MODAL ANALYSIS
The equation of motion does not include applied loads nor damping.
M is the mass matrix, K is the stiffness matrix
To solve this equation assume a harmonic solution.
Φ eigenvector or mode shape
ω circular natural frequency
Aside from this harmonic solution being the key to the numerical solution of the problem, this form also has a physical importance: all dofs in the vibrating structure move in a synchronous manner, the basic shape during motion does not change, only its amplitude changes.
Differentiating and substituting we obtain a set of homogenous equations, this is an eigenvalue problem. A non trivial solution is obtained by making the determinant equal to zero. The solution is a set of discrete frequencies, and its corresponding eigenvectors or mode shapes.
Mode shapes are orthogonal. Any vibration of the structure subjected to any loading is a linear combination of its mode shapes.
Solutions with ω=0 represent rigid body motion, they appear in unconstrained structures.