Dynamic Responses of Planar Multilink Mechanism considering Mixed Clearances

Translational and revolute joints are the main kinds of joints in planar multilink mechanisms. Translational and revolute clearance joints have great influence on dynamical responses of planar mechanisms. Most research studies mainly focused upon revolute clearance of planar mechanisms based upon the modified Coulomb friction model, some studies investigated clearance of the pinslot joint, and few studies researched mixed clearances (considering both translational clearance and revolute clearance) based on the LuGre frictionmodel. Dynamic response of the 2-DOF nine-bar mechanism consideringmixed clearances based on the LuGre model is investigated in this work. *e dynamic model with mixed clearances is built by the Lagrange multipliers. Dynamic responses including motion output of the slider, driving torques, contact forces, shaft center trajectories at revolute clearance pairs, and slider trajectory inside the guide are analyzed, respectively. Influences of different frictionmodels on dynamic responses are studied, such as LuGre and modified Coulomb’s friction models. Effects of different clearance values and different driving speeds on dynamic responses with mixed clearances are both analyzed. Virtual prototype model considering mixed clearances is carried out through ADAMS to verify correctness.


Introduction
Planar multilink mechanisms are widely applied in engineering, such as mechanical press, agricultural equipment, medical apparatus and instruments, and precision instrument [1][2][3][4]. Translational and revolute joints are important parts to connect each component of planar multilink mechanisms. Translational and revolute clearances will cause impact between components, which will lead to a deviation between the expected output and the real output, and it also degrades the accuracy and stability [5][6][7][8][9]. Dynamic characteristics of multilink mechanisms become worse. So, it is important to study the effect of mixed clearances on dynamic responses of planar multilink mechanisms.
In the recent decades, many scholars have studied the dynamic behavior of mechanical systems containing revolute pair clearances. Most of the researches used the modified Coulomb friction model to build friction force at revolute clearance joints [10][11][12][13][14][15][16][17][18][19][20][21][22]. Wang et al. [10] put forward a nonpenetration method for the planar rigid slidercrank mechanism considering revolute clearance. Zhang et al. [11] analyzed dynamic responses of the planar parallel mechanism considering six revolute clearance pairs. Varedi et al. [12] put forward an optimization model for a slidercrank mechanism to optimize mass distribution of the linkages to reduce contact forces at revolute clearances. Chen et al. [13] discussed the effect of various factors on dynamical response of the crank-slider mechanism through ADAMS. Chen et al. [14] analyzed the dynamics of a hydraulic rock-breaker mechanism containing revolute clearances. Li-Xin et al. [15] analyzed effects of revolute clearances on the dynamical behavior of a planar 2-DOF parallel manipulator. Hou et al. [16] developed a dynamical equation of the RU-RPR mechanism containing single clearance. Li et al. [17] investigated influence of the harmonic drive on dynamics of the mechanism considering clearances. Erkaya and Dogan [18] constituted a pseudorigid body model of compliant crank-slider mechanism containing revolute clearance. Wang et al. [19] analyzed nonlinear dynamics of the flexible crank-slider mechanism considering clearance. Farahan et al. [20,21] studied nonlinear dynamic behavior of simple mechanism considering revolute clearance by using Poincaré portraits and the bifurcation diagram. Flores et al. [22] discussed the effect of various contact force models on dynamics of a simple mechanism containing single clearance.
Some scholars studied dynamic responses of planar simple mechanisms with revolute clearances by using the LuGre model to build friction, such as slider-crank mechanism. Studies on dynamical responses of planar multilink mechanisms containing multiple revolute clearances by combining the LuGre friction model are few [23][24][25][26]. Muvengei et al. [23] analyzed dynamical responses of the crank-slider mechanism with clearances considering stickslip friction. Jiang et al. [24] studied on the wear characteristics of the multilink mechanism with multiple revolute pair clearances. Marques et al. [25] examined and compared 21 different friction force models on dynamics of the crankslider mechanism with clearance. Muvengei et al. [26] have carried out detailed analysis to the movement state of the clearance of two rotation pairs in a crank-slider mechanism.
In the past few years, some research studies were conducted to study the effect of translational clearance and pinslot clearance joint on dynamic responses of mechanical systems, and the existing literatures mainly considered the crank-slider mechanism as the research object. Studies on dynamic responses of planar multilink mechanisms containing mixed clearances by the LuGre model have not been found in the existing literature. Skrinjar et al. [27,28] discussed the influence of pin-slot clearance joint on kinematic and dynamic characteristics. Flores et al. [29] put forward a methodology for rigid planar mechanism containing translational clearance joints. Several possible motion cases between the slider and guide were both analyzed. Zhuang and Wang [30] presented an approach for analyzing and modeling the rigid multibody mechanism considering translation clearance and driving constraints. Zhang and Wang [31] put forward an FEM for the multibody system containing a translational clearance. Flores et al. [32] presented a method based upon the nonsmooth dynamical approach to build and analyze the dynamic model for the rigid crank-slider mechanism considering translational clearance. Wang et al. [33] analyzed the impact of revolute clearance and translational clearance on dynamics of crankslider mechanism. Zhan et al. [34] proposed the motion reliability analysis methodology of planar parallel manipulators containing revolute and prismatic joint clearances.
Previous studies investigated the effects of translational or revolute clearances on dynamic behavior independently, rather than considering effects of these two kinds of clearances on the dynamics of planar multilink mechanisms simultaneously. Effects of mixed clearances on dynamical responses of planar multilink mechanisms by the LuGre model are rarely investigated, and previous studies mainly use the modified Coulomb friction model as the friction force model. erefore, this paper studied the dynamical responses of 2-DOF nine-bar mechanism considering mixed clearances based on the LuGre model. Dynamics model containing mixed clearances was developed by Lagrange multipliers. Dynamic response, such as motion output of the slider, driving torques, contact forces, shaft center trajectories at revolute clearance pairs, and slider trajectory inside the guide, is obtained. Influences of the different friction models, clearance values, and driving speeds on dynamic responses with mixed clearances are analyzed. Virtual prototype model containing mixed clearances has been modeled through ADAMS to verify correctness.

Kinematic Model.
Translational clearance model is shown in Figure 1. Figures 1(a) and 1(b) are the front view and side view of the slider. Length and width of the slider are L and W, total distance between guide surfaces is H, and slider thickness is T. a is half of the perimeter of the rectangular contact side surface of the slider, a � T + L [29,33]; c is the clearance value, and its expression is written as When the slider moves inside the guide, four position cases between the slider and the guide are displayed, as in Figure 2. e translational clearance model is displayed in Figure 3  Set P, Q, R, and T as the geometric constraint points on the guide that may make contact with the slider. A m , B m , C m , and D m the corners of the slider, A n , B n , C n , and D n are the points on the guide that are close to the slider's corner. Because the location equation of each corner on the slider is similar to each other, only the mathematical model of point A is described.
Position vector of any point H on component k in the global coordinate system is where r k and A k are, respectively, the position vector and the transformation matrix of component k and s H k is the position vector of point H.
Vector that connects slider corner A m to the point A n on the guide surface is Normal vector n of the guide surface is where t x and t y are the components of the tangential vector in the direction of X and Y. When the slider and guide are pressed into each other, vector δ is parallel to normal vector n of the guide surface, but the direction is opposite.

2
Shock and Vibration erefore, the penetration condition for the contact is Magnitude of penetration depth can be given by  Figure 4(a). Contact force could be expressed as [5,29] F n � Kδ, where stiffness parameter K could be written as [29]. where a is half of the perimeter of the rectangular contact surface of the slider, σ l � (1 − ] 2 l /E l )(l � m, n), E l is the elastic modulus, and ] l is Poisson's ratio.
As shown in Figure 4(b), when one or two opposite slider corners contact with the guide surface, it is assumed that the contact is between a spherical surface and a plane, and the contact force model can be given by where K is the generalized stiffness parameter, K � (4/3(σ m + σ n )) ��� � R mn , and R mn is a small curvature radius, assumed on the contact corner [5,29]. e power exponent n depends on material and geometric properties of the contact area [20]. D is the damping coefficient, and it has high accuracy for describing the contact process of clearance elements [35,36], and its expression is defined by where c e means the coefficient of restitution and _ δ is the initial contact velocity.

Tangential Force
Model. Superimposing Coulomb, viscous, stiction, and Stribeck models, we could get a complete static friction model that could be used to build stick-slip motions. However, discontinuity of force at zero speed for static friction models poses some numerical challenges and does not model real friction phenomenon. erefore, some dynamic friction models were researched, such as Dahl friction model and Karnopp friction model. But, in these dynamic friction models, normal force was not considered to result from contact-impact forces arising from influence of clearance and friction models are strongly coupled with the rest of the equations of motion of the system [23,37]. LuGre law was proposed by the Canudas de Wit et al. e LuGre model uses the microscopic average bristle deflection of the contacting surfaces as the internal state. In this model, friction is visualized as forces produced by bending bristles. e LuGre model could capture variation of friction force with slip velocity, thus making it suitable for studies involving stick-slip motions. Besides, the LuGre method could be observed to capture the Stribeck effect, which is a phenomenon related to the stick-slip friction [25,38]. However, it is very difficult to determine the parameters, and it needs to take a smaller step to solve, so the calculation efficiency is low. Based on classical definition of friction, friction force can be written as [26] F t � μF n .
Instantaneous coefficient of friction μ could be given by [4,23] where ξ 0 ξ 1 , ξ 2 , and v are, respectively, the bristle stiffness, microscopic damping coefficient, viscous friction coefficient, and tangential velocity. Evolution differential equation for average bristle deflection [4,6,23] is written as where μ k , μ s , c, and v s are, respectively, kinetic friction coefficient, static friction coefficient, gradient of friction decay, and Stribeck velocity. Substituting equation (13) into equation (12), we can get Choice of z is based on following assumptions [4].
Because simulations at steady-state condition are needed, z was supposed to be constant for a particular value of relative tangential speed of the members. erefore, in the steady state, the first derivative of z is equal to zero, that is, equation (13) � 0, and z can be obtained at steady state as follows [4,23]: Force and moment acting on the guide n are

Kinematic Model.
Revolute clearance model is displayed in Figure 5. Component i and component j are the bearing and the shaft. Eccentricity vector could be given by Penetration depth could be given by where c is the clearance size, c � R i − R j , in which R i and R j are the radii of the bearing and the shaft. Criteria for judging whether collision occurs between elements of revolute pair with clearance are as follows [15,39]:

Normal Force Model
. L-N model is more suitable for general mechanical impact with a high coefficient of restitution, and it is also in well conformity with experimental results. It not only involves energy loss in impact process, but also considers material properties, local elastic deformation, and collision velocity of the collision body; the L-N contact force model is applied [18,36,[40][41][42], and it could be expressed by equation (9), where K is the stiffness parameter, , ] k is Poisson's ratio, E k is the elastic modulus, and D is the damping coefficient, which can also be expressed by equation (10).

Tangential Force Model.
Friction force might well appear in the revolute clearance joints in the multibody system. e LuGre model is also used in revolute clearance joints to study clearance influence on dynamic response. e friction force acting on the revolute clearance can also be expressed by equations (11)-(15).

Force Analysis of Revolute Clearance.
Force and moment acting on the bearing and the shaft can be written by equations (16)-(19).

Establishment of Dynamics Model of 2-DOF
Nine-Bar Mechanism with Mixed Clearances e 2-DOF nine-bar mechanism is assembled of frame, cranks 1 and 4, links 2, 3, 6, and 8, triangular member 7, and slider 9. Link 5 is frame [43]. Crank 1 and link 2 are linked by a revolute clearance pair A. Crank 4 and link 3 are linked by a revolute clearance pair B. Translational clearance is directly related with the slider, which is denoted by C. erefore, clearances at A, B, and C can better reflect impact of clearances on dynamic response. Two cranks are driven by motors. Geometry of 2-DOF nine-bar mechanism with mixed clearances is illustrated in Figure 6.
Because the nine-bar mechanism has good quick return characteristics and the speed of the slider at the bottom dead center is low and stable, the nine-bar mechanism is applied to the multilink mechanical press as the executive mechanism. It can greatly improve the quality of components, ensure the production efficiency of workpiece processing, reduce the vibration and noise in production and improve the service life of mechanisms and moulds, and so on.
Global generalized coordinates of each member are defined as where θ i is the angle of component i and x i and y i are the coordinates of the centroid of component i in X and Y direction. rough the establishment of the kinematic model of the 2-DOF nine-bar mechanism, the expressions of the rotation angle and the coordinates of the centroid in the X and Y directions of each component are obtained. e initial values of generalized coordinates can be obtained by bringing t � 0 Velocity of global generalized coordinates can be given by When t � 0, the initial values of velocity of generalized coordinates are defined as When the mechanism contains mixed clearances, constraint equations could be obtained as When t � 0, the initial centroid coordinate of the component, angle of component, structural size, and driving speed are brought into equation (26), and the vector of numerical values of constraint equations in initial configuration could be written as the following equation: Taking the first derivative of equation (26) with respect to time, the following is obtained: where Φ q is the Jacobian matrix and Φ q � zΦ/zq, Φ t is the derivative of constraint equation to time: Taking the second derivative of equation (26) with respect to time, the following is obtained: Dynamic equations could be written as [44][45][46] M€ q + Φ T q λ � g, where M, λ, and g are the mass matrix, Lagrange multiplier vector, and generalized force vector of the whole system, respectively. e generalized force consists of gravity and collision force at the clearance pairs. e normal contact force and tangential contact force at translational clearance Figure 6: Geometry of 2-DOF nine-bar mechanism considering mixed clearances. adopt the force model described in Section 2.2. e normal contact force and tangential contact force at revolute clearances are modeled as described in Section 3.2. e multibody dynamical equation with a single translational clearance and two revolute clearances in differential algebraic form is given by A more effective method is the default stability algorithm proposed by Baumgarte. In this method, displacement constraints and velocity constraints are introduced into the acceleration constraints equation by feedback control theory so as to reduce the default of the system [11,26,33,47]. So it is applied where α and β are the correction parameters and _ Φ � dΦ/dt.

Analysis of Dynamic Response
e equation of motion (equation (33)) of the 2-DOF ninebar multilink mechanism with mixed clearances is solved through the Runge-Kutta method. Effects of various friction models, clearance value, and driving velocity on dynamical responses are all researched.

Effect of Different Friction Models on Dynamics Responses
with Mixed Clearances. Driving speeds of cranks are set as ω 1 � − 2π(rad/s) and ω 4 � 2π(rad/s) and clearance values of pairs A, B, and C are all set as 0.4 mm. For investigating the effect of different friction models on dynamics, the modified Coulomb friction and LuGre models are discussed. Acceleration, contact force of the slider, contact forces of revolute clearances, driving torques, and slider trajectory are shown in Figure 7.
Instability may occur in the first two periods of calculation. erefore, the data of the first two periods are discarded, and then, the data of two periods are taken to draw the dynamic response curves.
As illustrated in Figure 7, influences of the LuGre and modified Coulomb friction models on dynamic responses for the 2-DOF nine-bar mechanism are slightly different. When the LuGre model is applied, vibration is more obvious than that of the modified Coulomb model. ere are also some differences in values of dynamic responses. When the LuGre model is applied, the value of dynamic responses is bigger than that of the modified Coulomb friction model.

Influence of Different Clearance Values on Dynamics
Responses with Mixed Clearances. For researching influence of clearance values on dynamic responses, slider's kinematic characteristics, contact forces at revolute clearance pairs, contact force of the slider, driving torques, shaft center trajectories at revolute clearance pairs, and slider trajectory inside the guide are all studied, as shown in Figures 8-15. Assume that clearance values of revolute pairs A and B and translational pair C are all same. Supposing that driving speeds of crank 1 ω 1 � − 2π(rad/s) and crank 4 ω 4 � 2π(rad/s), the    As illustrated in Figure 8(a), the clearances have little effect on the displacement curve, which is very close to the situation that all ideal joints are without clearances. As     shown in Figure 8(b), as there is an increase in the clearance value at revolute joints and translational joints, the vibration of the velocity curve becomes more and more severe. Existence of clearance has a relatively high effect on acceleration, contact forces at revolute pairs, contact force of the slider, and driving torque of cranks, as displayed in Figures 8(c)-12. Moreover, its peak and vibration frequency both increase with increase of the clearance value. Driving torques of cranks are mainly affected by collision forces at clearance joints A and B. e vibration of driving torque is consistent with contact force vibration at A and B clearance pairs. As displayed in Figures 8(c) and 11, the frequency of vibration and the time point of peak value of slider's acceleration are basically consistent with the contact force of the slider. It could be seen that contact force of the slider directly affects the slider's acceleration.
If the clearance values 0.1 mm, 0.2 mm, 0.5 mm, and 1 mm are considered to discuss the effect of clearance value on the shaft center trajectory and slider trajectory inside the guide, there are too many pictures in this paper, so 0.1 mm and 0.5 mm are selected. From Figures 13-15, with increase of clearances, chaos of trajectories become more and more serious. e reason is that the bigger the clearance value is, the more serious impact happens and the more chaotic slider trajectory inside the guide is. When the clearance value is small, the dynamic response is smooth and close to situation  Shock and Vibration that all ideal joints are without clearances. It is suggested that continuous contact state between the slider and the guide is longer; therefore, fewer impacts exist on the slider and the guide.

Effect of Different Driving Velocity on Dynamics Responses
with Mixed Clearances. Clearance values of revolute A, revolute B, and translational pair C are all defined as 0.25 mm. Driving speed of cranks 1 and 4 is assumed as − 2.5π(rad/s) and 2.5π(rad/s), − 5π(rad/s) and 5π(rad/s), respectively. Velocity, velocity error, and acceleration are displayed in Figures 16-18. As displayed in Figure 16, velocity is less affected by clearances and it is closer to the situation that all ideal joints are without clearances. However, according to Figure 17, it is shown that the velocity error increases with increase in the driving speed. As displayed in Figure 18, vibration of acceleration is more intense than slider's velocity. Moreover, the peak value of acceleration increases with increase in the driving speed.
e contact forces at pairs A, B, and contact force of the slider are shown in Figures 19-21. From Figures 19-21, it can be seen that, with the increase in the driving speeds of driving components, the peak value of contact forces at pairs A and B and contact force of the slider are both increased. e reason is that the greater the driving speed is, the more     severe the impact is and the greater the contact force is. erefore, the maximum value of dynamic responses becomes larger and larger. e shaft center trajectory of A, shaft center trajectory of B, and slider trajectory inside the guide are shown in Figures 22-24  speed increases, shaft center trajectory of A, shaft center trajectory of B, and slider trajectory inside the guide become more and more chaotic, and the collisions at clearance joints are more and more serious.

Virtual Simulation Results.
For conforming correctness of the numerical result, a virtual prototype model of 2-DOF nine-bar mechanism with mixed clearances has been built through ADAMS. Clearance sizes of translational clearance and two revolute clearances are all assumed as 0.1 mm, and   driving velocity of cranks 1 and 4 is set as − 2π(rad/s) and 2π(rad/s). Slider's displacement, velocity, acceleration, and contact force of the slider are shown in Figures 25-28.
From Figures 25-28, the result of ADAMS is slightly different from MATLAB. is is mainly due to the different modeling methods, solution methods, and integration errors [48], and they are also different in the way to identify contact between the slider and the guide. From Figures 25-28, it can be seen that the curves obtained by MATLAB and ADAMS are same in trend and have similar regularity, and the difference between the two curves is smaller; then, the correctness of the theoretical model can be proved.

Conclusions
is paper researches the dynamic responses of the 2-DOF nine-bar mechanism containing mixed clearances based on the LuGre model. With the increase in clearance sizes, peak, and vibration frequencies of dynamic responses, and slider trajectory inside the guide becomes more chaotic. (4) Effects of different driving velocities on dynamic responses with mixed clearances are studied. With the increase in driving speeds, peak of contact force at revolute clearance pairs, contact force, and acceleration of the slider are increased, and the shaft center trajectories at revolute clearance pairs and slider trajectory become more and more chaotic. Data Availability e data used to support the findings of this study are included within the article.

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Shock and Vibration