Abstract
The parallel mechanism has advantages of the high speed, high precision, strong carrying capacity, and high structural rigidity. Most of the previous studies concerning the dynamic modeling focused on planar mechanisms with revolute clearance joints or spatial mechanisms with one spherical joint clearance, while few studies focused on spatial parallel mechanisms with multi-spherical joint clearances. In this article, a general dynamic modeling method for spatial parallel mechanism with multi-spherical joint clearances based on Lagrange multiplier method is proposed. Taking 4universal joint-prismatic joint-spherical joint/universal joint-prismatic joint- universal joint (4UPS-UPU) spatial parallel mechanism as an example, the constraint equations of common kinematic pairs in spatial parallel mechanism, such as universal joint, spherical joint, and prismatic joint, are derived in detail. The dynamic model of the parallel mechanism with two spherical joint clearances combining the Flores contact force model and the LuGre friction model is established. The correctness of model has also been verified by comparing the analysis results of MATLAB with those of ADAMS. It can be seen that dynamic model of spatial parallel mechanism with multi-spherical joint clearances could be easily established by this method, which provides a theoretical reference to establish the dynamic model of other parallel mechanism with multi-clearance in the future.
Highlights
The parallel mechanism has advantages of the high speed, high precision, strong carrying capacity, and high structural rigidity
From the viewpoint of constraints among components of mechanism system, the constraint equation of joints for mechanism system have been described through generalized coordinates, and the constraint reaction between the components of multi-body systems can be obtained by calculating Lagrange multiplier
The curve with clearance by MATLAB and the curve with clearance by ADAMS have the same overall movement trend, which shows the correctness of the established dynamic model to a certain extent
Summary
Schematic diagram of the clearance model of spherical joint is shown in Figure 1, the body i and body j connected through spherical joint. oi À xiyizi and oj À xjyjzj, respectively, represent local coordinate systems. ri and rj represent position vectors of coordinate origins of local coordinate systems oi À xiyizi and oj À xjyjzj in the global coordinate system, respectively. Oi À xiyizi and oj À xjyjzj, respectively, represent local coordinate systems. Ri and rj represent position vectors of coordinate origins of local coordinate systems oi À xiyizi and oj À xjyjzj in the global coordinate system, respectively. Its unit vector can be given by n 1⁄4 e =e ð2Þ pffiffiffiffiffiffiffiffi where e is the eccentric amplitude, e 1⁄4 e Te. The penetration depth of socket and ball in collision can be expressed as d 1⁄4 eÀc ð3Þ where c denotes clearance size between socket and ball, c 1⁄4 Ri À Rj, Ri and Rj are the radius of the socket and ball, respectively. V n 1⁄4 1⁄2ðr_ Qj À r_ QiÞTn n v t 1⁄4 ðr_ Qj À r_ QiÞT À v n ð6Þ where r_ Qi and r_ Qj represent the velocities of collision point relative to the global coordinate system
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