Damping Models for Structural Vibration

This dissertation reports a systematic study on analysis and identification of multiple parameter damped mechanical systems. The attention is focused on viscously and non-viscously damped multiple degree-offreedom linear vibrating systems. The non-viscous damping model is such that the damping forces depend on the past history of motion via convolution integrals over some kernel functions. The familiar viscous damping model is a special case of this general linear damping model when the kernel functions have no memory. The concept of proportional damping is critically examined and a generalized form of proportional damping is proposed. It is shown that the proportional damping can exist even when the damping mechanism is non-viscous. Classical modal analysis is extended to deal with general non-viscously damped multiple degree-offreedom linear dynamic systems. The new method is similar to the existing method with some modifications due to non-viscous effect of the damping mechanism. The concept of (complex) elastic modes and nonviscous modes have been introduced and numerical methods are suggested to obtain them. It is further shown that the system response can be obtained exactly in terms of these modes. Mode orthogonality relationships, known for undamped or viscously damped systems, have been generalized to non-viscously damped systems. Several useful results which relate the modes with the system matrices are developed. These theoretical developments on non-viscously damped systems, in line with classical modal analysis, give impetus towards understanding damping mechanisms in general mechanical systems. Based on a first-order perturbation method, an approach is suggested to the identify non-proportional viscous damping matrix from the measured complex modes and frequencies. This approach is then further extended to identify non-viscous damping models. Both the approaches are simple, direct, and can be used with incomplete modal data. It is observed that these methods yield non-physical results by breaking the symmetry of the fitted damping matrix when the damping mechanism of the original system is significantly different from what is fitted. To solve this problem, approaches are suggested to preserve the symmetry of the identified viscous and non-viscous damping matrix. The damping identification methods are applied experimentally to a beam in bending vibration with localized constrained layer damping. Since the identification method requires complex modal data, a general method for identification of complex modes and complex frequencies from a set of measured transfer functions have been developed. It is shown that the proposed methods can give useful information about the true damping mechanism of the beam considered for the experiment. Further, it is demonstrated that the damping identification methods are likely to perform quite well even for the case when noisy data is obtained. The work conducted here clarifies some fundamental issues regarding damping in linear dynamic systems and develops efficient methods for analysis and identification of generally damped linear systems.