Nonlinear finite elements/Natural vibration

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Special case : natural vibrations[edit]

Recall that the finite element system of equations has the form

We could also have written this equation as

For natural vibrations, the forces and the displacements are assumed to be periodic in time, i.e.,

and

Then, the accelerations take the form

Plugging these into the FE system of equations, we get

After simplification, we get

If there is no forcing, the right hand side is zero, and we get the finite element system of equations for free vibrations

The above equation is similar to the eigenvalue problem of the form

Since the right hand side is zero, the finite element system of equations has a solution only if

For a two noded element,

Therefore,

The determinant is

This gives us a quadratic equation in which can be solved to find the natural frequencies of the element.