Damped and Forced Oscillations: Simple Harmonic Motion, Resonance, and Quality Factor

Simple Harmonic Motion and Damping

Linearly Damped Oscillator

When an oscillator is subjected to a drag force proportional to velocity, it is said to be linearly damped. The equation of motion for a mass-spring system with damping is:

• Equation: $m \frac{d^2s}{dt^2} + b \frac{ds}{dt} + ks = 0$

• Drag force: $F_d = -bv$

• General solution: $s(t) = A_0 e^{-\gamma t} \cos(\omega' t)$, where $\gamma = \frac{b}{2m}$ and $\omega' = \sqrt{\frac{k}{m} - \frac{b^2}{4m^2}}$

Types of Damping

The nature of the solution depends on the relationship between the damping constant and the mass and spring constant:

• Underdamped: Oscillatory motion with gradually decreasing amplitude ($b^2 < 4mk$)

• Critically damped: Returns to equilibrium as quickly as possible without oscillating ($b^2 = 4mk$)

• Overdamped: Returns to equilibrium without oscillating, but slower than critical ($b^2 > 4mk$)

Example: Shock absorbers in vehicles are designed to be critically damped to prevent oscillations after a disturbance.

Quality Factor (Q) and Energy Loss

Definition and Physical Meaning

The quality factor Q is a dimensionless measure of how underdamped an oscillator is and describes the sharpness of resonance:

• Q is inversely proportional to energy loss per cycle.

• Formula: $Q = \frac{\omega_0 m}{b} = \omega_0 \tau$

• Energy decay: $E(t) = E_0 e^{-t/\tau}$, where $\tau = \frac{m}{b}$ is the decay time.

• Fractional energy loss per period: $\frac{\Delta E}{E} = \frac{2\pi}{Q}$

Example: The low E string of a guitar loses half its energy after 2.25 s, allowing calculation of its decay time and Q factor.

Forced Oscillations and Resonance

Driven Damped Oscillator

When a damped oscillator is subjected to an external periodic force, its equation of motion becomes:

• Equation: $m \frac{d^2s}{dt^2} + b \frac{ds}{dt} + ks = F_0 \cos(\omega t)$

• Steady-state solution: $s(t) = A_0 \cos(\omega t - \phi_0)$

• Amplitude: $A_0 = \frac{F_0}{\sqrt{(k - m\omega^2)^2 + (b\omega)^2}}$

• Phase constant: $\tan \phi_0 = \frac{b\omega}{k - m\omega^2}$

Resonance

Resonance occurs when the driving frequency matches the natural frequency of the oscillator, resulting in maximum amplitude and energy transfer:

• Resonant frequency: $\omega_0 = \sqrt{\frac{k}{m}}$

• At resonance: Amplitude is maximized, and the system absorbs energy most efficiently from the driving force.

• Resonance curve width: $\Delta \omega = \frac{\omega_0}{Q}$

Important Equations Summary

• Period of mass-spring oscillator: $T = 2\pi \sqrt{\frac{m}{k}}$

• Period of simple pendulum: $T = 2\pi \sqrt{\frac{l}{g}}$

• Damped oscillator solution: $s(t) = A_0 e^{-\gamma t} \cos(\omega' t)$

• Damping coefficient: $\gamma = \frac{b}{2m}$

• Damped frequency: $\omega' = \sqrt{\frac{k}{m} - \frac{b^2}{4m^2}}$

• Quality factor: $Q = \frac{\omega_0 m}{b}$

• Energy loss per period: $\frac{\Delta E}{E} = \frac{2\pi}{Q}$

• Forced oscillator amplitude: $A_0 = \frac{F_0}{\sqrt{(k - m\omega^2)^2 + (b\omega)^2}}$

• Resonance curve width: $\Delta \omega = \frac{\omega_0}{Q}$

Applications and Examples

• Shock absorbers: Designed for critical damping to prevent oscillations after a disturbance.

• Musical instruments: The Q factor determines how long a string vibrates after being plucked.

• Engineering: Resonance must be avoided in bridges and buildings to prevent structural failure.

Additional info: The notes cover all major aspects of damped and forced oscillations, including mathematical derivations, physical interpretations, and practical applications. The included images directly illustrate the physical phenomena and concepts discussed.