1. MUSICAL ACOUSTICS Chapter 2 VIBRATING SYSTEMS

2. SIMPLE HARMONIC MOTION A simple vibrator consisting of a mass and a spring. At equilibrium (center), the upward force exerted by the spring and the force of gravity balance each other, and the net force F on the mass is zero.

3. Simple Harmonic Motion Graphs of simple harmonic motion: (a) Displacement versus time (b) Speed versus Time. Note that speed reaches its maximum when displacement is zero and vice versa.

4. Vibratory motion: y,v, and a all change with time.

5. Displacement of a damped vibrator whose amplitude decreases with time

6. Representing a Vibrating System Waveform Spectrum

7. EVERY VIBRATING SYSTEM HAS Inertia (mass) Elasticity (spring) For a mass/spring Hooke’s Law F = Ky In Chapter 1 we learned that KE= ½ mv 2 Similarly, it can be shown that PE = ½ Ky 2 If the vibrator has damping:

8. A mass hangs from a spring. You raise the mass 1 cm, hold it there for a short time and then let it drop Make a graph of its motion Make a graph of its total energy.

9. SIMPLE VIBRATING SYSTEMS A simple pendulum

10. A mass-spring system vibrates at a frequency f If the mass is doubled: a)The frequency will be 2 f b)The frequency will be √2 f c)The frequency will remain f d)The frequency will be f /√2 e) The frequency will be f /2

11. A mass-spring system vibrates at a frequency f If the mass is doubled: a)The frequency will be 2 f b)The frequency will be √2 f c)The frequency will remain f d)The frequency will be f /√2 e) The frequency will be f /2 A mass swings on the end of a string at frequency f If the mass is doubled: a)The frequency will be 2 f b) The frequency will be √2 f c)The frequency will remain f d)The frequency will be f /√2 e) The frequency will be f/ 2

12. SIMPLE VIBRATING SYSTEMS A piston free to vibrate in a cylinder

13. SIMPLE VIBRATING SYSTEMS A piston free to vibrate in a cylinder A Helmholtz resonator

14. SIMPLE VIBRATING SYSTEMS A piston free to vibrate in a cylinder A Helmholtz resonator m= ρ ɑ l K=ρ ɑ 2 l 2 /V

15. SYSTEMS WITH TWO MASSES

16. Longitudinal vibrations of a three-mass vibrator Transverse vibration of a three-mass vibrator Transverse vibrations for spring systems with multiple masses

17. LINEAR ARRAY OF OSCILLATORS

18. MODES OF CIRCULAR MEMBRANES

19. BASS DRUM SNARE DRUM TIMPANI

20. VIBRATING BARS Both ends freeOne end clamped Arrows locate the nodes

21. CHLADNI PATTERNS OF A CIRCULAR PLATE SALT COLLECTS AT THE NODES

22. CHLADNI PATTERNS JOE WOLFE’S PHYSCLIPS ON MODES OF VIBRATION AND CHLADNI PATTERN CAN BE ACCESSED AT p://www.phys.unsw.edu.au/jw/chladni.html#modes http://www.phys.unsw.edu.au/jw/chladni.html#modes p://www.phys.unsw.edu.au/jw/chladni.html#modes

23. HOLOGRAPHIC INTERFEROMETRY

24. VIBRATIONAL MODES OF A CYMBAL (top) AND A CIRCULAR PLATE (bottom)

25. CYMBALS GONG TAM TAM

26. VIBRATIONS OF A TUNING FORK

27. ANIMATIONS OF TUNING FORK VIBRATIONS AT DAN RUSSELL’S WEBSITE http//www.acs.psu.edu/drussell/Demos/TuningFork/fork- modes.html http://www.acs.psu.edu/drussell/Demos/TuningFork/fork- mohttp://www.acs.psu.edu/drussell/Demos/TuningFork/fork-modes.html des.html http://www.acs.psu.edu/drussell/Demos/TuningFork/fork- mohttp://www.acs.psu.edu/drussell/Demos/TuningFork/fork-modes.html des.html HTT ttp://www.acs.psu.edu/drussell/De mos/TuningFork/fork-modes.html http://www.acs.psu.edu/drussehttp:// www.acs.psu.edu/drussell/Demos/Tu ningFork/fork-modes.html ll/Demos/TuningFork/fork- modes.hthhttp://www.acs.psu.edu/dru ssell/Demos/TuningFork/fork- mohttp://www.acs.psu.edu/drussell/D emos/TuningFork/fork-modes.html des.html http://www.acs.psu.edu/drussell/Dem os/TuningFork/fork-modes.html ttp://www.acs.psu.edu/drussell/Demo s/TuningFork/fork-modes.html

28. Vibration spectrum of a plucked string A Spectrum is a graph of Amplitude vs Frequency

29. ASSIGNMENT FOR MONDAY, Jan. 11 READ CHAPTER 3 EXERCISES IN CHAPTER 2: 1-7