53/58:153 Lecture 6 Fundamental of Vibration ______________________________________________________________________________ - 1 - Lecture 6: Modal Superposition Reading materials: Section 2.3 1.Introduction Exact solution of the free vibration problems is where coefficients can be determined from the initial conditions. The method is not practical for large systems since two unknown coefficients must be introduced for each mode shape. Modal superposition is a powerful idea of obtaining solutions. It is applicable to both free vibration and forced vibration problems. The basic idea To use free vibrations mode shapes to uncouple equations of motion. The uncoupled equations are in terms of new variables called the modal coordinates. Solution for the modal coordinates can be obtained by solving each equation independently. A superposition of modal coordinates then gives solution of the original equations. Notices It is not necessary to use all mode shapes for most practical problems. Good approximate solutions can be obtained via superposition with only first few mode shapes.
53/58:153 Lecture 6 Fundamental of Vibration ______________________________________________________________________________ - 2 - 2.Orthogonality of undamped free vibration mode shapes An ndegree of freedom system has nnatural frequencies and ncorresponding mode shapes.