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

$c_{c}=2m\omega$

(6)

Undercritically-Damped Systems

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

$\xi ={\frac {c}{c_{c}}}={\frac {c}{2m\omega }}$

(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

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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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