C4-3. Resonance and Damping

1. Free and Forced Oscillations

  • Free Oscillations: Occur when an oscillating system is displaced and allowed to vibrate naturally without any ongoing transfer of energy to or from the surroundings. A free vibration always oscillates precisely at its natural frequency ($f_0$).
  • Forced Oscillations: A periodic force (external driving force) is continuously applied to an oscillating system to replace the mechanical energy lost to damping. The system is forced to vibrate at the driving frequency ($f_d$).

2. The Nature of Resonance

  • Resonance is the condition achieved when the external driving frequency precisely matches the natural frequency of the oscillating system ($f_d = f_0$).
  • At resonance, the external driver performs maximum work on the system, transferring kinetic energy at absolute peak efficiency.
  • This results in the maximum steady-state amplitude of oscillations.
  • Phase Shift (HL Core): At resonance, the displacement of the oscillator explicitly lags the driving force by exactly $\dfrac{\pi}{2}$ radians ($90^\circ$).

3. The Effect of Damping

Damping is the continuous reduction in energy and amplitude of oscillations due to resistive forces (drag, friction). Crucially, the frequency of damped simple harmonic oscillations remains perfectly constant even as the amplitude decays.

  • Light Damping: The system completes many successive oscillations, and its amplitude decays exponentially with time ($A = A_0 e^{-\gamma t}$). Example: A pendulum swinging in air.
  • Critical Damping: The resistive force is tuned to bring the system back to its original equilibrium position in the shortest possible time without oscillating or overshooting. Example: Car suspension systems.
  • Heavy Damping: The system is so restricted that it completely fails to oscillate, taking an excessively long time to creep back to equilibrium. Example: Heavy hydraulic door dampers.
Time (t) Displacement Light Damping Critical Heavy

4. Effects of Damping on a Resonance Curve (HL)

A resonance curve plots the driving frequency ($f_d$) against the steady-state amplitude of forced oscillations. Increasing the degree of damping warps the profile of the curve in three distinct, mathematically predictable ways:

  • Peak Lowering: The amplitude of resonance vibrations decreases dramatically because energy is being continuously extracted by resistive forces.
  • Peak Broadening: The curve becomes wider and flatter, indicating the system responds substantially to a wider band of frequencies. (Lower Q-factor).
  • Leftward Shift (Crucial HL Property): As heavy damping is introduced, the maximum forced amplitude occurs at a driving frequency slightly lower than the theoretical undamped natural frequency. The resonant peak shifts to the left ($f_{r} < f_0$), even though $f_0$ itself remains unchanged.
Driving Frequency (f) Steady-State Amplitude f₀ (Natural Freq) Light Damping: High, Sharp Heavy Damping: Lower, Broader, Shifted Left

Example 4: Quantitative Energy Analysis at Resonance (HL)

Problem: A spring-mass system with mass $m = 0.400\text{ kg}$ is driven by an external motor. The natural angular frequency of the system is $\omega_0 = 50.0\text{ rad s}^{-1}$. Under conditions of very light damping, the system is driven exactly at resonance and reaches a steady-state amplitude of $A = 0.080\text{ m}$.

(a) Calculate the total maximum kinetic energy of the system at resonance.
(b) The system is placed into a highly viscous fluid, heavily increasing damping. It is again driven at $50.0\text{ rad s}^{-1}$. Explain quantitatively what happens to the maximum kinetic energy.

Solution:

(a) Maximum Kinetic Energy
For an oscillator undergoing SHM, the maximum kinetic energy is equal to the total energy of the system. The formula for maximum kinetic energy is:

$$E_{k(max)} = \dfrac{1}{2} m \omega^2 A^2$$

Substitute the known values at resonance ($\omega = \omega_0$):

$$E_{k(max)} = \dfrac{1}{2} (0.400\text{ kg}) (50.0\text{ rad s}^{-1})^2 (0.080\text{ m})^2$$ $$E_{k(max)} = 0.200 \times 2500 \times 0.0064 = \mathbf{3.20\text{ J}}$$

(b) Effect of High Viscosity (Heavy Damping)
Two distinct factors will cause the maximum kinetic energy to decrease severely:
1. Amplitude Collapse: The viscous fluid will drain energy from the system continually, vastly reducing the steady-state amplitude ($A \ll 0.080\text{ m}$). Since $E_k \propto A^2$, the energy drops quadratically with this reduction.
2. Shift of Resonant Peak: Because heavy damping shifts the resonant peak to the left ($f_r < f_0$), driving the system at the original $50.0\text{ rad s}^{-1}$ means the system is no longer being driven at its peak efficiency. It is operating "off-resonance", compounding the loss in amplitude and energy.