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

Examples:– a3 a1/9

– a5 a1/25

2n 1a (1/n ) a

a1

a2

a3

a4

First Four Normal Modes

           

 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)

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

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

Wide Plectra

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

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

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

< 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

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

Compression

Fo

rce

F = kx

F = Kx^p

Preferred range of values for p is 2 - 3

Force in Space and Time

Fmax

Wh

Force

½ Fmax

Distance Along String

Fmax

Th

Force

½ Fmax

Time

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.

Vibrating Bars

Mode 1

Positions of Supports

Other Modes

Mode 2 Mode 3

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

Length Modes

Width Modes

Mode 1

Mode 2

Mode Shapes

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

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

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

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.