understand that a resistive force acting on an oscillating system causes damping

Damped and Forced Oscillations, Resonance

Learning Objective

Understand that a resistive force acting on an oscillating system causes damping, and explore the consequences for natural and forced oscillations.

1. Simple Harmonic Motion (Review)

For an undamped mass‑spring system the equation of motion is

$$m\ddot{x}+kx=0$$

with solution $x(t)=A\cos(\omega_0 t+\phi)$ where $\omega_0=\sqrt{k/m}$.

2. Introducing a Resistive (Damping) Force

A resistive force proportional to the velocity, $F_{\text{d}}=-b\dot{x}$, is common in real systems (air resistance, internal friction, electrical resistance in an RLC circuit). Adding this term gives the damped equation of motion

$$m\ddot{x}+b\dot{x}+kx=0$$

where $b$ is the damping coefficient (units kg s⁻¹).

3. Types of Damping

Condition	Relation of $b$ to $2\sqrt{mk}$	Behaviour
Underdamped	$b<2\sqrt{mk}$	Oscillatory motion with exponentially decaying amplitude.
Critically damped	$b=2\sqrt{mk}$	Returns to equilibrium as quickly as possible without overshoot.
Overdamped	$b>2\sqrt{mk}$	Non‑oscillatory return to equilibrium, slower than critical.

4. Solution for the Underdamped Case

Define the damping ratio $\zeta =\dfrac{b}{2\sqrt{mk}}$ and the damped angular frequency $\omega_d = \omega_0\sqrt{1-\zeta^2}$.

The displacement is

$$x(t)=A e^{-\zeta\omega_0 t}\cos(\omega_d t+\phi)$$

The amplitude decays as $A(t)=A e^{-\zeta\omega_0 t}$, illustrating how the resistive force reduces the oscillation amplitude over time.

5. Energy Considerations

The mechanical energy of the oscillator is

$$E(t)=\frac{1}{2}kA^2 e^{-2\zeta\omega_0 t}$$

Because the damping force does negative work, energy is continuously removed from the system at a rate

$$\frac{dE}{dt} = -b\dot{x}^2$$

6. Forced Oscillations

When an external periodic driving force $F_{\text{ext}} = F_0\cos(\omega t)$ acts on the damped system, the equation becomes

$$m\ddot{x}+b\dot{x}+kx = F_0\cos(\omega t)$$

The steady‑state (particular) solution is

$$x(t)=X(\omega)\cos(\omega t - \delta)$$

with amplitude

$$X(\omega)=\frac{F_0/m}{\sqrt{(\omega_0^2-\omega^2)^2+(2\zeta\omega_0\omega)^2}}$$

and phase lag

$$\tan\delta = \frac{2\zeta\omega_0\omega}{\omega_0^2-\omega^2}$$

7. Resonance

• Resonance condition: The amplitude $X(\omega)$ reaches a maximum when the driving frequency $\omega$ is close to the natural frequency $\omega_0$, shifted slightly by damping.
• Resonant frequency: For light damping ($\zeta\ll1$) the peak occurs at $\omega_r \approx \omega_0\sqrt{1-2\zeta^2}$.
• Maximum amplitude: $X_{\max}= \dfrac{F_0}{2m\omega_0\zeta}$, showing that weaker damping (smaller $b$) leads to a larger resonant response.

8. Practical Implications

1. Engineering structures (bridges, buildings) must include sufficient damping to limit resonant amplitudes.
2. In musical instruments, controlled damping shapes the sound quality.
3. In electrical circuits, the analogue of $b$ is the resistance $R$ in an RLC circuit; the same mathematics predicts resonance and bandwidth.

9. Summary Checklist

• Resistive force $F_{\text{d}}=-b\dot{x}$ introduces a term $b\dot{x}$ in the equation of motion.
• Damping reduces amplitude exponentially; the rate is set by $b$ (or $\zeta$).
• Three damping regimes are distinguished by the value of $b$ relative to $2\sqrt{mk}$.
• In forced oscillations, the steady‑state amplitude depends on both driving frequency $\omega$ and damping coefficient $b$.
• Resonance occurs near $\omega_0$; the peak amplitude is inversely proportional to the damping coefficient.