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 
.
