Single degree of freedom damped free vibrations: Difference between revisions
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| < | | <math>m\ddot{u}(t) + c\dot{u}(t) + ku(t) = 0</tex> | ||
| align = "right" | (1) | | align = "right" | (1) | ||
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| < | | <math>\ddot{u}(t) + \frac{c}{m}\dot{u}(t) + \frac{k}{m}u(t) = 0</tex> | ||
| align = "right" | (2) | | align = "right" | (2) | ||
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After substitution of < | After substitution of <math>\omega = \sqrt{\frac{k}{m}}</tex> Eqn. (2) becomes | ||
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| < | | <math>\ddot{u}(t) + \frac{c}{m} \dot{u}(t) + \omega^2 u(t) = 0</tex> | ||
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| < | | <math>\lambda^2 + \frac{c}{m} \lambda + \omega^2 = 0</tex> | ||
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| < | | <math>\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> | ||
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=== Critically-Damped Systems === | === Critically-Damped Systems === | ||
If the radical term in Eqn. (5)is set equal to zero then < | If the radical term in Eqn. (5)is set equal to zero then <math>c/2m = \omega</tex> and the critical value of damping coefficient, <math>c_c</tex> can be expressed as | ||
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| < | | <math>c_c = 2 m \omega</tex> | ||
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=== Undercritically-Damped Systems === | === Undercritically-Damped Systems === | ||
If damping is less than critical, i.e. if < | If damping is less than critical, i.e. if <math>c < c_c</tex> (in other words if <math>c < 2 m \omega</tex>), it is convenient to define the damping in the terms of damping ratio, which is defined as | ||
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| < | | <math>\xi = \frac{c}{c_c} = \frac{c}{2 m \omega}</tex> | ||
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Introducing < | Introducing <math>\xi</tex> into Eq. (5) yields | ||
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| < | | <math>\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</tex> | ||
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| < | | <math>\omega_D \equiv \omega \sqrt{1 - \xi^2} </tex> | ||
| align = "right" | (9) | | align = "right" | (9) | ||
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Revision as of 02:13, 22 October 2024
Governing Equation
Governing equation for the motion of single of freedom oscilator can be written as:
| <math>m\ddot{u}(t) + c\dot{u}(t) + ku(t) = 0</tex> | (1) |
This is a homogeneous second-order linear ordinary differential equation with constant coefficients that can be rearranged as follows:
| <math>\ddot{u}(t) + \frac{c}{m}\dot{u}(t) + \frac{k}{m}u(t) = 0</tex> | (2) |
After substitution of <math>\omega = \sqrt{\frac{k}{m}}</tex> Eqn. (2) becomes
| <math>\ddot{u}(t) + \frac{c}{m} \dot{u}(t) + \omega^2 u(t) = 0</tex> | (3) |
which yields the following characteristic equation
| <math>\lambda^2 + \frac{c}{m} \lambda + \omega^2 = 0</tex> | (4) |
with a solution
| <math>\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 <math>c/2m = \omega</tex> and the critical value of damping coefficient, <math>c_c</tex> can be expressed as
| <math>c_c = 2 m \omega</tex> | (6) |
Undercritically-Damped Systems
If damping is less than critical, i.e. if <math>c < c_c</tex> (in other words if <math>c < 2 m \omega</tex>), it is convenient to define the damping in the terms of damping ratio, which is defined as
| <math>\xi = \frac{c}{c_c} = \frac{c}{2 m \omega}</tex> | (7) |
Introducing <math>\xi</tex> into Eq. (5) yields
| <math>\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</tex> | (8) |
where
| <math>\omega_D \equiv \omega \sqrt{1 - \xi^2} </tex> | (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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