-Mechanical Vibration-Lect-07-Two-Degree-of-Freedom Systems

Lecture 7

Two-Degree-of-Freedom Systems
Contents
1. Introduction
2. Free-Vibration Analysis of an Undamped System

Introduction
Systems that require two independent coordinates to describe their motion are called two degree-of-freedom systems. We shall consider only two-degree-of-freedom systems in this lecture, so as to provide a simple introduction to the behavior of systems with an arbitrarily large number of degrees of freedom, which will be discussed later. Consider the automobile shown in Fig.1 (a). For the vibration of the automobile in the vertical plane, a two-degree-of freedom model shown in Fig. 1(b) can be used. Here the body is idealized as a bar of mass m and mass moment of inertia supported on the rear and front wheels (suspensions) of stiffness and The displacement of the automobile at any time can be specified by the linear coordinate x(t) denoting the vertical displacement of the C.G. of the body and the angular coordinate indicating the rotation (pitching) of the body about its C.G. Alternately, the motion of the automobile can be specified using the independent coordinates x_(1(t)), and x_(2(t)), of points A and B.

Thus the system has one point mass m and two degrees of freedom, because the mass has two possible types of motion (translations along the x and y directions). The general rule for the computation of the number of degrees of freedom can be stated as follows:
Number of degrees of freedom of the system =
Number of masses of the system x number of possible types of motion of each mass
There are two equations of motion for a two-degree-of-freedom system, one for each mass (more precisely, for each degree of freedom). They are generally in the form of coupled differential equations that is; each equation involves all the coordinates. If a harmonic solution is assumed for each coordinate, the equations of motion lead to a frequency equation that gives two natural frequencies for the system. If we give suitable initial excitation, the system vibrates at one of these natural frequencies. During free vibration at one of the natural frequencies, the amplitudes of the two degrees of freedom (coordinates) are related in a specific manner and the configuration is called a normal mode, principal mode, or natural mode of vibration. Thus a two-degree-of-freedom system has two normal modes of vibration corresponding to the two natural frequencies.