# Probabilistic Approach to System Reliability of Mechanism with Correlated Failure Models

Authors: Xianzhen Huang; Yimin Zhang
Journal: Mathematical Problems in Engineering
 (2012)

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## 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.


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## Body

## 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](2.1)Q=Q(V,L,U),
where Qs×1is the performance parameter vector. For example, for a function generator, Qs×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. Vm×1is the input (independent) variable vector, Lp×1is the effective dimension variable vector, and Un×1 is the output (dependent) variable vector which can be obtained by solving the loop closure equations of planar linkages(2.2)F(L,U,V)=0.The performance errors of the mechanism under consideration can be obtained as(2.3)ΔQ=∂Q∂LTΔL+∂Q∂UTΔU+∂Q∂VTΔV,
where ∂Q/∂LT, ∂Q/∂UTand ∂Q/∂VTare Jacobian matrices, whose values are got at the mean values of the random variables. ΔLp×1,ΔUn×1,andΔVm×1are the tolerance vectors of random design variables. ΔVm×1and ΔLp×1are determined by several objective factors such as the machining accuracy, the assembly accuracy and the operation precision. From (2.3),ΔUn×1can be obtained:(2.4)ΔU=-[∂F∂UT]-1(∂F∂VTΔV+∂F∂LTΔL).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 are(2.5)gU(X)=ε-ΔQ=ε-JΔX,gL(X)=ΔQ+ε=JΔX+ε,
where(2.6)J=[∂Q∂VT-∂Q∂UT(∂F∂UT)-1∂F∂VT,∂Q∂LT-∂Q∂UT(∂F∂UT)-1∂F∂LT].gUand gL are called the upper and lower limit state functions, X=[V1,…,Vm,L1,…,Lp]T is the basic variables, ε is the allowable errors, and ΔX=[ΔV1,…,ΔVm,ΔL1,…,ΔLp]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 thekth dependent variable, Qk, is defined as:(2.7)RL(k)=∫gL(k)(X)>0f(X)dX,
where f(X)is the joint probability density function of multidimensional basic random variables, X and gL(k)(X)=ΔQk+εk=JkΔX+εk is the limit state function of the kth dependent variable,Qk. Note that Jkis the kth row of matrix J.The mean value,μL(k), and variance,(σL(k))2, of the limit state function, gL(k)(X), can be expressed as(2.8)μL(k)=E[gL(k)(X)]=JkE(ΔX)+εk,(σL(k))2=Var⁡[gL(k)(X)]=Jk[2]cs(Cov(ΔX)),
whereE(ΔX)andCov(ΔX) are the mean value vector and covariance matrix of primary errors, respectively, Jkis the kth row of matrix J, (·)[2]=(·)⊗(·)is the second-order Kronecker power of (·), and⊗represents Kronecker product [11](2.9)Ap×q⊗Bs×t=[a11Ba12B…a1qBa21Ba22B…a2qB⋮⋮⋱⋮ap1Bap2B…apqB]ps×qt,cs(·)is the column string of (·), the column sequence(2.10)cs(Ap×q)=∑j=1q(eq×1j⊗Ip×p)Ap×qeq×1j,
whereIp×pis identity matrix withp×pdimensions, and eq×1jis 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:(2.11)βL(k)=μL(k)σL(k),
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:(2.12)RL(k)=Φ(βL(k)),
where Φ(·) is the standard normal distribution function.The correlation coefficient between performance functionsgL(k) and gL(t) is(2.13)Cov(gL(k),gL(t))=E[(gL(k)-g̅L(k))(gL(t)-g̅L(t))]=∑i=1m+l∑j=1m+l∂gL(k)∂Xi∂gL(t)∂XjCov(Xi,Xj).
The correlation coefficient between performance functions gL(k) and gL(t) is(2.14)ρL(k,t)=Cov(gL(k),gL(t))Var⁡(gL(k))Var⁡(gL(t)).
Then the joint reliability of gL(k) and gL(t) can be estimated by the joint normal distribution function:(2.15)RL(k,t)=1-∬-∞0fkt(gL(k),gL(t))dgL(k)dgL(t)=1-∫-∞-βLt∫-∞-βLkϕL(k,t)(u,v)dudv=1-∫-∞-βLtΦ[-βLk-ρL(k,t)v1-(ρL(k,t))2]ϕ(v)dv,
where(2.16)fkt(gL(k),gL(t))=12πσL(k)σL(t)[1-(ρL(k,t))2]×exp⁡{-12[1-(ρL(k,t))2][(gL(k)-μL(k))2Var⁡(gL(k))-2ρL(k,t)(gL(k)-μL(k))(gL(t)-μL(t))σL(k)σL(t)+(gL(t)-μL(t))2Var⁡(gL(t))]}
is the joint PDF of gL(k) and gL(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 as(2.17)Rk=RL(k)+RU(k)-1,R(k,t)=RL(k,t)+RU(k,t)-1.


## 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:(3.1)min⁡(max⁡)cTps.t.a1p=b1a2p≥b2,
wherepis the design variable vector of the linear programming, cis a vector of coefficients, cTpis 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 into2n 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 Qi meets the requirement of performance quality as eventEi. The space S is divided into 8 MECE events, {e1=E1E2E3,e2=E̅1E2E3,e3=E1E̅2E3,e4=E1E2E̅3,e5=E̅1E̅2E3,e6=E̅1E2E̅3,e7=E1E̅2E̅3,ande8=E̅1E̅2E̅3}. Let pi=P(ei),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,
(3.2)∑i=18pi=1,(3.3)pi≥0,i=(1,2,…,8).
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 Figure1,(3.4)P(Ei)=Pi=∑r:er⊆Eipr,(3.5)P(EiEj)=Pij=∑r:er⊆EiEjpr,scilicet,(3.6)P(E1)=P1=p1+p3+p4+p7,P(E2)=P2=p1+p2+p4+p6,P(E3)=P3=p1+p2+p3+p5,P(E1E2)=P12=p1+p4,P(E1E3)=P13=p1+p3,P(E2E3)=P23=p1+p2.
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 are(3.7)a1=[11111111101100101101010011101000100100001010000011000000],b1=[1P1P2P3P12P13P23]T,a2=I8×8,b2=[00000000]T,
whereI8×8is identity matrix with8×8dimensions.According to the relationship between failure modes (series or parallel), there are four different kinds of systems. As shown in Table1, the coefficient ci (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 cTp.
c1c2c3c4c5c6c7c8E1∪E2∪E311111110E1∩E2∩E310000000E1∩E2∪E311111000(E1∪E2)∩(E2∪E̅3)11100010


## 4. Numerical Examples

Consider the vector loop as shown in Figure2, 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 Figure1, 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,Mx,My]T, then the closure equations of the planar linkage are(4.1)F=[r6cos⁡α6+r7cos⁡α7-r8cos⁡α2-r5r6sin⁡α6+r7sin⁡α7-r8sin⁡α2r2cos⁡α2+r3cos⁡α3-r4cos⁡α4-r1r2sin⁡α2+r3sin⁡α3-r4sin⁡α4].
The kinematic performance functions of the linkage are(4.2)Q=[α3+α9r2cos⁡α2+r9cos⁡(α3+α9)r2sin⁡α2+r9sin⁡(α3+α9)],
Suppose thatX=[r1,r2,r3,r4]Tis the random variable vector, then the Jacobian matrices are derived as:(4.3)∂F∂XT=[00000000-1cos⁡α2cos⁡α3-cos⁡α40sin⁡α2sin⁡α3-sin⁡α4],∂F∂UT=[r8sin⁡α200-r7sin⁡α7-r8cos⁡α200r7cos⁡α7-r2sin⁡α2-r3sin⁡α3r4sin⁡α40r2cos⁡α2r3cos⁡α3-r4cos⁡α40],∂Q∂XT=[00000cos⁡α2000sin⁡α200],∂Q∂UT=[0100-r2sin⁡α2-r9sin⁡(α3+α9)00r2cos⁡α2-r9cos⁡(α3+α9)00].
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.



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