Matrix of Constraints for the Motion of the Planar Kinematic Chains with Rotational Links with Clearances

Abstract

Our goal is to present a general approach for the planar chains with rotational links with clearances. This approach is realized in a multibody style, the main problem being the determination of the matrix of constraints. Introduction The application of the multibody type methods [1-13], made possible the elaboration of certain general algorithms for the numerical, dynamical and elasto-dynamical calculation of the kinematical and dynamical parameters for the mechanical systems’ motions. These algorithms can be applied both to the determined kinematical mechanical systems with and without friction [4,9,14], respectively with one degree of mobility, and to the systems with one degree of mobility and a single motor element as, for instance, [2,7,12], the mechanical convertor of torque created by G. Constantinescu. The applications of the multibody type methods for the studying of the planar mechanical systems having articulations with clearances (rotational kinematical joints) [3,4,6,10,14], assumes the inserting of certain virtual without mass elements, which leads to singular matrix of inertia and, consequently, it does not permit the separation of the general system of equations in two systems from which result, in order, the time history of the reactions and then the time history of the kinematical parameters. In this paper we elaborate a multibody type method based on a new form of the matrix of constraints, method that permits the numerical dynamical study of the planar systems with rotational joints with or without clearance, with one or several degrees of freedom. 1. General Aspects We consider the planar kinematical chain from the Figure 1 at which the elements denoted by 1, 2, ... are linked one to another by rotational kinematical links with or without clearance 1 O , 2 O Denoting by i C the centre of weight of an element i , which is either a bar, or a shell, and denoting by i i i y x C the proper reference system, Figure 2, then the position of this elements, relative to the general fixed reference system OXY , is defined by the coordinates i X , i Y of the centre of weight, and by the angle i θ between the axes i i x C and OX . This element, linked to the next element j by the rotational joint k O , can have a point ) ~ , ~ ( ~ l l l Y X O with known motion, ) ( ~ ~ t X X l l = , ) ( ~ ~ t Y Y l l = . If the rotational kinematical joint k O is with clearance, Figure 3, with permanent contact between the elements i and j , then the clearance can be defined by the difference ) ( ) ( j k i k k O O r = , ) ( ) ( j k i k k r r r − = , between the radius ) (i k r of the rim and the radius ) ( j k r of the shaft, and by the angle k α between ) ( ) ( j k i k O O and OX . S c i e n c e s | 33 Figure 1. Planar kinematical chain.