\[
F_{\text{damp}}=-b\,\dot x
\]
where b is the damping coefficient (kg s\(^{-1}\)).
\[
m\ddot x + b\dot x + kx = 0
\]
with
\[
\gamma=\frac{b}{2m}\qquad\text{(damping constant, s\(^{-1}\))}
\]
and the natural angular frequency of the undamped system
\[
\omega_{0}= \sqrt{\frac{k}{m}}\qquad\text{(rad s\(^{-1}\))}.
\]
The characteristic equation of the differential equation is
\[
mr^{2}+br+k=0.
\]
Its roots are
\[
r{1,2}= -\gamma\pm\sqrt{\gamma^{2}-\omega{0}^{2}}.
\]
When the discriminant is zero the two roots are equal; this defines the critical damping coefficient:
\[
b{c}=2\sqrt{mk}=2m\omega{0},\qquad\gamma{c}=\frac{b{c}}{2m}=\omega_{0}.
\]
At \(b=b_{c}\) the system returns to equilibrium as fast as possible without overshooting.
| Regime | Condition (relative to \(b_{c}\) or \(\gamma\)) | Typical motion | Displacement‑time sketch |
|---|---|---|---|
| Light (underdamped) | \(b | Oscillatory motion whose amplitude decays exponentially. | fill="none" stroke="#0066cc" stroke-width="2"/> |
| Critical | \(b=b{c}\) or \(\gamma=\omega{0}\) | Monotonic return to equilibrium, fastest possible without overshoot. | critical damping |
| Heavy (over‑damped) | \(b>b{c}\) or \(\gamma>\omega{0}\) | No oscillation; displacement decays monotonically and more slowly than in the critical case. | heavy (over‑damped) |
| Regime | General solution \(x(t)\) | Key parameters |
|---|---|---|
| Light (underdamped) | \[ x(t)=A\,e^{-\gamma t}\cos\!\bigl(\omega_{d}t+\phi\bigr) \] | \(\displaystyle \gamma=\frac{b}{2m},\qquad \omega{d}= \sqrt{\omega{0}^{2}-\gamma^{2}},\qquad \omega_{0}= \sqrt{\frac{k}{m}}\) |
| Critical | \[ x(t)=\bigl(A+Bt\bigr)\,e^{-\gamma t} \] | \(\displaystyle \gamma=\omega_{0}\) |
| Heavy (over‑damped) | \[ x(t)=A\,e^{-\lambda{1}t}+B\,e^{-\lambda{2}t}, \qquad \lambda{1,2}= \gamma\pm\sqrt{\gamma^{2}-\omega{0}^{2}} \] | \(\displaystyle \gamma>\omega_{0}\) |
For an under‑damped system the ratio of successive amplitudes is
\[
\delta = \ln\!\left(\frac{x{n}}{x{n+1}}\right)=\frac{2\pi\gamma}{\omega_{d}}.
\]
It provides a quick experimental way to determine the damping constant \(\gamma\).
\[
m\ddot x + b\dot x + kx = F_{0}\cos(\omega t).
\]
\[
x(t)=X(\omega)\cos\!\bigl(\omega t-\delta\bigr)
\]
where
\[
X(\omega)=\frac{F{0}/m}{\sqrt{(\omega{0}^{2}-\omega^{2})^{2}+(2\gamma\omega)^{2}}},\qquad
\tan\delta=\frac{2\gamma\omega}{\;\omega_{0}^{2}-\omega^{2}\;}.
\]
\[
\boxed{\;\omega{\text{res}}=\sqrt{\;\omega{0}^{2}-2\gamma^{2}\;}}\qquad(b\neq0).
\]
When \(b\to0\) this reduces to \(\omega{\text{res}}\approx\omega{0}\).
\[
Q=\frac{\omega{0}}{2\gamma}=\frac{\omega{0}m}{b}.
\]
A high‑\(Q\) system (light damping) gives a narrow, high peak; a low‑\(Q\) system (heavy damping) gives a broad, low peak.
\[
\Delta\omega=\frac{\omega_{0}}{Q}=2\gamma.
\]
This relation is often required in exam questions on resonance.
| Feature | Behaviour | Effect of Damping |
|---|---|---|
| Amplitude peak | Maximum at \(\omega_{\text{res}}\) | Peak height ↓ and width ↑ as \(\gamma\) (or \(b\)) increases. |
| Phase lag \(\delta\) | \(0^{\circ}\) (low \(\omega\)) → \(90^{\circ}\) at \(\omega_{\text{res}}\) → \(180^{\circ}\) (high \(\omega\)) | Transition becomes less abrupt with larger \(\gamma\). |
| Quality factor \(Q\) | \(Q=\omega_{0}/(2\gamma)\) | Higher damping → lower \(Q\) → broader resonance. |
| Energy loss per cycle | \(\Delta E = 2\pi\,\frac{b}{m}\,X^{2}\) (for the steady‑state amplitude) | Proportional to the damping coefficient. |
| Light (underdamped) | Critical | Heavy (over‑damped) | Forced (resonance) | |
|---|---|---|---|---|
| Condition on \(b\) | \(b| \(b=b_{c}\) | \(b>b_{c}\) | Variable; peak at \(\omega_{\text{res}}\) | |
| Displacement behaviour | Decaying sinusoid: \(A e^{-\gamma t}\cos(\omega_{d}t+\phi)\) | Monotonic: \((A+Bt)e^{-\gamma t}\) | Two exponentials: \(A e^{-\lambda{1}t}+B e^{-\lambda{2}t}\) | Steady‑state sinusoid: \(X(\omega)\cos(\omega t-\delta)\) |
| Energy loss | Exponential, rate \(\propto b\) | Maximum rate compatible with non‑oscillation | Rapid initial loss, then very slow | Continuous supply from driver; average loss \(\propto bX^{2}\) |
| Typical applications | Vehicle suspension, seismometers | Door closers, instrument pointers | Heavy‑machinery shock absorbers | Musical instruments, radio/TV circuits, tuning forks |
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