Location of Nodes and Antinodes for Selected Bars of La Favre Marimba
Jeff La Favre
jlafavre@gmail.com
The data in the table below serve as a reference for tuning the transverse modes of marimba bars. Keep in mind that values for marimba bars with different dimensions or made from different wood may have different values. Nevertheless, the values below can be helpful in a tuning effort. In order to tune a transverse mode, it is necessary to know the approximate location of the antinodes and nodes. It is even better if you know exactly where the nodes and antinodes are located. In the arch area, removing wood from an antinode reduces the vibration rate for the mode, while removing wood from a node does not change the vibration rate.
The exact locations of the nodes for the La Favre marimba were determined as follows. The bar was placed on two blocks of foam, the blocks located in the vicinity of the nodes of the fundamental. Salt was sprinkled across the entire surface of the bar (except for fundamental determination, where salt was not sprinkled on center of bar). To determine the nodes of the fundamental (first transverse mode), the bar was repeatedly tapped lightly in the center with a mallet. For the higher modes, the bar was exposed to sound from a speaker, which was connected to a tone generator. For each mode, the tone frequency was selected to match the frequency of the vibration for that mode. By exposing the bar to sound at a frequency that matches the frequency of a mode, the bar is excited into vibration in that mode. As the bar vibrates, the salt is cleared from the antinode areas and accumulates at the nodes.
|
Bar
|
Length |
Thickness
(inches) |
Location
along length of bar (x/l)*
|
|||||||
|
Nodes
(measured**)
|
Antinodes
(calculated**)
|
|||||||||
|
1st
***
|
2nd
|
3rd
|
4th
|
1st
|
2nd
|
3rd
|
4th
|
|||
|
C2
|
23 3/32
(23.094)
|
1.55
|
0.122
|
0.108
0.500 |
0.092 |
0.087
0.300 0.500 |
0.500
|
0.351
|
0.261
0.500 |
0.194
0.412 |
|
F2
|
20 9/16
(20.562)
|
1.36
|
0.128
|
0.109
0.500 |
0.093
0.354 |
0.085
0.298 0.500 |
0.500
|
0.351
|
0.263
0.500 |
0.192
0.411 |
|
C3
|
17 11/32
(17.344)
|
1.14
|
0.166
|
0.123
0.500 |
0.106
0.357 |
0.090
0.303 0.500 |
0.500
|
0.357
|
0.269
0.500 |
0.197
0.413 |
|
F3
|
16 1/8
(16.125)
|
0.88
|
0.186
|
0.132
0.500 |
0.109 |
|
0.500
|
0.360
|
0.277
0.500 |
|
|
C4
|
14 7/16
(14.438)
|
0.87
|
0.193
|
0.136
0.500 |
0.113
0.364 |
|
0.500
|
0.362
|
0.276
0.500 |
|
|
F4
|
13 1/2
(13.500)
|
0.85
|
0.204
|
0.139
0.500 |
|
|
0.500
|
0.363
|
|
|
|
C5
|
12 5/32
(12.156)
|
0.88
|
0.195
|
0.144
0.500 |
|
|
0.500
|
0.365
|
|
|
|
* x = distance from end of bar to location of node or antinode, l = length of bar, therefore, x/l values are the fraction of the total bar length were a node or antinode occurs. For example, a node occurring at 2 inches from the end of a bar that is 8 inches long would have a x/l value of 0.250 (i.e., 2/8 = 0.250). In all cases for node pairs (all nodes are paired except a central node) the nodes on both sides of the bar had the same or close values. When the values were not the same, the average was calculated. ** location of nodes was determined by the salt method (sprinkle salt on bar and expose bar to sound from tone generator or for first transverse mode, strike center of bar with mallet). The salt accumulates at the nodes, which allows direct measurement. The antinodes are located approximately halfway between adjacent nodes. To calculate a more accurate location for the antinodes, the following formulas were adapted from Bork et al. (1999): 2nd transverse mode, (0.500 - x/l of outer node)(0.62) + x/l of outer node; 3rd transverse mode, the central antinode is located halfway between the inner nodes and since these nodes were very close to the same value for each side of the bar in all bars, a value of 0.500 is used for the central antinode, for the outer antinodes, the value was calculated as follows (x/l inner node - x/l outer node)(0.65) + x/l outer node; for the 4th transverse mode the inner antinodes value was calculated as follows (0.500 - x/l node adjacent to central node)(0.56) + x/l node adjacent to central node; the outer antinodes value was calculated as follows (x/l node adjacent to central node - x/l outer node)(0.50) + x/l outer node. *** 1st = first transverse mode (fundamental), 2nd = second transverse mode, 3rd = third transverse mode, 4th = fourth transverse mode |
||||||||||
The Fundamental (first transverse mode)
The nodes for the fundamental had x/l values that ranged from 0.122 for the C2 bar to 0.204 for the F4 bar. These are the approximate locations where the bars should be drilled for the cord. You may find sources that give a formula for calculating the nodes of the fundamental. The data here demonstrates the need to avoid such formulas if you want to do accurate work in fabricating marimba bars. The true antinode of the fundamental is always at or very near the center of the bar. Therefore, if you want to lower the frequency of the fundamental, remove wood from the center of the bar. However, keep in mind that the third transverse mode also has an antinode at the center of the bar. Therefore, the third transverse mode will also be lowered when removing wood from the center of the bar.
The Second Transverse Mode
The second transverse mode has two antinodes, relatively close to the center of the bar (one on each side of the center). In the La Favre marimba, the location of these antinodes occurs at x/l values of 0.351 to 0.365, depending on the bar. Therefore, removing wood at these locations will have the greatest effect in lowering the second transverse mode of vibration. At the same time, removing wood at a second transverse antinode will have little effect on third transverse vibration since it has nodes very close to the antinodes of the second transverse mode. However, removal of wood at a antinode of the second transverse mode will lower the fundamental since this location is not near the nodes of the fundamental. This is an important point to keep in mind during tuning. Removing wood at a specific place usually affects more than one mode. You select a specific place for wood removal based on which mode needs to be lowered most, with the knowledge that other modes may be lowered to a lesser extent
The Third Transverse Mode
The third transverse mode has three antinodes, one in the center (x/l = 0.500) of the bar and two located at x/l values ranging from 0.261 to 0.277, depending on the bar. When you need to lower the third transverse mode, but not the fundamental, then you remove wood from the outer antinodes located at 0.261 to 0.277. This will lower the fundamental slightly and the second transverse mode even more. But the third transverse mode will be lowered the most.
The Fourth Transverse Mode
The fourth transverse mode has four antinodes, two located one side of the center and two on the other. The antinode pairs are located at x/l values of 0.411 to 0.413 for the inner pair and 0.192 to 0.197 for the outer pair. I am somewhat hesitant to suggest a tuning strategy for the fourth transverse mode since I have not done any tuning of the mode up to this point in time. Nevertheless, the data suggests that the fourth transverse mode could be tuned by removing wood at the outer pair of antinodes. This would also lower the third transverse mode somewhat since it has the closest antinodes of the other three transverse modes. Removing wood at the outer antinodes of the fourth transverse mode should not have much effect on the fundamental and second transverse mode. Tuning the fourth transverse mode could be done in the lowest octave of a 5 octave marimba.
Photos of salt patterns on F2 bar (nodes located where salt accumulates)
Fundamental
Second transverse mode (look
carefully for salt at center of bar)
Third transverse mode
Fourth transverse mode
Cited References
Bork, I, A. Chaigne, L.-C. Trebuchet, M. Kosfelder and D Pillot, 1999. Comparison between Modal Analysis and Finite Element Modelling of a Marimba Bar. Acust. Acta Acust. 85: 258-266.
Last update 2/3/07