Vibration - Multiple Degrees of Freedom Systems and Mode Shapes - Eigenvalue Problem

Eigenvalue Problem

This is referred to an eigenvalue problem in mathematics and can be put in the standard format by pre-multiplying the equation by

and if we let and

The solution to the problem results in N eigenvalues (i.e. ), where N corresponds to the number of degrees of freedom. The eigenvalues provide the natural frequencies of the system. When these eigenvalues are substituted back into the original set of equations, the values of that correspond to each eigenvalue are called the eigenvectors. These eigenvectors represent the mode shapes of the system. The solution of an eigenvalue problem can be quite cumbersome (especially for problems with many degrees of freedom), but fortunately most math analysis programs have eigenvalue routines.

The eigenvalues and eigenvectors are often written in the following matrix format and describe the modal model of the system:

and

A simple example using our 2 DOF model can help illustrate the concepts. Let both masses have a mass of 1 kg and the stiffness of all three springs equal 1000 N/m. The mass and stiffness matrix for this problem are then:

and

Then

The eigenvalues for this problem given by an eigenvalue routine will be:

The natural frequencies in the units of hertz are then (remembering ) and .

The two mode shapes for the respective natural frequencies are given as:

Since the system is a 2 DOF system, there are two modes with their respective natural frequencies and shapes. The mode shape vectors are not the absolute motion, but just describe relative motion of the degrees of freedom. In our case the first mode shape vector is saying that the masses are moving together in phase since they have the same value and sign. In the case of the second mode shape vector, each mass is moving in opposite direction at the same rate.