Chapter 8

Real Hammers and Plectra

Simulating Broad Plectra

Simulating Complex Shapes

• Use a collection of narrow plectra to simulate the broad one in any pattern.

• Initial vibrational pattern can be deduced by taking the individual patterns of the narrow plectra and combining.

Example Case

½ ¼

Fundamental Mode Excitations

• For the fundamental mode the central plectrum is at the antinode (maximum excitation).o Amplitude a1 if it acts alone

• The plectrum at the ¼ point excites the fundamental 0.707 as much.o Amplitude b1 = 0.707 a1 if it acts alone

• When both act together, the displacement is constant between the plucking points and equal to

a1 + (⅔)b1

Note: I’m not quite sure where the ⅔ comes from, but clearly the second position should get less weight in predicting the excitation of this mode

Net Mode 1 Excitation

• Mode 1 is excited to an amplitude of

a1 + (⅔)[(0.707)a1] = 1.47a1

Changing the Method of Plucking

a1 - (⅔)(0.707)a1 = 0.53a1

Notice the minus sign – the second plectra gives a negative contribution for this mode

Another Example

Mode 7 Oscillation (3½ wavelengths)p’s mark positive maximaq’s mark negative maximaBoth the magenta and red plectra grab the string at p positions

Constructive Use of Plectra

• Two plectra at p’s should give twice the initial amplitude as one at a p position.

• The following situation should also produce twice the initial amplitude as one plectrum.

To see this ask yourself which way the string would move when the plectra are removed. Notice that both plectra reinforce the mode

Destructive Use of Plectra

• If only the left plectrum were there, then after release all p’s move down and all q’s move up.

• If only the right plectrum were there, then after release all q’s move down and all p’s move up.

• With both plectra in place, mode 7 oscillations are destroyed.

Generalize(Four Points)

1. The result of using several plectra is given by combining the results of each plectrum acting alone (call this the narrow plectra substitution model).

2. We must take proper account of the sign of the individual results. Some plectrum positions may cancel others.

Continued

3. For two plectra close together (less than ⅓ rd the spacing of mode maxima) and pulling in the same direction, we get nearly the same effect as pulling one back twice as far.

4. Two plectra pulling in the same direction separated by exactly ½ wavelength will cancel out that mode (the destructive example two slides ago). If the separation is approximately ½ wavelength, that mode is only weakly excited.

Application to Wide Plectra

• Recall that a wide plectrum (width W) acts the same as two narrow plectra (separted by W).

W

Finger Picking a Guitar

• Use a thumb and forefinger on a guitar string (separation W about 2 cm) o modes are weakly excited whose wavelength is around

4 cm (W then would be ½ and point 4 applies).

• Since the length of the string is 60 cm (L), modes that fit this condition are

n = L/W = 60 cm/4 cm = 15

(that is the 15th harmonic would have a wavelength of 4 cm)

Results of model

• Mode 15 is absent and near 15 are weak.• Point three tells us that modes of wavelength

greater than three times W (>6 cm) are excited by a large amount and are accurately predicted by this narrow plectra substitution model

n = L/(3W) = 60 cm/6 cm = 10 • Modes up to 10 are predicted well• Modes between 10 and 15 are not predicted

Using a Pick

• Pick width 0.2 cm

• Harmonics around mode 150 are missing and simple theory carries us to mode 100 o Since the amplitudes fall off with n (use of a

pick is like striking the string), these are all the usable modes

Hammer Width for Struck Strings

• Force is not uniformly distributed

• Imagine the hammer hitting a wax block

• The indentation gives an accurate view of the forces at work

Restoring Force

• 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 use.o Larger force must be applied to produce the

same compression. F = Kxp

o p is the non-linearity coefficient

Comparing forces

Compression (x)

Fo

rce

(F)

F = kx

F = Kx^p

Effective Nonlinearity Exponent

• Hammers taken from pianoso p = 2.2 – 3.5

• Unused hammerso p = 1.5 – 2.8

• Preferred range of values is 2 - 3o p < 1 hammer is too linear and loud notes are simply

amplified soft noteso p > 3 there is too much contrast - fortissimos are too

harsh and pianissimos are too bland

Hammer ForcesSpatial Variation of the Force

Fmax

Wh

Force

½ Fmax

Distance Along String

Observations

• For Wh < ¼ wavelength () vibrational

modes are excited just like a narrow plectrum.

• For Wh ½ excitations are about half as

strong as a narrow plectrum

• For Wh > that mode receives very little

excitation.

Impact DurationTemporal Variation of the Force

Fmax

Th

Force

½ Fmax

Time

Observations

• When Th < P/4 (¼ Period), vibrational

modes are excited that are the same as an instantaneous hammer strike (impulse).

• For Th P/2 are excited at about half the

strength as an instantaneous hammer strike.

• For Th > P that mode received little

excitation.

Hard and Soft Hammers(expected linear response)

Soft Hammer Hard Hammer

• Soft hammers apply the force over a longer period of time.

• Soft and hard blows last the same time

Soft Blows

Hard Blows

Real Piano Hammers

• Solid curves show results

• Dotted curves show attempts to fit data to various models

Soft and Hard Blows(real piano)

Hard blow results from higher impact speed

String Stiffness

• So far we have considered flexible strings

• Stiffness makes the string shape roundedo Higher frequency modes are blocked

Summary

• Major assumptions of the method: o Excitation of a mode depends on the relative

amplitude of the mode at the point of striking or plucking.

o Extended plectra or hammers may be treated by adding the result of several narrow plectra or hammers arranged to give the same initial shape.

Continued

• More assumptions:o Modes whose wavelength are comparable to the width of the plectrum or hammer are not excited. The same holds if the time interval of the hammer strike is comparable to the period of the mode.

o Plucked string modes decrease in initial amplitude by 1/n2. Struck string modes decrease in initial amplitude by 1/n.

Examples

• Pluck guitar string 6 as normal and then damp at the mid-point. o The odd-numbered modes have antinodes at the mid-

point and are damped. o Only the even-numbered modes remain. o The normal 329.6 Hz tone (E4 ) is replaced by 2*329.6

Hz = 659.2 Hz or E5.

• If the damper is applied at the 1/3rd point, then modes 3, 6, 9, 12 survive, giving a tone close to B5.