A closed-form solution procedure to the vibration of non-classically damped systems subjected to harmonic loads
The dynamic response of a multi-degree-of-freedom (MDF) system with non-proportional damping subjected to harmonic loads is considered. Modal substitution is employed to transform the coupled differential equations of motion from geometric to modal coordinates. As might be expected, the modal transformation does not uncouple the differential equations of motion because of the inherent non-proportional damping, but transforms them into a system of coupled algebraic equations of convenient form for solution. The modal coordinates are then easily determined with the help of conventional techniques for solving systems of coupled equations. The method presented gives a closed form solution without the need to resort to iterative procedures, which would otherwise be necessary for other types of more irregular excitations like earthquake ground motions. Besides, it presented a technique of compiling the damping matrix of MDF systems exhibiting different mechanisms of energy dissipation in different regions. The application of the method is illustrated on a two-mass system, whose foundation interacts with the supporting soil - a phenomenon resulting totally to a four-degree-of-freedom system.
Keywords: Non-proportional damping, geometric damping, modal transformation, harmonic loading