Part I: Oscillations | Chapter 2

Damped & Driven Oscillators

Real oscillators lose energy — and gain it from external driving forces

2.1 The Damped Harmonic Oscillator

Derivation 1: Equation of Motion with Damping

Real oscillators experience friction or resistive forces that dissipate energy. For a viscous damping force proportional to velocity, $F_{\text{damp}} = -b\dot{x}$, Newton's second law gives:

$$m\ddot{x} + b\dot{x} + kx = 0$$$$\ddot{x} + 2\gamma\dot{x} + \omega_0^2 x = 0$$

where $\gamma = b/(2m)$ is the damping rate and $\omega_0 = \sqrt{k/m}$

Trying $x = e^{\alpha t}$, we get the characteristic equation:

$$\alpha^2 + 2\gamma\alpha + \omega_0^2 = 0 \quad \Longrightarrow \quad \alpha = -\gamma \pm \sqrt{\gamma^2 - \omega_0^2}$$

The nature of the solution depends critically on the ratio $\gamma/\omega_0$, which defines three qualitatively different regimes.

2.2 Three Damping Regimes

Figure. Three damping regimes: underdamped (blue, oscillatory decay), critically damped (yellow, fastest non-oscillatory return), and overdamped (red, sluggish exponential decay).

Derivation 2: Classification of Solutions

Underdamped: $\gamma < \omega_0$

The discriminant is negative, giving complex roots. The system oscillates with exponentially decaying amplitude:

$$x(t) = Ae^{-\gamma t}\cos(\omega_d t + \phi) \quad \text{where } \omega_d = \sqrt{\omega_0^2 - \gamma^2}$$

The damped frequency $\omega_d$ is always less than $\omega_0$. This is the most common case in practice: the system rings down with decaying oscillations.

Critically Damped: $\gamma = \omega_0$

The discriminant is zero, giving a repeated root $\alpha = -\gamma$. The general solution requires a second, linearly independent solution:

$$x(t) = (C_1 + C_2 t)e^{-\gamma t}$$

This is the fastest return to equilibrium without oscillation. Critical damping is used in door closers, car suspensions, and galvanometer movements.

Overdamped: $\gamma > \omega_0$

Both roots are real and negative. The system returns to equilibrium sluggishly without oscillation:

$$x(t) = C_1 e^{\alpha_+ t} + C_2 e^{\alpha_- t} \quad \text{where } \alpha_\pm = -\gamma \pm \sqrt{\gamma^2 - \omega_0^2}$$

Both exponentials decay, but the slower one ($\alpha_+$) dominates at late times. The return to equilibrium is always slower than critical damping.

2.3 The Quality Factor

Derivation 3: Energy Decay and Q

The quality factor $Q$ measures how many oscillations occur before the energy decays significantly. For the underdamped case, the energy decays as:

$$E(t) = E_0 e^{-2\gamma t} = E_0 e^{-t/\tau}$$

where $\tau = 1/(2\gamma)$ is the energy decay time constant

The Q factor is defined as $2\pi$ times the ratio of stored energy to energy lost per cycle:

Quality Factor

$$Q = 2\pi \frac{E}{\Delta E_{\text{per cycle}}} = \frac{\omega_0}{2\gamma} = \frac{\omega_0 \tau}{1}$$

A high-Q oscillator rings for many cycles. Typical values: tuning fork $Q \sim 10^3$, quartz crystal $Q \sim 10^5$, optical cavity $Q \sim 10^{10}$.

The number of oscillations before the amplitude drops to $1/e$ of its initial value is:

$$N_{1/e} = \frac{Q}{\pi} \approx \frac{Q}{3}$$

2.4 Forced Oscillations and Resonance

Derivation 4: Steady-State Response

Now add a sinusoidal driving force $F(t) = F_0\cos(\omega t)$:

$$\ddot{x} + 2\gamma\dot{x} + \omega_0^2 x = \frac{F_0}{m}\cos(\omega t)$$

Using the complex notation $\tilde{x} = \tilde{A}e^{i\omega t}$ for the steady-state (particular) solution:

$$(-\omega^2 + 2i\gamma\omega + \omega_0^2)\tilde{A} = \frac{F_0}{m}$$$$\tilde{A} = \frac{F_0/m}{(\omega_0^2 - \omega^2) + 2i\gamma\omega}$$

The amplitude and phase of the steady-state oscillation are:

Resonance Response

$$A(\omega) = \frac{F_0/m}{\sqrt{(\omega_0^2 - \omega^2)^2 + (2\gamma\omega)^2}}$$$$\tan\phi = \frac{-2\gamma\omega}{\omega_0^2 - \omega^2}$$

Resonance Peak

The amplitude is maximized at the resonance frequency:

$$\omega_{\text{res}} = \sqrt{\omega_0^2 - 2\gamma^2} \approx \omega_0 \text{ for } Q \gg 1$$$$A_{\text{max}} = \frac{F_0}{2m\gamma\omega_d} \approx \frac{QF_0}{m\omega_0^2} \text{ for } Q \gg 1$$

The full width at half maximum (FWHM) of the power resonance curve is:

$$\Delta\omega_{\text{FWHM}} = 2\gamma = \frac{\omega_0}{Q}$$

Resonance is everywhere: radio tuning (selecting one station from many), MRI (exciting nuclear spins), laser cavities, musical instruments, and even the destructive resonance of the Tacoma Narrows Bridge (1940).

2.5 Transient and Steady-State

Derivation 5: Complete Solution

The complete solution to the driven, damped oscillator is the sum of the homogeneous (transient) and particular (steady-state) solutions:

Complete Solution

$$x(t) = \underbrace{Be^{-\gamma t}\cos(\omega_d t + \psi)}_{\text{transient}} + \underbrace{A(\omega)\cos(\omega t + \phi)}_{\text{steady state}}$$

The transient dies away after a few decay times $\tau = 1/(2\gamma)$, leaving only the steady-state oscillation at the driving frequency $\omega$.

The constants $B$ and $\psi$ are determined by initial conditions. After the transient has died out (typically $t \gg 1/\gamma$), the system oscillates at the driving frequency regardless of the natural frequency.

Power Absorption

The time-averaged power absorbed from the driving force is:

$$\langle P \rangle = \frac{F_0^2}{2m} \frac{\gamma\omega^2}{(\omega_0^2 - \omega^2)^2 + (2\gamma\omega)^2}$$

This Lorentzian line shape peaks at $\omega = \omega_0$ (exactly, not shifted) with a maximum value of $F_0^2/(4m\gamma)$. The FWHM of the power curve is$\Delta\omega = 2\gamma$, confirming that $Q = \omega_0/\Delta\omega$.

2.5b Impedance and Transfer Functions

The complex notation provides the most elegant treatment of the driven oscillator. The mechanical impedance relates the complex force to the complex velocity:

$$Z_{\text{mech}}(\omega) = \frac{\tilde{F}}{\tilde{v}} = b + i\left(m\omega - \frac{k}{\omega}\right)$$

At resonance ($\omega = \omega_0$), the reactive part vanishes and$Z = b$: the system is purely resistive. This is exactly analogous to the electrical impedance $Z = R + i(\omega L - 1/\omega C)$ of an RLC circuit.

The transfer function $H(\omega) = \tilde{x}/\tilde{F}$fully characterizes the system's linear response:

Transfer Function

$$H(\omega) = \frac{1/m}{(\omega_0^2 - \omega^2) + 2i\gamma\omega}$$

The magnitude $|H(\omega)|$ gives the amplitude response, and$\arg H(\omega)$ gives the phase shift. The transfer function contains complete information about the system and is the central concept in signal processing and control theory.

For any periodic driving force, Fourier decompose it and apply $H(\omega)$to each component. For an arbitrary driving force $F(t)$, the response is the convolution with the impulse response $h(t)$ (the inverse Fourier transform of $H(\omega)$):

$$x(t) = \int_{-\infty}^{t} h(t - t')F(t')\,dt'$$

2.5c Parametric Resonance

Besides direct forcing, oscillations can be excited by periodically modulating a parameter of the system. Consider a pendulum whose length varies periodically:

$$\ddot{\theta} + \omega_0^2[1 + \epsilon\cos(\Omega t)]\theta = 0$$

This is the Mathieu equation. Parametric resonance occurs primarily when the modulation frequency is near $\Omega \approx 2\omega_0$ (twice the natural frequency), with instability bands at $\Omega \approx 2\omega_0/n$for integer $n$.

Everyday example: A child on a swing pumps by standing and squatting twice per swing period — this is parametric resonance at $\Omega = 2\omega_0$. The energy input grows exponentially until limited by nonlinearity. Parametric amplification is also used in microwave electronics (parametric amplifiers) and quantum optics (optical parametric oscillators).

2.5d Beyond Linearity: Anharmonic Effects

Real oscillators are only approximately harmonic. Including the next Taylor expansion term gives the Duffing oscillator:

$$\ddot{x} + 2\gamma\dot{x} + \omega_0^2 x + \alpha x^3 = F_0\cos(\omega t)$$

Key nonlinear effects include:

Amplitude-Dependent Frequency

The oscillation frequency shifts with amplitude:$\omega \approx \omega_0(1 + \frac{3\alpha A^2}{8\omega_0^2})$. The pendulum period increasing with amplitude is an example.

Bistability and Hysteresis

The driven Duffing oscillator can have two stable response amplitudes at the same driving frequency. Sweeping the frequency up vs down gives different response curves (hysteresis), used in MEMS sensors and nonlinear circuits.

Harmonic Generation

The $x^3$ nonlinearity generates frequency components at$3\omega$ (third harmonic generation). In optics, this is the basis of nonlinear frequency conversion in crystals.

Chaos

For sufficiently strong driving, the Duffing oscillator exhibits deterministic chaos: sensitive dependence on initial conditions. This was one of the first systems studied in nonlinear dynamics.

2.6 Applications

RLC Circuits

Adding a resistor R to the LC circuit gives $L\ddot{Q} + R\dot{Q} + Q/C = V(t)$, exactly the damped driven oscillator with $\gamma = R/(2L)$. RLC filters are the backbone of radio receivers and signal processing.

Seismometers

A damped mass-spring system responds to ground motion. The displacement of the mass relative to the frame records seismic waves. Modern broadband seismometers use feedback to extend the frequency response while maintaining critical damping.

Laser Linewidths

The natural linewidth of an atomic transition is determined by the radiative damping rate. The Lorentzian line shape we derived for the power absorption is exactly the line shape of a classical radiating atom (Lorentz model).

Suspension Bridges

The 1940 collapse of the Tacoma Narrows Bridge was a dramatic demonstration of resonance. Wind-driven vortex shedding provided periodic forcing near a torsional mode frequency, with insufficient damping to prevent catastrophic oscillation.

2.6b The RLC Circuit: Electrical Resonance

The damped driven RLC series circuit is the exact electrical analog of the mechanical oscillator. With an AC voltage source $V(t) = V_0\cos(\omega t)$:

$$L\ddot{Q} + R\dot{Q} + \frac{Q}{C} = V_0\cos(\omega t)$$

The impedance formalism gives the current amplitude and phase directly:

RLC Impedance

$$Z(\omega) = R + i\left(\omega L - \frac{1}{\omega C}\right)$$$$|I_0| = \frac{V_0}{|Z|}, \quad Q = \frac{1}{R}\sqrt{\frac{L}{C}} = \frac{\omega_0 L}{R}$$

At resonance ($\omega = 1/\sqrt{LC}$), the reactive impedance vanishes and the current is maximum. The voltage across the capacitor or inductor can be $Q$times the source voltage — this is the basis of the tank circuit used in radio tuning.

Bandwidth: The 3 dB bandwidth (half-power points) of the RLC circuit is$\Delta\omega = R/L = \omega_0/Q$. For a radio receiver tuned to 1 MHz with $Q = 100$, the bandwidth is 10 kHz — enough for an AM radio station. FM radio requires $Q \sim 10$ for its 200 kHz bandwidth.

2.7 Historical Context

George Gabriel Stokes (1851): Studied the resistance of fluids to the motion of pendulums, establishing the viscous damping model$F = -b\dot{x}$ that we use today. His work was motivated by improving the accuracy of pendulum clocks.

Hendrik Lorentz (1878): Developed the classical electron model of atomic absorption, treating bound electrons as damped, driven oscillators. This Lorentz oscillator model correctly predicts the frequency-dependent dielectric constant and the shape of absorption lines.

Lord Rayleigh (1894): Introduced the concept of the quality factor in his study of acoustic resonators. He showed that the sharpness of a resonance peak is inversely proportional to the damping, formalized as the Q factor.

Heinrich Hertz (1887): Used resonant LC circuits to generate and detect electromagnetic waves, confirming Maxwell's predictions. The tuning of the receiver circuit to the transmitter frequency is a direct application of driven oscillator resonance.

2.8 Python Simulation

This simulation visualizes the three damping regimes, the resonance curve, the transient response to a driving force, and the power absorption spectrum.

Damped and Driven Oscillators: Regimes, Resonance, and Power Absorption

Python

script.py141 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
from scipy.integrate import odeint

# ============================================================
# Damped and Driven Oscillators
# ============================================================

fig, axes = plt.subplots(2, 2, figsize=(12, 10))
fig.suptitle('Damped and Driven Oscillators', fontsize=16, fontweight='bold', color='white')
fig.patch.set_facecolor('#0a0a0a')

for ax in axes.flat:
    ax.set_facecolor('#0a0a0a')
    ax.tick_params(colors='white')
    ax.xaxis.label.set_color('white')
    ax.yaxis.label.set_color('white')
    ax.title.set_color('white')
    for spine in ax.spines.values():
        spine.set_color('#333')

omega0 = 2 * np.pi  # natural frequency
t = np.linspace(0, 5, 2000)

# --- Panel 1: Three damping regimes ---
ax1 = axes[0, 0]

# Underdamped
gamma_u = 0.3 * omega0
omega_d = np.sqrt(omega0**2 - gamma_u**2)
x_under = np.exp(-gamma_u * t) * np.cos(omega_d * t)
ax1.plot(t, x_under, color='#22d3ee', linewidth=2, label='Underdamped (gamma=0.3 omega0)')

# Critically damped
gamma_c = omega0
x_crit = (1 + gamma_c * t) * np.exp(-gamma_c * t)
ax1.plot(t, x_crit, color='#f97316', linewidth=2, label='Critical (gamma=omega0)')

# Overdamped
gamma_o = 2 * omega0
alpha_p = -gamma_o + np.sqrt(gamma_o**2 - omega0**2)
alpha_m = -gamma_o - np.sqrt(gamma_o**2 - omega0**2)
C1 = alpha_m / (alpha_m - alpha_p)
C2 = -alpha_p / (alpha_m - alpha_p)
x_over = C1 * np.exp(alpha_p * t) + C2 * np.exp(alpha_m * t)
ax1.plot(t, x_over, color='#a855f7', linewidth=2, label='Overdamped (gamma=2 omega0)')

# Envelope for underdamped
ax1.plot(t, np.exp(-gamma_u * t), color='#22d3ee', linewidth=1, linestyle='--', alpha=0.5)
ax1.plot(t, -np.exp(-gamma_u * t), color='#22d3ee', linewidth=1, linestyle='--', alpha=0.5)

ax1.axhline(y=0, color='white', linewidth=0.3, alpha=0.3)
ax1.set_xlabel('Time (s)')
ax1.set_ylabel('x(t) / x(0)')
ax1.set_title('Three Damping Regimes')
ax1.legend(fontsize=7, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 2: Resonance curves for different Q ---
ax2 = axes[0, 1]
omega = np.linspace(0.01, 3 * omega0, 1000)

for Q_val, color in [(2, '#22d3ee'), (5, '#f97316'), (10, '#a855f7'), (50, '#4ade80')]:
    gamma_val = omega0 / (2 * Q_val)
    A_resp = 1.0 / np.sqrt((omega0**2 - omega**2)**2 + (2 * gamma_val * omega)**2)
    A_resp_norm = A_resp / np.max(A_resp)
    ax2.plot(omega / omega0, A_resp_norm, linewidth=2, color=color, label='Q = ' + str(Q_val))

ax2.set_xlabel('omega / omega_0')
ax2.set_ylabel('Amplitude (normalized)')
ax2.set_title('Resonance Curves')
ax2.legend(fontsize=8, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')
ax2.axvline(x=1, color='white', linewidth=0.5, alpha=0.3, linestyle=':')

# --- Panel 3: Transient + steady state ---
ax3 = axes[1, 0]
gamma_tr = 0.15 * omega0
omega_drive = 0.9 * omega0
F0_over_m = omega0**2

def driven_osc(y, t_val):
    x_val, v_val = y
    dxdt = v_val
    dvdt = -2 * gamma_tr * v_val - omega0**2 * x_val + F0_over_m * np.cos(omega_drive * t_val)
    return [dxdt, dvdt]

t_long = np.linspace(0, 8, 3000)
y0 = [0.0, 0.0]
sol = odeint(driven_osc, y0, t_long)

# Steady state amplitude
A_ss = F0_over_m / np.sqrt((omega0**2 - omega_drive**2)**2 + (2 * gamma_tr * omega_drive)**2)

ax3.plot(t_long, sol[:, 0], color='#22d3ee', linewidth=1.5, label='Full solution')
ax3.plot(t_long, A_ss * np.ones_like(t_long), color='#f97316', linewidth=1, linestyle='--', alpha=0.7, label='Steady-state amplitude')
ax3.plot(t_long, -A_ss * np.ones_like(t_long), color='#f97316', linewidth=1, linestyle='--', alpha=0.7)
ax3.axhline(y=0, color='white', linewidth=0.3, alpha=0.3)
ax3.set_xlabel('Time (s)')
ax3.set_ylabel('x(t)')
ax3.set_title('Transient + Steady State (omega_d = 0.9 omega_0)')
ax3.legend(fontsize=8, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 4: Power absorption (Lorentzian) ---
ax4 = axes[1, 1]
for Q_val, color in [(3, '#22d3ee'), (10, '#f97316'), (30, '#a855f7')]:
    gamma_val = omega0 / (2 * Q_val)
    P_avg = gamma_val * omega**2 / ((omega0**2 - omega**2)**2 + (2 * gamma_val * omega)**2)
    P_avg_norm = P_avg / np.max(P_avg)
    ax4.plot(omega / omega0, P_avg_norm, linewidth=2, color=color, label='Q = ' + str(Q_val))

ax4.set_xlabel('omega / omega_0')
ax4.set_ylabel('Power (normalized)')
ax4.set_title('Power Absorption (Lorentzian)')
ax4.legend(fontsize=8, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')
ax4.axvline(x=1, color='white', linewidth=0.5, alpha=0.3, linestyle=':')

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a0a')
plt.close()
print("Plot saved to output.png")

print("\n" + "=" * 60)
print("DAMPED AND DRIVEN OSCILLATORS: RESULTS")
print("=" * 60)
print("\n--- Damping Parameters ---")
print("  omega_0 = " + str(round(omega0, 4)) + " rad/s")
print("  Underdamped: gamma = " + str(round(gamma_u, 4)) + ", omega_d = " + str(round(omega_d, 4)))
print("  Q_underdamped = " + str(round(omega0 / (2 * gamma_u), 2)))
print("\n--- Resonance Peak ---")
for Q_val in [2, 5, 10, 50]:
    gamma_val = omega0 / (2 * Q_val)
    omega_res = np.sqrt(omega0**2 - 2 * gamma_val**2)
    A_peak = 1.0 / (2 * gamma_val * omega_res)
    FWHM = 2 * gamma_val
    print("  Q = " + str(Q_val) + ": omega_res/omega_0 = " + str(round(omega_res / omega0, 4)) + ", FWHM/omega_0 = " + str(round(FWHM / omega0, 4)))
print("\n--- Steady-State Amplitude ---")
print("  Driving at omega = 0.9 omega_0")
print("  Steady-state amplitude = " + str(round(A_ss, 4)))
print("=" * 60)

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Summary

Core Equations

EOM: $\ddot{x} + 2\gamma\dot{x} + \omega_0^2 x = F_0\cos(\omega t)/m$
Q factor: $Q = \omega_0/(2\gamma)$
Resonance amplitude: $\propto Q$
FWHM: $\Delta\omega = \omega_0/Q$
Lorentzian line shape for power

Key Insights

Three regimes: under/critical/overdamped
Critical damping = fastest non-oscillatory return
High Q = sharp resonance = slow decay
Steady state oscillates at driving frequency
Phase shifts by $\pi$ across resonance

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