Mechanism and Machine Theory Vol. 17, No. 6, pp. 397-403, 1982 0094-114X/82/060397-07503.00/0 Printed in Great Britain. Pergamon Press Ltd. DYNAMIC ANALYSIS OF PLANE MECHANISMS WITH LOWER PAIRS IN BASIC COORDINATES MIGUEL ANGEL SERNA Cfitedra de Mec~inica, Escuela Superior de IngenierosIndustrialesde Bilbao,Espafia and RAFAEL AVILI~S and JAVIER GARCIA de JALON CAtedrade Cinem~itica y Dimimica de M~iquinas, Escuela Superiorde IngenierosIndustrialesde Bilbao, Espafia (Received 8 July 1980; in revised[otto 15 April 1981) Abstract--ln this article, we shall present the numerical solution to the dynamic problem of planar mechanisms with lower pairs. This method is based on the basic coordinates and link constraints. We shall begin by describing the matrix formulation of the inertial forces on a link and creating the dynamic equilibrium equations in three different ways: in Lagrangian coordinates, generalized coordinates and coordinates with Lagrange Multipliers. Finally,some examples,obtained by numericalintegrationof the equations of movement, willbe given of problems of evolutionin time of the configuration of a mechanicalsystem. INTRODUCTION The dynamic analysis of mechanisms is of fundamental importance for the design of a large variety of mechani- cal systems used widely in industrial society: textile machinery, graphic arts equipment, agricultural and pub- lic works machinery, suspensions, etc. This analysis makes it possible to determine input forces, reactions in the pairs, tensions in the links, and also the temporal evolution of the position of the machine in function of the applied forces and of the inertial characteristics of its links. For the design process, dynamic analysis is of much greater interest than kinematic analysis. It could be said that the principal application of the later is that it is a prior condition to the former, since dynamic analysis always implies, in some way, kinematic analysis. Dynamic study is usually much more complicated and, in practice, should always be solved with the aid of a computer. Several programs for the dynamic analysis of mechanisms were developed during the last decade[l- 12]. Some of these programs have been widely circulated and used. An excellent comparative review of their principal possibilities and characteristics has been done by Paul [13]. Recently, Paul published an important book on the Kinematic and Dynamic Analysis of planar movement [14]. In two previous articles[15, 16], the authors of this study presented a new method for the kinematic analysis of mechanisms, based on the choice of the coordinates of some concrete points as Lagrangian coordinates and on the use of the concept of link constraints. This method combines excellent conditions of simplicity and ease of programming. In this article, we shall present the extension of the aforementioned method to the solution of the characteristic problems of dynamic analysis: the kinetostatic problem or inverse problem, which consists in the determination of the input forces and of the reactions in the pairs for a known movement of the mechanism and the direct problem, which is the cal- culation of othe movement of the mechanism and its evolution in time, in function of the applied forces. In this article, the main characteristics of the method are described in a certain amount of detail; we shall begin with the matrix formulation of the inertial forces and go on to introduce the dynamic equilibrium equa- tions in three different versions by means of the Prin- ciple of Virtual Work. Following this, we shall describe the calculation of the reactions in pairs by the Lagrange Multipliers Method and shall finish with a brief remark about the numerical integration of the differential equa- tions of the movement and some examples. INERTIAL FORCES OF A LINK All of the methods reviewed by Paul[13] use the center of gravity as a key point to determine the inertial forces. It is obvious that, when the acceleration of this point and the angular acceleration of the link are known, the inertial forces are formed directly. However, this approach is not coherent with the basic co-ordinates used by the GAS method and, for this reason, a different formulation has been adopted, based on the concept of equivalent masses: the mass characteristics of any link are reduced to punctual masses located on the base points and to masses uniformly distributed along the lines which join these points. The equivalent masses-- punctual and distributed--should have the same total mass, the same center of gravity and the same inertial moment as the actual link. These conditions make it possible to determine the values of the equivalent mas- ses, although, obviously, not unequivocally. A link with two basic points and the center of gravity located on the straight line which joins them can be substituted by two punctual masses and a mass uni- formly distributed between them; in this case, there are three conditions and three values to be determined. When the center of gravity is not on the straight line that 397