Damping, Forcing, Resonance 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 -Other forms of energy loss include sound, light

Under-damping Small resistive forces cause a gradual, exponential drop in amplitude The period of oscillation increases with the degree of damping -Example: the gradual damping of a pendulum due to air resistance

Critical Damping Resistance returns the system to equilibrium as quickly as possible without oscillation Critical damping is a particular case of damping -Example: door closing mechanisms -Example: gun recoil mechanisms

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

A different kind of 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

Natural Frequency, f0 If allowed to move freely, oscillations tend to occur at a natural frequency, denoted f0 Natural frequency may also be referred to as “characteristic frequency” or “resonant frequency” More than one natural frequency may exist for a system…the lowest is called the “fundamental” frequency and the higher ones are “harmonics”

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

Resonance What happens if the driving frequency fD 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? -Pendulum X swings at its natural frequency -Pendulum X applies forcing to the other pendula via the horizontal string -The natural frequencies of A, B, D, E are different from X, so they only oscillate weakly -Pendulum C has the same natural frequency as X, so it experiences resonance -Because of damping, the amplitude of C does not increase forever -Furthermore, the amplitudes of X and C will cycle since energy is conserved

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 f0 for H2O 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 kHz

Summary Damping Forcing Natural Frequency Mismatched Forcing Resonance 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