The method of transmission matrix for investigating planar mechanisms

Abstract

In this paper the method of transmission matrix is applied for investigating planar mechanisms [5-7]. As is known, in many engineering problems, for example, in the problem of synthesis and analysis of motion, the following problem is more and more interested. There is an object (a particle or body) connected with a frame of reference, which moves with respect to another frame. It is necessary to define the configuration (the position, velocity, acceleration) of the object under consideration in the latter frame. There are two types of the frame of reference. A frame that is fixed in time named inertia frame or global frame of reference and the frame attached to each body of the system- the body frame of reference. The problem is stated now as follows: Determine the configuration in global frame of a particle fixed in any body of the system.In this paper a method is constructed for solving such a problem by means of transmission matrices.The obtained results are illustrated by examples.

Motion of mechanical systems with non-ideal constraints

“…Its origin moves along the trajectory. The components of the reaction in the coordinate system (Mxyz) can be expressed in term of its component in the (Mtnb) coordinate axes by means of the transmission matrix V [8], where Mt is oriented by the tangent direction, Mn -by the normal direction, but Mb -by the bi-normal direction. By such a way, we have…”

Method of transmission matrix for investigating planar relative motions

“…By (1.4) the components of the velocity of the point Min the global fran-, is calculated by the formula (1.5) where [2] ti. (*) , v(*), cp(*) are the (3x3) diagonal matrices of the form uoo voo r.poo Similarly, the acceleration of the point M in the inertia frame is defined by the formula 8) where the matrices ii(*),v(*) , cf'(*) are of the same form of (1.6), in which the quantities u, v, cp are substituted by ii, v, (/; respectively, but the matrix r; is of the form:…”

“…2) . Hereafter for simplicity the frames of reference Yi-I are expressed by the vectors· located along the xaxis [2] Let us consider the point M rigidly conneted to the (i+ l)th_link, which is the last one in the group.…”

“…The absolute angular velodty of the crank CB denoted by w 2 will be [2] c* r . * · * _ 2 r 1i-i* = c2w1.…”

Kinematics of a rigid about a fixed point

“…Reference [2] presents a systematic investigation of this general subj ect though the methods used are com plex. This paper describes the use of transmission matrices [5,6] as a simple but powerful approach with the added advantage of being in a form well suited for use with the Mathcad and Mathlab software packages.…”