Oscillations Waves Damping Forcing Resonance Damping So far

Oscillations & Waves Damping, Forcing, Resonance

Damping • So far we have assumed that wave propagation is lossless • In reality, friction and other resistive forces exist • The energy of oscillation is eventually dissipated as heat • 3 categories of damping

Under-damping • Small resistive forces cause a gradual, exponential drop in amplitude • The period of oscillation increases with the degree of damping

Critical Damping • Resistance returns the system to equilibrium as quickly as possible without oscillation • Critical damping is a particular case of damping

Over-damping • The resistance is so great that no oscillations occur (as in critical damping) • The return to equilibrium is slower than in the case of critical damping

Damping Summary

Forcing • To counteract resistive forces, one can force an oscillation • Forcing usually involves the application of a periodic force • The oscillator eventually adopts the forced frequency A different kind of forcing!!

Natural Frequency, f 0 • If allowed to move freely, oscillations tend to occur at a natural frequency, denoted f 0

Mismatched Forcing • What happens if the driving frequency f. D is different from the natural frequency? • Oscillations will occur at the driving frequency, but with limited amplitude

Resonance • What happens if the driving frequency f. D matches the natural frequency? • The driving force is synchronized with the oscillation • Amplitude is magnified with each wave cycle

Putting it All Together • How are damping, forcing, and resonance related?

Examples

Earthquake Preparedness • An earthquake may excite a building’s resonant frequency, with catastrophic results

Wheel Balancing

Microwave Cooking • Multi-atomic molecules can be visualized as mass-spring systems • Each molecule has natural frequencies • f 0 for H 2 O is 2. 45 GHz

Timekeeping • Clocks contain small quartz crystal tuning forks which oscillate at known frequencies • In most watches, the crystal oscillates at 32. 768 k. Hz

Summary • Damping – Under-damping, critical damping, over-damping • • • Forcing Natural Frequency Mismatched Forcing Resonance Examples

Homework • In Tsokos: – Ch 4. 1 - #3, 6, 8, 9, 12, 15, 17, 25, 29, 31, 37 – Ch 4. 2 - #1, 3, 6, 7, 12, 15, 17 – Ch 4. 3 - #1, 4, 5, 7, 10 – Ch 4. 4 - #1, 4, 7