1. Damping, Forcing, Resonance
Oscillations & Waves
Damping, Forcing, Resonance

2. 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

3. 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

4. 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

5. 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

6. Damping Summary

7. 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

8. 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”

9. 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

10. 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

11. 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

12. Examples

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

14. Wheel Balancing

15. Microwave Cooking
Multi-atomic molecules can be visualized as mass-spring systems
Each molecule has natural frequencies
f0 for H2O is 2.45 GHz

16. Timekeeping
Clocks contain small quartz crystal tuning forks which oscillate at known frequencies
In most watches, the crystal oscillates at kHz

17. Summary Damping Forcing Natural Frequency Mismatched Forcing Resonance
Under-damping, critical damping, over-damping
Forcing
Natural Frequency
Mismatched Forcing
Resonance
Examples

18. Homework In Tsokos: Ch 4.1 - #3, 6, 8, 9, 12, 15, 17, 25, 29, 31, 37