Damped Free Vibration

Damped free vibration refers to unconstrained oscillation where energy is lost over time. When a real-world system vibrates, energy is continuously lost due to friction, air resistance or internal material damping.

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To account for energy loss, we introduce a damping force proportional to velocity ($c\dot{x}$).

Damping Coefficient

$c$ is the damping coefficient (also known as viscous damping coefficient). It is a measure of how strongly a damper or resistive medium opposed the velocity of a moving object.

$m\ddot{x}+c\dot{x}+kx=0⟹\ddot{x}+2\zeta {\omega }_n\dot{x}+{\omega }_n^{2}x=0$

Assuming $x=e^{rt}$, the characteristic equation $r^2+2\zeta {\omega }_{n}r+{\omega }_n^2=0$ gives:
$r_{1,2}=-\zeta {\omega }_n\pm i{\omega }_n\sqrt{1-{\zeta }^2}$

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As we can see, we get a different solution form depending on the term underneath the square root.

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Damping Ratio

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Damping ratio ($\zeta$) is a dimensionless parameter that dictates the behaviour of the system.

${\omega }_n=\sqrt{\frac{k}{m}}\zeta =\frac{c}{2{\omega }_{n}m}=\frac{c}{2\sqrt{mk}}$

Case	Condition	Physical Behaviour
Underdamped	$\zeta <1$	Oscillates while amplitude decays gradually over time.
Critically Damped	$\zeta =1$	Returns to equilibrium as fast as possible without oscillating.
Overdamped	$\zeta >1$	Sluggish return to equilibrium without oscillating.

Critically damped vibration solution:
$x(t)=[x_0+(v_0+{\omega }_n{x}_0)t]e^{-{\omega }_{n}t}$

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Underdamped Vibration ($\zeta <1$)

The general solution for underdamped vibration is:
$x(t)=A_m{e}^{-\zeta {\omega }_{n}t}\sin ({\omega }_{d}t+\varphi )$

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The peak-to-peak amplitude decays exponentially due to damping, governed by the envelop $e^{-\zeta {\omega }_{n}t}$.

The system oscillates at a slightly lower frequency (${\omega }_d$), due to resistance, than its undamped natural frequency (${\omega }_n$):
${\omega }_d={\omega }_n\sqrt{1-{\zeta }^2}$

Amplitude ($A_m$)
$A_m=\sqrt{x_0^2+(\frac{v_0+\zeta {\omega }_n{x}_0}{{\omega }_d}{)}^2}$

Phase Angle ($\varphi$)
$\varphi ={\tan }^{-1}\frac{{\omega }_d{x}_0}{v_0+\zeta {\omega }_n{x}_0}$

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Calculating Damping Ratio

When working with real-world test data featuring amplitude decay curves, we can use specific metrics to calculate $\zeta$.

Logarithmic Decrement

Logarithmic decrement ($\delta$) compares the heights of two peak amplitudes separated by exactly one period $T_d=2\pi /{\omega }_d$. The peak ratio is expressed as:
$SR=\frac{x(t_m)}{x(t_m+T_d)}=e^{\zeta {\omega }_n{T}_d}$

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Taking the natural log gives $\delta =\ln (SR)=\zeta {\omega }_n{T}_d$.

Damping ratio can then be found using an exact or approximate formula:
$\zeta =\frac{\delta }{\sqrt{4{\pi }^2+{\delta }^2}}\approx \frac{\delta }{2\pi }$

Reverberation Time

Reverberation time ($T_{60}$) is the time it takes for a vibration level to drop by 60 dB, which relates to damping through the following equation:
$\zeta =\frac{60}{8.686T_{60}{\omega }_n}$

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Initial Velocity from an Impact

For a short force $F_0$ acting for $\Delta t$ on a system at rest, impulse-momentum gives $F_0\Delta t=m{v}_0$. We can then solve as free vibration with:
$x(0)=0\dot{x}(0)=\frac{F_0\Delta t}{m}$

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This applies to both undamped and damped free vibration.

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