Illustration of the normal modes

Free-free bar

• Consider a perfectly elastic small bar that is completely free to vibrate, that is, not constrained at any point.
• This physical system describes well the vibrating parts of some percussion instruments such as the marimba and the glockenspiel.
• Its normal modes are labelled by a single integer.
• This number gives directly the number of nodes that the bar forms in its oscillation, minus one.
• For example, mode {2} means that there are 3 nodes.
• Once the dimensions, the shape of the bar and the material it is made of are fixed, the natural frequency of each mode depends on the mode number, but not in a simple way: the frequency of oscillation of each mode is not directly proportional to the mode number, as it is instead in the case of the string.

	barra_ff_freq_1.mp3 mode {1}, ${\displaystyle f=f_{0}\!}$
	barra_ff_freq_2.mp3 mode {2}, ${\displaystyle f=2.75f_{0}\!}$
	barra_ff_freq_3.mp3 mode {3}, ${\displaystyle f=5.40f_{0}\!}$
	barra_ff_freq_4.mp3 mode {4}, ${\displaystyle f=8.93f_{0}\!}$

Clamped-free bar

• Consider a perfectly elastic small bar rigidly clamped at one end and free to vibrate at the other.
• This physical system describes well the reed of some wind instruments such as the clarinet and the harmonica, or the tongues of a music box or of a jaw harp.
• The shape and the frequencies of the normal modes are different from those of the free-free bar and, again, they are not in harmonic ratios with the fundamental.

	barra_ff_freq_1.mp3 mode {1}, ${\displaystyle f=f_{0}\!}$
	barra_cf_freq_2.mp3 mode {2}, ${\displaystyle f=6.27f_{0}\!}$
	barra_cf_freq_3.mp3 mode {3}, ${\displaystyle f=17.55f_{0}\!}$
	barra_cf_freq_4.mp3 mode {4}, ${\displaystyle f=34.39f_{0}\!}$