A study on dynamics of planar multibody mechanical systems with multiple revolute clearance joints

European Journal of Mechanics A/Solids 60 (2016) 95e111

Contents lists available at ScienceDirect

European Journal of Mechanics A/Solids journal homepage: www.elsevier.com/locate/ejmsol

A study on dynamics of planar multibody mechanical systems with multiple revolute clearance joints Zheng Feng Bai a, *, Yi Sun b a b

Department of Mechanical Engineering, Harbin Institute of Technology at Weihai, No.2 Wen-hua West Road, Weihai, 264209 Shandong, PR China Department of Astronautic Science and Mechanics, Harbin Institute of Technology, Harbin, 150001 Heilongjiang, PR China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 23 July 2014 Received in revised form 2 March 2015 Accepted 25 June 2016 Available online 28 June 2016

In this work, the dynamic responses of multibody mechanical systems including multiple revolute clearance joints are investigated numerically. The joint components of an imperfect joint with clearance are modeled as contact bodies. The normal force in the revolute clearance joint is modeled using a continuous contact force model, which is based on the elastic Hertz theory together with a dissipative term. The friction effect is considered using a modified Coulomb friction model. Then the dynamics equation of multibody systems with clearance joint is established. Finally, a planar four-bar mechanical system with multiple revolute clearance joints is utilized as demonstrative application example to perform the investigation. The main results of this work can be drawn that clearances in joints play a crucial role in predicting accurately the dynamic responses of multibody mechanical systems and the multiple clearance joints interact strongly with one another. © 2016 Elsevier Masson SAS. All rights reserved.

Keywords: Multibody mechanical system Multiple clearances Contact and impact Dynamics analysis Numerical simulation

1. Introduction In general dynamic analysis of multibody mechanical systems it is assumed that the kinematic joints are ideal or perfect, that is, clearance effects are neglected. However, in a real mechanical kinematical joint a clearance is always existence due to assemblage, manufacturing errors and wear. Moreover, the clearance occurs in each joint with the movement of mechanism. The movement of the real mechanism is deflected from the ideal mechanism and the motion accuracy decreases due to the joint clearances. These clearances in joints also cause impact loads, modify the dynamic response of the system, justify the deviations between the numerical predictions and the experimental measurements and eventually lead to important deviations between the projected behavior of the mechanism and their real outcome (Garcia, 2005; Flores et al., 2006a; Flores, 2009, 2010; Bai and Zhao, 2012a; Erkaya and Uzmay, 2008, Erkaya, 2012, Muvengei et al., 2012a, 2012b, 2013). Over the last few decades, many works were carried out to study

* Corresponding author. E-mail address: [email protected] (Z.F. Bai). http://dx.doi.org/10.1016/j.euromechsol.2016.06.009 0997-7538/© 2016 Elsevier Masson SAS. All rights reserved.

the dynamic responses of particular mechanisms with clearance theoretically and experimentally. Rhee and Akay (1996) investigated the dynamic responses of a revolute joint with clearance. A four-bar mechanism was implemented as an example and the response of the four-bar mechanism is examined for the case of a clearance at one of its connections by modeling the three phases of motion: contact, free-flight, and impact. And the impact model is based on the momentum-exchange principle. It also indicated that mechanisms with clearances can have nonlinear dynamic behavior. Stoenecu and Marghitu (2003) investigated the dynamic responses of a planar, rigid-link mechanism with a sliding joint clearance and the response of the system with clearance was chaotic at relatively high crank speeds and low values of the coefficient of restitution. Flores et al. (2004) presented a dynamic analysis of planar multibody systems with revolute joint clearances, including dry contact and lubricant effect. It showed that lubricant can improve the dynamic behavior of system with clearance joint. Bauchau and Ju (2006) studied the development of methodologies for the analysis of unilateral contact conditions in joints with clearance and of the resulting normal and friction forces. Two joint configurations were developed, the planar and spatial clearance joints that can deal with typical configurations where contact and clearance are likely to occur. Khemili and Romdhane (2008) investigated the

96

Z.F. Bai, Y. Sun / European Journal of Mechanics A/Solids 60 (2016) 95e111

dynamic behavior of a planar flexible slider-crank mechanism having joint with clearance. Simulation and experimental tests were carried out for this goal. This work considered one revolute clearance joint with clearance and the results showed that clearance has important effects on dynamic behavior of the system. Bing and Ye (2008) presented a general methodology for dynamic characteristics of a reheat-stop-valve mechanism with revolute clearance joints, in which the leading ingredients of the model proposed were the contact force model in consideration of the manufacturing tolerance and the thermal effects of the high temperature steam in working condition. Erkaya and Uzmay (2008, 2009a, 2009b) studied the kinematics and dynamics of planar mechanisms having revolute joints with clearance. A neural network-genetic algorithm procedure has been proposed to decrease the deviations arising from joint clearance on kinematic and dynamic characteristics of mechanisms. Also, an experimental investigation has been implemented to analyze the bearing vibrations of mechanism with joint clearances. Flores (2010) presented a parametric study on the dynamic response of planar multibody systems with clearance joints. In this investigation, it observed that multibody systems with clearance joints are well known as nonlinear dynamic systems that, under certain conditions, exhibit a chaotic response. And it was found that the dynamics of the revolute clearance joint in mechanical systems is sensitive to the clearance size value operating conditions. Zhao and Bai (2011) studied the dynamics characteristics of a space robot manipulator with one joint clearance. The nonlinear equivalent spring-damper model is established for the contact model in joint clearance. Also, the friction effect is considered using the Coulomb friction model. The results showed that clearance leaded to the motion accuracy and stability decrease. Muvengei et al. (2012a, 2012b) presented a study of a typical slider-crank mechanism with revolute clearance joint by using the LuGre friction law. The results showed that the LuGre friction law can capture the sliding and station friction together with stick-slip motions inside a revolute clearance joint. More recently, Koshy et al. (2013) presented a computational and experimental study on the contact forces developed in revolute clearance joints. For this purpose, a wellknown slider-crank mechanism with a revolute clearance joint between the connecting rod and slider is analyzed. It indicated that contact force model in clearance joint has a significant effects on the dynamics characteristics of mutibody system with clearance. Muvengei et al. (2013) investigated a slider-crank mechanism with two revolute joints with clearance and studied the different motion modes due to the clearance joints. There will be nine possible motions due to the two clearance joints. A discussion on the dynamic interaction of the motions inside the two clearance revolute joints was presented. It showed that the motion mode in one revolute clearance joint will determine the motion mode in the other clearance joints, and this will consequently affect the dynamic behavior of the system. In order to capture accurately the dynamic behavior of a multibody system, all the joints in it should be modeled as clearance joints. In conclusion, a great deal of researches on dynamic characteristics of mechanism with clearance are progressing and lots of productions are obtained, which have played a positive role in dynamic design, optimization analysis and performance improvement of mechanism with joint clearance. All the researches indicate that clearance had important effects on the dynamics characteristics of multibody mechanical systems. However, some of these works were limited to the single clearance joint. These researches are less focus on the interaction between the multiple clearances. Therefore, this work focuses on the effects of multiple clearances on the dynamic responses of the mutibody mechanical systems.

The objective of this paper is to study the dynamic characteristics of planar multibody mechanical systems with multiple revolute clearance joints using a computational methodology. The normal and frictional forces are modeled using the continuous contact force model and modified Coulomb friction model respectively once contact occurs between the journal and the bearing of the clearance revolute joint. The dynamics equation of multibody systems with clearance joint is established using a dynamic segmentation modeling method. The planar four-bar mechanism is used as demonstrative application example. Numerical results for four-bar mechanisms with multiple revolute clearance joints are presented and discussed. The effects of multiple clearances on dynamics characteristics of mechanism are investigated. The dynamics responses of mechanical systems are discussed after steady-state has been reached. Additionally, the different motion modes of mechanical systems during unsteadystate with three clearances are analyzed. This paper is organized as follows. Section 2 defines the clearance and presents the model of revolute joint with clearance. Section 3 presents the contact force model as well as the friction force model of revolute joint with clearance. Section 4 establishes the dynamics equations of multibody system with clearance. In section 5 the planar four-bar mechanism with multiple revolute clearance joints is used as numerical example to investigate the dynamic characteristics of multibody mechanical system with multiple revolute clearance joints. Finally, section 6 ends the paper with the concluding remarks.

2. Model of revolute joint with clearance 2.1. Vector model in joint clearance The clearance in joint of mechanical system is necessary to allow the relative motion of connected bodies, as well as to permit the assemblage of the mechanical system. Clearance exists also due to manufacturing tolerances, imperfections, wear and material deformation (Mukras et al., 2010; Flores and Ambrosio, 2004; Tian et al., 2010; Bai et al., 2013). It is known that the performance of mechanisms is degraded by the presence of clearance due to the contact-impact forces. These clearances modify the dynamic responses of the system and eventually lead to important deviations between the expected behavior of the mechanisms and their real outcome as well as energy dissipation and unwanted shake responses. In general, a clearance joint can be included in mechanism much like a revolute joint. The classical approach, known as zeroclearance approach, considers that the connecting points of two

Bearing

Journal eij

Oj

Oi

Fig. 1. Sketch map of clearance vector model in clearance joint.

Z.F. Bai, Y. Sun / European Journal of Mechanics A/Solids 60 (2016) 95e111

97

Bearing

bodies linked by a revolute joint are coincident. The introduction of the clearance in a joint separates these two points. The study performs the dynamic analysis of multibody mechanical system with clearance joint based on the clearance vector model, which is developed by introducing a clearance vector, eij, in a revolute joint as shown in Fig. 1. Clearance vector represents the potential real movement and the relative position between journal and bearing. Clearance vector is defined in a local floating Cartesian coordinate frame. The origin of clearance vector fixes at the center of bearing and ends at the center of journal as shown in Fig. 1. It shows that the clearance vector must be within the clearance circle, whose radius is determined by the tolerances of journal and bearing diameters. The clearance vector can identify whether the journal and bearing of clearance joint contact or not. Furthermore, it should be noted that the proposed clearance vector does not depend on the local configuration of the revolute joint because clearance is unavoidable existent in revolute joint.

Journal ey

eij

Oj

Oi ex r jo rio

Y X O

Fig. 3. Clearance model in revolute joint of mechanical system.

2.2. Definition of clearance

2.3. Mathematic model of revolute joint with clearance

Fig. 2 depicts a revolute joint with clearance. The difference in radii between the bearing and journal defines the size of the radial clearance. Although, a revolute joint with clearance does not constraint any degree of freedom from the mechanical system like the ideal joint, it imposes some kinematic restrictions, limiting the journal to move within the bearing. Thus, when clearance is present in a revolute joint, the two kinematic constraints are removed and two degrees of freedom are introduced instead. The dynamics of the joint are then controlled by forces working on the journal and bearing. Thus, whilst a perfect revolute joint in a mechanical system imposes kinematic constraints, a revolute clearance joint leads to force constraints. When contact exists between the journal and bearing, a contact force is applied perpendicular to the plane of collision. Therefore, the motion of mechanical system with clearance always includes contact-impact process. The key point of dynamics model for mechanical system with clearance is to incorporate the clearance model and dynamics model, which needs accurate modeling of the contact-impact process in joint clearance. The radial clearance is defined as follow,

c ¼ RB  RJ

In Fig. 3, Oi andOj define the centers of bearing and journal, respectively. r oi and r oj are the vectors donating the positions of bearing and journal in the global inertia coordinate, respectively. Thus, in Fig. 3, the clearance vector is represented as Eq. (2):

eij ¼ roj  roi

(2)

where eij represents the eccentric vector of journal relative to bearing. So the eccentricity of clearance vector can be given as following:

eij ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e2x þ e2y

(3)

The unit normal vector of bearing and journal at the point of their contact is represented as Eq. (4):

(1)

where RB and RJ are the radii of bearing and journal, respectively.

Bearing

RJ RB

c

Journal Fig. 2. Sketch map of revolute joint with clearance.

Fig. 4. Revolute clearance joint with contact.

98

Z.F. Bai, Y. Sun / European Journal of Mechanics A/Solids 60 (2016) 95e111

 n ¼ eij eij

(4)

eij That is n ¼ pffiffiffiffiffiffiffiffiffiffi . e2x þe2y Fig. 4 describes the impact between bearing and journal. r Q and i rQ are the positional vectors of the bearing and journal, respecj tively, which are described in global coordinate frame. The contact deformation caused by impact between bearing and journal can be calculated as Eq. (5) (Flores et al., 2004, 2006a, 2006b; Bing and Ye, 2008; Zhao and Bai, 2011; Bai and Zhao, 2012a; Muvengei et al., 2013):

d ¼ eij  c

(5)

where c represents the radial clearance and it is a constant. Thus, d can be used to decide whether the bearing and journal have contacted. The kinematic contact condition between the bearing and journal can be given by Eq. (6):

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d ¼ e2x þ e2y  c  0:

(6)

It is important to estimate the loss of energy in the course of contact between the bearing and journal, so it is necessary to calculate the relative speed between the surfaces of them. When we project d to the facet of contact, we can get the normal and tangential speed of the potential points of contact between the bearing and journal, as shown in Eq. (7):

 T vn ¼ d_ n  T : vt ¼ d_ t

(7)

where the unit tangential vector t can be achieved by reversing the unit normal vector n for 90 . 3. Contact force model of revolute joint with clearance 3.1. Normal contact force model Contact and impact, such as it happens in a revolute clearance joint, is one of the most common types of dynamic loading conditions that give rise to impulsive forces. The contact and impact forces in clearance joint excite higher vibration modes and affect the dynamic characteristics of the mechanical systems (Hunt and Crossley, 1975; Liu et al., 2007; Bai and Zhao, 2012b, 2013; Erkaya, 2012; Flores et al., 2006a, 2006b; Lankarani and Nikravesh, 1990; Machadom et al., 2012). The contact-impact model of multibody system with clearance is mainly focus on the discrete analysis method and continuous contact analysis method. The former assumes that the contactimpact is very short and did not change the overall configuration of the objects. Then, the contact-impact process is divided into two stages, before and after impact. The relative sliding, viscous stagnation and reverse movement will occur between two objects after the impact. The latter assumes that interaction forces between the impact objects are continuous in the entire contactimpact process. This approach tallies with real contact-impact behavior of objects. Therefore, the continuous contact force model is widely used for contact-impact analysis of mechanism with clearance joints. Various types of constitutive laws were suggested in the literature, being one of the more prominent proposed by Hertz. However, this law is purely elastic in nature and can not explain the energy loss during the impact process. Lankarani and Nikravesh

(1990) overcame this difficulty by separating the normal contact force into elastic and dissipative components. Thus, in this work, for revolute joint with clearance, the dry contact between the journal and bearing is modeled using the continuous contact force model proposed by Lankarani and Nikravesh and the expression of the continuous contact force model is expressed as: n Fn ¼ K d þ Dd_

(8)

where elastic deformation force is represented by the first item of the right side of Eq. (8) and the energy losing is represented by the second item. d is the deformation, d_ is the relative deformation velocity. K is the contact stiffness coefficient of the impact body, which is obtained from impact experiment of two spheres. K is obtained from the following:

#1 " Ri Rj 2 4   K¼ 3p si þ sj Ri  Rj

si ¼ sj ¼

1  v2i pEi

(9)

(10)

1  v2j

pEj

where v and E are Poisson ratio and Young modulus, respectively. Ri and Rj are radii of the two spheres. Coefficient, D, in Eq. (8) is damping coefficient and d_ is relative impact velocity in impact process. The expression of D is shown in Eq. (11):

D¼

  n 3K 1  c2e d ðÞ

4d_

(11)

ðÞ where ce is coefficient of restitution and d_ is initial relative velocity of the impact point.

3.2. Friction force model The tangential contact characteristic of clearance joint is represented using tangential friction force model. In presented work, the friction model in clearance joint is considered as dry contact and there is no lubrication. The most famous one of friction model is Coulomb friction model, which is used to represent the friction response in impact and contact process. In this paper, a modified Coulomb friction model is used to represent the friction response between the journal and bearing. Friction coefficient, which is not a constant, is introduced in the modified Coulomb friction model. Friction coefficient is a function of tangential sliding velocity, which can represent the friction response in impact and contact process as well as the viscous and micro-slip phenomenon in relative lowvelocity case more accurately. And also, the modified Coulomb friction model can avoid the case of abrupt change of friction in numerical calculation as the change of velocity direction. The expression tangential friction force is shown as Eq. (12): (Bai and Zhao, 2012a, 2012b; Bai et al., 2013).

Ft ¼ mðvt ÞFn

vt jvt j

(12)

where, friction coefficient m(vt) is a function of tangential sliding velocity and which can be expressed as:

Z.F. Bai, Y. Sun / European Journal of Mechanics A/Solids 60 (2016) 95e111

8 md signðvt Þ > > > > 

2 

) > > jvt j  vs > <  m þ ðm  m Þ jvt j  vs 32 signðvt Þ d s d vd  vs vd  vs mðvt Þ ¼ > > >

2

> > > vt þ vs > m  2m vt þ vs : 3 s s 2vs vs

where vt is relative sliding velocity in the collision point of journal and bearing, which is the velocity component in tangential direction. md is dynamic friction coefficient. ms is static friction coefficient. vs is critical velocity of static friction. vd is critical velocity of the maximum dynamic friction. The function curve of dynamic friction coefficient is shown as Fig. 5.

4. Dynamics equation of multibody system with clearance joint

A kinematic joint imposes certain conditions on the relative motion between the adjacent bodies that it comprises. When these conditions are expressed in analytical form, they are called constraint equations. In a simple way, a constraint is any condition that reduces the number of degrees of freedom in a system. The revolute joint is a bearing and journal type of joint that constrains the relative translation between the two bodies i and j, allowing only the relative rotations, as it is illustrated in Fig. 6. The kinematic conditions for the revolute joint require that two and distinguish points, each one belonging to a different body, share the same position in space all the time. This means that the global position of a point P in body i is coincident with the global position of a point P in body j. Such condition is expressed by two algebraic equations that can be obtained from the following vector loop equation,

which is re-written as,

jvt j > vd

for

vs  jvt j  vd

for

jvt j < vs

(13)

F ¼ ri þ Ai sPi  rj  Aj sPj ¼ 0

(15)

Thus, there is only one relative degree of freedom (DOF) between two bodies that are connected by a planar revolute joint. For a constrained multibody system, the kinematical joints are described by a set of holonomic algebraic constraints and F can be written in a compact form as, (Nikravesh, 1988)

Fðq; tÞ ¼ 0

(16)

where q is the vector of generalized coordinates and t is the time variable, in general associated with the driving elements.

4.1. Constraint equation for revolute joints

ri þ sPi  rj  sPj ¼ 0

for

99

(14)

4.2. Motion equations of mechanism with clearance The dynamics model of the mutibody mechanical system is established considering the clearance model. As a joint clearance embedded in the multibody system, two different motion phases of bodies connected with the clearance joints are considered, one is the bodies move freely in the clearance; the other is the bodies interact as contact or impact. Therefore the mechanism system with clearances between bodies is a variable topology system and there are three motion modes: free-flight mode, impact mode and contact mode. In the continuous contact mode, the journal follows the bearing walls and this mode is ended when the journal and bearing separate from each other to enter free-flight mode. In freeflight mode, the journal moves freely inside the bearing hence no contact force is developed at the joint. In the impact mode which occurs at the termination of the free-flight mode, impact forces are applied and removed in the system, and this mode is characterized by discontinuities in the kinematic and dynamic characteristics of the system (Muvengei et al., 2013). Since the clearance exists in revolute joint, the dynamics of joint is controlled by force constraints instead of kinematic constraints as it happens in an ideal revolute joint.

yi (i )

Oi

.

. P

siP

sPj

xi

yj

.

Oj

ri

xj

rj

Y

.

O Fig. 5. Coefficient of friction vs. slip velocity.

X

Fig. 6. Planar revolute joint connecting bodies i and j.

( j)

100

Z.F. Bai, Y. Sun / European Journal of Mechanics A/Solids 60 (2016) 95e111

The variable topology system of multibdoy mechanical system with clearance is solved by using dynamic segmentation modeling method. The equation of motion is obtained using Lagrange multiplier method and the constraints are added to the equations of motion. Therefore, the set of equations that describe the motion of the multibody system with clearance is written as Eq. (17) and Eq. (18), (Bai et al., 2013). 1) In the free motion phase, the journal and bearing separate from each other, and the journal can move freely inside the bearing boundaries, that is, the journal and the bearing are not in contact. Consequently, no reaction force is developed at the joint. Thus, the equations of motion and algebraic kinematic independent holonomic constraints for multi-body system can be written as

Mq€ þ C q_ þ Kq þ FTq l ¼ f Fðq; tÞ ¼ 0

2) In interact phase, the bodies contact or impact. The contact forces exist in the clearances. Therefore, the contact-impact force expressed by Eq. (8) and non-linear friction expressed by Eq. (12) are applied in the system. To avoid constraints violation during numerical integration, Eq. (17) can be modified as

(18)

where Fc is the generalized force vector relative to the q, which contains both normal contact-impact force, Fn expressed as Eq. (8), and tangential friction force, Ft, expressed as Eq. (12) in contact mode or impact mode. The above dynamic models are developed respectively by introducing generalized force for contact-separate mode of the clearance joints, and adding and/or deleting kinematic constraint equations for contact-impact-separate description to facilitate the global numerical simulation.

4.3. Contact-separate detection for clearance joints

Crank

B

l2

l3

l1

Follower

y O

x

C

l4

Ground

Fig. 7. Four-bar mechanism with multiple clearance joints.

5. Demonstrative application to a planar four-bar mechanism 5.1. Description of the four-bar mechanism In this section, the academic planar four-bar mechanism is used to illustrate the dynamics of multibody system with multiple revolute clearance joints. Fig. 7 depicts the kinematic configuration of the four-bar mechanism. The four-bar mechanism consists of four rigid bodies that represent the crank, coupler, follower, ground, and one ideal revolute joint connecting the ground to the crank. Three revolute joints with clearance exist between the crank and the coupler (Joint A), between the couple and follower (Joint B), and between the ground and the follower (Joint C). In the dynamic simulation, the crank is the driving link and rotates at a constant angular velocity of 600 r/min. The initial configuration corresponding to crank and ground is vertical and the initial angular velocity is zero. Initially, the journal and bearing centers are coincident. In order to analyze the dynamic responses of the system, long time simulations are performed and the results presented below are plotted against two full crank rotations after steady-state has been reached. Table 1 provides the parameters which were used in the simulation of the four-bar mechanism with multiple revolute clearance joints. 5.2. Results and discussion

In order to evaluate the contact-impact force and friction, it is necessary to require the knowledge of the pre-impact conditions. Thus, one of the most critical aspects in the dynamic simulation of the multibody system with impact is the detection of the precise presence of contact. Positive values of d given by Eq. (5) mean that the contact/impact mode is presence between the journal and the bearing. Thus, the detection of the presence of contact and impact mode is the value of d changes between the two discrete moments, t and t þ Dt, that is

dðqðtÞÞT dðqðt þ DtÞÞ < 0

A

(17)

where q is the generalized coordinate column matrix, M, C and K are the generalized mass matrix, generalized damp matrix and generalized stiffness matrix respectively. Fq is the Jacobin matrix of constraint equation. f is the generalized force matrix. l is the Lagrange multiplier column matrix.

Mq€ þ C q_ þ Kq þ FTq l ¼ f þ F c Fðq; tÞ ¼ 0

Couple

(19)

Therefore, contact is detected when Eq. (19) is verified. An alternative way to determine the presence of contact for multiple time-step size integrators uses the characteristics of the integration algorithm selected. The step size is proposed to reach smaller values to keep the integration tolerance error. Thus, the impact velocity and the direction of the plane of collision can be predicted in the overall system motion.

5.2.1. Effects of the clearance size Firstly, in this section, the effects of clearance on the dynamic Table 1 Parameters used in the dynamic simulation of the four-bar mechanism. Length of crank Length of couple Length of follower Length of ground Mass of crank Mass of couple Mass of follower Moment of inertia of crank Moment of inertia of couple Moment of inertia of follower Young’s modulus Restitution coefficient Dynamic friction coefficient Static friction coefficient Input speed of the crank

0.55 m 0.36 m 0.64 m 0.21 m 3.6235 kg 2.4394 kg 4.1894 kg 0.10214 kgm2 0.031428 kgm2 0.15796 kgm2 207 GPa 0.9 0.1 0.3 600 r/min

Z.F. Bai, Y. Sun / European Journal of Mechanics A/Solids 60 (2016) 95e111 250

7500

Without clearance One clearance,c=0.25mm

200

Without clearance One clearance,c=0.25mm 6500 Follower velocity / (°/s)

150

Follower angle / °

101

100 50 0 -50

5000

3500

-100 -150 -200 0

180

360 Crank angle / °

540

2000 0

720

180

360 Crank angle / °

(a)

540

720

(b)

5

4

x 10

10000 Without clearance One clearance,c=0.25mm

Without clearance One clearance,c=0.25mm

8000

Reac tion c rank m om ent / Nm

Follower acceleration / (°/s 2)

3 2 1 0 -1 -2

4000

0

-4000

-3 -4 0

180

360 Crank angle / °

540

720

-8000 0

180

360 Crank angle / °

(c)

720

(d) -4

4

2.5

540

x 10

4 ideal joint One clearance,c=0.25mm

x 10

Clearance Circle One clearance,c=0.25mm

3

2

1

1.5

Y/m

Joint reaction force / N

2

1

0 -1 -2

0.5 -3 0 0

180

360 Crank angle / °

540

720

(e)

-4 -4

-3

-2

-1

0 X/m

1

2

3

4 -4

x 10

(f)

Fig. 8. Dynamic characteristics of four-bar mechanism with one clearance joint ((a) Follower angular displacement; (b) Follower angular velocity; (c) Follower angular acceleration; (d) Crank moment required to maintain the crank angular velocity constant; (e) Joint reaction force at the clearance joint; (f) Journal center orbit relative to the bearing center in unsteady state).

responses of the four-bar mechanism are analyzed. In the simulation of the mechanism, only one joint is modeled as clearance joint, namely, the joint that connects the couple and follower (Joint B). The dynamic characteristics of the four-bar mechanism is

obtained and represented in Fig. 8. Two different kinds of results are presented. One is all the joints of four-bar mechanism are considered as ideal, in which no clearances exist. The other is the mechanism is simulated with a revolute clearance joint, which

102

Z.F. Bai, Y. Sun / European Journal of Mechanics A/Solids 60 (2016) 95e111

4

x 10

between couple and follower and the clearance size is 0.25 mm. Fig. 8(a)e(c) are the angular displacement, velocity and acceleration of the follower, respectively. The moment acting on the crank, which is required to maintain the crank angular velocity constant, is presented in Fig. 8(d). The joint reaction force at the clearance joint is presented in Fig. 8(e). The relative motion between the journal and bearing centers is plotted in Fig. 8(f). Fig. 8(a) shows that the angular displacement curves of follower are almost completely a coincidence between the ideal mechanism without clearance and real mechanism with clearance. Fig. 8(b) shows that the angular velocity of follower is slight shaky and not greatly affected by the existence of the joint clearance. It indicates that clearance has less effect on angular velocity of follower. However, in contrast to the angular displacement and angular velocity, the angular acceleration of the follower shows significant differences between the dynamic response of the mechanism when modeled with and without joint clearance, as shown in Fig. 8(c). The existence of clearance affects the angular acceleration of follower by leading to severe vibration. The sudden changes in angular acceleration of follower are due to the impact between the journal and bearing for the clearance joint. The angular acceleration of follower with clearance is obviously shaky and presents high peaks

-4

3 2

Y/m

1 0 -1 -2 -3 -4 -4

-2

0

2

4

X/m

x 10

-4

Fig. 9. Journal center orbit relative to the bearing center in unsteady state.

5

5

4

x 10

4 Without clearance One clearance,c=0.05mm

Without clearance One clearance,c=0.1mm

3

2

Follower acceleration / (°/s 2)

Follower acceleration / (°/s 2)

3

x 10

1 0 -1 -2

2 1 0 -1 -2 -3

-3 -4 0

180

360 Crank angle / °

540

-4 0

720

180

(a) Without clearance One clearance,c=0.25mm

x 10

3

Follower acceleration / (°/s 2)

Follower acceleration / (°/s 2)

4

2 1 0 -1 -2

Without clearance One clearance,c=0.4mm

2 1 0 -1 -2

-3 -4 0

720

(b)

x 10

3

540

5

5

4

360 Crank angle / °

180

360 Crank angle / °

(c)

540

720

-3 0

180

360 Crank angle / °

540

(d)

Fig. 10. Acceleration of follower with different clearance sizes ((a) c ¼ 0.05 mm; (b) c ¼ 0.1 mm; (c) c ¼ 0.25 mm; (d) c ¼ 0.4 mm).

720

Z.F. Bai, Y. Sun / European Journal of Mechanics A/Solids 60 (2016) 95e111 4

1

4

x 10

1

x 10

Without clearance One clearance,c=0.1mm

Reaction crank moment / Nm

Reaction crank moment / Nm

Without clearance One clearance, c=0.05mm 0.5

0

-0.5

-1 0

180

360 Crank angle / °

540

0.5

0

-0.5

-1 0

720

180

(a) 1

720

540

720

x 10

Without clearance One clearance,c=0.4mm

Reaction crank moment / Nm

Without clearance One clearance,c=0.25mm

Reaction crank moment / Nm

540

4

x 10

0.5

0

-0.5

-1 0

360 Crank angle / °

(b)

4

1

103

180

360 Crank angle / °

540

720

(c)

0.5

0

-0.5

-1 0

180

360 Crank angle / °

(d)

Fig. 11. Crank moment with different clearance sizes ((a) c ¼ 0.05 mm; (b) c ¼ 0.1 mm; (c) c ¼ 0.25 mm; (d) c ¼ 0.4 mm).

of its values, which indicates that, when considering the clearance, the existence of clearances will cause vibration of the four-bar mechanism and influence the dynamic performance of the system. The same phenomena can be observed in the curve of crank moment, which is required to maintain the crank angular velocity constant, presented by Fig. 8(d). Furthermore, the system’s response repeats itself from cycle to cycle clearly. Fig. 8(e) shows that the existence of clearance leads to the impact force in joint increase and the impact force is high-frequency vibration. The contact force changes periodically, repeatedly and severely. In each cycle, the contact force is very large in certain range of crank angle, which explains the acceleration characteristics in Fig. 8(c). Fig. 8(f) shows that the journal is always in contact with the bearing wall. This observation is logical since the two bodies are moving in the same direction. And the journal and bearing contact frequently in some special regions. It can be see that the journal and bear is in continuous contact mode when the steady-state of mechanism with clearance has been reached. Fig. 8 shows the dynamic responses of mechanism with clearance and the journal and bear is in continuous contact mode when the steady-state of the system has been reached. Further,

considering unsteady-state of the system, such as initial state, the motion of the mechanism due to one clearance is in three modes, that is, continuous contact mode, the free-flight mode and the impact mode, as shown in Fig. 9, where the journal center orbit relative to the bearing center is plotted in unsteady-state. Clearance size is one of the most important parameters that affect the dynamic responses of mechanical system. Further, the effects of clearance size on the dynamics characteristics of mechanical system are studied. The dynamic simulation of four-bar mechanism with different clearance sizes is carried out. The values of the clearance size are considered as 0.05 mm, 0.1 mm 0.25 mm and 0.4 mm. Fig. 10 presents the effects of clearance size on the angular acceleration of follower. The angular accelerations tend to be close to those obtained for ideal joint when clearance size is reduced. However, when the clearance size is increased, the dynamic responses of mechanism are changed obviously, which is represented by shaky with higher peaks. It also shows that the higher size of clearance will lead to the more obvious shake and higher peaks of angular acceleration. The same conclusion can be drawn from Fig. 11, where the crank moment, which is required to maintain the crank angular velocity constant, is presented. Fig. 12

104

Z.F. Bai, Y. Sun / European Journal of Mechanics A/Solids 60 (2016) 95e111 4

4

3

x 10

ideal joint One clearance,c=0.05mm

x 10

ideal joint One clearance,c=0.1mm

2.5

Joint reaction force / N

2.5

Joint reaction force / N

3

2

1.5

1

0.5

2

1.5

1

0.5

0 0

180

360 Crank angle / °

540

0 0

720

180

(a) ideal joint One clearance,c=0.25mm

540

720

x 10

2.5

Joint reaction force / N

Joint reaction force / N

3

2

1.5

1

0.5

0 0

720

4

x 10

2.5

540

(b)

4

3

360 Crank angle / °

ideal joint One clearance,c=0.4mm

2

1.5

1

0.5

180

360 Crank angle / °

540

720

(c)

0 0

180

360 Crank angle / °

(d)

Fig. 12. Joint reaction force with different clearance sizes ((a) c ¼ 0.05 mm; (b) c ¼ 0.1 mm; (c) c ¼ 0.25 mm; (d) c ¼ 0.4 mm).

shows the joint reaction force with different clearance sizes. As shown in Fig. 12, it indicates that the higher size of clearance induces the higher peaks of the contact force, which explains the acceleration characteristics in Fig. 10. Therefore, the level of the contact peaks increases with the increase of the clearance size. The behavior of the revolute clearance joint is illustrated by plotting the orbit of the journal inside the bearing boundaries, as shown in Fig. 13. It shows that the journal is always in contact with the bearing wall. However, the contact behavior of the revolute clearance joint is different when steady-state has been reached. Further, the simulation results are compared to other studies from previous literature (Erkaya and Uzmay, 2010; Bauchau and Rodriguez, 2002; Khemili and Romdhane, 2008; Flores and Ambrosio, 2004), in which the research results also showed that clearance had important effects on the dynamics characteristics of multibody mechanical systems and bigger size of clearance caused higher peaks of impact force and worse characteristics of the systems. So the simulation results are validated by other data published on the field on dynamics of multibody systems with clearance joints. The fact that the existence of the clearance joint

has an important effect on the system supports the idea that the model of clearance joints must be considered in the analysis and design of the real mechanical system. 5.2.2. Effects of the number of clearance joints In this section, the effects of the number of clearance joints on the dynamic responses of the multibody mechanical system are discussed. The simulation of the four-bar mechanism is performed with one, two and three clearance joints respectively. The four-bar mechanism consists of four rigid bodies that represent the crank, coupler, follower, ground, and one ideal revolute joint connecting the ground to the crank. Three revolute joints with clearance exist between the crank and the coupler (Joint A), between the couple and follower (Joint B), and between the ground and the follower (Joint C), as shown in Fig. 7. Here, the clearance size of each revolute clearance joint is considered as 0.1 mm and the crank is the driving link and rotates at a constant angular velocity of 600 r/ min. Four different cases are studied to investigate the effects of the number of clearance joints on the dynamic response of the

Z.F. Bai, Y. Sun / European Journal of Mechanics A/Solids 60 (2016) 95e111 -4

1

105

-4

x 10

2

x 10

Clearance Circle One clearance,c=0.05mm

Clearance Circle One clearance,c=0.1mm

Y/m

Y/m

1

0

0

-1

-1 -1

0 X/m

-2 -2

1

-1

-4

x 10

(a)

2 -4

x 10

(b)

x 10

3

1

-4

-4

4

0 X/m

6

x 10

Clearance Circle One clearance,c=0.4mm

Clearance Circle One clearance,c=0.25mm

3

2

Y/m

Y/m

1 0

0

-1

-3

-2 -3 -4 -4

-2

0 X/m

2

4 -4

x 10

(c)

-6 -6

-3

0 X/m

3

6 -4

x 10

(d)

Fig. 13. Journal center orbit relative to the bearing center with different clearance sizes ((a) c ¼ 0.05 mm; (b) c ¼ 0.1 mm; (c) c ¼ 0.25 mm; (d) c ¼ 0.4 mm).

multibody mechanical system. Therefore, the four-bar mechanism is simulated without clearance joint and with one, two and three clearance joints, respectively. So, the first case is all the joints of four-bar mechanism are considered as ideal, in which no clearances exist. The second case is the mechanism is simulated with one clearance joint, which exists between couple and follower (Joint B). The third case is the mechanism is simulated with two clearance joints, which exist between the couple and follower (Joint B) and between the ground and follower (Joint C). The fourth case is the mechanism is simulated with three clearance joints, which exist between the crank and the coupler (Joint A), between the couple and follower (Joint B) and between the ground and follower (Joint C). The dynamic characteristics of the four-bar mechanism with different number of clearance joints are obtained. The simulation results presented below are plotted against two full crank rotations after steady-state has been reached. Figs. 14 and 15 present the angular velocity and acceleration respectively. The moment acting on the crank, which is required to maintain the crank angular

velocity constant, is presented in Fig. 16. The joint reaction forces at the clearance joints are represented in Figs. 17 and 18. Fig. 14 shows that the angular velocity of follower is not greatly affected by the number of clearance joints. However, from the partial enlarged drawing, these figures show that the effects of multiple clearances on the angular velocity are different with that of different clearance number. The angular velocity of mechanism is vibration both with one clearance and multiple clearances, but the vibration peaks and locations are different. Therefore, the dynamic responses of mechanism are different with different number of clearance joints. Fig. 15 shows that the angular acceleration of the follower presents significant differences between the dynamic behavior of the mechanism when modeled with one, two and three clearance joints. When more joints are modeled as clearance joints, the dynamic responses of the mechanism are changed more obviously, which is represented by vibration with higher frequency and peaks. It indicates that when the number of clearance joints increases, the angular acceleration is more obvious vibration, which is represented by higher frequency shake

106

Z.F. Bai, Y. Sun / European Journal of Mechanics A/Solids 60 (2016) 95e111 7500

7500

Without clearance Two clearance joints,c=0.1mm

Without clearance One clearance,c=0.1mm 6500

Follower velocity / (°/s)

Follower velocity / (°/s)

6500

5000

3500

2000 0

180

360 Crank angle / °

540

5000

3500

2000 0

720

180

360 Crank angle / °

(a)

540

720

(b) 6300

7500 Without clearance Three clearance joints,c=0.1mm

Follower velocity / (°/s)

Follower velocity / (°/s)

6500

5000

3500

2000 0

180

360 Crank angle / °

540

720

(c)

6100

5900

5700 240

Without clearance One clearance,c=0.1mm Two clearance,c=0.1mm Three clearance,c=0.1mm

245

250

255 260 Crank angle / °

265

270

(d)

Fig. 14. Angular velocity of follower with different number of clearance joints ((a) One clearance joint exist between the couple and follower (Joint B); (b) Two clearance joints: one clearance joint between the couple and follower (Joint B) and one clearance joint between the ground and follower (Joint C); (c) Three clearance joints: one clearance joint between the crank and the coupler (Joint A), one clearance joint between the couple and follower (Joint B) and one clearance joint between the ground and follower (Joint C); (d) Partial zoom-in and comparison of the results).

and higher vibration peaks. The same conclusion can also be drawn from Fig. 16, where the crank moment, which is required to maintain the crank angular velocity constant, is presented. Therefore, it is well visible that the characteristics of mechanism are not affected in the same manner when considering different number of clearance joints. More clearance joints will lead to higher frequency vibration and higher vibration peaks. The reason is that the clearance joints interact strongly with one another. So, when more joints are considered as clearance joints, the effects on the dynamics characteristics of multibody mechanical systems are more obvious. Figs. 17 and 18 show the joint reaction forces of mechanism with different number of clearance joints. As shown in Fig. 17, the reaction force of joint B is significant differences when the mechanism was modeled with one, two and three clearance joints. It shows that when joint B is modeled as clearance joint and other joints are modeled as ideal joint, the contact force is less shaky. However, when Joint A and Joint C are also modeled as clearance joints, the reaction force of joint B is obvious vibration and presents the higher peaks of the contact force. It indicates that when more

joints modeled as clearance joints, the contact force at each clearance joint is more obvious vibration with higher frequency and higher peaks, which explains the acceleration characteristics in Fig. 15. In addition, the same phenomena can be observed in the curve of contact force presented by Fig. 18, in which the reaction force of clearance Joint C is presented. Therefore, the level of the contact peaks in each clearance joint increase with the increase of the number of clearance joints. From the simulation results presented, the clearance joints have been found to interact strongly with one another. Therefore, when more joints are considered as clearance joints, the reaction force of each clearance joints of multibody mechanical systems is changed more obviously. It also can be drawn that the journal and bear is in continuous contact mode when the steady-state of mechanism with clearance has been reached. Additionally, considering unsteady-state of the system, a mechanism with n-number of revolute clearance joints will exhibit 3n motions which are combinations of the three modes in a single clearance joint (Muvengei et al., 2013). Therefore, in this paper, a mechanism with three clearance joints will exhibit 27

Z.F. Bai, Y. Sun / European Journal of Mechanics A/Solids 60 (2016) 95e111 6

1

Follower acceleration / (°/s 2)

6

x 10

1.2 Without clearance One clearance,c=0.1mm

0.6

0.2 0 -0.2

Without clearance Two clearance joints,c=0.1mm

0.6

0.2

-0.2

-0.6

-0.6

-1 0

x 10

1

Follower acceleration / (°/s 2)

1.2

107

180

360 Crank angle / °

540

-1 0

720

180

360 Crank angle / °

(a)

540

720

(b) 6

1.2

x 10

Follower acceleration / (°/s 2)

1

Without clearance Three clearance joints,c=0.1mm

0.6

0.2

-0.2

-0.6

-1 0

180

360 Crank angle / °

540

720

(c) Fig. 15. Acceleration of follower with different number of clearance joints ((a) One clearance joint exist between the couple and follower (Joint B); (b) Two clearance joints: one clearance joint between the couple and follower (Joint B) and one clearance joint between the ground and follower (Joint C); (c) Three clearance joints: one clearance joint between the crank and the coupler (Joint A), one clearance joint between the couple and follower (Joint B) and one clearance joint between the ground and follower (Joint C)).

motions which are combinations of the three modes in a single clearance joint. The 27 motions are as Table 2. Fig. 19 shows the contact forces in revolute clearance joints corresponding to the time interval [0.182 0.202] seconds for mechanism with three clearance joints. From Fig. 19, we can identify some motion modes of the system, such as free-contact-contact motion, contactcontact-contact motion and other motion modes. It indicates that when considering multiply clearance joints, the mechanism has a variety of motions. In summary, from the dynamics analysis and discussion of mechanical system with multiple clearances, we can find that the output responses of mechanical systems are obviously vibration and presents higher peaks of its values. It indicates that the effects of clearance on the dynamic characteristics of mechanical system can not be ignored and the existence of clearance causes the dynamics characteristics of mechanical systems changed and worse. In addition, it also can be found that the clearance joints interact strongly with one another. It indicates that as the number of the clearance joints increase, more joints are controlled by force constraints instead of kinematic constraints as it happens in an ideal

revolute joint. Therefore, when all the joints of a mechanism considered as imperfect joints, the system will transfer to a pure force-based system from an ideal-jointed system. It will cause more clearance joints interact strongly with one another. The effects of clearance on the dynamics characteristics of mechanism will more significant. Therefore, when more joints are modeled as clearance joints, the dynamic responses of the mechanism are changed more obviously, which is represented by vibration with higher frequency and peaks. The fact that multiple clearances have important effects on the dynamics responses supports the idea that modeling all the joints in multibody mechanical system as clearance joints must be considered in the analysis and design of the real mechanical system. 6. Conclusions In this work, the effects of multiple clearances on dynamic responses of multibody mechanical systems are investigated. The normal force in the revolute clearance joints is modeled using a continuous contact force model and the friction effect is considered

108

Z.F. Bai, Y. Sun / European Journal of Mechanics A/Solids 60 (2016) 95e111 4

4

x 10

Reaction crank moment / Nm

1.5

2 Without clearance One clearance,c=0.1mm

1 0.5 0 -0.5 -1

Without clearance Two clearance joints,c=0.1mm

1 0.5 0 -0.5 -1

-1.5 -2 0

x 10

1.5

Reaction crank moment / Nm

2

180

360 Crank angle / °

540

-1.5 0

720

180

360 Crank angle / °

(a)

540

720

(b) 4

2

x 10

Reaction crank moment / Nm

1.5

Without clearance Three clearance joints,c=0.1mm

1 0.5 0 -0.5 -1 -1.5 0

180

360 Crank angle / °

540

720

(c) Fig. 16. Crank moment with different number of clearance joints ((a) One clearance joint exist between the couple and follower (Joint B); (b) Two clearance joints: one clearance joint between the couple and follower (Joint B) and one clearance joint between the ground and follower (Joint C); (c) Three clearance joints: one clearance joint between the crank and the coupler (Joint A), one clearance joint between the couple and follower (Joint B) and one clearance joint between the ground and follower (Joint C)).

using a modified Coulomb friction model. Finally, a planar four-bar mechanical system with multiple revolute clearance joints is utilized as demonstrative application example to perform the studied. The dynamics responses of mechanical systems are discussed after steady-state has been reached. Additionally, the different motion modes of mechanical systems during unsteady-state with three clearances are analyzed. The effects of clearance on the dynamic characteristics of mechanism can not be ignored and the existence of clearance causes vibration of the acceleration and decrease of the motion stability. The angular acceleration of mechanism with clearance is obviously shaky and presents high peaks of its values, which indicates that, when considering the clearance, the existence of clearances will cause vibration of the mechanism and influence the dynamic performance of the system. It also can be found that the journal and bear is in continuous contact mode when the steadystate of mechanism with clearance has been reached. However, during the unsteady-state of mechanism with one clearance, there are three motion modes of the system, namely continuous contact mode, the free-flight mode and the impact mode. In addition, the higher size of clearance will lead to the more obvious vibration and higher peaks of the acceleration of mechanism.

Further, when considering multiple clearance joints, the angular acceleration of the mechanism is significant differences when the mechanism was modeled with one, two and three clearance joints. Therefore, the characteristics of mechanism are not affected in the same manner when considering different number of clearance joints. It indicates that when more joints are modeled as clearance joints, the dynamic responses of the mechanism are changed more obviously, which is represented by vibration with higher frequency and peaks. It also can be found that the clearance joints interact strongly with one another, which leads to worse of the performance of the mechanical system. The journal and bear is in continuous contact mode when the steady-state of mechanism with clearance has been reached. However, when considering multiply clearance joints, the mechanism has a variety of motions in unsteady-state. In this work, a mechanism with three clearance joints will exhibit 27 motions. Clearances in joints play a crucial role in predicting accurately the dynamic responses of multibody systems. The dynamics modeling and analysis of multibody systems considering multiple clearance joints is the basis of precision analysis and design of mechanism. Therefore, in order to capture the dynamic behavior of a multibody mechanical system accurately, all the joints in the

Z.F. Bai, Y. Sun / European Journal of Mechanics A/Solids 60 (2016) 95e111 4

4

4

x 10

4 ideal joint One clearance,c=0.1mm

3.5

x 10

ideal joint Two clearance joints,c=0.1mm

3.5 3

Joint reaction force / N

3

Joint reaction force / N

109

2.5 2 1.5

2.5 2 1.5

1

1

0.5

0.5

0 0

180

360 Crank angle / °

540

0 0

720

180

360 Crank angle / °

(a)

540

720

(b) 4

4

x 10

ideal joint Three clearance joints,c=0.1mm

3.5

Joint reaction force / N

3 2.5 2 1.5 1 0.5 0 0

180

360 Crank angle / °

540

720

(c) Fig. 17. Reaction force of clearance joint B for mechanism with different number of clearance joints ((a) one clearance joint; (b) two clearance joints; (c) three clearacne joints).

4

4

x 10

Joint reaction force / N

5

6 ideal joint Two clearance joints,c=0.1mm

4

3

2

ideal joint Three clearance joints,c=0.1mm

4

3

2

1

1

0 0

x 10

5

Joint reaction force / N

6

180

360 Crank angle / °

(a)

540

720

0 0

180

360 Crank angle / °

540

720

(b)

Fig. 18. Reaction force of clearance joint C for mechanism with different number of clearance joints ((a) two clearance joints; (b) three clearacne joints).

110

Z.F. Bai, Y. Sun / European Journal of Mechanics A/Solids 60 (2016) 95e111

Table 2 The 27 motion modes of the four-bar mechanism with three clearance joints. Motions

Mode descriptions

Free-free-free motion Free-free-contact motion Free-free-impact motion Free-contact-free motion Free-contact-contact motion Free-contact-impact motion Free-impact-free motion Free-impact-contact motion Free-impact-impact motion Contact-free-free motion Contact-free-contact motion Contact-free-impact motion Contact-contact-free motion Contact-contact-contact motion Contact-contact-impact motion Contact-impact-free motion Contact-impact-contact motion Contact-impact-impact motion Impact-free-free motion Impact -free-contact motion Impact -free-impact motion Impact -contact-free motion Impact -contact-contact motion Impact -contact-impact motion Impact -impact-free motion Impact -impact-contact motion Impact -impact-impact motion

Free-flight mode in clearance joint A, free-flight mode in clearance joint B and free-flight mode in clearance joint C. Free-flight mode in clearance joint A, free-flight mode in clearance joint B and contact mode in clearance joint C. Free-flight mode in clearance joint A, free-flight mode in clearance joint B and impact mode in clearance joint C. Free-flight mode in clearance joint A, contact mode in clearance joint B and free-flight mode in clearance joint C. Free-flight mode in clearance joint A, contact mode in clearance joint B and contact mode in clearance joint C. Free-flight mode in clearance joint A, contact mode in clearance joint B and impact mode in clearance joint C. Free-flight mode in clearance joint A, impact mode in clearance joint B and free mode in clearance joint C. Free-flight mode in clearance joint A, impact mode in clearance joint B and contact mode in clearance joint C. Free-flight mode in clearance joint A, impact mode in clearance joint B and impact mode in clearance joint C. Contact mode in clearance joint A, free-flight mode in clearance joint B and free-flight mode in clearance joint C. Contact mode in clearance joint A, free-flight mode in clearance joint B and contact mode in clearance joint C. Contact mode in clearance joint A, free-flight mode in clearance joint B and impact mode in clearance joint C. Contact mode in clearance joint A, contact mode in clearance joint B and free-flight mode in clearance joint C. Contact mode in clearance joint A, contact mode in clearance joint B and contact mode in clearance joint C. Contact mode in clearance joint A, contact mode in clearance joint B and impact mode in clearance joint C. Contact mode in clearance joint A, impact mode in clearance joint B and free mode in clearance joint C. Contact mode in clearance joint A, impact mode in clearance joint B and contact mode in clearance joint C. Contact mode in clearance joint A, impact mode in clearance joint B and impact mode in clearance joint C. Impact mode in clearance joint A, free-flight mode in clearance joint B and free-flight mode in clearance joint C. Impact mode in clearance joint A, free-flight mode in clearance joint B and contact mode in clearance joint C. Impact mode in clearance joint A, free-flight mode in clearance joint B and impact mode in clearance joint C. Impact mode in clearance joint A, contact mode in clearance joint B and free-flight mode in clearance joint C. Impact mode in clearance joint A, contact mode in clearance joint B and contact mode in clearance joint C. Impact mode in clearance joint A, contact mode in clearance joint B and impact mode in clearance joint C. Impact mode in clearance joint A, impact mode in clearance joint B and free mode in clearance joint C. Impact mode in clearance joint A, impact mode in clearance joint B and contact mode in clearance joint C. Impact mode in clearance joint A, impact mode in clearance joint B and impact mode in clearance joint C.

2.5

x 10

4

Clearance joint B Clearance joint C Clearance joint A

2

Contact forces / N

1.5

1

0.5

0 Free-contact-Contact mode -0.5 0.182

0.184

0.186

Contact-contact-impact mode 0.188

0.19

0.192

Contact-contact-free mode 0.194

0.196

Contact-contact-contact mode 0.198

0.2

0.202

Time / s

Fig. 19. Contact forces for revolute clearance joint A, B and C of mechanism with three clearance joints.

system should be modeled as clearance joints in future works. When all the joints of a mechanism considered as imperfect joints, the system will transfer to a pure force-based system from an idealjointed system. It will cause more clearance joints interact strongly with one another. If all the four revolute joints of the 4-bar mechanism are considered as clearance joint, it will cause 81 motions. Therefore, the effects of clearance on the dynamics characteristics of mechanism will be more and more significant, especially the nonlinear characteristics of the system, which will be researched in our future work. This work presents an investigation of mechanical systems with multiple clearance joints, which puts forward up dynamics analysis of multibody mechanical system

with clearance joints and improves the engineering application. Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant No.: 51305093), the Natural Science Foundation of Shandong Province (Grant No.: ZR2013EEQ004). Project supported by the “China Postdoctoral Science Foundation funded project (Grant Nos. 2012M520723; 2014T70317)” are also gratefully acknowledged. This work was also supported by the National Key Basic Research Program of China (No. 2013CB733000).

Z.F. Bai, Y. Sun / European Journal of Mechanics A/Solids 60 (2016) 95e111

References Bai, Z.F., Zhao, Y., 2012a. Dynamics modeling and quantitative analysis of multibody systems including revolute clearance joint. Precis. Eng. 36, 554e567. Bai, Z.F., Zhao, Y., 2012b. Dynamic behaviour analysis of planar mechanical systems with clearance in revolute joints using a new hybrid contact force model. Int. J. Mech. Sci. 54, 190e205. Bai, Z.F., Zhao, Y., 2013. A hybrid contact force model of revolute joint with clearance for planar mechanical system. Int. J. Nonlinear Mech. 48, 15e36. Bai, Z.F., Zhao, Y., Wang, X.G., 2013. Wear analysis of revolute joints with clearance in multibody systems. Sci. China Phys. Mech. Astron. 56, 1581e1590. Bauchau, O.A., Ju, C.K., 2006. Modeling friction phenomena in flexible multibody dynamics. Comp. Method Appl. M 195, 6909e6924. Bauchau, O.A., Rodriguez, J., 2002. Modeling of joints with clearance in flexible multi-body system. Int. J. Solids Struct. 34, 41e63. Bing, S., Ye, J., 2008. Dynamic analysis of the reheat-stop-valve mechanism with revolute clearance joint in consideration of thermal effect. Mech. Mach. Theory 43 (12), 1625e1638. Erkaya, S., 2012. Investigation of joint clearance effects on welding robot manipulators. Robot. Comput. Integr. Manuf. 28, 449e457. Erkaya, S., Uzmay, I., 2008. A neural-genetic (NN-GA) approach for optimizing mechanisms having joints with clearance. Multibody Syst. Dyn. 20 (1), 69e83. Erkaya, S., Uzmay, I., 2009a. Investigation on effect of joint clearance on dynamics of four-bar mechanism. Nonlinear Dyn. 58, 179e198. Erkaya, S., Uzmay, I., 2009b. Determining link parameters using genetic algorithm in mechanisms with joint clearance. Mech. Mach. Theory 44, 222e234. Erkaya, S., Uzmay, I., 2010. Experimental investigation of joint clearance effects on the dynamics of a slider-crank mechanism. Multibody Syst. Dyn. 24, 81e102. Flores, P., 2009. Modeling and simulation of wear in revolute clearance joints in multibody systems. Mech. Mach. Theory 44, 1211e1222. Flores, P., 2010. Parametric study on the dynamic response of planar multibody systems with multiple clearance joints. Nonlinear Dyn. 61, 633e653. Flores, P., Ambrosio, J., 2004. Revolute joints with clearance in multibody systems. Comput. Struct. 82, 1359e1369. sio, J., Claro, J.P., 2004. Dynamic analysis for planar multibody Flores, P., Ambro mechanical systems with lubricated joints. Multibody Syst. Dyn. 12, 47e74. Flores, P., Ambrosio, J., Claro, J.C.P., Lankarani, H.M., Koshy, C.S., 2006a. A study on dynamics of mechanical systems including joints with clearance and lubrication. Mech. Mach. Theory 41, 247e261. Flores, P., Ambrosio, J., Claro, J.C.P., Lankarani, H.M., 2006b. Influence of the contactimpact force model on the dynamic response of multibody systems. P. I. Mech. Eng. K-J Multibody Dyn. 220 (1), 21e34.

111

Garcia, O.J.C., 2005. Analysis of clearance in multibody system. Multibody Syst. Dyn. 13, 401e420. Hunt, K.H., Crossley, F.R.E., 1975. Coefficient of restitution interpreted as damping in vibroimpact. J. Appl. Mech. 42, 440e445. Khemili, I., Romdhane, L., 2008. Dynamic analysis of a flexible slider-crank mechanism with clearance. Eur. J. Mech. A Solid 2, 882e898. Koshy, C.S., Flores, P., Lankarani, H.M., 2013. Study of the effect of contact force model on the dynamic response of mechanical systems with dry clearance joints-computational and experimental approaches. Nonlinear Dyn. 73,325e73,338. Lankarani, H.M., Nikravesh, P.E., 1990. A contact force model with hysteresis damping for impact analysis of multibody systems. ASME J. Mech. Des. 112, 369e376. Liu, C.S., Zhang, K., Yang, R., 2007. The FEM analysis and approximate model for cylindrical joints with clearances. Mech. Mach. Theory 42, 183e197. Machadom, M., Moreira, Pe, Flores, P., Lankarani, H.M., 2012. Compliant contact force models in multibody dynamics evolution of the Hertz contact theory. Mech. Mach. Theory 53, 99e121. Mukras, S., Kim, N.H., Mauntler, N.A., Schmitz, T.L., Sawyer, W.G., 2010. Analysis of planar multibody systems with revolute joint wear. Wear 268, 643e652. Muvengei, O., Kihiu, J., Ikua, B., 2012a. Computational implementation of LuGre friction law in a relute joint with clearance. In: Proceedings of the 2012 Mechanical Engineering Conference on Sustainable Research and Innovation, vol. 4, pp. 99e108. Muvengei, O., Kihiu, J., Ikua, B., 2012b. Dynamic analysis of planar multi-body systems with LuGre friction at differently located revolute clearance joints. Multibody Syst. Dyn. 28, 369e393. Muvengei, O., Kihiu, J., Ikua, B., 2013. Dynamic analysis of planar rigid-body mechanical systems with two-clearance revolute joints. Nonlinear Dyn. 73,259e73,273. Nikravesh, P.E., 1988. Computer-Aided Analysis of Mechanical Systems. PrenticeHall, Englewood Cliffs, New Jersey. Rhee, J., Akay, A., 1996. Dynamic response of a revolute joint with clearance. Mech. Mach. Theory 31 (1), 121e134. Stoenecu, E.D., Marghitu, D.B., 2003. Dynamic analysis of a planar rigid-link mechanism with rotating slider joint and clearance. J. Sound. Vib. 266, 94e404. Tian, Q., Zhang, Y.Q., Chen, L.P., Yang, J.Z., 2010. Simulation of planar flexible multibody systems with clearance and lubricated revolute joints. Nonlinear Dyn. 60, 489e511. Zhao, Y., Bai, Z.F., 2011. Dynamics analysis of space robot manipulator with joint clearance. Acta Astronaut. 68, 1147e1155.