| A novel analysis is proposed for the vibration of an elastically suspended rigid body. This new analysis is based on recent discoveries about the structure of stiffness and inertia. For a single planar body, vibration centers are used to describe the modes shapes and are shown to be constrained to regions specified by the center of elasticity, center of mass, and stiffness principal directions. Responses are classified by the number of pure translation modes and conditions for existence are given. Necessary or sufficient conditions for the existence of pure translation and pure couple modes are also given for spatial articulated and rigid bodies. It is shown that these two types of modes can be used to design a multi-degree of freedom vibration absorber. For non-proportionally damped planar vibrations of a single rigid body, the modes shapes are shown to be rotations about a point which is either stationary or traveling along a straight line depending on the type of damping (undamped, underdamped, critically damped, overdamped). Similarly, the spatial mode shapes are rotations and parallel translations about an axis which is either stationary or traveling along a cylindroid. An explanation of the transition between types of damping is also given. Analysis of the forced (damped and undamped) vibrations for planar and spatial motion shows how to avoid exciting a particular mode. Other results include some properties associated with the stiffness matrix and a decomposition of the damping matrix based on two singular eigenvalue problems. The research makes significant contributions to understanding the relationship between constitutive properties (stiffness and inertia) and modal characteristics (natural frequencies and mode shapes). Finally, the application of the results to the design of mode shapes and to the inverse mode shape problem are outlined. Numerical examples illustrate the results. |
