Single degree of freedom damped free vibrations

Governing Equation

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

<tex>m\ddot{u}(t) + c\dot{u}(t) + ku(t) = 0</tex>

(1)

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

<tex>\ddot{u}(t) + \frac{c}{m}\dot{u}(t) + \frac{k}{m}u(t) = 0</tex>

(2)

After substitution of <tex>\omega = \sqrt{\frac{k}{m}}</tex> Eqn. (2) becomes

<tex>\ddot{u}(t) + \frac{c}{m} \dot{u}(t) + \omega^2 u(t) = 0</tex>

(3)

which yields the following characteristic equation

<tex>\lambda^2 + \frac{c}{m} \lambda + \omega^2 = 0</tex>

(4)

with a solution

<tex>\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}</tex>

(5)

Critically-Damped Systems

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

<tex>c_c = 2 m \omega</tex>

(6)

Undercritically-Damped Systems

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

<tex>\xi = \frac{c}{c_c} = \frac{c}{2 m \omega}</tex>

(7)

Introducing <tex>\xi</tex> into Eq. (5) yields

<tex>\lambda_{1,2} = - \xi \omega \pm \sqrt{(\xi \omega)^2 - \omega^2} = - \xi \omega \pm \omega i \sqrt{1 - \xi^2}</tex>

(8)

where

<tex>\omega_D \equiv \omega \sqrt{1 - \xi^2} </tex>

(9)

is the free vibration frequency of damped system.

Derivation

       To Do: Provide solution for single degree of freedom damped free vibrations (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

External Links

• SDOF Damped Free Vibrations at eFunda

 

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