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Dynamic modelling and properties analysis of 3RSR parallel mechanism considering spherical joint clearance and wear

考虑球副间隙及磨损的3RSR 并联机构动力学建模与特性分析

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Abstract

The collision and wear caused by inevitable clearance in kinematic pair have an effect on the dynamic characteristics of the mechanism. Therefore, we established the dynamic model of a 3RSR (R is the revolute joint and S is the spherical joint) parallel mechanism with spherical joint clearance based on the modified Flores contact force model and the modified Coulomb friction model using Newton-Euler method. The standard quaternion was introduced in the constraint equation, and the four-order Runge-Kutta method was adopted to solve the 3RSR dynamic model. The simulation results were compared and analyzed with the numerical results. The geometrical parameters of the worn ball socket were solved based on the Archard wear model, and the geometrical reconstruction of the worn surface was carried out. The geometric reconstruction parameters were substituted into the dynamic model, which was to analyze the dynamic response of the 3RSR parallel mechanism with wear and spherical joint clearance. The simulation results show that the irregular wear occurs in the spherical joint with clearance under the presence of the impact and friction force. The long-term wear will increase the fluctuation of the contact force, thereby decreasing the movement stability of the mechanism.

摘要

运动副难以避免地存在间隙, 间隙的存在将引起碰撞与磨损, 而运动副间隙和磨损会影响机构 动力学特性。考虑3RSR(R 为转动副, S 为球副)并联机构中球副间隙, 基于改进的Flores 接触力模型 和修**的Coulomb 摩擦力模型, 利用牛顿-欧拉法建立机构的动力学模型, 引入标准四元数搭建约束 方程, 并采用四阶Runge-Kutta 法进行求解, 将仿真与数值求解结果进行分析对比。基于Archard 磨 损模型, 求解磨损球窝的几何参数, 利用其对磨损的表面进行几何重构, 并将几何重构参数代入动力 学模型, 分析考虑磨损特性的含球副间隙3RSR 并联机构的动力学响应。仿真结果表明, 在冲击力和 摩擦力存在的情况下, 球副出现不规则磨损, 长期磨损会增加接触力的波动, 从而降低机构的运动稳 定性。

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Abbreviations

A k :

The attitude transformation matrix of the local coordinate system located on the component k

c :

The clearance radius

c d :

The coefficient of dynamic correction

c e :

The coefficient of restitution

c f :

The coefficient of sliding friction

dt :

The time integration step

e :

The clearance vector between the ball head and the ball socket

E i, E j :

The elastic modulus

ε i, ε j :

The Poisson ratio

F :

The collision force of ball head on the bearing

F n :

The normal contact force between the ball head and the ball socket

F t :

The expression of the tangential contact force of the ball head on the ball socket

H :

The rigidity of the softer material

h :

The wear depth in the element

I xk :

The moment of inertia of the drive rod or the transmission rod around the xk

I yk :

The moment of inertia of the drive rod or the transmission rod around the yk

I zk :

The moment of inertia of the drive rod or the transmission rod around the zk

I x7 :

The moment of inertia of the moving platform around the x7

I y7 :

The moment of inertia of the moving platform around the y7

I z7 :

The moment of inertia of the moving platform around the z7

K :

The stiffness coefficient

l k :

Length of kth drive rod or kth transmission rod

M :

The mass matrix of the system

m k :

Mass of the drive rod or the transmission rod

m 7 :

Mass of the moving platform

n :

The normal vector of the contact surface

p k :

The standard quaternion of the local coordinate system of the component k in the fixed coordinate system

Q :

The generalized force vector including the external forces and moment of external forces

q :

The kinematic state of the mechanism

q̇ :

The generalized vector of velocity

q̈ :

The generalized vector of acceleration

R :

Radius of the joint distribution circle on the moving platform

R i :

The radius of the ball head

R j :

The radius of the ball socket

r :

Radius of the joint distribution circle on the fixed platform

r k :

The coordinate of local coordinate origin on component k in the fixed coordinate system

S :

The area of the element

s :

The distance of the relative slip

Δs τ :

The slip distance in the τ-th time integration step

t :

The tangential vector of the contact surface

V :

The volume of the wear

ΔV τ :

The wear loss in the τ-th time integration step

ν t :

The relative tangential velocity of collision

α(β):

The stability correction coefficient

δ :

The penetration depth of the ball socket

η :

The wear coefficient

λ :

The Lagrange multiplier vector

θ k :

The drive angle of kth drive rod or kth transmission rod

ρ :

The curvature radius of the element

\({{\bf{\Phi}}^{{D_k}}}\) :

The driving constraint equations in a 3RSR parallel mechanism

\({{\bf{\Phi}}^{{E_k}}}\) :

The constrained equation form of the standard quaternion

Φ R :

The constraint equation of the revolute joints in a 3RSR parallel mechanism

Φ S :

The constraint equation of each spherical joint in a 3RSR parallel mechanism

Φ(q):

The kinematic constraint equation

Φ q :

The Jacobian matrix of the constraint equation

Φ t :

The derivative of the constraint equation versus time

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Authors and Affiliations

  1. School of Mechanical Engineering, Yanshan University, Qinhuangdao, 066004, China

    Yu-lei Hou  (侯雨雷) & Yun-jiao Deng  (邓云蛟)

  2. School of Mechanical Engineering, Dongguan University of Technology, Dongguan, 523015, China

    Da-xing Zeng  (曾达幸)

Authors
  1. Yu-lei Hou  (侯雨雷)
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  2. Yun-jiao Deng  (邓云蛟)
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  3. Da-xing Zeng  (曾达幸)
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Corresponding author

Correspondence to Da-xing Zeng  (曾达幸).

Additional information

Foundation item

Project(2018YFB1307900) supported by the National Key R&D Program of China; Project(51775473) supported by the National Natural Science Foundation of China; Projects(E2018203140, E2019203109) supported by the Natural Science Foundation of Hebei Province, China; Project(ZD2019020) supported by the Key Research Project in Higher Education Institutions of Hebei Province, China; Project(2017KSYS009) supported by the Key Laboratory of Robotics and Intelligent Equipment of Guangdong Regular Institutions of Higher Education, China; Project(KCYCXPT2017006) supported by the Innovation Center of Robotics and Intelligent Equipment of Dongguan University of Technology, China

Contributors

The overarching research goals were developed by HOU Yu-lei, DENG Yun-jiao and ZENG Da-xing. HOU Yu-lei conducted the literature review and established the dynamic models. ZENG Da-xing and DENG Yun-jiao solved the geometrical parameters and analyzed the dynamic response. The initial draft of the manuscript was written by HOU Yu-lei, DENG Yun-jiao and ZENG Da-xing. All authors replied to reviewers’ comments and revised the final version.

Conflict of interest

HOU Yu-lei, DENG Yun-jiao and ZENG Da-xing declare that they have no conflict of interest.

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Hou, Yl., Deng, Yj. & Zeng, Dx. Dynamic modelling and properties analysis of 3RSR parallel mechanism considering spherical joint clearance and wear. J. Cent. South Univ. 28, 712–727 (2021). https://doi.org/10.1007/s11771-021-4640-y

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