SHM

13-10 Resonance and Driven Harmonic Oscillators

Introduction to Oscillation

• Natural Frequency (fo): The inherent frequency at which a mechanical system oscillates.

• Energy Loss: An oscillating system like a swing loses energy due to friction and eventually comes to rest unless sustained by external energy.

• Driving Mechanism: External forces can sustain oscillations, such as pushing a swing.

Relationship Between Driving Frequency and Natural Frequency

• Matching Frequencies: If the time between pushes (tpush) matches the period of the oscillation (T=1/fo), the swing resonates.

  • Formula: When driving frequency (fdriving) = natural frequency (fo), it achieves resonance (fdriving = fo).

• Other Frequencies: The system can absorb energy at other ratios of the natural frequency (fo/2, fo/3, etc.).

Examples of Mechanical Resonance

• Common Examples:

  • Resonating strings (e.g., violin)

  • Drum membranes

  • Ear's basilar membrane

  • Shattering wine glasses with sound waves.

• Electromagnetic Resonance: In radios, matching the circuit frequency with the broadcast frequency increases signal amplitude.

Damped Mass-Spring System and Driven Oscillator

• Definitions:

  • Driven Oscillator: A damped system subjected to external harmonic forces.

  • Forced Oscillations: The result of the external driving force on the oscillator.

• Driving Force Equation:

  • Form: [ F_{driven} = F_0 ext{cos}( \omega_{driven} t ) ]

• Net Force Equation:

  • [ F = F_{spring} + F_{damping} + F_{driven} ]

  • Where: ( F_{spring} = -kx ) (restoring force) and ( F_{damping} = -bv_x ) (damping force).

Displacement of the Driven Oscillator

• Displacement Equation: After transient effects die down,

  • Displacement: [ x(t) = A_{driven} ext{cos}( \omega_{driven} t + eta_{driven} ) ]

  • ( A_{driven} ) (amplitude) and ( \beta_{driven} ) (phase constant) depend on system parameters.

Properties of Driven Oscillations

1. Frequency Dependence: The forced oscillator oscillates at the driving frequency, not its natural frequency (e.g., human ear responds to 500 Hz, not 3000 Hz).

2. Amplitude and Phase Constants: These depend on system parameters (natural frequency, driving frequency, force magnitude, damping constant, and mass) rather than initial conditions.

Amplitude Response to Driving Frequency

• As ( \omega_{driven} ) approaches ( \omega_0 ) (natural frequency), amplitude increases, peaking at ( \omega_0 ).

• Beyond ( \omega_0 ), amplitude decreases:

  • If damping ( b ) is small, amplitude can become significantly large as ( \omega_{driven} ) approaches ( \omega_0 ).

  • In practical scenarios, damping is never zero, so amplitude becomes large but finite.

Real-World Applications

• Wine Glass Shattering: Caused by sound waves at the glass's natural oscillation frequency.

• Earthquake-Resistant Structures: Must incorporate sufficient damping to prevent large amplitude oscillations near natural frequencies.

Amplitude Dependence on Damping

• A plot of amplitude vs. damping shows resonance effects; increasing damping shifts peak amplitude away from the natural frequency.

• Formula for Q-value: ( Q = \frac{\omega_0}{b/m} ) indicates how sharp the resonance peak is.