Single degree of freedom damped free vibrations
Contents |
Governing Equation
Governing equation for the motion of single of freedom oscilator can be written as:
(1) |
This is a homogeneous second-order linear ordinary differential equation with constant coefficients that can be rearranged as follows:
(2) |
After substitution of Eqn. (2) becomes
(3) |
which yields the following characteristic equation
(4) |
with a solution
(5) |
Critically-Damped Systems
If the radical term in Eqn. (5)is set equal to zero then and the critical value of damping coefficient, can be expressed as
(6) |
Undercritically-Damped Systems
If damping is less than critical, i.e. if (in other words if ), it is convenient to define the damping in the terms of damping ratio, which is defined as
(7) |
Introducing into Eq. (5) yields
(8) |
where
(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
External Links
- SDOF Damped Free Vibrations at eFunda
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