Acoustics of Music Week 4: Semester 2 Energy Systems

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Acoustics of Music Week 4: Semester 2 Energy Systems Aims: To begin modelling musical instruments by considering energy Objectives: Forced and Free Response Describe simple oscillating system Transient and steady state response Energy

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### Acoustics of MusicWeek 4: Semester 2Energy Systems

Aims:

To begin modelling musical instruments

by considering energy

Objectives:

Forced and Free Response

Describe simple oscillating system

Energy
• To develop model we need to investigate energy transfer
• Where is comes from and where does it go
• Direct energy input (acoustic instruments) trumpet, violin, drum, acoustic guitar, etc

Forced Response Oscillator

Free Responses oscillator

• Impulsive source starts the oscillator which is allowed to oscillate freely dissipating it's energy.
• The attack component important feature of sound
• Plucked strings, Stuck strings, Percussion
• Driver adds constantly to oscillator
• The sustain component is the most important feature of the sound.
• Reed, Brass, Bowed Strings
Forced Response (Power)

Power Out

Power In

Consider intensity

Mechanical Power

Power

Omni-directional

p is average pressure oscillation, 0 is density of air, c is speed of sound

Free Response (e.g. String)

Energy in

Energy Out

impulsive - decays w.r.t. time

Force is tension in string

Omni-directional output

assume sin = tan = 2x/L for small angles

What Happens in Between?

What are the forces on the oscillating system?

Our Simplest Instrument

Free Response

1 degree of freedom

Lumped Parameters

Viscous Damping

Inertia

Stiffness

Damping

Hence equation of motion

Solving Equation of Motion

Assume Solution

Substituting into

equation of motion

Assume true for all

(auxiliary equation)

Hence two roots

So general solution

A and B constants

Implications of solution

Decay Term

Potential Oscillation Term

Real Roots - System just decays

Complex Roots – Oscillation!!!

Frequency of oscillation given by

Real Instruments – Radiate to Air

flattens tuning since 4mk is the larger term

Decay increase

Ok for guitar but will have to force tuba

Forced Response

Consider harmonic driver

For steady state, assume particular integral of form

Substitute in equation of motion

Hence particular integral

Complete Solution

Will have a transient (F=0) and a steady state solution

After the transient decays we have a stead state

Can you see a complex number?

Mechanical Impedance

Where Reactance

So can write steady state solution as…

Differentiating to give velocity

Resonance

Maximum Power when Mechanical Reactance is zero

i.e. when

Q Factor

Sharpness of Peak

Half power either side

• Large Q -> little damping and large response
• Low Q -> large damping small resonant peak
• There are three types of way a system can respond to resonance
• Respond well to single frequency, sharp resonance,

little damping, Z small close to resonant frequency. e.g. Tuning fork

• Respond well to discrete set of frequencies. e.g. Trumpet
• Flat response (loudspeakers and microphones)