Rigid-Flexible Coupling Dynamics Analyses and Experiments of Spatial Parallel Mechanisms with Clearances

doi:10.3969/j.issn.1004-132X.2026.03.008

Abstract

Abstract:

In order to accurately predict the dynamics characteristics of the spatial parallel mechanisms under the combined action of joint clearances and component elasticity， a 3-RRPaR spatial parallel mechanism was taken as the research object， and rigid-flexible coupling dynamics model of the mechanisms was established with clearances. The fourth-order Runge-Kutta method and the generalized $α$ algorithm were used to solve the dynamics model. The dynamic output responses such as displacement， velocity and acceleration of the moving platform of rigid body dynamics model with clearances and the rigid-flexible coupling dynamics model with clearances were compared and analyzed. An experimental platform was built to verify the correctness of the dynamics analysis results. The results show that the elastic deformations of the components may aggravate the velocity and acceleration fluctuation of the mechanisms， and the increase of the clearance values may lead to the increase of fluctuation of dynamic output response.

Key words: spatial parallel mechanism, motion pair clearance, rigid-flexible coupling, dynamics characteristics

摘要：

为了精确预测空间并联机构在运动副间隙与构件弹性共同作用下的动力学特性，以3-RRPaR空间并联机构为研究对象，建立了含间隙机构刚柔耦合动力学模型，并采用四阶龙格-库塔法和广义- $α$ 算法对动力学模型进行求解；对比分析了含间隙刚体动力学模型与含间隙刚柔耦合动力学模型的动平台位移、速度和加速度等动态输出响应，并搭建试验平台验证了动力学分析结果的正确性。研究结果表明，构件的弹性变形会加剧机构的速度和加速度波动，间隙值的增大会导致动态输出响应的波动增大。

关键词: 空间并联机构, 运动副间隙, 刚柔耦合, 动力学特性

CLC Number:

TH112

CHEN Xiulong, SUN Chuijun, DENG Yu. Rigid-Flexible Coupling Dynamics Analyses and Experiments of Spatial Parallel Mechanisms with Clearances[J]. China Mechanical Engineering, 2026, 37(3): 586-594.

陈修龙, 孙垂军, 邓昱. 含间隙空间并联机构刚柔耦合动力学分析与试验[J]. 中国机械工程, 2026, 37(3): 586-594.

Figures/Tables 13

Fig.1 Model of dry clearance

Fig. 2 Three-dimensional two-node beam element model

Fig.3 Structure diagram of 3-RRPaR spatial parallel mechanism

Fig.4 Branch chain 1 rigid-flexible coupling structure diagram

Tab.1 Structural parameters of 3-RRPaR spatial parallel mechanism

构件	参数	值
主动臂	l_i1长度/mm	255
Pa支链短杆	l_sh长度/mm	105
末端执行器	x₄距离/mm	60
末端执行器	y₅距离/mm	80
Pa支链长杆	l_s长度/mm	640
机架	x₁距离/mm	100
	y₂距离/mm	720
	z₃距离/mm	600

Tab.2 Gap parameters of 3-RRPaR spatial parallel mechanism with clearance

参数	数值	参数	数值
修正系数 $α$ ， $β$	5	泊松比 $ν k$	0.3
恢复系数 $c e$	0.9	滑动摩擦系数c_f	0.05
轴体半径 $R t / m$	0.014	轴套半径 $R s / m$	0.015
弹性模量 $E k / G P a$	207	极限速度 $v S$ ， $v D / (m ⋅ s - 1)$	0.0001， 0.000 001

Tab.2 Gap parameters of 3-RRPaR spatial parallel mechanism with clearance

参数	数值	参数	数值
修正系数 $α$ ， $β$	5	泊松比 $ν k$	0.3
恢复系数 $c e$	0.9	滑动摩擦系数c_f	0.05
轴体半径 $R t / m$	0.014	轴套半径 $R s / m$	0.015
弹性模量 $E k / G P a$	207	极限速度 $v S$ ， $v D / (m ⋅ s - 1)$	0.0001， 0.000 001

Tab.3 Flexible component parameters of 3-RRPaR spatial parallel mechanism with clearance

参数	数值	参数	数值
从动臂长杆横截面积 $S / m m 2$	16×16	从动臂长杆弹性模量 $E / G P a$	116
从动臂长杆密度 $ρ / (k g · m - 3)$	1700	谱半径 $p$	0.7
允许误差 $t o l$	10^-7	步长 $s$	0.001

Tab.3 Flexible component parameters of 3-RRPaR spatial parallel mechanism with clearance

参数	数值	参数	数值
从动臂长杆横截面积 $S / m m 2$	16×16	从动臂长杆弹性模量 $E / G P a$	116
从动臂长杆密度 $ρ / (k g · m - 3)$	1700	谱半径 $p$	0.7
允许误差 $t o l$	10^-7	步长 $s$	0.001

Fig.5 Comparison of dynamic response of rigid body with clearances and rigid-flexible coupling dynamic model with clearances （c=0.5 mm）

Fig.6 Comparison of dynamic response of rigid-flexible coupling dynamic model under different clearances （c1=0.5 mm， c2=0.1 mm）

Fig.7 3-RRPaR spatial parallel mechanism experimental bench

Tab.4 Parameters of revolute joint with clearance

运动副元素	内径/mm	外径/mm	长度/mm	材料
轴套	15	24	50	铝合金
间隙轴销		14/14.8	98	钢

Tab.5 The test trajectory of 3-RRPaR spatial parallel mechanism

轨迹Ⅰ/m	轨迹Ⅱ/m
$X = - 0.2 + 0.05 c o s (3 π t) Y = 0.08 - 0.05 s i n (3 π t) Z = - 0.54$	$X = - 0.18 + 0.03 c o s (3 π t) Y = 0.08 - 0.03 s i n (3 π t) Z = - 0.54$

Tab.5 The test trajectory of 3-RRPaR spatial parallel mechanism

轨迹Ⅰ/m	轨迹Ⅱ/m
$X = - 0.2 + 0.05 c o s (3 π t) Y = 0.08 - 0.05 s i n (3 π t) Z = - 0.54$	$X = - 0.18 + 0.03 c o s (3 π t) Y = 0.08 - 0.03 s i n (3 π t) Z = - 0.54$

Fig.8 Comparison of the experimental results of the acceleration response of the end effector with the calculation results of the dynamic model （c1=0.5 mm， c2=0.1 mm）

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