Relative Motion as a Fundamental Characteristic of Two Degree of Freedom Systems

Abstract

Two-degree-of-freedom (2DOF) systems play an important role in many areas of mechanical engineering. There are two groups of structural compositions of 2DOF systems. One of these groups consists of two rigid bodies (masses) connected to each other by one or two (in parallel) links (springs and/or dashpots). In this paper this group is considered as the basic 2DOF system. The other structural composition of a 2DOF system consists of a basic system, in which one or both masses are connected to non-movable supports. It should be stressed that these connecting links allow relative motion between the two masses. Each mass in a 2DOF system moves independently according to its law of motion. Therefore, in 2DOF systems the values of the forces that the connecting links exert toward the masses are proportional to the differences between the absolute values of the displacements and/or velocities of the masses. This is contrary to 1DOF systems, where these links exert forces proportional to the absolute values of the corresponding parameters of motion. A survey of published sources from the last seventy-five years shows that all published pairs of differential equations attempting to describe the motion of 2DOF systems actually contain a mix of mathematical terms representing the absolute values of the forces exerted by the connecting links as well as the differences of these forces. None of the surveyed sources contained any rigorous solutions of any pair of simultaneous differential equations of motion of 2DOF systems. Accounting for the relative motion between the masses of 2DOF systems, I revised the differential equations from the surveyed publications and also composed the additional pairs of differential equations for all practically relevant 2DOF systems and the possible varieties of their structures. Applying the Laplace Transform methodology to these differential equations of motion, I obtained rigorous mathematical solutions for all of them. Several examples are presented in this paper.