Real and simulated marimba bars

Krista Ehinger and Justin Kao

Overview

We carved marimba bars and simulated them in Matlab.

The Bars

The marimba bars were made from 1''x2''x12'' and 1''x2''x18'' pieces of East Indian Rosewood. The curves of the tuned bars were determined experimentally by making shallow cuts and then measuring the pitch. The 12'' turned bar has a fundamental of A5 and a second partial of about E7. The 18'' tuned bar has a fundamental of E4 and a second partial of E6.

The Model

Since a marimba bar is much longer than its width and much wider than its height, we model the marimba bar in one dimension only: the vertical displacement from equilibrium as a function of distance along the length. In this model of the bar, the mallet exerts a force on the center of the bar causing it to bend, and elasticity provides the force causing the bar to vibrate thereafter. The vibration decays in time due to damping forces.

Equations for this were adapted from Chaigne and Doutaut [1], and predict the vibration of the marimba bar given only physical constants [2] and measurements of the marimba bar and mallet. ¹

The sound radiation of the vibrating bar is modeled as radiation of a vibrating line of spheres, as described in Doutaut, Matignon and Chaigne [3].

We deviate from [1] and [3] in our bar's supporting force. We model this as two localized forces, representing the strings holding a real marimba bar in place.

Simulation

The simulation treats the bar as a string of rectangular slices taken along the length. This is a discrete version of our model's differential equations. The discretized equations can then be solved by computer at each time step, yielding the motion of the mallet and marimba bar.

The code for this was written in Matlab.

We created several virtual bars: an ideal bar given in [4], two curved bars based on our real bars, and two flat bars based on our real bars. Since wood is variable in its physical properties, we estimated density from dimensional measurements and weight of the flat bars. Using a formula² for the fundamental frequency of a flat bar cited in [5], we calculated the Young's modulus of our wood from its fundamental frequency.

Sound radiation is approximated by considering each slice of the bar as a vibrating sphere. The simulation calculates the actual sound pressure at the observer's location.

Results

Our simulation approximates the fundamentals of the bars very well, but error increases for the higher peaks. Part of the problem may be that the simulation generally yeilds fewer peaks than the real bar, so it can be difficult to identify and match up peaks.

Table 1: Results for 12'' curved bar

	Real Bar	Simulation	Percent Error
Fundamental	879 Hz	904 Hz	2.84
Second partial	2745 Hz	2539 Hz	7.50
Third partial	5045 Hz	6294 Hz	24.76
Plots	png	png

Table 2: Results for 12'' flat bar

	Real Bar	Simulation	Percent Error
Fundamental	1112 Hz	1109 Hz	0.27
Second partial	2328 Hz	2512 Hz	7.90
Third partial	3562 Hz	5984 Hz	68.00
Plots	png	png

Table 3: Results for 18'' curved bar

	Real Bar	Simulation	Percent Error
Fundamental	328 Hz	334 Hz	1.83
Second partial	1314 Hz	none
Third partial	2351 Hz	2531 Hz	7.66
Plots	png	png

Table 4: Results for 18'' flat bar

	Real Bar	Simulation	Percent Error
Fundamental	483 Hz	479 Hz	0.83
Second partial	971 Hz	none
Third partial	2470 Hz	2587 Hz	4.74
Plots	png	png

Table 5: Sound comparison

	Real Bar	Simulation
Curved 12'' bar	wav	wav
Flat 12'' bar	wav	wav
Curved 18'' bar	wav	wav
Flat 18'' bar	wav	wav

The model that we used does not take into account non-vertical vibrations, so this may account for some of the error. Non-vertical vibrations can be ignored in a long, flat bar, but they are significant in a short, thick bar such as the 12'' bar. In addition, there are probably variations in height, density, and elasticity along the length of the real bars, which would also contribute to error.

Suggestions for Future Projects

An obvious extension would be to consider the marimba resonator as modeled in [3]. The accuracy of the simulation could be improved by better measurements of the rosewood's elasticity, density, and other properties.

Bibliography

1

Antoine Chaigne and Vincent Doutaut.
Numerical simulations of xylophones. i. time-domain modeling of the vibrating bars.
Journal of the Acoustical Society of America, 101(1):539-557, 1997.

2

D. Holz.
Acoustically important properties of xylophone-bar materials: Can tropical woods be replaced by european species?
Acta Acustica, 82:878-884, 1996.

3

Vincent Doutaut, Denis Matignon, and Antoine Chaigne.
Numerical simulations of xylophones. ii. time-domain modeling of the resonator and of the radiated sound pressure.
Journal of the Acoustical Society of America, 104(3):1633-1647, 1998.

4

Felipe Ordu na Bustamante.
Nonuniform beams with harmonically related overtones for use in percussion instruments.
Journal of the Acoustical Society of America, 90(6):2935-2941, 1991.

5

Greg Merrill.
The Marimba.
http://faculty.smu.edu/ttunks/projects/merrill/Marimba.H.html.

About this document ...

Real and simulated marimba bars

This document was generated using the LaTeX2HTML translator Version 2002-1 (1.68)

Copyright © 1993, 1994, 1995, 1996, Nikos Drakos, Computer Based Learning Unit, University of Leeds.
Copyright © 1997, 1998, 1999, Ross Moore, Mathematics Department, Macquarie University, Sydney.

The command line arguments were:
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The translation was initiated by Justin Kao on 2003-06-06

Footnotes

... mallet.¹

The motion of the bar itself is described by the Euler-Bernoulli equation

$$M(x,t) = -E I(x) \left[ 1 + \eta \frac{\partial}{\partial t} \right] \frac{\partial^2 u}{\partial x^2} (x,t)$$

$$\frac{\partial^2 u}{\partial t^2} (x,t) = \frac{1}{\rho S(x)} \frac{\partial^2 M}{\partial x^2} (x,t) + f(x,t)$$

where $u(x,t)$ is the displacement of the bar from equilibrium, $M$ is the bending moment, and $f$ is external forcing. The external forcing term accounts for the impact of the mallet, the support of the marimba bar, and damping forces. This gives us a system of differential equations in time and one space dimension. The force due to the mallet is further described by Hertz's law of contact

$$f_{mal} = K \vert \eta(t) - u(x_0,t) \vert^{3/2}$$

and the mallet is modeled as a mass with a given initial velocity impacting the marimba bar.

... formula²

$f=1.03 H L \sqrt{\frac{E}{\rho}}$ for the fundamental frequency of a flat bar, where $f$ is the frequency in Hz, $H$ and $L$ are thickness and length, $E$ is the Young's modulus, and $\rho$ is density.