Transcript Review – Exam II

Review – Exam II
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.
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.
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.
Amplitude Ratios for Plucked
Strings
a n  (1/n ) a1
2
 Examples:
– a3  a1/9
– a5  a1/25
First Four Normal Modes
a1
a2
a3
a4
Amplitudes of First Eight Modes of a Plucked String (1/4 point)
Mode Number
1
Normalized
Mode
Amplitude (a)
0.707
Mode Number
Squared (b)
Initial
Amplitude
(a/b)
Normalized
Amplitude
1
2
1.00
4
3
0.707
9
4
0.00
16
5
0.707
6
1.00
7
0.707
25
36
49
8
0.00
64
0.707
0.25
0.079
0.00
0.028
0.028
0.014
0.00
1.00
0.353
0.111
0.00
0.040
0.039
0.020
0.00
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.
Striking a String
 Striking gives a 1/n dependence
Amplitudes of the Normal Modes
Relative Amplitude
1
0.8
0.6
Plucked
Struck
0.4
0.2
0
0
2
4
6
Mode Number
8
10
Wide Plectra
 A wide plectrum can be simulated with a
series of narrow plectra
Example Case
The plectrum at the ¼ point excites
the fundamental 0.707 as much.
For the
fundamental
mode the
central
plectrum is at
the antinode
(maximum
excitation).
When both act together, the displacement is constant
between the plucking points and equal to a1 + (2/3)b1
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)
Two Narrow Plectra Results
Separation
Notes
 < 1/3rd l pulling in
 Same as one pulling
same direction
 One wavelength
pulling in same
direction
 About one wavelength
twice as hard
 That mode is canceled
 mode is only weakly
excited.
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
Comparing forces
F = kx
Force
F = Kx^p
Compression
Preferred range of values for p is 2 - 3
Force in Space and Time
Force
Fmax
Wh
½ Fmax
Distance Along String
Force
Fmax
Th
Time
½ Fmax
Force notes
 Wh < ¼ l vibrational
modes are excited just
like a narrow
plectrum.
 Wh  ½ l excitations
are about half as
strong.
 Wh > l 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.
Vibrating Bars
Mode 1
Positions of
Supports
Other Modes
Mode 2
Mode 3
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.
Mode Shapes
Length
Modes
Width Modes
Mode 1
Mode 2
Width modes will have higher frequency
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
Tuning a Plate – Changing Mass
f = constant* S M
 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
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 
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.