Published by Patrick Mutisya · 8 days ago
Damping is the gradual loss of mechanical energy of an oscillating system due to a resistive force (usually proportional to velocity).
The equation of motion for a damped simple harmonic oscillator is
\$m\ddot{x}+b\dot{x}+kx=0,\$
where \$m\$ is the mass, \$k\$ the spring constant and \$b\$ the damping coefficient.
Depending on the magnitude of \$b\$ relative to the critical damping coefficient \$b_c=2\sqrt{mk}\$, three regimes are defined:
Below are qualitative sketches of \$x(t)\$ for each damping regime. The amplitude \$A\$ is shown decreasing with time.
Light damping (underdamped):
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Critical damping:
A
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Heavy damping (over‑damped):
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For each regime the solution of the differential equation can be written as:
| Regime | Solution \$x(t)\$ | Behaviour |
|---|---|---|
| Light (underdamped) | \$x(t)=A e^{-\gamma t}\cos(\omega_d t+\phi)\$ | Oscillatory with exponentially decreasing amplitude; \$\gamma=b/(2m)\$, \$\omegad=\sqrt{\omega0^2-\gamma^2}\$. |
| Critical | \$x(t)=(A+Bt)e^{-\gamma t}\$ | Returns to equilibrium as quickly as possible without overshooting. |
| Heavy (over‑damped) | \$x(t)=A e^{-\lambda1 t}+B e^{-\lambda2 t}\$ | Two exponential terms with \$\lambda{1,2}=\gamma\pm\sqrt{\gamma^2-\omega0^2}\$; no oscillation. |
When an external periodic driving force \$F(t)=F_0\cos(\omega t)\$ acts on the system, the equation becomes
\$m\ddot{x}+b\dot{x}+kx=F_0\cos(\omega t).\$
The steady‑state solution is
\$x(t)=X\cos(\omega t-\delta),\$
where the amplitude \$X\$ and phase lag \$\delta\$ depend on the driving frequency \$\omega\$.
Resonance occurs when the driving frequency \$\omega\$ is close to the natural frequency \$\omega_0=\sqrt{k/m}\$ of the undamped system.
The amplitude of the steady‑state motion is
\$X(\omega)=\frac{F0/m}{\sqrt{(\omega0^2-\omega^2)^2+(2\gamma\omega)^2}}.\$
The maximum amplitude \$X_{\max}\$ is reached at
\$\omega{\text{res}}=\sqrt{\omega0^2-2\gamma^2}\;( \text{for }b\neq0).\$
Key features of resonance:
| Feature | Light Damping | Critical Damping | Heavy Damping | Resonance (Forced) |
|---|---|---|---|---|
| Damping coefficient \$b\$ | \$b | \$b=b_c\$ | \$b>b_c\$ | Varies; optimum at \$\omega_{\text{res}}\$ |
| Displacement behaviour | Oscillatory, amplitude \$\propto e^{-\gamma t}\$ | Monotonic return, fastest without overshoot | Monotonic return, slower than critical | Steady‑state oscillation with large amplitude |
| Energy loss | Gradual, exponential | Maximum rate without overshoot | Very rapid initial loss, then slow | Energy supplied continuously by driver |
| Typical applications | Vehicle suspension (comfort) | Door closers, instrument pointers | Shock absorbers in heavy machinery | Musical instruments, radio circuits |