Use of Four-Pole Parameters in Vibration Calculations

Abstract

Linear elastic systems which have a single input point and a single output point can be characterized by a pair of simple linear equations, namely Finput=α11Foutput + α12Voutput,Vinput=α21Foutput + α22Voutput, where the F's are forces, the V's are velocities, and the α's are the four-pole parameters for the system. The use of four-pole parameters permits combinations of mechanical elements to be handled as a single entity, and, as the parameters belong only to the elastic system for which they are defined, they do not depend upon what precedes that particular system or follows it in a mechanical network. This permits the statement of results in a rather general form. These parameters have been used for many years in the analysis of electrical circuits, and it is the purpose of this paper to show their utility in the field of mechanical vibrations. In particular, the four-pole parameters for the basic mechanical elements mass, spring, and resistance will be obtained. Also, a description of mechanical sources will be given. The rules for the connection of four poles and a method of experimentally measuring four-pole parameters will be presented. A few specific problems illustrating the use of four poles will be set forth, namely: a generalized vibration isolation problem, the response of an elastically mounted mass on an electrodynamic shake table, shock excitation analysis, and some distributed parameter systems (i.e., helical springs and rubber in shear).