5.4.2. Undamped System and Damped System

RecurDyn offers two solutions of an undamped system and a damped system.

5.4.2.1. Undamped System

Undamped system focuses on the result of undamped natural frequency. To get the solution, RecurDyn solver ignores the damping matrix (\(\mathbf{C}\)) of the system. Therefore, Eigensolver computes with (5.25) in spite of using (5.12).

(5.25)\[\mathbf{M\ddot{x}}+\mathbf{Kx}=\mathbf{0}, \mathbf{C}=\mathbf{0}\]

In undamped system, the damping is ignored. Always the real (\({\lambda }'\)) of eigenvalue is computed in undamped system. The undamped natural frequency (\({{f}_{n}}\)) as follows :

(5.26)\[{{f}_{n}}\equiv \sqrt{{{\lambda }'}}/2\pi [Hz]\]

If the computed the real eigenvalue (\({\lambda }'\)) is positive value then the mode doesn’t have an undamped natural frequency. At this time the reported value is defined as follows :

(5.27)\[{{\omega }_{n}}\equiv \sqrt{{{\lambda }'}} [rad/time]\]

Also, in the 1-DOF undamped system, we can define the undamped natural frequency from the analytic solution as follows:

(5.28)\[{{\omega }_{n}}\equiv \sqrt{\frac{k}{m}}\]

5.4.2.2. Damped System

Damped system includes the damping of the system. From the result (5.24), we call the \({{\lambda }_{i}}\) as the damped natural frequency (\({\omega}_d\)) as follows:

(5.29)\[{{\omega }_{d}}\equiv {{\lambda }_{i}}\]

And also, in the 1-DOF damped system, we can define the damping ratio (\(\zeta\)) and critical damping coefficient (\(c_c\))as follows:

(5.30)\[\zeta \equiv \frac{c}{{{c}_{c}}}\]

(5.31)\[{{c}_{c}}\equiv 2\sqrt{mk}=2m{{\omega }_{n}}\]

Here, the \(c\) is the damping coefficient. Also, in the damped system, if the damping coefficient is less than the critical damping coefficient (\(c<{{c}_{c}}\)) then, we can also calculate the undamped natural frequency (\({{\omega }_{n}}\)) and damping ratio (\(\zeta\)) as follows:

(5.32)\[{{\omega }_{n}}=\sqrt{{{\lambda }_{r}}^{2}+{{\lambda }_{i}}^{2}}\,\,\,,\,\,\zeta =\left| \frac{{{\lambda }_{r}}}{{{\omega }_{n}}} \right|\]

Note

When imaginary value of an eigenvalue is close to zero, the undamped natural frequency cannot be defined like as (5.32). In that time the undamped natural frequency is set zero value.

5.4.2.3. Additional Information on Undamped System

Generally, the eigenvalue(\(\lambda\)) can be obtained as follows.

(5.33)\[\lambda =\frac{-C\pm \sqrt{{{C}^{2}}-4MK}}{2M}\]

where, \(\zeta =\frac{C}{{{C}_{c}}}=\frac{C}{2M{{\omega }_{n}}}, C_c=2M{\omega}_n=2\sqrt{KM}\)

Hence, the general solution is given by following equation as shown (5.34) below.

(5.34)\[\lambda ={{\lambda }_{r}}+i{{\lambda }_{i}}\]

where \({{\lambda }_{r}}=-{{\omega }_{n}}\zeta, {\lambda}_i=\pm \omega_n\sqrt{1-\zeta^2}\)

At this point, in case of the undamped system which has the real eigenvalue(\({{\lambda }_{r}}\)) as zero, the eigenvalue should be came from just the imaginary eigenvalue(\({{\lambda }_{i}}\)). So, because the imaginary eigenvalue(\({{\lambda }_{r}}\)) has two roots which are conjugate complex value, if the imaginary eigenvalue(\({{\lambda }_{i}}\)) is zero, the solver should show two eigenvalue as zero.