Dynamics investigation of spatial parallel mechanism considering rod flexibility and spherical joint clearance

Mechanism and Machine Theory 137 (2019) 83–107

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Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmachtheory

Research paper

Dynamics investigation of spatial parallel mechanism considering rod flexibility and spherical joint clearance Gengxiang Wang a,∗, Liang Wang b a b

Faculty of Mechanical and Precision Instrument Engineering, Xi’an University of Technology, P.O.Xi’an, Shaanxi 710048, China Development Engineer, MSC. Adams/Solver, 201 Depot Street, #100, Ann Arbor, MI 48104, USA

a r t i c l e

i n f o

Article history: Received 12 February 2019 Revised 9 March 2019 Accepted 10 March 2019

Keywords: Parallel mechanism Flexible actuated rod Reference conditions Normal mode approach Clearance spherical joint Contact force model

a b s t r a c t The dynamic model of 4-SPS/PS parallel mechanism with a flexible actuated rod and clearance spherical joint is established based on the equation of motion. Firstly, the flexible actuated rod is modeled using finite element (FE) 3D beam based on the floating frame of reference (FFR) formulation since small deformation occurs in the rod. In order to build the actual dynamic model, some rigid joints are introduced to connect the related components in 4-SPS/PS parallel mechanism. Secondly, two different sets of reference conditions are imposed at the FE analysis stage for the 3D beam to obtain the free-free modes and fixed-fixed modes based on the normal mode approach, which circumvents the fact that the free-free modes are not suitable for some instances. Further, the dimension of the whole dynamic model is reduced by the normal mode approach as well. Besides, clearance is introduced to one of the spherical joints in this parallel mechanism. The normal and tangential contact forces are estimated based on the Lankarani–Nikravesh contact force model and a modified Coulomb friction model, respectively. Finally, the effects of the free-free modes and fixed-fixed modes from the flexible actuated rod on the dynamic responses of the parallel mechanism with clearance spherical joint are firstly discussed in this investigation. The comparative analysis between considering flexible actuated rod (FAR model), considering clearance spherical joint (CSJ model) and simultaneously considering flexible actuated rod and clearance spherical joint (FAR-CSJ model) has been carried out. The simulation results showed that a reliable and comprehensive solution requires both appropriate reference conditions, and taking the clearance spherical joint and the flexible moving platform into consideration. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction During the past three decades, the research on parallel mechanism has attracted lots of attention from the industry communities because it has many advantages such as the large stiffness, high accuracy, heavy-load capability, high-speed, excellent dynamic performance and so on [1]. However, the parallel mechanism has hardly been applied in actual industry field until now. The main reason is that its motion precision is hard to achieve the requirement of the manufacturing and assembling [2], which hinders its development and application. The two primary reasons that affect the accuracy of the parallel mechanism are: (i) the parallel mechanism operates under the heavy-load and high-speed environment, thereby, the

∗

Corresponding author. E-mail addresses: [email protected] (G. Wang), [email protected] (L. Wang).

https://doi.org/10.1016/j.mechmachtheory.2019.03.017 0094-114X/© 2019 Elsevier Ltd. All rights reserved.

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component’s deformation caused by the large inertial forces is inevitable [3], which has an adverse effect on the accuracy and stability of the parallel mechanism. (ii) the presence of the clearance joint not only guarantees the relative motion between joint elements but also leave enough space for the lubrication to reduce the wear phenomenon [4]. Therefore, the clearance joint in the parallel mechanism is inevitable as well. Nevertheless, the impact behavior caused by the clearance joint severely degenerates the dynamic property of the parallel mechanism, such as wear phenomenon, noise, vibration, energy dissipation and so on [5]. Therefore, the flexible components and clearance joints should be considered simultaneously in the parallel mechanism when formulating its dynamics model, which is more beneficial to enhance the accuracy of the dynamic model [6]. Moreover, in the dynamic model of the parallel mechanism with flexible body and clearance joint, the flexible component in the parallel mechanism can dissipate the energy [7,8]. To this end, there are many scholars who make massive contributions in trying to build a high fidelity dynamics model. The research progress of non-ideal parallel mechanism can be classified as three different phases: (i) only the flexible components is considered; (ii) only the clearance joints are considered; (iii) both the flexible components and clearance joints are considered. As for the flexible components in the parallel mechanism, Jiang et al. [9] analyzed the dynamics model of the cabledriven parallel mechanism with roll and pitch movement using the coupling between the global motion and small deformation. The nonlinear control method is proposed to suppress the vibration caused by the deformation, the feasibility of the proposed control method is verified by the numerical method and experiments. Long et al. [10] built a dynamic model of Stewart with the flexible actuated rod based on the FFR formulation, in which the moving platform consists of the flexible beams. The dynamic response of the flexible actuated rod is calculated using the first ten modes based on the free-free boundary conditions. Zhang et al. [11] designed a novel 3-DOF parallel mechanism with three smart links that have the abilities of self-sensing and self-control, therefore, the undesired vibration in modal space is suppressed using an adaptive sliding mode. Furthermore, the first and second modes of each link can be suppressed effectively by the numerical analysis. The proposed vibration control method is validated by both simulation and experimental results. Zhang et al. [12] formulated the dynamic model of planar 3-PRR parallel mechanism based on the sub-structuring and assumed modes method, the assumed modes of the flexible link are verified by the experimental modal tests. A variable structural control method is used to suppress the undesired vibration caused by the flexible links effectively. Wang [8] investigated the dynamic performance of a 4-DOF parallel mechanism with the flexible actuated rod based on the absolute nodal coordinate formulation (ANCF), in which the flexible actuated rod is treated as a 3D beam. Besides, the effect of Young’s modulus on the trajectory of the parallel mechanism is tested. Simulation results proved that the deformation in the parallel mechanism could not be ignored when the moving platform is made of soft materials. Liang et al. [13] built the dynamic model of a 2-DOF parallel mechanism based on the equation of motion, the influence of the rigid-flexible coupling terms in its dynamic performance is analyzed based on assumed modes synthesis. Simulation results were validated by the virtual prototype built by SimMechanics. Subsequently, they [14] used the theory of flexible multibody dynamics to formulate the dynamic model of this parallel mechanism with multiple actuation modes. The proportional-differential feedback controller is proposed to achieve the end-effector’s trajectory, and the strain rate feedback controller is applied to suppress the vibration caused by the flexible links. Sharifnia et al. [15] implemented the dynamic response and vibration analysis of 3-PSP parallel mechanism with the flexible actuated rod. The moving platform consisted of three Euler-Bernoulli beams to form a star shape. The inverse dynamics and vibration analysis of this parallel mechanism are solved based on the constrained assumed modes method. The simulation results are verified by finite element software. Tian et al. [16] developed the dynamic performance of a 3DOF micro parallel mechanism with the flexure hinges, and its dynamic performance is validated by ANSYS, the conclusions proved that this parallel mechanism could be utilized for high-speed micromanipulation with high accuracy. Wang et al. [17] investigated the dynamic performance of 3-PRR parallel mechanism with the flexible links using a sub-structuring approach, the dimension of the dynamics model of this parallel mechanism is reduced based on CB method. Moreover, the simulation results were validated using Adams software. Šalinic and Nikolic [18] modeled the quasi-static response of the 3-RRR compliant parallel mechanism using a new pseudo-rigid-body model approach, in which the adjacent rigid bodies were connected by 3-DOF joints. Lu and Hu [19] formulated the total stiffness matrices of 3-5DOF elastic parallel kinematic machines based on their equivalent parallel kinematic machines and calculated the elastic deformation using their total stiffness matrices. When the moving platform located at the initial posture which the extensions of active legs are the smallest, the parallel kinematic machines had the largest stiffness. Sharifnia et al. [15] implemented the dynamic response and vibration analysis of 3-PSP parallel mechanism with the flexible actuated rod. The moving platform consisted of three Euler-Bernoulli beams to form a star shape. The inverse dynamics and vibration analysis of this parallel mechanism are solved based on the constrained assumed modes method. The simulation results are verified by finite element software. Rezaei and Akbarzadeh [20] formulated the overall stiffness of a 3-PRUP compliant parallel mechanism where a continuous modeling method which is based on the energy method and Castigliano’s second theorem is used, which can obtain a highly accurate analytical stiffness model. This model can be used to evaluate the effects of the joints and bodies flexibility on the position accuracy of this parallel mechanism. Rezaei et al. [21] formulated the stiffness model of the 3-PSP parallel mechanism using two analytical methods, the results were validated by ANSYS software, moreover, its stiffness property is calculated using the kinematic stiffness index in the workspace. Zhou et al. [22] modeled the full flexible 3-PRS parallel mechanism to analyze the vibration caused by the deformation, the moving platform is modeled using a triangular membrane combining with a bending plate, the legs are taken as the finite elements of a spatial beam, in addition, the joint complication is replaced by the virtual springs. The dynamic performances of this parallel mechanism were validated by

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the experiments. Ebrahimi et al. [3] built the elastodynamics model of planar 3-RPR parallel mechanism with the flexible links based on the assumed mode method. The simulation proved that the accuracy and stiffness property of this planar parallel mechanism are better than the 1-RPR serial mechanism. Zhang et al. [23] taken the link in the 3-PRR planar parallel mechanism as the 2D Euler–Bernoulli beam. The high-frequencies modes caused by the flexible body in its dynamic model are truncated using the component mode synthesis (CMS), the dynamic model of this parallel mechanism was verified by the test rig. Piras et al. [24] built the dynamic model of planar 3-PRR parallel mechanism with the flexible links based on Lagrange equation of motion, simulation results implied that the configuration of this parallel mechanism has a severe effect on the deflection of the flexible link. As for considering the clearance joints in the parallel mechanism, Zhang et al. [25] implemented a comparative analysis between 3-RRR and 4-RRR mechanism with clearance joints, firstly, they used the mass-less virtual link to describe the clearance joint, and then, based on the contact force model to analyze the dynamic performance of the parallel mechanism. The research conclusively proved that the 4-RRR has excellent combined performance compared to the 3-RRR parallel mechanism. Zhang et al. [26] developed a dynamics model of a 3-PRR parallel mechanism with clearance joint based on Newton-Euler equations and proposed a concept that the root means square error of the angular acceleration could be used to quantify the effect of the joint clearance on the dynamics behavior. In order to improve the performance of this parallel mechanism, the effect of the clearance size, restitution coefficient of the material and movement trajectory on the dynamic response of this parallel mechanism are considered, the simulation results showed that the material with a moderate value of coefficient restitution is beneficial to the dynamics response. Farajtabar et al. [27] investigated the pick and place trajectory of a planar 3-RRR parallel mechanism with clearance joint. In order to build the kinematics model in a convenient way, the clearance in the joint is treated as a massless virtual link. Finally, the trajectory error caused by the clearance joint is compensated by changing the inputs appropriately. Cammarata [28] proposed a novel method that used a single elastostatic analysis and the principle of virtual work to find the displacements at each node of the overconstrained mechanism with the flexibility and clearance errors considered. The reaction force of the 3-RRR spherical parallel mechanism is calculated based on this method, which is verified by the Adams software. Varedi-Koulaei et al. [29] analyzed the dynamic behavior of the planar 3-RRR parallel mechanism with revolute clearance joint. Considering that the clearance joint adversely degenerates its dynamics performance, an optimization algorithm which lengths and mass distribution of the links are treated as design variable was proposed to reduce the undesirable effects caused by the clearance joint, which was proved by the Adams software. Hou et al. [30] studied the chaos phenomenon of 2-DOF parallel mechanism with clearance joint and friction, one of the revolute joints is treated as the clearance joint. The dynamics model of this parallel mechanism is formulated based on the Newton-Euler equations, which is solved using the fourth adaptive Runge-Kutta method. The simulation shows that the clearance joint leads to the chaotic phenomenon and affects the stability of this parallel mechanism. Xu and Li [31] formulated the dynamics model of a planar 2-DOF pick-and-place parallel mechanism with clearance joint based on the multibody system dynamics equation. The contact forces are computed based on the normal contact force model with hysteretic damping and modified Coulomb friction model. The simulation showed that the clearance joint degenerated the kinematic and dynamic accuracy of the manipulator’s end-effector. Chen et al. [32]studied the chaotic phenomenon of spatial parallel mechanism with clearance spherical joint by phase diagram, Poincare section mapping method and Lyapunov exponent. The simulation shown that the chaotic phenomenon appeared when the clearance size increased. As for considering the flexible components and clearance joints in the parallel mechanism at the same time, Shiau et al. [33] formulated a nonlinear dynamics model of a 3-PRS parallel mechanism with the flexible body and clearance joint based on the Newtonian method. This dynamic model is solved using the Runge–Kutta method and contact verification criterion. The simulation showed that the clearance joint decreased the natural frequencies of the flexible body, and the contact force increased with the clearance size and friction coefficient increasing. Chen et al. [34] researched the chaos and bifurcation phenomenon of the spatial parallel mechanism with clearance joint and flexible links based on the KED and Lagrange method. Wang and Liu [35] analyzed the dynamic performance of the 4-DOF parallel mechanism with flexible actuated rod and clearance joint, in which the dynamics model of the flexible actuated rod is formulated based on the (ANCF). The comparison analysis between considering flexible actuated rod, considering clearance joint and considering flexible actuated rod and clearance joint was implemented, which proved that the flexible actuated rod could be treated as the suspension to alleviate the impact caused by the clearance joint. Considering the universality of the spherical joint applied in the spatial parallel mechanism, ones who interested in the clearance spherical joint can refer to these literature [36,37–39]. Furthermore, Tian et al. [40] studied the effect of the clearance spherical joint on the flexible multibody system, wherein the dynamics model of the flexible body is formulated by ANCF. Subsequently, they [41] proposed a new elastohydrodynamic lubricated spherical joint model in the flexible multibody system, wherein the flexible spherical socket is meshed by the isoparametric fifteen-node pentahedron finite element. The pressure between socket and ball is evaluated by solving the Reynolds’ equation. Finally, some simulation results are validated by the commercial software ADINA. Besides, the reader can refer to other 3D models and applications [42–47] when developing the dynamics model of the spatial mechanism.

1.1. Limitations and main contributions Based on the reviews above, regarding the research of the dynamic model of the parallel mechanism with flexible components and clearance joints, there are at least two limitations can be summarized as follows:

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(1) When considering the flexible components and clearance joints in the spatial parallel mechanism simultaneously [33], its dynamic model is simplified based on the linear theory of elastodynamics [48] to neglect the rigid-flexible coupling terms or high-order inertial terms [23] for the sake of solving its dynamics equations. In addition, the dynamic model of the flexible component is formulated based on the (ANCF) [49]. Although ANCF can be used to study the small deformation of the flexible components, the strain-displacement is non-linear [48]. This nonlinearity leads to a highly non-linear stiffness matrix which would increase the difficulty in solving of dynamic model of parallel mechanism [50]. In addition, the time-varying stiffness matrix is not beneficial to use the CMS to reduce the dimension of the dynamic model. (2) The mainly researched object of the parallel mechanism is the 3-DOF planar parallel mechanism rather than spatial parallel mechanism [15,26,51]. In addition, the investigations above primarily focused on the effect of the flexible body on the stiffness characteristic and kinematic accuracy of parallel mechanism [2], however, the dynamic model considering the flexible component is seldom studied. Even if the dynamic model of the spatial parallel mechanism is investigated based on the assumed modal approach, there is hardly any researcher to analyze the effect of different sets of reference conditions on its dynamics model [23]. Because of that, in some cases, the inappropriate set of reference conditions can lead to an unreliable solution [52,53]. Therefore, it is necessary to test the effect of different modal sets on its dynamics response in order to get a more accurate solution. In light of the above research features, the main contributions of this investigation can be obtained according to the limitations of the current research background: (1) In order to build a high fidelity parallel mechanism which can be applied into the engineering industry, the facts that the kinematic chains do not directly connect to the moving platform and base platform are adopted. Actually, the kinematic chains firstly connect to the ball, and then, the socket is connected with the moving platform or base platform. Finally, the ball and socket comprise as the spherical joint, which ensures the relative motion between the kinematic chain and the moving platform or base platform. The connection manner can be seen in Fig. 1 in detail. Therefore, the rigid joints are introduced between the socket and moving platform or base platform. The contact event occurs between the ball and socket rather than between ball and moving platform [35]. (2) The dynamic model of spatial 4-DOF parallel mechanism with a flexible actuated rod and clearance spherical joint is established without neglecting any rigid-flexible coupling terms and high-order inertial terms. The flexible actuated rod is treated as 3D beam based on the floating frame of reference (FFR) formulation rather than ANCF because the strain-displacement relationship is linear based on FFR, which lead to that the stiffness matrix is constant. Also, two different reference conditions are selected to test the effect of the different modal sets (free-free modes and fixedfixed modes) on the dynamic response of this parallel mechanism with clearance joint. In addition, the normal mode approach is used to reduce the dimension of the dynamics model of the parallel mechanism in order to improve computational efficiency. (3) The first time to discuss the effect of the flexible actuated rod on the ideal model using free-free modes and fixedfixed modes is implemented. It is more important that, in this investigation, the effect of using free-free modes and fixed-fixed modes on the dynamic response of the parallel mechanism with clearance spherical joint is implemented. Comparing with the FAR model, the dynamic’s error between free-free modal bases and fixed-fixed modal bases are unacceptable in the FAR-CSJ model, which proves that the correct reference conditions must be imposed to remove the rigid body modes at the finite element analysis stage. 1.2. Structure of this investigation The structure of this investigation can be organized as follows: In Section 2, the structure characteristic of a 4-DOF parallel mechanism is briefly introduced. In Section 3, the dynamics model of the flexible actuated rod is formulated based on the FFR formulation, two different sets of reference conditions (free-free reference conditions and fixed-fixed reference conditions) are selected. The natural frequencies and mode shapes are calculated based on the normal approach. In Section 4, the contact kinematic between the socket and ball is formulated, the normal and tangential contact forces are evaluated based on the Lankarani–Nikravesh contact force model and a modified Coulomb friction model, respectively. In Section 5, the dynamics model of 4-SPS/PS parallel mechanism with the flexible actuated rod and clearance spherical joint is formulated based on the equation of motion. In Section 6, the comparative analysis between FAR model, CSJ model, and FAR-CSJ model is implemented. In Section 7, the main conclusions are summarized. 2. Structure of 4-SPS/PS parallel mechanism The structure of 4-SPS/PS parallel mechanism (S-spherical joint, P-prismatic joint) can be referred to the literature [49]. However, in order to have better consistency with the actual structure applied in engineering as displayed in Fig. 1(a), regarding the kinematic chain, the spherical joint is separated from all the kinematic chains and the constrained chain as two independent components that consist of the socket and ball. Hence, the socket is connected to the base or moving platform using the rigid joint that limits all the motions between the connected bodies. The ball is connected to the socket

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Fig. 1. Structure of 4-SPS/PS the parallel mechanism.

Table 1 Physical parameters of 4-SPS/PS the parallel mechanism. Member

Mass/kg

Length/mm

IXX / kg.mm2

IYY /kg mm2

IZZ /kg mm2

Cylinder Actuated rod Constrained cylinder Constrained rod Socket on moving platform Ball Socket on base platform Moving platform Based platform

2.08 0.21 1.96 0.59 0.30 0.36 0.30 0.31 –

212.0 240.0 200.0 240.0 – – – 100 200

7991.531 996.802 6731.411 2837.976 127.400 85.866 – 2080.267 –

7991.531 996.802 6731.411 2837.976 98.769 85.866 – 1040.159 –

415.648 10.330 392.121 29.409 98.769 46.728 – 1040.159 –

using the spherical joint. The kinematic chain comprises the actuated rod and cylinder, and two components are connected by the prismatic joint. Regarding the constrained chain, the socket is connected to the moving platform using the rigid joint. The constrained cylinder is connected to the base platform using the rigid joint; the ball is connected to the constrained rod by the rigid joint; the socket is connected to the ball by the spherical joint; the prismatic joint comprises the constrained rod and constrained cylinder. The detailed structure of this parallel mechanism can be seen in the Fig. 1(b). Its structure parameters are presented in Table 1. The actuated rod in the kinematic chain l1 is treated as the flexible body, and its material parameters can be seen in Table 2. In addition, the clearance is introduced in the spherical joint, which is assumed at point A1 in the kinematic chain l1 . The parameters of the clearance spherical joint can be seen in Table 3.

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G. Wang and L. Wang / Mechanism and Machine Theory 137 (2019) 83–107 Table 2 Material parameter of the actuated rod of kinematic chain l1. Parameter

Value

Density Poison’s ratio Young’s modulus Radius Length

2740 kg/m3 0.33 7.17 × 1010 Pa 0.01 m 0.240 m

Table 3 Physical parameters of spherical clearance joint. Parameters

Socket

Ball

Radius Poison’s ratio Young’s modulus Density

0.0205 m 0.29 2.068 × 1011 Pa 7900 kg/m3

0.0200 m 0.29 2.068 × 1011 Pa 7900 kg/m3

Fig. 2. 3D beam element.

3. Three-dimensional beam According to above structure characteristic of 4-SPS/PS parallel mechanism, in this investigation, the actuated rod is treated as the flexible body that is discretized using finite element 3D beam elements and integrated with the whole system based on the FFR formulation. 3.1. Kinematic equations of an arbitrary point p The 3D beam element [54] shown in Fig. 2, each element has two nodes, and each node has six degrees of freedom. The nodal coordinates of a 3D beam element [48] are expressed as

e = [e1 , e2 , e3 , e4 , e5 , e6 , e7 , e8 , e9 , e10 , e11 , e12]T

(1)

As shown in Fig. 2, the position vector rp of the arbitrary point p of the element i in the global coordinate system π 0 can be expressed as

r p = R + Au¯ ip (i = 1, 2, . . . , n)

(2)

where R is the position vector of the origin of the body coordinate system (BCS) π l in the global coordinate system π 0 , A is the transformation matrix between BCS π l and the global coordinate system π 0 . The deformation of the 3D beam element is depicted concerning the BCS π l which is the reference frame of all the elements. u¯ ip is the position vector of the arbitrary point P on the element i in the BCS π l , and n is the total number of the elements in the flexible actuated rod. As described in the previous paragraph, one can write the position vector of point P on the element i as

u¯ ip = Se

(3)

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in which, S is the shape function of 3D beam element i, which can be seen in Appendix A. In addition, the nodal coordinates e of element i can be written in terms of the total vector of nodal coordinates of the flexible actuated rod as

e = Bb eb

(4)

in which Bb is the constant Boolean matrix that defines the connectivity between finite elements forming the flexible body, eb is the total vector of nodal coordinates of flexible actuated rod resulting from the finite-element discretization. When the deformation of flexible actuated rod occurred, the nodal coordinate can be written as

eb = eb0 + eb f

(5)

where eb0 is the nodal coordinates vector in the un-deformed configuration of the flexible actuated rod, and ebf is the nodal coordinates vector in the deformed configuration. In order to eliminate the rigid body modes in the shape function matrix and define a unique displacement field, therefore, a new nodal coordinate vector qf of the flexible actuated rod can be written as

eb f = Brf q f

(6)

where Brf is a linear transformation matrix that arises from imposing the reference conditions [48]. Thereby, the position vector of the arbitrary point P on the element i of the flexible actuated rod in the BCS π l can be expressed as





u¯ ip = SBb eb0 + Brf q f = u¯ i0 + u¯ if

(7)

u¯ i0

u¯ if

where is the position of point P in the undeformed state, and is the position of the deformation vector. Substituting Eq. (7) into Eq. (2), the position vector of an arbitrary point P on element i of the flexible actuated rod in the global coordinate system π 0 can be expressed as



r p = R + A u¯ i0 + u¯ if



(8)



In the FFR formulation, the vector of flexible actuated rod generalized coordinates is assumed as q = qTr

qTf

T

, qr =

[R θ ]. θ is the flexible body reference rotational coordinates. In order to obtain the mass matrix, the definition of the kinetic energy can be written as using the position vector rp

T =

1 2



V

1 2

ρ r˙ Tp r˙ p dV = q˙ T Mq˙

(9)

in which M is the mass matrix of the flexible actuated rod, the specified matrix form can refer to the literature [48]. The virtual work of the elastic force of the flexible actuated rod can be expressed as

δW = −



V

σ T δε dV

(10)

where σ and ɛ are, respectively, the stress and strain vectors, since the small deformation assumption is implemented in the FFR formulation, the linear relationship between strain and displacement can be expressed as

ε = Du f = DSq f

(11)

where D is a differential operator that relates the strains and displacements. Concerning a linear isotropic material, the constitutive equations relating the stress and strains can be written as σ = Eε, where E is the symmetric matrix of elastic coefficients. Substituting Eq. (11) to Eq. (10), the virtual work can be rewritten as

δW = −



V

qTf (DS )T EDSδ q f dV = −qTf



V

 (DS )T EDSdV δ q f = −qTf Kq f

(12)

where K is the symmetric positive definite stiffness matrix associated with the elastic coordinates of the flexible actuated rod in the parallel mechanism. The stiffness matrix can be expressed as [48]



K=

V

(DS )T EDSdV

(13)

3.2. Dynamic model of flexible beam The dynamic model of the flexible actuated rod can be obtained based on the FFR formulation, which is expressed as

Mq¨ + Kq = Qe + Qν

(14)

where M is the mass matrix of the flexible actuated rod, and q¨ is the vector of generalized acceleration. Qe is the vector of externally applied forces, and Qν is the quadratic velocity vector that contains the gyroscopic and Coriolis forces, which dependents on the inertial shape integrals [48].

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G. Wang and L. Wang / Mechanism and Machine Theory 137 (2019) 83–107 Table 4 Frequencies of 3D Euler–Bernoulli beam based on fixed-fixed reference conditions (Hz). Fixed-fixed modes (ANSYS)

Fixed-fixed modes (Normal mode approach)

Error percentage

1553.7 1553.7 4229.7 4229.7 6391.0 8356.4

1576.8 1576.8 4314.2 4314.2 6403.6 8362.7

1.49% 1.49% 2.00% 2.00% 0.20% 0.07%

In order to extract the locally linear relationship from the Eq. (14), the generalized coordinates of the flexible actuated rod are partitioned as the reference coordinates and elastic coordinates, hence, the equation of motion of flexible actuated rod can be written as



Mrr Mfr

Mr f Mf f



q¨ r q¨ f



+

0 0

0 K



qr qf



=

(Qe )r + (Qv )r ( Qe ) f + ( Qv ) f



(15)

where Mrr is the mass matrix corresponding to the reference motion, Mr f = MTf r is the mass matrix corresponding to the coupling between the reference motion and elastic deformation, Mff is the mass matrix corresponding to the elastic deformation. 3.3. Normal mode approach In order to calculate the frequencies of 3D beam element, the generalized method that is the normal mode approach [48] is introduced in this section. Based on Eq. (15), the linear equations corresponding to the free vibration of the flexible body can be written as

M f f q¨ f + Kq f = 0

(16)

The eigenvalue problem of this equation can be directly defined as





K − ω2j M f f A j = 0 ( j = 1, 2, . . . , m)

(17)

The natural frequencies and mode shapes Aj can be obtained by solving Eq. (17), where m is the number of modes. After truncating the high-frequency modes, the modal transformation matrix can be written as

=



A1

A2

...

Ak



(k < m )

(18)

In the normal mode approach, any set of reference conditions can be imposed during the calculation process of the mass matrix and stiffness matrix of the flexible actuated rod. Therefore, this method is a generalized method, which can be used for all applications [52]. 3.4. Frequency estimation under two different sets of reference conditions In order to select a more proper set of reference conditions, two different reference conditions (fixed-fixed reference conditions and free-free reference conditions) are selected to test the effect of different modal sets on the dynamic characteristic of the parallel mechanism. The flexible actuated rod is meshed by 3D beam element, and a convergence study has been carried out to determine the number of the elements used in this investigation is 12. The reference conditions of fixed-fixed modes can be represented by xi = yi = zi = θx = θy = θz = 0(i = 1, 13), where i is the nodal number. The first six frequencies corresponding to the fixed-fixed modes are shown in Table 4, which is validated by ANSYS. The first twelve frequencies corresponding to the free-free modes are shown in Table 5, which is verified by ANSYS as well. Also, the first four mode shapes and associated frequencies that result from imposing two different sets of reference conditions can be seen in Figs. 3 and 4, respectively. 4. Clearance spherical joint and contact force model The presence of the clearance is inevitable in the joint because of the manufacturing and assembling error, which ensures the relative motion between the joint elements, and leaves the space to the lubrication in order to reduce the wear phenomenon. Therefore, in order to formulate a more accurate dynamic model of this parallel mechanism, it is necessary to introduce the clearance into the spherical joint located at point A1 in the kinematic chain l1 . As displayed in Fig. 5, A1 is the center of the socket, and b1 is the center of the ball. The radial clearance between socket and ball is assumed as c = Ri − Rj (Rj is the radius of the ball, Ri is the radius of the socket). The unit normal vector n between the socket and ball can be calculated by the eccentricity vector d. The unit vector t denotes the tangential vector perpendicular to the unit normal vector n.

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Table 5 Frequencies of 3D Euler–Bernoulli beam based on free-free reference conditions (Hz). Free-free modes (ANSYS)

Free-free modes (Normal mode approach)

Error percentage

0 0 0 0 0 0 1565.3 1565.3 4257.2 4257.2 6453.0 8349.7

0 0 0 0 0 0 1563.7 1563.7 4254.9 4254.9 6403.6 8206.5

0 0 0 0 0 0 0.10% 0.10% 0.05% 0.05% 0.77% 1.72%

Fig. 3. Mode shape and associated frequencies corresponding to fixed-fixed modes.

4.1. Kinematics model of the clearance spherical joint The eccentricity vector d can be written as in the global coordinate system π 0



d =rb1 − rA1 = dx

dy

dz

T

(19)

where the position vectors of points A1 and b1 in the global coordinate system π 0 can express as



rA1 =r1 + R1 s1A1 rb1 =r6 + R6 s6b

1

(20)

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Fig. 4. Mode shape and associated frequencies corresponding to free-free modes.

Fig. 5. Kinematics model of spherical joint with clearance.

where r1 , r6 are the position vector of the original of local coordinate systems π 1 , π 6 in the global coordinate system π 0 , and s6b is the position vector of the center point b1 of the ball in the local coordinate system π 6 . s1A is the position vector 1

1

of the center point A1 of the socket in the local coordinate system π 1 , and R1 , R6 is the transformation matrix between the local coordinate system π 1 , π 6 and the global coordinate system π 0 . The magnitude of eccentricity vector d can be written as,

d=



dT d =

dx2 + dy2 + dz2

(21)

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Therefore, the contact detection can be carried out based on the penetration depth δ = d − c. The judgment can be referred to in the literature [55] in detail. A unit normal vector n is expressed as

n=

d d

(22)

The position vector of the contact point P and Q can be computed in the global coordinate system π 0 [35]



rP = rA1 + nRi rQ = rb1 + nR j

(23)

The relative normal and tangential velocities between contact bodies can be written as [55],



vn =





r˙ P − r˙ Q

T 

n n



(24)

vt = r˙ P − r˙ Q − vn 4.2. Contact force model

Based on the literature [56], the normal contact force models are classified as the pure elastic contact force model and the contact force model considering energy dissipation. However, the pure elastic contact force model does not coincide with the actual contact event. Therefore, the contact force model with energy dissipation is always used to imitate the energy transformation of the contact process by the hysteresis damping factor. The Lankarani–Nikravesh contact force model [57] extended the Hertz contact model to include the energy dissipation caused by the internal damping, and this model exhibits excellent capability in obtaining the accurate results, which has been validated by experimental test [58].



FN = K δ

n

1+



3 1 − cr2 4





.

δ

(25)

. (− )

δ

where FN is the contact force; δ is the penetration depth; cr is the coefficient of restitution; δ˙ is the penetration velocity; in general the power exponent n is equal to 3/2 when the contact pressure between contact surfaces is considered as parabolic distribution; δ˙ (− ) is the initial impact velocity; K is a constant depending on the material and geometry properties of contact bodies, and it can be expressed as

K=



4

  3 π σi + σ j

Ri R j Ri − R j

1 2

, σ(k=i, j ) =

1 − υk2

(26)

π Ek

where υk is Poisson’s Ratio of the contact bodies; Ek is Young’s modulus of the contact bodies. In order to calculate the tangential contact force between contact bodies, and considering that the original Coulomb friction model has a discontinuity phenomenon in the vicinity of the zero contact velocity, the modified Coulomb friction model is used to calculate the tangential contact force, and a dynamic friction coefficient is used to circumvent the discontinuity, which is the function of the tangential velocity rather than a constant parameter. The modified Coulomb friction model can be written as [59]

Ft = −μ(vt )Fn

vt

(27)

|vt |

where Fn is the normal contact force, vt is the tangential velocity. As displayed in Fig. 6, μ(vt ) is the dynamic friction coefficient, which can be expressed as



−sign(vt ) · μd μ(vt ) = −step(|vt |, vd , μd , vs , μs ) · sign(vt ) step(vt , −vs , μs , vs , −μs )

|vt > vd | vs ≤ |vt | ≤ vd −vs < vt < vs

(28)

where sign( · ) is the symbol function; μd is the slip friction coefficient; μs is the static friction coefficient; vs is the critical velocity of the static friction; vd is the critical velocity of the maximum dynamic friction. The step function is written as

step(x, x0 , h0 , x1 , h1 ) =

⎧ ⎨h0

 0 2   0  h + (h1 − h0 ) xx1−x 3 − 2 xx1−x −x0 −x0 ⎩ 0 h1

x ≤ x0 x0 < x < x1 x ≥ x1

(29)

When the contact forces are obtained, it should be noted that the contact forces must be transformed to the center of mass of the contact bodies because the dynamics model of this parallel mechanism is formulated based on the BCS and assembled in the global coordinate system [35].

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Fig. 6. Modified Coulomb friction model.

5. Dynamics model The constraint equations of 4-SPS/PS parallel mechanism with the flexible actuated rod and clearance spherical joint are formulated while formulating the Lagrangian equation of motion of the system. There are two kinds of constraint equations need to be formulated in detail: (i) constraint equations between the flexible actuated rod and rigid bodies. Since the spherical joint is separated from the kinematic chain, and the actuated rod is treated as the flexible body. Thereby, the rigid joint is used to connect the ball with the flexible actuated rod. Also, the translation joint is used to connect the cylinder with the flexible actuated rod. (ii) constraint equations of the rigid joint between rigid bodies, since all spherical joints are separated from the kinematic and constrained chain as two components consisting of the socket and ball. Hence, the rigid joints are introduced between the socket, moving platform, and base platform. Likewise, both ends of all the kinematic chains and constrained chain are connected to the ball by the rigid joint as well, which can be seen in Fig. 1. 5.1. Constraint equation between flexible body and rigid body Regarding the constraint equation of the prismatic joint in the kinematic chain l1 , as displayed in Fig. 7. In order to consider the effect of the deformation of the flexible actuated rod on the constraint equations of the prismatic joint [35], an intermediate joint coordinate system xi1 yi1 z1i rigidly attached to the prismatic joint definition at point P on the flexible actuated rod is introduced. Also, the BCS of the flexible actuated rod is x1 y1 z1 , which needs not to be rigidly attached to a material point P on the flexible actuated rod. The BCS of the cylinder rod is on its center of mass x2 y2 z2 . The constant vector v¯ 11 is defined along the prismatic joint in the intermediate coordinate system xi1 yi1 z1i , and two constant vectors v¯ 12 and v¯ 13 are perpendicular to v¯ 11 . The other constant vector v¯ 21 is defined along the prismatic joint in the BCS x2 y2 z2 . Likewise, two constant vectors v¯ 22 and v¯ 23 are perpendicular to v¯ 21 . The vectors v¯ 11 , v¯ 12 and v¯ 13 can be defined in the global coordinate system x0 y0 z0 as v1k = A1 Ai v¯ 1k (k = 1, 2, 3), in which Ai is the transformation matrix between the BCS x1 y1 z1 and the global coordinate system x0 y0 z0 , and Ai is the transformation matrix between the intermediate coordinate system xi1 yi1 z1i and the BCS x1 y1 z1 . The v¯ 21 , v¯ 22 and v¯ 23 can be defined in the global coordinate system x0 y0 z0 as v2k = A2 v¯ 2k (k = 1, 2, 3), in which A2 is the transformation matrix between the BCS x2 y2 z2 and the global coordinate system x0 y0 z0 . Therefore, the translation joint between the actuated rod and cylinder rod in the kinematic chain l1 can be written as

⎡

T

⎤

v13 v22 ⎢ 1T 2 ⎥  1 2  ⎢ v2 T v1 ⎥ 1 2⎥ C q ,q = ⎢ ⎢ v3T v1 ⎥ = 0 ⎣v1 r12 ⎦ 2 p

(30)

T

v13 r12 p 1 2 1 1 ¯ 1p ,r2p =R2 + A2 u¯ 2p . R1 is the position vector of the original point of the BCS x1 y1 z1 in the where r12 p = r p − r p , r p = R + A1 u global coordinate system x0 y0 z0 . u¯ 1p is the position vector from joint location P to the original point of the BCS x1 y1 z1 . Since

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Fig. 7. Kinematic chain l1.

the actuated rod is a flexible body, the position vector u¯ 1p can be rewritten as u¯ 1p = u¯ 10 + S1 q1f . u¯ 10 is the undeformed vector in the BCS x1 y1 z1 . S1 is the shape function matrix of 3D Euler-Bernoulli beam elements, q1f is the elastic coordinates of the flexible actuated rod. R2 is the position vector of the original point of the BCS x2 y2 z2 in the global coordinate system x0 y0 z0 , and u¯ 2p is the position vector from joint location P to the original point of the BCS x2 y2 z2 . Regarding the rigid joint between the flexible actuated rod and ball from the spherical joint [35], the constraint equation can be formulated based on Eq. (30) because the rigid joint has one more translational constraint along the axis of the joint. Thereby, the constraint equation of the rigid joint between the flexible actuated rod and the ball can be expressed as

⎡

T

⎤

v13 v32  1 3  ⎢v 1 T v 3 ⎥ C q , q = ⎢ 2T 1⎥ = 0 ⎣v 1 v 3 ⎦ 3 1 r13

(31)

where r13 has three constraint equations for limiting the translational motion of the joint. q3 is the generalized coordinates vector of the ball; v31 and v32 are the unit vectors on the ball in the global coordinate system π 0 (3 is the number of the ball in the kinematic chain l1 ). Moreover, the constraint equations of other joints are not affected by the deformation, thereby, the constraint equations of other joints between rigid bodies in this parallel mechanism can refer to the literature [60]. 5.2. Dynamics model When the constraint equations of all joints are formulated, the dynamic model of this parallel mechanism with the flexible actuated rod and clearance spherical joint can be written as



Mrr Mfr Cqr

Mr f Mf f Cq f

CTqr CTq f 0

  q¨ r q¨ f

λ

=

  (Qr)e +(Qr)ν +Qc Qf

e

+ Q f − Qs ν Qd

(32)

where Cq is the Jacobian matrix; Qd is the quadratic velocity; λ is the Lagrange multipliers, and Qc is the contact forces from the impact event between socket and ball. In order to reduce the dimension of the dynamics model using the normal mode approach, Eq. (32) is transformed into the modal space using the modal transformation matrix , the relationship between the elastic coordinates qf and modal coordinates pf can be written as

q f = p f

(33)

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Fig. 8. Trajectory of moving platform around y0 -axis. Table 6 Parameters for the numerical simulations. Parameters

Value

Time span Time step Integration tolerance error Power exponent Static friction coefficient Slip friction coefficient Critical velocity of the static friction Critical velocity of the maximum dynamic friction

0.18 s 0.0 0 0 01s 0.004 mm 1.5 0.15 0.1 0.2 mm/s 10 mm/s

Substituting Eq. (33) into Eq. (32), the dynamics model of this parallel mechanism with the flexible actuated rod and clearance joint can be rewritten as using the modal coordinates [48]

⎡

Mrr ¯ fr ⎣M

¯ rf M ¯ ff M

Cqr

Cp f

⎤ 

CTqr q¨ r CTp f ⎦ p¨ f λ 0

 =

(Qr)e +(Qr)ν +Qc ¯f Q

e



¯s ¯ f −Q + Q ν Qd

(34)

where T ¯ ¯T ¯ Mr f = M f r T= Mr f , Mf f = T M f f, Cp f = CTq f , ¯ f =  Qf , Q ¯ f =  Qf , Q ¯ s =  Kp f Q e e ν ν

6. Simulation strategy and results The trajectory of the moving platform is assumed as in Fig. 8 for 4-SPS/PS parallel mechanism. The dynamic performance of the ideal parallel mechanism can be obtained according to the assumed trajectory. The trajectory of the moving platform is obtained by giving a displacement for kinematic chains l1 and l2 . The displacement function of l1 and l2 is expressed as







l1 step(t, 0, 0, 0.09, 60) + step(t, 0.09, 0, 0.18, −60) = l2 step(t, 0, 0, 0.09, 60) + step(t, 0.09, 0, 0.18, −60)

(35)

The displacement, velocity, and acceleration along the z0 axis of the center of mass of moving platform shown in Fig. 9(a)–(c), Fig. 9(d) are the reaction forces of spherical joint located at point A1 displayed in Fig. 1. These solutions of ideal 4-SPS/PS parallel mechanism are taken as the reference solutions to measure the effect of the flexible actuated rod and clearance spherical joint on its dynamics performance. The numerical parameters can be seen in Table 6. The calculation process of the dynamic model of 4-SPS/PS parallel mechanism with the flexible actuated rod and clearance spherical joint can be seen in Fig. 10. The specified descriptions of the dynamic simulation are explained in the following section. 6.1. Simulation strategy Step 1. The inertia shape integrals are calculated in the pre-processing stage, which dependents on the assumed displacement field. Because the mass matrix and inertial forces of the whole system are defined using these inertial shape integrals.

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Fig. 9. Dynamic response of ideal the parallel mechanism: (a) displacement; (b) velocity; (c) acceleration; (d) reaction force at A1 joint.

Also, the constant mass matrix Mff in Eq. (32) and the stiffness matrix K in Eq. (13) can be calculated in advance. Simultaneously, the modal characteristics (frequencies and corresponding mode shapes) of the flexible actuated rod can be determined by solving the eigenvalue problem of free vibration in the case of imposing reference conditions matrix Brf . Finally, the modal transformation matrix  is obtained. Step 2. The initial configurations qr0 , q f0 , q˙ r0 and q˙ f0 of 4-SPS/PS parallel mechanism are estimated, the constraint equations corresponding all joints in this parallel mechanism are formulated. This step is the pre-condition for the dynamics simulation of the whole system. Step 3. Based on the initial configurations and model transformation matrix , the elastic coordinates q f0 are transformed into the modal space, and the number of the elastic coordinates is reduced using the modal transformation matrix after truncating high-frequencies, which can be written as p f0 = −1 q f0 . Likewise, the modal velocity can be obtained as p˙ f0 = −1 q˙ f0 . (Since the modal transformation matrix is not a square matrix, which leads to that the inverse matrix is hard to be T T solved, hence, the inverse matrix can be calculated using this method, −1 = M−1 p  M, M p = M f f ) [61].

Step 4. Evaluated the Jacobian matrix and based on the constrained equation, identify the dependent and independent coordinates, and use Newton-Raphson method to solve the nonlinear constraint equations for the dependent coordinates, consequently, recompute the constraint Jacobian to solve the dependent velocities. Step 5. Based on the contact detection criterion, the contact deformation δ between socket and ball is checked, if the δ ≤ 1 × 10-10 , there is no contact occurred in the clearance spherical joint. And then, go to step 7. This threshold value helps enhance the computational efficiency, because the dynamics model is the ideal model without high-frequency caused by clearance spherical joint when the penetration depth is 0 < δ < 1 × 10-10 rather than δ < 0, therefore, the time step could be increased [62]. Step 6. When the contact is detected, firstly, the penetration depth δ, normal contact velocity vn , and tangential contact velocity vt can be calculated based on the kinematic model of the clearance joint. Subsequently, the normal contact force and tangential contact force are calculated based on Eqs. (25) and (27), in order to assemble the dynamics model based on

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Fig. 10. Computational strategy for the dynamics model of 4-SPS/PS the parallel mechanism with flexible actuated rod and clearance spherical joint.

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Eq. (32) [63], the contact forces are transformed from the contact point to the center of mass of the contact bodies as the external force Qc . Step 7. If the contact is undetected, according to the inertial shape integrals from Step 1, the mass matrix in Eq. (32) of the whole system, the modal elastic force Qs , the quadratic velocity vector Qν from the kinematic energy can be calculated. Also, the quadratic velocity Qd from constraint equations and the Jacobian matrices Cqr and Cq f can be obtained using the initial configuration of this parallel mechanism. Further, the dynamics model of the whole system is transferred from the physical space to the modal space using Eq. (33), thereby, the dynamics model without the contact event can be assembled based on Eq. (34). Step 8. Solving the linear equations Eq. (34) to obtain the acceleration q¨ r0 and p¨ f0 at time t, thereby, the state vector can



be assembled as y˙ t0 = q˙ Tr0

p˙ Tf

0

q¨ Tr0

p¨ Tf

0

T

using independent coordinates and velocities. Subsequently, the state vector



y˙ t0 is integrated using a predictor-corrector method to obtain yt1 = qTr1 is the time step.

pTf

1

q˙ Tr1

p˙ Tf

1

T

at next time t+ t, in which t

Step 9. Steps 4–8 are repeated until the simulation time is exceeded. 6.2. Simulation results In this section, four different scenarios are implemented: (i) comparison analysis between ideal model and FAR model; (ii) comparison analysis between ideal model and CSJ model; (iii) comparison analysis between ideal model, CSJ model, FAR-CSJ model based on free-free modes and FAR-CSJ model based on fixed-fixed modes; (iv) comparison analysis between ideal model, CSJ model, and FAR-CSJ model based on fixed-fixed modes. 6.2.1. Comparison analysis between the ideal model and FAR model With respect to the FAR model, Fig. 11 shows the dynamic performance of 4-SPS/PS parallel mechanism with flexible actuated rod using free-free modes and fixed-fixed modes, respectively. Fig. 11(a) shows the trajectory of the moving platform around y0 axis, Fig. 11(b)–(d) are the displacement, velocity, and acceleration along the z0 axis of the center of mass of moving platform. Fig. 11(e) shows the reaction force of the spherical joint located at point A1 . Fig. 11(e) shows the trajectory of the center of the ball in the global coordinate system. In Fig. 11(a)–(d), the trajectory, displacement, velocity and acceleration of 4-SPS/PS parallel mechanism with flexible actuated rod based on the free-free modes are the same with the solution obtained using the fixed-fixed modes, and these solutions are consistent with the ideal model. This conclusion not only shows the deformation caused by the flexible actuated rod has no effect on the kinematic properties of this parallel mechanism, but also proved that the free-free modes of the flexible actuated rod could be used to obtain the right solution. However, in Fig. 11(e), the reaction force located as A1 of parallel mechanism with flexible actuated rod using free-free modes or fixed-fixed modes has a slight difference from the ideal model. It is more important that the vibration caused by the flexible actuated rod can be observed, which increases the difficulty of the selection of the driver unit. Further, the trajectories of the center of the ball obtained using the free-free modes or fixed-fixed modes are almost the same with the ideal model, although the trajectory obtained from the FAR model has a litter difference from the ideal model. This difference between the ideal model and FAR model can be ignored because this error is tiny. From the simulation results of the FAR model, the deformation of the flexible actuated rod has almost no effect on the dynamic performance of this parallel mechanism. There are two main reasons at least: (i) the deformation caused by the flexible actuated rod is very small; (ii) the non-overconstrained parallel mechanism could eliminate the small error automatically. Also, the free-free modes can be used to this application that the 4-SPS/PS parallel mechanism only considers the flexible actuated rod, and it is validated by the fixed-fixed modes. 6.2.2. Comparison analysis between the ideal model and CSJ model With respect to the CSJ model, the effect of the clearance spherical joint on the dynamic response of this parallel mechanism can be seen in Fig. 12. The clearance spherical joint has almost no influence on the trajectory and displacement properties as shown in Fig. 12(a) and (b). That is mainly because that the size of the clearance is very small comparing with the size of the components in this parallel mechanism, which has a negligible effect on the trajectory and displacement of the moving platform. Also, the 4-SPS/PS parallel mechanism is a non-overconstrained parallel mechanism that can automatically adjust the slight error caused by the manufacturing and assembling [64]. On the contrary, in Fig. 12(d), the effect of the clearance spherical joint on the acceleration property is very conspicuous since the contact forces in Fig. 12(e) caused by the fact that the impact event in the clearance joint is treated as the external force added on the right side in Eq. (32). That is why the effect of the clearance joint on the acceleration is obvious, which leads to reducing the transmission stability of this parallel mechanism and degenerating the operation accuracy. While the velocity property in Fig. 12(c) does not suffer from the influence of the clearance joint because the velocity has no time to change in a short time. However, in Fig. 12(f), the trajectory of the center of the ball in the global coordinate system is significantly different from the ideal model, which proved that whether or not this parallel mechanism can be used to the mechanical engineering with high-accuracy is mainly dependent on the precision of manufacturing of these spherical joints.

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Fig. 11. Dynamic performance of parallel mechanism with flexible actuated rod: (a) trajectory of moving platform; (b) displacement along z0 axis of the center of mass of moving platform; (c) velocity along z0 axis of the center of mass of moving platform; (d) acceleration along z0 axis of the center of mass of moving platform; (e) reaction force of spherical joint located at A1 point; (f) trajectory of the center of the ball in the global coordinate system.

6.2.3. Comparison analysis between the ideal model, CSJ model, and FAR-CSJ model Concerning the FAR-CSJ model, due to that in CSJ model and FAR model, both clearance spherical joint and flexible actuated rod have no effect on the trajectory, displacement and velocity, hence, in FAR-CSJ model, the dynamic behaviors including the trajectory, displacement, and velocity as shown in Fig. 13(a)–(c) are the same as the FAR model and CSJ model. However, the acceleration and contact force properties as shown in Fig. 13(d) and (e) have a significant change comparing with the CSJ model. Moreover, it is observed that the contact force and acceleration obtained using the free-free modes are obviously different from the solutions obtained using the fixed-fixed modes, which shows the fact that the impact occurred in the clearance spherical joint inspires the effect of the different displacement field on these dynamic responses. Meanwhile, this conclusion implied that the correct reference conditions should be imposed to define the deformed basis vector of the flexible actuated rod in the FAR-CSJ model. Although the reference conditions can be neglected in FAR model, the reference conditions must be imposed in the FAR-CSJ model. This situation can be proved again in Fig. 13(f), in which the trajectory of the center of the ball has a slightly different from the solution obtained using the fixed-fixed model. Moreover, in the FAR-CSJ model, the deformation of the flexible actuated rod can be treated as the dashpot to dissipate the energy and damp out the high-frequency vibration caused by the clearance spherical joint [35,65]. Thereby, the magnitude of the acceleration and contact force properties are decreased comparing with the CSJ model. Actually, from Eq. (32), the elastic force generated by the flexible actuated rod is a negative item on the right side, which leads to reducing the effect of the

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Fig. 12. Dynamic performance of parallel mechanism with clearance spherical joint: (a) trajectory of moving platform; (b) displacement along z0 axis of the center of mass of moving platform; (c) velocity along z0 axis of the center of mass of moving platform; (d) acceleration along z0 axis of the center of mass of moving platform; (e) reaction force of spherical joint located at A1 point; (f) trajectory of the center of the ball in the global coordinate system.

contact force on the dynamic performances of this parallel mechanism. That is why the flexible actuated rod has a positive effect on the dynamics property of this parallel mechanism, which not only reduces the vibration and noise caused by the clearance spherical joint, but also makes this parallel mechanism more stable and reliable comparing with the CSJ model. Therefore, the more accurate and more reliable solutions can be obtained in the FAR-CSJ model by not only imposing the appropriate reference conditions, but also meanwhile considering the clearance spherical joint and flexible moving platform. Considering that the effect of the clearance spherical joint and flexible actuated rod on the angle, displacement, and velocity is not obvious in the FAR model, the CSJ model and FAR-CSJ model. In order to understand which is the dominant factor to degenerate the accuracy of the system, the maximum error between the ideal model better, FAR model and FARCSJ model is analyzed. In Table 7, the effect of the flexible actuated rod on these dynamic responses can be neglected no matter free-free modes or fixed-fixed modes are used. However, in Table 8, the effect of the clearance spherical joint on these dynamic responses cannot be ignored. Comparing with the flexible actuated rod, the clearance spherical joint is the main culprit to degenerate the accuracy of this parallel mechanism. Further, in Table 9, comparing with the CSJ model, the effect of the non-ideal factors on these dynamic responses is further enlarged because of the coupling between the clearance

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Fig. 13. Dynamic performance of parallel mechanism with flexible actuated rod and clearance spherical joint: (a) trajectory of moving platform; (b) displacement along z0 axis of the center of mass of moving platform; (c) velocity along z0 axis of the center of mass of moving platform; (d) Acceleration along z0 axis of the center of mass of moving platform; (e) reaction force of spherical joint located at A1 point; (f) trajectory of the center of the ball in the global coordinate system.

spherical joint and flexible actuated rod. Comparing with the FAR model, the clearance spherical joint is the dominant factor to lose the accuracy of the system, the coupling between the clearance spherical joint and the flexible actuated rod is the second reason. Further, Fig. 14 shows the angle and velocity on the Poincaré maps, and Fig. 15 shows the displacement and velocity on the Poincaré maps. Since the trajectory, displacement and velocity are not sensitive to the flexible actuated rod and clearance spherical joint. Thereby, the results obtained from the CSJ model or the FAR-CSJ model are consistent with the ideal model. However, the velocity and acceleration on the Poincaré map as shown in Fig. 16, the dynamic behavior of this parallel mechanism changes from periodic to quasi-periodic, and then to chaotic [66]. Although the solutions obtained using free-free modes or using fixed-fixed modes in the FAR-CSJ model can keep the same trend comparing with the ideal model and CSJ model, the solutions obtained using free-free modes are significantly different from the results obtained using fixedfixed modes. Fig. 17 is the trajectory of the center of the ball in the socket. The trajectory obtained using free-free modes are significantly different from the solution obtained using fixed-fixed modes. This conclusion further proves that the reference

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Table 7 Maximum error evaluation in FAR model. Dynamic properties Angle (rad) Displacement (mm) Velocity (mm/s)

Maximum Error Ideal model and free-free model

Ideal model and fixed-fixed model

1.900 × 10−5 1.0 0 0 × 10−4 0.190

1.900 × 10−5 1.0 0 0 × 10−4 0.200

Table 8 Maximum error evaluation in CSJ model. Dynamic properties

Maximum error between ideal model and CSJ model

Angle (rad) Displacement (mm) Velocity (mm/s)

1.654 × 10−3 1.962 × 10−1 13.570

Table 9 Maximum error evaluation in FAR-CSJ model. Dynamic properties Angle (rad) Displacement (mm) Velocity (mm/s)

Maximum error Ideal model and free-free model

Ideal model and fixed-fixed model

1.733 × 10−3 2.002 × 10−1 13.726

1.697 × 10−3 2.002 × 10−1 13.901

Fig. 14. Poincaré maps consisting of angle versus velocity along the z-axis.

Fig. 15. Poincaré maps consisting of displacement versus velocity along the z-axis.

103

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Fig. 16. Poincaré maps consisting of velocity versus acceleration along the z-axis.

Fig. 17. The trajectory of the center of the ball regarding the socket. Table 10 Computational efficiency of the different modes. Model types

Ideal model

FAR model Free-free modes

Fixed-fixed modes

Free-free modes

CSJ-FAR model Fixed-fixed modes

CPU time

1

1.21

1.15

39.74

39.63

CSJ model 19.95

conditions should be imposed into the shape function of flexible actuated rod despite that the free-free modes can be used to obtain the rational solutions in this parallel mechanism. From this simulation results, the clearance spherical joint and flexible actuated rod must be considered simultaneously in the 4-SPS/PS parallel mechanism. Any dynamic models that only considering the flexible actuated rod or independently considering the clearance spherical joint can lead to simulation error. Although the deformation of the flexible actuated rod has a slight effect on the trajectory, displacement, velocity of moving platform in the FAR-CSJ model, the flexible actuated rod plays an important role to dissipate the energy caused by the impact. It is treated as a dashpot to alleviate the vibration and noise and damp out the high-frequency components comparing with the CSJ model. Namely, the coupling relationship between clearance spherical joint and flexible actuated rod cannot be neglected in the 4-SPS/PS parallel mechanism. Through the above four different model analysis, their computational efficiency can be seen in Table 10. The ideal model that is a rigid body model is treated as the reference in evaluating efficiency. In the FAR model, although the actuated rod is treated as the flexible body, its computational efficiency is very close to the ideal model no matter what using free-free modes or fixed-fixed modes. That means the coupling between the rigid body and the flexible body has a slight effect on the simulation efficiency. In the CSJ model, the high-frequency components caused by the contact event in a short time, which need a very small time step to capture it. Thereby, its computational efficiency is lower than the ideal model in around twenty times. In CSJ-FAR model, although the flexible actuated rod is taken as a suspension to dampen high-frequency components from the clearance spherical joint, the coupling relationship between clearance spherical joint and the flexible

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actuated rod is a dominant factor to make the simulation inefficiency no matter what using free-free modes or fixed-fixed modes. Its computational efficiency is the lowest among these four different models, which is lower than the ideal model in around forty times. This conclusion illuminates that the coupling relationship between flexible body and clearance joint cannot be ignored, which proves that the flexible actuated rod and clearance spherical joint should be considered simultaneously rather than independently in this investigation. 7. Conclusions In this investigation, in order to build a high fidelity dynamic model of the 4-SPS/PS parallel mechanism, the model is formulated in the case of considering clearance spherical joint and flexible moving platform, simultaneously. In order to systematically analyze the effect of the clearance spherical joint and flexible moving platform, four kinds of models that are the ideal model, the FAR model, the CSJ model, and FAR-CSJ model, are implemented in this paper. In the FAR model, when only considering the flexible actuated rod in the 4-SPS/PS parallel mechanism, its dynamics performances are not sensitive to the deformation of flexible actuated rod, in addition, the solutions obtained using freefree modes are consistent with the solutions obtained using fixed-fixed modes, which implies that the free-free modes are suitable for this model. Namely, the unreliable solution will not happen when the reference conditions are not imposed in the FAR model. However, the flexible actuated rod brings the elastic fluctuation to the FAR model which is significant. In the CSJ model, since the clearance size is very small, the effect of the clearance spherical joint on the angle, displacement, and velocity is not conspicuous, but the error caused by the clearance spherical joint cannot be accepted for this parallel mechanism applied into a high precision engineering. However, in actual engineering application, the real physical clearance size of the spherical joint is much smaller than 0.5 mm. Therefore, the effect of the real spherical joint on the angle, displacement, and velocity could be neglected. Despite this, the impact occurred in the clearance spherical joint makes this parallel mechanism generate a dramatic tremble so that this parallel mechanism is hard to control, and its dynamics performance is unstable. However, in the FAR-CSJ model, although the trajectory, displacement, and velocity of the moving platform is not sensitive to whether or not imposing the reference conditions, these solutions (the acceleration and contact force) obtained using the free-free modes are significantly different from the results obtained using the fixed-fixed modes because the impact activates the differences between using these two modal bases, which illuminates that the reference conditions must be imposed in the finite element stage of the flexible actuated rod to define a unique displacement field. Otherwise, the unreliable solution will still be obtained [52], and the source is hard to trace. Finally, the flexible actuated rod is taken as a dashpot to absorb the energy to alleviate the vibration and noise caused by the clearance spherical joint, which makes this parallel mechanism more stable and reliable. More important, if the researcher wants to obtain a more realistic and comprehensive dynamic performance of this parallel mechanism, not only the appropriate reference conditions need to be imposed, but also the clearance spherical joint and flexible actuated rod should be meanwhile considered in this system. Declaration of conflicting interests The author declares no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. Funding The author received no financial support for the research, authorship, and/or publication of this article. Appendix A The shape function can be seen as follows: ⎡

1−ξ

⎢ ⎢ S=⎢ 0 ⎣ 0

 6

 ξ − ξ2 η

 6

 ξ − ξ2 ζ

 0

1 − 3ξ 2 + 2ξ 3

0

− ( 1 − ξ )l ζ

0

1 − 3ξ 2 + 2ξ 3

(1 − ξ )l η





1 − 4ξ + 3ξ 2 l ζ 0



−ξ + 2 ξ 2 − ξ 3 l





−1 + 4ξ − 3ξ 2 l η





ξ − 2ξ 2 + ξ 3 l 0

    ξ 6 −ξ + ξ 2 η 6 −ξ + ξ 2 ζ

 0

0

3ξ 2 − 2ξ 3

0

−l ξ ζ

0

0

3ξ 2 − 2ξ 3

lξ η



−2ξ + 3ξ 2 l ζ 0





ξ2 − ξ3 l



 ⎤

2ξ − 3ξ 2 l η



 ⎥ ⎥ ⎦

−ξ 2 + ξ 3 l ⎥ 0

where ξ = x/l , η = y/l , ζ = z/l . References [1] Y. Pan, F. Gao, Position model computational complexity of walking robot with different parallel leg mechanism topology patterns, Mech. Mach. Theory 107 (2017) 324–337, doi:10.1016/j.mechmachtheory.2016.09.016. [2] Y. Jiang, T. Li, L. Wang, F. Chen, Improving tracking accuracy of a novel 3-DOF redundant planar parallel kinematic machine, Mech. Mach. Theory 119 (2018) 198–218, doi:10.1016/j.mechmachtheory.2017.09.012. [3] S. Ebrahimi, A. Eshaghiyeh-Firoozabadi, Dynamic performance evaluation of serial and parallel RPR manipulators with flexible intermediate links, Iran. J. Sci. Technol. Trans. Mech. Eng. 40 (2016) 169–180, doi:10.1007/s40997- 016- 0019- 3.

106

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[4] Q. Tian, P. Flores, H.M. Lankarani, A comprehensive survey of the analytical, numerical and experimental methodologies for dynamics of multibody mechanical systems with clearance or imperfect joints, Mech. Mach. Theory 122 (2018) 1–57, doi:10.1016/j.mechmachtheory.2017.12.002. [5] A. Gummer, B. Sauer, Modeling planar slider-crank mechanisms with clearance joints in RecurDyn, Multibody Syst. Dyn. 31 (2014) 127–145, doi:10. 1007/s11044- 012- 9339- 2. [6] O.A. Bauchau, J. Rodriguez, Modeling of joints with clearance in flexible multibody systems, Int. J. Solids Struct. 39 (2002) 41–63. [7] S.M. Megahed, A.F. Haroun, Analysis of the Dynamic Behavioral Performance of Mechanical Systems With Multi–Clearance Joints, J. Comput. Nonlinear Dyn. 7 (2012) 1–11, doi:10.1115/1.4004263. [8] G. Wang, Elastodynamics modeling of 4-SPS/CU parallel mechanism with flexible moving platform based on absolute nodal coordinate formulation, Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci. 232 (2018) 3843–3858, doi:10.1177/0954406217744814. [9] L. Jiang, B. Gao, Z. Zhu, Design and nonlinear control of a 2-DOF flexible parallel humanoid arm joint robot, Shock Vib. (2017). https://doi.org/10.1155/ 2017/2762169. [10] P. Long, W. Khalil, P. Martinet, Dynamic modeling of parallel robots with flexible platforms, Mech. Mach. Theory 81 (2014) 21–35, doi:10.1016/j. mechmachtheory.2014.06.009. [11] Q. Zhang, C. Li, J. Zhang, J. Zhang, Smooth adaptive sliding mode vibration control of a flexible parallel manipulator with multiple smart linkages in modal space, J. Sound Vib. 411 (2017) 1–19, doi:10.1016/j.jsv.2017.08.052. [12] Q. Zhang, J.K. Mills, W.L. Cleghorn, J. Jin, C. Zhao, Trajectory tracking and vibration suppression of a 3-PRR parallel manipulator with flexible links, Multibody Syst. Dyn. 33 (2015) 27–60, doi:10.1007/s11044- 013- 9407- 2. [13] D. Liang, Y. Song, T. Sun, X. Jin, Rigid-flexible coupling dynamic modeling and investigation of a redundantly actuated parallel manipulator with multiple actuation modes, J. Sound Vib. 403 (2017) 129–151, doi:10.1016/j.jsv.2017.05.022. [14] D. Liang, Y. Song, T. Sun, X. Jin, Dynamic modeling and hierarchical compound control of a novel 2-DOF flexible parallel manipulator with multiple actuation modes, Mech. Syst. Signal Process. 103 (2018) 413–439, doi:10.1016/j.ymssp.2017.10.004. [15] M. Sharifnia, A. Akbarzadeh, Dynamics and vibration of a 3-PSP parallel robot with flexible moving platform, J. Vib. Control 22 (2016) 1095–1116, doi:10.1177/1077546314538882. [16] Y. Tian, B. Shirinzadeh, D. Zhang, Design and dynamics of a 3-DOF flexure-based parallel mechanism for micro/nano manipulation, Microelectron. Eng. 87 (2010) 230–241, doi:10.1016/j.mee.20 09.08.0 01. [17] X. Wang, J.K. Mills, Dynamic modeling of a flexible-link planar parallel platform using a substructuring approach, Mech. Mach. Theory 41 (2006) 671–687, doi:10.1016/j.mechmachtheory.20 05.09.0 09. ´ A. Nikolic, ´ A new pseudo-rigid-body model approach for modeling the quasi-static response of planar flexure-hinge mechanisms, Mech. [18] S. Šalinic, Mach. Theory 124 (2018) 150–161, doi:10.1016/j.mechmachtheory.2018.02.011. [19] Y. Lu, B. Hu, Analysis of stiffness and elastic deformation for some 3–5-DOF PKMs With SPR or RPS-type legs, J. Mech. Des. 130 (2008) 102307, doi:10.1115/1.2918918. [20] A. Rezaei, A. Akbarzadeh, Influence of joints flexibility on overall stiffness of a 3-PRUP compliant parallel manipulator, Mech. Mach. Theory 126 (2018) 108–140, doi:10.1016/j.mechmachtheory.2018.03.011. [21] A. Rezaei, A. Akbarzadeh, M.R. Akbarzadeh-T., An investigation on stiffness of a 3-PSP spatial parallel mechanism with flexible moving platform using invariant form, Mech. Mach. Theory 51 (2012) 195–216, doi:10.1016/j.mechmachtheory.2011.11.011. [22] Z. Zhou, J. Xi, C.K. Mechefske, Modeling of a fully flexible 3PRS manipulator for vibration analysis, J. Mech. Des. 128 (2006) 403–412, doi:10.1115/1. 2167655. [23] X. Zhang, J.K. Mills, W.L. Cleghorn, Dynamic modeling and experimental validation of a 3-PRR parallel manipulator with flexible intermediate links, J. Intell. Robot. Syst. Theory Appl 50 (2007) 323–340, doi:10.1007/s10846- 007- 9167- 4. [24] G. Piras, W.L. Cleghorn, J.K. Mills, Dynamic finite-element analysis of a planar high-speed, high-precision parallel manipulator with flexible links, Mech. Mach. Theory 40 (2005) 849–862, doi:10.1016/j.mechmachtheory.20 04.12.0 07. [25] X. Zhang, X. Zhang, A comparative study of planar 3-RRR and 4-RRR mechanisms with joint clearances, Robot. Comput. Integr. Manuf. 40 (2016) 24–33, doi:10.1016/j.rcim.2015.09.005. [26] H. Zhang, X. Zhang, X. Zhang, J. Mo, Dynamic analysis of a 3-PRR parallel mechanism by considering joint clearances, Nonlinear Dyn. 90 (2017) 405–423, doi:10.1007/s11071- 017- 3672- 1. [27] M. Farajtabar, H.M. Daniali, S.M. Varedi, Pick and place trajectory planning of planar 3-RRR parallel manipulator in the presence of joint clearance, Robotica 35 (2017) 241–253, doi:10.1017/S0263574714002768. [28] A. Cammarata, A novel method to determine position and orientation errors in clearance-affected overconstrained mechanisms, Mech. Mach. Theory 118 (2017) 247–264, doi:10.1016/j.mechmachtheory.2017.08.012. [29] S.M. Varedi-Koulaei, H.M. Daniali, M. Farajtabar, B. Fathi, M. Shafiee-Ashtiani, Reducing the undesirable effects of joints clearance on the behavior of the planar 3-RRR parallel manipulators, Nonlinear Dyn. 86 (2016) 1007–1022, doi:10.1007/s11071- 016- 2942- 7. [30] Y. Hou, Y. Wang, G. Jing, Y. Deng, D. Zeng, X. Qiu, Chaos phenomenon and stability analysis of RU-RPR parallel mechanism with clearance and friction, Adv. Mech. Eng. 10 (2018) 1–11, doi:10.1177/1687814017746253. [31] L.X. Xu, Y.G. Li, Investigation of joint clearance effects on the dynamic performance of a planar 2-DOF pick-and-place parallel manipulator, Robot. Comput. Integr. Manuf. 30 (2014) 62–73, doi:10.1016/j.rcim.2013.09.002. [32] X. Chen, W. Gao, Y. Deng, Chaotic characteristic analysis of spatial parallel mechanism with clearance in spherical joint, Nonlinear Dyn. 94 (2018) 2625–2642. https://doi.org/10.1007/s11071- 018- 4513- 6. [33] T.-N. Shiau, Y.-J. Tsai, M.-S. Tsai, Nonlinear dynamic analysis of a parallel mechanism with consideration of joint effects, Mech. Mach. Theory 43 (2008) 491–505, doi:10.1016/j.mechmachtheory.20 07.03.0 08. [34] X. Chen, S. Jiang, Y. Deng, Q. Wang, Dynamics analysis of 2-DOF complex planar mechanical system with joint clearance and flexible links, Shock Vib. 93 (2018) 1009–1034, doi:10.1007/s11071- 018- 4242- x. [35] G. Wang, H. Liu, Dynamics model of 4-SPS/CU parallel mechanism with spherical clearance joint and flexible moving platform, J. Tribol. 140 (2018) 1–10, doi:10.1115/1.4037463. [36] S. Erkaya, S. Dog˘ an, E. S¸ efkatlıog˘ lu, Analysis of the joint clearance effects on a compliant spatial mechanism, Mech. Mach. Theory 104 (2016) 255–273, doi:10.1016/j.mechmachtheory.2016.06.009. [37] F. Marques, F. Isaac, N. Dourado, A.P. Souto, P. Flores, H.M. Lankarani, A study on the dynamics of spatial mechanisms with frictional spherical clearance joints, J. Comput. Nonlinear Dyn. 12 (2017) 51013-1–10, doi:10.1115/1.4036480. [38] P. Flores, H.M. Lankarani, Spatial rigid-multibody systems with lubricated spherical clearance joints: modeling and simulation, Nonlinear Dyn. 60 (2010) 99–114, doi:10.1007/s11071- 009- 9583- z. [39] E. Askari, P. Flores, D. Dabirrahmani, R. Appleyard, Nonlinear vibration and dynamics of ceramic on ceramic artificial hip joints: a spatial multibody modelling, Nonlinear Dyn. 76 (2014) 1365–1377, doi:10.1007/s11071- 013- 1215- y. [40] Q. Tian, Y. Zhang, L. Chen, P. Flores, Dynamics of spatial flexible multibody systems with clearance and lubricated spherical joints, Comput. Struct. 87 (2009) 913–929, doi:10.1016/j.compstruc.2009.03.006. [41] Q. Tian, J. Lou, A. Mikkola, A new elastohydrodynamic lubricated spherical joint model for rigid-flexible multibody dynamics, Mech. Mach. Theory 107 (2017) 210–228, doi:10.1016/j.mechmachtheory.2016.09.006. [42] J. Ambrósio, J. Pombo, A unified formulation for mechanical joints with and without clearances/bushings and/or stops in the framework of multibody systems, Multibody Syst. Dyn. 42 (2018) 317–345, doi:10.1007/s11044- 018- 9613- z. [43] J. Costa, J. Peixoto, P. Moreira, P. Flores, A. Pedro Souto, H.M. Lankarani, Influence of the hip joint modeling approaches on the kinematics of human gait, J. Tribol. 138 (2015) 31201-1–10, doi:10.1115/1.4031988.

G. Wang and L. Wang / Mechanism and Machine Theory 137 (2019) 83–107

107

[44] C. Quental, J. Folgado, J. Ambrósio, J. Monteiro, A new shoulder model with a biologically inspired glenohumeral joint, Med. Eng. Phys. 38 (2016) 969–977, doi:10.1016/j.medengphy.2016.06.012. [45] E. Askari, P. Flores, D. Dabirrahmani, R. Appleyard, A computational analysis of squeaking hip prostheses, J. Comput. Nonlinear Dyn. 10 (2014) 245021–7, doi:10.1115/1.4028109. [46] E. Askari, P. Flores, D. Dabirrahmani, R. Appleyard, Dynamic modeling and analysis of wear in spatial hard-on-hard couple hip replacements using multibody systems methodologies, Nonlinear Dyn. 82 (2015) 1039–1058, doi:10.1007/s11071- 015- 2216- 9. [47] F. Marques, F. Isaac, N. Dourado, P. Flores, An enhanced formulation to model spatial revolute joints with radial and axial clearances, Mech. Mach. Theory 116 (2017) 123–144, doi:10.1016/j.mechmachtheory.2017.05.020. [48] A.A. Shabana, Dynamics of Multibody Systems Fourth Edition, Cambridge University Press, 2013. [49] G. Wang, H. Liu, Three-Dimensional Wear Prediction of Four-Degrees-of- Freedom Parallel Mechanism With Clearance Spherical Joint and Flexible Moving Platform, J. Tribol. 140 (2018) 1–14, doi:10.1115/1.4038806. [50] A.A. Shabana, Flexible Multibody Dynamics: review of Past and Recent Develoments, Multibody Syst. Dyn. 1 (1997) 189–222. [51] X. Zhang, X. Zhang, Robotics and Computer-Integrated Manufacturing A comparative study of planar 3-RRR and 4-RRR mechanisms with joint clearances, Robot. Comput. Integr. Manuf. 40 (2016) 24–33, doi:10.1016/j.rcim.2015.09.005. [52] A.A. Shabana, G. Wang, Durability analysis and implementation of the floating frame of reference formulation, Proc. Inst. Mech Eng. Part K J Multibody Dyn. 232 (2018) 295–313, doi:10.1177/1464419317731707. [53] J.J. O’Shea, P. Jayakumar, D. Mechergui, A.A. Shabana, L. Wang, Reference conditions and substructuring techniques in flexible multibody system dynamics, J. Comput. Nonlinear Dyn. 13 (2018) 41007-1–12, doi:10.1115/1.4039059. [54] H. Zohoor, S.M. Khorsandijou, Enhanced nonlinear 3D Euler-Bernoulli beam with flying support, Nonlinear Dyn. 51 (2008) 217–230, doi:10.1007/ s11071-007-9205-6. [55] G. Wang, H. Liu, P. Deng, Dynamics analysis of spatial multibody system with spherical joint wear, J. Tribol. 137 (2015) 1–10, doi:10.1115/1.4029277. [56] M. MacHado, P. Moreira, P. Flores, H.M. Lankarani, Compliant contact force models in multibody dynamics: evolution of the Hertz contact theory, Mech. Mach. Theory 53 (2012) 99–121, doi:10.1016/j.mechmachtheory.2012.02.010. [57] H.M. Lankarani, P.E. Nikravesh, A contact force model with hysteresis damping for impact analysis of multibody systems, J. Mech. Des. 112 (1990) 369–376, doi:10.1115/1.2912617. [58] C.S. Koshy, P. Flores, H.M. Lankarani, 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 (2013) 325–338, doi:10.1007/s11071- 013- 0787- x. [59] E. Pennestrì, V. Rossi, P. Salvini, P.P. Valentini, Review and comparison of dry friction force models, Nonlinear Dyn. 83 (2015) 1785–1801, doi:10.1007/ s11071-015-2485-3. [60] A.A. Shabana, Computational Dynamics Third Edition, A John Wiley and Sons, Ltd., 2010. [61] A.A. Shabana, Vibration of Discrete and Continuous Systems, Springer Science & Business Media, 2012, doi:10.1115/1.2888197. [62] P. Flores, J. Ambrósio, On the contact detection for contact-impact analysis in multibody systems, Multibody Syst. Dyn. 24 (2010) 103–122, doi:10. 1007/s11044-010-9209-8. [63] E. Zheng, R. Zhu, S. Zhu, X. Lu, A study on dynamics of flexible multi-link mechanism including joints with clearance and lubrication for ultra-precision presses, Nonlinear Dyn. 83 (2016) 137–159, doi:10.1007/s11071- 015- 2315- 7. [64] M.C. Palpacelli, L. Carbonari, G. Palmieri, M. Callegari, A family of non-overconstrained 3-DoF reconfigurable parallel manipulators, (2016) 1–9. doi:10. 1007/978- 3- 319- 56802- 7. [65] M.A. Ben Abdallah, I. Khemili, N. Aifaoui, Numerical investigation of a flexible slider-crank mechanism with multijoints with clearance, Multibody Syst. Dyn. 38 (2016) 173–199, doi:10.1007/s11044- 016- 9526- 7. [66] Z. Wang, Q. Tian, H. Hu, P. Flores, Nonlinear dynamics and chaotic control of a flexible multibody system with uncertain joint clearance, Nonlinear Dyn. 86 (2016) 1571–1597, doi:10.1007/s11071- 016- 2978- 8.