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A simplified finite element method for studying acoustic characteristics inside a car cavity
Journal of Sound and Vibration (1979) 63( I), 6 1-72
A SIMPLIFIED FINITE ELEMENT METHOD FOR STUDYING ACOUSTIC CHARACTERISTICS INSIDE A CAR CAVITY T. L. RICHARDS AND S. K. JHA? School of’Automotive
Studies, Cranfield Institute oj’Technology, England
(Received 9 March 1978, and in revisedform
Cranfield, Bedjord MK43 OAL,
7 September
1978)
A simplified finite element method has been developed for analyzing the acoustic resonances of a prismatic car cavity. Use is made of the non-variant geometric and material properties of such a cavity in one direction (across the width of the car) and the solution of the full three-dimensional problem is related very simply to the solution of a single twodimensional problem, with a consequent reduction of computing effort. Some experimental results obtained for a half-size model of a car cavity are compared with the finite element results. The accuracy 2f the finite element solution is assessed by analyzing a rectangular cavity and a cylindrical cavity, for which analytical solutions are already available.
1.
INTRODUCTION
Excessive noise and vibration are among the major problems facing today’s vehicle designers, and at the School of Automotive Studies some basic work is being carried out aimed at improving our understanding of the relation between noise and the source characteristics. Noise inside a vehicle is primarily caused by the vibrating surfaces enclosing the passenger compartment. Since this is a confined space, its air resonances (normally referred to as cavity resonances) have considerable influence in the noise generation process, especially as they are fairly widely spaced in the frequency range of interest. A knowledge of the acoustic resonances and their modes (i.e., sound pressure distributions) is of considerable value in studying the interaction between the vibrating surfaces and the enclosed cavity, and ultimately in determining the resultant interior noise level. Some experimental work [l] has been done in the past to determine the acoustic resonances of a car cavity, and the associated sound pressure distributions in one plane. Several authors have made an effort to apply numerical techniques such as finite elements [24] and finite differences [S] to determine the resonances of an irregularly shaped cavity, and have been reasonably successful. This paper describes a finite element method for obtaining the acoustic resonances of a car cavity which differs from previous methods in that use is made of the fact that such cavities are approximately prismatic, The two-dimensional solutions are found (by using a two-dimensional finite element subdivision) and from these the three-dimensional solutions are easily calculated. The computation of a two-dimensional finite element solution requires fewer degrees of freedom than a full three-dimensional solution, and is thus more economical in both computer storage and time. Also formulation of two-dimensional elements is usually easier than that of three-dimensional elements, and in this case fairly simple quadratic triangular elements have been used. The paper also includes some experimental results obtained for a half-size model of a car cavity made of iinch thick perspex sheet. The finite element results compare well with these. tNow at Central Research Laboratory, Firestone Tire & Rubber Company, 1200 Firestone Parkway. Akron, Ohio 44317, U.S.A. 61 0022460X/79/050061
+ 12 $02,00/O
@ 1979 Academic
Press Inc. (London)
Limited
62
T. L. RICHARDS
AND S. K. JHA
2. ACOUSTIC FINITE ELEMENTS
The acoustic pressure field in a homogeneous medium is governed by the wave equation [6] v2p = (l/2)
a2jqat2,
(1)
where c is the speed of sound in the medium. For cavity resonance, the pressure is assumed to vary harmonically with time, such that p = P, exp jut, and substitution into equation (1) gives the Helmholtz equation VP,
= - (02/c’) P,.
(2)
Equation (2) is an eigenvalue problem, and solution (with the appropriate boundary conditions) will yield a number of eigenvalues (equal to - w’/c’) and their corresponding eigenfunctions P. The natural frequencies of the cavity can thus be obtained from the eigenvalues, while the eigenfunctions are simply the acoustic modes. In the majority of practical cases, it is necessary to solve equation (2) numerically, and the finite element method is well suited for this purpose. It can be shown (see the Appendix) that a finite element formulation leads to a matrix eigenvalue problem [K](P)
- (co'/c")[A4]{P} = 0.
(3)
{P}isa vector listing the values of the pressure amplitude P, at the element nodes, which uniquely defines P, throughout the cavity. Solution of equation (3) again gives a number (in fact equal to the number of nodal values of P)of resonant frequencies o, and corresponding eigenvectors {P}which describe the mode shapes. Many algorithms for use with digital computers have been devised for solving the matrix eigenvalue problem, and since most subroutine libraries contain a selection of these this subject is not discussed further here.
3. PRISM-SHAPED CAVITIES
In many physical problems the geometry and material properties do not vary along one co-ordinate direction, and if the co-ordinate system is Cartesian one is therefore dealing with prismatic structures or volumes. Zienkiewicz [7] has given an account of how the finite element solution of such problems can be made far more economical than that of general three-dimensional problems. If the problem is described in the co-ordinate system (x, y, z), with geometry and material properties not changing in the z direction, then a two-dimensional finite element subdivision is made in the x-y plane, with variation of the unknown quantity in the z direction being expressed as a sum of Fourier components. Thus, instead of having a single degree of freedom at each node, one now has as many degrees of freedom as there are Fourier components. At first sight it appears that the number of degrees of freedom required will still be approximately equal to that required for a full three-dimensional subdivision, hence requiring the same amount of computing effort. However, because of the mutual orthogonality of some of the Fourier components, the matrix equation describing the whole system often decouples into a number (usually equal to the number of Fourier components) of smaller matrix equations, which can then be solved separately. For most solution routines the storage and time required are proportional to the square and the cube of the number of degrees of freedom, respectively, and so the advantages of the procedure outlined above become immediately apparent. If this method is applied to acoustic cavity resonance. it is found that not only does the
FINITE ELEMENT METHOD FOR A CAR CAVITY
63
overall matrix equation decouple into a number of smaller problems, but each of these is very simply related to the others. This makes it necessary to solve a single two-dimensional problem only, from which all the three-dimensional modes can be calculated. While this derivation is not particularly difficult, it is not included here as the same result can be obtained rather more directly by assuming a separable solution of equation (2) of the form P&C, y)Z(z). It is then found that if a two-dimensional eigenfunction (i.e., pressure amplitude distribution) P,(x, y) and resonant frequency o,, are found which satisfy equation (2). then P,(x, y, z) and w,, are also solutions if On = c &%c~/a~)
+ (w$c2). n = 0,l. 2, . . . ,
(4)
P” = P, cos (nnz/a)
n even,
(5)
Pn = P, sin (nrcz/a)
n odd,
(6)
where a is the length of the cavity in the z direction. The solutions P&c, y) described here have been obtained by using the finite element method, but there is no reason in principle why they should not be evaluated by using some other method.
4. TRIANGULAR
FINITE
ELEMENTS
Although the choice of element shape is more or less arbitrary, and can be made to suit specific problems, triangular elements are used here because (unlike, for example, rectangular elements) they can be made to lit the most general of two-dimensional regions. The simplest triangular element has nodes only at the vertices, and is usually referred to as a “linear triangle”. This is because the variation of the unknown quantity P, is a linear function of the spatial co-ordinates. Thus P, is specified uniquely everywhere within the triangle by the three nodal values, and contours are always straight lines. Triangular elements of higher orders can also be devised [7]. For example, a quadratic in two dimensions needs six parameters to define it, so a quadratic triangle has six nodes (three at the vertices and three on the sides of the triangle), while a cubic triangle has ten nodes. Although the linear triangle requires the simplest formulation, the quadratic triangle has been chosen because it is capable of a better representation of the pressure amplitude distribution P,. The boundary conditions require that dP&?n (the normal component of grad PO) is zero at the surface of the cavity, and this means that contours of P, must be normal to the surface where they meet it. In some cases (for example, when two sides of an element lie on the boundary) the linear triangle cannot satisfy this requirement because contours of P, within it are necessarily straight lines. It is also widely held that, for a given number of degrees of freedom, subdivision into a few complex elements achieves better results than subdivision into a larger number of simple elements [7]. Details of the quadratic triangular element with mid-side nodes are given in the Appendix, along with the element matrices which are assembled to form equation (3). 5. RESULTS
A rectangular cavity of 3 m x 2 m was used as a test case for the finite element computer program, with c = 250 ms-‘. The resonant frequencies for such a two-dimensional cavity
64
T. L. RICHARDS AND S. K. JHA
are given by [6] f = W&/(~x/~x)’ + (q$YY
(7)
where IZ, and Al,,are integers, and Ix and lYare the dimensions of the cavity in the x and Y directions, respectively. Two finite element subdivisions were used, and these are shown in Figure 1. The results are shown in Table 1, and illustrate the increased accuracy of the finer mesh.
X2
Elements Nodes
4x 4 32 Elements 81 Nodes
Figure 1. Finite element subdivisions for the 3 m x 2 m rectangular cavity.
TABLE
The resonant frequencies
1
of a rectangular cavity of section 3 m x 2 m (c = 250 m/s) Resonant
Mode
r
frequencies
(Hz)
Finite element
5
(2 x 2)
(4 x 4)
Exact
1
0
0 1 2 2 0 1
1 1 0 1 2 2
41.8 62.7 76.4 91.0 110.8 135.1 141.3
41.7 62.5 75.2 83.6 105.0 125.4 132.7
41.7 62.5 75.1 83.3 104.2 125.0 131.8
nX
FINITE ELEMENT METHOD FOR A CAR CAVITY
65
Also considered was the case of resonance of a cylindrical cavity. Exact results can be obtained fairly simply by treating the axisymmetric modes only, such that the pressure amplitude P, is a function only of the distance I from the axis of the cylinder, and the resonant frequencies of these are given by f, =
(8)
(U27WlR)B,.
where R is the radius of the cavity and the j?, are the roots of J,(p), the Bessel function of the first kind of order one [S]. The finite element mesh for a cylindrical cavity of 0.4 m radius is shown in Figure 2.
Elements Nodes
Y
_
A
Y
O-4m
60”/ //I
Figure 2. Finite element subdivision for the cylindrical
cavity.
Because we are considering only the axisymmetric modes, attention may be confined to any radial segment of the cylinder and in this case a segment of 60” was arbitrarily chosen. Table 2 shows the results for c = 343 ms- ‘, along with the exact results according to equation (8). It can be seen that accuracy of the first two resonant frequencies is good, while the second two results are rather poor. This decrease in accuracy is to be expected because the modes invariably become more complicated with increasingfrequency, and the finite element mesh is less able to provide a good approximation. When this happens the computed mode shapes tend to exhibit sharp discontinuities at the boundaries between elements, and a poor approximation to the boundary conditions on the surface. Thus by inspecting these modes TABLE 2
The resonant frequencies of a cylindrical cavity (axisymmetric modes only: R = 0.4 m, c = 343 m/s) ~
Finite element (Hz) -__ 526.1 966.8 1414.8 1982.8
Exact (Hz)
-- _____
~~ .~__ 522.9 957.5 1388.4 1818.4
66
T. L. RICHARDS AND S. K. JHA
it is usually possible to gain some idea of the validity of the solution, and if it is suspect a finer mesh must be used. Figure 3 shows the finite element subdivision for a car cavity (half-size). The two-dimensional resonances were computed and these frequencies are shown (marked with asterisks) in Table 3 along with the three-dimensional resonant frequencies obtained by using equation (4) with a = 0.65 m and c = 343 ms- ‘.
44 Elements 109
Figure
Nodes
3. Finite element
subdivision
TABLE
of the car cavity.
3
Naturalfrequencies of a car cavity calculated by usingfinite elements
o.ot
0 1 2
263.8 527.7
0 1 2
175.17 316.7 556.0
0 1 2
309.6.t 406-8 611.8
0 1 2
330.2t 422.7 622.5
0 1 2
427.3t 502.2 679.0
0 1 2
455.17 526.1 696.8
7 Two-dimensional
frequencies.
FINITE ELEMENT METHOD
67
FOR A CAR CAVITY
The first five two-dimensional modes are shown in Figures 4-8 (these are contour plots of pressure amplitude in arbitrary units) and the three-dimensional modes are related to these by equations (5) and (6). Some experimental results were obtained for a cavity of this shape [9], and a photograph
Figure 4. Mode at 175.1 Hz. *, Measured
2
position
2
I I I I I
of nodal plane.
0
-I : I I I I
i /I /
//
i I I I I I \ , I
Figure
5. Mode at 309.6 Hz
Figure
6. Mode at 330.2 Hz
68
T. L. RICHARDS -I
0
AND S. K. JHA I I I
Figure
7. Mode at 427.3 Hz
Figure
8. Mode
at 4551 Hz.
of the experimental model (made of i inch thick perspex) is shown in Figure 9. Figure 10 shows the sound pressure response at the rear head position due to a vibrating disc of 55 cm diameter on the floor of the cavity. For comparison the vertical lines are drawn in Figure 10 to identify the eleven lowest resonant frequencies listed in Table 3, and Table 4 lists the calculated and measured values. Also shown in Figure 4 are measured positions of the nodal plane for the mode at 175 Hz.
6. CONCLUSIONS
1. The acoustic resonances of a car cavity have been calculated by using a method based on finite elements. Considerable simplification has been achieved by taking into account the prismatic geometry of the cavity, and the solution to the full three-dimensional problem can be obtained easily from the solution to a smaller two-dimensional problem. 2. Six-noded triangular elements with quadratic shape functions were used in the analysis, with subdivision irrthe x-y plane of the cavity. Good agreement was found with experimental results. 3. Test cases with a rectangular cavity show that the first order modes (only one node in any co-ordinate direction) can be accurately calculated with a coarse 2 x 2 mesh,
FOR A CAR CAVITY
FINITE ELEMENTMETHOD
69
Figure 9. Cavity with rigid walls showing source and receiver.
while for higher order modes better accuracy can be obtained by using a finer 4 x 4 mesh. 4. The method was also applied to a cylindrical cavity to determine the axisymmetric modes. Agreement between these results and the exact solution is good for the first two modes, but rather poor for the higher modes. Such deterioration with increasing frequency is an inherent feature of the finite element method, and accuracy can be maintained only by using finer meshes. It is concluded that the use of finite elements to solve acoustic cavity
s:
I
&so-
TJ
330.2 316.7
s
iz 263.8 309 6 50 100
I 200
427.: 422.s 406.1 455.
: 3 00 Frequency
(Hz)
Figure 10. Sound pressure response inside the model car compared with theoretical resonant frequencies
T. L. RICHARDS AND S. K. JHA
70
TABLE 4
Comparison of calculated and measured natural frequencies of the car cavity Finite/Element (Hz)
Measured (Hz)
175.1 263.8 309.6 316.7 330.2 406.8 422.1 427.3 455.1 502.2 526.1
175 264 309 317 322 407 421 426 452 500 522
resonance problems is most economical
when the frequency range of interest covers only
a few of the lowest resonances.
ACKNOWLEDGMENTS
The authors gratefully acknowledge the financial support given by the Science Research Council (U.K.) for interior noise study of vehicles. The work reported here forms a part of the above investigation.
REFERENCES
1. T. PRIEDE and S. K. JHA 1970 Journal of Automotive Engineering l(5), 17-21. Low frequency noise in cars-its origin and elimination. 2. T. SHUKU and K. ISHIHARA 1973 Journal of Sound and Vibration 29, 67-76. The analysis of the acoustic field in irregularly shaped rooms by the finite element method. 3. M. PETYT, J. LEA and G. H. KOOPMANN1976 Journal of Sound and Vibration 45, 495-502. A finite element method for determining the acoustic modes of irregular shaped cavities. 4. A. CRAGGS1972 Journal of Soundand Vibration 23,331-339. The use of simple three-dimensional acoustic finite elements for determining the natural modes and frequencies of complex shaped enclosures. 5. G. JENNEQUIN1971 Proceedings of the Joint Symposium on Vibration and Noise in Motor Vehicles, Institution of Mechanical Engineers, London, Jury 1971, Paper No. C108/71, 132-137. Is the computation of noise level inside a car feasible? 6. L. B. KINSLERand A. R. FREY 1962 Fundamental of Acoustics. New York : John Wiley & Sons, Inc. See chapter 7. 7. 0. C. ZIENKIEWICZ1971 The Finite Element Method in Engineering Science. London: McGrawHill Book Company, Limited, second edition. 8. M. ABRAMoWlTzand I. A. STEGuN(Editors) 1968 Handbook of Mathematical Functions. New York : Dover Publications, Inc. 9. S. K. JHA and N. CHEILAS1976 American Society of Mechanical Engineers Paper No. 76-WA/DE-l. Acoustic characteristics of a car cavity and estimation of interior sound field produced by vibrating panel.
FINITE
ELEMENT
METHOD
71
FOR A CAR CAVITY
APPENDIX: FINITE ELEMENT FORMULATION RESONANCE
FOR ACOUSTIC CAVITY
Acoustic cavity resonance is governed by the equation v?;
= (oZ/c2) P,.
(Al)
Since the surface of the cavity is assumed to be rigid, then the normal component of particle velocity here must be zero. In the case of harmonic vibration the particle velocity is proportional to the vector VP,, and if LIP,/& is the normal component of VP, at the surface, then
ap,pn
= 6
642)
everywhere on the surface of the cavity. Equations (Al) and (A2) constitute an eigenvalue problem, the solution of which yields a number of discrete eigenvalues CD,and their corresponding eigenfunctions (or modes) P,. Euler’s theorem of variational calculus shows how a variational statement can be expressed as a differential equation with boundary conditions, and it can be shown that equations (Al) and (A2) are equivalent to a variational statement that requires minimization of the functional X = t
(VP;VP,)do s”
- +$
P;du. s I,
(A3)
The integrations are carried out over the volume of the cavity. In this form the problem is suitable for solution by finite element techniques, resulting in the matrix equation [K](P)
- (fD’/c”)[M]{P)
= 0.
(A4)
The matrices [K] and [M] are obtained as an assembly of individually calculated element matrices in a procedure identical to those used in matrix structural analysis, and many algorithms (for use with digital computers) are available for solving equation (A4). Zienkiewicz [7] has given an account of both Euler’s theorem and the finite element formulation of field problems of this nature. Quadratic triangular finite elements have six nodes and associated with each element are the element matrices [k]” and [ml=, which are assembled to form equation (A4). The matrix [k]’ can be split into two separate matrices, such that [k]’ = [kJ
+ [k$
and the [kJ’ matrix is given by
=---
4:
_
0
$bb,&
$bP3b2 $b; + b,b, + b:)
$b,b,+b,b,+b:+2b,b,)
$b,b,+b,b,+b;+2b,b,)
;blb,
0
:b,b,
$b,b,+b,b,+b:+2b,b,)
%b:+b,b,+b;)
$b,bz+b,b,+b;+26,b2)
:b,b2
:b,b,
0
$b,b,+b,b,+b:+2b,b,)
$b,b,+b,b,+b;+2b,b,)
$(b;+b,b,+b;)
(4
72
T. L. RICHARDS
AND S. K. JHA
where A is the area of the triangular element. a is the width of the cavity, and b, = Y, - y3>b, = y3 - yr, b, = y1 - yz. The [k,]” matrix has exactly the same form, but the entries are expressed in terms of etc., rather than b,, etc., where
c,,
c1 = xj - x2, c2 = x1 - x3, c3 = x2 - x1. (x1, YJ (x2, YJ and (xS9Y,) are the co-ordinates of the vertices of the triangle. The [ml’ matrix is given by -l/180
-l/180
-l/45
0
0
l/30
-l/180
0
0
-l/45
-;/“’
i/30
;,45
y,45
- 1/45 4145
- l/45
0
4/45
8/45
4/45
0
- l/45
4145
4145
8145
646)
A SIMPLIFIED FINITE ELEMENT METHOD FOR STUDYING ACOUSTIC CHARACTERISTICS INSIDE A CAR CAVITY T. L. RICHARDS AND S. K. JHA? School of’Automotive
Studies, Cranfield Institute oj’Technology, England
(Received 9 March 1978, and in revisedform
Cranfield, Bedjord MK43 OAL,
7 September
1978)
A simplified finite element method has been developed for analyzing the acoustic resonances of a prismatic car cavity. Use is made of the non-variant geometric and material properties of such a cavity in one direction (across the width of the car) and the solution of the full three-dimensional problem is related very simply to the solution of a single twodimensional problem, with a consequent reduction of computing effort. Some experimental results obtained for a half-size model of a car cavity are compared with the finite element results. The accuracy 2f the finite element solution is assessed by analyzing a rectangular cavity and a cylindrical cavity, for which analytical solutions are already available.
1.
INTRODUCTION
Excessive noise and vibration are among the major problems facing today’s vehicle designers, and at the School of Automotive Studies some basic work is being carried out aimed at improving our understanding of the relation between noise and the source characteristics. Noise inside a vehicle is primarily caused by the vibrating surfaces enclosing the passenger compartment. Since this is a confined space, its air resonances (normally referred to as cavity resonances) have considerable influence in the noise generation process, especially as they are fairly widely spaced in the frequency range of interest. A knowledge of the acoustic resonances and their modes (i.e., sound pressure distributions) is of considerable value in studying the interaction between the vibrating surfaces and the enclosed cavity, and ultimately in determining the resultant interior noise level. Some experimental work [l] has been done in the past to determine the acoustic resonances of a car cavity, and the associated sound pressure distributions in one plane. Several authors have made an effort to apply numerical techniques such as finite elements [24] and finite differences [S] to determine the resonances of an irregularly shaped cavity, and have been reasonably successful. This paper describes a finite element method for obtaining the acoustic resonances of a car cavity which differs from previous methods in that use is made of the fact that such cavities are approximately prismatic, The two-dimensional solutions are found (by using a two-dimensional finite element subdivision) and from these the three-dimensional solutions are easily calculated. The computation of a two-dimensional finite element solution requires fewer degrees of freedom than a full three-dimensional solution, and is thus more economical in both computer storage and time. Also formulation of two-dimensional elements is usually easier than that of three-dimensional elements, and in this case fairly simple quadratic triangular elements have been used. The paper also includes some experimental results obtained for a half-size model of a car cavity made of iinch thick perspex sheet. The finite element results compare well with these. tNow at Central Research Laboratory, Firestone Tire & Rubber Company, 1200 Firestone Parkway. Akron, Ohio 44317, U.S.A. 61 0022460X/79/050061
+ 12 $02,00/O
@ 1979 Academic
Press Inc. (London)
Limited
62
T. L. RICHARDS
AND S. K. JHA
2. ACOUSTIC FINITE ELEMENTS
The acoustic pressure field in a homogeneous medium is governed by the wave equation [6] v2p = (l/2)
a2jqat2,
(1)
where c is the speed of sound in the medium. For cavity resonance, the pressure is assumed to vary harmonically with time, such that p = P, exp jut, and substitution into equation (1) gives the Helmholtz equation VP,
= - (02/c’) P,.
(2)
Equation (2) is an eigenvalue problem, and solution (with the appropriate boundary conditions) will yield a number of eigenvalues (equal to - w’/c’) and their corresponding eigenfunctions P. The natural frequencies of the cavity can thus be obtained from the eigenvalues, while the eigenfunctions are simply the acoustic modes. In the majority of practical cases, it is necessary to solve equation (2) numerically, and the finite element method is well suited for this purpose. It can be shown (see the Appendix) that a finite element formulation leads to a matrix eigenvalue problem [K](P)
- (co'/c")[A4]{P} = 0.
(3)
{P}isa vector listing the values of the pressure amplitude P, at the element nodes, which uniquely defines P, throughout the cavity. Solution of equation (3) again gives a number (in fact equal to the number of nodal values of P)of resonant frequencies o, and corresponding eigenvectors {P}which describe the mode shapes. Many algorithms for use with digital computers have been devised for solving the matrix eigenvalue problem, and since most subroutine libraries contain a selection of these this subject is not discussed further here.
3. PRISM-SHAPED CAVITIES
In many physical problems the geometry and material properties do not vary along one co-ordinate direction, and if the co-ordinate system is Cartesian one is therefore dealing with prismatic structures or volumes. Zienkiewicz [7] has given an account of how the finite element solution of such problems can be made far more economical than that of general three-dimensional problems. If the problem is described in the co-ordinate system (x, y, z), with geometry and material properties not changing in the z direction, then a two-dimensional finite element subdivision is made in the x-y plane, with variation of the unknown quantity in the z direction being expressed as a sum of Fourier components. Thus, instead of having a single degree of freedom at each node, one now has as many degrees of freedom as there are Fourier components. At first sight it appears that the number of degrees of freedom required will still be approximately equal to that required for a full three-dimensional subdivision, hence requiring the same amount of computing effort. However, because of the mutual orthogonality of some of the Fourier components, the matrix equation describing the whole system often decouples into a number (usually equal to the number of Fourier components) of smaller matrix equations, which can then be solved separately. For most solution routines the storage and time required are proportional to the square and the cube of the number of degrees of freedom, respectively, and so the advantages of the procedure outlined above become immediately apparent. If this method is applied to acoustic cavity resonance. it is found that not only does the
FINITE ELEMENT METHOD FOR A CAR CAVITY
63
overall matrix equation decouple into a number of smaller problems, but each of these is very simply related to the others. This makes it necessary to solve a single two-dimensional problem only, from which all the three-dimensional modes can be calculated. While this derivation is not particularly difficult, it is not included here as the same result can be obtained rather more directly by assuming a separable solution of equation (2) of the form P&C, y)Z(z). It is then found that if a two-dimensional eigenfunction (i.e., pressure amplitude distribution) P,(x, y) and resonant frequency o,, are found which satisfy equation (2). then P,(x, y, z) and w,, are also solutions if On = c &%c~/a~)
+ (w$c2). n = 0,l. 2, . . . ,
(4)
P” = P, cos (nnz/a)
n even,
(5)
Pn = P, sin (nrcz/a)
n odd,
(6)
where a is the length of the cavity in the z direction. The solutions P&c, y) described here have been obtained by using the finite element method, but there is no reason in principle why they should not be evaluated by using some other method.
4. TRIANGULAR
FINITE
ELEMENTS
Although the choice of element shape is more or less arbitrary, and can be made to suit specific problems, triangular elements are used here because (unlike, for example, rectangular elements) they can be made to lit the most general of two-dimensional regions. The simplest triangular element has nodes only at the vertices, and is usually referred to as a “linear triangle”. This is because the variation of the unknown quantity P, is a linear function of the spatial co-ordinates. Thus P, is specified uniquely everywhere within the triangle by the three nodal values, and contours are always straight lines. Triangular elements of higher orders can also be devised [7]. For example, a quadratic in two dimensions needs six parameters to define it, so a quadratic triangle has six nodes (three at the vertices and three on the sides of the triangle), while a cubic triangle has ten nodes. Although the linear triangle requires the simplest formulation, the quadratic triangle has been chosen because it is capable of a better representation of the pressure amplitude distribution P,. The boundary conditions require that dP&?n (the normal component of grad PO) is zero at the surface of the cavity, and this means that contours of P, must be normal to the surface where they meet it. In some cases (for example, when two sides of an element lie on the boundary) the linear triangle cannot satisfy this requirement because contours of P, within it are necessarily straight lines. It is also widely held that, for a given number of degrees of freedom, subdivision into a few complex elements achieves better results than subdivision into a larger number of simple elements [7]. Details of the quadratic triangular element with mid-side nodes are given in the Appendix, along with the element matrices which are assembled to form equation (3). 5. RESULTS
A rectangular cavity of 3 m x 2 m was used as a test case for the finite element computer program, with c = 250 ms-‘. The resonant frequencies for such a two-dimensional cavity
64
T. L. RICHARDS AND S. K. JHA
are given by [6] f = W&/(~x/~x)’ + (q$YY
(7)
where IZ, and Al,,are integers, and Ix and lYare the dimensions of the cavity in the x and Y directions, respectively. Two finite element subdivisions were used, and these are shown in Figure 1. The results are shown in Table 1, and illustrate the increased accuracy of the finer mesh.
X2
Elements Nodes
4x 4 32 Elements 81 Nodes
Figure 1. Finite element subdivisions for the 3 m x 2 m rectangular cavity.
TABLE
The resonant frequencies
1
of a rectangular cavity of section 3 m x 2 m (c = 250 m/s) Resonant
Mode
r
frequencies
(Hz)
Finite element
5
(2 x 2)
(4 x 4)
Exact
1
0
0 1 2 2 0 1
1 1 0 1 2 2
41.8 62.7 76.4 91.0 110.8 135.1 141.3
41.7 62.5 75.2 83.6 105.0 125.4 132.7
41.7 62.5 75.1 83.3 104.2 125.0 131.8
nX
FINITE ELEMENT METHOD FOR A CAR CAVITY
65
Also considered was the case of resonance of a cylindrical cavity. Exact results can be obtained fairly simply by treating the axisymmetric modes only, such that the pressure amplitude P, is a function only of the distance I from the axis of the cylinder, and the resonant frequencies of these are given by f, =
(8)
(U27WlR)B,.
where R is the radius of the cavity and the j?, are the roots of J,(p), the Bessel function of the first kind of order one [S]. The finite element mesh for a cylindrical cavity of 0.4 m radius is shown in Figure 2.
Elements Nodes
Y
_
A
Y
O-4m
60”/ //I
Figure 2. Finite element subdivision for the cylindrical
cavity.
Because we are considering only the axisymmetric modes, attention may be confined to any radial segment of the cylinder and in this case a segment of 60” was arbitrarily chosen. Table 2 shows the results for c = 343 ms- ‘, along with the exact results according to equation (8). It can be seen that accuracy of the first two resonant frequencies is good, while the second two results are rather poor. This decrease in accuracy is to be expected because the modes invariably become more complicated with increasingfrequency, and the finite element mesh is less able to provide a good approximation. When this happens the computed mode shapes tend to exhibit sharp discontinuities at the boundaries between elements, and a poor approximation to the boundary conditions on the surface. Thus by inspecting these modes TABLE 2
The resonant frequencies of a cylindrical cavity (axisymmetric modes only: R = 0.4 m, c = 343 m/s) ~
Finite element (Hz) -__ 526.1 966.8 1414.8 1982.8
Exact (Hz)
-- _____
~~ .~__ 522.9 957.5 1388.4 1818.4
66
T. L. RICHARDS AND S. K. JHA
it is usually possible to gain some idea of the validity of the solution, and if it is suspect a finer mesh must be used. Figure 3 shows the finite element subdivision for a car cavity (half-size). The two-dimensional resonances were computed and these frequencies are shown (marked with asterisks) in Table 3 along with the three-dimensional resonant frequencies obtained by using equation (4) with a = 0.65 m and c = 343 ms- ‘.
44 Elements 109
Figure
Nodes
3. Finite element
subdivision
TABLE
of the car cavity.
3
Naturalfrequencies of a car cavity calculated by usingfinite elements
o.ot
0 1 2
263.8 527.7
0 1 2
175.17 316.7 556.0
0 1 2
309.6.t 406-8 611.8
0 1 2
330.2t 422.7 622.5
0 1 2
427.3t 502.2 679.0
0 1 2
455.17 526.1 696.8
7 Two-dimensional
frequencies.
FINITE ELEMENT METHOD
67
FOR A CAR CAVITY
The first five two-dimensional modes are shown in Figures 4-8 (these are contour plots of pressure amplitude in arbitrary units) and the three-dimensional modes are related to these by equations (5) and (6). Some experimental results were obtained for a cavity of this shape [9], and a photograph
Figure 4. Mode at 175.1 Hz. *, Measured
2
position
2
I I I I I
of nodal plane.
0
-I : I I I I
i /I /
//
i I I I I I \ , I
Figure
5. Mode at 309.6 Hz
Figure
6. Mode at 330.2 Hz
68
T. L. RICHARDS -I
0
AND S. K. JHA I I I
Figure
7. Mode at 427.3 Hz
Figure
8. Mode
at 4551 Hz.
of the experimental model (made of i inch thick perspex) is shown in Figure 9. Figure 10 shows the sound pressure response at the rear head position due to a vibrating disc of 55 cm diameter on the floor of the cavity. For comparison the vertical lines are drawn in Figure 10 to identify the eleven lowest resonant frequencies listed in Table 3, and Table 4 lists the calculated and measured values. Also shown in Figure 4 are measured positions of the nodal plane for the mode at 175 Hz.
6. CONCLUSIONS
1. The acoustic resonances of a car cavity have been calculated by using a method based on finite elements. Considerable simplification has been achieved by taking into account the prismatic geometry of the cavity, and the solution to the full three-dimensional problem can be obtained easily from the solution to a smaller two-dimensional problem. 2. Six-noded triangular elements with quadratic shape functions were used in the analysis, with subdivision irrthe x-y plane of the cavity. Good agreement was found with experimental results. 3. Test cases with a rectangular cavity show that the first order modes (only one node in any co-ordinate direction) can be accurately calculated with a coarse 2 x 2 mesh,
FOR A CAR CAVITY
FINITE ELEMENTMETHOD
69
Figure 9. Cavity with rigid walls showing source and receiver.
while for higher order modes better accuracy can be obtained by using a finer 4 x 4 mesh. 4. The method was also applied to a cylindrical cavity to determine the axisymmetric modes. Agreement between these results and the exact solution is good for the first two modes, but rather poor for the higher modes. Such deterioration with increasing frequency is an inherent feature of the finite element method, and accuracy can be maintained only by using finer meshes. It is concluded that the use of finite elements to solve acoustic cavity
s:
I
&so-
TJ
330.2 316.7
s
iz 263.8 309 6 50 100
I 200
427.: 422.s 406.1 455.
: 3 00 Frequency
(Hz)
Figure 10. Sound pressure response inside the model car compared with theoretical resonant frequencies
T. L. RICHARDS AND S. K. JHA
70
TABLE 4
Comparison of calculated and measured natural frequencies of the car cavity Finite/Element (Hz)
Measured (Hz)
175.1 263.8 309.6 316.7 330.2 406.8 422.1 427.3 455.1 502.2 526.1
175 264 309 317 322 407 421 426 452 500 522
resonance problems is most economical
when the frequency range of interest covers only
a few of the lowest resonances.
ACKNOWLEDGMENTS
The authors gratefully acknowledge the financial support given by the Science Research Council (U.K.) for interior noise study of vehicles. The work reported here forms a part of the above investigation.
REFERENCES
1. T. PRIEDE and S. K. JHA 1970 Journal of Automotive Engineering l(5), 17-21. Low frequency noise in cars-its origin and elimination. 2. T. SHUKU and K. ISHIHARA 1973 Journal of Sound and Vibration 29, 67-76. The analysis of the acoustic field in irregularly shaped rooms by the finite element method. 3. M. PETYT, J. LEA and G. H. KOOPMANN1976 Journal of Sound and Vibration 45, 495-502. A finite element method for determining the acoustic modes of irregular shaped cavities. 4. A. CRAGGS1972 Journal of Soundand Vibration 23,331-339. The use of simple three-dimensional acoustic finite elements for determining the natural modes and frequencies of complex shaped enclosures. 5. G. JENNEQUIN1971 Proceedings of the Joint Symposium on Vibration and Noise in Motor Vehicles, Institution of Mechanical Engineers, London, Jury 1971, Paper No. C108/71, 132-137. Is the computation of noise level inside a car feasible? 6. L. B. KINSLERand A. R. FREY 1962 Fundamental of Acoustics. New York : John Wiley & Sons, Inc. See chapter 7. 7. 0. C. ZIENKIEWICZ1971 The Finite Element Method in Engineering Science. London: McGrawHill Book Company, Limited, second edition. 8. M. ABRAMoWlTzand I. A. STEGuN(Editors) 1968 Handbook of Mathematical Functions. New York : Dover Publications, Inc. 9. S. K. JHA and N. CHEILAS1976 American Society of Mechanical Engineers Paper No. 76-WA/DE-l. Acoustic characteristics of a car cavity and estimation of interior sound field produced by vibrating panel.
FINITE
ELEMENT
METHOD
71
FOR A CAR CAVITY
APPENDIX: FINITE ELEMENT FORMULATION RESONANCE
FOR ACOUSTIC CAVITY
Acoustic cavity resonance is governed by the equation v?;
= (oZ/c2) P,.
(Al)
Since the surface of the cavity is assumed to be rigid, then the normal component of particle velocity here must be zero. In the case of harmonic vibration the particle velocity is proportional to the vector VP,, and if LIP,/& is the normal component of VP, at the surface, then
ap,pn
= 6
642)
everywhere on the surface of the cavity. Equations (Al) and (A2) constitute an eigenvalue problem, the solution of which yields a number of discrete eigenvalues CD,and their corresponding eigenfunctions (or modes) P,. Euler’s theorem of variational calculus shows how a variational statement can be expressed as a differential equation with boundary conditions, and it can be shown that equations (Al) and (A2) are equivalent to a variational statement that requires minimization of the functional X = t
(VP;VP,)do s”
- +$
P;du. s I,
(A3)
The integrations are carried out over the volume of the cavity. In this form the problem is suitable for solution by finite element techniques, resulting in the matrix equation [K](P)
- (fD’/c”)[M]{P)
= 0.
(A4)
The matrices [K] and [M] are obtained as an assembly of individually calculated element matrices in a procedure identical to those used in matrix structural analysis, and many algorithms (for use with digital computers) are available for solving equation (A4). Zienkiewicz [7] has given an account of both Euler’s theorem and the finite element formulation of field problems of this nature. Quadratic triangular finite elements have six nodes and associated with each element are the element matrices [k]” and [ml=, which are assembled to form equation (A4). The matrix [k]’ can be split into two separate matrices, such that [k]’ = [kJ
+ [k$
and the [kJ’ matrix is given by
=---
4:
_
0
$bb,&
$bP3b2 $b; + b,b, + b:)
$b,b,+b,b,+b:+2b,b,)
$b,b,+b,b,+b;+2b,b,)
;blb,
0
:b,b,
$b,b,+b,b,+b:+2b,b,)
%b:+b,b,+b;)
$b,bz+b,b,+b;+26,b2)
:b,b2
:b,b,
0
$b,b,+b,b,+b:+2b,b,)
$b,b,+b,b,+b;+2b,b,)
$(b;+b,b,+b;)
(4
72
T. L. RICHARDS
AND S. K. JHA
where A is the area of the triangular element. a is the width of the cavity, and b, = Y, - y3>b, = y3 - yr, b, = y1 - yz. The [k,]” matrix has exactly the same form, but the entries are expressed in terms of etc., rather than b,, etc., where
c,,
c1 = xj - x2, c2 = x1 - x3, c3 = x2 - x1. (x1, YJ (x2, YJ and (xS9Y,) are the co-ordinates of the vertices of the triangle. The [ml’ matrix is given by -l/180
-l/180
-l/45
0
0
l/30
-l/180
0
0
-l/45
-;/“’
i/30
;,45
y,45
- 1/45 4145
- l/45
0
4/45
8/45
4/45
0
- l/45
4145
4145
8145
646)