Chapter 6

The Modes of Oscillation of Simple and Composite Systems

Simple Harmonic Oscillator

• Mass on a Spring

• Projection of Uniform Circular Motion

Both produce sine waves and sine waves can describe sound

Forces Involved

• Newton showed that SHM results from a restoring force

F -x

Viscous (Drag) Forces

• Always oppose the motion and proportional to the velocity of the flow

Fv -v

• As you stir a cup of honey, notice that it is much harder to stir fast than to stir slowly.

Damped Sinusoidal Motion

• By experiment we get

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.00 0.25 0.50 0.75 1.00

Time

Amplitude

Or, in symbols

mass moving

tcoefficien stiffness frequency

M

S f

Halving Time

moving masshalving time =

damping coefficient

D

MT

21

Oscillations of a Mass Supported by Springs

• From above for one band call the stiffness S – since we use two bands, the frequency is…

Frequency of Oscillation

f = constant 2S/M

f = constant 3S/M

2

Decreasing the mass (Mass enters as the square root), so decreasing mass by half increases the frequency by

•To increase the frequencyIncreasing the stiffness (Adding a band amounts to

increasing the stiffness) – if three bands are used then

The Effect of Stiffness

A Simplified Mass and Spring Oscillator

• Make bands much stiffer and the nut mass much smaller (both increasing the frequency).

• Construction of the system determines the frequencies it emits when struck The stiffness coefficient S is determined by the

materials and the construction.• How the system is struck makes a difference

Recall the previous slide on changing the positions of the bands and how the transverse frequencies were affected.

Simple Model of Oscillation

• Single mass, single plane, transverse mode

Two Mass Model• Single plane, transverse modes• We consider small amplitudes and get damped

sinusoidal motion.

Notes on Two Mass Model

• Central chain in mode one stretches very littleo Overall stiffness should be less than mode twoo Less stiffness means lower frequency for mode

one

Three Mass Model

Three normal modes

Four Mass Model

Four normal modes

Notes on the Model

• Number of mass gives the number of modes

• Mode 1 has one amplitude maximum, mode 2 has two maxima, and so forth

Model with Many Masses

• Lowest mode shows ½-wavelength

• Each mode is ½-wavelength different from its neighbors

Piano

• Many mass model with each mass a molecule in a stringo Thousands of normal modes should be presento Only the lowest few dozen are related to music

Mass Distribution

The waveforms are now skewed, but not the number of wavelengths included in each mode

Other Simple Models

• Longitudinal waves

• Torsional waves

• Air columns (wind instruments)

• Two-dimensional surfaces (percussion, sounding boards)