review – exam ii. normal modes collection of natural frequencies for object if the initial shape...
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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
2n 1a (1/n ) a
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a1
a2
a3
a4
First Four Normal Modes
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Mode Number 1 2 3 4 5 6 7 8
Normalized Mode Amplitude (a)
0.707 1.00 0.707 0.00 0.707 1.00 0.707 0.00
Mode Number Squared (b) 1 4 9 16 25 36 49 64
Initial Amplitude (a/b)
0.707 0.25 0.079 0.00 0.028 0.028 0.014 0.00
Normalized Amplitude 1.00 0.353 0.111 0.00 0.040 0.039 0.020 0.00
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
Amplitudes of the Normal Modes
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10
Mode Number
Re
lati
ve
Am
plit
ud
e
Plucked
Struck
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Wide Plectra
A wide plectrum can be simulated with a series of narrow plectra
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Example CaseFor 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
< 1/3rd pulling in same direction
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.
Separation Notes
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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 (Hooke’s Law)
When a steady force is applied to the felt of a piano hammer, the felt becomes stiffer with more compressions.o Larger force must be applied to produce the
same compression. F = Kxp
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Comparing forces
Compression
Fo
rce
F = kx
F = Kx^p
Preferred range of values for p is 2 - 3
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Force in Space and Time
Fmax
Wh
Force
½ Fmax
Distance Along String
Fmax
Th
Force
½ Fmax
Time
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Force notes
Wh < ¼ vibrational modes are excited just like a narrow plectrum.
Wh ½ excitations are about half as strong.
Wh > that mode receives very little excitation.
Th < P/4 vibrational modes are excited that are 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
Positions of Supports
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Other Modes
Mode 2 Mode 3
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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.
Finding Modes
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Length Modes
Width Modes
Mode 1
Mode 2
Mode Shapes
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
f = constant* S M
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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
Effect of Thinning the Plate
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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.