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Damped Harmonic Motion

Damped harmonic motion is the oscillation of a system whose amplitude diminishes continuously with time due to dissipative forces such as friction or viscous drag. Energy is gradually converted into non-recoverable thermal energy, reducing the mechanical energy of the oscillator.

UP TGT ScienceUP PGT PhysicsStudy Notes + Practice

Scope and prerequisites

TGT core

Understanding of differential equations for damped oscillators, definition of damping coefficient, relaxation time, quality factor, and qualitative graphs of displacement-time under light, critical, and heavy damping.

PGT extension

Rigorous derivation of underdamped, critically damped, and overdamped motion solutions, phase space trajectories, logarithmic decrement, and power dissipation in driven-damped systems.

Prerequisites

Simple Harmonic Motion (SHM) kinematics and dynamics
Linear second-order homogeneous differential equations with constant coefficients
Conservation of mechanical energy and non-conservative forces

Equation of Motion and Restoring Force

The net force acting on a damped oscillator comprises the restoring force (-kx) and the damping resistive force (-bv).
The parameter b represents the damping constant, depending on the medium viscosity and geometry of the moving object.
The resulting differential equation is m(d²x/dt²) + b(dx/dt) + kx = 0.

Classification of Damping Regimes

Light damping occurs when the damping force is small compared to the restoring and inertial forces, leading to oscillatory decay.
Critical damping occurs when the system returns to equilibrium in the shortest possible time without oscillating.
Overdamping (heavy damping) occurs when the medium is highly viscous, causing the system to creep slowly back to equilibrium without oscillation.

Logarithmic Decrement and Relaxation Time

Logarithmic decrement (λ) is the natural logarithm of the ratio of two successive amplitudes separated by one time period.
Relaxation time (τ) is the time interval in which the amplitude of a damped oscillator falls to 1/e of its initial value.
Higher damping leads to a larger logarithmic decrement and a shorter relaxation time.

Quality Factor (Q-factor)

The Quality Factor Q is a dimensionless parameter characterizing how underdamped an oscillator is relative to its energy dissipation.
It is defined as 2π times the ratio of total stored energy to energy dissipated per cycle.
High Q oscillators exhibit very sharp resonance curves and slow amplitude decay rates.

Derivations

General Solution for Underdamped Harmonic Motion

Conditions: b² < 4mk (Light damping)

Write the standard differential equation as d²x/dt² + 2γ(dx/dt) + ω₀²x = 0, where $\gamma = b/(2m)$ and ω₀ = sqrt(k/m).
Assume a trial solution of the form x(t) = e^rt and substitute to obtain the auxiliary equation r² + 2γr + ω₀² = 0.
Solve for roots using the quadratic formula: $r = -\gamma ± \sqrt(\gamma^{2} - \omega_{0}^{2})$. For underdamped motion, let ω₁ = sqrt(ω₀² - γ²).
Combine the complex exponential roots to form the general real solution: x(t) = A e^-γt cos(ω₁t + φ).

Result: x(t) = A e^{-γt} cos(ω_1 t + φ)

Logarithmic Decrement Derivation

Conditions: Underdamped motion regime

Write the expression for the amplitude at time t as a(t) = A e^-γt.
Find the amplitude after one complete period T₁: a(t + T₁) = A e^{-γ(t + T₁}).
Take the natural logarithm of the ratio of two successive amplitudes: $\lambda = \ln(a(t) / a(t + T_{1}))$.
Substitute T₁ = 2π/ω₁ to obtain $\lambda = \gammaT_{1} = \gamma(2\pi/\omega_{1})$.

Result: $\lambda = \frac{2\pi \gamma}{\omega_1}$

Formula sheet

Formula / law	Equation	Variables and units	Conditions
Differential Equation of Damped Motion	$$m \frac{d^2 x}{dt^2} + b \frac{dx}{dt} + kx = 0$$	m = mass, b = damping coefficient, k = spring constant N (Force)	Linear regime
Damping Factor and Natural Frequency	$$\gamma = \frac{b}{2m}, \quad \omega_0 = \sqrt{\frac{k}{m}}$$	$\gamma = damping factor$, ω₀ = undamped angular frequency s^-1 (rad/s)	Standard linear oscillator
Underdamped Angular Frequency	$$\omega_1 = \sqrt{\omega_0^2 - \gamma^2}$$	ω₁ = frequency of damped oscillation rad/s	ω₀ > γ
Displacement in Underdamped Motion	$$x(t) = A e^{-\gamma t} \cos(\omega_1 t + \phi)$$	A = initial amplitude, $\phi = phase constant$ m (meters)	b² < 4mk
Critically Damped Condition	$$b_c = 2\sqrt{mk} = 2m\omega_0$$	b_c = critical damping coefficient kg/s	Fastest non-oscillatory return
Logarithmic Decrement	$$\lambda = \ln\left(\frac{x(t)}{x(t + T_1)}\right) = \gamma T_1$$	$\lambda = logarithmic decrement$, T₁ = time period Dimensionless	Successive peaks
Quality Factor Q	$$Q = 2\pi \frac{E}{\Delta E} = \frac{\omega_0}{2\gamma}$$	E = total energy, ΔE = energy lost per cycle Dimensionless	Weak damping limit
Relaxation Time	$$\tau = \frac{1}{\gamma} = \frac{2m}{b}$$	$\tau = time constant$ for amplitude decay s (seconds)	Amplitude drops to 1/e

Graphs and experiments

Displacement vs Time for Three Damping Regimes

Draw: Three curves on an x-t plot: an exponentially decaying sinusoidal wave (underdamped), a single sharp non-oscillatory approach to zero (critically damped), and a slow sluggish asymptotic curve (overdamped).

Interpret: Contrasts the oscillatory decay with fast non-oscillatory and sluggish returns to equilibrium.

Exam use: Direct identification questions and matching regimes to b² vs 4mk relations.

Envelope of Amplitude Decay

Draw: A sinusoidal oscillation bounded symmetrically by upper and lower exponential decay curves +Ae^-γt and -Ae^-γt.

Interpret: Shows that the oscillation frequency remains constant while the peak amplitudes decrease exponentially.

Exam use: Used in problems calculating relaxation time and logarithmic decrement from successive peak values.

Torsional Pendulum in Viscous Medium Experiment

Draw: A schematic of a disc suspended by a wire oscillating inside a fluid container, plotting angular deflection against time.

Interpret: Demonstrates practical measurement of damping constants and verification of exponential amplitude decay.

Exam use: VIVA and conceptual questions regarding laboratory determination of Q-factor.

Solved numericals

TGT core

A particle of mass 0.5 kg executes damped harmonic motion with a damping constant $b = 0.1 kg/s$ and spring constant $k = 2.0 N/m$. Calculate its angular frequency of damped oscillation.

Identify given values: m = 0.5 kg, $b = 0.1 kg/s$, $k = 2.0 N/m$.
Calculate undamped angular frequency ω₀ = sqrt(k/m) = sqrt(2.0 / 0.5) = sqrt(4) = 2.0 rad/s.
Calculate damping factor $\gamma = b / (2m) = 0.1 / (2 × 0.5) = 0.1 rad/s$.
Calculate damped frequency ω₁ = sqrt(ω₀² - γ²) = sqrt(2.0² - 0.1²) = sqrt(4 - 0.01) = sqrt(3.99) ≈ 1.997 rad/s.

Answer: 1.997 rad/s

PGT extension

The amplitude of a damped harmonic oscillator falls to 1/e of its initial value in 50 seconds. If the mass of the oscillator is 2 kg, determine the damping coefficient b.

Identify relaxation time $\tau = 50 s$ and mass m = 2 kg.
Use the relation for relaxation time: $\tau = 1/\gamma = 2m/b$.
Substitute the values: $50 = (2 × 2) / b$.
Solve for b: $b = 4 / 50 = 0.08 kg/s$.

Answer: 0.08 kg/s

PGT extension

For an underdamped oscillator, the amplitude decreases by 50% after 10 complete oscillations. If the time period is 2 seconds, find the logarithmic decrement and the damping factor γ.

Initial amplitude A₀, amplitude after n = 10 cycles is A₁₀ = 0.5 A₀.
Logarithmic decrement per cycle $\lambda = (1/n) \ln(A_{0} / A_{n}) = (1/10) \ln(1 / 0.5) = \ln(2) / 10 = 0.6931 / 10 = 0.0693$.
Time period T₁ = 2 s, so damping factor $\gamma = \lambda / T_{1} = 0.0693 / 2 = 0.03465 s^{-1}$.

Answer: $\lambda = 0.0693$, $\gamma = 0.03465 s^-1$

Assumptions and limitations

The damping force is assumed to be directly proportional to the velocity (Stokes' drag), which holds strictly for small velocities in laminar flow regimes.
The restoring force obeys Hooke's law perfectly, neglecting elastic hysteresis and anharmonic terms at large amplitudes.

Common misconceptions

Common error: Damping changes the time period of oscillation significantly when damping is very small.

Correct: In light damping (γ << ω₀), the time period remains practically unchanged compared to undamped SHM.

The frequency correction term depends on γ², which is negligible for small damping coefficients.

Common error: A critically damped system oscillates infinitely fast around the equilibrium position.

Correct: A critically damped system does not oscillate at all; it approaches equilibrium asymptotically in minimum time.

By definition, critical damping eliminates oscillations while providing the fastest non-overshooting return.

Quick revision

Damping force acts opposite to velocity, causing exponential decay of mechanical energy and amplitude: x(t) = A e^-γt cos(ω₁t + φ).
Differential Equation of Damped Motion: m \frac{d^2 x}{dt^2} + b \frac{dx}{dt} + kx = 0
Damping Factor and Natural Frequency: \gamma = \frac{b}{2m}, \quad \omeg$a_0 = \sqrt{\frac{k}{m}}$

Must remember

The differential equation governing a damped harmonic oscillator is $m(d^2x/dt^2) + b(dx/dt) + kx = 0$.
The damping factor is defined as $\gamma = b/(2m)$ and the undamped natural frequency as $\omega_0 = \sqrt{k/m}$.
In light damping ($b^2 < 4mk$), the angular frequency of the oscillator becomes $\omega_1 = \sqrt{\omega_0^2 - \gamma^2}$.
The critical damping coefficient is given by $b_c = 2\sqrt{mk} = 2m\omega_0$, resulting in the fastest non-oscillatory return to equilibrium.
Logarithmic decrement $\lambda = \ln(x(t)/x(t + T_1)) = \gamma T_1$ measures the rate of amplitude decay per cycle.
Relaxation time $\tau = 1/\gamma = 2m/b$ represents the time in which the amplitude falls to $1/e$ of its initial value.
The Quality Factor $Q$ is related to the damping factor by $Q = \omega_0 / (2\gamma)$ and signifies energy retention capacity.

Glossary

Damping Constant ($b$)A parameter representing the magnitude of resistive force proportional to velocity in a moving system.

Logarithmic Decrement ($\lambda$)The natural logarithm of the ratio of two successive amplitudes separated by one time period.

Relaxation Time ($\tau$)The time interval in which the amplitude of a damped oscillator decays to $1/e$ of its initial value.

Quality Factor ($Q$)A dimensionless parameter characterizing the sharpness of resonance and energy dissipation relative to total energy.

Critical Damping ($b_c$)The boundary damping value where the system returns to equilibrium in the shortest time without oscillating.

Damping Regimes Comparison

Damping Regime	Condition	Motion Characteristics
Light Damping	$b^2 < 4mk$	Oscillatory decay with exponentially decreasing amplitude
Critical Damping	$b^2 = 4mk$	Fastest non-oscillatory return to equilibrium
Overdamping	$b^2 > 4mk$	Sluggish, non-oscillatory creeping return to equilibrium

Practice Quiz

Choose an option to reveal the explanation.

1. What is the primary physical origin of the damping force in a typical mechanical oscillator moving through a fluid?

Explanation: The damping force in a resistive medium is typically modeled as a viscous force proportional to the velocity of the moving object, represented as $-bv$.

2. The differential equation of a linear damped harmonic oscillator is given by $m \frac{d^2 x}{dt^2} + b \frac{dx}{dt} + kx = 0$. What is the dimension of the damping coefficient $b$?

Explanation: Since the damping force term is $b(dx/dt)$, the units are $\text{N} = b \cdot (\text{m/s})$, meaning $b$ has units of $\text{N s m}^{-1}$ or $\text{kg s}^{-1}$.

3. For a lightly damped harmonic oscillator, the angular frequency of oscillation $\omega_1$ is related to the undamped natural frequency $\omega_0$ and damping factor $\gamma$ by:

Explanation: Solving the auxiliary equation yields the damped angular frequency $\omega_1 = \sqrt{\omega_0^2 - \gamma^2}$ under the light damping condition $\omega_0 > \gamma$.

4. Which of the following conditions defines critical damping for a system with mass $m$, spring constant $k$, and damping coefficient $b$?

Explanation: Critical damping occurs when the discriminant of the auxiliary equation vanishes, leading to $b_c = 2\sqrt{mk}$.

5. How does an overdamped system return to its equilibrium position?

Explanation: An overdamped (heavy damping) system is dominated by viscous resistance, causing it to creep slowly back to equilibrium without any oscillations.

6. What is the expression for the relaxation time $\tau$ of a damped harmonic oscillator in terms of the damping factor $\gamma$?

Explanation: Relaxation time $\tau$ is defined as the time interval in which the amplitude falls to $1/e$ of its initial value, which gives $\tau = 1/\gamma = 2m/b$.

7. The logarithmic decrement $\lambda$ of a damped oscillator is defined as $\ln(x(t) / x(t + T_1))$. It is equal to:

Explanation: Logarithmic decrement represents the natural log of amplitude reduction over one period $T_1$, mathematically evaluated as $\lambda = \gamma T_1$.

8. What is the physical significance of a high Quality Factor ($Q$) in a damped oscillator?

Explanation: A high $Q$-factor indicates that the oscillator retains its energy for many cycles, resulting in slow amplitude decay and sharp resonance curves.

9. In the weak damping limit, the Quality Factor $Q$ is approximately given by:

Explanation: In weak damping, the Quality Factor is expressed as $Q = 2\pi (E / \Delta E) = \omega_0 / (2\gamma)$.

10. If the damping coefficient $b$ of a system is increased while keeping $m$ and $k$ constant, what happens to the relaxation time $\tau$?

Explanation: Since $\tau = 2m/b$, an increase in the damping coefficient $b$ leads to a smaller relaxation time, meaning faster amplitude decay.

11. Which damping regime is desired in automobile suspension systems to prevent bouncing while ensuring a quick return to equilibrium?

Explanation: Automobile shock absorbers are designed to be nearly critically damped so that the vehicle returns to equilibrium rapidly without continuous bouncing or sluggish recovery.

12. What is the nature of the displacement-time graph for a critically damped harmonic oscillator released from a non-zero displacement with zero initial velocity?

Explanation: A critically damped system approaches equilibrium asymptotically without oscillating or crossing the zero-displacement axis.

13. In light damping, why does the time period remain practically unchanged compared to undamped SHM?

Explanation: Since $\omega_1 = \sqrt{\omega_0^2 - \gamma^2}$, for $\gamma \ll \omega_0$, the $\gamma^2$ term is extremely small, making the difference in frequency negligible.

14. Which of the following represents the correct unit for the Quality Factor $Q$?

Explanation: The Quality Factor $Q$ is a ratio of energies ($2\pi E / \Delta E$) or frequencies, making it a dimensionless quantity.

15. If the total energy of a weakly damped oscillator decreases by $5%$ per cycle, what can be inferred about its $Q$-factor?

Explanation: Using $Q = 2\pi (E / \Delta E)$, a small fractional energy loss per cycle ($\Delta E / E$) corresponds to a high $Q$-factor.

PYQ Patterns

These are pattern-based practice questions. Authentic year labels are only used when verified from a source.

1. The amplitude of a damped oscillator drops to $1/e$ of its initial value in $10\text{ s}$. What is the damping factor $\gamma$?

Explanation: The relaxation time $\tau = 1/\gamma$. Given $\tau = 10\text{ s}$, the damping factor is $\gamma = 1/10 = 0.1\text{ s}^{-1}$

2. For a critically damped oscillator of mass $m = 1\text{ kg}$ and spring constant $k = 100\text{ N m}^{-1}$, what is the value of the critical damping coefficient $b_c$?

Explanation: Critical damping coefficient is $b_c = 2\sqrt{mk} = 2\sqrt{1 \times 100} = 2 \times 10 = 20\text{ kg s}^{-1}$.

3. If the logarithmic decrement of a damped harmonic oscillator is $0.05$, what is the ratio of successive amplitudes separated by one time period?

Explanation: Logarithmic decrement is $\lambda = \ln(x(t) / x(t + T_1)) = 0.05$. Therefore, $x(t) / x(t + T_1) = e^{0.05}$.

4. A damped oscillator has an undamped frequency $\omega_0 = 50\text{ rad/s}$ and a damping factor $\gamma = 30\text{ s}^{-1}$. What is the damped angular frequency $\omega_1$?

Explanation: Using $\omega_1 = \sqrt{\omega_0^2 - \gamma^2} = \sqrt{50^2 - 30^2} = \sqrt{2500 - 900} = \sqrt{1600} = 40\text{ rad/s}$.

5. Which of the following differential equations correctly represents forced and damped harmonic motion?

Explanation: The inclusion of an external driving force $F_0 \cos(\omega t)$ along with the restoring and damping terms completes the forced-damped oscillator equation.

6. The Quality Factor of a damped oscillator is $50$, and its undamped natural frequency is $1000\text{ rad/s}$. What is the damping factor $\gamma$?

Explanation: Using $Q = \omega_0 / (2\gamma)$, we get $\gamma = \omega_0 / (2Q) = 1000 / (2 \times 50) = 1000 / 100 = 10\text{ s}^{-1}$.

Timed Topic Test

Attempt these questions under a ten-minute limit using a separate timer.

1. In a damped harmonic oscillator, the presence of damping always results in:

Explanation: Damping forces perform negative work, causing continuous dissipation of mechanical energy into thermal energy.

2. What happens to the displacement amplitude $A e^{-\gamma t}$ of an underdamped oscillator as time $t$ approaches infinity?

Explanation: Due to the exponential decay factor $e^{-\gamma t}$, the amplitude approaches zero as $t \to \infty$.

3. If the damping constant $b$ exceeds the critical damping value $b_c$, the roots of the auxiliary equation for the differential equation are:

Explanation: For overdamping ($b^2 > 4mk$), the discriminant is positive, yielding two distinct real and negative roots corresponding to non-oscillatory exponential decay.

4. How is the relaxation time $\tau$ related to the logarithmic decrement $\lambda$ and the time period $T_1$?

Explanation: Since $\lambda = \gamma T_1$ and $\tau = 1/\gamma$, substituting gives $\tau = T_1 / \lambda$.

5. Which of the following parameters does NOT affect the undamped natural frequency $\omega_0$ of a spring-mass oscillator?

Explanation: The undamped natural frequency is $\omega_0 = \sqrt{k/m}$, which is independent of the damping coefficient $b$.

6. If an oscillator has a very high Quality Factor ($Q = 500$), what can be said about the sharpness of its resonance peak?

Explanation: High $Q$ oscillators feature very little energy loss per cycle, resulting in extremely sharp and narrow resonance response curves.

7. In the context of damped motion, what physical condition separates oscillatory motion from non-oscillatory motion?

Explanation: The condition $b^2 = 4mk$ (critical damping) is the exact mathematical boundary separating underdamped (oscillatory) and overdamped (non-oscillatory) regimes.

8. If the mass of a damped oscillator is quadrupled while keeping $b$ and $k$ constant, what happens to the damping factor $\gamma$?

Explanation: Since $\gamma = b/(2m)$, quadrupling $m$ results in $\gamma$ being reduced by a factor of 4.

9. Which term in the differential equation $m \frac{d^2 x}{dt^2} + b \frac{dx}{dt} + kx = 0$ accounts for inertia?

Explanation: The term $m(d^2x/dt^2)$ represents mass times acceleration, which constitutes the inertial force term according to Newton's second law.

10. Why is a critically damped system particularly useful in sensitive measuring instruments like galvanometer needles or pointer gauges?

Explanation: Critical damping ensures the instrument pointer settles down at its equilibrium position quickly without oscillating or overshooting, ensuring fast and accurate readings.