Impedance at Boundaries

1
Impedance at Boundaries

• Acoustic of Music
• Semester 2 Week 7

• Aims
• To explain how realistic boundary conditions
  modify harmonic content
• Learning Outcomes
• To be able to derive equation for impedance at a
  realistic (bridge) boundary condition
• To be able to show how mass and stiffness effects
  harmonics
• Appreciate importance of characteristic impedance
  and input impedance

2
Impedance

• Factors work against flow of energy such as the
  inertia, stiffness and resistance
• Impedance Potential / Flow
• ImpedanceForce/Velocity

3
Impedance at Bridge

• Assume impedance at x0 infinite (RIGID)
  
• Impedance at bridge xL is ZL
• Assume no movement of nut or bridge in x
  direction

4
Give expression for Bridge Impedance
Resolve vertically forces on bridge
Vertical Speed
Hence bridge Impedance
5
Assume Solution
Where k is wave number
Some partial Differentiation
Simplifying
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What does this tell us?
Remember mechanical impedance for forced, lumped,
1d freedom
mmass, Kstiffness,Rresistance
Impedance influenced by mass, stiffness and
resistance
7
What if system mass dominated
8
Mass Dominated Sharpened Harmonics
Increasing mass detunes harmonics towards ideal
rigid case.
9
What if system stiffness dominated
10
Stiffness dominated Flattened Harmonics
Increasing stiffness sharpens harmonics towards
ideal rigid case.
11
Implications for Musical Instruments

• rigidly terminated string - harmonic series
• bridge impedance sharpens or flattens harmonics
• The bridge of a violin must be able to flex to
  communicate oscillations to the soundboard.
  Hence, when plucked the natural frequencies of a
  violin are slightly non-harmonic.
• When the violin is bowed the slip-stick mechanism
  produces well defined periodic behaviour
  (saw-tooth) and hence more consonant harmonics.

12
Dissonant Harmonics

• Transients give distinct timbre
• Helps us recognise instruments
• Interesting sounds
• Variation between notes
• e.g. piano strings with different masses and
  stiffness each give unique timbre

13
String Impedance

• Given a string that isnt fixed
• Forced oscillation at one end
• What works against energy flow?

14
Characteristic Impedance of String
Harmonic driver
Waves move in one direction
Resolve vertically
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Partial Differentiation
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Characteristic impedance and bridge impedance
Maximum Energy Flow, No modes
Get modes
Get modes of different frequencies