Dynamic response and sensitivity analysis for mechanical systems with clearance joints and parameter uncertainties using Chebyshev polynomials method

Mechanical Systems and Signal Processing 138 (2020) 106596

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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp

Dynamic response and sensitivity analysis for mechanical systems with clearance joints and parameter uncertainties using Chebyshev polynomials method Wuweikai Xiang a, Shaoze Yan b,⇑, Jianing Wu c, Wendong Niu b a

Institute of Systems Engineering, China Academy of Engineering Physics (CAEP), Mianyang 621999, China State Key Laboratory of Tribology, Department of Mechanical Engineering, Tsinghua University, Beijing 100084, China c School of Aeronautics and Astronautics, Sun Yat-Sen University, Guangzhou 510006, China b

a r t i c l e

i n f o

Article history: Received 29 April 2019 Received in revised form 11 September 2019 Accepted 21 December 2019

Keywords: Clearance joints Uncertainty Mechanical system Dynamics

a b s t r a c t A mechanical system with clearance joints exhibits non-linear characteristics. Even a small variation of one parameter may lead to a drastic change of the overall system response. The coupling of clearance and uncertainty would therefore significantly influence the system performance. In this paper, an analysis method for dynamic response and parameter sensitivity of mechanical systems is proposed with the consideration of both clearance joints and uncertainties. In this method, elements of a revolute clearance joint are modeled as colliding bodies, where continuous contact force model and modified friction force model are employed to describe the impact-contact behavior. The clearance joint model is then coupled with the system motion equations to obtain the system dynamic response. Furthermore, by introducing the multi-dimensional Chebyshev polynomials approach, dependence of the system dynamic response on its parameters can be established. The bounds of the dynamic response could be obtained by using the interval operations, and the parameter sensitivity is interpreted as the rate of the change of the system dynamic response with respect to the parameter variation, revealing the contribution of each parameter on the dynamic performance. Finally, dynamics of a crank-slider mechanism with clearance and uncertainties is investigated. Results show that a larger clearance would cause severe oscillation in the response bounds and larger expansion of the interval length, and clearance effect restraint zones can be obtained through the sensitivity analysis, where the expansion of the response interval length could be reduced. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction With the increasing requirement for high-speed and precise mechanisms, machines and manipulators, more issues are considered in the dynamic modeling of mechanical systems to improve the prediction accuracy, among which the clearance joint is a main factor affecting the system dynamic response [1–3]. The presence of clearance in a mechanical system leads to impact, friction, deformation and other phenomena in joints, and justifies the deviations between the numerical predictions and experimental measurements. It may further cause vibration, noise and even random overall behavior of the mechanical system [4–7]. ⇑ Corresponding author. E-mail address: [email protected] (S. Yan). https://doi.org/10.1016/j.ymssp.2019.106596 0888-3270/Ó 2019 Elsevier Ltd. All rights reserved.

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W. Xiang et al. / Mechanical Systems and Signal Processing 138 (2020) 106596

Therefore, the kinematics and dynamics of mechanical systems with clearance have gained much attention during the past few decades, and a considerable amount of theoretical and experimental research has been conducted to discuss modeling of clearance joints. Ravn [8] presented a continuous analysis method for planar multibody systems with joint clearance. This method not only quantifies the overall behavior of the system, but also analyzes the impact mechanics in the clearance joint, and thus has been extensively used in the modeling and analyzing of clearance effect in mechanical systems later. Flores et al. [9] presented a general methodology for dynamic analysis of mechanical systems with joint clearance where a continuous contact force model was adopted to represent the impact forces, and provided an experimental verification and validation of this methodology. Khemili et al. [10] carried out simulation and experimental tests to study the dynamic behavior of a planar slider-crank mechanism. In this study, three motion modes of the journal inside the bearing were observed, namely the free motion, the continuous contact motion and the impact motion. Erkaya et al.[11] also presented an experimental investigation of the vibration behaviors of a compliant mechanism under different driving speeds, clearance values and multi-axes flexible pivot diameters, revealing increase in vibration characteristics due to different motion modes between joint elements. Based on these researches, models of revolute joints [12–13], cylindrical joints [14–15] and spherical joints [16–18] have been established, and different contact models have been further compared [19–21]. In addition, the dynamic performance of mechanical systems with clearance joints has been extensively studied, and more issues have been considered in the analysis, such as flexibility, lubrication, wear and the effect of multiple clearance joints. Erkaya and Uzmay [22] investigated the dynamics of a four-bar mechanism with two clearance joints, where a massless link approach was adopted to analyze the dynamic response of the mechanism, and a genetic algorithm was used to determine the optimal values of design variables for reducing the vibration. They also proposed a versatile method to carry out the optimum synthesis of planar mechanisms with clearance [23]. Flores [24] performed a parametric study on the influence of the clearance size, input crank speed, and number of clearance joints on the dynamic response of multibody systems with multiple clearance joints. It can be concluded that the dynamics of multibody systems with clearance joints is sensitive to the system parameters. A subtle change of the parameters would lead to the transition of the dynamic response from periodic to chaotic. Besides, by the introduction of a modified contact force model and the use of Archard wear model, Xiang et al. [25] proposed a comprehensive method for joint wear prediction in planar mechanical systems, and the effects of clearance size and input power on joint wear were investigated and discussed respectively. Lai et al. [26] proposed an efficient method for the prediction of the clearance wear in planar low-velocity mechanisms. The effectiveness of the method was further verified by experiments. Muvengei et al. [27] studied nine simultaneous motion modes at two revolute clearance joints and their influence on the dynamic behavior of a slider-crank mechanism, which indicated that generally the revolute clearance joint nearer to the input link would experience the largest values of contact-impact forces. As a compliant mechanism has fewer joints than a classic mechanism and could reduce the effects of clearance, Erkaya et al. [28] studied the joint clearance effects on partly compliant mechanisms where a pseudo-rigid body model of compliant mechanism was constituted, and the suspension effect of the small-length flexural pivot in compliant mechanism was revealed. Zheng et al. [29] established a comprehensive model of flexible multi-link mechanism with clearance and studied the dynamics of the mechanism considering revolute and spherical clearance joints, lubrication, and flexibility. It indicated that due to the spherical joint clearances, link flexibility, lubrication and strong nonlinearity of the mechanism, the permanent contact mode can be ignored. Considering the time-varying pressure angle and gear backlash, Yi et al. [30] proposed a nonlinear dynamic model for a spur gear system and revealed characteristics different from previous models. Results showed that the system may turn into chaotic motion and impact state could alter under the influence of backlash. Fernandez-del-Rincon et al. [31] proposed a model to simultaneously consider the internal excitations and excitations consequence of the bearing variable compliance including clearances in gear transmissions. Effects of bearing clearances and friction forces were studied in four cases with different combinations of the factors. More recently, Farahan et al. [32] investigated the bifurcation in a planar four-bar mechanism with a revolute clearance joint. Results showed that an increase in the size of the clearance might not always lead to chaotic behavior, but can change the periodicity number of the stable limit cycle. Besides, dynamics of multibody systems with uncertain parameters were studied by Wang et al. [33] and Li et al. [34] using a non-intrusive computation methodology and Monte Carlo method, respectively. Chen et al. [35] investigated the dynamic behavior of a spatial parallel mechanism with clearance by combining KED method and Lagrange method, revealing the sensitivity of the system motion to the joint clearance and flexible links. Wang et al. [36] proposed a method to calculate the wear of clearance spherical joint in four-degree-of-freedom parallel mechanism and the non-uniform distribution of 3D wear was further studied. In addition, Brogliato [37] reviewed several strategies for feedback control of mechanisms with joint clearance, namely impactless trajectories with persistent contact, control through collisions, the stabilization of equilibrium points, and trajectory tracking control. The dynamic performance of a polydyne cam and roller follower mechanism was investigated by Yousuf et al. [38] numerically and experimentally. In the case of clearance equal to 2 mm, the system exhibits more chaotic characteristics than others, and the increase in the follower guides’ clearances would further increase the non-periodicity of the follower motion. Cao et al. [39–40] analyzed the error motion, vibration and stability of rotary systems with clearances. The rise in clearance size could lead to the increase of the radial motion error and the change of the system response from periodic to quasi-periodic. Besides, the dynamic performance of a planar rigid-flexible coupled solar array with multiple clearance joints was investigated by Li et al [41], which reveals the domination of elastic vibration property of flexible panels in the system shock in impact phase. From the literature survey, it can be found that previous researches on the modeling of clearance joints and the analysis of the dynamics of mechanical systems with clearance are mainly based on the assumption that all parameters are determin-

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istic, and the uncertainties are rarely stated. Variations in material properties and geometry in these researches are considered by introducing the extreme or mean values and/or the application of safety factors. Thus, the dynamic responses of these mechanical systems are uniquely determined. However, due to manufacturing tolerances, measurement variability and thermal alternating, uncertainties are inevitable in mechanical systems. And it should be noticed that the dynamics of mechanical systems with clearance joints is sensitive to the system parameters. A subtle change of the parameters would lead to the drastic change of the system dynamic response. Therefore, in the dynamic analysis of mechanical systems, both the clearance joints and the uncertainties should be considered. The primary objective of this work is to present a method for dynamic analysis of mechanical systems considering both clearance joints and parametric uncertainties. This method presented could not only analyze the dynamic response of mechanical systems with clearance and uncertainty, but could also investigate the parameter sensitivity, which quantifies the effect of each uncertain parameter on the system dynamic response. In this paper, the model for revolute clearance joints is described briefly. And dynamic modelling of deterministic mechanical systems considering clearance joints is presented. Furthermore, the approximation of system dynamic response is established using the Chebyshev polynomials method, by which the relations between parameters and system response could be established. Based on the approximation, the analysis method for dynamic response of mechanical systems with clearance and uncertainty is proposed. Finally, the dynamics of a slider-crank mechanism with clearance and uncertain length is studied using this method. The effects of clearance joint and uncertain length on the system dynamic response are revealed. And the sensitivity of the system response on the uncertain parameter is also investigated.

2. Model for revolute clearance joints A mechanical system with deterministic parameters consists of a number of components which are interconnected by joints. In the case of an ideal revolute joint, the journal and bearing centers coincide throughout the motion, where only relative rotation is allowed between them. However, when clearance is considered in the joint, the journal could move freely inside the bearing, and two extra degrees of freedom are added into the system, whose performance and reliability would be adversely influenced. Thus, a proper model for revolute joints with clearance is necessary to analyze the system dynamic response.

2.1. Kinematic model for a revolute joint with clearance Fig. 1 shows the configuration of a revolute joint with clearance, where the journal and the bearing are rigidly attached to body j and body b respectively. XY is the global coordinate system, while XjYj and XbYb are the local coordinate systems. The eccentric vector e which connects the geometric centers of the journal and the bearing can be expressed as

e ¼ rpj  rpb

 ¼ Rmj þ Aj upj  Rmb þ Ab upb

ð1Þ

where rpj and rpb represent the position vectors of the journal and bearing centers in the global coordinate system, Rmj and Rmb denote the position vectors of the mass centers of the two bodies in the global coordinate system respectively, Aj and Ab are the matrices that transform a vector from the local coordinate system to the global coordinate system, and upj and upb

Fig. 1. Configuration of a revolute joint with clearance.

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represent the position vectors of the journal and bearing centers in local coordinate systems XjYj and XbYb respectively. It should be noticed that once the journal is in contact with the bearing, the contact force will be applied along the direction of the eccentric vector. The penetration depth between the journal and the bearing during impact and contact can be given as

d ¼ je j  c

ð2Þ

where c is defined as the radial clearance, and it is the difference between the bearing radius (Rb) and the journal radius (Rj). As shown in Fig. 1, the points of contact on the journal and the bearing are defined as Cj and Cb, and their locations in the global coordinate system can be expressed as



rcj ¼ Rmj þ Aj upj þ Rj n

ð3Þ

rcb ¼ Rmb þ Ab upb þ Rb n where n is the unit vector in the direction of the eccentric vector which is given as

n ¼ e=e

ð4Þ

Based on the position vectors rcj and rcb, velocity of the contact points can be calculated as

(

r_ cj ¼ R_ mj þ A_ j upj þ Rj n_ r_ cb ¼ R_ mb þ A_ b upb þ Rb n_

ð5Þ

And projecting the velocity of the contact points to the normal and tangential planes of contact respectively, the components of the relative velocity can be expressed as



v N ¼ ðr_ cj  r_ cb Þ  n v T ¼ ðr_ cj  r_ cb Þ  t

ð6Þ

where t is the unit vector in the tangential direction, which could be obtained by rotating the vector n anticlockwise by 

90 . 2.2. Force estimation for a revolute joint with clearance Once the journal is in contact with the bearing, a force normal to the plane of contact should be applied. As the LankaraniNikravesh contact force model accounts for both elastic nature of colliding bodies and energy dissipation during impact [42], it is widely used in the dynamic analysis of mechanical systems with clearance. According to the Lankarani-Nikravesh contact force model, the normal contact force in terms of the penetration depth is expressed as [42]

F N ¼ Kdn þ Dd_

ð7Þ

where FN is the normal contact force, Kdn represents the elastic force, Dd_ accounts for the energy dissipation due to internal damping, K is the generalized stiffness parameter, D is the damping coefficient, and the exponent n is set to be 1.5. The generalized stiffness parameter K depends on the physical and geometric properties of the contact bodies which is given by

K¼

 1 4 Rj Rb 2 3ðrj þ rb Þ Rb  Rj

where Rj and Rb are the journal radius and the bearing radius respectively, and

r¼

ð8Þ

r is the material parameter given as

1m pE

2

ð9Þ

where E and m are the Young’s modulus and Possion’s ratio associated with each sphere respectively. The damping coefficient proposed by Lankarani-Nikravesh is expressed as

D¼

3Kð1  c2e Þdn 4d_ ðÞ

ð10Þ

ðÞ where ce is the restitution coefficient, and d_ represents the penetration velocity before impact. Furthermore, the Coulomb friction model is the most fundamental and simplest model to describe the sliding friction force between dry contact surfaces. However, this model does not account for stiction phenomenon, and it may lead to numerical challenges in simulations of mechanical systems. In order to avoid such difficulties, a modified Coulomb friction model is adopted. The friction coefficient l can be written as [43–44]

W. Xiang et al. / Mechanical Systems and Signal Processing 138 (2020) 106596

8 ld jv T j > V d > >   > < jv T j p jv T j < V s l ¼ ls sin 2 V s > h
 i >  > ls þld 1 s : þ 2 ls  ld cos p jVv T jV V s 6 jv T j 6 V d 2 V s

5

ð11Þ

d

where ls and ld are the static and sliding friction coefficients, and V T , V s and V d are the relative tangential velocity, stick–slip switch velocity and static-sliding friction switch velocity respectively. 3. Modelling of deterministic mechanical systems with clearance joints Ideal joints in a mechanical system impose constraints on the relative motion of the bodies, and the description of these constraints is expressed as algebraic equations which can be given as [24]

Uðq; tÞ ¼ 0

ð12Þ

where q is the vector of generalized coordinates that describes the configuration of the system, and t denotes time. Based on the position equations, the velocity equations and acceleration equations can be obtained by differentiating Eq. (12) once and twice on time respectively, which are shown as

Uq q_ ¼ Ut € ¼ ðUq qÞ _ q q_  2Uqt q_  Utt  c Uq q

ð13Þ

where the subscripts q and t represent partial differentiations with respect to the generalized coordinates and time, respectively. Furthermore, in order to determine the dynamic response of a constrained mechanical system, the equations of motion should be established which can be expressed as

€ ¼ g þ gðcÞ Mq

ð14Þ (c)

where M represents the generalized mass matrix, g is the generalized force vector, and g and can be expressed as

gðcÞ ¼ UTq k

is the vector of joint reaction force

ð15Þ

where k is the Lagrange multipliers. Combining Eq. (12), Eq. (14) and Eq. (15), the system differential algebraic equations of motion could be given as

(

€ þ UTq ðq; tÞk ¼ gðq; q; _ tÞ Mq Uðq; t Þ ¼ 0

ð16Þ

It should be highlighted that in mechanical systems where clearance joints are present, the clearance joints are no longer kinematic constraints but impose force constraints on the system. Therefore, the joint reaction forces in these joints should be explicated in the generalized force vector in Eq. (16). Based on Eq. (16), if the set of the initial condition is given as

h i  y ¼ yðt0 Þ ¼ qð0ÞT ; q_ ð0ÞT

ð17Þ

the dynamic response of the system is uniquely determined which is shown as



y ¼ qT ; q_ T

ð18Þ

In general, Eq. (18) describes the system motion in a deterministic way, and thus the dynamic response of the mechanical system considering clearance joints and deterministic parameters can be uniquely determined. 4. Approximation of system dynamic response According to the Weierstrass approximation theorem, every real-valued continuous function f ðnÞ defined on a closed interval can be uniformly approximated as closely as desired by a polynomial function pðnÞ. And the Chebyshev polynomials could provide an approximation that is close to the polynomial of best approximation to a continuous function under the maximum norm. It has also been proved that the Chebyshev polynomials method could control the overestimation in interval computations [45]. Therefore, it is employed to approximate the system dynamic response in this study. As a welldeveloped method in uncertainty analysis, Refs. [45–48] provide details of the Chebyshev polynomials method. In this section, the Chebyshev polynomials method is briefly introduced, and expressions that are related to the uncertainty analysis with consideration of clearance are presented. For a one-dimensional problem, the polynomial of degree n of Chebyshev polynomials is expressed as [48]

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 T n ðnÞ ¼ cosðnhÞ h ¼ arccos

 2n 

   = v v  



vþ v 

where the variable n is defined in the interval



ð19Þ



v ; v and h 2 ½0; p. Based on the Chebyshev polynomials, the function f ðnÞ can



be approximated as

f ðnÞ  pn ðnÞ ¼

n X 1 f0 þ f l T l ðnÞ 2 l¼1

ð20Þ

where f l is the constant coefficient, and n represents the highest order in the Chebyshev polynomials. The constant coefficient can be calculated numerically by the Mehler integral method based on the orthogonality of Chebyshev polynomials as [45]

fl ¼

2

p

Z p

f ðnÞcoslhdh 

0

m   2 X f nj coslhj m j¼1

ð21Þ

where 

2j  1 p j h ¼ n ¼ m 2 j



vþ v

v v

2

2

þ

 coshj ; j ¼ 1; 2; :::; m

ð22Þ

where m is the number of the interpolation points to calculate the constant coefficient and it should be no less than n + 1. It can be observed from Eq. (21) that the constant coefficient is a linear combination of the function and the polynomial values in different interpolation points. Furthermore, for a multi-dimensional problem, the polynomial is the tensor product of each polynomial in onedimensional case. For instance, the polynomial of a k-dimensional Chebyshev polynomials is defined as [45–47]

T i1 ;:::;ik ðnÞ ¼ T i1 ;i2 ;:::;ik ðn1 ; n2 ; :::; nk Þ ¼ cosði1 h1 Þcosði2 h2 Þ:::cosðik hk Þ; ik ¼ 0; 1; 2; :::; n

ð23Þ 

where ik represents the order, nk 2 ½1; 1 and hk ¼ arccosðnk Þ. For a more general case of nk 2





v ; vk , similar expressions k could be obtained by linear transformation as in the one-dimensional case. According to the polynomial defined, the function f ðnÞ including k variables can be approximated as [45–47] f ðnÞ 

n X

:::

i1 ¼0

n  s X 1 ik ¼0

2

f i1 ;:::;ik T i1 ;:::;ik ðnÞ

ð24Þ

and the constant coefficient in the multi-dimensional problem can be calculated as [45–47]

f i1 ;i2 ;:::;ik ¼

 k Z p 2

p

:::

0

Z p

f ðn1 ; :::; nk Þcosi1 h1 :::cosik hk dh1 :::dhk 

0

 k X m m   X 2 j j j j ::: f n11 ; :::; nkk cosi1 h11 :::cosik hkk m j ¼1 j ¼1 1

ð25Þ

k

where 

j hkk

2j  1 p j ¼ k ; jk ¼ 1; 2; :::; m nkk ¼ m 2

vk þ v

k

2



þ

vk  v

k

2

j

coshkk ; jk ¼ 1; 2; :::; m

ð26Þ

Based on the polynomial approximation method mentioned above, assuming a mechanical system includes k parameters, then the relation between the parameters and the system dynamic response can be established by using the Chebyshev polynomials as

yðn; tÞ 

n X i1 ¼0

:::

n  s X 1 ik ¼0

2

yi1 ;:::;ik ðtÞT i1 ;:::;ik ðnÞ

ð27Þ

where n is the vector of the parameters, yðn; t Þ is the system dynamic response, yi1 ;:::;ik ðtÞ is the vector of the constant coefficients of the Chebyshev polynomials, and s represents the count of zeros in the subscripts i1 ; :::; ik . The vector of coefficients in Eq. (27) can be calculated as

yi1 ;i2 ;:::;ik ðtÞ 

 k X m m   X 2 j j j j ::: y t; cosh11 ; :::; coshkk cosi1 h11 :::cosik hkk m j ¼1 j ¼1 1

ð28Þ

k

It should be noticed that in Eq. (28) the vector of the constant coefficients of the Chebyshev polynomials is only the function of time, and the polynomial T i1 ;:::;ik ðnÞ is a function of parameters which does not vary with time.

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W. Xiang et al. / Mechanical Systems and Signal Processing 138 (2020) 106596

By introducing the Chebyshev polynomials, the implicit relation between the dynamic response of a mechanical system and the parameters can be established, which provides a feasible method to analyze the dynamic response of a mechanical system considering both clearance and uncertainty. 5. The analysis method for system dynamic response and parameter sensitivity 5.1. Dynamic response of mechanical systems with clearance and uncertainty For planar mechanical systems with deterministic parameters, the presence of a revolute joint introduces two extra degrees of freedom into the system. The relative motion between the journal and the bearing could be determined according to the kinematic analysis. And by using the contact force model in Eq. (7) and the friction force model in Eq. (11), the joint reaction forces in the local joint could be calculated. In this case, the clearance joint proposes force constraints on the system instead of kinematic constraints. The joint reaction forces in the clearance joint are external forces applied to the system, which should be included in the generalized force in Eq. (16). In this way, the effect of the clearance joint is introduced into the system equations. And dynamics of deterministic mechanical systems with clearance joints could be analyzed. Besides, according to the approximation of system dynamic response stated in Section 4, the dynamic response of mechanical systems with clearance could be expressed by using the Chebyshev polynomials (Eq. (27)), which explicitly reveals the relation between the system dynamic response and the parameters. Furthermore, as the statistic information of uncertain parameters can hardly be obtained, and the bounds of them can be known, the system uncertain parameters could be expressed as interval vectors shown as 

½n ¼ ½ n ; n 

ð29Þ 

where ½n is the interval vector whose component in each dimension corresponds to an uncertain parameter, n and n are the  lower and upper bound vectors of the uncertain parameters which are shown as

 T n ¼ v1 ; v2 ; :::; vk     h  i   T n ¼ v1 ; v2 ; :::; vk

ð30Þ

By substituting the interval vector ½n for the vector of the parameters n in Eq. (27), the dynamic response of mechanical systems with clearance and uncertainty could be analyzed. The dynamic response bounds could be obtained by using the interval operations. It should be highlighted that since the Chebyshev polynomials can be expressed as the cosine function where the variable h is defined on the interval of ½0; p. When the interval variable ½n is substituted into the Chebyshev polynomial, it has T n ð½nÞ ¼ cosðn½hÞ ¼ cosðn½0; pÞ ¼ ½1; 1 for any n > 0 and T 0 ð½nÞ ¼ cosð0 ½hÞ ¼ 1 for n = 0 according to the interval method. Based on the Eq. (27), the bounds of the system dynamic response could be calculated as

y ðt Þ ¼  

y ðt Þ ¼


1k 2


1k 2

y0;:::;0 ðt Þ 

y0;:::;0 ðt Þ þ

n P i1 ¼0 n P i1 ¼0

::: :::

 n 
 P   1 s  2 yi1 ;:::;ik ðtÞ

ik ¼0

 n 
 P  1 s   2 yi1 ;:::;ik ðt Þ

ð31Þ

ik ¼0



where y ðt Þ and y ðt Þ are the lower and upper bounds of the system dynamic response, respectively, and i1,i2,. . .,ik are not all  zero. In general, considering the uncertain parameters as interval vectors whose upper and lower bounds are known, and constructing the Chebyshev polynomials to approximate the system dynamic response where the interval vector is taken as the variable, the dynamic response of a mechanical system with both clearance and uncertainty could be obtained. 5.2. Parameter sensitivity analysis for mechanical systems with clearance and uncertainty According to the method provided in Section 5.1, the dynamic response of a mechanical system with clearance and uncertainty could be analyzed and the bounds of the system dynamic response could be calculated. However, the effect of each parameter on the dynamic response differs from others, that the response interval caused by each uncertain parameter is dissimilar. Thus, the sensitivity of the system response to the uncertain parameter is quite different. As the system dynamic response is approximated by using the Chebyshev polynomials, the explicit relation between the system dynamic response and the parameters can be established. Hence, according to the Eq. (27), the sensitivity of the system response to one uncertain parameter could be interpreted as the rate of change of the system dynamic response with respect to the uncertain parameter [49–50] which is expressed as

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su;np ðt Þ ¼

@yu ðn; t Þ jn¼nc ; p ¼ 1; 2; :::; k @np

ð32Þ


 where su;np ðt Þ represents the sensitivity of the system response to the uncertain parameter np np 2 n , yu and np are the u-th component of the system dynamic response y and the p-th component of the vector of the uncertain parameters, k is the number of the uncertain parameters, and nc is the vector of the mean value of the uncertain parameters. Particularly, the polynomial in Chebyshev polynomials is presented as cosine function, and therefore the Eq. (32) could be conveniently calculated as

su;np ðt Þ ¼ 

n P

i1 ¼0

...

 s n
 P 1 yu;i1 ;:::;ik ðtÞip cos ði1 h1 Þ cos ði2 h2 Þ::: sin ip hp ::: cos ðik hk Þ 2 ik ¼0

ð33Þ

From Eq. (33), it can be observed that since the sensitivity is the function of time, it varies with the motion of the system. 5.3. The analysis method for dynamic response of mechanical systems with clearance and uncertainty The analysis method for dynamic response of mechanical systems with clearance and uncertainty mainly consists of three steps, namely the construction of the Chebyshev polynomials, the analysis of the dynamic response and the parameter sensitivity analysis. The algorithm is outlined in Fig. 2. Firstly, the interpolation points are produced by Eq. (26), and the motion of the mechanical system is divided into a number of sub-intervals. In each sub-interval, based on the model of clearance joints presented in Section 2 and the motion equations of the system shown in Eq. (16), the corresponding dynamic response with respect to every combination of interpolation points could be calculated. It should be noticed that as the interpolation points are chosen according to

Fig. 2. The analysis method for mechanical systems with clearance and uncertainty.

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9

Eq. (26), the parameters in the equations at this step are deterministic. Based on the results, the coefficients of the Chebyshev polynomials can be determined by Eq. (28), and the relations between the system dynamic response and the uncertain parameters can be established by using the Chebyshev polynomials (Eq. (27)). Then, the dynamic response of mechanical systems with clearance and uncertainty could be obtained by using the interval algorithm, and the dynamic response bounds could further be calculated by Eq. (31). Finally, according to Eq. (33), the sensitivity of the system response to every parameter can be analyzed respectively, and the contribution of each parameter on the dynamic response could be revealed independently. Although, the motion equations of the system are stiff due to the sudden change in kinematic configuration, where special numerical algorithms should be introduced in solving the derivative and algebraic equations, the analysis method proposed in this paper is an external iterative method which does not modify the equations themselves. Therefore, the predictor–corrector algorithm with both variable step size and order in solving the motion equations of mechanical systems with clearance can be used. Besides, the polynomial of the Chebyshev polynomials in high dimension is constructed as the tensor product of each polynomial in one-dimensional case, so the coupling of the parameters is also considered. 6. Case study: A slider-crank mechanism with clearance and uncertain length In this section, a planar slider-crank mechanism is performed as an example to demonstrate the method presented in this paper. Fig. 3 shows the configuration of the selected slider-crank mechanism. The mechanism consists of two ideal revolute joints, one between the crank and the ground, and the other between the crank and the coupler, an ideal transitional joint between the slider and the ground, and a clearance revolute joint between the coupler and the slider. Besides, the lengths of the crank and the coupler are uncertain, which are defined on two given intervals respectively. The nominal values (mean values) of the geometric and inertial properties of the slider-crank mechanism are listed in Table 1. In Table 2, the parameters associated with the clearance joint are provided. The lengths of the crank l1 and the coupler l2 are uncertain parameters which are assumed as

½li  ¼ li ð1 þ 0:1fi Þ; fi 2 ½1; 1; i ¼ 1; 2

ð34Þ

where li represents the mean value of the link length. Due to the uncertain link lengths, the moments of inertia of the crank and coupler also become uncertain parameters which are expressed as

½I i  ¼

1 2 mi ½li ð1 þ 0:1fi Þ ; fi 2 ½1; 1; i ¼ 1; 2 3

ð35Þ

where I1 and I2 are the moments of inertia of the crank and coupler respectively. 6.1. Effect of uncertain length on the dynamic response In this section, the effect of uncertain length on the system dynamic response is investigated. The radial clearance size of the imperfect joint is c = 0.02 mm, and the angular speed of the crank is 2000 r/min which remains constant during the motion of the mechanism. In addition, to balance the computational cost and the precision, the third-order Chebyshev polynomials are adopted to approximate the dynamic response of the mechanism (its computational cost and precision are analyzed below). Meanwhile, similar to Refs. [45–47], the scanning method is introduced as a comparison method to determine the precise bounds. To ensure the precision, 10 symmetrical sampling points are applied to each uncertain parameter. In the scanning method, the deterministic parameters are extracted from the interval of the uncertain parameter, and directly input to the system motion equation to obtain the dynamic response and further to form the response bounds with their envelope. When multiple uncertain parameters are considered, all possible combinations of the deterministic parameters will be calculated.

Fig. 3. Configuration of the slider-crank mechanism with clearance.

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Table 1 Geometric and inertial properties of the slider-crank mechanism. No.

Body

Length (m)

Mass (kg)

Moment of inertia (kg  m2 )

1 2 3

Crank Connecting rod Slider

0.05 0.12 –

0.30 0.21 0.14

0.00025 0.00100 0.00010

Table 2 Parameters used in the dynamic simulation. Parameter

Value

Parameter

Value

Journal radius Coefficient of restitution Stick-slip switch velocity Static-sliding friction switch velocity

9.8 mm 0.9 0.001 m/s 0.0015 m/s

Static friction coefficient Sliding friction coefficient Young’s modulus Poisson’s ratio

0.04 0.03 207GPa 0.3

The dynamic response of the slider-crank mechanism is illustrated in Fig. 4 where the results of the Chebyshev polynomials method and the scanning method are compared respectively. It can be observed that the bounds of the slider displacement and velocity predicted by the two methods agree quite well. Although in part of the prediction for the slider acceleration and the relative penetration depth overestimation can be observed, the bounds predicted by the scanning method are wrapped by the Chebyshev polynomials method and the deviation between them is small compared with the interval length. Table 3 shows the relative errors between the two methods, in which root mean square (RMS) values of the responses are adopted. It can be observed that the third-order Chebyshev polynomials method shows accurate prediction for the slider displacement and velocity, where relative error is below 0.5%. The maximum errors for the slider acceleration

Fig. 4. Dynamic response of the slider-crank mechanism with clearance and uncertainty (c = 0.02 mm): (a) slider displacement (b) slider velocity (c) slider acceleration (d)relative penetration depth.

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W. Xiang et al. / Mechanical Systems and Signal Processing 138 (2020) 106596 Table 3 Relative error of the dynamic response bounds. Dynamic response Lower bound

Upper bound

Relative error (%) Slider displacement Slider velocity Slider acceleration Penetration depth Slider displacement Slider velocity Slider acceleration Penetration depth

0.08 0.37 11.78 7.89 0.22 0.24 14.23 8.81

and penetration depth are less than 15% and 9%, respectively, which are also of good precision. According to previous researches [1,3], contact-impact and friction are typical nonlinear phenomena in clearance joints. The influence of the local nonlinear behavior on the slider acceleration would be stronger than on the displacement. The prediction error is thus larger for the slider acceleration than the displacement owing to the complex impact between the journal and the bearing, and the dynamic response to be predicted therefore becomes one of the factors that influence the prediction error. Besides, the prediction error is also affected by the polynomial order. With a lower polynomial order, it may result in a large error in the prediction of dynamic response that is sensitive to the local behavior in the clearance joint. In addition, assuming that sampling points are r in the scanning method, it means that r points would be extracted from the interval of an uncertain parameter. As a k-dimensional problem corresponds to k uncertain parameters, it requires rk iterations to solve the system motion equation and obtain the system dynamic response. To ensure sufficient precision, r is usually set to be large. For the Chebyshev polynomials method, the use of the n-th order polynomials requires at least n + 1 interpolation points for each parameter. The number of iterations to solve the motion equation is therefore (n + 1)k. As n < r, the computational cost can be reduced exponentially. In this example, 10 sampling points are used in the scanning method, and third order Chebyshev polynomials are adopted corresponding to 4 interpolation points. As two parameters are considered to be uncertain, the number of iterations for the scanning method and the Chebyshev polynomials method are 102 and 42, respectively. The computational cost could be obviously reduced by the use of the Chebyshev polynomials method. Therefore, it can be reasonably concluded that the Chebyshev polynomials method could be effectively applied to estimate the dynamic response of mechanisms considering both clearance and uncertainty. Furthermore, comparisons of the dynamic responses between the mechanisms with the nominal parameters and the uncertain parameters are also illustrated in Fig. 4. The responses of the displacement and velocity of the slider in the mechanism with nominal parameters are single-valued and smooth. However, when the uncertainties are considered in the mechanism, the responses exhibit interval characteristics where the specific responses cannot be known and only the bounds can be predicted. Besides, the interval length of the response also varies with the motion of the mechanism. For example, the interval length of the slider displacement attains its maximum value when the crank and the coupler are collinear, however, the interval length of the slider velocity attains its minimum value at this moment, which may not be zero as in the nominal case. It should be highlighted that the presence of the clearance in the revolute joint would cause the impact and friction between the journal and the bearing which would further lead to the oscillation of the slider acceleration and the relative penetration depth of the two elements. Hence, different from the slider displacement and velocity, the slider acceleration (Fig. 4(c)) and the relative penetration depth (Fig. 4(d)) in the mechanism with nominal parameter are both non-smooth where peaks are quite obvious in the responses, and they oscillate within their envelopes which are similar to the bounds of the responses of the uncertain mechanism. To make a clear illustration and further reduce the wrapping effect, envelops of the bounds calculated are used in this study. It is obvious in Fig. 4(c) and (d) that the variation of the interval length of the slider acceleration in the uncertain mechanism is in accordance with that of the amplitude of the slider acceleration in the 

nominal mechanism, and so is the relative penetration depth. For instance, when h1 equals 465 , the amplitudes of the slider acceleration and the relative penetration depth in the nominal mechanism attain the maximum values, and similarly, the interval lengths of the uncertain mechanism responses also have maximum values at this moment, which decrease subsequently. In order to further investigate the effect of the uncertain link length on the dynamic response, two extra mechanisms with deterministic parameters are simulated which comprise the maximum and minimum values of the link length according to Eq. (34) respectively. It is defined that mechanism I has the longest link, and mechanism II has the shortest link. The relative penetration depth between the journal and the bearing in the mechanism with different link length is illustrated in Fig. 5, where peaks represent impact-contact between the two elements in the joint. It can be observed that variations of the amplitude and frequency of the results are quite similar, which indicates that the contact behavior of the journal and bearing is not essentially influenced by the uncertain link length. As the contact force is directly determined by the contact behavior of the two elements in the clearance joint, it can be reasonably inferred that the slider acceleration is also slightly affected by the uncertain link length, which can be proved by Fig. 6. And when the uncertainty is considered, the variation of the interval length of the dynamic response of the uncertain

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Fig. 5. Relative penetration depth between the journal and bearing in the deterministic mechanism: (a) mechanism with the longest links (b) mechanism with the shortest links.

mechanism corresponds to the acceleration amplitude of each deterministic mechanism. In addition, the time–frequency analysis for the slider acceleration of each deterministic mechanism (Figs. 4(c) and 6) is illustrated in Fig. 7. Despite of the difference of the link length, the distribution and amplitude of each slider acceleration resembles another. Generally, it can be concluded that as the relative penetration depth and the slider acceleration of each mechanism are quite similar, the contact behavior between the journal and the bearing in the clearance joint is not essentially influenced by the uncertain link length. Thus, the envelopes of the relative penetration depth and the slider acceleration of each deterministic mechanism with different link length resemble the interval bounds of the corresponding responses of the uncertain mechanism. However, when the link length uncertainty is considered, the slider displacement and velocity exhibit interval characteristics which are no longer single-valued as in the deterministic mechanism and would lead to the decrease of the motion precision of the mechanism.

6.2. Effect of clearance size on the dynamic response In order to analyze the effect of clearance joint on the dynamic response, the clearance size in the uncertain mechanism is altered as c = 0.01 mm in this section, and the uncertainty remains the same. The dynamic responses of this mechanism are illustrated in Fig. 8. Comparing Figs. 4 with 8, it can be concluded that the slider displacement changes little due to the variation of the clearance size, and the bounds of the slider velocity oscillate slightly. When the clearance size is 0.01 mm, the maximum interval length of the slider velocity is 3.21 m/s, and when the clearance size increases to 0.02 mm, the maximum interval length of the slider velocity becomes 3.36 m/s, which indicates the expansion of the response interval length due to a larger clearance. Furthermore, it can be inferred that the contact behavior between the journal and the bearing changes with the variation of

Fig. 6. Slider acceleration of the deterministic mechanism: (a) mechanism with the longest links (b) mechanism with the shortest links.

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Fig. 7. Time-frequency analysis for the slider acceleration of each deterministic mechanism:(a) mechanism with the longest link (b) mechanism with the shortest link (c) mechanism with the link of nominal values.

the clearance size, where the interval length of the slider acceleration decreases 20% when the clearance size reduces from 0.02 mm to 0.01 mm. According to the parameter sensitivity analysis method presented in this paper, the sensitivity of the slider displacement and velocity to the variation of the link length could be studied, and the results are shown from Figs. 9–11. In the analysis, a positive value corresponds to a positive correlation between the parameter and the dynamic response, while a negative value means the opposite. For example, it could be observed in Fig. 9 that the slider displacement sensitivity to the length of the coupler l2 is always positive which means that with the increase of the coupler length, the range of the slider motion becomes wider. However, determined by the position of the crank, the sensitivity of the slider displacement to the length of 

the crank l1 alters around zero. It is because the joint connecting the crank and the ground rotates 360 in a motion period, but the coupler only swings in a small angle. The sensitivity of the slider displacement to the variation of the link length is illustrated in Fig. 9. In general, the sensitivity is smooth and not obviously influenced by the variation of the clearance size, and it coincides with the conclusion drawn before. In sharp contrast to the displacement, peaks occur in the velocity sensitivity in Figs. 10 and 11, and with the increase of the clearance size, the sensitivity oscillates around the ideal curve more severe. The mean square of residual (MSR) for each velocity sensitivity is listed in Table 4, and the value of the MSR directly represents the effect of the clearance size on the sensitivity. It can be inferred that, with the increase of the clearance size, the slider velocity is more sensitive to the link length, and the motion precision of the mechanism reduces. When the clearance size doubles, the MSR of the velocity sensitivity to the coupler length increases by 4.5 times. Furthermore, the sensitivity of the slider velocity in the ideal mechanism is smooth (Figs. 10 and 11), indicating smooth bounds of the slider velocity in the

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Fig. 8. Dynamic response of the slider-crank mechanism with clearance and uncertainty (c = 0.01 mm): (a) slider displacement (b) slider velocity (c) slider acceleration (d) relative penetration depth.

Fig. 9. Sensitivity of the slider displacement to the variation of the link length: (a) c = 0.01 mm (b) c = 0.02 mm.

ideal mechanism where the effect of clearance joint is ignored. As analyzed before, the presence of the clearance joint introduces impact and contact between the joint elements. The bounds of the slider velocity oscillate owing to the nonlinear behavior in the clearance joint. Therefore the sensitivity of the slider velocity in the imperfect mechanism where clearance joint is considered would oscillate around the curve of the ideal mechanism. As the sensitivity is the rate of change of the system dynamic response with respect to one parameter, the deviation between them is caused by the clearance joint. The deviation changes with the motion of the system, and a smaller deviation corresponds to a narrower interval length expansion caused by the clearance. Define the zone in which the sensitivity deviation is 50% less than its maximum value

W. Xiang et al. / Mechanical Systems and Signal Processing 138 (2020) 106596

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Fig. 10. Sensitivity of the slider velocity to the variation of the crank length (l1): (a) c = 0.01 mm (b) c = 0.02 mm.

Fig. 11. Sensitivity of the slider velocity to the variation of the length of the connecting rod (l2): (a) c = 0.01 mm (b) c = 0.02 mm.

Table 4 Mean square of residual of velocity sensitivity. Clearance size (mm)

MSR of slider velocity sensitivity to l1

MSR of slider velocity sensitivity to l2

0.01 0.02

0.0074 0.0266

0.0076 0.0341

as the clearance effect restraint zone. Three clearance effect restraint zones could be found in a motion period as shown in Figs. 10 and 11, where the sensitivity oscillates more slightly, and the expansion of the interval length of the velocity response caused by the clearance could be reduced. To further investigate the effect of the uncertain clearance on the system dynamic performance, the joint clearance size is set to be an uncertain parameter. The nominal value for the clearance size is cn = 0.2 mm and cn = 0.5 mm. The uncertainty for the clearance size is 10%, resulting in two clearance size intervals, namely [0.18, 0.22] mm and [0.45, 0.55] mm, respectively. Fig. 12 illustrates the response bounds of the slider displacement and velocity which are directly associated with the motion precision. As the clearance size is order of magnitude smaller than the link length, the effect of uncertain clearance size is not obvious on the slider displacement. Although, the interval length is much smaller compared to the motion of the slider, the maximum interval length of the slider displacement is larger than the nominal clearance size (Table 5), indicating that the uncertainty in clearance size would degrade the displacement by the amplitude larger than its size. On the contrary, effects of uncertain clearance are obvious on the slider velocity. The interval length of the slider velocity is larger than the displacement, and there is oscillation in its bounds. Since the contact and impact behavior in the clearance joint is affected by the clearance, the interval length increases and the oscillation becomes more sever with the increase of the nominal clearance size. This conclusion can also be demonstrated by the sensitivity analysis shown in Table 5. The sensitivity of slider velocity is orders of magnitude larger than the displacement. And when the nominal clearance size becomes 2.5 times larger,

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Fig. 12. Dynamic response of the slider-crank mechanism with uncertain clearance (a) slider displacement (b) slider velocity.

Table 5 Dynamic response with uncertain clearance size. Nominal clearance size (mm)

Maximum interval length of slider displacement (mm)

Maximum interval length of slider velocity (m/s)

Sensitivity of slider displacement (RMS)

Sensitivity of slider velocity (RMS)

0.2 0.5

0.35 1.25

0.55 1.73

0.0001 0.0003

0.2540 0.7511

the maximum interval length of the slider displacement and velocity and their sensitivity on clearance size triple, indicating a nonlinear increase in the dynamic response. 7. Conclusions An analysis method for dynamic response of mechanical systems with both clearance joints and uncertainties is proposed in this paper, which could be used to predict the system dynamic behavior and analyze the parameter sensitivity. In this method, the relative motion of the joint elements and the reaction forces could be determined through the kinematic and dynamic analyses according to the model of clearance joint, where the continuous contact force model and the modified friction model are employed to describe the normal and tangential forces respectively. The joint reaction forces are then introduced into the system motion equations to substitute the kinematic constraints of the ideal joint, and the dynamic model for deterministic systems considering clearance can be established. Furthermore, introducing the uncertain parameters as interval vectors, relations between the system dynamic response and the uncertain parameters could be explicated with the use of the multi-dimensional Chebyshev polynomials method. Based on the approximation, the dynamic response of the mechanical systems with clearance joints and uncertain parameters can be obtained by using the interval operations. And the parameter sensitivity could be analyzed revealing the contribution of each parameter on the system dynamic response. In addition, a slider-crank mechanism with clearance and uncertain lengths is used as an example to demonstrate this method. The effects of the clearance joints and uncertainties on the system dynamic response are investigated. It can be concluded that the contact behavior between the journal and the bearing in the clearance joint is not essentially influenced by the uncertain link lengths. The envelopes of the relative penetration depth and the slider acceleration of the deterministic mechanisms with different link lengths resemble the interval bounds of the corresponding responses of the uncertain mechanism. But when the link length is uncertain, the slider displacement and velocity exhibit typical interval characteristics and joint clearance would further cause the oscillation of the response bounds and the expansion of the interval length, which directly leads to the decrease of the motion precision of the mechanism. According to the parameter sensitivity analysis, however, there exist clearance effect restraint zones during the system motion, where the parameter sensitivity oscillates more slightly and the expansion of the response interval length could be reduced. Future work may consider the evolution of prediction precision on the number of the uncertain parameters, especially parameters that are directly associated with nonlinear phenomena such as contact stiffness or damping effect in the clearance joint. Acknowledgements This work is supported by the National Natural Science Foundation of China (Grant No. 11872033) and the Beijing Natural Science Foundation (Grant No. 3172017).

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