# Dynamic Response Analysis of Planar Multilink Mechanism considering Wear in Clearances

Authors: Shuai Jiang; Xiulong Chen; Yu Deng
Journal: Shock and Vibration
 (2019)

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## Abstract

Clearance wear is one of the factors that affects dynamics for mechanical systems. A numerical methodology suitable for modeling and calculation of wear at multiple revolute clearance pairs in the field of the planar multilink mechanism is proposed. In this paper, the 2-DOF nine-bar mechanism considering two revolute clearance joints is regarded as the study object. Normal contact force and friction force models of revolute clearance joints used Lankarani–Nikravesh (L-N) and LuGre models, respectively. The iterative wear prediction process based upon Archard’s model has been applied to calculate wear characteristics. The wear prediction procedure is integrated with multibody dynamics, wear depths at revolute clearance joints are calculated, and the surface of shaft and bearing is reconstructed twice. The dynamic responses of mechanism considering two nonregular revolute clearances caused by wear are studied in depth. The nonlinear characteristics of the mechanism after wear are studied by the phase diagram and Poincaré map. Influences of different initial constant clearance values and different driving speeds on wear of two revolute joints are also researched. The results show that it is necessary to consider the factor of irregular clearances caused by wear in analysis of dynamics of precision mechanisms. Initial constant clearance values and driving speeds have some influence on wear phenomenon. This research provides a theoretical basis for studying dynamics of the planar multilink mechanism considering wear in multiple clearances.


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## Body

## 1. Introduction

In the actual mechanical mechanism, the clearance of the motion pair is unavoidable [1–4]. With the development of precision mechanical engineering, it is increasingly necessary to accurately predict the wear phenomenon of mechanical system. In the process of contact movement of pair elements, the surface material of pair element will be constantly losing, which will increase the clearance value at motion pair. And the dynamic responses of the mechanism will be changed accordingly [5–7]. In particular, wear prediction of the mechanism system containing multiple clearance joints is one of the crucial questions to be settled urgently. Therefore, it is of great practical value to study dynamic wear phenomena of motion pairs.In the last few years, many scholars have done plenty of researches on dynamics behavior for mechanical systems considering constant clearances. However, there are few studies on dynamic responses for multibody mechanical systems considering nonregular clearances. Tan et al. [8] explored dynamics response for the slider crank mechanism containing clearances. And the dynamics model was built through combining the Newton–Euler method and improved the Coulomb friction force model together, and the Baumgarte stabilization approach was used to enhance numerical stability. Wang et al. [9] proposed a nonpenetration methodology for frictional contact analysis for modeling clearance of the planar crank-slider mechanism. Chen et al. [10] conducted the dynamic simulations of the crank-slider mechanism considering clearance, flexibility, and friction by using ADAMS. Influences of link flexibility and clearance value on dynamic response were researched. Wang et al. [11] discussed nonlinear dynamics for the flexible crank-slider mechanism considering clearance at one revolute pair and modeled flexible dynamic equations by utilizing ANCF. Zheng et al. [12] established a flexible dynamics model for high-precision press mechanism considering clearance and lubrication. Effects of clearance size, blanking force, and input crank speed on dynamics behavior of the multilink mechanism have been also studied. Muvengei et al. [13] studied nine motion modes of two rotating clearance pairs and the influence of clearances on the dynamic response for the crank-slider mechanism. Normal force and frictional force at revolute clearances have been, respectively, modeled by the L-N contact force model and LuGre model. Flores et al. [14] presented a method to discuss influence of the clearance value and friction coefficient on dynamic performance for the planar four-bar mechanism containing revolute clearance joints. Through utilizing Poincaré maps, periodic responses and chaotic responses for the four-bar mechanism were both observed. Chen et al. [15] developed the nonlinear dynamic model for 2-DOF nine-bar mechanism containing clearance and flexibility rods through using the Lagrange equation and FEM. Yaqubi et al. [16] analyzed dynamics behavior of a crank-slider mechanism containing clearance pairs and investigated nonlinear dynamic by Poincaré maps and bifurcation diagrams. Influences of the friction on dynamic responses and nonlinear characteristics are also discussed. Muvengei et al. [17] discussed dynamic response of the crank-slider mechanism existing stick-slip friction at revolute clearance pairs. And the LuGre model was presented to express stick-slip friction at revolute clearance pairs.As is well known, existence of clearance at kinematic pair causes wear, which will directly reduce the life of mechanism. However, the research studies on wear of planar mechanisms are mainly concentrated upon simple planar mechanisms containing single revolute clearance pair and mainly concentrated on wear depth and surface reconstruction. Few studies have been done on dynamic response for the planar complex mechanism considering multiple clearances after many times wear by combining the L-N model with the LuGre friction model. Flores [18] developed a method for researching and quantifying wear at revolute pair including clearance. The simple planar four-bar mechanism is utilized to perform numerical simulations. Li et al. [19] analyzed wear phenomenon at two revolute clearance pairs in the crank-slider mechanism through coupling dynamic with tribology. And the effect of diverse constant clearance sizes on wear depth was also studied. Bai et al. [20] investigated wear prediction at revolute pair for four-bar mechanism by using a computational method. A novel hybrid contact force and a modified Coulomb friction model were, respectively, used as normal contact and friction models. And it discussed the dynamics performance and wear phenomenon on the four-bar mechanism considering single revolute clearance joint. Zhao et al. [21] put forward a numerical methodology to predict the wear of revolute pairs considering clearances for the flexible planar crank-slider mechanism. The normal contact force of the clearance joint used the L-N model, and friction force adopted the LuGre model. Archard’s model was applied to forecast wear, and wear depth was also calculated. Mukras et al. [22] put forward a procedure to discuss the dynamic of the crank-slider mechanism. Wear prediction at revolute clearance pair is presented based upon Archard’s model. The procedure was validated by comparing wear prediction with the wear experiment. Bai et al. [23] researched dynamic response and wear phenomenon at single revolute joint of the four-bar mechanism by using a computational method. An integrated method of dynamic response analysis and wear analysis was proposed. Xiang et al. [24] proposed and discussed the comprehensive method to predict wear in the planar crank-slider mechanism containing clearance joints. The wear process has been integrated into a differential equation to perform coupling iteration analysis of dynamics response and wear prediction. Wang et al. [25] predicted the effect of spherical clearance caused by the wear on dynamic behavior for spatial four-bar mechanism based upon Archard’s model.Through above description, in the research field of planar multibody dynamics system considering clearance wear, previous literatures mainly concentrated upon study of the simple mechanism (such as the crank-slider mechanism and four-bar mechanism) considering single revolute clearance joint and the research of wear depth and surface reconstruction of the shaft and bearing after wear. And few studies have been done on dynamic responses and nonlinear dynamic characteristics for the planar multilink mechanism containing multiple revolute clearances by adopting the LuGre model after many times wear. In this paper, the dynamic model of the multilink mechanism considering wear at multiple clearance joints by using the LuGre model is developed, and dynamic responses of the multilink mechanism containing multiple clearance joints after many times wear are studied systematically and in detail. The nonlinear characteristics of the mechanism after wear are studied by the phase diagram and Poincaré map. The influences of different factors on wear at multiple clearance joints are researched.This paper is organized as followed: In Section1, the clearance model of the revolute joint is established. In Section 2, the dynamic model of 2-DOF nine-bar mechanism considering two revolute clearance joints is established. In Section 3, based upon Archard’s model, the wear model of the clearance of the rotating pair is established. In Section 4, wear depth of the two revolute clearance joints is forecasted, and the surface of shaft and bearing is reconstructed twice. The dynamic responses after twice wear of 2-DOF nine-bar mechanism considering constant clearance and nonregular clearance caused by wear are both discussed, which include kinematic characteristics of the slider, shaft center trajectories, the contact forces at clearance joints, and the input driving torques of cranks. The nonlinear characteristics of the mechanism after wear are studied by the phase diagram and Poincaré map. The influences of different initial constant clearance sizes and different driving speeds on wear are researched. Finally, main conclusions are drawn.


## 2. Establishment of Clearance Model

### 2.1. Kinematic Model of Revolute Clearance Joint

As illustrated in Figure1, it is shown that bearing i and shaft j are connected via a revolute clearance pair. Ri and Rj, respectively, represent the radius of bearing and shaft. Clearance size is defined as c = Ri−Rj. Global coordinate system is XOY. Local coordinate systems of the bearing and shaft are xioiyi and xjojyj, respectively. Pi and Pj are the rotating center of bearing and the shaft, and the position vector in the fixed coordinate system is riP and rjP.Figure 1
Revolute clearance model.Eccentricity vector and eccentricity between bearing and shaft are(1)eij=rjP−riP,eij=eijTeij.Unit normal vector could be written as(2)nij=eijeij.While shaft and bearing impact with each other, penetration depth can be expressed as(3)δij=eij−c.Whenδij<0, bearing and shaft are in the free flight mode. When δij=0, bearing and shaft are in the continuous contact mode. When δij>0, bearing and shaft are in the impact mode.Contact pointQi and Qj in the global coordinate system could be written as(4)rkQ=rkP+Rknij,k=i,j.Velocity of contact pointQi and Qj is(5)r˙kQ=r˙kP+Rkn˙ij,k=i,j.Relative velocities in the normal direction and tangential direction could be given by(6)vn=r˙jQ−r˙iQTnij,vt=r˙jQ−r˙iQTtij.


### 2.2. The Normal Force Model

The Lankarani–Nikravesh contact force model is more suitable for general mechanical contact with the high coefficient of restitution, particularly while energy dissipation is relatively few [13, 22]. The L-N model is well conformity to experimental results. And it is straightforward for the numerical integration algorithm [8, 14, 21]. Therefore, the L-N model has been applied in this paper:(7)Fn=Kδijn+Dδ˙ij,where Krepresents the stiffness parameter; D represents the damping coefficient; n represents a constant decided by the material property of contact surface; for metal surface, n = 1.5; and δ˙ij is the penetration velocity.Stiffness parameterK can be given by(8)K=43πσi+σjRiRjRi+Rj1/2,where σi=1−νi2/πEi and σj=1−νj2/πEj, in which νi and νj represent Poisson’s ratio of the bearing and shaft, respectively. Ei and Ej represent the elastic modulus of bearing and shaft. Radius is negative for concave surfaces and positive for the convex surface.Damping coefficientD could be expressed by(9)D=3K1−ce2δijn4δ˙ij−,where ce means the restitution coefficient; δ˙ij− is the initial impact velocity; if δijtnδijtn+1≤0, δijtn<0 and δijtn+1>0; contact occurs between two discrete times of tn and tn+1; and penetration velocity at tn+1 represents the initial impact velocity δ˙ij−.


### 2.3. The Tangential Force Model of Revolute Clearance

LuGre model was presented by the Canudas de Wit et al. It can capture the change of friction force with the slip speed. Therefore, it is suitable for the study of stick-slip motion. The LuGre model can also be utilized to capture the Stribeck phenomenon [26]. According to classical definition of friction, the friction force could be expressed as follows [17, 21]:(10)Ft=μFn.The instantaneous coefficient of frictionμ is viewed to vary as a function of tangential velocity and an average bristle deflection z, and we can get the following [10]:(11)μ=σ0z+σ1z˙+σ2v,where σ0 represents the bristle stiffness, σ1 represents the microscopic damping coefficient, σ2 represents the viscous friction coefficient, and v is the tangential velocity.Evolution differential equation of average bristle deflection [13, 27] could be given by(12)z˙=v−σ0vμk+μs−μke−v/vsγz,where μk represents the kinetic friction coefficient that is a measure of Coulomb friction force, μs represents the static friction coefficient which is a measure of stiction force, γ represents the gradient of friction decay, and vs represents the stribeck velocity.Substituting equation (12) into equation (11), then(13)μ=σ0z1−σ0vμk+μs−μke−v/vsγ+σ1+σ2v.In this paper, choice ofz was based upon following assumptions [17]: because simulations at the steady-state condition are required, average bristle deflection z was supposed to be a constant for a particular value of relative tangential velocity of components. Hence, at steady state, z could be given by(14)z˙=v−σ0vμk+μs−μke−v/vsγz=0,z=vv⋅μk+μs−μke−v/vsγσ0.


### 2.4. Force Analysis of the Revolute Clearance

The force and moment acting on the centroid of bearingi are written as(15)Fij=Fnn+Ftt=FijxFijyT,Mij=−yiQ−yiFijx+xiQ−xiFijy.The force and moment acting on the centroid of shaftj are expressed as(16)Fji=−Fji,Mji=−yjQ−yjFjix+xjQ−xjFjiy.


## 2.1. Kinematic Model of Revolute Clearance Joint

As illustrated in Figure1, it is shown that bearing i and shaft j are connected via a revolute clearance pair. Ri and Rj, respectively, represent the radius of bearing and shaft. Clearance size is defined as c = Ri−Rj. Global coordinate system is XOY. Local coordinate systems of the bearing and shaft are xioiyi and xjojyj, respectively. Pi and Pj are the rotating center of bearing and the shaft, and the position vector in the fixed coordinate system is riP and rjP.Figure 1
Revolute clearance model.Eccentricity vector and eccentricity between bearing and shaft are(1)eij=rjP−riP,eij=eijTeij.Unit normal vector could be written as(2)nij=eijeij.While shaft and bearing impact with each other, penetration depth can be expressed as(3)δij=eij−c.Whenδij<0, bearing and shaft are in the free flight mode. When δij=0, bearing and shaft are in the continuous contact mode. When δij>0, bearing and shaft are in the impact mode.Contact pointQi and Qj in the global coordinate system could be written as(4)rkQ=rkP+Rknij,k=i,j.Velocity of contact pointQi and Qj is(5)r˙kQ=r˙kP+Rkn˙ij,k=i,j.Relative velocities in the normal direction and tangential direction could be given by(6)vn=r˙jQ−r˙iQTnij,vt=r˙jQ−r˙iQTtij.


## 2.2. The Normal Force Model

The Lankarani–Nikravesh contact force model is more suitable for general mechanical contact with the high coefficient of restitution, particularly while energy dissipation is relatively few [13, 22]. The L-N model is well conformity to experimental results. And it is straightforward for the numerical integration algorithm [8, 14, 21]. Therefore, the L-N model has been applied in this paper:(7)Fn=Kδijn+Dδ˙ij,where Krepresents the stiffness parameter; D represents the damping coefficient; n represents a constant decided by the material property of contact surface; for metal surface, n = 1.5; and δ˙ij is the penetration velocity.Stiffness parameterK can be given by(8)K=43πσi+σjRiRjRi+Rj1/2,where σi=1−νi2/πEi and σj=1−νj2/πEj, in which νi and νj represent Poisson’s ratio of the bearing and shaft, respectively. Ei and Ej represent the elastic modulus of bearing and shaft. Radius is negative for concave surfaces and positive for the convex surface.Damping coefficientD could be expressed by(9)D=3K1−ce2δijn4δ˙ij−,where ce means the restitution coefficient; δ˙ij− is the initial impact velocity; if δijtnδijtn+1≤0, δijtn<0 and δijtn+1>0; contact occurs between two discrete times of tn and tn+1; and penetration velocity at tn+1 represents the initial impact velocity δ˙ij−.


## 2.3. The Tangential Force Model of Revolute Clearance

LuGre model was presented by the Canudas de Wit et al. It can capture the change of friction force with the slip speed. Therefore, it is suitable for the study of stick-slip motion. The LuGre model can also be utilized to capture the Stribeck phenomenon [26]. According to classical definition of friction, the friction force could be expressed as follows [17, 21]:(10)Ft=μFn.The instantaneous coefficient of frictionμ is viewed to vary as a function of tangential velocity and an average bristle deflection z, and we can get the following [10]:(11)μ=σ0z+σ1z˙+σ2v,where σ0 represents the bristle stiffness, σ1 represents the microscopic damping coefficient, σ2 represents the viscous friction coefficient, and v is the tangential velocity.Evolution differential equation of average bristle deflection [13, 27] could be given by(12)z˙=v−σ0vμk+μs−μke−v/vsγz,where μk represents the kinetic friction coefficient that is a measure of Coulomb friction force, μs represents the static friction coefficient which is a measure of stiction force, γ represents the gradient of friction decay, and vs represents the stribeck velocity.Substituting equation (12) into equation (11), then(13)μ=σ0z1−σ0vμk+μs−μke−v/vsγ+σ1+σ2v.In this paper, choice ofz was based upon following assumptions [17]: because simulations at the steady-state condition are required, average bristle deflection z was supposed to be a constant for a particular value of relative tangential velocity of components. Hence, at steady state, z could be given by(14)z˙=v−σ0vμk+μs−μke−v/vsγz=0,z=vv⋅μk+μs−μke−v/vsγσ0.


## 2.4. Force Analysis of the Revolute Clearance

The force and moment acting on the centroid of bearingi are written as(15)Fij=Fnn+Ftt=FijxFijyT,Mij=−yiQ−yiFijx+xiQ−xiFijy.The force and moment acting on the centroid of shaftj are expressed as(16)Fji=−Fji,Mji=−yjQ−yjFjix+xjQ−xjFjiy.


## 3. Dynamic Model of 2-DOF Nine-Bar Mechanism considering Revolute Clearances

A 2-DOF nine-bar mechanism is taken as a research object, which consists of the frame, crank 1, rod 2, rod 3, crank 4, rod 6, triangular panel 7, rod 8, and slider 9. Rod 5 is frame [15]. Two cranks are driven by the motors, and the DOF of mechanism is 2.Crank 1 and rod 2 are connected through a revolute clearance pairA, and rod 3 and crank 4 are connected through a revolute clearance pair B. Crank 1 and crank 4 are driven by motors directly. Therefore, clearance at joint A and joint B could better reflect influence of revolute clearances on dynamics responses. Structure diagram of 2-DOF nine-bar mechanism considering two revolute clearance joints is displayed in Figure 2.Figure 2
Structure diagram of 2-DOF nine-bar mechanism with revolute clearances.Local coordinate systems are built on centroid of each component. Generalized coordinates of the each member can be given by(17)qi=xiyiθiT,i=1,2,3,4,6,7,8,9,where xi and yi are the coordinates of centroid for component i in the global coordinate system and θi represents the angle for component i in the global coordinate system. Velocity and acceleration of generalized coordinates could be expressed as(18)q˙i=x˙iy˙iθ˙iT,q¨i=x¨iy¨iθ¨iT,i=1,2,3,4,6,7,8,9.When the clearance exists in the revolute pair, the constraints at the revolute pair are lost; that is, the corresponding constraints in theX and Y directions are deleted. Compared with the constraint equation without clearances, the number of constraint equations of the 2-DOF nine-bar mechanism containing revolute clearance joint A and revolute clearance joint B is reduced to 20, and the constraint equations considering revolute clearance joint A and revolute clearance joint B could be expressed as(19)Φq,t=x1−Ls1cosθ1y1−Ls1sinθ1x4−Ls4cosθ4−L5y4−Ls4sinθ4x7+Ls72cosθ7−β1−x2−Ls2cosθ2y7+Ls72sinθ7−β1−y2−Ls2sinθ2x7+Ls72cosθ7−β1−x3−Ls3cosθ3y7+Ls72sinθ7−β1−y3−Ls3sinθ3x6−Ls6cosθ6+Hxy6−Ls6sinθ6−Hyx7−Ls71cosθ7+β−x6−Ls6cosθ6y7−Ls71sinθ7+β−y6−Ls6sinθ6x7+Ls73cosθ7+β12+w12−x8+Ls8cosθ8y7+Ls71sinθ7+β12+w12−y8+Ls8sinθ8x9+Hx−L6cosθ6−L72cosθ7+β12−Ls8cosθ8y9−Hy−L6sinθ6−L72sinθ7+β12−Ls8sinθ8y9−Hyθ9θ1−ω1t−60.28°θ4−ω4t+52.86°=0.Taking the first derivative of equation (19) with respect to time, the velocities of constraint equations containing revolute clearance joint A and revolute clearance joint B could be written as(20)Φqq˙=−Φt≡υ,where Φq represents the Jacobian matrix of constraint equations, Φq=∂Φ/∂q, and Φt represents the derivative of constraint equation to time, Φt=∂Φ/∂t.Taking the first derivative of equation (20) with respect to time, accelerations of constraint equations containing two revolute clearances could be written as(21)Φqq¨=−Φqq˙qq˙−2Φqtq˙−Φtt≡γ.Dynamics equations of the 2-DOF nine-bar mechanism considering revolute clearance jointA and revolute clearance joint B could be written as(22)Mq¨+ΦqTλ=Fg,where M represents the system mass matrix, λ represents the vector of Lagrange multiplier, and Fg represents the vector of generalized force vector.Multibody dynamic equations containing revolute clearance jointA and revolute clearance joint B in the differential algebraic form could be formed by equations (21) and (22):(23)MΦqTΦq0q¨λ=FgγT.Equation (23) does not directly apply the displacement constraint and velocity constraint equations of the system, which bring about default of equations. At present, the most common and effective method is the default stability algorithm proposed by Baumgarte [8]. In this method, displacement constraints and velocity constraint are introduced into the acceleration constraints equation by feedback control theory so as to reduce the default of system. Therefore, it could be written as follows:(24)MΦqTΦq0q¨λ=Fgγ′T,where γ′=γ−2αΦ˙−β2Φ, α and β are correction parameters greater than 0, and Φ˙=dΦ/dt.Since equation of system (equation (24)) is complex, it is nearly impossible to be solved analytically. Therefore, in order to guarantee accuracy and efficiency, Runge–Kutta methodology has been used to solve equations through MATLAB.


## 4. Wear

### 4.1. Wear Modeling in Revolute Clearance Joint

In the multibody mechanical system, while two components of revolute pair are in contact with each other or in relative motion, the surface of the two components will be worn due to the contact force produced by the collision, which makes the surface no longer smooth. Therefore, it is very important to forecast the wear phenomenon at the revolute clearance pairs.Archard’s model connects wear volume with some physical and geometrical properties of sliding components, such as sliding distance, hardness, and applied load [18, 21]. Moreover, its construction is simple and it is most close to the adhesive wear mechanism, and Archard’s model is validated by experimental results. Therefore, it is widely applied in wear prediction for the multibody mechanical system [5, 6, 24]. So, it could be written as(25)Vs=kFnH,where V represents the wear volume, k represents the dimensionless wear coefficient, Fn represents the normal contact force, H represents the Brinell hardness of soft materials, and s is the sliding distance.Contact form and geometric relationship of revolute pair considering clearance are shown in Figure3. eij=PiPj and β¯ is the contact angle.Figure 3
Contact form and geometric relationship of revolute joint considering clearance.According to the cosine theorem, the contact angle can be expressed as(26)β¯=arccoseij2+Ri2−Rj22eijRi.InΔDPiC,(27)DC=Risinπ−β¯.In fact,∠DPjC is far bigger than the real because penetration depth δij is far smaller than Ri and Rj. So the length of contact arc CE along the circumferential direction is equal approximately to DC; that is, DC≈CE [7].Therefore, actual contact areaAa could be expressed as(28)Aa≈DC⋅w,where w represents contact width between shaft and bearing along the axial direction.In practical engineering applications, wear depth can describe wear characteristics more conveniently than wear volume. Therefore, dividing equation (25) by the actual contact area Aa, we can get the following [5, 24]:(29)hs=kpH=kdp,where kd is the linear wear coefficient, kd=k/H, h is the wear depth, and p is the contact stress.Equation (25) is the most commonly used expression of Archard’s model. However, the wear procedure is generally regarded as a dynamic process. Therefore, contact point of pair continuously changes during motion of mechanism, and frequent contact areas usually occur in some special contact locations, which result in irregular contact wear rather than uniform wear of elements in clearance joints. In addition, the sliding distance of contact collision points of contact elements of clearance joint changes with the dynamic contact force. Therefore, the dynamic wear characteristics of the multibody system with clearances are generally studied in differential form [5, 19, 24]:(30)dh=kdpds.Equation (30) is updated via a temporal discretization of relative motion of contact components, and the wear depth of m cycles can be written as follows [18, 22, 25]:(31)hm=hm−1+Δhm,where hm−1 represents the wear depth at the previous cycle and Δhm represents the wear depth at the mth cycle.


### 4.2. Surface Reconstruction after Wear for Revolute Clearance Joint

Based upon thought of the finite element method, the geometric surface of shaft and bearing are discretized. In every integral time step, the contact condition between shaft and bearing is judged. When shaft contacts with bearing, the location (discrete region) where the shaft contacts with bearing is stored and wear depth at this location is also calculated. Finally, the amount of wear depth accumulated on a discrete region is the sum of all partial wear depths at each integral time step of this discrete region. By this method, new geometry of the joint surface caused by wear can be calculated.Therefore, the total amount of the wear depth could be written as follows [18, 23]:(32)h=∑hn,where n represents the code name of discrete region in which the joint surface is divided and hn is the wear depth at the nth discrete region.It is assumed that the surface material of the shaft and bearing is same and wear depth in shaft is equal to wear depth in bearing. Wear depth values of shaft and wearing are both set as half of the total wear depth. Based upon revolute pair geometric properties, new journal and bearing radius at each discrete region after wear can be given by the following equation [20, 23]:(33)Ri∗=Ri−h2,Rj∗=Rj+h2.


### 4.3. Dynamic Analysis and Wear Analysis Integration

The main calculation processes for the mechanics system containing revolute pair wear include dynamics research and wear analysis. In order to forecast the effect of wear at revolute pair on dynamic characteristics, it is important to integrate the wear model with the dynamic model considering clearance and analyzing the effect of wear on dynamic responses.It is noteworthy that the clearance value is no longer a constant after wear, and it depends on the radius of the new bearing and shaft at this contact region. The clearance in this contact region after wear could be expressed as(34)c∗n=Ri∗n−Rj∗n.Once clearance value at this contact region is obtained, penetration depth of this region could be obtained by using equation (3). In addition, the contact force model could be updated with new sizes of the radius of bearing and shaft; that is, the stiffness coefficient defined in equation (8) can be updated according to new radius of bearings and shafts. The stiffness coefficient after wear could be expressed as follows:(35)K∗n=43πσi+σjRi∗nRj∗nRi∗n+Rj∗n1/2.The dynamic and wear integration frame is shown in Figure4.Figure 4
The dynamic and wear integration frame.


## 4.1. Wear Modeling in Revolute Clearance Joint

In the multibody mechanical system, while two components of revolute pair are in contact with each other or in relative motion, the surface of the two components will be worn due to the contact force produced by the collision, which makes the surface no longer smooth. Therefore, it is very important to forecast the wear phenomenon at the revolute clearance pairs.Archard’s model connects wear volume with some physical and geometrical properties of sliding components, such as sliding distance, hardness, and applied load [18, 21]. Moreover, its construction is simple and it is most close to the adhesive wear mechanism, and Archard’s model is validated by experimental results. Therefore, it is widely applied in wear prediction for the multibody mechanical system [5, 6, 24]. So, it could be written as(25)Vs=kFnH,where V represents the wear volume, k represents the dimensionless wear coefficient, Fn represents the normal contact force, H represents the Brinell hardness of soft materials, and s is the sliding distance.Contact form and geometric relationship of revolute pair considering clearance are shown in Figure3. eij=PiPj and β¯ is the contact angle.Figure 3
Contact form and geometric relationship of revolute joint considering clearance.According to the cosine theorem, the contact angle can be expressed as(26)β¯=arccoseij2+Ri2−Rj22eijRi.InΔDPiC,(27)DC=Risinπ−β¯.In fact,∠DPjC is far bigger than the real because penetration depth δij is far smaller than Ri and Rj. So the length of contact arc CE along the circumferential direction is equal approximately to DC; that is, DC≈CE [7].Therefore, actual contact areaAa could be expressed as(28)Aa≈DC⋅w,where w represents contact width between shaft and bearing along the axial direction.In practical engineering applications, wear depth can describe wear characteristics more conveniently than wear volume. Therefore, dividing equation (25) by the actual contact area Aa, we can get the following [5, 24]:(29)hs=kpH=kdp,where kd is the linear wear coefficient, kd=k/H, h is the wear depth, and p is the contact stress.Equation (25) is the most commonly used expression of Archard’s model. However, the wear procedure is generally regarded as a dynamic process. Therefore, contact point of pair continuously changes during motion of mechanism, and frequent contact areas usually occur in some special contact locations, which result in irregular contact wear rather than uniform wear of elements in clearance joints. In addition, the sliding distance of contact collision points of contact elements of clearance joint changes with the dynamic contact force. Therefore, the dynamic wear characteristics of the multibody system with clearances are generally studied in differential form [5, 19, 24]:(30)dh=kdpds.Equation (30) is updated via a temporal discretization of relative motion of contact components, and the wear depth of m cycles can be written as follows [18, 22, 25]:(31)hm=hm−1+Δhm,where hm−1 represents the wear depth at the previous cycle and Δhm represents the wear depth at the mth cycle.


## 4.2. Surface Reconstruction after Wear for Revolute Clearance Joint

Based upon thought of the finite element method, the geometric surface of shaft and bearing are discretized. In every integral time step, the contact condition between shaft and bearing is judged. When shaft contacts with bearing, the location (discrete region) where the shaft contacts with bearing is stored and wear depth at this location is also calculated. Finally, the amount of wear depth accumulated on a discrete region is the sum of all partial wear depths at each integral time step of this discrete region. By this method, new geometry of the joint surface caused by wear can be calculated.Therefore, the total amount of the wear depth could be written as follows [18, 23]:(32)h=∑hn,where n represents the code name of discrete region in which the joint surface is divided and hn is the wear depth at the nth discrete region.It is assumed that the surface material of the shaft and bearing is same and wear depth in shaft is equal to wear depth in bearing. Wear depth values of shaft and wearing are both set as half of the total wear depth. Based upon revolute pair geometric properties, new journal and bearing radius at each discrete region after wear can be given by the following equation [20, 23]:(33)Ri∗=Ri−h2,Rj∗=Rj+h2.


## 4.3. Dynamic Analysis and Wear Analysis Integration

The main calculation processes for the mechanics system containing revolute pair wear include dynamics research and wear analysis. In order to forecast the effect of wear at revolute pair on dynamic characteristics, it is important to integrate the wear model with the dynamic model considering clearance and analyzing the effect of wear on dynamic responses.It is noteworthy that the clearance value is no longer a constant after wear, and it depends on the radius of the new bearing and shaft at this contact region. The clearance in this contact region after wear could be expressed as(34)c∗n=Ri∗n−Rj∗n.Once clearance value at this contact region is obtained, penetration depth of this region could be obtained by using equation (3). In addition, the contact force model could be updated with new sizes of the radius of bearing and shaft; that is, the stiffness coefficient defined in equation (8) can be updated according to new radius of bearings and shafts. The stiffness coefficient after wear could be expressed as follows:(35)K∗n=43πσi+σjRi∗nRj∗nRi∗n+Rj∗n1/2.The dynamic and wear integration frame is shown in Figure4.Figure 4
The dynamic and wear integration frame.


## 5. Dynamic Response Analysis of 2-DOF Nine-Bar Mechanism with Revolute Clearances

When the number of cycles for the mechanism is small, wear depth and clearance value change little. If the surface of the shaft and bearing is reconstructed after each contact, the calculation cost will be greatly increased and the efficiency will be very low. Therefore, assumptions are set as follows:(1)
It is assumed that the surface of the shaft and bearing does not change with time in a certain operating periods. Supposing that the mechanism runs every 100 cycles, wear is calculated once and surface of shaft and bearing is reconstructed once. Because the wear amount produced in 100 cycles is too little, the surface reconstruction is not clear, and if mechanism runs two million cycles, it will waste a lot of time, so the 100 cycles are magnified 20,000 times to approximate the mechanism running 2 million cycles in this paper.(2)
It is assumed that shaft and bearing surfaces are divided into 1000 discrete regions.(3)
It is assumed that the surface material of shaft and bearing is the same, and the wear depth in shaft is equal to wear depth in bearing, which is half of the total wear depth.Based upon above, the wear phenomenon of revolute clearance joints of 2-DOF nine-bar mechanism is studied and the surface of shaft and bearing after wear is also reconstructed. The dynamic responses of 2-DOF nine-bar mechanism considering constant and nonregular revolute clearances caused by wear are also both researched. Influences of different initial constant clearance values and different driving speeds on wear are also researched. Ode15s function in MATLAB is used to solve the dynamic equation.


### 5.1. System Parameters

The system parameters are illustrated in Tables1 and 2.Table 1
Dimensions and mass properties.
Component
Crank 1
Rod 2
Rod 3
Crank 4
Rod 5
Rod 6
Triangular panel 7
Rod 8
Slider 9
Lengthmm
L
1
L
2
L
3
L
4
L
5
L
6
L
71
L
72
L
73
L
8
—
45
326
497
95
430
230
45
326
497
95
L
s
1
L
s
2
L
s
3
L
s
4
—
L
s
6
L
s
7
L
s
8
23
163
249
48
115
147
168
Massg
148
805
603
265
—
581
4334
827
801
Moment of inertia10−3kg⋅m2
0.2382
8.001
13.37
1.210
—
12.12
3.802
8.663
—Table 2
Clearance joints’ parameters.
Designed parameter
Parameter values
Bearing radiusR1mm
15
Restitution coefficientce
0.9
Elastic moduliEiandEjGPa
200
Poisson’s ratiosυiandυj
0.3
Baumgarte stabilization coefficientα
5
Baumgarte stabilization coefficientβ
5
Bristle stiffnessσ0 (N/m)
100,000
Microscopic damping coefficientσ1 (Ns/m)
400
Viscous friction coefficientσ2
0
Stribeck velocityvs
1% of maximumvt
Gradient of friction decayγ
2
Linear wear coefficient
8
×
10
−
14
Pa
−
1
Integral step (s)
0.0001
Relative tolerance
0.000001


### 5.2. Wear Prediction and Dynamic Analysis

It is assumed that speeds of crank 1 and crank 4 are, respectively,−2πrad/s and 2πrad/s. And the constant clearance size before wear of two revolute clearance joints are both set as 0.5 mm.


#### 5.2.1. Wear Prediction

Wear depth at jointA and joint B is displayed in Figures 5 and 6. The reconstructed surface of shaft A, shaft B, bearing A, and bearing B after wear is shown in Figures7–10, respectively. Local enlarged drawing of the reconstructed surface of shaft A, shaft B, bearing A, and bearing B is shown in Figures 11–14, respectively.Figure 5
Wear depth of jointA.Figure 6
Wear depth of jointB.Figure 7
Shaft surface of jointA.Figure 8
Shaft surface of jointB.Figure 9
Bearing surface of jointA.Figure 10
Bearing surface of jointB.Figure 11
Local enlarged drawing of the shaft surface of jointA.Figure 12
Local enlarged drawing of the shaft surface of jointB.Figure 13
Local enlarged drawing of the bearing surface of jointA.Figure 14
Local enlarged drawing of the bearing surface of jointB.After the first wear cycle of the mechanism, according to the wear depth of the first wear cycle, the process of reconstructing the surface of the shaft and bearing for the first time is called the first wear. After the second wear cycle of the mechanism, the wear depth corresponding to the second wear cycle is generated. On the surface of the shaft and bearing formed after the first wear, the process of reconstructing the surface of the shaft and bearing for the second time is called the second wear.From Figures5 and 6, it can be seen that the wear depth of the second wear is obviously higher than that of the first wear, wear depth at revolute joint B is greater than that of revolute joint A, the wear is more serious, and the wear range is more widely.The wear depths of first wear and second wear at revolute jointA are 5.768×10−6m and 1.079×10−5m, respectively. The wear depths of first wear and second wear at revolute joint B are 6.965×10−6m and 3.336×10−5m, respectively. As can be seen from Figure 5, the wear area of joint A is mainly in [32°, 120°] and [246°, 328°]. As shown in Figure 7, the wear area of joint B is mainly focused on [0°, 20°], [123°, 180°], and [260°, 360°]. This is mainly due to frequent contact between shaft and bearing in these regions. From Figures 7–14, the wear is not obvious on the surface of bearings and shafts due to the wear depths of two revolute clearances are too small. It can be seen from the local enlargement diagram that the surface damage of shaft and bearing after the second wear is more serious than that of the first wear.


#### 5.2.2. Influence of Clearance Wear on Dynamic Responses

The dynamics analyses of 2-DOF nine-bar mechanism with constant revolute clearance values and nonregular revolute clearance values caused by wear (including first wear and second wear) are developed. The motion characteristics of the end slider are shown in Figures15–20, including the slider’s displacement, slider’s displacement error, slider’s velocity, slider’s velocity error, slider’s acceleration, and slider’s acceleration error. Input torques of cranks, contact forces at revolute clearance joints, and shaft trajectories of revolute clearance pairs are illustrated in Figures 21–26.Figure 15
Slider’s displacement.Figure 16
Slider’s displacement error.Figure 17
Slider’s velocity.Figure 18
Slider’s velocity error.Figure 19
Slider’s acceleration.Figure 20
Slider’s acceleration error.Figure 21
Input torque of crank 1.Figure 22
Input torque of crank 4.Figure 23
Contact force of jointA.Figure 24
Contact force of jointB.Figure 25
Shaft trajectory of jointA.Figure 26
Shaft trajectory of jointB.According to Figures15 and 16, influence of the revolute clearances on displacement of the slider is very small, which is in good agreement with the displacement curve under mechanism without clearances. As can be seen from Figures 17 and 18, clearance joints have influence on the speed of slider. Compared with the case without clearance, obvious vibration can be seen in Figure 17. It can be concluded from the velocity error diagram (Figure 18) that the vibration frequency of velocity increases with the increase of wear times. However, from Figures 19–24, the revolute clearances have a great influence on slider’s acceleration, contact forces, and input torques, which produce huge peak value and violent vibration, mainly due to impact between shaft and bearing at clearance pairs.It can be seen from the figure that the peak values of dynamic responses and vibration after wear increased and intensified compared with those without wear. Moreover, dynamic responses of second wear are more obvious than that of first wear. The peak value of acceleration of slider, contact force of jointA, contact force of joint B, input torque of crank 1, and input torque of crank 4 of no wear are 372.4 m/s2, 2072 N, 2314 N, −52.33 N ∗ m, and −164.3 N ∗ m. The peak value of acceleration of slider, contact force of joint A, contact force of joint B, input torque of crank 1, and input torque of crank 4 of first wear are −606.1 m/s2, 2741 N, 3062 N, −53.38 N ∗ m, and −177.5 N ∗ m. The peak value of acceleration of slider, contact force of joint A, contact force of joint B, input torque of crank 1, and input torque of crank 4 of second wear are 708.7 m/s2, 3885 N, 4489 N, −79.34 N ∗ m, and −420.2 N ∗ m. As displayed in Figures 25 and 26, with the increase of wear times, shaft trajectories of clearance joints become more and more chaotic. It is shown that wear has a greater impact on dynamic response, which will accelerate damage and failure for mechanism. Reasons are that clearance pair wear is nonuniform, and the clearance value is changed larger and nonregularly after clearance joint wear.


#### 5.2.3. Influence of Clearance Wear on Nonlinear Dynamic Characteristics

Clearance wear will destroy the revolute joint with clearance, so it has great influence on the nonlinear dynamics of the mechanism. The nonlinear characteristics of the mechanism before wear and after second wear are both studied by using the phase diagram and Poincaré map. The phase diagram and Poincaré map are shown in Figures27 and 28.Figure 27
Phase diagram of (a) jointA in the X direction, (b) joint A in the Y direction, (c) joint B in the X direction, and (d) joint B in the Y direction.
(a)
(b)
(c)
(d)Figure 28
Poincaré map of (a) jointA in the X direction, (b) joint A in the Y direction, (c) joint B in the X direction, and (d) joint B in the Y direction.
(a)
(b)
(c)
(d)It can be seen from the phase diagram and Poincaré map that wear enlarges the area of the phase diagram and makes the range of Poincaré map point diffuse. Moreover, the distribution of mapping points after wear is more confused. The chaotic phenomena at the clearance joint are enhanced by the wear. The reason is that wear makes the surface of shaft and bearing not smooth and increases the impact and vibration between elements of clearance pair.


### 5.3. The Influence of Different Factors on Wear

#### 5.3.1. The Effect of Different Initial Constant Clearance Sizes on Wear

The effect of three different initial constant clearance values on wear analysis is investigated in this section. It is assumed that revolute pairA and revolute pair B have the same clearance value. It is supposed that driving speeds of crank 1 ω1=−2πrad/s and crank 4 ω4=2πrad/s. Initial constant clearance values 0.3 mm, 0.5 mm, and 0.8 mm are all selected to study the wear analysis. Wear depths at joint A and joint B are displayed in Figures 29 and 30. The reconstructed surface of bearing A and bearing B after wear is shown in Figures 31 and 32. Local enlarged drawing of the reconstructed surface of bearing A and bearing B is shown in Figures 33 and 34. The figures are drawn on the basis of the data after the second wear in this section.Figure 29
Wear depth at jointA.Figure 30
Wear depth at jointB.Figure 31
Bearing surface of jointA.Figure 32
Local enlarged drawing of bearing surface of jointA.Figure 33
Bearing surface of jointB.Figure 34
Local enlarged drawing of bearing surface of jointB.From Figures29–34, it can be seen that, because of frequent collisions in some areas, wear phenomenon on bearing surface is not uniformly distributed. And the wear of revolute joint B is more serious than revolute joint A. As shown in Figures 29 and 30, the greater the constant clearance before wear is, the greater the wear depth is, the enlarger the wear area is, and the worse the wear of shaft and bearing are. When the initial constant clearance values are 0.3 mm, 0.5 mm, and 0.8 mm, the corresponding wear depths of revolute A are 1.824×10−5m, 1.079×10−5m, and 7.182×10−6m, and the corresponding wear depths of revolute B are 3.881×10−5m, 3.336×10−5m, and 1.492×10−5m, respectively. It can be guessed that proper reduction of the clearance value at the rotating pair is beneficial to the reduction of wear phenomenon. The wear at revolute clearance joints seriously degenerate precision and reliability of the multibody mechanism system.


#### 5.3.2. The Effect of Different Driving Speeds on Wear

Influences of three different driving speeds on wear analyses are researched in this section. It is supposed that constant clearance values at two revolute clearance joints before wear are both set as 0.3 mm. Driving speed of crank 1 and crank 4 is chosen asω1=−2πrad/s and ω4=2πrad/s, ω1=−3πrad/s and ω4=3πrad/s, and ω1=−4πrad/s and ω4=4πrad/s. Wear depth at joints A and B is displayed in Figures 35 and 36. The reconstructed surface of bearing A and bearing B after wear is shown in Figures 37 and 38. Local enlarged drawing of the reconstructed surface of bearing A and bearing B is shown in Figures 38 and 39. The figures are also drawn on the basis of the data after the second wear in this section.Figure 35
Wear depth at jointA.Figure 36
Wear depth at jointB.Figure 37
Bearing surface of jointA.Figure 38
Bearing surface of jointB.Figure 39
Local enlarged drawing of bearing surface of jointB.With the increase of driving speed, the clearances at the revolute joints will make the collision of the mechanism more intense, accelerate the damage of mechanism, and reduce life of the mechanism. According to Figures35–40, the wear of revolute joint B is more severe than that of revolute joint A, the greater the driving speed is, the greater the wear depth is, the worse wear on the bearing surface is, and the faster failure of the mechanism is. When the driving speeds are ω1=−2πrad/s and ω4=2πrad/s, ω1=−3πrad/s and ω4=3πrad/s, and ω1=−4πrad/s and ω4=4πrad/s, the corresponding wear depths of revolute A are 9.673×10−5m, 1.237×10−5m, and 7.182×10−6m, respectively. When the clearance values are ω1=−2πrad/s and ω4=2πrad/s, ω1=−3πrad/s and ω4=3πrad/s, and ω1=−4πrad/s and ω4=4πrad/s, the corresponding wear depths of revolute B are 1.465×10−4m, 1.599×10−5m, and 1.492×10−5m, respectively. After the clearance wear, clearance values of revolute joints are increased and not a constant value, which will cause the high-frequency vibration of dynamic responses. The driving speed will directly affect the relative motion state between elements of clearance pair, and it has impact on the wear of clearance pair.Figure 40
Local enlarged drawing of bearing surface of jointA.


## 5.1. System Parameters

The system parameters are illustrated in Tables1 and 2.Table 1
Dimensions and mass properties.
Component
Crank 1
Rod 2
Rod 3
Crank 4
Rod 5
Rod 6
Triangular panel 7
Rod 8
Slider 9
Lengthmm
L
1
L
2
L
3
L
4
L
5
L
6
L
71
L
72
L
73
L
8
—
45
326
497
95
430
230
45
326
497
95
L
s
1
L
s
2
L
s
3
L
s
4
—
L
s
6
L
s
7
L
s
8
23
163
249
48
115
147
168
Massg
148
805
603
265
—
581
4334
827
801
Moment of inertia10−3kg⋅m2
0.2382
8.001
13.37
1.210
—
12.12
3.802
8.663
—Table 2
Clearance joints’ parameters.
Designed parameter
Parameter values
Bearing radiusR1mm
15
Restitution coefficientce
0.9
Elastic moduliEiandEjGPa
200
Poisson’s ratiosυiandυj
0.3
Baumgarte stabilization coefficientα
5
Baumgarte stabilization coefficientβ
5
Bristle stiffnessσ0 (N/m)
100,000
Microscopic damping coefficientσ1 (Ns/m)
400
Viscous friction coefficientσ2
0
Stribeck velocityvs
1% of maximumvt
Gradient of friction decayγ
2
Linear wear coefficient
8
×
10
−
14
Pa
−
1
Integral step (s)
0.0001
Relative tolerance
0.000001


## 5.2. Wear Prediction and Dynamic Analysis

It is assumed that speeds of crank 1 and crank 4 are, respectively,−2πrad/s and 2πrad/s. And the constant clearance size before wear of two revolute clearance joints are both set as 0.5 mm.


### 5.2.1. Wear Prediction

Wear depth at jointA and joint B is displayed in Figures 5 and 6. The reconstructed surface of shaft A, shaft B, bearing A, and bearing B after wear is shown in Figures7–10, respectively. Local enlarged drawing of the reconstructed surface of shaft A, shaft B, bearing A, and bearing B is shown in Figures 11–14, respectively.Figure 5
Wear depth of jointA.Figure 6
Wear depth of jointB.Figure 7
Shaft surface of jointA.Figure 8
Shaft surface of jointB.Figure 9
Bearing surface of jointA.Figure 10
Bearing surface of jointB.Figure 11
Local enlarged drawing of the shaft surface of jointA.Figure 12
Local enlarged drawing of the shaft surface of jointB.Figure 13
Local enlarged drawing of the bearing surface of jointA.Figure 14
Local enlarged drawing of the bearing surface of jointB.After the first wear cycle of the mechanism, according to the wear depth of the first wear cycle, the process of reconstructing the surface of the shaft and bearing for the first time is called the first wear. After the second wear cycle of the mechanism, the wear depth corresponding to the second wear cycle is generated. On the surface of the shaft and bearing formed after the first wear, the process of reconstructing the surface of the shaft and bearing for the second time is called the second wear.From Figures5 and 6, it can be seen that the wear depth of the second wear is obviously higher than that of the first wear, wear depth at revolute joint B is greater than that of revolute joint A, the wear is more serious, and the wear range is more widely.The wear depths of first wear and second wear at revolute jointA are 5.768×10−6m and 1.079×10−5m, respectively. The wear depths of first wear and second wear at revolute joint B are 6.965×10−6m and 3.336×10−5m, respectively. As can be seen from Figure 5, the wear area of joint A is mainly in [32°, 120°] and [246°, 328°]. As shown in Figure 7, the wear area of joint B is mainly focused on [0°, 20°], [123°, 180°], and [260°, 360°]. This is mainly due to frequent contact between shaft and bearing in these regions. From Figures 7–14, the wear is not obvious on the surface of bearings and shafts due to the wear depths of two revolute clearances are too small. It can be seen from the local enlargement diagram that the surface damage of shaft and bearing after the second wear is more serious than that of the first wear.


### 5.2.2. Influence of Clearance Wear on Dynamic Responses

The dynamics analyses of 2-DOF nine-bar mechanism with constant revolute clearance values and nonregular revolute clearance values caused by wear (including first wear and second wear) are developed. The motion characteristics of the end slider are shown in Figures15–20, including the slider’s displacement, slider’s displacement error, slider’s velocity, slider’s velocity error, slider’s acceleration, and slider’s acceleration error. Input torques of cranks, contact forces at revolute clearance joints, and shaft trajectories of revolute clearance pairs are illustrated in Figures 21–26.Figure 15
Slider’s displacement.Figure 16
Slider’s displacement error.Figure 17
Slider’s velocity.Figure 18
Slider’s velocity error.Figure 19
Slider’s acceleration.Figure 20
Slider’s acceleration error.Figure 21
Input torque of crank 1.Figure 22
Input torque of crank 4.Figure 23
Contact force of jointA.Figure 24
Contact force of jointB.Figure 25
Shaft trajectory of jointA.Figure 26
Shaft trajectory of jointB.According to Figures15 and 16, influence of the revolute clearances on displacement of the slider is very small, which is in good agreement with the displacement curve under mechanism without clearances. As can be seen from Figures 17 and 18, clearance joints have influence on the speed of slider. Compared with the case without clearance, obvious vibration can be seen in Figure 17. It can be concluded from the velocity error diagram (Figure 18) that the vibration frequency of velocity increases with the increase of wear times. However, from Figures 19–24, the revolute clearances have a great influence on slider’s acceleration, contact forces, and input torques, which produce huge peak value and violent vibration, mainly due to impact between shaft and bearing at clearance pairs.It can be seen from the figure that the peak values of dynamic responses and vibration after wear increased and intensified compared with those without wear. Moreover, dynamic responses of second wear are more obvious than that of first wear. The peak value of acceleration of slider, contact force of jointA, contact force of joint B, input torque of crank 1, and input torque of crank 4 of no wear are 372.4 m/s2, 2072 N, 2314 N, −52.33 N ∗ m, and −164.3 N ∗ m. The peak value of acceleration of slider, contact force of joint A, contact force of joint B, input torque of crank 1, and input torque of crank 4 of first wear are −606.1 m/s2, 2741 N, 3062 N, −53.38 N ∗ m, and −177.5 N ∗ m. The peak value of acceleration of slider, contact force of joint A, contact force of joint B, input torque of crank 1, and input torque of crank 4 of second wear are 708.7 m/s2, 3885 N, 4489 N, −79.34 N ∗ m, and −420.2 N ∗ m. As displayed in Figures 25 and 26, with the increase of wear times, shaft trajectories of clearance joints become more and more chaotic. It is shown that wear has a greater impact on dynamic response, which will accelerate damage and failure for mechanism. Reasons are that clearance pair wear is nonuniform, and the clearance value is changed larger and nonregularly after clearance joint wear.


### 5.2.3. Influence of Clearance Wear on Nonlinear Dynamic Characteristics

Clearance wear will destroy the revolute joint with clearance, so it has great influence on the nonlinear dynamics of the mechanism. The nonlinear characteristics of the mechanism before wear and after second wear are both studied by using the phase diagram and Poincaré map. The phase diagram and Poincaré map are shown in Figures27 and 28.Figure 27
Phase diagram of (a) jointA in the X direction, (b) joint A in the Y direction, (c) joint B in the X direction, and (d) joint B in the Y direction.
(a)
(b)
(c)
(d)Figure 28
Poincaré map of (a) jointA in the X direction, (b) joint A in the Y direction, (c) joint B in the X direction, and (d) joint B in the Y direction.
(a)
(b)
(c)
(d)It can be seen from the phase diagram and Poincaré map that wear enlarges the area of the phase diagram and makes the range of Poincaré map point diffuse. Moreover, the distribution of mapping points after wear is more confused. The chaotic phenomena at the clearance joint are enhanced by the wear. The reason is that wear makes the surface of shaft and bearing not smooth and increases the impact and vibration between elements of clearance pair.


## 5.2.1. Wear Prediction

Wear depth at jointA and joint B is displayed in Figures 5 and 6. The reconstructed surface of shaft A, shaft B, bearing A, and bearing B after wear is shown in Figures7–10, respectively. Local enlarged drawing of the reconstructed surface of shaft A, shaft B, bearing A, and bearing B is shown in Figures 11–14, respectively.Figure 5
Wear depth of jointA.Figure 6
Wear depth of jointB.Figure 7
Shaft surface of jointA.Figure 8
Shaft surface of jointB.Figure 9
Bearing surface of jointA.Figure 10
Bearing surface of jointB.Figure 11
Local enlarged drawing of the shaft surface of jointA.Figure 12
Local enlarged drawing of the shaft surface of jointB.Figure 13
Local enlarged drawing of the bearing surface of jointA.Figure 14
Local enlarged drawing of the bearing surface of jointB.After the first wear cycle of the mechanism, according to the wear depth of the first wear cycle, the process of reconstructing the surface of the shaft and bearing for the first time is called the first wear. After the second wear cycle of the mechanism, the wear depth corresponding to the second wear cycle is generated. On the surface of the shaft and bearing formed after the first wear, the process of reconstructing the surface of the shaft and bearing for the second time is called the second wear.From Figures5 and 6, it can be seen that the wear depth of the second wear is obviously higher than that of the first wear, wear depth at revolute joint B is greater than that of revolute joint A, the wear is more serious, and the wear range is more widely.The wear depths of first wear and second wear at revolute jointA are 5.768×10−6m and 1.079×10−5m, respectively. The wear depths of first wear and second wear at revolute joint B are 6.965×10−6m and 3.336×10−5m, respectively. As can be seen from Figure 5, the wear area of joint A is mainly in [32°, 120°] and [246°, 328°]. As shown in Figure 7, the wear area of joint B is mainly focused on [0°, 20°], [123°, 180°], and [260°, 360°]. This is mainly due to frequent contact between shaft and bearing in these regions. From Figures 7–14, the wear is not obvious on the surface of bearings and shafts due to the wear depths of two revolute clearances are too small. It can be seen from the local enlargement diagram that the surface damage of shaft and bearing after the second wear is more serious than that of the first wear.


## 5.2.2. Influence of Clearance Wear on Dynamic Responses

The dynamics analyses of 2-DOF nine-bar mechanism with constant revolute clearance values and nonregular revolute clearance values caused by wear (including first wear and second wear) are developed. The motion characteristics of the end slider are shown in Figures15–20, including the slider’s displacement, slider’s displacement error, slider’s velocity, slider’s velocity error, slider’s acceleration, and slider’s acceleration error. Input torques of cranks, contact forces at revolute clearance joints, and shaft trajectories of revolute clearance pairs are illustrated in Figures 21–26.Figure 15
Slider’s displacement.Figure 16
Slider’s displacement error.Figure 17
Slider’s velocity.Figure 18
Slider’s velocity error.Figure 19
Slider’s acceleration.Figure 20
Slider’s acceleration error.Figure 21
Input torque of crank 1.Figure 22
Input torque of crank 4.Figure 23
Contact force of jointA.Figure 24
Contact force of jointB.Figure 25
Shaft trajectory of jointA.Figure 26
Shaft trajectory of jointB.According to Figures15 and 16, influence of the revolute clearances on displacement of the slider is very small, which is in good agreement with the displacement curve under mechanism without clearances. As can be seen from Figures 17 and 18, clearance joints have influence on the speed of slider. Compared with the case without clearance, obvious vibration can be seen in Figure 17. It can be concluded from the velocity error diagram (Figure 18) that the vibration frequency of velocity increases with the increase of wear times. However, from Figures 19–24, the revolute clearances have a great influence on slider’s acceleration, contact forces, and input torques, which produce huge peak value and violent vibration, mainly due to impact between shaft and bearing at clearance pairs.It can be seen from the figure that the peak values of dynamic responses and vibration after wear increased and intensified compared with those without wear. Moreover, dynamic responses of second wear are more obvious than that of first wear. The peak value of acceleration of slider, contact force of jointA, contact force of joint B, input torque of crank 1, and input torque of crank 4 of no wear are 372.4 m/s2, 2072 N, 2314 N, −52.33 N ∗ m, and −164.3 N ∗ m. The peak value of acceleration of slider, contact force of joint A, contact force of joint B, input torque of crank 1, and input torque of crank 4 of first wear are −606.1 m/s2, 2741 N, 3062 N, −53.38 N ∗ m, and −177.5 N ∗ m. The peak value of acceleration of slider, contact force of joint A, contact force of joint B, input torque of crank 1, and input torque of crank 4 of second wear are 708.7 m/s2, 3885 N, 4489 N, −79.34 N ∗ m, and −420.2 N ∗ m. As displayed in Figures 25 and 26, with the increase of wear times, shaft trajectories of clearance joints become more and more chaotic. It is shown that wear has a greater impact on dynamic response, which will accelerate damage and failure for mechanism. Reasons are that clearance pair wear is nonuniform, and the clearance value is changed larger and nonregularly after clearance joint wear.


## 5.2.3. Influence of Clearance Wear on Nonlinear Dynamic Characteristics

Clearance wear will destroy the revolute joint with clearance, so it has great influence on the nonlinear dynamics of the mechanism. The nonlinear characteristics of the mechanism before wear and after second wear are both studied by using the phase diagram and Poincaré map. The phase diagram and Poincaré map are shown in Figures27 and 28.Figure 27
Phase diagram of (a) jointA in the X direction, (b) joint A in the Y direction, (c) joint B in the X direction, and (d) joint B in the Y direction.
(a)
(b)
(c)
(d)Figure 28
Poincaré map of (a) jointA in the X direction, (b) joint A in the Y direction, (c) joint B in the X direction, and (d) joint B in the Y direction.
(a)
(b)
(c)
(d)It can be seen from the phase diagram and Poincaré map that wear enlarges the area of the phase diagram and makes the range of Poincaré map point diffuse. Moreover, the distribution of mapping points after wear is more confused. The chaotic phenomena at the clearance joint are enhanced by the wear. The reason is that wear makes the surface of shaft and bearing not smooth and increases the impact and vibration between elements of clearance pair.


## 5.3. The Influence of Different Factors on Wear

### 5.3.1. The Effect of Different Initial Constant Clearance Sizes on Wear

The effect of three different initial constant clearance values on wear analysis is investigated in this section. It is assumed that revolute pairA and revolute pair B have the same clearance value. It is supposed that driving speeds of crank 1 ω1=−2πrad/s and crank 4 ω4=2πrad/s. Initial constant clearance values 0.3 mm, 0.5 mm, and 0.8 mm are all selected to study the wear analysis. Wear depths at joint A and joint B are displayed in Figures 29 and 30. The reconstructed surface of bearing A and bearing B after wear is shown in Figures 31 and 32. Local enlarged drawing of the reconstructed surface of bearing A and bearing B is shown in Figures 33 and 34. The figures are drawn on the basis of the data after the second wear in this section.Figure 29
Wear depth at jointA.Figure 30
Wear depth at jointB.Figure 31
Bearing surface of jointA.Figure 32
Local enlarged drawing of bearing surface of jointA.Figure 33
Bearing surface of jointB.Figure 34
Local enlarged drawing of bearing surface of jointB.From Figures29–34, it can be seen that, because of frequent collisions in some areas, wear phenomenon on bearing surface is not uniformly distributed. And the wear of revolute joint B is more serious than revolute joint A. As shown in Figures 29 and 30, the greater the constant clearance before wear is, the greater the wear depth is, the enlarger the wear area is, and the worse the wear of shaft and bearing are. When the initial constant clearance values are 0.3 mm, 0.5 mm, and 0.8 mm, the corresponding wear depths of revolute A are 1.824×10−5m, 1.079×10−5m, and 7.182×10−6m, and the corresponding wear depths of revolute B are 3.881×10−5m, 3.336×10−5m, and 1.492×10−5m, respectively. It can be guessed that proper reduction of the clearance value at the rotating pair is beneficial to the reduction of wear phenomenon. The wear at revolute clearance joints seriously degenerate precision and reliability of the multibody mechanism system.


### 5.3.2. The Effect of Different Driving Speeds on Wear

Influences of three different driving speeds on wear analyses are researched in this section. It is supposed that constant clearance values at two revolute clearance joints before wear are both set as 0.3 mm. Driving speed of crank 1 and crank 4 is chosen asω1=−2πrad/s and ω4=2πrad/s, ω1=−3πrad/s and ω4=3πrad/s, and ω1=−4πrad/s and ω4=4πrad/s. Wear depth at joints A and B is displayed in Figures 35 and 36. The reconstructed surface of bearing A and bearing B after wear is shown in Figures 37 and 38. Local enlarged drawing of the reconstructed surface of bearing A and bearing B is shown in Figures 38 and 39. The figures are also drawn on the basis of the data after the second wear in this section.Figure 35
Wear depth at jointA.Figure 36
Wear depth at jointB.Figure 37
Bearing surface of jointA.Figure 38
Bearing surface of jointB.Figure 39
Local enlarged drawing of bearing surface of jointB.With the increase of driving speed, the clearances at the revolute joints will make the collision of the mechanism more intense, accelerate the damage of mechanism, and reduce life of the mechanism. According to Figures35–40, the wear of revolute joint B is more severe than that of revolute joint A, the greater the driving speed is, the greater the wear depth is, the worse wear on the bearing surface is, and the faster failure of the mechanism is. When the driving speeds are ω1=−2πrad/s and ω4=2πrad/s, ω1=−3πrad/s and ω4=3πrad/s, and ω1=−4πrad/s and ω4=4πrad/s, the corresponding wear depths of revolute A are 9.673×10−5m, 1.237×10−5m, and 7.182×10−6m, respectively. When the clearance values are ω1=−2πrad/s and ω4=2πrad/s, ω1=−3πrad/s and ω4=3πrad/s, and ω1=−4πrad/s and ω4=4πrad/s, the corresponding wear depths of revolute B are 1.465×10−4m, 1.599×10−5m, and 1.492×10−5m, respectively. After the clearance wear, clearance values of revolute joints are increased and not a constant value, which will cause the high-frequency vibration of dynamic responses. The driving speed will directly affect the relative motion state between elements of clearance pair, and it has impact on the wear of clearance pair.Figure 40
Local enlarged drawing of bearing surface of jointA.


## 5.3.1. The Effect of Different Initial Constant Clearance Sizes on Wear

The effect of three different initial constant clearance values on wear analysis is investigated in this section. It is assumed that revolute pairA and revolute pair B have the same clearance value. It is supposed that driving speeds of crank 1 ω1=−2πrad/s and crank 4 ω4=2πrad/s. Initial constant clearance values 0.3 mm, 0.5 mm, and 0.8 mm are all selected to study the wear analysis. Wear depths at joint A and joint B are displayed in Figures 29 and 30. The reconstructed surface of bearing A and bearing B after wear is shown in Figures 31 and 32. Local enlarged drawing of the reconstructed surface of bearing A and bearing B is shown in Figures 33 and 34. The figures are drawn on the basis of the data after the second wear in this section.Figure 29
Wear depth at jointA.Figure 30
Wear depth at jointB.Figure 31
Bearing surface of jointA.Figure 32
Local enlarged drawing of bearing surface of jointA.Figure 33
Bearing surface of jointB.Figure 34
Local enlarged drawing of bearing surface of jointB.From Figures29–34, it can be seen that, because of frequent collisions in some areas, wear phenomenon on bearing surface is not uniformly distributed. And the wear of revolute joint B is more serious than revolute joint A. As shown in Figures 29 and 30, the greater the constant clearance before wear is, the greater the wear depth is, the enlarger the wear area is, and the worse the wear of shaft and bearing are. When the initial constant clearance values are 0.3 mm, 0.5 mm, and 0.8 mm, the corresponding wear depths of revolute A are 1.824×10−5m, 1.079×10−5m, and 7.182×10−6m, and the corresponding wear depths of revolute B are 3.881×10−5m, 3.336×10−5m, and 1.492×10−5m, respectively. It can be guessed that proper reduction of the clearance value at the rotating pair is beneficial to the reduction of wear phenomenon. The wear at revolute clearance joints seriously degenerate precision and reliability of the multibody mechanism system.


## 5.3.2. The Effect of Different Driving Speeds on Wear

Influences of three different driving speeds on wear analyses are researched in this section. It is supposed that constant clearance values at two revolute clearance joints before wear are both set as 0.3 mm. Driving speed of crank 1 and crank 4 is chosen asω1=−2πrad/s and ω4=2πrad/s, ω1=−3πrad/s and ω4=3πrad/s, and ω1=−4πrad/s and ω4=4πrad/s. Wear depth at joints A and B is displayed in Figures 35 and 36. The reconstructed surface of bearing A and bearing B after wear is shown in Figures 37 and 38. Local enlarged drawing of the reconstructed surface of bearing A and bearing B is shown in Figures 38 and 39. The figures are also drawn on the basis of the data after the second wear in this section.Figure 35
Wear depth at jointA.Figure 36
Wear depth at jointB.Figure 37
Bearing surface of jointA.Figure 38
Bearing surface of jointB.Figure 39
Local enlarged drawing of bearing surface of jointB.With the increase of driving speed, the clearances at the revolute joints will make the collision of the mechanism more intense, accelerate the damage of mechanism, and reduce life of the mechanism. According to Figures35–40, the wear of revolute joint B is more severe than that of revolute joint A, the greater the driving speed is, the greater the wear depth is, the worse wear on the bearing surface is, and the faster failure of the mechanism is. When the driving speeds are ω1=−2πrad/s and ω4=2πrad/s, ω1=−3πrad/s and ω4=3πrad/s, and ω1=−4πrad/s and ω4=4πrad/s, the corresponding wear depths of revolute A are 9.673×10−5m, 1.237×10−5m, and 7.182×10−6m, respectively. When the clearance values are ω1=−2πrad/s and ω4=2πrad/s, ω1=−3πrad/s and ω4=3πrad/s, and ω1=−4πrad/s and ω4=4πrad/s, the corresponding wear depths of revolute B are 1.465×10−4m, 1.599×10−5m, and 1.492×10−5m, respectively. After the clearance wear, clearance values of revolute joints are increased and not a constant value, which will cause the high-frequency vibration of dynamic responses. The driving speed will directly affect the relative motion state between elements of clearance pair, and it has impact on the wear of clearance pair.Figure 40
Local enlarged drawing of bearing surface of jointA.


## 6. Conclusions

The planar 2-DOF nine-bar mechanism is utilized as an example to investigate dynamic responses of the mechanical system considering two revolute clearance joints after twice wear. Main contents are as follows:(1)
Dynamic equation of 2-DOF nine bar mechanism considering two revolute clearance joints by combining the L-N model and LuGre model is established.(2)
Based upon Archard’s model, wear depths of two revolute clearance joints are predicted, and the surface of shaft and bearing is reconstructed twice.(3)
By integrating the wear model with the dynamic model, the dynamic responses of 2-DOF nine-bar mechanism considering two revolute clearances wear are both discussed. The nonlinear characteristics of the mechanism after wear are studied by the phase diagram and Poincaré map.(4)
The influences of different initial constant clearance values and driving speed of cranks on wear of two revolute clearance joints are also investigated.



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*Source: 5389732-2019-08-28.xml*