Damped and forced oscillations, resonance

1. Overview

In an ideal simple harmonic oscillator, energy is conserved and the system oscillates with a constant amplitude indefinitely. However, real-world macroscopic systems are subject to resistive forces (such as friction, air resistance, or viscous drag) that perform work against the motion. This work done by the system is converted into thermal energy, which is dissipated into the surroundings. This process is known as damping, and it results in a continuous decrease in the amplitude of the oscillations over time.

When a system is left to oscillate on its own after an initial displacement, it vibrates at its natural frequency ($f_0$). If a periodic external force is applied, the system undergoes forced oscillations at the frequency of the external driver ($f$). The interaction between the driving frequency and the natural frequency leads to resonance—a state of maximum energy transfer where the amplitude of oscillation reaches its peak. Understanding how to control damping and resonance is fundamental to engineering stable structures, designing vehicle suspension, and utilizing medical technologies like Magnetic Resonance Imaging (MRI).

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Key Definitions

• Damping: The dissipation of total energy from an oscillating system over time due to resistive forces (e.g., friction or viscous drag) acting in the opposite direction to motion, which results in a reduction in amplitude.
• Light Damping (Under-damping): A condition where the resistive forces are small, allowing the system to oscillate many times with an amplitude that decreases exponentially over time. The period remains approximately constant.
• Critical Damping: The specific degree of damping that allows an oscillating system to return to its equilibrium position in the shortest possible time without overshooting or oscillating.
• Heavy Damping (Over-damping): A condition where the resistive forces are very large, preventing oscillation and causing the system to return to its equilibrium position more slowly than in critical damping.
• Natural Frequency ($f_0$): The frequency at which an object oscillates when it is allowed to vibrate freely without any external driving force or significant damping.
• Forced Oscillations: Oscillations that occur when a system is subjected to an external periodic driving force, causing the system to vibrate at the frequency of the driver ($f$) rather than its natural frequency.
• Resonance: A phenomenon occurring when the driving frequency of an external force is equal to the natural frequency of the system ($f = f_0$), resulting in maximum amplitude and maximum rate of energy transfer from the driver to the oscillator.

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Content

3.1 The Physics of Damping

Damping occurs because a resistive force $F_r$ acts on the oscillator. In most A-Level contexts (like air resistance or viscous drag), this force is proportional to the velocity ($F_r \propto -v$).

Energy Dissipation: The total energy $E$ of an oscillator is directly proportional to the square of its amplitude $A$: $$E = \frac{1}{2} m \omega^2 A^2$$ $$E \propto A^2$$

As the system moves, the resistive force does work. The power $P$ dissipated at any instant is given by $P = F_r v$. Since energy is being removed from the system, the total energy $E$ decreases, and consequently, the amplitude $A$ must decrease.

3.2 Graphical Representation of Damping Types

In the exam, you must be able to sketch and identify displacement-time ($x-t$) graphs for the three types of damping.

1. Light Damping

• Graph Characteristics: A sine or cosine wave where the peaks (maxima) and troughs (minima) follow an exponential decay envelope.
• Key Feature: The system crosses the equilibrium ($x=0$) line multiple times.
• Period: The time period $T$ is slightly longer than the undamped period but is treated as constant for most calculations.

2. Critical Damping

• Graph Characteristics: The displacement drops from its maximum value to zero very quickly.
• Key Feature: The curve does not cross the time axis. It reaches $x=0$ in the minimum time $t$.
• Application: Used in car suspension (shock absorbers) and analog meters (like a moving-coil galvanometer) to ensure the pointer stops at the correct reading immediately without wobbling.

3. Heavy Damping

• Graph Characteristics: The displacement returns to zero very slowly.
• Key Feature: Like critical damping, there is no oscillation, but the gradient of the $x-t$ graph is much shallower.
• Application: Heavy fire doors or hydraulic closers that prevent slamming.

3.3 Forced Oscillations and the Driver-Driven Relationship

When a system is "forced," it eventually stops oscillating at its natural frequency $f_0$ and adopts the frequency of the driver $f$.

• Transient State: The initial period where the system's natural frequency and the driving frequency "fight," often resulting in irregular motion.
• Steady State: The system oscillates purely at the driving frequency $f$ with a constant amplitude.

3.4 Resonance and the Resonance Curve

Resonance is the specific case of forced oscillation where $f = f_0$.

The Resonance Graph (Amplitude vs. Driving Frequency): If you plot the amplitude $A$ of the driven system against the frequency $f$ of the driver, you get a resonance curve.

1. At $f < f_0$: The amplitude is small. The driver and oscillator are mostly in phase.
2. At $f = f_0$: The amplitude reaches its maximum. The phase difference between the driver and the oscillator is $\frac{\pi}{2}$ rad (90°).
3. At $f > f_0$: The amplitude decreases. At very high frequencies, the oscillator cannot keep up, and the amplitude approaches zero. The driver and oscillator are $\pi$ rad (180°) out of phase.

3.5 The Effect of Damping on Resonance

Damping significantly changes the shape of the resonance curve. As damping increases:

1. The peak amplitude decreases: More energy is dissipated as heat, so the maximum displacement is lower.
2. The peak becomes broader (flatter): The system becomes less sensitive to the exact matching of frequencies.
3. The resonant frequency shifts slightly to the left: The frequency at which maximum amplitude occurs becomes slightly lower than the true natural frequency $f_0$.

3.6 Barton's Pendulums: A Classic Demonstration

A heavy "driver" pendulum and several lighter "driven" pendulums of varying lengths are hung from a common string.

• The driver pendulum is set in motion.
• It provides a periodic force to the others via the string.
• The driven pendulum with the same length as the driver will have the same natural frequency ($T = 2\pi\sqrt{L/g}$).
• This pendulum will oscillate with the largest amplitude because it is in resonance.
• Pendulums much shorter or longer than the driver will have very small amplitudes.

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Key Equations

1. Total Energy of an Oscillator $$E = \frac{1}{2} m \omega^2 A^2$$

• $E$: Total energy (J)
• $m$: Mass (kg)
• $\omega$: Angular frequency (rad s⁻¹)
• $A$: Amplitude (m)
• Note: Not explicitly on the formula sheet, but derived from $v_{max} = \omega A$ and $E_k = \frac{1}{2}mv^2$.

2. Angular Frequency $$\omega = 2\pi f = \frac{2\pi}{T}$$

• Status: On Data Sheet.

3. Relationship between Energy and Amplitude $$E \propto A^2 \implies \frac{E_1}{E_2} = \left(\frac{A_1}{A_2}\right)^2$$

• Status: Must be memorized for ratio questions.

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5. Worked Examples

Worked Example 1 — Energy Loss in a Damped System

A mass of $200\text{ g}$ is attached to a spring and undergoes light damping. The initial amplitude of oscillation is $5.0\text{ cm}$. After $20$ complete cycles, the amplitude has decreased to $2.0\text{ cm}$. The frequency of oscillation is constant at $1.5\text{ Hz}$. Calculate the average power dissipated by the resistive forces during these $20$ cycles.

Step 1: Convert all units to SI. $m = 0.20\text{ kg}$ $A_1 = 0.05\text{ m}$ $A_2 = 0.02\text{ m}$ $f = 1.5\text{ Hz}$

Step 2: Calculate angular frequency $\omega$. $$\omega = 2\pi f = 2 \times \pi \times 1.5 = 3\pi \approx 9.425\text{ rad s}^{-1}$$

Step 3: Calculate initial and final total energy. $$E_1 = \frac{1}{2} m \omega^2 A_1^2 = 0.5 \times 0.20 \times (3\pi)^2 \times (0.05)^2$$ $$E_1 = 0.1 \times 88.826 \times 0.0025 = 0.02221\text{ J}$$

$$E_2 = \frac{1}{2} m \omega^2 A_2^2 = 0.5 \times 0.20 \times (3\pi)^2 \times (0.02)^2$$ $$E_2 = 0.1 \times 88.826 \times 0.0004 = 0.00355\text{ J}$$

Step 4: Calculate total energy lost. $$\Delta E = E_1 - E_2 = 0.02221 - 0.00355 = 0.01866\text{ J}$$

Step 5: Calculate the time taken for 20 cycles. $$T = \frac{1}{f} = \frac{1}{1.5} = 0.667\text{ s}$$ $$t_{total} = 20 \times T = 20 \times 0.667 = 13.33\text{ s}$$

Step 6: Calculate average power ($P = \Delta E / t$). $$P = \frac{0.01866}{13.33} = 0.0014\text{ W}$$ Answer: $1.4 \times 10^{-3}\text{ W}$

Worked Example 2 — Interpreting Resonance Curves

A student performs an experiment to investigate the resonance of a bridge model. They apply a driving force of varying frequency and record the maximum amplitude. The natural frequency of the model is $10\text{ Hz}$. (a) Describe the change in amplitude as the driving frequency increases from $2\text{ Hz}$ to $20\text{ Hz}$. (b) If the student adds a thick oil to the joints of the model, sketch how the resonance curve changes.

Solution: (a) As the frequency increases from $2\text{ Hz}$ to $10\text{ Hz}$, the amplitude increases. At $10\text{ Hz}$ ($f = f_0$), the amplitude reaches its maximum value (resonance). As the frequency increases from $10\text{ Hz}$ to $20\text{ Hz}$, the amplitude decreases. (b) Adding oil increases the damping. The new curve should have:

1. A lower peak (lower maximum amplitude).
2. A broader peak (wider resonance region).
3. A peak position shifted slightly to the left (lower frequency).

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Common Mistakes to Avoid

• ❌ Wrong: Stating that damping reduces the frequency of oscillation significantly.
  • ✓ Right: In light damping, the frequency/period is considered constant for calculations. Only the amplitude decreases.
• ❌ Wrong: Confusing Critical and Heavy damping on a graph.
  • ✓ Right: Critical damping is the "fastest" way back to zero. Heavy damping is a "slow" crawl back to zero. Neither one oscillates.
• ❌ Wrong: Thinking resonance occurs when the "driving force is at its maximum."
  • ✓ Right: Resonance is about frequency matching ($f_{driver} = f_{natural}$), not the magnitude of the force itself.
• ❌ Wrong: Forgetting the square in the energy-amplitude relationship.
  • ✓ Right: If the amplitude is halved, the energy is one-quarter ($1/2^2$) of the original, not half.
• ❌ Wrong: Using degrees for phase difference in resonance descriptions.
  • ✓ Right: Use radians. At resonance, the phase difference is $\pi/2$ rad.

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Exam Tips

1. Keywords for Resonance: When defining resonance, you must mention:
   • Driving frequency = Natural frequency.
   • Maximum amplitude.
   • Maximum rate of energy transfer.
2. Graph Precision: When sketching light damping, ensure the time period (distance between zero-crossings) looks constant. If the "waves" get wider or narrower, you will lose marks.
3. Phase Relationships: Remember the "Phase Trio" for forced oscillations:
   • Below resonance: Driver and oscillator are in phase (0 rad).
   • At resonance: Oscillator lags driver by $\pi/2$ rad.
   • Above resonance: Oscillator lags driver by $\pi$ rad (antiphase).
4. Real-world Contexts: Be prepared to apply these concepts to:
   • Buildings/Bridges: Damping is added (tuned mass dampers) to prevent resonance during earthquakes or high winds.
   • Radio Tuning: A radio circuit is adjusted so its natural electrical frequency matches the frequency of the broadcast station (resonance).
   • MRI: Atomic nuclei are forced to resonate by radio-frequency pulses.
5. Unit Check: Always convert mass to kg and amplitude to m before using the energy equation $E = \frac{1}{2}m\omega^2A^2$. Squaring a value in cm leads to a factor of $10,000$ error.