MUSICAL ACOUSTICS
Chapter 2
VIBRATING 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 versus
Time. Note that
speed reaches its
maximum when
displacement 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 HAS
Inertia (mass)
Elasticity (spring)
Hooke’s Law
F = Ky
For a mass/spring
In Chapter 1 we learned that
Similarly, it can be shown that
KE= ½ mv2
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 simple
pendulum
A mass-spring system
vibrates at a frequency f
If the mass is doubled:
a) The frequency will be 2f
b) The frequency will be √2f
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 2f
b) The frequency will be √2f
c)The frequency will remain f
d) The frequency will be f/√2
e) The frequency will be f/2
SIMPLE VIBRATING SYSTEMS
A piston free to
vibrate in a
cylinder
SIMPLE VIBRATING SYSTEMS
A piston free to
vibrate in a
cylinder
A Helmholtz
resonator
SIMPLE VIBRATING SYSTEMS
A piston free to
vibrate in a
cylinder
A Helmholtz
resonator
m=ρɑl
K=ρɑ2l 2/V
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
SALT COLLECTS AT THE NODES
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
HOLOGRAPHIC INTERFEROMETRY
VIBRATIONAL
MODES OF A
CYMBAL (top)
AND A
CIRCULAR
PLATE
(bottom)
CYMBALS
GONG
TAM TAM
VIBRATIONS OF A TUNING FORK
ANIMATIONS OF TUNING FORK VIBRATIONS AT DAN
RUSSELL’S WEBSITE
http//www.acs.psu.edu/drussell/Demos/TuningFork/forkmodes.htmlhttp://www.acs.psu.edu/drussell/Demos/TuningFork/forkmohttp://www.acs.psu.edu/drussell/Demos/TuningFork/fork-modes.html
des.html
HTTttp://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-
ASSIGNMENT FOR MONDAY, Jan. 12
• READ CHAPTER 3
• EXERCISES IN CHAPTER 2: 1-7