Chapter 7 The Plucked String

Combinations of Modes when a system is struck and left to its own devices, any possible motion is made up of a collection of the natural frequencies.

Normal Modes of a Two Mass Chain

Two Mass Normal Modes Mode 1 Mode 2

Normal Mode Notes If we start the system in a normal mode, it persists. Let us get to a new configuration in steps

New Starting Configuration Method 1 Move to normal mode 1 first, followed by mode two.

New Starting Configuration Method 2 Move to normal mode 2 first, followed by mode one.

Resulting Motion Original shape not maintained

Conclusions 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.

The Plucked String Now the model is increased to a large number of balls (like in Chapter 6). The first three normal modes are…

Standing Waves When two sets of waves of equal amplitude and wavelength pass through each other in opposite directions, it is possible to create an interference pattern that looks like a wave that is “standing still.” –Waves on a violin string reflect off of the bridge. The reflected wave acts as the second wave moving in the opposite direction from the original.

Standing Waves Click inside the box

Notes on Standing Waves There is no vibration at a node. There is maximum vibration at an antinode.   is twice the distance between successive nodes or successive antinodes.

Mode Excitation of a Plucked String The position at which the string is plucked determines which of the partials (normal modes or standing waves) will be excited.

Place the Plectrum Here Notice that each mode has some amplitude at the dotted line Mode 1 Mode 2 Mode 3

Change Plectrum Position Modes one and three are at maximum here Mode two is at a zero. o Can’t expect to excite a mode that has no amplitude

Notes Notice that modes one and three are symmetrical about the mid-point Mode two is antisymmetrical 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 excitaiton.

Generalize In general, the excitation of a mode is proportional to the amplitude of the mode at the plucking point. Notice that the previous statements are included here.

Harmonic Amplitudes for Plucked Strings Thus even if we pluck a string at the best possible point to excite mode 4, the initial amplitude will be 1 / 16 of a 1 The amplitudes of the higher harmonics are related to the amplitude of the fundamental

Harmonic Amplitudes Plucked Strings HarmonicAmplitude 1a1a1 2¼ a 1 3 a 1 4 5

String Plucking “Recipe” Consider a string that is plucked ¼ of the way from one end The first four modes are shown next, where the amplitudes are the same for simplicity

First Four Normal Modes a1a1 a2a2 a3a3 a4a4 with a little trig…

Amplitudes a 1 = a 2 = 1.00 a 3 = a 4 = 0.00 First four modes set the pattern for the others Every fourth mode has initial amplitude of zero Every other even mode has amplitude one All odd modes have amplitude 0.707

Amplitudes of First Eight Modes of a Plucked String (1/4 point) Mode Number Normalized Mode Amplitude (a) Mode Number Squared (b) Initial Amplitude (a/b) Normalized Amplitude (Initial Amplitude/0.707)

Notes The absence of multiples of 4 is the result of the plucking point. Had it been at the ⅓ position, then modes that are multiples of three are missing. To remove the n th mode and its multiples, pluck at the 1/n th position. String ½ ⅓ ¼

More Notes Plucking near one of these positions weakens the corresponding modes. High order modes are weak because of the 1/n 2 dependence.

Striking a String Plucking gave a 1/n 2 dependence of the amplitudes of the normal modes Striking gives a 1/n dependence o The higher modes have more relative amplitude

Amplitudes of First Eight Modes of a Struck String (¼ point) Mode Number Normalized Mode Amplitude (a) Mode Number (b) Initial Amplitude (a/b) Normalized Amplitude (Initial Amplitude/0.707)

Comparison of Amplitudes

Tuning Forks Striking a tuning fork at a point ¼ to ½ of the way from the end excites only mode 1 The end of the tuning fork is not fixed as in the examples above o This puts an antinode at that end For a tuning fork struck at the ¼ point, normal modes might look like…

Tuning Fork Normal Modes Fixed End Open End

Observations Mode 1 is near its max Mode 2 is close to a node Mode 3 is close to a node Mode 4 is close to a max o Higher modes have rather low amplitude (Mode 4 would have only 1/16 th the initial amplitude of Mode 1).

Guitar Pick-up Locations Stimulated by large amplitudes of the string Modes that are amplified are those whose amplitudes are near maximum at the location of the pickup Place the pickup at the ¼ point and plucked the string at the center.

Normal modes of the Guitar String

Observations Only the odd modes are excited and they are at their antinode. Even modes are at their nodes

Amplitudes of First Eight Modes of a Plucked String (1/2 point) Mode Number Normalized Mode Amplitude (a) Mode Number Squared (b) Initial Amplitude (a/b) Normalized Amplitude

Now to the Pickup We have to now modify the amplitude table to get the amplitude that the excited modes have at the pickup position.

Moving the pickup We change the amplitude of the excited modes and therefore the mix of the normal modes. Also changing notes on the same string means using different frets, changing the length of the string

Comparing Plucking Positions