MUSICAL ACOUSTICSMUSICAL ACOUSTICS

Chapter 2

VIBRATING SYSTEMSVIBRATING SYSTEMS

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.

Simple Harmonic Motion

Graphs of simple harmonic motion:

(a) Displacement versus time(b) Speed versusTime. Note that speed reaches itsmaximum whendisplacement is zero and vice versa.

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

Displacement of a damped vibrator whose amplitude decreases with time

EVERY VIBRATING SYSTEM HASInertia (mass)Elasticity (spring)

For a mass/springHooke’s LawF = Ky

In Chapter 1 we learned that KE= ½ mv2

Similarly, it can be shown that PE = ½ Ky2

If the vibrator has damping:

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.

SIMPLE VIBRATING SYSTEMS

A simplependulum

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A mass-spring system vibrates at a frequency f If the mass is doubled:a) The frequency will be 2fb) The frequency will be √2fc) The frequency will remain fd) The frequency will be f/√2e) 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 2fb) The frequency will be √2fc)The frequency will remain fd) The frequency will be f/√2e) The frequency will be f/2

SIMPLE VIBRATING SYSTEMS

A piston free tovibrate in a Cylinder

A Helmholtz Resonator

SYSTEMS WITH TWO MASSES

Longitudinal vibrations of a three-mass vibrator

Transverse vibration of a three-mass vibrator

Transverse vibrations for spring systems with multiple masses

LINEAR ARRAY OF OSCILLATORS

MODES OF CIRCULAR MEMBRANES

BASS DRUM

SNARE DRUM

TIMPANI

VIBRATING BARS

Both ends free One end clamped

Arrows locate the nodes

CHLADNI PATTERNS OF A CIRCULAR PLATE

SaltS

SALT COLLECTS AT THE NODES

VIBRATIONAL MODES OF A CYMBAL (top)

AND A CIRCULAR

PLATE (bottom)

CYMBALS GONG TAM TAM

VIBRATIONS OF A TUNING FORK