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. 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:

7. 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.

8. SIMPLE VIBRATING SYSTEMS A simple pendulum ƒaƒƒƒƒƒƒaƒƒƒƒƒ

9. 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

10. SIMPLE VIBRATING SYSTEMS A piston free to vibrate in a Cylinder A Helmholtz Resonator

11. SYSTEMS WITH TWO MASSES

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

13. LINEAR ARRAY OF OSCILLATORS

14. MODES OF CIRCULAR MEMBRANES

15. BASS DRUM SNARE DRUM TIMPANI

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

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

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

19. CYMBALS GONG TAM TAM

20. VIBRATIONS OF A TUNING FORK