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Acoustics of Music Week 4: Semester 2 Energy Systems

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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  1. 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

  2. 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

  3. 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

  4. 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

  5. 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

  6. 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

  7. 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

  8. Implications of solution Decay Term Potential Oscillation Term Real Roots - System just decays Complex Roots – Oscillation!!! Frequency of oscillation given by

  9. Real Instruments – Radiate to Air flattens tuning since 4mk is the larger term Decay increase Ok for guitar but will have to force tuba

  10. Forced Response Consider harmonic driver For steady state, assume particular integral of form Substitute in equation of motion Hence particular integral

  11. 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?

  12. Mechanical Impedance Where Reactance So can write steady state solution as… Differentiating to give velocity

  13. Resonance Maximum Power when Mechanical Reactance is zero i.e. when

  14. 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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