }_{-\infty }^{\infty }f\left(r_0\right){\prod }_{i=1}^N{\int }_{-\infty }^{r\left(i\right)}{f}_{max}\left(\sigma \left(L\left(N_i\right),\phi \left(N_i\right),E\left(N_i\right)\right)\right)d\sigma \left(L\left(N_i\right),\phi \left(N_i\right),E\left(N_i\right)\right)df\left(r_0\right)$$

(20)

Failure rate is the probability that a component or system that has worked to a moment where it has not yet failed will fail per unit of time after that moment. Hence, the failure rate of the mechanism can be calculated from the time-varying reliability as follows:

$$\begin{array}{c}\hfill \lambda \left(N\right)=\left\{{\int }_{-\infty }^{\infty }f\left(r_0\right)\prod _{i=1}^N{\int }_{-\infty }^{r\left(i\right)}{f}_{max}\left(\sigma \left(L\left(N_i\right),\phi \left(N_i\right),E\left(N_i\right)\right)\right)\right.d\sigma \left(L\left(N_i\right),\phi \left(N_i\right),E\left(N_i\right)\right)df(r_0)-\\ \hfill {\int }_{-\infty }^{\infty }f\left(r_0\right){\displaystyle \prod _{i=1}^{N+1}}{\int }_{-\infty }^{r\left(i\right)}\left.f_{max}\left(\sigma \left(L\left(N_i\right),\phi \left(N_i\right),E\left(N_i\right)\right)\right)d\sigma \left(L\left(N_i\right),\phi \left(N_i\right),E\left(N_i\right)\right)df(r_0)\right\}\\ \hfill /\left\{{\int }_{-\infty }^{\infty }f\left(r_0\right)\prod _{i=1}^N{\int }_{-\infty }^{r\left(i\right)}{f}_{max}\left(\sigma \left(L\left(N_i\right),\phi \left(N_i\right),E\left(N_i\right)\right)\right)\right.d\sigma \left(L\left(N_i\right),\phi \left(N_i\right),E\left(N_i\right)\right)df(r_0)\end{array}$$

(21)

The proposed time-varying reliability models of the mechanism are able to take into account the effects of changes in clearance wear, local structural stiffness degradation, random loading and strength degradation on the time-varying reliability of the mechanism under the fatigue failure mode. The models not only take into account the nonlinear characteristics of the system caused by changes in clearance and local stiffness, but also consider analytically characterizing the effects of these factors on the reliability of the mechanisms over time. The proposed reliability models can be used to analyze the effects of system failure factors on the reliability and remaining life of the mechanism and can also be used for dynamic fault diagnosis of the mechanism systems.

3.2. Reliability Model Validation

In order to verify the correctness and validity of the proposed reliability models, the Monte Carlo simulation method is used in this paper. In the Monte Carlo method, the working process is simulated based on the working mechanism of the mechanisms and the stochastic characteristics of the system parameters. The proposed reliability models are not used in the simulation process. In this way, it is possible to make the simulation process independent of the proposed computational models, thus accomplishing the validation of the reliability analytical models. The specific Monte Carlo simulation process is shown in Figure 5. As shown in Figure 5, during each simulation of the system, in each TP, random loads are generated according to the random distribution characteristics of the loads and the corresponding stress responses are obtained. Through the damage caused by the stress and the maximum stress, it is judged whether the residual strength can resist the working stress. When the system can work normally in all TPs, the system is considered to be reliable, otherwise the system is considered to fail. Finally, the system reliability is obtained by counting the number of system failures.

4. Numerical Examples

In this section, the proposed analytical models for reliability assessment will be validated, and then the effect of mechanism clearance and local stiffness variation together on the time-varying reliability of the mechanism will be analyzed. Finally, the effect of different strength dispersion on the time-varying reliability of a mechanism containing a clearance is considered.

(1)

The parameters of the mechanism system are shown in

Table 1. Mechanism system parameters.

Parameter	Value	Unit
$l_1$	1	m
$l_2$	1	m
$l_3$	0.1	m
$l_4$	1	m
Mean value of load $\mu \left(L\right)$	1000	N
Standard deviation of working load $\mathsf{\sigma }\left(L\right)$	120	N
Modulus of elasticity E in flexible element	$5\times {10}^9$	Pa
Density $\rho$	7800	kg/m3
Angular velocity	6.283	Rad/s
Diameter of bearing	0.020	m
Diameter of journal	0.018	m
Mean value of initial strength $\mu \left({\mathrm{r}}_0\right)$	1.3	MPa
Standard deviation of initial strength $\mathsf{\sigma }\left({\mathrm{r}}_0\right)$	0.2	MPa

. $\mu \left(∎\right)$ denotes the mean value of a random variable. In this section, the proposed reliability model is verified by the Monte Carlo simulation method. The obtained Monte Carlo simulation results and reliability calculation results are shown in Figure 6.

The Monte Carlo simulation is based on the system working mechanism for simulation and does not depend on the proposed reliability models. As can be seen in Figure 6, the proposed time-varying reliability model of the mechanism is consistent with the Monte Carlo simulation results. Additionally, the reliability does not decrease linearly or in the form of an exponential function under the traditional assumption of constant failure rate. The rate of reliability decline is slower at the beginning of the operational duration, but as the strength continues to decrease, the effect of the mechanism clearance on the mechanism becomes more obvious, accelerating the rate of reliability decline and thus accelerating the failure of the mechanism system.

In practice, material parameters need to be obtained from random samples. For example, using $x_i$ to denote the ith sample of a material parameter x and the total number of samples is $N_{\mathrm{s}}$, the standard deviation of the parameter $\sigma \left(x\right)$ can be expressed as follows.

$$\mathit{\sigma }\left(\mathit{x}\right)=\sqrt{{\displaystyle \frac{1}{{\mathit{N}}_{\mathbf{s}}}}\sum _{\mathit{i}=1}^{{\mathit{N}}_{\mathbf{s}}}{\left({\mathit{x}}_{\mathit{i}}-\mathit{\mu }\left(\mathit{x}\right)\right)}^2}$$

where $\mu \left(x\right)$ is the mean value of $x$.

(2)

In order to consider the combined effect of the mechanism clearance and the local stiffness of the mechanism on the time-varying reliability of the mechanism, the results of the time-varying reliability calculations for the mechanism with different values of clearance and local stiffness shown in

Table 2. Structural parameters.

	Modulus of Elasticity E in Flexible Element (Pa)	Diameter of Journal (m)
Case 1	$1\times {10}^{10}$	0.0176
		0.0178
		0.018
Case 2	$5\times {10}^9$	0.0176
		0.0178
		0.018

are shown in Figure 7 and Figure 8.

From Figure 7 and Figure 8, it can be seen that the mechanism clearance and the local stiffness of the members have very important effects on the time-varying reliability of the mechanism. At the same local stiffness, as the clearance increases, the time-varying reliability of the mechanism system is lower and decreases faster. Due to the increase in clearance, the collision force of the system clearance is larger and the additional stress as well as the damage is also larger. Thus, it may not only cause a larger maximum stress, but also accelerate the system’s strength degradation rate, resulting in a rapid decline in the time-varying reliability of the mechanism system. Increased clearance will significantly reduce the service life of the mechanism and affect the safety and reliability of the mechanism system.

When the stiffness of the mechanism is high, the collision forces and stresses on the members of the mechanism are greater. The increase in member flexibility slows down the rate of damage accumulation and strength decline. The ability of the system to resist the effects of external forces increases. This slows down the rate of decrease in the time-varying reliability of the system. Overall, when the local stiffness is the same, the trend of the effect of clearance on the time-varying reliability of the mechanism is the same. However, the effect of clearance on the reliability of the mechanism is greater when the local stiffness is higher because the increase in the flexibility of the components can slow down the rate of decrease in the reliability of the mechanism. The difference in the time-varying reliability of the mechanism under the two different clearance conditions is more pronounced at higher stiffness and smaller at greater flexibility.

(3)

In order to consider the effects of different initial strength dispersion and different clearance on the time-varying reliability of the mechanism, the time-varying reliability of the mechanism with different values of standard deviation of initial strength and clearances are shown in Figure 9, Figure 10 and Figure 11 when the diameter of the journal is 0.0176 m, 0.0178 m, and 0.018 m, respectively.

The combined effect of initial strength dispersion and clearance on the time-varying reliability of the mechanism is demonstrated in Figure 9, Figure 10 and Figure 11. It can be seen that the initial strength dispersion has different effects on the time-varying reliability of the mechanism at different TP when the clearances are the same. When the initial strength dispersion increases, the mechanism reliability decreases in the earlier operating phase and increases in the later part of the operating time. This is due to the fact that an increase in the initial strength increases the likelihood that the system will have a smaller value of initial strength in the early work phase, while increasing the likelihood of a larger residual strength in the latter half of the work period. In addition, for the same initial strength dispersion condition, the reliability of the system is lower when the clearance is larger, which is due to the fact that the increase in clearance causes an increase in the stresses and cumulative damage in the system. Furthermore, the increase in initial strength dispersion will weaken the effect of clearance on the time-varying reliability of the mechanism. Decreasing the value of clearance can also weaken the effect of initial strength dispersion on the time-varying reliability of the mechanism.

5. Conclusions

In this paper, time-varying reliability models of the mechanism systems under fatigue failure mode are developed by considering the combined effects of clearance, local structural stiffness, random load and random strength degradation. The traditional mechanism reliability models pay more attention to the reliability change of the mechanism accuracy under the influences of failure factors. However, in practical engineering, the fatigue damage, time-varying reliability and remaining life of the mechanisms are also important factors affecting the safe operation and durability of the systems. By establishing a time-varying reliability model under the combined influences of multiple failure and degradation factors, a quantitative analysis method that can take into account the influence of nonlinear factors and strength degradation factors on the time-varying reliability of the mechanism systems is proposed, which is validated by the Monte Carlo simulation method. The results show that the mechanism clearance, local stiffness, and initial strength dispersion have a significant effect on the time-varying reliability of the mechanisms. The increase in clearance causes an increase in the maximum stress matrix and cumulative damage matrix, which reduces the time-varying reliability of the mechanism. The increase in local stiffness attenuated the effect of clearance on the reliability of the mechanisms. In addition, the initial strength dispersion has different effects on mechanism reliability at different stages. Larger initial strength dispersion decreases early mechanism reliability and increases later mechanism reliability. Meanwhile, the effect of clearance on the time-varying reliability of the mechanisms will be weakened with the increase in initial strength dispersion. The effect of initial strength dispersion on the time-varying reliability of the mechanisms will be weakened with the decrease in clearance. The proposed models can quantitatively analyze the time-varying effects of failure factors on the reliability of the mechanisms and can also be used for the stochastic remaining life evaluation and fault diagnosis of the mechanism systems. The proposed method provides a basis for time-varying reliability analysis of mechanisms in fatigue failure mode. In future work, more in-depth research will be carried out for the analysis of mechanism dynamics and reliability evaluation of complex mechanisms with multiple failure modes.