Massenersatz durch punktmassen in räumlichen Getrieben

Abstract

Dynamic and kinetostatic analysis of three-dimensional mechanisms (i.e. mechanisms with intersecting or skew axes) can be brought about by using mass points. The objective of this paper is to give simple ways of determining such mass points with respect to their quantity and position by means of matrix calculus. Two rigid mass systems will be in quadratic equivalence, if their total masses are equal and their centers of mass coincide and their second moments of mass are equal with respect to a common system of coordinates. Under equal initial conditions, the two bodies will then perform equal motions if equal forces are acting upon them. Thus the total mass of a given three-dimensional rigid body may be substituted for by a finite number of mass points in rigid interconnection (Fig. 1). A mass point of quantity m i at a point P i ( x i , y i , z i ) is determined by four defining quantities ( m i , x i , y i , z i ). The condition of equality of the total masses leads to one equation, the coincidence of the centers of mass results in three equations, and six equations follow from the equality of the second moments of mass. Thus the condition of quadratic equivalence will result in a system of non-linear equations (equation (4)) consisting of 1 + 3 + 6 = 10 scalar equations. These must be satisfied by four defining quantities per mass point. Nevertheless a three-dimensional body cannot be substituted for by three mass points with 3 × 4 = 12 defining quantities, since these three mass points are always on the same plane and so can only substitute for a mass system on one plane. The minimum number of mass points necessary is four. Of the total 4 × 4 = 16 defining quantities, 16 − 10 = 6 can be chosen freely. After introducing suitable values for these six quantities, the ten unknown quantities can be calculated in a definite and explicit way (equations (6)–(15)) in spite of the above-mentioned system of equations being non-linear. It is not necessary to calculate the principle axes of inertia. As two further instances the substitution of a mass system on one plane by three mass points and the substitution of a rectilinear mass system by two mass points is discussed.