# Single degree of freedom damped free vibrations

## Governing Equation

Governing equation for the motion of single of freedom oscilator can be written as:

 $m\ddot{u}(t) + c\dot{u}(t) + ku(t) = 0$ (1)

This is a homogeneous second-order linear ordinary differential equation with constant coefficients that can be rearranged as follows:

 $\ddot{u}(t) + \frac{c}{m}\dot{u}(t) + \frac{k}{m}u(t) = 0$ (2)

After substitution of $\omega = \sqrt{\frac{k}{m}}$ Eqn. (2) becomes

 $\ddot{u}(t) + \frac{c}{m} \dot{u}(t) + \omega^2 u(t) = 0$ (3)

which yields the following characteristic equation

 $\lambda^2 + \frac{c}{m} \lambda + \omega^2 = 0$ (4)

with a solution

 $\lambda_{1,2} = \frac{-\frac{c}{m} \pm \sqrt{(\frac{c}{m})^2 - 4 \omega^2}}{2} = -\frac{c}{2m} \pm \sqrt{(\frac{c}{2m})^2 - \omega^2}$ (5)

### Critically-Damped Systems

If the radical term in Eqn. (5)is set equal to zero then $c/2m = \omega$ and the critical value of damping coefficient, $c_c$ can be expressed as

 $c_c = 2 m \omega$ (6)

### Undercritically-Damped Systems

If damping is less than critical, i.e. if $c < c_c$ (in other words if $c < 2 m \omega$), it is convenient to define the damping in the terms of damping ratio, which is defined as

 $\xi = \frac{c}{c_c} = \frac{c}{2 m \omega}$ (7)

Introducing $\xi$ into Eq. (5) yields

 $\lambda_{1,2} = - \xi \omega \pm \sqrt{(\xi \omega)^2 - \omega^2} = - \xi \omega \pm i \omega \sqrt{1 - \xi^2} = - \xi \omega \pm i \omega_D$ (8)

where

 $\omega_D \equiv \omega \sqrt{1 - \xi^2}$ (9)

is the free-vibration frequency of damped system.

### Overcritically-Damped Systems

To Do: Provide solution for single degree of freedom damped free vibrations, over-critically damped system (who: user:ok; priority: 100; hours: 0) (all) (cat)

## References

• Ray W. Clough and Joseph Penzien: Dynamics of Structures, 2nd edition, McGraw-Hill, New York, 1993. 738 pages. ISBN 0-07-011394-7, section 2-6, p. 26