Probabilistic Approach to System Reliability of Mechanism with Correlated Failure Models

Abstract

In this paper, based on the kinematic accuracy theory and matrix-based system reliability analysis method, a practical method for system reliability analysis of the kinematic performance of planar linkages with correlated failure modes is proposed. The Taylor series expansion is utilized to derive a general expression of the kinematic performance errors caused by random variables. A proper limit state function (performance function) for reliability analysis of the kinematic performance of planar linkages is established. Through the reliability theory and the linear programming method the upper and lower bounds of the system reliability of planar linkages are provided. In the course of system reliability analysis, the correlation of different failure modes is considered. Finally, the practicality, efficiency, and accuracy of the proposed method are shown by a numerical example.

1. Introduction

Mechanisms are the skeletons of modern mechanical products and devices. The kinematic accuracy of mechanisms greatly influences the performance and reliability of the mechanical products and devices. Traditionally, in mechanism synthesis, a designer often tries to choose proper mechanism configurations and component dimensions to make the designed mechanism meet prespecified requirements. However, in the physical realization of any constituent member, primary errors always occur due to technological features of production. Once a theoretical solution is translated into physical reality, a theoretically feasible mechanism might be unable to meet practical requirements because of the effects of uncertain factors (e.g. manufacturing tolerances, elastic deformations and joint clearances). Since these uncertain factors are inevitable, it is necessary to build a proper mode to quantify the effects of the uncertain factors on the accuracy of mechanisms and optimally allocate the working ranges of the mechanisms [1–3].

In recent years, with the continual increase of the demands of consumers on the kinematic and dynamic performance of mechanical products, the theory of mechanical reliability is more and more widely applied in mechanism analysis and synthesis. Mechanism reliability can simply be defined as the capability that a mechanism performs its prespecified movement accurately, timely, and coordinately throughout its lifetime. Based on the analysis of the sources of original errors, Sergeyev [4] clarified the main failure modes of mechanisms and presented an analytical method for reliability analysis of mechanisms preliminarily. Subsequently, there have been various attempts to derive the reliability of the kinematic and dynamic accuracy of mechanisms, such as the linear regression method [5], the mean value first-order second-moment method [6], the advanced first-order second-moment method [7], the hybrid dimension reduction method [8], and the Monte-Carlo simulation method [9].

As shown in practical engineering, most performance deficiencies of mechanical products are found in the stage of systematic analysis, therefore it’s important to build a proper system reliability analysis model to evaluate the performance quality of mechanical equipments and products. Recently, Zhang et al.[9] studied the method for system reliability analysis of mechanisms without considering the interactions of failure modes. However, to the best of the authors’ knowledge, system reliability analysis of mechanisms with correlated failure modes has not been reported yet. Combining the mechanism theory and system reliability analysis method, this paper proposes a general method for system reliability analysis of planar mechanisms with correlated failure modes.

2. Reliability Analysis for Kinematic Performance of Planar Linkages

The kinematic performance function of planar linkages can be expressed as [10]𝐐=𝐐(𝐕,𝐋,𝐔), (2.1) where 𝐐_𝑠×1is the performance parameter vector. For example, for a function generator, 𝐐_𝑠×1may be referred to the positions of the output link, and for a path generator, it may be the coordinates of a point on the output link. 𝐕_𝑚×1is the input (independent) variable vector, 𝐋_𝑝×1is the effective dimension variable vector, and 𝐔_𝑛×1 is the output (dependent) variable vector which can be obtained by solving the loop closure equations of planar linkages𝐅(𝐋,𝐔,𝐕)=𝟎. (2.2)

The performance errors of the mechanism under consideration can be obtained asΔ𝐐=𝜕𝐐𝜕𝐋^𝑇Δ𝐋+𝜕𝐐𝜕𝐔^𝑇Δ𝐔+𝜕𝐐𝜕𝐕^𝑇Δ𝐕, (2.3) where 𝜕𝐐/𝜕𝐋^𝑇, 𝜕𝐐/𝜕𝐔^𝑇and 𝜕𝐐/𝜕𝐕^𝑇are Jacobian matrices, whose values are got at the mean values of the random variables. Δ𝐋_𝑝×1,Δ𝐔_𝑛×1,andΔ𝐕_𝑚×1are the tolerance vectors of random design variables. Δ𝐕_𝑚×1and Δ𝐋_𝑝×1are determined by several objective factors such as the machining accuracy, the assembly accuracy and the operation precision. From (2.3),Δ𝐔_𝑛×1can be obtained:Δ𝐔=−𝜕𝐅𝜕𝐔^𝑇−1𝜕𝐅𝜕𝐕^𝑇Δ𝐕+𝜕𝐅𝜕𝐋^𝑇Δ𝐋. (2.4)

A limit state is defined as a condition in which a mechanism becomes unsuitable for its intended motion (i.e., a violation of the serviceability limit state). The corresponding limit state functions (performance functions) when the mechanism meets the requirements of the upper and lower limits are𝐠_𝑈(𝐗)𝐠=𝜺−Δ𝐐=𝜺−𝐉Δ𝐗,_𝐿(𝐗)=Δ𝐐+𝜺=𝐉Δ𝐗+𝜺, (2.5) where𝐉=𝜕𝐐𝜕𝐕^𝑇−𝜕𝐐𝜕𝐔^𝑇𝜕𝐅𝜕𝐔^𝑇−1𝜕𝐅𝜕𝐕^𝑇,𝜕𝐐𝜕𝐋^𝑇−𝜕𝐐𝜕𝐔^𝑇𝜕𝐅𝜕𝐔^𝑇−1𝜕𝐅𝜕𝐋^𝑇. (2.6)𝐠_𝑈and 𝐠_𝐿 are called the upper and lower limit state functions, 𝐗=[𝑉₁,…,𝑉_𝑚,𝐿₁,…,𝐿_𝑝]^𝑇 is the basic variables, 𝜺 is the allowable errors, and Δ𝐗=[Δ𝑉₁,…,Δ𝑉_𝑚,Δ𝐿₁,…,Δ𝐿_𝑝]^𝑇 are used to represent the random error vector of basic variables.

The kinematic reliability of a mechanism is the probability that the mechanism realizes its required motion within a specified tolerance limit. The lower limit reliability of the 𝑘th dependent variable, 𝑄_𝑘, is defined as:𝑅_𝐿^(𝑘)=_{𝑔𝐿(𝑘)(𝐗)>0}𝑓(𝐗)d𝐗, (2.7) where 𝑓(𝐗)is the joint probability density function of multidimensional basic random variables, 𝐗 and 𝑔_𝐿^(𝑘)(𝐗)=Δ𝑄_𝑘+𝜀_𝑘=𝐉_𝑘Δ𝐗+𝜀_𝑘 is the limit state function of the 𝑘th dependent variable,𝑄_𝑘. Note that 𝐉_𝑘is the 𝑘th row of matrix 𝐉.

The mean value,𝜇_𝐿^(𝑘), and variance,(𝜎_𝐿^(𝑘))², of the limit state function, 𝑔_𝐿^(𝑘)(𝐗), can be expressed as𝜇_𝐿^(𝑘)=𝐸𝑔_𝐿^(𝑘)(𝐗)=𝐉_𝑘𝐸(Δ𝐗)+𝜀_𝑘,𝜎_𝐿^(𝑘)2=Var𝑔_𝐿^(𝑘)(𝐗)=𝐉_𝑘^[2]cs(Cov(Δ𝐗)), (2.8) where𝐸(Δ𝐗)andCov(Δ𝐗) are the mean value vector and covariance matrix of primary errors, respectively, 𝐉_𝑘is the 𝑘th row of matrix 𝐉, (⋅)^[2]=(⋅)⊗(⋅)is the second-order Kronecker power of (⋅), and⊗represents Kronecker product [11]𝐀_𝑝×𝑞⊗𝐁_𝑠×𝑡=𝑎₁₁𝐁𝑎₁₂𝐁…𝑎_1𝑞𝐁𝑎₂₁𝐁𝑎₂₂𝐁…𝑎_2𝑞𝐁𝑎⋮⋮⋱⋮_𝑝1𝐁𝑎_𝑝2𝐁…𝑎_𝑝𝑞𝐁_{𝑝𝑠×𝑞𝑡}, (2.9)cs(⋅)is the column string of (⋅), the column sequencecs𝐀_𝑝×𝑞=^𝑞_𝑗=1𝐞^𝑗_𝑞×1⊗𝐈_𝑝×𝑝𝐀_𝑝×𝑞𝐞^𝑗_𝑞×1, (2.10) where𝐈_𝑝×𝑝is identity matrix with𝑝×𝑝dimensions, and 𝐞^𝑗_𝑞×1is the 𝑗th elementary vector with𝑞×1dimensions, all zeros except 1 in the 𝑗th position.

In the mechanism literature, the distribution of the random variables is always assumed independent normal [4–8]. The distance from the “minimum” tangent plane to the failure surface may be used to approximate the actual failure surface, and the reliability index of the 𝑖th output variable is defined as:𝛽_𝐿^(𝑘)=𝜇_𝐿^(𝑘)𝜎_𝐿^(𝑘), (2.11) which can be used to reflect the position (the distance from the original point) and dispersion degree of the safety margin. When the primary errors are normally and independently distributed, the unary estimator of the kinematic performance reliability of planar linkages is represented as follows:𝑅_𝐿^(𝑘)=Φ𝛽_𝐿^(𝑘), (2.12) where Φ(⋅) is the standard normal distribution function.

The correlation coefficient between performance functions 𝑔_𝐿^(𝑘) and 𝑔_𝐿^(𝑡) isCov𝑔_𝐿^(𝑘),𝑔_𝐿^(𝑡)=𝐸𝑔_𝐿^(𝑘)−𝑔_𝐿^(𝑘)𝑔_𝐿^(𝑡)−𝑔_𝐿^(𝑡)=^𝑚+𝑙_𝑖=1^𝑚+𝑙_𝑗=1𝜕𝑔_𝐿^(𝑘)𝜕𝑋_𝑖𝜕𝑔_𝐿^(𝑡)𝜕𝑋_𝑗Cov𝑋_𝑖,𝑋_𝑗. (2.13) The correlation coefficient between performance functions 𝑔_𝐿^(𝑘) and 𝑔_𝐿^(𝑡) is𝜌_𝐿^(𝑘,𝑡)=Cov𝑔_𝐿^(𝑘),𝑔_𝐿^(𝑡)Var𝑔_𝐿^(𝑘)Var𝑔_𝐿^(𝑡). (2.14) Then the joint reliability of 𝑔_𝐿^(𝑘) and 𝑔_𝐿^(𝑡) can be estimated by the joint normal distribution function:𝑅_𝐿^(𝑘,𝑡)=1−⁰_−∞𝑓_𝑘𝑡𝑔_𝐿^(𝑘),𝑔_𝐿^(𝑡)𝑑𝑔_𝐿^(𝑘)𝑑𝑔_𝐿^(𝑡)=1−^{−𝛽𝑡𝐿}_−∞^{−𝛽𝑘𝐿}_−∞𝜙_𝐿^(𝑘,𝑡)(𝑢,𝑣)d𝑢d𝑣=1−^{−𝛽𝑡𝐿}_−∞Φ−𝛽^𝑘_𝐿−𝜌_𝐿^(𝑘,𝑡)𝑣1−𝜌_𝐿^(𝑘,𝑡)2𝜙(𝑣)d𝑣, (2.15) where𝑓_𝑘𝑡𝑔_𝐿^(𝑘),𝑔_𝐿^(𝑡)=12𝜋𝜎_𝐿^(𝑘)𝜎_𝐿^(𝑡)1−𝜌_𝐿^(𝑘,𝑡)2−1×exp21−𝜌_𝐿^(𝑘,𝑡)2𝑔_𝐿^(𝑘)−𝜇_𝐿^(𝑘)2Var𝑔_𝐿^(𝑘)−2𝜌_𝐿^(𝑘,𝑡)𝑔_𝐿^(𝑘)−𝜇_𝐿^(𝑘)𝑔_𝐿^(𝑡)−𝜇_𝐿^(𝑡)𝜎_𝐿^(𝑘)𝜎_𝐿^(𝑡)+𝑔_𝐿^(𝑡)−𝜇_𝐿^(𝑡)2Var𝑔_𝐿^(𝑡) (2.16) is the joint PDF of 𝑔_𝐿^(𝑘) and 𝑔_𝐿^(𝑡).

In the same way, the reliability corresponding to each failure modes and the joint reliability between each two failure modes while the mechanism satisfies the upper limits could be obtained. Then the reliability corresponding to each failure model and the joint reliability between two failure models while the mechanism meets the upper and lower limits can be derived as𝑅^𝑘=𝑅_𝐿^(𝑘)+𝑅_𝑈^(𝑘)𝑅−1,_(𝑘,𝑡)=𝑅_𝐿^(𝑘,𝑡)+𝑅_𝑈^(𝑘,𝑡)−1. (2.17)

3. System Reliability of Linkage Performance

For convenience system reliability analysis of structures with multi-failure modes is often performed by consuming that the failure modes are independent between each other. In most cases, however, the failure modes of a mechanism (e.g., the position and pose of a rigid-body guidance mechanism) are correlated. Consequently, it is of great meaning to propose an accurate and efficient system reliability analysis method to evaluate the working state of the mechanism. Ditlevsen [12] presented the well-known “narrow bounds theory” for computing system reliability. The correlation between each of the two failure modes is considered in Ditlevsen’s method, making it more physically reasonable. And then the bounds method in which the system reliability is estimated by computing the bound values developed continuously and received wide acceptance [13–15]. In this section, a practical method for system reliability analysis of mechanisms is proposed by using the linear programming.

Linear programming solves the problem of minimizing or maximizing a linear function, whose variables are subject to linear equality and inequality constraints. And the linear programming for solving the possible bounds on the system reliability of linkages can be presented as follows:min(max)𝐜^𝑇𝐩s.t.𝐚₁𝐩=𝐛₁𝐚₂𝐩≥𝐛₂, (3.1) where𝐩is the design variable vector of the linear programming, 𝐜is a vector of coefficients, 𝐜^𝑇𝐩is the linear objective function, and𝐚₁,𝐚₂,𝐛₁,and𝐛₂ are the coefficient matrices and vectors that represent the equality and inequality constraints, respectively.

In the proposed system reliability analysis method, the kinematic failure space of a mechanism can be divided into 2^𝑛 mutually exclusive and collectively exhaustive (MECE) events according to the number of failure modes, 𝑛. Typically, for a system with three failure modes, the performance sample space can be depicted as Figure 1 by defining kinematic safety of the mechanism as event 𝑆 and defining the performance 𝑄_𝑖 meets the requirement of performance quality as event𝐸_𝑖. The space 𝑆 is divided into 8 MECE events, {𝑒₁=𝐸₁𝐸₂𝐸₃,𝑒₂=𝐸₁𝐸₂𝐸₃,𝑒₃=𝐸₁𝐸₂𝐸₃,𝑒₄=𝐸₁𝐸₂𝐸₃,𝑒₅=𝐸₁𝐸₂𝐸₃,𝑒₆=𝐸₁𝐸₂𝐸₃,𝑒₇=𝐸₁𝐸₂𝐸₃,and𝑒₈=𝐸₁𝐸₂𝐸₃}. Let 𝑝_𝑖=𝑃(𝑒_𝑖),𝑖=1,2,…,8 denotes the probability of the 𝑖th basic MECE event. These probabilities serve as the design variables in the linear programming problem to be formulated. According to the basic definition of probability, ⁸_𝑖=1𝑝_𝑖=1, (3.2)𝑝_𝑖≥0,𝑖=(1,2,…,8). (3.3) The constraint (3.2) is analogous to the equality constraints in linear programming (3.1) with 𝐚₁ being a row vector of 1′s and 𝐛₁=1, whereas (3.3) is analogous to the inequality constraints with𝐚₂being an 8×8 identity matrix and 𝐛₂ a 8×1 vector of 0′s.

Figure 1 

Sample space for the linkage.

As can be seen from Figure 1,𝑃𝐸_𝑖=𝑃_𝑖=_{𝑟∶𝑒𝑟⊆𝐸𝑖}𝑝_𝑟,𝑃𝐸_𝑖𝐸_𝑗=𝑃_𝑖𝑗=_{𝑟∶𝑒𝑟⊆𝐸𝑖𝐸𝑗}𝑝_𝑟, (3.4)(3.5)

scilicet,𝑃𝐸₁=𝑃₁=𝑝₁+𝑝₃+𝑝₄+𝑝₇,𝑃𝐸₂=𝑃₂=𝑝₁+𝑝₂+𝑝₄+𝑝₆,𝑃𝐸₃=𝑃₃=𝑝₁+𝑝₂+𝑝₃+𝑝₅,𝑃𝐸₁𝐸₂=𝑃₁₂=𝑝₁+𝑝₄,𝑃𝐸₁𝐸₃=𝑃₁₃=𝑝₁+𝑝₃,𝑃𝐸₂𝐸₃=𝑃₂₃=𝑝₁+𝑝₂. (3.6) Equations (3.4) and (3.5) provide linear equality constraints on the design variable 𝐩 with 𝐚₁ a matrix having elements of 0 or 1 and 𝐛₁ a vector listing the known reliability. With the increase of the known or computed reliability, such as the uni-, bi-, and sometimes trimode reliability, the upper and lower bounds of the system reliability obtained by the linear programming become increasingly accuracy. However, there is always a tradeoff between complexity and accuracy, and with the increase of the constraints, the convergence of the linear programming becomes more and more difficult.

By now, the coefficient matrices and vectors of the constraint functions of linear programming (3.1) to obtain the upper and lower bounds of the system reliability of the mechanism are completely established, which are𝐚₁=,𝐛11111111101100101101010011101000100100001010000011000000₁=1𝑃₁𝑃₂𝑃₃𝑃₁₂𝑃₁₃𝑃₂₃^𝑇,𝐚₂=𝐈_8×8,𝐛₂=00000000^𝑇, (3.7) where𝐈_8×8is identity matrix with8×8dimensions.

According to the relationship between failure modes (series or parallel), there are four different kinds of systems. As shown in Table 1, the coefficient 𝑐_𝑖 (𝑖=1,2,…,8) of the object functions of linear programming (3.1) for each kind of system can be determined, respectively. So far, the linear programming to derive the lower and upper bounds of the system reliability of a mechanism with random parameters is completely established.

Table 1 

Coefficients of the object functions c^Tp.

4. Numerical Examples

Consider the vector loop as shown in Figure 2, the nominal geometry characteristics of the double-rocker four-bar linkage with driving crank are shown as: 𝑟₁=2.36cm, 𝑟₂=1.33cm, 𝑟₃=5.08cm, 𝑟₄=3.94cm, 𝑟₅=1.00cm, 𝑟₆=0.45cm, 𝑟₇=1.50cm, 𝑟₈=1.00cm, 𝑟₉=6.00cm, and𝛼₉=30^∘. Among them,𝑟₁,𝑟₂,𝑟₃,and𝑟₄ are random variables, which are normally and independently distributed, and the variation coefficient of the random variables are supposed to be 𝑐=0.001. All other variables are deterministic parameters. According to the working condition, the maximum allowable values of the kinematic performance errors vector are𝜀=[0.015rad,0.8mm,0.6mm]^𝑇. The double-rocker four-bar linkage can work normally, only if all the kinematic performance quality requirements are satisfied (i.e., the linkage system is series). It is required to solve the system reliability of the double-rocker four-bar linkage in its working range (𝛼₆=150^∘∼270^∘).

Figure 2 

Double-rocker four-bar linkage with driving crank.

As shown in Figure 1, in the double-rocker four-bar linkage with driving crank, the effective dimension variable vector is𝐋=[𝑟₁,𝑟₂,𝑟₃,𝑟₄,𝑟₅,𝑟₆,𝑟₇,𝑟₈,𝑟₉,𝛼₉]^𝑇, the input (independent) and output (dependent) variable vectors are 𝑉=[𝛼₆]and 𝐔=[𝛼₂,𝛼₃,𝛼₄,𝛼₇]^𝑇, respectively, the performance parameter vector is 𝐐=[𝛼₃,𝑀_𝑥,𝑀_𝑦]^𝑇, then the closure equations of the planar linkage are𝑟𝐅=⁶cos𝛼₆+𝑟₇cos𝛼₇−𝑟₈cos𝛼₂−𝑟₅𝑟₆sin𝛼₆+𝑟₇sin𝛼₇−𝑟₈sin𝛼₂𝑟₂cos𝛼₂+𝑟₃cos𝛼₃−𝑟₄cos𝛼₄−𝑟₁𝑟₂sin𝛼₂+𝑟₃sin𝛼₃−𝑟₄sin𝛼₄. (4.1) The kinematic performance functions of the linkage are𝛼𝐐=³+𝛼₉𝑟₂cos𝛼₂+𝑟₉cos𝛼₃+𝛼₉𝑟₂sin𝛼₂+𝑟₉sin𝛼₃+𝛼₉, (4.2) Suppose that𝐗=[𝑟₁,𝑟₂,𝑟₃,𝑟₄]^𝑇is the random variable vector, then the Jacobian matrices are derived as:𝜕𝐅𝜕𝐗^𝑇=00000000−1cos𝛼₂cos𝛼₃−cos𝛼₄0sin𝛼₂sin𝛼₃−sin𝛼₄,𝜕𝐅𝜕𝐔^𝑇=𝑟₈sin𝛼₂00−𝑟₇sin𝛼₇−𝑟₈cos𝛼₂00𝑟₇cos𝛼₇−𝑟₂sin𝛼₂−𝑟₃sin𝛼₃𝑟₄sin𝛼₄0𝑟₂cos𝛼₂𝑟₃cos𝛼₃−𝑟₄cos𝛼₄0,𝜕𝐐𝜕𝐗^𝑇=00000cos𝛼₂000sin𝛼₂,00𝜕𝐐𝜕𝐔^𝑇=0100−𝑟₂sin𝛼₂−𝑟₉sin𝛼₃+𝛼₉𝑟00₂cos𝛼₂−𝑟₉cos𝛼₃+𝛼₉.00 (4.3) By substituting (4.3) into (2.3), the kinematic performance error vector,Δ𝐐, of the linkage can be obtained. Then (2.12) can be used to obtain the reliability corresponding to each failure mode. The covariance matrix of the limit sate functions of the planar linkage can be derived from (3.5), and then the joint reliability between each two failure modes can also be derived from (2.15).

The upper and lower bounds of the system reliability of the double-rocker four-bar linkage can be obtained by solving the linear program (3.1), and the results are, respectively, shown as the pan dash line and triangle dash dot line in Figure 3. Besides the system reliability of the manipulator using Monte-Carlo simulation with 10⁶ samples is shown as the point solid line. What needs to be specially notified is that, in order to demonstrate the proposed mechanism system reliability analysis method, the truncation errors caused by the first-order Taylor expansion are omitted in the Monte-Carlo simulation. Comparing with the results of numerical simulation, the kinematic performance system reliability of the double-rocker four-bar linkage obtained by the proposed method is of high accuracy.

Figure 3 

System reliability of the double-rocker four-bar linkage with driving crank.

5. Conclusions

Using the mechanism accuracy theory and (system) reliability analysis method, this paper proposes a general method for system reliability analysis of planar linkages with correlated failure modes. The proposed method is applicable to any system defined as a logical expression of kinematic failure modes of planar linkages. This includes series and parallel systems, as well as general systems. Utilization of the first-order Taylor expansion technique in error estimation of kinematic performance of mechanisms must result in a certain degree of truncation errors. And these errors will increase with the increase of sensitivity of performance functions to design parameters. The accuracy of system reliability analysis can be improved by increasing of the order of Taylor expansion. However, in this process, the complexity of calculation will greatly increase. Further studies are needed to provide a more precise and robust method for reliability analysis of kinematic accuracy of mechanisms.

Acknowledgments

The authors gratefully acknowledge the support of National Natural Science Foundation of China (51105062 & 51135003). Moreover, the authors would like to give their thanks to the anonymous referees and the editor for their valuable comments and suggestions leading to an improvement of the paper.