Review Exam II

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Review Exam II
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Normal Modes

• Collection of natural frequencies for object
• If the initial shape agrees with a normal mode,
  the system will retain its shape.
• If the initial shape is not one of the normal
  modes, the system will not retain its shape.
• By using various amounts of the normal modes, we
  can construct any initial pattern we like.

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Plucked Strings

• Plucking a string at the node of any mode will
  not excite that mode.
• Plucking a string at the antinode of a mode gives
  the strongest excitation.

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Plucking Position

• Modes one and three are symmetric
• Mode two is anti-symmetric
• Modes 1 and 3 excited, not 2

The excitation of a mode is proportional to the
amplitude of the mode at the plucking point.
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Amplitude Ratios for Plucked Strings

• Examples
• a3 ? a1/9
• a5 ? a1/25

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First Four Normal Modes
a1
a2
a3
a4
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Amplitudes of First Eight Modes of a Plucked
String (1/4 point)
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Removing Modes

• To remove the nth mode and its multiples, pluck
  at the 1/nth position.
• Plucking near one of these positions weakens the
  corresponding modes.
• High order modes are weak because of the 1/n2
  dependence.

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Striking a String

• Striking gives a 1/n dependence

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Wide Plectra

• A wide plectrum can be simulated with a series of
  narrow plectra

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Example Case

• For the fundamental mode the central plectrum is
  at the antinode (maximum excitation).

The plectrum at the ¼ point excites the
fundamental 0.707 as much.
When both act together, the displacement is
constant between the plucking points and equal to
a1 (2/3)b1
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Net Mode 1 Excitation

• Mode 1 is excited to an amplitude of
• a1 (2/3)(0.707)b1 1.47a1 (constructive)

a1 - (2/3)(0.707)b1 0.53a1 (destructive)
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Two Narrow Plectra Results
Separation
Notes

• One wavelength pulling in same direction
• About one wavelength

• Same as one pulling twice as hard
• That mode is canceled
• mode is only weakly excited.

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Hammer Strike

• Must consider spatial and temporal distribution
  of the forces.
• The simple model uses a linear restoring force F
  -kx (Hookes Law)
• When a steady force is applied to the felt of a
  piano hammer, the felt becomes stiffer with more
  compressions.
• Larger force must be applied to produce the same
  compression. F Kxp

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Comparing forces
Preferred range of values for p is 2 - 3
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Force in Space and Time
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Force notes

• Wh a narrow plectrum.
• Wh ? ½ l excitations are about half as strong.
• Wh l that mode receives very little excitation.

• Th the same as an impulse.
• Th ? P/2 are excited at about half the strength
  as an impulse.
• Th P that mode receives little excitation.

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Vibrating Bars
Mode 1
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Other Modes
Mode 2
Mode 3
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Finding Modes

• Motion on one side of a node is opposite from the
  other side of the node.
• Tapping at the node does nothing to stimulate
  that mode.
• Tapping near antinode gives maximum stimulation
  of that mode.

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Mode Shapes
Length Modes
Width Modes
Width modes will have higher frequency
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Types of Plate Edges

• Free Edge antinodes always appear at the edges
• Clamped Edge ends are merging into nodes rather
  slowly
• Hinged Edge ends come more rapidly into nodes

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Tuning a Plate Changing Mass

• Adding mass will decrease the frequency
• Positioned near a node has no effect on that mode
• Positioned near an antinode has maximum effect on
  that mode

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Effect of Thinning the Plate

• Changing the plate thickness affects the plate
  stiffness
• Since f ? (S/M)½, thinning the plate decreases
  the mass (raising the frequency) M ? means f ?
• Thinning the plate also lowers the stiffness
  (lowering the frequency) S ? means f ?

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Net effect

• Rayleigh finds that the change in frequency
  caused by thinning the plate is about three times
  the effect caused by mass but acting in opposite
  senses.
• The craftsman finds the places where he can add
  wax to get the frequencies he wants.
• Wax adds mass without affecting stiffness.
• The change in stiffness dominates in the other
  direction
• Cut away wood at the positions of the wax.
• The amount of wood mass removed is half the mass
  of the wax.