# Dynamics Analysis of Spatial Parallel Mechanism with Irregular Spherical Joint Clearance

Authors: Xiulong Chen; Yuewen Li
Journal: Shock and Vibration
 (2019)

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## Abstract

Clearances caused by machining accuracy and assembly requirements are regular, but they will be irregular due to the wear of the kinematic pairs. At present, there are few studies on wear of space kinematic pairs. In order to grasp the effect of irregular spherical joint clearance after wear on the dynamic response, a method for solving irregular clearance problems based on the Newton–Euler method is proposed, and the dynamic response of 4-UPS-UPU spatial parallel mechanism with irregular spherical joint clearance is investigated. The kinematic model and contact force model of the clearance of the spherical joint are derived. The dynamic model of the mechanism with spherical joint clearance is established by the Newton–Euler method. Based on the Archard model, the three-dimensional dynamic wear model for spherical joint with clearances is developed. The wear depth and wear position of the spherical joint are obtained by the numerical solution. The method of reconstructing the geometric morphology after wear is proposed based on the finite element thought. The solution of the irregular clearance problem is put forward, and the dynamic response of the mechanism after wear is also analyzed. The results show that the dynamic response curves of the mechanism fluctuate around the ideal curves whether before wear or after wear. Compared with the regular clearance before wear, the results of the irregular clearance after wear have a greater impact on acceleration and contact force, and the vibration of the acceleration and contact force curve become more intense than before. Moreover, the displacement, velocity, and acceleration curves of the irregular clearance show some hysteresis than that before wear. Therefore, it can be inferred that the irregular clearance has more adverse effects on the mechanism and aggravates the wear between the elements of the kinematic joint; in addition, the stability and the reliability of the mechanism can be reduced.


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## Body

## 1. Introduction

Due to factors such as manufacturing error and assembly error, there will be clearances between joints [1]. At the same time, friction caused by relative motion between the joint elements with clearance will inevitably lead to joint wear, which is also a major reason to cause the equipment failure [2, 3]. Also, with the increase of equipment use time, mechanical performance degradation caused by wear of joints will gradually show up. Therefore, the study on dynamic responses of mechanism with irregular joint clearance after wear is very important for increasing the reliability, improving product quality, and prolonging the service life of the mechanical system.So far, researchers have done a lot of studies on the dynamic characteristics of the multibody system with regular clearances [4], including revolute joints [5, 6], spherical joints [7–9], cylindrical joints [10, 11], and so on. However, there are few studies on the dynamic performance of multibody systems with irregular clearances. Ma and Qian [12] presented a general procedure for dynamic modeling and simulation when multiple revolute joint clearances are considered. Then, ADAMS is used to confirm the validity of the model. Varedi-Koulaei et al. [13] studied the dynamic characteristics of 3-RRR planar parallel manipulator with six revolute joint clearances by using a continuous contact force model based on the elastic Hertz theory with a dissipative term. Erkaya et al. [14] took a fully articulated 3D slider-crank mechanism as the research object, the effects of spherical joint clearance on partly compliant were studied by a numerical method, and the advantages and disadvantages of flexible connection are analyzed. Marques et al. [15] presented a new formulation to model spatial revolute joints with radial and axial clearances, and the numerical calculation of the planar and spatial crank-slider mechanism is taken as an example. Tian et al. [16] put forward a new approach to model and analyze flexible spatial multibody systems with cylindrical joint clearance that considered the dry friction and lubrication. Two cases are used to verify the validity of the method. Chen et al. [17] established the dynamic model of 4-UPS-RPU parallel mechanism with regular clearance based on the Lagrange method, and the influence of regular clearance on the dynamic response and chaotic characteristics of the system is analyzed.Also, scholars have done a lot of research studies on the irregular clearances after joint wear [18–22], but most of them are focused on revolute joints, and the study on spatial joint, such as spherical joint, is still rare. Pei et al. [23] took the crank-slider mechanism as the research object, and the clearance model of two revolute joints and the dynamic model of the system are established. The wear of the two revolute joints in the multibody system is studied by combining the dynamics and tribology. Lai et al. [24] proposed an effective method for calculating the wear of revolute joint of low-speed planar mechanisms, and the method can accurately predict the wear of joints when the wear depth increments are not big; meanwhile, the method was verified by experiments using planar four-bar mechanisms. Zhu et al. [25] put forward to a nonlinear contact pressure distribution model by combining dynamic analysis with wear calculation and verified it by experiments. This work not only describes the nonlinear relationship between contact pressure and penetration depth, but also avoids the complexity of contact pressure calculation. Sun et al. [26] established a dynamic wear model to predict the wear volume in a mechanism, involving aleatory and epistemic uncertainty. The result shows that when both aleatory and epistemic uncertainties are considered, the wear volume boundary is wider and better than that when only aleatory uncertainty is considered. Shankar and Nithyaprakash [27] studied the effect of different radial clearance values on contact pressure and wear of artificial hip under normal walking conditions with the finite element method. Wang et al. [28] established the dynamic model of spatial four-bar mechanism with spherical joint clearance and carried out the wear prediction. The results are verified by finite element simulation.The main aim of this paper is to propose a method of reconstructing the geometric morphology of the spherical joint after wear and a method for establishing the dynamic equation of the mechanism with irregular spherical joint clearance after wear based on the Newton–Euler method. Also, the influence of irregular spherical joint clearance on the dynamic response of the spatial parallel mechanism is studied. A spatial 4-UPS-UPU parallel mechanism (Figure1) that can be used as a parallel machine tool is taken as the research object in this study. In Section 2, the kinematic model and contact force model of spherical joint clearance are established. In Section 3, the dynamic model of parallel mechanism with clearance is developed and the dynamic equation of the mechanism is derived based on the Newton–Euler method. In Section 4, the wear model of the joint clearance is established and the geometric morphology of the spherical joint is reconstructed based on the finite element thought. A method for solving the dynamic response of irregular clearance problems is proposed. In Section 5, a simulation example is used to reconstruct the surface of the spherical joint, and the dynamic characteristics of irregular clearance after wear are analyzed by comparing with regular clearance.Figure 1
4-UPS-UPU parallel mechanism.


## 2. Kinematic Analysis of Spherical Joints with Clearance and Establishment of Contact Force Model

### 2.1. Kinematic Model of Spherical Joint with Clearance

In Figure2, the fixed coordinate system, the moving coordinate system, and the branch coordinate system are established on the center of the fixed platform, the center of the moving platform, and each branch chain, respectively. The three coordinate systems are recorded as A, B, and Ci, where i=1,2,3,4,5. It is assumed that there is a clearance at the spherical joint between the 3rd driving limb and the moving platform. As shown in Figure 3, S3 is the center of the ball sleeve and b3 is the center of the sphere. n and t represent the normal vector and the tangent vector, respectively.Figure 2
Coordinate systems of 4-UPS-UPU parallel mechanism.Figure 3
Spherical joint clearance with contact.The eccentricity between the spherical elements is as follows:(1)eA=PAb3−PAS3∗=PABO+RBABPb3−PABO∗−RBA∗PBS3,where PABO is the coordinate of the moving platform center in the fixed system when the clearance is not considered, PABO∗ is the coordinate of the moving platform center in the fixed system when the clearance is considered, RBA is the rotation matrix of the moving coordinate system relative to the fixed coordinate system when the clearance is not considered, and RBA∗ is the rotation matrix of the moving coordinate system relative to the fixed coordinate system when the clearance is considered.The scalar of the eccentricity is expressed as(2)e=eATeA=eA1,12+eA2,12+eA3,12.The normal unit vector is written as(3)nAn=eAe.Contact deformation of the two contact bodies when the sphere and the ball sleeve are in contact can be expressed as(4)δ=e−c,where c=RS3−Rb3 is the radius of clearance, RS3 is the radius of the ball sleeve, and Rb3 is the radius of the sphere.According to the geometric relationship between the ball sleeve and the sphere, whether the clearance elements are in contact can be judged as follows:(5)δ<0,no contact,δ=0,start contact or start separation,δ>0,contact and elastic deformation,C3 and D3 are the contact points on the ball sleeve and sphere when the ball sleeve and sphere are in contact. The position coordinate of two points in A can be written as(6)PAc3=PAS3∗+RS3nAn,PAD3=PAb3+Rb3nAn.Then, the velocity of contact points can be expressed as(7)VAC3=VAS3∗+RS3n˙An,VAD3=VAb3+Rb3n˙An,where(8)VS3∗A=ωAB∗×rAS3∗+VABO∗,VAb3=VAS3=ωAB×rAS3+VABO.The relative normal velocityVn and the relative tangential velocity Vt of the joint elements with clearance can be obtained by projecting the relative velocity onto the contact plane and the normal plane, respectively: (9)VAn=VAD3−VAC3TnAnnAn,VAt=VAD3−VAC3−VAn.The scalar of the relative tangential velocity is expressed as(10)Vt=VAtTVAt.


### 2.2. Contact Force Model of Spherical Joint with Clearance

#### 2.2.1. Establishment of Normal Contact Force Model

Lankarani–Nikravesh contact force model has been verified by experimental results [29, 30] and applied in many studies. Also, the damping hysteresis effect and the energy loss due to internal damping during the impact process are considered in L-N model. Therefore, the L-N model is chosen in this study, and the normal contact force can be expressed as(11)Fn=Kδn1+31−ce24δ˙δ˙0,where(12)K=43σS3+σb3RS3Rb3RS3+Rb3,(13)σS3=1−νS32ES3,(14)σb3=1−νb32Eb3.In formulas (11)–(14), n is the power exponent; δ is the collision depth; ce is the coefficient of restitution; δ˙ is the relative permeation velocity, that is, velocity of the center of the ball sleeve relative to the center of the sphere; δ˙0 is the initial collision velocity; K is the stiffness of the material; VS3 and Vb3 are Poisson’s ratio of the ball sleeve and the sphere, respectively; ES3 and Eb3 are the elastic modulus of the ball sleeve and the sphere; and RS3 and Rb3 are the radius of the ball sleeve and the sphere.


#### 2.2.2. Establishment of the Tangential Contact Force Model

The Coulomb friction model is more common in the tangential force model of multibody system dynamics and it can better describe the friction phenomenon in collision. However, the problem of the instability of numerical integration will occur when the value of the tangential velocity approaches zero. The modified Coulomb’s friction model proposed by Ambrósio [31] can avoid that and it is used in this study. The expression of tangential contact force is as follows:(15)Ft=−cfcdFnvtvt,where cf is the sliding friction coefficient and cd is the coefficient of dynamic correction. The expression of cd is written as(16)cd=0,vt<v0,vt−v0v1−v0,v0≤vt≤v1,1,vt>v1,where v0 and v1 represent the limit value of given velocity, respectively. According to [32], the limit velocity is selected as v0 = 0.0001 m/s and v1 = 0.01 m/s.When joint elements are in contact, the contact force can be expressed as(17)Ff=FnAnn+FtAtt.For convenience, the dynamic force from the third driving limb to the moving platform is expressed byfA3∗ that is fA3∗=Ff.


## 2.1. Kinematic Model of Spherical Joint with Clearance

In Figure2, the fixed coordinate system, the moving coordinate system, and the branch coordinate system are established on the center of the fixed platform, the center of the moving platform, and each branch chain, respectively. The three coordinate systems are recorded as A, B, and Ci, where i=1,2,3,4,5. It is assumed that there is a clearance at the spherical joint between the 3rd driving limb and the moving platform. As shown in Figure 3, S3 is the center of the ball sleeve and b3 is the center of the sphere. n and t represent the normal vector and the tangent vector, respectively.Figure 2
Coordinate systems of 4-UPS-UPU parallel mechanism.Figure 3
Spherical joint clearance with contact.The eccentricity between the spherical elements is as follows:(1)eA=PAb3−PAS3∗=PABO+RBABPb3−PABO∗−RBA∗PBS3,where PABO is the coordinate of the moving platform center in the fixed system when the clearance is not considered, PABO∗ is the coordinate of the moving platform center in the fixed system when the clearance is considered, RBA is the rotation matrix of the moving coordinate system relative to the fixed coordinate system when the clearance is not considered, and RBA∗ is the rotation matrix of the moving coordinate system relative to the fixed coordinate system when the clearance is considered.The scalar of the eccentricity is expressed as(2)e=eATeA=eA1,12+eA2,12+eA3,12.The normal unit vector is written as(3)nAn=eAe.Contact deformation of the two contact bodies when the sphere and the ball sleeve are in contact can be expressed as(4)δ=e−c,where c=RS3−Rb3 is the radius of clearance, RS3 is the radius of the ball sleeve, and Rb3 is the radius of the sphere.According to the geometric relationship between the ball sleeve and the sphere, whether the clearance elements are in contact can be judged as follows:(5)δ<0,no contact,δ=0,start contact or start separation,δ>0,contact and elastic deformation,C3 and D3 are the contact points on the ball sleeve and sphere when the ball sleeve and sphere are in contact. The position coordinate of two points in A can be written as(6)PAc3=PAS3∗+RS3nAn,PAD3=PAb3+Rb3nAn.Then, the velocity of contact points can be expressed as(7)VAC3=VAS3∗+RS3n˙An,VAD3=VAb3+Rb3n˙An,where(8)VS3∗A=ωAB∗×rAS3∗+VABO∗,VAb3=VAS3=ωAB×rAS3+VABO.The relative normal velocityVn and the relative tangential velocity Vt of the joint elements with clearance can be obtained by projecting the relative velocity onto the contact plane and the normal plane, respectively: (9)VAn=VAD3−VAC3TnAnnAn,VAt=VAD3−VAC3−VAn.The scalar of the relative tangential velocity is expressed as(10)Vt=VAtTVAt.


## 2.2. Contact Force Model of Spherical Joint with Clearance

### 2.2.1. Establishment of Normal Contact Force Model

Lankarani–Nikravesh contact force model has been verified by experimental results [29, 30] and applied in many studies. Also, the damping hysteresis effect and the energy loss due to internal damping during the impact process are considered in L-N model. Therefore, the L-N model is chosen in this study, and the normal contact force can be expressed as(11)Fn=Kδn1+31−ce24δ˙δ˙0,where(12)K=43σS3+σb3RS3Rb3RS3+Rb3,(13)σS3=1−νS32ES3,(14)σb3=1−νb32Eb3.In formulas (11)–(14), n is the power exponent; δ is the collision depth; ce is the coefficient of restitution; δ˙ is the relative permeation velocity, that is, velocity of the center of the ball sleeve relative to the center of the sphere; δ˙0 is the initial collision velocity; K is the stiffness of the material; VS3 and Vb3 are Poisson’s ratio of the ball sleeve and the sphere, respectively; ES3 and Eb3 are the elastic modulus of the ball sleeve and the sphere; and RS3 and Rb3 are the radius of the ball sleeve and the sphere.


### 2.2.2. Establishment of the Tangential Contact Force Model

The Coulomb friction model is more common in the tangential force model of multibody system dynamics and it can better describe the friction phenomenon in collision. However, the problem of the instability of numerical integration will occur when the value of the tangential velocity approaches zero. The modified Coulomb’s friction model proposed by Ambrósio [31] can avoid that and it is used in this study. The expression of tangential contact force is as follows:(15)Ft=−cfcdFnvtvt,where cf is the sliding friction coefficient and cd is the coefficient of dynamic correction. The expression of cd is written as(16)cd=0,vt<v0,vt−v0v1−v0,v0≤vt≤v1,1,vt>v1,where v0 and v1 represent the limit value of given velocity, respectively. According to [32], the limit velocity is selected as v0 = 0.0001 m/s and v1 = 0.01 m/s.When joint elements are in contact, the contact force can be expressed as(17)Ff=FnAnn+FtAtt.For convenience, the dynamic force from the third driving limb to the moving platform is expressed byfA3∗ that is fA3∗=Ff.


## 2.2.1. Establishment of Normal Contact Force Model

Lankarani–Nikravesh contact force model has been verified by experimental results [29, 30] and applied in many studies. Also, the damping hysteresis effect and the energy loss due to internal damping during the impact process are considered in L-N model. Therefore, the L-N model is chosen in this study, and the normal contact force can be expressed as(11)Fn=Kδn1+31−ce24δ˙δ˙0,where(12)K=43σS3+σb3RS3Rb3RS3+Rb3,(13)σS3=1−νS32ES3,(14)σb3=1−νb32Eb3.In formulas (11)–(14), n is the power exponent; δ is the collision depth; ce is the coefficient of restitution; δ˙ is the relative permeation velocity, that is, velocity of the center of the ball sleeve relative to the center of the sphere; δ˙0 is the initial collision velocity; K is the stiffness of the material; VS3 and Vb3 are Poisson’s ratio of the ball sleeve and the sphere, respectively; ES3 and Eb3 are the elastic modulus of the ball sleeve and the sphere; and RS3 and Rb3 are the radius of the ball sleeve and the sphere.


## 2.2.2. Establishment of the Tangential Contact Force Model

The Coulomb friction model is more common in the tangential force model of multibody system dynamics and it can better describe the friction phenomenon in collision. However, the problem of the instability of numerical integration will occur when the value of the tangential velocity approaches zero. The modified Coulomb’s friction model proposed by Ambrósio [31] can avoid that and it is used in this study. The expression of tangential contact force is as follows:(15)Ft=−cfcdFnvtvt,where cf is the sliding friction coefficient and cd is the coefficient of dynamic correction. The expression of cd is written as(16)cd=0,vt<v0,vt−v0v1−v0,v0≤vt≤v1,1,vt>v1,where v0 and v1 represent the limit value of given velocity, respectively. According to [32], the limit velocity is selected as v0 = 0.0001 m/s and v1 = 0.01 m/s.When joint elements are in contact, the contact force can be expressed as(17)Ff=FnAnn+FtAtt.For convenience, the dynamic force from the third driving limb to the moving platform is expressed byfA3∗ that is fA3∗=Ff.


## 3. Dynamic Modeling of Spatial Parallel Mechanism with Clearance of Spherical Joints

### 3.1. Kinematic Analysis of Parallel Mechanisms with Spherical Clearance

According to the coordinates of the Hooke joint and spherical joint on each branch, the length of the fifth driving limb can be written as(18)li∗=Li∗=PASi∗−PAUi=PASix∗−PAUix2+PASiy∗−PAUiy2+PASiz∗−PAUiz21/2,l3=L3=PAb3−PAU3=PAb3x−PAU3x2+PAb3y−PAU3y2+PAb3z−PAU3z21/2.By using the single-open-chain method, the mathematical model of the two rotation angles of the Hooke joint on the fixed platform with clearance considered can be obtained:(19)ψ1i∗=arctanPASix∗−PAUixcφi+PASiy∗−PAUiysφiPASiz∗−PAUiz,ψ2i∗=arcsinPASix∗−PAUixsφi−PASiy∗−PAUiycφili∗,ψ13=arctanPAb3x−PAU3xcφ3+PAb3y−PAU3ysφ3PAb3z−PAU3z,ψ23=arcsinPAb3x−PAU3xsφ3−PAb3y−PAU3ycφ3l3.The branch chain coordinate systemCi∗ on the first, second, fourth, and fifth driving limb is established. The rotation matrix Ci∗ to A is written as(20)RAi∗φi,ψ1i∗,ψ2i∗=RZ,φiRY,ψ1i∗RX,ψ2i∗.The rotation matrix of the branch chain coordinate systemCi∗ relative to the moving coordinate system B∗ can be expressed as(21)RBi∗=RBA∗−1RAi∗=B1∗,B2∗,B3∗.The angular velocity of the moving platform in the fixed coordinate system can be expressed as(22)ωAB∗=100cosα∗0sinα∗α˙∗β˙∗.The angular acceleration of the moving platform can be expressed as(23)εAB∗=100cosα∗0sinα∗α¨∗β¨∗+000−sinα∗⋅α˙∗0cosα∗⋅α˙∗α˙∗β˙∗.The six-axis velocity of the moving platform in the fixed coordinate system can be expressed as(24)VABO∗ωAB∗=Tω∗P˙ABOx∗P˙ABOy∗P˙ABOz∗α˙∗β˙∗,where(25)Tω∗=100000100000100000100000cosα∗0000sinα∗,VABO∗ is the linear velocity of the center of the moving platform.The velocityVASi∗ of each hinge point Si on the moving platform can be expressed as(26)VASi∗=ωAB∗×rASi∗+VABO∗,VAS3=ωAB×rAS3+VABO.The driving velocity of each driving limb inA can be expressed as(27)VAbi∗=nAi∗l˙i∗+ωAi∗×nAi∗li∗,VAb3=nA3l˙3+ωA3×nA3l3.Then, the linear velocity of each driving limb can be written as(28)l˙i∗=nAi∗⋅ωAB∗×rASi∗+nAi∗⋅VABO∗=nAi∗Tr^ASi∗nAi∗TVABO∗ωAB∗T,l˙3=nAi⋅ωAB×rAS3+nA3⋅VABO=nA3Tr^AS3nA3TVABOωABT.After that, the linear acceleration of the driving limb can be expressed as(29)l¨i∗=nAi∗Tr^ASi∗nAi∗TaABO∗εAB∗T+ωAB∗Tr^ASi∗n^Ai∗ωAB∗+1li∗VABO∗ωAB∗TI−r^ASi∗TI−r^ASi∗VABO∗ωAB∗T−1li∗VABO∗TωAB∗TnAi∗r^ASi∗nAi∗TnAi∗Tr^ASi∗nAi∗TVABO∗ωAB∗T,l¨3=nA3Tr^AS3nA3TaABOεABT+ωABTr^AS3n^A3ωAB+1l3VABOωABTI−r^AS3TI−r^AS3VABOωABT−1l3VABOTωABTnA3−r^AS3nA3TnA3Tr^AS3nA3TVABOωABT,where nAi∗ represents the unit direction vector of each driving limb; rASi∗ is the radius vector of the center of moving platform OB to each spherical joint Si; aABO∗ is the acceleration of the center of the moving platform with spherical joint clearance; r^ASi∗ is the antisymmetric operator of rASi∗; n^Ai∗ is the antisymmetric operator of nAi∗; and I is a three-order unit matrix.The angular velocity of each driving limb is expressed as(30)ωAi∗=nAi∗li∗×ωAB∗×rASi∗+VABO∗=1li∗n^Ai∗−n^Ai∗r^ASi∗VABO∗ωAB∗T,ωA3=nA3l3×ωAB×rAS3+VABO=1l3n^A3−n^A3r^AS3VABOωABT.The angular acceleration of each driving limb is written as(31)εAi∗=1li∗n^Ai∗−n^Ai∗r^ASi∗aABO∗εAB∗T−n^Ai∗ω^AB∗Tr^ASi∗ωAB∗−2li∗2n^Ai∗−n^Ai∗r^ASi∗VABO∗ωAB∗TnAi∗Tr^ASi∗nAi∗TVABO∗ωAB∗T,εA3=1l3n^A3−n^A3r^AS3aABOεABT−n^A3ω^ABTr^AS3ωAB−2l32n^A3−n^A3r^AS3VABOωABTnA3Tr^AS3nA3TVABOωABT.The linear velocity of the center of mass of the oscillating rod in the fixed coordinate system is expressed as(32)VAZUi∗=ωAi∗×nAi∗lUi,VAZU3=ωA3×nA3lUi.The linear acceleration of the center of mass of the oscillating rod in the fixed coordinate system is expressed as(33)aAZUi∗=εAi∗×nAi∗lUi+ωAi∗×ωAi∗×nAi∗lUi,aAZU3=εA3×nA3lUi+ωA3×ωA3×nA3lUi.In the same way, the linear velocity of the center of mass of the telescopic rod in the fixed coordinate system is expressed as(34)VAZSi∗=nAi∗l˙i∗+ωAi∗×nAi∗li∗−lSi,VAZS3=nA3l˙3+ωA3×nA3l3−lSi.The linear acceleration of the center of mass point of the telescopic rod in the fixed coordinate system is expressed as(35)aAZSi∗=nAi∗l¨i∗+εAi∗×nAi∗li∗−lSi+ωAi∗×ωAi∗×nAi∗li∗−lSi+2ωAi∗×nAi∗l˙i∗,aAZS3=nAZS3l¨3+εA3×nA3l3−lSi+ωA3×ωA3×nA3l3−lSi+2ωA3×nA3l˙3.


### 3.2. Force Analysis of Hooke Joint

The structure of Hooke hinge is shown in Figure4, which can bear certain torque, and the force analysis diagram is shown in Figure 5. The torque of the moving platform to the driving limbs MUi∗ can be expressed as(36)MUi∗=00MUiZ∗T.Figure 4
The 3D model of the Hooke joint.Figure 5
The force diagram of the Hooke joint.The torque of the moving platform to the driving limbs transformed from the Hooke joint coordinate system to the moving coordinate system can be expressed as(37)MB1i∗=RotY,ξ∗MUi∗,where ξ∗=β∗−ψ1i∗.In the same way, the torque of the fixed platform to the driving limbsMUi∗′ can be expressed as(38)MUi∗′=00MUiZ∗′T.The torque of the fixed platform to the driving limbs transformed from the Hooke joint coordinate system to the branch chain coordinate system can be expressed as(39)Mi2i∗=RotX,ψ2i∗−1MUi∗′.


### 3.3. The Establishment of Dynamic Equation of Limbs

Supposing the vectorfii∗=fiix∗fiiy∗fiiz∗T is the force of each driving limb acting on the moving platform in Ci∗ when i = 1, 2, 4, 5. gA=00gT is the gravitational acceleration in A.Resultant momentMiCi of each driving limb relative to hinge points on the fixed platform in the branch chain coordinate system can be expressed as(40)τi+mSigRAi∗3,3−fiiz∗=mSiV˙iZSiz∗.For the first, second, fourth, and fifth limbs, according to the Euler equation(41)Mi1i∗+Mi2i∗+li∗nii∗×−fii∗+mUilUi+mSili∗−lSinii∗×gi∗=mUilUinii∗×V˙iZUi∗+mSili∗−lSinii∗×V˙iZSi∗+IiUiω˙ii∗+ω˙ii∗×IiUiωii∗+IiSiω˙ii∗+ωii∗×IiSiωii∗,where Mi1i∗ is the torque of the moving platform to each driving limb in Ci∗. When i = 1, Mi1i∗=Mi1ix∗,Mi1iy∗,Mi1iz∗T, and when i = 2, 4, 5, Mi1i∗=0,0,0T. Mi2i∗ is the torque of the fixed platform to each driving limb in Ci∗. When i = 1, Mi2i∗=Mi2ix∗,Mi2iy∗,Mi2iz∗T, and when i = 2, 4, 5, Mi2i∗=0,0,0T. V˙iZUi∗ and V˙iZSi∗ are the linear acceleration of the center of mass of the oscillating rod and telescopic rod, and ωii∗ and ω˙ii∗ are the angular velocity and the angular acceleration of the oscillating rod, respectively.For the first, second, fourth, and fifth limbs, according to the Euler equation(42)M313∗+M323∗+RA3−1PAD3−PAU3×RA3−1−fA3+mUilUi+mSil3−lSin33×g3=mUilUin33×V˙3ZU3+mSil3−lSin33×V˙3ZS3+IiUiω˙33+ω33×IiUiω33+IiSiω˙33+ω33×IiSiω33,where M313∗=M313x∗M313y∗M313z∗T, M323∗=000T and fA3∗ is the force of the 3rd driving limb acting on the moving platform. V˙iZU3∗ and V˙iZS3∗ are the linear acceleration of the center of mass of the oscillating rod and telescopic rod, and ωi3∗ and ω˙i3∗ are the angular velocity and the angular acceleration of the oscillating rod, respectively.


### 3.4. Dynamic Equation of the Moving Platform with Clearance

The force balance equation of the moving platform is derived according to Newton’s second law, and its expression in the fixed coordinate systemA is written as(43)mBgA+∑i=15fAi∗+FAB=mBV˙ABO∗,where fAi∗ is the force of each chain to the moving platform in A. When i=1,2,4,5, its expression is fAi∗=RAi∗fii∗, and when i = 3, its expression is fA3∗=Ff.According to the Euler equation, the moment equilibrium equation of the centroid of the moving platform is derived in the moving coordinate systemB∗, and the expression is written as(44)∑i=1,2,4,5rBSi∗×fBi∗+PBC3×fB3∗−MB11∗+MBF−NBB∗=0,where NBB∗=IBBω˙BB∗+ωBB∗×IBBωBB∗ and rBSi∗ is the position vector of the center of the moving platform to the center of the spherical joint in B∗. fBi∗ is the force of each limb to the moving platform in B∗. When i=1,2,4,5, its expression is fBi∗=RAi∗fii∗, and when i=3, its expression is fB3∗=RBA∗−1fA3∗.MB11∗ is the torque of the moving platform to each driving limb in B∗.According to (40), (41), (43), and (44), a more simplified expression can be obtained as(45)P¨ABOx∗=f1t,PABOx∗,PABOy∗,PABOz∗,α∗,β∗,P˙ABOx∗,P˙ABOy∗,P˙ABOz∗,α˙∗,β˙∗,P¨ABOy∗=f2t,PABOx∗,PABOy∗,PABOz∗,α∗,β∗,P˙ABOx∗,P˙ABOy∗,P˙ABOz∗,α˙∗,β˙∗,P¨ABOz∗=f3t,PABOx∗,PABOy∗,PABOz∗,α∗,β∗,P˙ABOx∗,P˙ABOy∗,P˙ABOz∗,α˙∗,β˙∗,α¨=f4t,PABOx∗,PABOy∗,PABOz∗,α∗,β∗,P˙ABOx∗,P˙ABOy∗,P˙ABOz∗,α˙∗,β˙∗,β¨=f5t,PABOx∗,PABOy∗,PABOz∗,α∗,β∗,P˙ABOx∗,P˙ABOy∗,P˙ABOz∗,α˙∗,β˙∗.Because the differential equations have strong nonlinearity, which makes its analytical solutions hard to deduce, so the numerical method is adopted to solve the above equation. The ODE113 which uses the variable order multistep Adams–Bashforth–Moulton PECE algorithm is used in MATLAB to solve it.


## 3.1. Kinematic Analysis of Parallel Mechanisms with Spherical Clearance

According to the coordinates of the Hooke joint and spherical joint on each branch, the length of the fifth driving limb can be written as(18)li∗=Li∗=PASi∗−PAUi=PASix∗−PAUix2+PASiy∗−PAUiy2+PASiz∗−PAUiz21/2,l3=L3=PAb3−PAU3=PAb3x−PAU3x2+PAb3y−PAU3y2+PAb3z−PAU3z21/2.By using the single-open-chain method, the mathematical model of the two rotation angles of the Hooke joint on the fixed platform with clearance considered can be obtained:(19)ψ1i∗=arctanPASix∗−PAUixcφi+PASiy∗−PAUiysφiPASiz∗−PAUiz,ψ2i∗=arcsinPASix∗−PAUixsφi−PASiy∗−PAUiycφili∗,ψ13=arctanPAb3x−PAU3xcφ3+PAb3y−PAU3ysφ3PAb3z−PAU3z,ψ23=arcsinPAb3x−PAU3xsφ3−PAb3y−PAU3ycφ3l3.The branch chain coordinate systemCi∗ on the first, second, fourth, and fifth driving limb is established. The rotation matrix Ci∗ to A is written as(20)RAi∗φi,ψ1i∗,ψ2i∗=RZ,φiRY,ψ1i∗RX,ψ2i∗.The rotation matrix of the branch chain coordinate systemCi∗ relative to the moving coordinate system B∗ can be expressed as(21)RBi∗=RBA∗−1RAi∗=B1∗,B2∗,B3∗.The angular velocity of the moving platform in the fixed coordinate system can be expressed as(22)ωAB∗=100cosα∗0sinα∗α˙∗β˙∗.The angular acceleration of the moving platform can be expressed as(23)εAB∗=100cosα∗0sinα∗α¨∗β¨∗+000−sinα∗⋅α˙∗0cosα∗⋅α˙∗α˙∗β˙∗.The six-axis velocity of the moving platform in the fixed coordinate system can be expressed as(24)VABO∗ωAB∗=Tω∗P˙ABOx∗P˙ABOy∗P˙ABOz∗α˙∗β˙∗,where(25)Tω∗=100000100000100000100000cosα∗0000sinα∗,VABO∗ is the linear velocity of the center of the moving platform.The velocityVASi∗ of each hinge point Si on the moving platform can be expressed as(26)VASi∗=ωAB∗×rASi∗+VABO∗,VAS3=ωAB×rAS3+VABO.The driving velocity of each driving limb inA can be expressed as(27)VAbi∗=nAi∗l˙i∗+ωAi∗×nAi∗li∗,VAb3=nA3l˙3+ωA3×nA3l3.Then, the linear velocity of each driving limb can be written as(28)l˙i∗=nAi∗⋅ωAB∗×rASi∗+nAi∗⋅VABO∗=nAi∗Tr^ASi∗nAi∗TVABO∗ωAB∗T,l˙3=nAi⋅ωAB×rAS3+nA3⋅VABO=nA3Tr^AS3nA3TVABOωABT.After that, the linear acceleration of the driving limb can be expressed as(29)l¨i∗=nAi∗Tr^ASi∗nAi∗TaABO∗εAB∗T+ωAB∗Tr^ASi∗n^Ai∗ωAB∗+1li∗VABO∗ωAB∗TI−r^ASi∗TI−r^ASi∗VABO∗ωAB∗T−1li∗VABO∗TωAB∗TnAi∗r^ASi∗nAi∗TnAi∗Tr^ASi∗nAi∗TVABO∗ωAB∗T,l¨3=nA3Tr^AS3nA3TaABOεABT+ωABTr^AS3n^A3ωAB+1l3VABOωABTI−r^AS3TI−r^AS3VABOωABT−1l3VABOTωABTnA3−r^AS3nA3TnA3Tr^AS3nA3TVABOωABT,where nAi∗ represents the unit direction vector of each driving limb; rASi∗ is the radius vector of the center of moving platform OB to each spherical joint Si; aABO∗ is the acceleration of the center of the moving platform with spherical joint clearance; r^ASi∗ is the antisymmetric operator of rASi∗; n^Ai∗ is the antisymmetric operator of nAi∗; and I is a three-order unit matrix.The angular velocity of each driving limb is expressed as(30)ωAi∗=nAi∗li∗×ωAB∗×rASi∗+VABO∗=1li∗n^Ai∗−n^Ai∗r^ASi∗VABO∗ωAB∗T,ωA3=nA3l3×ωAB×rAS3+VABO=1l3n^A3−n^A3r^AS3VABOωABT.The angular acceleration of each driving limb is written as(31)εAi∗=1li∗n^Ai∗−n^Ai∗r^ASi∗aABO∗εAB∗T−n^Ai∗ω^AB∗Tr^ASi∗ωAB∗−2li∗2n^Ai∗−n^Ai∗r^ASi∗VABO∗ωAB∗TnAi∗Tr^ASi∗nAi∗TVABO∗ωAB∗T,εA3=1l3n^A3−n^A3r^AS3aABOεABT−n^A3ω^ABTr^AS3ωAB−2l32n^A3−n^A3r^AS3VABOωABTnA3Tr^AS3nA3TVABOωABT.The linear velocity of the center of mass of the oscillating rod in the fixed coordinate system is expressed as(32)VAZUi∗=ωAi∗×nAi∗lUi,VAZU3=ωA3×nA3lUi.The linear acceleration of the center of mass of the oscillating rod in the fixed coordinate system is expressed as(33)aAZUi∗=εAi∗×nAi∗lUi+ωAi∗×ωAi∗×nAi∗lUi,aAZU3=εA3×nA3lUi+ωA3×ωA3×nA3lUi.In the same way, the linear velocity of the center of mass of the telescopic rod in the fixed coordinate system is expressed as(34)VAZSi∗=nAi∗l˙i∗+ωAi∗×nAi∗li∗−lSi,VAZS3=nA3l˙3+ωA3×nA3l3−lSi.The linear acceleration of the center of mass point of the telescopic rod in the fixed coordinate system is expressed as(35)aAZSi∗=nAi∗l¨i∗+εAi∗×nAi∗li∗−lSi+ωAi∗×ωAi∗×nAi∗li∗−lSi+2ωAi∗×nAi∗l˙i∗,aAZS3=nAZS3l¨3+εA3×nA3l3−lSi+ωA3×ωA3×nA3l3−lSi+2ωA3×nA3l˙3.


## 3.2. Force Analysis of Hooke Joint

The structure of Hooke hinge is shown in Figure4, which can bear certain torque, and the force analysis diagram is shown in Figure 5. The torque of the moving platform to the driving limbs MUi∗ can be expressed as(36)MUi∗=00MUiZ∗T.Figure 4
The 3D model of the Hooke joint.Figure 5
The force diagram of the Hooke joint.The torque of the moving platform to the driving limbs transformed from the Hooke joint coordinate system to the moving coordinate system can be expressed as(37)MB1i∗=RotY,ξ∗MUi∗,where ξ∗=β∗−ψ1i∗.In the same way, the torque of the fixed platform to the driving limbsMUi∗′ can be expressed as(38)MUi∗′=00MUiZ∗′T.The torque of the fixed platform to the driving limbs transformed from the Hooke joint coordinate system to the branch chain coordinate system can be expressed as(39)Mi2i∗=RotX,ψ2i∗−1MUi∗′.


## 3.3. The Establishment of Dynamic Equation of Limbs

Supposing the vectorfii∗=fiix∗fiiy∗fiiz∗T is the force of each driving limb acting on the moving platform in Ci∗ when i = 1, 2, 4, 5. gA=00gT is the gravitational acceleration in A.Resultant momentMiCi of each driving limb relative to hinge points on the fixed platform in the branch chain coordinate system can be expressed as(40)τi+mSigRAi∗3,3−fiiz∗=mSiV˙iZSiz∗.For the first, second, fourth, and fifth limbs, according to the Euler equation(41)Mi1i∗+Mi2i∗+li∗nii∗×−fii∗+mUilUi+mSili∗−lSinii∗×gi∗=mUilUinii∗×V˙iZUi∗+mSili∗−lSinii∗×V˙iZSi∗+IiUiω˙ii∗+ω˙ii∗×IiUiωii∗+IiSiω˙ii∗+ωii∗×IiSiωii∗,where Mi1i∗ is the torque of the moving platform to each driving limb in Ci∗. When i = 1, Mi1i∗=Mi1ix∗,Mi1iy∗,Mi1iz∗T, and when i = 2, 4, 5, Mi1i∗=0,0,0T. Mi2i∗ is the torque of the fixed platform to each driving limb in Ci∗. When i = 1, Mi2i∗=Mi2ix∗,Mi2iy∗,Mi2iz∗T, and when i = 2, 4, 5, Mi2i∗=0,0,0T. V˙iZUi∗ and V˙iZSi∗ are the linear acceleration of the center of mass of the oscillating rod and telescopic rod, and ωii∗ and ω˙ii∗ are the angular velocity and the angular acceleration of the oscillating rod, respectively.For the first, second, fourth, and fifth limbs, according to the Euler equation(42)M313∗+M323∗+RA3−1PAD3−PAU3×RA3−1−fA3+mUilUi+mSil3−lSin33×g3=mUilUin33×V˙3ZU3+mSil3−lSin33×V˙3ZS3+IiUiω˙33+ω33×IiUiω33+IiSiω˙33+ω33×IiSiω33,where M313∗=M313x∗M313y∗M313z∗T, M323∗=000T and fA3∗ is the force of the 3rd driving limb acting on the moving platform. V˙iZU3∗ and V˙iZS3∗ are the linear acceleration of the center of mass of the oscillating rod and telescopic rod, and ωi3∗ and ω˙i3∗ are the angular velocity and the angular acceleration of the oscillating rod, respectively.


## 3.4. Dynamic Equation of the Moving Platform with Clearance

The force balance equation of the moving platform is derived according to Newton’s second law, and its expression in the fixed coordinate systemA is written as(43)mBgA+∑i=15fAi∗+FAB=mBV˙ABO∗,where fAi∗ is the force of each chain to the moving platform in A. When i=1,2,4,5, its expression is fAi∗=RAi∗fii∗, and when i = 3, its expression is fA3∗=Ff.According to the Euler equation, the moment equilibrium equation of the centroid of the moving platform is derived in the moving coordinate systemB∗, and the expression is written as(44)∑i=1,2,4,5rBSi∗×fBi∗+PBC3×fB3∗−MB11∗+MBF−NBB∗=0,where NBB∗=IBBω˙BB∗+ωBB∗×IBBωBB∗ and rBSi∗ is the position vector of the center of the moving platform to the center of the spherical joint in B∗. fBi∗ is the force of each limb to the moving platform in B∗. When i=1,2,4,5, its expression is fBi∗=RAi∗fii∗, and when i=3, its expression is fB3∗=RBA∗−1fA3∗.MB11∗ is the torque of the moving platform to each driving limb in B∗.According to (40), (41), (43), and (44), a more simplified expression can be obtained as(45)P¨ABOx∗=f1t,PABOx∗,PABOy∗,PABOz∗,α∗,β∗,P˙ABOx∗,P˙ABOy∗,P˙ABOz∗,α˙∗,β˙∗,P¨ABOy∗=f2t,PABOx∗,PABOy∗,PABOz∗,α∗,β∗,P˙ABOx∗,P˙ABOy∗,P˙ABOz∗,α˙∗,β˙∗,P¨ABOz∗=f3t,PABOx∗,PABOy∗,PABOz∗,α∗,β∗,P˙ABOx∗,P˙ABOy∗,P˙ABOz∗,α˙∗,β˙∗,α¨=f4t,PABOx∗,PABOy∗,PABOz∗,α∗,β∗,P˙ABOx∗,P˙ABOy∗,P˙ABOz∗,α˙∗,β˙∗,β¨=f5t,PABOx∗,PABOy∗,PABOz∗,α∗,β∗,P˙ABOx∗,P˙ABOy∗,P˙ABOz∗,α˙∗,β˙∗.Because the differential equations have strong nonlinearity, which makes its analytical solutions hard to deduce, so the numerical method is adopted to solve the above equation. The ODE113 which uses the variable order multistep Adams–Bashforth–Moulton PECE algorithm is used in MATLAB to solve it.


## 4. Geometric Reconstruction of Spherical Joint and Dynamics Analysis with Irregular Clearance

Usually, the clearance generated by processing error, manufacturing accuracy, and assembly requirements is regular. In the process of equipment operation, the elements of the motion pair will wear each other. The wear depth is not uniform: some parts are serious, some are lighter, and some do not wear. Therefore, the clearance after wear is irregular. The following are all studies on irregular spherical joint clearance after wear.


### 4.1. Wear Model

At present, there are two kinds of wear models in tribology: Archard wear model and Reye’s hypothesis wear model [33]. In this paper, the former is used to calculate the wear depth. The Archard model takes many parameters into account such as contact material, sliding speed, material hardness, and wear volume. The reliability of that was verified through a series of experiments. The results are more accurate and have been approved by most scholars and widely used. Its expression can be written as(46)Vs=KFnH,where V is the wear volume, K is the dimensionless wear coefficient, Fn is the normal contact force, H is the Brinell hardness of soft materials, and s is the relative slip distance.In practical engineering applications, since the wear depth is more advantageous to engineering applications than the wear volume, the Archard wear model can be written as [33](47)hs=kdP,where kd is the linear wear coefficient kd=k/H, h is the wear depth, and P is the normal contact stress.According to the relationship between tangential velocity and slip distance, the form can also be written as(48)dh=kdPvtdt.


### 4.2. Approximate Contact Area of the Spherical Joint with Clearance

Figure6(a) depicts the contact form between the spherical joint elements with clearance. The approximate contact area of the spherical joint with clearance is derived according to the geometric relation of the contact body. The contact area is divided into N equal-height parts and each part is approximated as a cylinder. The sum of each cylinder’s lateral area is the approximate contact area between the spherical joint elements. In Figures 6(b) and 6(c), PS3 and Pb3 are the center of the ball sleeve and the center of the sphere, λ is the contact angle, a is the contact radius, h is the contact height, H is the position of maximum contact deformation, the height of each cylinder is Δy, and the radius is Δx.Figure 6
Contact form and geometric relationship of the spherical joint with clearance.
(a)
(b)
(c)According to the cosine theorem, the contact angle can be expressed as(49)λ=arccose2+Rb32−Rs322eRb3.The contact radius can be expressed as(50)a=Rb3sinπ−λ.Contact height can be expressed as(51)h=Rb31−cosπ−λ.Therefore, when the contact occurs between the elements of spherical joints, the approximate contact area is as follows:(52)A=2π∫0hdy∫0adx=2πah.


### 4.3. Analysis and Calculation of Wear Depth

Because the position and slip distance of the contact point change with the motion of the mechanism, the continuous contact process must be discretized in order to achieve a stable wear state. According to the Archard model, the expression of the wear depth can be expressed as(53)hi=kdpivtΔti,where hi is the wear depth at each time point, pi is the contact stress at each time point, and Δti is the time steps.Contact stress can be expressed as(54)pi=FNAnAn.In order to predict the three-dimensional wear volume of spherical joints with clearance, the wear depth is decomposed in three directions:(55)hxi=kdpxivtFnxΔti,hyi=kdpyivtFnyΔti,hzi=kdpzivtFnzΔti,where pxi, pyi, and pzi are components of normal contact stress in three directions and vtFnx, vtFny, and vtFnz are the tangential velocities corresponding to the component of the normal contact force in three directions.According to the preceding analysis, the relative contact velocity between the sphere and the ball sleeve can be obtained:(56)VAt=VAD3−VAC3−VAn=vtx,vty,vtzT.Corresponding relative contact velocity of normal contact force in X direction is(57)vtFnx=vty2+vtz2.In the same manner, corresponding relative contact velocities of normal contact force in Y and Z directions are as follows:(58)vtFny=vtx2+vtz2,vtFnz=vtx2+vty2.It is assumed that both the sphere and the ball sleeve will wear since the material of the ball sleeve and the sphere is the same:(59)RSxi=Rxi+hxi,RSyi=Ryi+hyi,RSzi=Rzi+hzi,Rbxi=Rxi−hxi,Rbyi=Ryi−hyi,Rbzi=Rzi−hzi.In summary, the wear depth component in three directions can be solved, and the analysis process is shown in Figure7.Figure 7
Wear analysis process.


### 4.4. Geometric Reconstruction and Wear Dynamic Solution

The wear depth between the clearance elements and the change of the clearance value are very small when the mechanism runs few cycles. If the surfaces of the sphere and the ball sleeve are reconstructed after each contact, the calculation cost will be greatly increased and the efficiency will be low. Therefore, it can be assumed that the clearance surface morphology does not change with time in a certain period of operation [34].According to the finite element thought, the geometric surface of the spherical joint is discretized. In each integral step, the contact between the sphere and the sleeve is judged. When the sphere and the ball sleeve are in contact, the position (discrete area) and wear depth of the collision between the sphere and the ball sleeve are calculated. At the end of the simulation, the total wear depth of each region can be obtained by accumulating the wear depth of each region. By using this method, the geometric morphology of the spherical joint clearance after wear can be obtained.According to the direction of longitude and latitude, the sphere is divided into several areas, forming a dense spherical mesh (the accuracy of calculation will be higher with the increase of partition number, but the amount of calculation will increase exponentially). The coordinates of the spherical mesh nodes are saved as a matrix to determine in which region the wear depth occurs. Then, the wear depth of the region is added or subtracted. Finally, the geometric shape of the clearance of the spherical joint is obtained, and the new grid node coordinates are preserved and the clearance radius of different regions is calculated.In order to study the effect of the spherical joint after wear on the dynamic characteristics of parallel mechanisms, that is, the effect of the irregular clearance on dynamics, a nonlinear dynamic variable stiffness coefficient is introduced:(60)Kij=43δS3+δb3RS3ijRijb3RS3ij+Rb3ij,i,j=1,2,3,…,360.In this case, the radius of the ball sleeve and the sphere is not constant. After discretization, different regions have different clearance radius, so different regions correspond to different stiffness coefficients. When solving the dynamics of the parallel mechanism, the first step is to determine which discrete region the collision will occur in. Then, calculate the stiffness coefficient of the region by taking the clearance radius in the region. Finally, the dynamic equation of clearance is solved. The dynamic solution process after wear is shown in Figure8.Figure 8
Dynamic solution process after wear.


## 4.1. Wear Model

At present, there are two kinds of wear models in tribology: Archard wear model and Reye’s hypothesis wear model [33]. In this paper, the former is used to calculate the wear depth. The Archard model takes many parameters into account such as contact material, sliding speed, material hardness, and wear volume. The reliability of that was verified through a series of experiments. The results are more accurate and have been approved by most scholars and widely used. Its expression can be written as(46)Vs=KFnH,where V is the wear volume, K is the dimensionless wear coefficient, Fn is the normal contact force, H is the Brinell hardness of soft materials, and s is the relative slip distance.In practical engineering applications, since the wear depth is more advantageous to engineering applications than the wear volume, the Archard wear model can be written as [33](47)hs=kdP,where kd is the linear wear coefficient kd=k/H, h is the wear depth, and P is the normal contact stress.According to the relationship between tangential velocity and slip distance, the form can also be written as(48)dh=kdPvtdt.


## 4.2. Approximate Contact Area of the Spherical Joint with Clearance

Figure6(a) depicts the contact form between the spherical joint elements with clearance. The approximate contact area of the spherical joint with clearance is derived according to the geometric relation of the contact body. The contact area is divided into N equal-height parts and each part is approximated as a cylinder. The sum of each cylinder’s lateral area is the approximate contact area between the spherical joint elements. In Figures 6(b) and 6(c), PS3 and Pb3 are the center of the ball sleeve and the center of the sphere, λ is the contact angle, a is the contact radius, h is the contact height, H is the position of maximum contact deformation, the height of each cylinder is Δy, and the radius is Δx.Figure 6
Contact form and geometric relationship of the spherical joint with clearance.
(a)
(b)
(c)According to the cosine theorem, the contact angle can be expressed as(49)λ=arccose2+Rb32−Rs322eRb3.The contact radius can be expressed as(50)a=Rb3sinπ−λ.Contact height can be expressed as(51)h=Rb31−cosπ−λ.Therefore, when the contact occurs between the elements of spherical joints, the approximate contact area is as follows:(52)A=2π∫0hdy∫0adx=2πah.


## 4.3. Analysis and Calculation of Wear Depth

Because the position and slip distance of the contact point change with the motion of the mechanism, the continuous contact process must be discretized in order to achieve a stable wear state. According to the Archard model, the expression of the wear depth can be expressed as(53)hi=kdpivtΔti,where hi is the wear depth at each time point, pi is the contact stress at each time point, and Δti is the time steps.Contact stress can be expressed as(54)pi=FNAnAn.In order to predict the three-dimensional wear volume of spherical joints with clearance, the wear depth is decomposed in three directions:(55)hxi=kdpxivtFnxΔti,hyi=kdpyivtFnyΔti,hzi=kdpzivtFnzΔti,where pxi, pyi, and pzi are components of normal contact stress in three directions and vtFnx, vtFny, and vtFnz are the tangential velocities corresponding to the component of the normal contact force in three directions.According to the preceding analysis, the relative contact velocity between the sphere and the ball sleeve can be obtained:(56)VAt=VAD3−VAC3−VAn=vtx,vty,vtzT.Corresponding relative contact velocity of normal contact force in X direction is(57)vtFnx=vty2+vtz2.In the same manner, corresponding relative contact velocities of normal contact force in Y and Z directions are as follows:(58)vtFny=vtx2+vtz2,vtFnz=vtx2+vty2.It is assumed that both the sphere and the ball sleeve will wear since the material of the ball sleeve and the sphere is the same:(59)RSxi=Rxi+hxi,RSyi=Ryi+hyi,RSzi=Rzi+hzi,Rbxi=Rxi−hxi,Rbyi=Ryi−hyi,Rbzi=Rzi−hzi.In summary, the wear depth component in three directions can be solved, and the analysis process is shown in Figure7.Figure 7
Wear analysis process.


## 4.4. Geometric Reconstruction and Wear Dynamic Solution

The wear depth between the clearance elements and the change of the clearance value are very small when the mechanism runs few cycles. If the surfaces of the sphere and the ball sleeve are reconstructed after each contact, the calculation cost will be greatly increased and the efficiency will be low. Therefore, it can be assumed that the clearance surface morphology does not change with time in a certain period of operation [34].According to the finite element thought, the geometric surface of the spherical joint is discretized. In each integral step, the contact between the sphere and the sleeve is judged. When the sphere and the ball sleeve are in contact, the position (discrete area) and wear depth of the collision between the sphere and the ball sleeve are calculated. At the end of the simulation, the total wear depth of each region can be obtained by accumulating the wear depth of each region. By using this method, the geometric morphology of the spherical joint clearance after wear can be obtained.According to the direction of longitude and latitude, the sphere is divided into several areas, forming a dense spherical mesh (the accuracy of calculation will be higher with the increase of partition number, but the amount of calculation will increase exponentially). The coordinates of the spherical mesh nodes are saved as a matrix to determine in which region the wear depth occurs. Then, the wear depth of the region is added or subtracted. Finally, the geometric shape of the clearance of the spherical joint is obtained, and the new grid node coordinates are preserved and the clearance radius of different regions is calculated.In order to study the effect of the spherical joint after wear on the dynamic characteristics of parallel mechanisms, that is, the effect of the irregular clearance on dynamics, a nonlinear dynamic variable stiffness coefficient is introduced:(60)Kij=43δS3+δb3RS3ijRijb3RS3ij+Rb3ij,i,j=1,2,3,…,360.In this case, the radius of the ball sleeve and the sphere is not constant. After discretization, different regions have different clearance radius, so different regions correspond to different stiffness coefficients. When solving the dynamics of the parallel mechanism, the first step is to determine which discrete region the collision will occur in. Then, calculate the stiffness coefficient of the region by taking the clearance radius in the region. Finally, the dynamic equation of clearance is solved. The dynamic solution process after wear is shown in Figure8.Figure 8
Dynamic solution process after wear.


## 5. Case Study of Spatial Parallel Mechanism with Clearance

### 5.1. Parameter Setting of 4-UPS-UPU Parallel Mechanism with Spherical Joint Clearance

The basic structural parameters and the hinge point coordinates of UPS-UPU parallel mechanism are shown in Tables1 and 2.Table 1
System parameters.
Design parameters
Value
Mass of the moving platform,mB (kg)
37.48
Mass of the oscillating rod,mUi (kg)
25.96
Mass of the telescopic rod,mSi (kg)
8.45
Length of the oscillating rod,l1i (m)
0.76
Length of the telescopic rod,l2i (m)
0.88
Length from the oscillating rod of the center of mass to the hinge point of the fixed platform,lUi (m)
0.38
Length from the telescopic rod of the center of mass to the hinge point of the moving platform,lSi (m)
0.45
Radius of the sphere spherical joint with clearance,Rb3 (m)
0.015
Power exponent of the metal surface,n
1.5
Coefficient of restitution,ce
0.9
Linear wear coefficient
8
×
10
−
14
Pa
−
1
Moment of inertia of the moving platform,IBB (kg·m2)
0.963
0
0
0
0.674
0
0
0
0.651
Moment of inertia of the oscillating rod,IiUi (kg·m2)
1.254
0
0
0
1.254
0
0
0
0.033
Moment of inertia of the telescopic rod,IiSi (kg·m2)
0.782
0
0
0
0.782
0
0
0
0.0019
Elastic modulus of the sphere and the ball sleeve,ES3,Eb3 (GPa)
200
Poisson’s ratio of the sphere and the ball sleeve,VS3,Vb3
0.3
The sliding friction coefficient,μd
0.05
Simulation step size
0.001 sTable 2
The coordinates of joints on the fixed platform and moving platform (unit: m).
The coordinates of the Hooke joint on the fixed coordinate system
The coordinates of the Hooke joint and spherical joint in the moving coordinate system
P
A
U
1
=
−
0.71
0
0
T
P
A
U
1
=
−
0.2284
0
−
0.0591
T
P
A
U
2
=
−
0.4596
−
0.4596
0
T
P
A
U
2
=
−
0.624
−
0.1921
0
T
P
A
U
3
=
0.4596
−
0.4596
0
T
P
A
U
3
=
0.1634
−
0.1187
0
T
P
A
U
4
=
0.4596
0.4596
0
T
P
A
U
4
=
0.1634
0.1187
0
T
P
A
U
5
=
−
0.4596
0.4596
0
T
P
A
U
5
=
−
0.624
0.1921
0
T


### 5.2. Geometry Reconstruction of Spherical Joint after Wear

The smaller the discrete area is, the higher the calculation accuracy is, but the amount of calculation will increase exponentially. In order to ensure the accuracy, the sphere is divided into 360 zones according to the direction of longitude and latitude to form a dense spherical mesh. The coordinates of spherical mesh nodes are saved as a matrix of 360 ∗ 360 to determine in which region the wear depth occurs, and then, the corresponding coordinates of the region are added or subtracted. Finally, the geometric shape of the clearance of the spherical joint is obtained, and the new mesh node coordinates are preserved and the clearance radius of different regions is calculated. Because the wear between joint elements is very small in a short time, the numerical simulation of tens of thousands and more cycles is unrealistic. Therefore, the simulation is carried out in 100 cycles, and the wear condition of the 100 cycles is scaled up 10000 times to approximate the wear condition of 106 cycles. Based on the above, the wear dynamics of the mechanism is analyzed. The parallel mechanism is not loaded at present, and the simulation trajectory is shown as follows:(61)X=0.10sinπt,Y=−0.10cosπt,Z=0.8+0.002t,α=0,β=0.The wear depth of each time step is related to the selected size of the simulation step. The smaller the step is, the smaller the wear depth in a time step is. The geometry of the ball sleeve can be obtained by adding the wear depth in each time step to the coordinates of the spherical mesh nodes according to the position relationship, as shown in Figure9(a). In order to observe the wear effect on the surface of the ball sleeve easily, the figure is obtained by accumulating the wear depth on the surface of a sphere with a radius of 1.5 × 10−6 m (that is, the wear depth is enlarged 10000 times). Figure 9(b) is an enlarged view of the geometric morphology of the wear area. According to the local enlarged view, it can be judged that some areas of the ball sleeve will be concave due to wear. Figure 9(c) shows more directly the wear position of the spherical sleeve. The wear mainly occurs in the lower right area of the spherical joint. In order to better describe the wear depth quantificationally, the surface is expanded to a two-dimensional plane. Figure 9(d) is the wear depth of the ball sleeve, and the wear depth can reach 7.2507 × 10−7 m. Figure 9(d) is the wear area of the expanded ball sleeve. It can be clearly seen that the wear area presents irregular shape, and the yellow area is the most serious wear, followed by the light blue area. The wear area is mainly concentrated in longitude 100∼200 degrees and latitude 0∼170 degrees.Figure 9
Wear condition of the spherical joint with clearance: (a) Geometric reconstruction surface, (b) enlarged view of the geometric reconstruction surface, (c) wear region, (d) wear depth, and (e) two-dimensional expansion diagram of wear surface.
(a)
(b)
(c)
(d)
(e)


### 5.3. Dynamic Response Analysis of Mechanism with Irregular Clearance

The displacement curves of the moving platform of parallel mechanism are shown in Figure10. It can be seen from the graph that the displacement and angular displacement curves of the moving platform basically coincide with the ideal displacement curves when the clearance is regular or irregular. The difference between the curves can be seen only in enlarged view, which shows that the influence of clearance on the displacement curve and angular displacement curve is very small, and the difference between regular clearance and irregular clearance is also very small. At the same time, the displacement and angular displacement after wear are slightly hysteresis than before wear.Figure 10
Comparison of displacement curves before and after wear. (a) Displacement curve inX direction, (b) enlarged view of displacement curve in X direction, (c) displacement curve in Y direction, (d) enlarged view of displacement curve in Y direction, (e) displacement curve in Z direction, (f) enlarged view of displacement curve in Z direction, (g) angular displacement curve in α direction, (h) enlarged view of angular displacement curve in α direction, (i) angular displacement curve in β direction, and (j) enlarged view of angular displacement curve in β direction.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)The velocity curve and angular velocity curve before and after wear are shown in Figure11. It can be seen from the diagram that regular clearance and irregular clearance have a greater influence on velocity and angular velocity of the mechanism. From Figures 11(a)–11(d), the velocity curve fluctuates around the ideal curve in these two cases in the X and Y directions. The influence of regular and irregular clearance on velocity is similar. In X direction, the maximum velocity before wear is 0.323 m/s (7.993s) and the minimum velocity is −0.3186 m/s (6.987s). The maximum velocity after wear is 0.3231 m/s (7.995s), and the minimum velocity is −0.3189 m/s (6.991s). In the Y direction, the maximum velocity before wear is 0.3335 m/s (6.514s) and the minimum velocity is −0.3252 m/s (7.503s). The maximum velocity after wear is 0.3333 m/s (6.516s), and the minimum velocity is −0.3248 m/s (7.506s). From Figures 11(e)–11(j), in the Z, α, and β directions, compared with the curve before wear, the curve after wear has local high-frequency vibration on the basis of the original fluctuation, but the trend is closer to the curves before wear. In Z direction, the maximum velocity before wear is 0.01171 m/s (6.536s) and the minimum velocity is −0.008329 m/s (8.558s). The maximum velocity after wear is 0.01183 m/s (6.538s), and the minimum velocity is −0.008368 m/s (8.561s). In α direction, the maximum angular velocity before wear is 0.05921 rad/s (6.542s) and the minimum angular velocity is −0.05606 m/s (8.496s). The maximum angular velocity after wear is 0.05873 rad/s (6.544s), and the minimum angular velocity is −0.05557 m/s (8.499s). In β direction, the maximum angular velocity before wear is 0.07063 rad/s (6.573s) and the minimum angular velocity is −0.07672 m/s (6.536s). The maximum angular velocity after wear is 0.07091 rad/s (6.575s), and the minimum angular velocity is −0.07757 m/s (6.538s). To sum up, it can be seen that there is little difference in the numerical value before and after wear, but the velocity and angular velocity curves after wear show a certain hysteresis than that before wear.Figure 11
Comparison of velocity curves before and after wear. (a) Velocity curve inX direction, (b) enlarged view of velocity curve in X direction, (c) velocity curve in Y direction, (d) enlarged view of velocity curve in Y direction, (e) velocity curve in Z direction, (f) enlarged view of velocity curve in Z direction, (g) angular velocity curve in α direction, (h) enlarged view of angular velocity curve in α direction, (i) angular velocity curve in β direction, and (j) enlarged view of angular velocity curve in β direction.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)The acceleration and angular acceleration curves before and after wear are shown in Figure12. It is easy to see from the curves that the influence of regular clearance and irregular clearance on acceleration is greater than velocity and displacement. At the same time, from Figures 12(a)–11(h), in X and Y directions, the fluctuation of curves after wear is slightly larger than that before wear. It can be seen from the enlarged view that the curves after wear are more unstable than those before wear, and the acceleration curves after wear also show a certain hysteresis compared with the curves before wear. The influence of irregular clearance on acceleration after wear is greater than that of regular clearance, especially in Z, α, and β directions. From Figures 12(e)–12(j), in Z direction, the maximum value of vibration before wear is 1.918 m/s2 (6.582s) and the minimum value is −0.667 m/s2 (6.491s). The maximum value of vibration after wear is 2.002 m/s2 (6.529s), and the minimum value is −1.149 m/s2 (7.225s). In α direction, the maximum value of vibration before wear is 6.388 rad/s2 (6.467s) and the minimum value is −6.705 rad/s2 (8.485s). The maximum value of vibration after wear is 8.187 rad/s2 (6.416s), and the minimum value is −8.908 m/s2 (9.596 s). In the direction, the maximum value of vibration before wear is 5.194 rad/s2 (6.491s), and the minimum value is −14.86 rad/s1 (6.582s). The maximum value of vibration after wear is 8.789 rad/s2 (7.225s), and the minimum value is −15.55 m/s2 (6.529s). Compared with the velocity curve, the peak value of the acceleration curve before and after wear varies greatly. At the same time, the curves after wear also show a certain hysteresis compared with that before wear. Also, it can be seen from the partial enlarged view that the irregular clearance makes the acceleration curve more jitter. The reason is that the surface of the sphere and the ball sleeve is nonuniform wear, and the surface curvature of the contact point of the spherical joint changes at all times during the operation. The curvature radius becomes larger in severe wear area, and the contact force between the clearance elements on the spherical surface varies more severely, which leads to the larger jitter of acceleration. Therefore, the dynamic effect of irregular wear on the mechanism should be considered during the operation of the mechanism. Otherwise, the prediction of dynamic characteristics will gradually become inaccurate with the increase of wear.Figure 12
Comparison of acceleration curves before and after wear. (a) Acceleration curve inX direction, (b) enlarged view of acceleration curve in X direction, (c) acceleration curve in Y direction, (d) enlarged view of acceleration curve in Y direction, (e) acceleration curve in Z direction, (f) enlarged view of acceleration curve in Z direction, (g) angular acceleration curve in α direction, (h) enlarged view of angular acceleration curve in α direction, (i) angular acceleration curve in β direction, and (j) enlarged view of angular acceleration curve in β direction.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)The contact force curves of the clearance between the spherical joints before and after wear are shown in Figure13. It can be seen from the graph that the irregular clearance after wear makes the collision force change more intensely, which shows similar characteristics to the acceleration, but the basic trend is consistent. The maximum value of contact force is 533.5 N (8.122s) before wear, and the minimum value is 117.2 N (7.04s). After wear, the fluctuation range of contact force becomes larger. The maximum value reaches 569.8 N (8.124s) and the minimum value reaches 90.05 N (704.3s). It can be concluded that the worn spherical joint directly affects the curvature radius of the contact point, thus affecting the contact force and ultimately leading to changes in the dynamic response.Figure 13
Comparison of contact force curve before and after wear: (a) contact force curve and (b) enlarged view of the contact force curve.
(a)
(b)In summary, it can be seen from the influence of irregular clearance on the dynamic characteristics of parallel mechanism that the irregular clearance after wear will make the acceleration and contact force curve more intense. In addition, the more intense jitter of the curves will aggravate the wear of the spherical joint. The two processes promote each other, and the wear will be more serious. Eventually, the vibration of the mechanism will be aggravated, and the stability and accuracy of the mechanism will be reduced. Therefore, it is necessary to consider the adverse effects of wear on the mechanism when analyzing the dynamic characteristics of the mechanism. Otherwise, it will not be able to accurately predict the dynamic performance of the equipment after long-time use.


## 5.1. Parameter Setting of 4-UPS-UPU Parallel Mechanism with Spherical Joint Clearance

The basic structural parameters and the hinge point coordinates of UPS-UPU parallel mechanism are shown in Tables1 and 2.Table 1
System parameters.
Design parameters
Value
Mass of the moving platform,mB (kg)
37.48
Mass of the oscillating rod,mUi (kg)
25.96
Mass of the telescopic rod,mSi (kg)
8.45
Length of the oscillating rod,l1i (m)
0.76
Length of the telescopic rod,l2i (m)
0.88
Length from the oscillating rod of the center of mass to the hinge point of the fixed platform,lUi (m)
0.38
Length from the telescopic rod of the center of mass to the hinge point of the moving platform,lSi (m)
0.45
Radius of the sphere spherical joint with clearance,Rb3 (m)
0.015
Power exponent of the metal surface,n
1.5
Coefficient of restitution,ce
0.9
Linear wear coefficient
8
×
10
−
14
Pa
−
1
Moment of inertia of the moving platform,IBB (kg·m2)
0.963
0
0
0
0.674
0
0
0
0.651
Moment of inertia of the oscillating rod,IiUi (kg·m2)
1.254
0
0
0
1.254
0
0
0
0.033
Moment of inertia of the telescopic rod,IiSi (kg·m2)
0.782
0
0
0
0.782
0
0
0
0.0019
Elastic modulus of the sphere and the ball sleeve,ES3,Eb3 (GPa)
200
Poisson’s ratio of the sphere and the ball sleeve,VS3,Vb3
0.3
The sliding friction coefficient,μd
0.05
Simulation step size
0.001 sTable 2
The coordinates of joints on the fixed platform and moving platform (unit: m).
The coordinates of the Hooke joint on the fixed coordinate system
The coordinates of the Hooke joint and spherical joint in the moving coordinate system
P
A
U
1
=
−
0.71
0
0
T
P
A
U
1
=
−
0.2284
0
−
0.0591
T
P
A
U
2
=
−
0.4596
−
0.4596
0
T
P
A
U
2
=
−
0.624
−
0.1921
0
T
P
A
U
3
=
0.4596
−
0.4596
0
T
P
A
U
3
=
0.1634
−
0.1187
0
T
P
A
U
4
=
0.4596
0.4596
0
T
P
A
U
4
=
0.1634
0.1187
0
T
P
A
U
5
=
−
0.4596
0.4596
0
T
P
A
U
5
=
−
0.624
0.1921
0
T


## 5.2. Geometry Reconstruction of Spherical Joint after Wear

The smaller the discrete area is, the higher the calculation accuracy is, but the amount of calculation will increase exponentially. In order to ensure the accuracy, the sphere is divided into 360 zones according to the direction of longitude and latitude to form a dense spherical mesh. The coordinates of spherical mesh nodes are saved as a matrix of 360 ∗ 360 to determine in which region the wear depth occurs, and then, the corresponding coordinates of the region are added or subtracted. Finally, the geometric shape of the clearance of the spherical joint is obtained, and the new mesh node coordinates are preserved and the clearance radius of different regions is calculated. Because the wear between joint elements is very small in a short time, the numerical simulation of tens of thousands and more cycles is unrealistic. Therefore, the simulation is carried out in 100 cycles, and the wear condition of the 100 cycles is scaled up 10000 times to approximate the wear condition of 106 cycles. Based on the above, the wear dynamics of the mechanism is analyzed. The parallel mechanism is not loaded at present, and the simulation trajectory is shown as follows:(61)X=0.10sinπt,Y=−0.10cosπt,Z=0.8+0.002t,α=0,β=0.The wear depth of each time step is related to the selected size of the simulation step. The smaller the step is, the smaller the wear depth in a time step is. The geometry of the ball sleeve can be obtained by adding the wear depth in each time step to the coordinates of the spherical mesh nodes according to the position relationship, as shown in Figure9(a). In order to observe the wear effect on the surface of the ball sleeve easily, the figure is obtained by accumulating the wear depth on the surface of a sphere with a radius of 1.5 × 10−6 m (that is, the wear depth is enlarged 10000 times). Figure 9(b) is an enlarged view of the geometric morphology of the wear area. According to the local enlarged view, it can be judged that some areas of the ball sleeve will be concave due to wear. Figure 9(c) shows more directly the wear position of the spherical sleeve. The wear mainly occurs in the lower right area of the spherical joint. In order to better describe the wear depth quantificationally, the surface is expanded to a two-dimensional plane. Figure 9(d) is the wear depth of the ball sleeve, and the wear depth can reach 7.2507 × 10−7 m. Figure 9(d) is the wear area of the expanded ball sleeve. It can be clearly seen that the wear area presents irregular shape, and the yellow area is the most serious wear, followed by the light blue area. The wear area is mainly concentrated in longitude 100∼200 degrees and latitude 0∼170 degrees.Figure 9
Wear condition of the spherical joint with clearance: (a) Geometric reconstruction surface, (b) enlarged view of the geometric reconstruction surface, (c) wear region, (d) wear depth, and (e) two-dimensional expansion diagram of wear surface.
(a)
(b)
(c)
(d)
(e)


## 5.3. Dynamic Response Analysis of Mechanism with Irregular Clearance

The displacement curves of the moving platform of parallel mechanism are shown in Figure10. It can be seen from the graph that the displacement and angular displacement curves of the moving platform basically coincide with the ideal displacement curves when the clearance is regular or irregular. The difference between the curves can be seen only in enlarged view, which shows that the influence of clearance on the displacement curve and angular displacement curve is very small, and the difference between regular clearance and irregular clearance is also very small. At the same time, the displacement and angular displacement after wear are slightly hysteresis than before wear.Figure 10
Comparison of displacement curves before and after wear. (a) Displacement curve inX direction, (b) enlarged view of displacement curve in X direction, (c) displacement curve in Y direction, (d) enlarged view of displacement curve in Y direction, (e) displacement curve in Z direction, (f) enlarged view of displacement curve in Z direction, (g) angular displacement curve in α direction, (h) enlarged view of angular displacement curve in α direction, (i) angular displacement curve in β direction, and (j) enlarged view of angular displacement curve in β direction.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)The velocity curve and angular velocity curve before and after wear are shown in Figure11. It can be seen from the diagram that regular clearance and irregular clearance have a greater influence on velocity and angular velocity of the mechanism. From Figures 11(a)–11(d), the velocity curve fluctuates around the ideal curve in these two cases in the X and Y directions. The influence of regular and irregular clearance on velocity is similar. In X direction, the maximum velocity before wear is 0.323 m/s (7.993s) and the minimum velocity is −0.3186 m/s (6.987s). The maximum velocity after wear is 0.3231 m/s (7.995s), and the minimum velocity is −0.3189 m/s (6.991s). In the Y direction, the maximum velocity before wear is 0.3335 m/s (6.514s) and the minimum velocity is −0.3252 m/s (7.503s). The maximum velocity after wear is 0.3333 m/s (6.516s), and the minimum velocity is −0.3248 m/s (7.506s). From Figures 11(e)–11(j), in the Z, α, and β directions, compared with the curve before wear, the curve after wear has local high-frequency vibration on the basis of the original fluctuation, but the trend is closer to the curves before wear. In Z direction, the maximum velocity before wear is 0.01171 m/s (6.536s) and the minimum velocity is −0.008329 m/s (8.558s). The maximum velocity after wear is 0.01183 m/s (6.538s), and the minimum velocity is −0.008368 m/s (8.561s). In α direction, the maximum angular velocity before wear is 0.05921 rad/s (6.542s) and the minimum angular velocity is −0.05606 m/s (8.496s). The maximum angular velocity after wear is 0.05873 rad/s (6.544s), and the minimum angular velocity is −0.05557 m/s (8.499s). In β direction, the maximum angular velocity before wear is 0.07063 rad/s (6.573s) and the minimum angular velocity is −0.07672 m/s (6.536s). The maximum angular velocity after wear is 0.07091 rad/s (6.575s), and the minimum angular velocity is −0.07757 m/s (6.538s). To sum up, it can be seen that there is little difference in the numerical value before and after wear, but the velocity and angular velocity curves after wear show a certain hysteresis than that before wear.Figure 11
Comparison of velocity curves before and after wear. (a) Velocity curve inX direction, (b) enlarged view of velocity curve in X direction, (c) velocity curve in Y direction, (d) enlarged view of velocity curve in Y direction, (e) velocity curve in Z direction, (f) enlarged view of velocity curve in Z direction, (g) angular velocity curve in α direction, (h) enlarged view of angular velocity curve in α direction, (i) angular velocity curve in β direction, and (j) enlarged view of angular velocity curve in β direction.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)The acceleration and angular acceleration curves before and after wear are shown in Figure12. It is easy to see from the curves that the influence of regular clearance and irregular clearance on acceleration is greater than velocity and displacement. At the same time, from Figures 12(a)–11(h), in X and Y directions, the fluctuation of curves after wear is slightly larger than that before wear. It can be seen from the enlarged view that the curves after wear are more unstable than those before wear, and the acceleration curves after wear also show a certain hysteresis compared with the curves before wear. The influence of irregular clearance on acceleration after wear is greater than that of regular clearance, especially in Z, α, and β directions. From Figures 12(e)–12(j), in Z direction, the maximum value of vibration before wear is 1.918 m/s2 (6.582s) and the minimum value is −0.667 m/s2 (6.491s). The maximum value of vibration after wear is 2.002 m/s2 (6.529s), and the minimum value is −1.149 m/s2 (7.225s). In α direction, the maximum value of vibration before wear is 6.388 rad/s2 (6.467s) and the minimum value is −6.705 rad/s2 (8.485s). The maximum value of vibration after wear is 8.187 rad/s2 (6.416s), and the minimum value is −8.908 m/s2 (9.596 s). In the direction, the maximum value of vibration before wear is 5.194 rad/s2 (6.491s), and the minimum value is −14.86 rad/s1 (6.582s). The maximum value of vibration after wear is 8.789 rad/s2 (7.225s), and the minimum value is −15.55 m/s2 (6.529s). Compared with the velocity curve, the peak value of the acceleration curve before and after wear varies greatly. At the same time, the curves after wear also show a certain hysteresis compared with that before wear. Also, it can be seen from the partial enlarged view that the irregular clearance makes the acceleration curve more jitter. The reason is that the surface of the sphere and the ball sleeve is nonuniform wear, and the surface curvature of the contact point of the spherical joint changes at all times during the operation. The curvature radius becomes larger in severe wear area, and the contact force between the clearance elements on the spherical surface varies more severely, which leads to the larger jitter of acceleration. Therefore, the dynamic effect of irregular wear on the mechanism should be considered during the operation of the mechanism. Otherwise, the prediction of dynamic characteristics will gradually become inaccurate with the increase of wear.Figure 12
Comparison of acceleration curves before and after wear. (a) Acceleration curve inX direction, (b) enlarged view of acceleration curve in X direction, (c) acceleration curve in Y direction, (d) enlarged view of acceleration curve in Y direction, (e) acceleration curve in Z direction, (f) enlarged view of acceleration curve in Z direction, (g) angular acceleration curve in α direction, (h) enlarged view of angular acceleration curve in α direction, (i) angular acceleration curve in β direction, and (j) enlarged view of angular acceleration curve in β direction.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)The contact force curves of the clearance between the spherical joints before and after wear are shown in Figure13. It can be seen from the graph that the irregular clearance after wear makes the collision force change more intensely, which shows similar characteristics to the acceleration, but the basic trend is consistent. The maximum value of contact force is 533.5 N (8.122s) before wear, and the minimum value is 117.2 N (7.04s). After wear, the fluctuation range of contact force becomes larger. The maximum value reaches 569.8 N (8.124s) and the minimum value reaches 90.05 N (704.3s). It can be concluded that the worn spherical joint directly affects the curvature radius of the contact point, thus affecting the contact force and ultimately leading to changes in the dynamic response.Figure 13
Comparison of contact force curve before and after wear: (a) contact force curve and (b) enlarged view of the contact force curve.
(a)
(b)In summary, it can be seen from the influence of irregular clearance on the dynamic characteristics of parallel mechanism that the irregular clearance after wear will make the acceleration and contact force curve more intense. In addition, the more intense jitter of the curves will aggravate the wear of the spherical joint. The two processes promote each other, and the wear will be more serious. Eventually, the vibration of the mechanism will be aggravated, and the stability and accuracy of the mechanism will be reduced. Therefore, it is necessary to consider the adverse effects of wear on the mechanism when analyzing the dynamic characteristics of the mechanism. Otherwise, it will not be able to accurately predict the dynamic performance of the equipment after long-time use.


## 6. Conclusion

In this paper, the dynamic response of 4-UPS-UPU parallel mechanism with irregular spherical joint clearance is studied. The main conclusions are as follows:(i)
The dynamic model of 4-UPS-UPU parallel mechanism with spherical joint clearance is established based on the Newton–Euler method. The wear model of clearance joint is derived based on the Archard model, and the wear depth is solved by tangential contact velocity.(ii)
A method of reconstructing the geometric morphology of the spherical joint with clearance is proposed based on the finite element thought. The geometric morphology of the spherical joint with clearance is reconstructed. The wear area and depth of the spherical joint are visually and accurately displayed, and the wear condition is predicted.(iii)
A method for analyzing dynamic responses of the mechanism with irregular clearance after wear is proposed, and the dynamics problem of irregular clearance is solved. The results show that the dynamic response curves of parallel mechanism with regular clearance and after wear with irregular clearance both fluctuate around ideal curves. Compared with regular clearance, irregular clearance after wear has the least effect on displacement and the greatest effect on acceleration. At the same time, the displacement, velocity, and acceleration curves with irregular clearance show some hysteresis compared with the regular clearance. In addition, irregular clearance can cause acceleration and contact force to produce more intense jitter and cause adverse effects on the mechanism. Compared with previous studies, the conclusions of this paper are basically similar to those in other literature studies.



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*Source: 6242971-2019-09-22.xml*