Damped system

To obtain Eigenvalue we reorganize matrices from linearization as Eq. (1).

                                (1)

Where,,  and  are displacement, velocity and acceleration of the independent coordinate, respectively. 

,  and  are the mass matrix, stiffness matrix, and damping matrix respectively.

The equation of motion can be expressed as follows:

                                        (2)

Where,  is independent coordinates.

In order to make the Eq. (2) as the Eigenvalue problem, let’s modify the Eq. (2) as follows:

,,      (3)

Therefore, we can express the Eq. (2) as follows:

                                                     (4)

               (5)

In order to solve the Eq. (4), let’s assume the solution as follows:

                                           (6)

Therefore, if we substitute  to the equation of motion,

                                      (7)

                                                (8)

If we multiply the inverse of modified mass matrix () to Eq. (4), then

                                                 (9)

If we define the  as follows,

                                                (10)

Then,

                                                    (11)

If we consider the standard form of Eigenvalue problem as follows:

                                           (12)

                                                  (13)

In Eq. (12), the standard form of Eigenvalue problem is exactly same with our equation form of Eq. (11). Therefore we can get the Eigenvalue () and Eigenvector () from the Eigensolver. Here, the Eigienvalue () can be defined as follows:

                                     (14)