Proceedings of The Canadian Society for Mechanical Engineering Forum 2010 CSME FORUM 2010 June 7-9, 2010, Victoria, British Columbia, Canada
COMPARISON OF THE ASSUMED SHAPE AND FINITE ELEMENT METHODS FOR A VIBRATING BRACE-PLATE SYSTEM Patrick Dumond and Natalie Baddour Department of Mechanical Engineering University of Ottawa Ottawa, Canada Abstract—although many would agree that the finite element method is the method of choice when modeling vibrating systems, new symbolic computational tools have made it possible to accurately model systems using other discrete methods that have become nothing more than methods of academic interest. This has revived interest in these methods. One such method is that of the assumed shape method.
over other methods, namely its ability to model complex systems and boundaries not possible with global function methods. It can also be easily programmed into a numeric solver in order to solve very large and complex modal solutions. However, the finite element solution will have errors due to the approximation of the solution as well as the geometry of the domain [1]. The finite element solution also requires a large number of functions in order for the solution to converge to decent results.
This work compares simplified models of stringed instrument soundboards built using both the finite element and assumed shape methods. The same model is built using both methods using typical approximations when necessary.
Global trial function methods offer insight into the building blocks of the solution since they are generally solved by superposition of trial functions. Since the solutions are built using the kinetic and potential energy of the given system, they do not have the black box stigma associated with the finite element method. One can clearly identify what parts of the solution are affecting different parts of the system. The global function methods do require that kinetic and potential energies be determined for the entire domain of the system. This limits complexity of possible models. In this case, again, errors are due to approximations of the solution. However, decent convergence of the solution generally requires a smaller number of functions than for the finite element method.
Results demonstrate that similar results are obtained using both methods. The finite element method requires a large number of elements in order to converge to decent results, but because it is solved numerically, it remains computationally less cumbersome then the assumed shape method. The assumed shape method uses much fewer functions in order to converge to reasonable results. It is also much easier to see modeling mistakes because of its intuitive approach and transparent solution methods. Keywords-finite element method; asumed shape method; vibration; modeling; brace-plate system
I.
INTRODUCTION
The debate between approximate vibration solution methods that use global trial functions which extend over the entire domain of a system and local functions, such as finite elements, which extend over small subdomains of the system, has long been debated between engineers. The finite element method has seemingly become the frontrunner because it lends itself well to numerical computer coding, which allows the computation of more precise solutions. However, recently the emergence of powerful symbolic computational software has made it possible to use methods with an increasing number of global trial functions, reiterating interest in these solution methods.
Many methods fall under the category of global trial functions, these include such methods as the Rayleigh-Ritz method, Galerkin method and assumed shaped method. The former two methods are not used because they involve solving the harmonic free response before the application of the chosen trial functions to the problem, whereas the assumed shaped method solves for the harmonic free response after the chosen trial functions are applied. This means that the assumed shape method can formulate the forced response problem, rather than just the free response [2]. The assumed shape method also gives the equation of motion of the system. While all three methods lead to the same eigenvalue problem, which generally converges rapidly, the assumed shape method allows for the development of a more intuitive solution on which this work is based.
Both methods present advantages and disadvantages. Using the finite element method offers significant advantages
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II.
MOTIVATION
z
The purpose of this comparison of vibration solution methods, is to verify the accuracy of a simple brace-plate model used in modeling the soundboard of a string musical instrument [3]. Since very little is known scientifically about the effect that changes in the brace’s thickness and plate’s stiffness has on the overall system, a robust model is required. A typical guitar soundboard can be seen in Fig. 1.
Ly
y
Brace
x
0 h1
Braces
0
h2 Plate
x1 x 2 Lx Figure 2. Test plate with brace across its width
Soundboard
Figure 1. Braced guitar soundboard
The results obtained from such a model will be used in developing an analytical model which can be used to incorporate acoustical consistency into the manufacturing of wooden stringed musical instruments. To do so, stiffness measurements would be taken across the grain for the wooden plate used in the soundboard assembly. These measurements would be input into the analytical model which would calculate optimized brace dimensions. These dimensions could then be input into a numerical milling machine which would cut custom braces for a given plate stiffness. It is therefore extremely important to ensure that developed models produce accurate results. Results obtained from multiple sources help confirm modeling accuracy. III.
METHODOLOGY
In order to build a robust model, the brace-plate interactions on the soundboard must first be understood and it is important to ensure that the model accurately represents theses vibration interactions. To do so, the model is simplified to a single brace mounted onto a plate. Once this simplified model gives light as to these interactions and verifies their accuracy, a more complex model can be built. The simplified model itself consists of an isotropic rectangular plate on which a brace has been positioned across its width as seen in Fig. 2. The plate is considered to be thin enough to use Kirchhoff’s plate theory [4]. 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 [5]. Since clamped edges prevent rotation at the edge, local stiffening occurs. This leads to an increase in the natural frequencies.
A. The Finite Element Method The finite element method can model a complex continuous system by reducing it to local elements for which solutions are known. To do so, the system’s geometry must first be specified. This is done by first specifying keypoints in the model of Fig. 2 from which can be built a three dimensional object. This three dimensional object must then be divided into elements by specifying divisions along all three axes and then meshed. For each element, an element type must be specified. For this work, an eight node brick element having three degrees of freedom at each node, such that there is translation along all three axes, is chosen. Since brick elements are chosen and in order to account for bending of the brace-plate system, multiple bricks must be specified across the thickness of the system. The boundary conditions which are stipulated as simply supported must also be included in the finite element model. In order to do so, translations along the boundary lines must be fixed but rotations must be allowed. Finally, a modal type solution is chosen for the analysis. ANSYS was used to create the finite element model. ANSYS is an engineering simulation software produced by ANSYS, Inc. It specializes in finite element analysis and more specifically for the purpose of this study, finite element modal analysis. The model’s geometry was built using ANSYS’ prep7 and material properties were then given. The results obtained with the finite element model are compared to those obtained using the assumed shape method. B. The Assumed Shape Method The assumed shape method is another technique used to model a continuous system as a discrete one, thereby simplifying the solution. The assumed shape method is used to determine the equation of motion of a system from which an eigenvalue solution may be determined.
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To use the assumed shape method, a finite series for the time-dependent displacement is assumed such that [2]
x Ly
V=
my
mx
w( x, y, t ) = ∑ ∑ φnx ny ( x, y ) ⋅ qnx ny (t )
x 1 + D1 ∫ ∫ U dydx 2 x2 0
nx =1 n y =1
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, ⎛
φn n = sin ⎜ nx ⋅ π ⋅ x y
⎝
⎛ x ⎞ y ⎟ ⋅ sin ⎜⎜ n y ⋅ π ⋅ Lx ⎠ Ly ⎝
⎞ ⎟⎟ ⎠
(2)
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 main strength of the assumed shape method is in the choice of trial functions. In this case, the plate and brace system is thought to be ‘similar’ to a rectangular plate and therefore intuitively, the modeshapes of the plate and brace system would be expected to be modifications of the modeshapes of the underlying system of the plate alone. This kind of intuitive insight can be made mathematically precise by building the solution for the plate and brace system to consist of combinations of building blocks that are modeshapes of the plate alone. In this manner, the way in which the presence of the brace modifies the resulting modeshapes of the combined system can be analyzed analytically, rather than purely numerically as with the finite element method. The assumed shape method is an energy method. Therefore, the expression in (1) will be used in the expression for kinetic and potential energy of the isotropic simply-supported rectangular plate [6]. The plate has been modified by the addition of a brace across its width. To account for the extra material thickness, the energy integrals have been divided into three sections as specified along the x-axis in Fig. 2. The kinetic energy can be expressed as x Ly
T=
x Ly
1 1 1 2 w 2 ρ1 dydx + ∫ ∫ ∫ 20 0 2 x1
∫ w ρ 2
2
0
Lx Ly
1 + ∫ 2 x2
∫ w ρ 2
1
dydx
(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
(4)
L Ly
(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 x-direction respectively and my and ny represent the same in the ydirection.
x Ly
1 2 1 1 D1 ∫ ∫ U dydx + D2 ∫ ∫ U dydx 2 0 0 2 x1 0
where U = ( wxx + wyy ) + 2 (1 −ν ) ( wxy2 − wxx wyy ) . 2
(5)
The subscripts on the displacement terms are partial derivatives in the direction defined by the subscript and the stiffness terms are
Di =
E hi3
12 (1 −ν 2 )
, i = 1, 2
(6)
where E is Young’s modulus and ν is Poisson’s ratio. Once the kinetic and potential energies have been determined, it remains to substitute them into Lagrange’s equations [2] such as, d ⎛ ∂T ⎞ ∂T ∂V ⎜ ⎟− + = Qnx ny , dt ⎜ ∂q nx ny ⎟ ∂qnx ny ∂qnx ny ⎝ ⎠ nx = 1, 2,..., mx and n y = 1, 2,..., m y .
(7)
Since no other forces are acting on the system, the generalized force is Q=0. Lagrange’s equations yield the equations of motion of the system which can be written in matrix form as G G G (8) M q + K q = 0 G where q = ⎡⎣ q11
q12
T
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 G JG q = A cos (ωt + φ ) (9) 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, JG G (K − ω 2 M ) A = 0 (10) 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 the system’s natural
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frequencies and modeshapes. Maple is a symbolic algebra system produced by Waterloo Maple Inc which has the benefit of allowing a user to solve problems symbolically rather than numerically. This gives much greater insight into the internal workings of the theory as it is solved for the given system. Once an algorithm is created satisfying the theory, Maple will create a symbolic solution to the eigenproblem. From there, various properties can be given a value and the solution can be obtained analytically. Both solutions use Cartesian coordinates in their solution approach in order to specify system dimensions.
These material properties are those used for developing both the finite element and assumed shape methods. For stringed musical instruments, it is generally thought that only the lowest natural frequencies are the most important in producing good tone in an instrument [8], therefore all efforts are concentrated on comparing the lower natural frequencies. The dimensions used for both modeling methods are found in table II. TABLE II.
IV.
RESULTS
The material used throughout the analysis is that of Sitka spruce. Material properties for Sitka spruce are obtained from the U.S. Department of Agriculture, Forest Products Laboratory [7]. Since properties between specimens of wood have a high degree of variability, the properties obtained from the Forest Products Laboratory are an average of specimen samplings. The model used in this study is isotropic, therefore the wood’s orthotropic properties are transformed into an isotropic material. To do so, the properties longitudinal to the wood’s grain are used as the material’s overall properties. The properties of wood as an isotropic material are seen in table I. TABLE I. Property μ (kg/m3) E (MPa) ν
MATERIAL PROPERTIES Value 403.2 10890 0.372
Dimension Lx (m) Ly (m) x1 (m) x2 (m) h1 (m) h2 (m)
MODEL DIMENSIONS Value 0.24 0.18 0.114 0.126 0.003 0.015
Using the equations, properties and dimensions presented, the analysis was done using the finite element method in ANSYS and the assumed shaped method in Maple. Results of the natural frequencies for the isotropic simply supported brace-plate model obtained via the assumed shape method for the first five modes are compared to those obtained using the finite element model in table III. Their associated modeshapes are also displayed. The finite element method uses just over 21000 nodes for the model while the assumed shape method uses 6×6 trial functions.
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TABLE III.
COMPARISON OF RESULTS FOR THE ASSUMED SHAPE VS. THE FINITE ELEMENT METHOD Assumed Shape Method
mx
Finite Element Method
my Natural Frequency (rad/s)
Modeshape
Natural Frequency (rad/s)
1
1
4257.89
5394.9
2
1
7132.40
5826.8
3
1
11015.53
10017.9
2
2
11611.43
10340.9
1
2
11831.71
10436.4
The dip in the center of the x-axis for the assumed shape method modeshapes is the location of the brace, which is clearly visible on the finite element method modeshapes. The brace stiffens this area and limits the amount of displacement that can occur. As seen in table III, lower estimates are obtained using the finite element method for all but the fundamental frequency of the isotropic plate. However, all the natural frequencies are within the same range. V.
DISCUSSION
Looking at these results, the first thing that is apparent is how both sets of natural frequencies for each case fall within the same range. This helps justify the similarity of both models. Differences in both models are primarily based on two factors. The first is the different assumptions made for
Modeshape
each case, given that both methods are approximate ones. The second is the level of convergence attained by each model based on the computational limits of the current setup. However, even though these two models use completely different approaches in building the given systems, they both seem to be converging to similar frequencies. It is also evident that the modeshape ordering for both methods has converged to the same result. The fact that both the assumed shape and finite element methods have converged to the same modeshape ordering indicates that a similar solution is obtained for both. This helps in comparing both methods because it demonstrates a rigorous order in the natural frequencies of the system. Although the assumed shape method has not converged to the level of precision in the values of the natural frequencies obtained through the finite element method, both models indicate that they are
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vibrating in the same manner. This follows since both methods are based on minimization theory which dictates that all solutions are over-approximations of the actual values. Therefore, any lower approximation is considered to be closer to the actual value of the natural frequency. In acoustics, it is difficult to get extremely precise frequency results from theoretical calculations because of the number of assumptions necessary to create models that are computationally manageable. It is clear that this aspect needs to be improved in order to produce models which can accurately predict the natural frequencies of a given system. This is ultimately necessary if the goal is to improve the manufacturing consistency of wooden musical instruments. Future work involving precision of the natural frequencies should look to the finite element method. This is clear in the results presented in table III, since the natural frequencies obtained through the finite element method are lower than those obtained via the assumed shape method. Once again, this is true because both methods are based on minimization theory which dictates that all solutions are over-approximations of the actual values. It is interesting to note that the fundamental frequency is found to be lower when calculated using the assumed shape method. This indicates that, in this case the assumed shape method converges faster than does the finite element method. A mesh refinement of the finite element method should help it converge to a more accurate value. An increase in the amount of trial functions used for the assumed shape method would also help to improve the convergence of the natural frequencies.
While producing the models for both methods, it also became obvious that the advantages of the assumed shape method are that any mistakes built into the model become very evident due to the fact that the assumed shape method is solved symbolically. Also, this method was used for the purpose of learning how to vary the thickness of the brace in order to compensate for variations in the plate’s stiffness. The assumed shape method made it much easier to see what needed to be done in order to modify given frequencies/modeshapes as was described in [3]. This is contrary to the black box analysis of the finite element method which gives results of the analysis without describing the details of how those results were achieved. While producing a numeric code for the finite element method specific to this application would probably help in this aspect, the assumed shape method’s intuitive approach and ease of use would still give it the advantage. This is the main reason that for many years, the assumed shape method was the preferred analysis method in the field of aeroelasticity [9]. REFERENCES [1] [2] [3]
[4] [5]
VI.
CONCLUSION
From the results obtained during this analysis, it has been shown that both the assumed shape method and finite element method produce similar results. This being said, the finite element method converges to more accurate results computationally quicker than does the assumed shape method, although the assumed shape method uses less functions than the finite element method and provides greater insight into the construction of the approximate solution.
[6] [7]
[8] [9]
J. N. Reddy, Energy Principles and Variational Methods in Applied Mechanics. Hoboken, NJ: John Wiley & Sons, Inc., 2002. L. Meirovitch, Fundamentals of Vibrations. New York: McGraw-Hill, 2001. P. Dumond and N. Baddour, "Towards improving the manufactured consistency of wooden musical instruments through frequency matching," Transactions of the North American Research Institution of SME, vol. 38, in press. L. Meirovitch, Principles and Techniques of Vibrations. Upper Saddle River, New Jersey: Prentice Hall, 2000. N. H. Fletcher and T. D. Rossing, The Physics of Musical Instruments, 2nd ed. New York: Springer, 1999. S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, 2nd ed. New York: McGraw-Hill, 1959. Forest Products Laboratory (US), Wood Handbook: Wood as an Engineering Material. Madison, Wisconsin.: U.S. Dept. of Agriculture, Forest Service, Forest Products Laboratory, 1999. C. M. Hutchins and D. Voskuil, "Mode Tuning for the Violin Maker," CAS Journal, vol. 2, pp. 5-9, November 1993. R. L. Bisplinghoff, H. Ashley, and R. L. Halfman, Aeroelasticity. Reading, Massachusetts: Addison-Wesley Publishing Company, 1955.
While both methods are good at solving such a model, it is clear that the advantages of the finite element method lie within its ability to model more complex systems. Systems that otherwise cannot be solved using the assumed shape method. In fact, in order to model exact soundboards rather than simplified model, a finite element type method may be necessary. The amount of elements that can be used in the finite element method is also much greater than the assumed shape method. This is in great part due to the fact that the finite element method is solved numerically rather than symbolically which still remains computationally less demanding. This makes it easier for the finite element method to converge to more accurate values.
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