Vibration Problem

Solution of a system of second-order homogeneous ordinary differential equations with constant coefficients of the form

where  is a positive definite matrix. To solve the vibration problem,

1. Solve the characteristic equation of  to get eigenvalues , ..., . Define .

2. Compute the corresponding eigenvectors , ..., .

3. The normal modes of oscillation are given by , ..., , where , ...,  and , ...,  are arbitrary constants.

4. The general solution is .