Nonlinear finite elements/Natural vibration

Special case : natural vibrationsEdit

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.