This paper presents a study on the dynamic modeling of a high-speed over-constrained press machine。 The main contribution of the paper is the development of an efficient approach to perform the dynamic analysis of a planner over-constrained mechanism。 The key idea is the establishment of a more general methodology, which is to gain a deformation compatibility equation for the over-constrained mechanism on the basis of the deformation compatibility analysis at each position of the mechanisms。 And this equation is then used together with the force/moment equilibrium equations obtained by the D’Alembert principle to form a total equation for the dynamics of the over-constrained mechanism。 The approach is applied to a particular press machine to validate the effectiveness of the approach, and in the meantime to provide some useful information for the improvement of the design of this press machine。76172

Keywords: Dynamics; Over-constrained mechanism; Deformation compatibility; High-speed press  machine

1。 Introduction

The over-constrained mechanism, especially with parallel structures, enjoys good stiffness and is thus used in machinery with high workload and high speed。 The dynamic force analy- sis of over-constrained mechanisms lacks systematics in that the particular geometric structure existing in such mechanisms, which represents over-constraints, depends on inpidual over- constrained mechanisms。 That said, a separate ad-hoc treat- ment for dynamic force analysis is needed for each over- constrained mechanism。 The foregoing difficulty does not disappear in the design problem of estimating the joint reac- tion force and/or moment and driving force  and/or  torque (with or without consideration of the inertia of the link) espe- cially when the link flexibility or deformation is considered。 Note that for machines with high workload, e。g。, press ma- chine, the deformation of the link is significant [1, 2]。

There are two situations for dynamic force analysis。  The first is that the motor is not included, and the motor is assumed to run at a constant speed。 In this case, the driving torque or force is unknown and to be determined for the selection of motor。 The second is that the motor system  is  included in force analysis; particularly, the relationship between the driv- ing torque or force and the motor’s motion will be included。 In this case, the motor’s motion is unknown and is governed by an ordinary differential equation: Solving the motor’s   motion

(as well as driving force or torque) and finding all the joint reaction forces or moments are carried out simultaneously。 Although the second situation is closer to the real situation, solving a set of ordinary differential equations can be lengthy and sometimes fails due to the computational issue [3-7]。 From a practical point of view, the first situation makes sense。 For instance, estimation of the maximum driving force or torque and joint reaction force or moment (then the maximum stress in the link) well falls into the domain of the first situa- tion and can be taken quickly。

Furthermore, in the first situation, the driving force  or torque can be carried out with the Newton-Euler formulation, Lagrange method, virtual work principle, Kane's method or energy equilibrium principle, and then solving the joint force and moment analysis problem by the force and moment equi- librium principle (indeed, in this case, the number of equations is equal to the number of unknown variables)。 However, it is not applicable to the over-constrained mechanism with the negative degree of freedom。 This is because after the driving force or moment is found, the “structure” is still indeterminate。 For the convenience of the following discussion, we use the term “kinematic motion” for the motion of the link subjected to the geometrical constraint only while calling it the real mo- tion of the link subject to both geometrical and force con- straints。 The study presented in this paper concerns the first situation (i。e。, the motor is assumed to be running constant) with consideration of the link flexibly or deformation。