1. Percussion Instruments
Classification
Idiophones (xylophone, marimba, chimes, cymbals, gongs…)
Membranophones (drams)
Aerophones (whistles, sirens)
Chordophones (piano, harpsichord)
Some have definite pitch, other do not have definite pitch
Some have harmonic overtones, other do not have it
Types of vibration
A) Mechanical:
bars
membranes
plates
strings (produce harmonic overtones)
B) Pneumatic:
air columns (produce harmonic overtones)
air chambers

2. Longitudinal vibrations Transverse vibrations
2. Vibrations of bars
Longitudinal vibrations
Transverse vibrations
A) Longitudinal vibrations (aluminum singing rods)
Harmonic modes (similar to a pipe with both ends open)
E – Young’s modulus, ρ – density.
It is possible selectively excite desired mode by clamping at the location of one of the nodes for the particular mode.
Longitudinal vibration are much higher in frequency then transverse.
Example:
L=61cm, v=5100m/s

3. B) Transverse vibrations (almost always)
Boundary conditions:
Free (almost always)
Clamped
Simply supported
Modes are not harmonic (because of that description is more complicated)
Frequency depends on thickness t, length L, and speed of sound v
Frequency of transverse vibrations in a bar with free ends:
for a rectangular bar.
for a tube with inner radius a, and outer radius b.
Frequency ratios: 1.00 : 2.76 : 5.40 : 8.90 … (not harmonic)
Note! Frequency
decrease as L2 rather then L
increase as m2
depend on shape (K – radius of gyration)

4. Physics Fairy performing the singing rod
2a. Aluminum singing rods
Longitudinal vibrations:
Transverse vibrations:
Question: Is it possible to have longitudinal
and transverse fundamentals coincide?
Question: Why from aluminum?
Physics Fairy performing the singing rod
Singing rod demo:

5. 2b. Rectangular bars: the glockenspiel
Rectangular steel bars 2.5 – 3.2 cm wide and 0.6 to 1.0 cm thick
What about length?
f = Hz (G5 – C8)
Modes:
Transverse
Torsional
Transverse exited edgewise

6. Tuned bars of rosewood or synthetic materials with tubular resonator
2c. Marimba
Tuned bars of rosewood or synthetic materials with tubular resonator
A deep arch is cut in the underside of bars to
reduce the length of bar required to reach the low pitches
allows tuning of overtones
Resonators:
Cylindrical pipes with one end open and one end closed.
These pipes are tuned to fundamental mode of the corresponding bar.
First overtone has frequency three times of the fundamental (closed pipe).
The resonators emphasize the fundamental and increase the loudness, which is done at the expense of shortening the decay time of the sound. (Think about the conservation of energy.)
Cover 3 to 3.5 octaves A2 – C7 (f =110 – 2093 Hz).
Some numbers:

7. (Wooden sound. In Greek: xylon. "wood" + phonē, "sound, voice“)
2d. Xylophone
(Wooden sound. In Greek: xylon. "wood" + phonē, "sound, voice“)
The term may be used generally, to include all such instruments, such as the marimba and balafon or, more specifically, to refer to an orchestral instrument of somewhat higher pitch range than the chromatic marimba. It is sometimes mistakenly used of similar lithophones and metallophone instruments of the glockenspiel type such as the pixiphone.
2e. Chimes
Chimes partials: 92:112:132 = 81:121:169 ~ 2:3:4
If partials are not harmonic, peach is picked up from the nearly harmonic partials near the center of audible range
Chimes do not have partials near the fundamental – pitch is purely subjective
Virtual pitch – missing fundamental (Similar situation in bells. However bells have weak fundamental)

8. T – surface tension (force divided by the length of the circle [N/m])
3. Vibrations of membranes
Membrane may be considered as two-dimensional string
Restoring force is tension applied from the edge
Can be tuned by changing tension
Mode frequencies are not harmonic (in ideal string they are harmonics)
Node lines (instead node points in strings)
3a. Circular membranes
Nodal lines are circles and diameters
Frequencies of vibrational modes:
a – radius of membrane
T – surface tension (force divided by the length of the circle [N/m])
σ – area density (mass divided by area [kg/m])
βmn – values for which Bessel functions becomes zero
Note! There are two indexes, because these are two-dimensional systems

9. Circular Membrane Modes
The four lowest vibrational modes for a circular membrane.

10. 4. Vibration of plates
Vibrations of plates relate to the vibrations of membranes the same way as vibrations of bars relate to the vibrations of strings
In string and membranes restoring force is tension
In bars and plates restoring force is stiffness of bulk material
Circular plates can vibrate in different modes often labeled as m and n
4a. Cymbals
Consist of thin, normally round plates
Could be excited in many different ways:
struck at various points
struck with wooden stick or soft beater
struck with another cymbal
The coupling between different modes in cymbals is strong.
Because of that, a large number of partials quickly appears in the spectrum
Cymbals exhibit nonlinear behavior
bifurcations and chaos
phase diagrams (plots)