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Acoustics of Music Week 4: Semester 2 Energy Systems PowerPoint PPT Presentation

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 Music Week 4: Semester 2 Energy Systems

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Acoustics of music week 4 semester 2 energy systems l.jpg

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

Transient and steady state response


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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


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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


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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


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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


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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


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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


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Implications of solution

Decay Term

Potential Oscillation Term

Real Roots - System just decays

Complex Roots – Oscillation!!!

Frequency of oscillation given by


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Real Instruments – Radiate to Air

flattens tuning since 4mk is the larger term

Decay increase

Ok for guitar but will have to force tuba


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Forced Response

Consider harmonic driver

For steady state, assume particular integral of form

Substitute in equation of motion

Hence particular integral


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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?


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Mechanical Impedance

Where Reactance

So can write steady state solution as…

Differentiating to give velocity


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Resonance

Maximum Power when Mechanical Reactance is zero

i.e. when


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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)


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