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VIBRATION PATTERNS AND SOUND ANALYSIS OF THE VIENNESE TIMPANI
http://bias.at/Forschung/pdf_dateien/2001e_MB_ISMA_timpani.pdf
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Mode Diagrams for Circular Membranes
[IMG]http://i224.photobucket.com/albums/dd269/AngeloClematide/01_ModeandDiagrams.jpg[/IMG]
Adjected segment are moving in opposite directions
Reference: Rossing, Thomas's D. "The Physics OF Kettledrums." Acoustics of Timpani: vibrations of membranes, history of timpani, membrane stiffness,Rossing, Thomas D, The Science of Sound, Massachusetts, Journal of Computational and Applied Mathematics.
The first 12 modes of an ideal circular membrane
Their nodal diagrams, mode designations, and their relative modal frequencies
[IMG]http://i224.photobucket.com/albums/dd269/AngeloClematide/02_ModeandDiagrams.jpg[/IMG]
Vibrational Modes of a Ideal Circular Membrane
The (0,1) Mode shows the fundamental mode shape of a vibrating circular membrane
[IMG]http://i224.photobucket.com/albums/dd269/AngeloClematide/The01Mode.gif[/IMG]
The mode number is designated as (0,1) since there are no nodal diameters, but one circlular node (the outside edge). Remember that a node is a point (or line) on a structure that does not move while the rest of the structure is vibrating. The (0,1) mode of a drum, such as a tympani, is excited when the drum head is struck at its center. When vibrating in this mode the membrane acts much like a monopole source, which radiates sound very effectively. Since it radiates sound so well when vibrating in this manner, the membrane quickly transfers its vibrational energy into radiated sound energy and the vibration dies away. The short duration (fraction of a second) of the (0,1) mode means that this mode does not contribute greatly to the musical tone quality of a drum. In fact, when struck at the center a tympani, or other large drum, produces a "thump" which decays quickly and with no definite pitch.
The (1,1) Mode with one nodal diameter and one circlular node (the outside edge)
[IMG]http://i224.photobucket.com/albums/dd269/AngeloClematide/The11Mode.gif[/IMG]
The exact location of the nodal diameter depends on the homogeneity of the membrane and the initial conditions when the vibration starts. The frequency of the (1,1) mode is 1.593 times the frequency of the (0,1) mode. When vibrating in the (1,1) mode a circular membrane acts much like a dipole source; instead of pushing air away from the membrane like the (0,1) mode does, in the (1,1) mode one half of the membrane pushes air up while the other half sucks air down resulting in air being pushed back and forth from side to side. As a result, the (1,1) mode radiates sound less effectively than the (0,1) mode which means that it does not transfer its vibrational energy into radiated sound energy as quickly as the (0,1) mode and therfore, the (1,1) mode takes longer to decay. Because the (1,1) mode "rings" for a while, it contributes to the musical sound or pitch of a drum. When a tympani, or other large drum, is struck somewhere between the center and outer edge, the sound has a definite pitch which lingers for several seconds.
The (2,1) Mode: The third mode of a circular membrane is the (2,1) mode which has two nodal diameters (at right angles to each other) and one nodal circle (the outside edge)
[IMG]http://i224.photobucket.com/albums/dd269/AngeloClematide/The21Mode.gif[/IMG]
The exact locations of the nodal diameters depend on the homogeneity of the membrane and the initial conditions when the vibration starts. The frequency of the (2,1) mode is 2.135 times the frequency of the (0,1) mode. When vibrating in the (2,1) mode a circular membrane acts much like a quadrupole source which is worse at radiating sound than the (1,1) dipole mode and much less effective at radiating sound than the (0,1) monopole mode. This means that the (2,1) transfers its vibrational energy into radiated sound energy much more slowly than the (1,1) and (0,1) modes and therefore takes longer to decay, and contributes to the musical pitch of a drum. In fact, the modes which most significantly determine the tone quality of a tympani drum are the (1,1), (2,1), (3,1), (4,1), and (5,1) modes.
The (0,2) Mode does not have any diameter nodes, but has two circular nodes - one at the outside edge and one at a distance of 0.436 a (a is the radius of the circular membrane) from the center of the membrane.
[IMG]http://i224.photobucket.com/albums/dd269/AngeloClematide/The02Mode.gif[/IMG]
The frequency of the (0,2) mode is 2.295 times the frequency of the (0,1) mode. Like the (0,1) mode, the (0,2) mode is excited when the membrane is struck at the center. The sound radiation characteristics of the (0,2) mode are more complicated than the first three modes -- it appears to be a mix between a monopole and a dipole. Its decay time is longer than the (0,1) mode, but shorter than the (1,1) mode. As a result, it contributes to the "thump" sound when a drum is hit at the center, but does not contribute much to the musical pitch of a drum when hit off center.
The (1,2) Mode[/B] has one nodal diameter and two nodal circles
[IMG]http://i224.photobucket.com/albums/dd269/AngeloClematide/The12Mode.gif[/IMG]
The frequency of the (1,2) mode is 2.917 times the frequency of the (0,1) mode. As you might expect after looking at the first several modes of the circular membrane, the (1,2) mode does not radiate sound very effectively. It has somewhat of a quadrupole type behavior. Thus, the (1,2) mode takes a relatively long time to decay. However, this mode doesn't seem to play a dominant role in the musical tone quality of a drum.
The (0,3) Mode[/B] has three circular nodes, but no diameter nodes
[IMG]http://i224.photobucket.com/albums/dd269/AngeloClematide/The03Mode.gif[/IMG]
The frequency of the (0,3) mode is 3.598 times the frequency of the (0,1) mode. Like the (0,1) and (0,2) modes, the (0,3) mode is excited when the membrane is struck at the center. The sound radiation characteristics of the (0,3) mode rather complicated. This mode is excited when the membrane is struck at the center, and it dies away fairly quickly. As a result, it contributes to the "thump" sound when a drum is hit at the center, but does not contribute much to the musical pitch of a drum when hit off center.
Reference and information: Dr. Dan Russell, Kettering University
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The Five Important Modes
Each of these modes of vibration has its own pattern of radiation. The first (0.1) mode radiates pretty much in all directions (see Fig. 1 below), but the other four (1.1) 2.1) 3.1) and (4.1) modes radiate most strongly in 2, 4, 6, or 8 directions. They generate sound fields which are said to exhibit monopole, dipole, quadrupole, hexapole, and octupole character.
When the membrane is struck, many different modes or patterns of vibrations are excited, including the six important ones shown in Fig. 1. Each of these modes of vibration has its own pattern of radiation. The first one radiates pretty much in all directions, at least in its own plane, but the other four radiate most strongly in 2, 4, 6, or 8 directions, respectively. They generate sound fields which are said to exhibit monopole, dipole, quadrupole, hexapole, and octupole character.
Fig. 1 Five modes of vibration of a drumhead. Arrows indicate the directions of maximum sound radiation in the plane of the membrane:
[IMG]http://i224.photobucket.com/albums/dd269/AngeloClematide/Fivemodesofvibration.gif[/IMG]
In a two-headed drum, such as a snare drum, the directional radiation patterns from the two vibrating membranes interact, and the directionality of the sound becomes even more complex. In fact, the two membranes interact strongly as they vibrate, and so the modes or patterns of the drum are quite different form what they are in a drum with a single membrane.
Four modes of vibration of a snare drum, in which the heads vibrate in much the same way as they do in the first two modes in Fig. 1, are shown in Fig. 2. The directions in which maximum sound is radiated are indicated by arrows:
Fig. 2 Four modes of vibration of a two-headed snare drum. The heads vibrate in much the same patterns as the first two modes in Fig. 1. Directions of maximum radiation for each mode are shown by the arrows:
[IMG]http://i224.photobucket.com/albums/dd269/AngeloClematide/rossing2b.gif[/IMG]
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Composing for Timpano
Laws Of Modes
1. Each mode is excited more or less, according to how closely the pattern of the disturbing force matches the pattern of the mode itself.
2. Striking the membrane at any given point excites each natural mode in proportion to how much that mode involves motion of that particular point.
3. If the striking point is on a nodal point or line of some particular mode, then that mode is completely left out of the recipe (overtones/partials).
4. Striking an entire region at once instead of just a point produces the same recipe as you get by adding together the recipes for striking at each of the points contained in that region.
5. If a striking force has finite duration T in time (i.e. a single stroke with a drum stick), then only modes whose frequencies are less than about 2/T are efficiently excited.
6. Localized frictional damping will affect each mode in proportion to how much motion that mode causes at the point of application of the damping; in particular, it leaves undisturbed any mode that has a node at the point of application.
7. When a drumhead is struck sharply near the center, most of the energy initially appears in the (0,1) and (0,2) modes. By the end of the 1st second, their spectral peaks narrow substantially, and the sound spectrum includes many partials radiated by modes that received energy from the (0,1) and (0,2) modes to which they are coupled.
Reference: Musical Acoustics, Donald E. Hall, Brooks/Cole Publishing Company, Pacific Grove, CA, 1980
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