Vibration Problem

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

$${d^2{\bf x}\over dt^2}+{\hbox{\sf B}}{\bf x}=0,$$

where B is a Positive Definite Matrix. To solve the vibration problem,

1. Solve the Characteristic Equation of B to get Eigenvalues $\lambda_1$, ..., $\lambda_n$. Define $\omega_i\equiv \sqrt{\lambda_i}$.

2. Compute the corresponding Eigenvectors ${\bf e}_1$, ..., ${\bf e}_n$.

3. The normal modes of oscillation are given by ${\bf x}_1 = A_1\sin(\omega_1 t+\alpha_1){\bf e}_1$, ..., ${\bf x}_n = A_n\sin(\omega_n t+\alpha_n){\bf e}_n$, where $A_1$, ..., $A_n$ and $\alpha_1$, ..., $\alpha_n$ are arbitrary constants.

4. The general solution is ${\bf x} = \sum_{i=1}^n {\bf x}_i$.