Single degree of freedom damped free vibrations

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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)


\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)


  • 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

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