2. Chapter 6 The Modes of Oscillation of Simple and Composite Systems

3. Simple Harmonic Oscillator Mass on a Spring Projection of Uniform Circular MotionUniform Circular Motion Both produce sine waves and sine waves can describe sound

4. Forces Involved Newton showed that SHM results from a restoring force F  -x

5. Viscous (Drag) Forces Always oppose the motion and proportional to the velocity of the flow F v  -v As you stir a cup of honey, notice that it is much harder to stir fast than to stir slowly.

6. Damped Sinusoidal Motion By experiment we get Or, in symbols

7. Halving Time

8. Oscillations of a Mass Supported by Springs

9. From above for one band call the stiffness S – since we use two bands, the frequency is… Frequency of Oscillation  Decreasing the mass (Mass enters as the square root), so decreasing mass by half increases the frequency by To increase the frequency  Increasing the stiffness (Adding a band amounts to increasing the stiffness) – if three bands are used then

10. The Effect of Stiffness

11. A Simplified Mass and Spring Oscillator Make bands much stiffer and the nut mass much smaller (both increasing the frequency). Construction of the system determines the frequencies it emits when struck  The stiffness coefficient S is determined by the materials and the construction. How the system is struck makes a difference  Recall the previous slide on changing the positions of the bands and how the transverse frequencies were affected.

12. Simple Model of Oscillation Single mass, single plane, transverse mode

13. Two Mass Model Single plane, transverse modes We consider small amplitudes and get damped sinusoidal motion.

14. Notes on Two Mass Model Central chain in mode one stretches very little o Overall stiffness should be less than mode two o Less stiffness means lower frequency for mode one

15. Three Mass Model Three normal modes

16. Four Mass Model Four normal modes

17. Notes on the Model Number of mass gives the number of modes Mode 1 has one amplitude maximum, mode 2 has two maxima, and so forth

18. Model with Many Masses Lowest mode shows ½-wavelength Each mode is ½-wavelength different from its neighbors

19. Piano Many mass model with each mass a molecule in a string o Thousands of normal modes should be present o Only the lowest few dozen are related to music

20. Mass Distribution The waveforms are now skewed, but not the number of wavelengths included in each mode

21. Other Simple Models Longitudinal waves Torsional waves Air columns (wind instruments) Two-dimensional surfaces (percussion, sounding boards)