TOWARDS IMPROVING THE MANUFACTURED CONSISTENCY OF WOODEN MUSICAL INSTRUMENTS THROUGH FREQUENCY MATCHING
Patrick Dumond and Natalie Baddour Department of Mechanical Engineering University of Ottawa Ottawa, Ontario
KEYWORDS Frequency matching, assumed shape method, musical instrument, consistency, manufacturing. ABSTRACT Although many improvements in the manufacturing of musical instruments have been made recently, one aspect that has often been overlooked is that of the acoustic consistency of the final manufactured product. The aim of this work is to create a method in which a soundboard can be frequency matched to a brace in order to meet a set standard after assembly. A simple analytical model is created in order to study the effect of the plate’s stiffness and brace thickness on the combined system. The assumed shape method is used in the analysis. Results show that by adjusting the thickness of the brace in order to compensate for the stiffness of the plate, one of the natural frequencies can be adjusted to meet a certain value. However, matching multiple natural frequencies cannot be done with a rectangular
brace. Therefore modifications to the shape of the brace are suggested. BACKGROUND The musical instrument manufacturing industry is one that likes to cling to tradition. Many customers of the industry still cite tradition as a key component in the manufacturing of superior instruments which, in many regards, holds a certain truth. This is primarily due to a lack of technology in the industry, which would permit the reproduction of the skills used by traditional instrument makers for building instruments in consistently good tone. Over the last decade or so, many larger manufacturers have started to look at ways to improve the consistency of their manufacturing processes. Most have taken the approach of increasing the accuracy of their tooling in order to improve their dimensional building consistency. To do so, many manufacturers have brought in numerically controlled machinery, lasers and robots as well as custombuilt jigs which ensure greater accuracy and consistency (French 2009). This has increased the number of “good” instruments that come out of the manufacturing process and decreased the
number of rejected instruments unsuitable for sale. However, the problem of acoustical consistency in the final product has yet to be solved when instruments are built of wood. This is because of wood’s inherently different properties from specimen to specimen within the same species and even within the same tree, even if a given board’s dimensions are exactly the same as another board. At most, a manufacturer will test a board’s stiffness across the grain in order to determine the sound quality that a board may produce and for which instrument line it should be used. The reason behind this inconsistency of properties is in the way a tree grows. Every year a tree grows, a new layer of wood is added to its girth. Depending on the time of year this wood is added, such as early or late in the growing season and because of climate variations, the wood can be more or less dense within these layers which are known as growth rings. Also, because seasons vary from year to year, width of the rings also vary. This leads to highly variable wood properties from specimen to specimen (Forest Products Laboratory (US) 1999). While structural engineers have solved the problem by overcompensating for these variations in properties using the lower end of the statistical distribution, acoustic engineers cannot use the same approach. Usually a specific natural frequency spectrum is sought. Therefore, one must understand how to compensate for these variations without damping the beautiful tone that a well built wooden instrument can produce. Over the last half-century, much research has been done to determine what makes an instrument sound good (Fletcher and Rossing 1999). Most people have yet to agree as to what makes an instrument sound good, even though there is a general agreement upon which instruments do sound good. Therefore, there is clearly a specific factor that sets good instruments apart from mediocre ones. Many theories have emerged including those based on the age of the wood, the skill of the instrument maker, the materials used and the method used to tune them. However, as of yet none have proven to be completely accurate. Some have tried to reproduce famously great sounding instruments by comparing and trying to match Chladni patterns (Hutchins and Voskuil 1993).
Others have tried to improve consistency by using the tap tuning method (Siminoff 2006). Both methods require the instrument to be built first then adjusted later so as to arrive at a certain standard set by the instrument maker. These methods try to scientifically reproduce what experienced instrument makers with finely tuned ears have been doing for centuries, namely fine-tuning an instrument to optimise the quality of its sound. There is general consensus in the scientific community that it is the lower frequencies that are the most important in creating good sound in an instrument (Hutchins and Voskuil 1993). While methods have emerged which enable instrument makers with less experience and without perfect pitch to build great sounding instruments, these techniques have not been able to cross over into production manufacturing. Even though many instrument manufacturers are not fully automated and still use certain traditional methods, tuning an instrument for sound is too labour intensive. This leads to manufactured instruments which cost less and which are dimensionally identical but which have a great degree of variation in their final sound. The purpose of this work is to present a method in which an instrument soundboard could be frequency matched with its braces before assembly, thereby improving the consistency in the sound of the final product without significantly increasing the cost of the manufacturing process. This work assumes that a good sounding instrument is known and which can be used for comparison in the frequency matching process. METHODOLOGY The most common way to tune or adjust an instrument soundboard in order to optimise the sound quality, is by adjusting the thickness of the inside soundboard braces as seen in figure 1 (Cumpiano and Natelson 1993). By removing material from the top of the braces, the brace stiffness is reduced, changing the overall natural frequency spectrum of the system which consists of the soundboard and braces.
Braces
Soundboard
is known as grain. In fact, wood’s properties vary in three different directions, relative to this grain as seen in figure 3. These directions are known as longitudinal, tangential and radial to the grain. Because wood’s properties are consistent along three orthogonal axes, wood can be considered an orthotropic material. Since instrument wood is quartersawn, the two important directions which lie along the x-y plane are the longitudinal and radial directions. Wood is strongest along its longitudinal direction which is why soundboards are generally reinforced across the grain, as seen in figure 2. R
FIGURE 1. BRACED GUITAR SOUNDBOARD.
To better understand how the system responds to a change of brace thickness and/or stiffness, a simple model has been developed. This model consists of a rectangular plate on which a brace has been positioned across its width as seen in figure 2. The plate is considered to be thin enough to use Kirchhoff’s plate theory (Meirovitch 2000). To simplify the model, the assumption was made that the soundboard is simply supported on all sides when in fact it is probably somewhere between simply supported and clamped (Fletcher and Rossing 1999). Since clamped edges prevent rotation at the edge, local stiffening occurs. This leads to an increase in the natural frequencies. z
Ly
y
Brace
x
0 h1 0
h2 r1
Plate
r2 Lx
FIGURE 2. TEST PLATE WITH BRACE ACROSS ITS WIDTH.
The nature of the wood’s properties must also be taken into account. Since the growth of a tree generally produces rings of different density wood season after season, wood contains what
T
L FIGURE 3. CONSTANT WOOD PROPERTIES RELATIVE TO THE GRAIN.
In order to properly study the system at hand, it is essential that a good model be created. No exact solution to the given system exists because of the extra complexity the brace adds to the system. Therefore an approximate method must be used. One such approximate method is that of the assumed shape method. The assumed shape method is a method in which a given system is made to be discrete by using the superposition of global trial functions in order to model the displacement of such a system. The global trial functions are defined over the entire spatial domain and so as to satisfy the geometric boundary conditions of the given system. This leads to a boundary-value problem that can be solved (Bisplinghoff, Ashley et al. 1955). The assumed shape method was chosen because of its inherent ability to be developed based on the physical properties of the given system while giving accurate results which can converge quickly. To use the assumed shape method, a finite series for the time-dependent displacement is assumed such that (Meirovitch 2001)
my
mx
w( x, y, t ) = ∑ ∑ φnx ny ( x, y ) ⋅ qnx ny (t )
(1)
nx =1 n y =1
where w is the displacement along the z-axis, ø are the chosen spatial trial functions and q(t) are the generalized time dependent coordinates. Additionally mx and nx represent the mode number and trial function number in the xdirection respectively and my and ny represent the same in the y-direction. To satisfy the simply supported boundary conditions, which represent a zero displacement around the perimeter of the plate, the trial functions are chosen to be similar to those obtained from the exact solution of a regular plate, ⎛ ⎛ x ⎞ y ⎞ φnx ny = sin ⎜ nx ⋅ π ⋅ ⎟ ⋅ sin ⎜ n y ⋅ π ⋅ ⎟ (2) ⎜ Lx ⎠ Ly ⎠⎟ ⎝ ⎝ where Lx and Ly are the lengths of the plate in their respective directions. It is important to note that the mode number is based on the maximum amount of trial functions chosen. The assumed shape method is an energy method. Therefore, the expression in equation (1) will be used in the expression for kinetic and potential energy of the orthotropic simplysupported rectangular plate (Timoshenko and Woinowsky-Krieger 1959). The plate has been modified by the addition of a brace across its width. To account for the extra material thickness and for the change of direction in the material properties, the energy integrals have been divided into three sections as specified along the x-axis in figure 2. The kinetic energy can be expressed as r Ly
r Ly
11 12 T = ∫ ∫ w& 2 ρ1 dydx + ∫ ∫ w& 2 ρ 2 dydx 20 0 2 r1 0 L Ly
1 x + ∫ 2 r2
∫ w& ρ 2
1
(3)
dydx
0
where ρi = μ·hi , i = 1,2, are the densities per unit area, μ the material density and hi the thickness of the different sections of the plate. The potential energy can be developed as r Ly
V=
r Ly
L Ly
11 12 1 x U1 dydx + ∫ ∫ U 2 dydx + ∫ ∫ U1 dydx (4) ∫ ∫ 20 0 2 r1 0 2 r2 0
where
2 U i = Dxi wxx2 + 2 Dxyi wxx wyy + Dyi wyy + 4 Dki wxy2 ,
(5) i = 1, 2 The subscripts on the displacement terms are partial derivatives in the direction defined by the subscript and the stiffness terms are S yy hi3 S h3 Dxi = xx i Dyi = 12 12 , i = 1, 2 (6) 3 S xy hi Gxy hi3 Dxyi = Dki = 12 12 where G is shear stress. The S’ are stiffness components which are defined as Ey ν yx Ex Ex S xx = S yy = S xy = (7) 1 −ν xyν yx 1 −ν xyν yx 1 −ν xyν yx and E is Young’s modulus and ν is Poisson’s ratio. Because of the method in which the brace thickness is added to that of the plate in the kinetic and strain energy equations, it was necessary to change the direction of the grain of the plate, in this region alone, to match that of the brace. This is a reasonable assumption because of how thin the plate is in comparison to the brace and the solid link of wood glue which bonds them together. Once the kinetic and potential energies have been determined, it remains to substitute them into Lagrange’s equations (Meirovitch 2001) d ⎛ ∂T ⎞ ∂T ∂V ⎜ ⎟− + = Qnx ny , dt ⎜⎝ ∂q&nx ny ⎟⎠ ∂qnx ny ∂qnx ny (8) nx = 1, 2,..., mx and n y = 1, 2,..., m y And since no other forces are acting on the system, the generalized force Q=0. Lagrange’s equations yield the equations of motion of the system which can be written in matrix form as r r r M q&& + K q = 0 (9) r T where q = ⎡⎣ q11 q12 q13 ... q21 ... qnx ny ⎤⎦ , M is the mass matrix and K is the stiffness matrix of the system, the dimensions and contents of which will vary depending on the number of trial functions or mode numbers chosen. This also leads to the number of degrees of freedom of the system. To solve for the natural frequencies and mode shapes of the system, a harmonic response is assumed such that
r ur q = A cos (ωt + φ )
(10)
where ω is the system’s natural frequencies, ø the phase shift and A is a magnitude vector of dimension mx x my by 1. By substituting this response into the equation of motion, an eigenvalue problem is obtained, ur r (K − ω 2 M ) A = 0 (11) from which it is possible to solve for the natural frequencies, ω, and the general coordinates, which are used to determine the mode shapes or displacement of the system. By creating an algorithm of the assumed shape method’s steps in a symbolic computational software package such as Maple, it is possible to determine how a plate reacts to changes in brace thickness and stiffness. RESULTS The material used for the test specimen is wood, more specifically, Sitka spruce. The important material properties of Sitka spruce are listed in table 1 (Forest Products Laboratory (US) 1999). TABLE 1. MATERIAL PROPERTIES OF SITKA SPRUCE.
Property μ (kg/m3) ER (MPa) EL (MPa) GLR (MPa) νLR νRL
Value 403.2 850 ER / 0.078 EL x 0.064 0.372 νLR x ER / EL
These material properties are those that were used throughout the problem analysis. As mentioned, the cross-grain stiffness is what seems to affect the sound quality the most. This is measured as Young’s modulus in the radial direction, ER. Therefore, it is varied in the analysis to see what effect it has on the system. The value given in table 1 is the statistical average only. However, a relationship exists between Young’s modulus in the radial direction and the other properties. It is also important to note that based on the setup of the test specimen seen in figure 2, the longitudinal direction of the wood’s grain follows along the xaxis from 0 to r1 and from r2 to Lx and the radial direction follows along the y-axis for the same
sections. Between r1 and r2 the grain is considered to have reversed directions. As mentioned previously, it is generally thought that only the lowest natural frequencies are the most important in producing good tone in an instrument, therefore all efforts are concentrated on frequency matching these lower frequencies rather than increasing the number of trial functions to improve the accuracy of the higher frequencies. Therefore, six trial functions are used in each of the x-y directions. This gives a solution of thirty six natural frequencies and modeshapes. Only the first and fourth natural frequencies are observed during the analysis because the second and third modeshapes have a node at the location of the brace and are not as affected by changes in its stiffness. The dimensions of the test specimen are found in table 2. TABLE 2. DIMENSIONS OF THE TEST SPECIMEN. Dimension Lx (m) Ly (m) r1 (m) r2 (m) h1 (m) h2 (m)
Value 0.24 0.18 0.114 0.126 0.003 0.015
Even though a thickness for the brace is specified as h2 in table 2, this is used only as a reference point. The brace thickness is adjusted in order to try and match soundboards that have different stiffness values. Using the equations, properties and dimensions presented, the analysis was done in Maple by first varying Young’s modulus in the radial direction and keeping a constant thickness of h2=0.015m as presented in table 3. TABLE 3. RESULTS FOR THE FIRST AND FOURTH NATURAL FREQUENCY WHEN VARYING ER (h2=0.015m).
ER (MPa) 750 800 850 900 950
ω1 (rad/s) 3492.00 3606.53 3717.52 3825.30 3930.12
ω4 (rad/s) 6272.83 6478.56 6677.94 6871.55 7059.84
A similar analysis was performed by varying the thickness h2 and keeping Young’s modulus constant at ER=850MPa which is shown in table 4.
Once again, forcing the fourth natural frequency to be more or less constant causes the first natural frequency to vary considerably from its value at ER =850MPa.
TABLE 4. RESULTS FOR THE FIRST AND FOURTH NATURAL FREQUENCY WHEN VARYING h2 (ER =850MPa).
The first four modes of vibration for the system of figure 2 are as follows.
h2 (m) 0.0140 0.0145 0.0150 0.0155 0.0160
ω1 (rad/s) 3458.57 3587.91 3717.52 3847.25 3976.95
ω4 (rad/s) 6481.33 6581.25 6677.94 6771.21 6860.91
From these results it is possible to see that as the radial stiffness of the plate or the thickness of the brace goes up, so do the first and fourth natural frequencies. In order to verify if it is possible to get consistency out of the natural frequencies, an analysis was performed in which an increase in the plate’s radial stiffness was compensated by reducing the thickness of the brace, table 5.
FIGURE 4. FIRST MODE OF VIBRATION OF THE MODEL.
TABLE 5. THE SYSTEM IS COMPENSATED SO THAT THE FIRST NATURAL FREQUENCY IS HELD CONSTANT.
ER (MPa) 750 800 850 900 950
h2 (m) 0.0159 0.0154 0.0150 0.0146 0.0142
ω1 (rad/s) 3711.34 3707.21 3717.52 3718.59 3711.02
ω4 (rad/s) 6428.13 6551.22 6677.94 6792.23 6894.64
It is clear from table 5 that although the first natural frequency has been held more or less constant, this did not result in a significant change in the fourth natural frequency from its original state. This led to a further analysis in which only the fourth natural frequency was held constant, table 6. TABLE 6. THE SYSTEM IS COMPENSATED SO THAT THE FOURTH NATURAL FREQUENCY IS HELD CONSTANT.
ER (MPa) 750 800 850 900 950
h2 (m) 0.0175 0.0161 0.0150 0.0140 0.0132
ω1 (rad/s) 4099.29 3883.36 3717.52 3558.84 3438.51
ω4 (rad/s) 6676.50 6673.04 6677.94 6669.23 6676.37
FIGURE 5. SECOND MODE OF VIBRATION OF THE MODEL.
compensate and keep the system’s frequency consistent, the stiffness of the brace must be decreased. This can be achieved by reducing its thickness.
FIGURE 6. THIRD MODE OF VIBRATION OF THE MODEL.
Once compensation has been completed so that all the first natural frequencies coincide, as in table 5, it is clear that the fourth natural frequencies do not coincide. Compensating for the fourth natural frequencies and looking at table 6, the first natural frequencies no longer coincide. This interesting result has led to the speculation that a rectangular brace alone cannot allow an instrument builder to frequency match a soundboard. This is why many builders opt to use shaped braces (Cumpiano and Natelson 1993), which seem to have a better overall affect on the sound of an instrument. One such shape is that of a scalloped brace as seen in figure 8.
Brace 1st mode of vibration
2nd mode of vibration
FIGURE 7. FOURTH MODE OF VIBRATION OF THE MODEL.
The dip in the center of the x-axis is the location of the brace. The brace stiffens this area and limits the amount of displacement that can occur. By increasing the number of trial functions, the order of the modes may be slightly different.
FIGURE 8. SCALLOPED BRACE WITH THE MODES OF VIBRATION IT AFFECTS.
DISCUSSION
The scalloped shape may in fact allow a builder to for example, modify a natural frequency having a mode number of two along the length of the brace, without altering the first natural frequency because of the location of the peaks along the brace. More research needs to be done on the shape of the brace for the purpose of frequency matching braces to a plate in the construction of a soundboard.
Based on the analysis performed, it is evident that as the stiffness of the plate in the radial direction and the thickness of the brace increase, the natural frequencies also increase. This coincides nicely with elementary vibration theory which states that as the stiffness of a system increases or its mass decreases, the system’s natural frequencies will increase and vice-versa (Meirovitch 2001). Therefore, when the stiffness of the plate increases, in order to
It is also interesting to note that the four first modes as seen in figure 4 to figure 7 are no longer in the same order as those for a regular rectangular plate. This is because the addition of a brace makes the plate significantly stiffer in one direction thereby making it more difficult for certain modes to vibrate and so shifts the order in which modes appear. Therefore, the location and magnitude of a brace’s stiffness has significant repercussions on the overall sound
that is emitted by an instrument. Further study on the location and range of stiffness of a brace would be necessary. In order to create a manufacturing process that would allow frequency matching of soundboards to a specified manufacturing standard, the cross-grain stiffness would first need to be measured. This is something that is already done is some factories. The only modification would be to note the exact stiffness instead of a range. Next, either a set of predetermined braces would have to be matched to the measured soundboard or a set of braces would need to be milled. Following this, the soundboard assembly would be done as normal using the frequency matched braces. This would ensure a greater acoustic consistency in the final product. Further research needs to be done in order to compare the results obtained in this analysis to those obtained experimentally. Also, the analysis could be made more accurate by taking into account certain things that have been neglected in order to make a simpler model. Such improvements include using higher order theories which take into account shear deformation and rotary inertia such as Mindlin plate theory, the effect of surrounding air on the system acting as extra inertia, all of which tends to lower the natural frequencies (Leissa 1969) and finally taking into account the internal damping of the wood. Thus far the system has been considered to be conservative because only the lower frequencies were considered. Internal damping has a much larger effect on the higher frequencies. CONCLUSION While it is clear that the simplified model used is basic compared to a real instrument soundboard, it has allowed a clear insight into how a soundboard’s frequency can be adjusted by changing the thickness of its braces. In fact, a brace could be preselected in a manufacturing process and paired with a given plate so that the final soundboard would be frequency matched to a given standard. This would greatly improve the consistency of the acoustic properties in a manufactured instrument. A lot of work still needs to be done in order to create a model complex enough and accurate enough to be implemented in a factory environment. However many research opportunities are clearly defined.
Over the years, Brazilian rosewood has become the favourite hardwood of instrument builders. Coincidentally, recent studies have shown that Brazilian rosewood has one of the most consistent properties from specimen to specimen. This suggests that perhaps consistency is one of the keys to building great sounding instruments. Even though this work has concentrated its efforts on the soundboard of a wooden musical instrument, the same principles are applicable to the other wooden parts of an instrument. REFERENCES Bisplinghoff, R.L., H. Ashley, et al. (1955). Aeroelasticity. Addison-Wesley Publishing Company, Reading, Massachusetts. Cumpiano, W.R. and J.D. Natelson (1993). Guitarmaking, Tradition and Technology: A Complete Reference for the Design & Construction of the SteelString Folk Guitar & the Classical Guitar. Chronicle Books, San Francisco. Fletcher, N.H. and T.D. Rossing (1999). The Physics of Musical Instruments. Springer, New York. Forest Products Laboratory (US) (1999). Wood Handbook: Wood as an Engineering Material. U.S. Dept. of Agriculture, Forest Service, Forest Products Laboratory, Madison, Wisconsin. French, R.M. (2009). Engineering the Guitar. Springer, New York. Hutchins, C.M. and D. Voskuil (1993). "Mode Tuning for the Violin Maker." CAS Journal, Vol. 2, (4 (Series II)), pp. 5-9. Leissa, A.W. (1969). Vibration of plates. Scientific and Technical Information Division, National Aeronautics and Space Administration, Washington. Meirovitch, L. (2000). Principles and Techniques of Vibrations. Prentice Hall, Upper Saddle River, New Jersey. Meirovitch, L. (2001). Fundamentals of Vibrations. McGraw-Hill, New York. Siminoff, R.H. (2006). The Art of Tap Tuning: How to Build Great Sound into Instruments. Hal Leonard, Milwaukee, Wisconsin. Timoshenko, S. and S. Woinowsky-Krieger (1959). Theory of Plates and Shells. McGraw-Hill, New York.