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Acoustic power radiation from a rectangular plate with mixed boundary conditions
T h i n - W a l l e d S t r u c t u r e s 18 (1994) 3-22
Acoustic Power Radiation from a Rectangular Plate with Mixed Boundary Conditions
M d . K. R a h m a n & A. Ertas Department of Mechanical Engineering, Texas Tech University, Lubbock, Texas, USA
ABSTRACT The total average acoustic power radiation from one side of a baffled rectangular flat plate, vibrating in its natural modes under two opposing ends fixed and the other two ends simply supported boundary conditions, has been investigated. Numerical results on the acoustic power radiation by the plate have been obtained for various mode orders, aspect ratios, and thickness ratios. Expressions for the acoustic power radiation by plates under specific edge restraints have been derived in terms of the far-field acoustic pressure distribution.
NOMENCLATURE Half of the length of the plate b = Half of the width of the plate C = Velocity of sound in air E = Modulus of elasticity of the plate material f = Frequency of vibration of the plate g -- Acceleration due to gravity G x = A function of m, factor of the dimensionless frequency factor a , = A function of n, factor of the dimensionless frequency factor H x = A function of m, factor of the dimensionless frequency factor A function of n, factor of the dimensionless frequency factor 4 = A function of m, factor of the dimensionless frequency factor Jy A function of n, factor of the dimensionless frequency factor a
3
4
Md. K. Rahman, A. Ertas
h K m n P Pw R Ra
= = = = = -= =
Rt
---
r t W w x, y, z 7
= = = -= = = -= = =
O(x) 0(y) 2 2f p
Pm
•
v = R, 0, ~ = {o =
Thickness of plate Acoustic wave number N u m b e r of nodal lines in x direction N u m b e r of nodal lines in y direction Acoustic pressure distribution Total acoustic power radiation Distance of the receiving point from the center of the plate Aspect ratio of the plate, b/a Thickness ratio of the plate, h/a Distance of the receiving point from the elemental area Time Amplitude of vibration Instantaneous displacement of the plate surface Cartesian coordinate system Roots of the equation, tan(7/2) T tanh(7/2) = 0 Displacement function in x direction Displacement function in y direction Wavelength of sound Dimensionless frequency factor Density of the surrounding medium Density of the plate material Poisson's ratio Spherical coordinate system Circular frequency of the plate vibration
INTRODUCTION With the growth of mechanization in modern life, the problem of noise is also growing steadily. The mechanical and electrical equipment which are an integral part of modern living are also the sources of many acoustical problems. Nearly every piece of mechanical equipment in a building is a source of sound and vibration. The size and complexity of machines, increasing road traffic, etc., contribute to the ever-increasing level of noise. The quality of life is being degraded by these noise sources. The problem is already recognized as one of primary importance and demands immediate attention. W o r k on the study of the vibration characteristics of rectangular plates began in the last part of the 19th century when it was realized that a large number of structural systems in civil, mechanical, and aerospace engineer-
Acoustic power radiation from a rectangular plate
5
ing are composed of these plate components. The dynamic behavior of these structures was first studied by Rayleigh. He developed the fundamental equations governing the vibration of plates. Since the middle of the 20th century, noise has been recognized as a source of environmental pollution which ultimately affects human comfort. From that time, research in this field has attracted much attention. The study of acoustic radiation characteristics of flat plates has not yet reached a satisfactory level. The vibration of plates was studied by many investigators in the past. Most of the initiators confined their study to the determination of the frequency of vibrating plates. In 1954, Warburton ~ derived approximate expressions for the frequencies of all the modes of vibration of isotropic plates subjected to any combination of free, simply supported or clamped edges. He applied the Rayleigh method, assuming that the deflections of the plates could be represented by suitable characteristic functions satisfying the boundary conditions. 2 In his analysis, Warburton first developed a set of beam functions satisfying the edge conditions of beams and applied these functions to the vibration of plates. He expressed the frequency in terms of a dimensionless frequency factor, which in turn is a function of the mode shapes of the vibrating plates. Although there has been much analysis of the vibration of solid structures such as rectangular plates, the study of acoustic radiation due to their vibration of flat plates is rather lacking. In 1954, Skudrzyk 3"4 obtained the solution to the radiation of power from an elastic plate for a time-harmonic point force. He used the classical plate theory. The solutions for the acoustic radiated pressure from a point-excited plate using the classical theory was given by Feit 5 and Skudrzyk. 4 The radiation impedance of membranes and plates of finite width, carrying a traveling wave, was calculated by Lowenthal and Tournois. 6 The method used led to a relating simple analytic expression. The problem of acoustic coupling of these strips with the fluid propagating medium was then examined. An expression was derived for the acoustic power radiated by an infinite uniform isotropic plate excited by a time-harmonic moment acting at a point on the plate by T h o m p s o n and Rattayya 7 in 1964. The effect of radiation loading was included. Results were obtained that were valid below the coincidence frequency. Tsakonas et al. 8 determined the acoustic radiation of an infinite plate excited by the field of a ship propeller both by theoretical and numerical methods. In 1967, Maidanik 9 computed the farfield acoustic pressure generated by an infinite plate driven by a point and a line force. The plate considered here was backed by a fluid medium that was terminated by a parallel infinite baffle. He also applied the same methodology to a coated infinite plate. Magrab and Reader 1° obtained
6
Md. K. Rahman, A. Ertas
an expression for the far-field pressure radiated by a thin elastic plate bounded on one side by a fluid medium when the plate was excited by a time-varying point load. In order to discuss the sound radiation from a direct-radiator loudspeaker in the high-frequency region, an elastically supported circular plate in an infinite baffle was used by Suzuki and Tichy. ~ The normal modes of this plate were consistently treated from the free boundary to the simply supported boundary, which were the limiting cases of the elastically supported boundary, by defining a dimensionless parameter representing the ratio of the edge stiffness to the bending stiffness of the plate. To improve the high-frequency prediction of elastic plates, Feit ~2employed the Timoshenko-Miudlin theory for such a prediction. This theory adds shear deformation and rotary inertia to the classical flexural theory. There are two dispersion curves for this plate. The flexural branch (acoustic) has phase and group velocities approaching the Rayleigh velocity of the plate material as the frequency increases. Sound radiation from a beam-reinforced plate, excited by point or line forces, has been investigated by a few authors. Feit and Saurenman ~3 analyzed the acoustic radiation of a point-excited plate reinforced with a beam, but confined their interests to high frequencies. Gorman t4 obtained the solution for a plate reinforced by many beams and excited by a line force parallel to the beam. His solution thus makes the beam's reaction on the plate as purely rotary and transverse impedances without a flexural wave traveling in the beams. Ahmed ~5 and Mandal ~6 studied the acoustic power radiation from one side of a baffled rectangular isotropic plate for different thickness ratio, aspect ratio, and mode orders. The problem of acoustic radiation from baffled finite elastic plates driven by multiple random point forces was analyzed by Peng and Keltie. j7 Expressions for the surface acoustic intensity and radiated sound power were derived neglecting the fluid loading. By assuming the plates to be lightly damped, and neglecting the modal coupling effects, approximate solutions for the acoustic intensity and sound power were obtained. An experimental study was also carried out to measure the surface acoustic intensity patterns and to compare them with the analytical results. As appears from the foregoing survey, the majority of the research works in the field of vibration noise relate to the natural frequency of rectangular plates. Not much work has been done to analyze the acoustic radiation characteristics of rectangular flat plates vibrating in their natural modes. Work on the natural frequency and acoustic radiation of rectangular plates is continuing and has a long way to go before full understanding is achieved.
Acoustic power radiation from a rectangular plate
7
The main objective of this research is to study the noise-generating characteristics of a rectangular flat plate subjected to two opposing ends fixed and the other two ends simply supported boundary conditions.
MATHEMATICAL MODELING In the analysis, the following assumptions have been made: 1. The thickness of the plate is small compared to the dimensions, and the thickness is uniform. 2. The plate is free from applied load. 3. The mid-plane remains neutral during bending. 4. The cross-sectional plane remains plane after deformation. 5. The plate is isotropic, elastic, and homogeneous. 6. Deflections are small in comparison with the thickness of the plate. 7. The normal stresses in the direction transverse to the plate can be neglected. These stresses must be zero at the free surfaces and, provided that the plate is thin, it is reasonable to assume that they are zero at any section. 8. The influence of shear and rotating inertia is neglected. 9. Only transverse displacement is considered. The vibrating rectangular plate of dimension 2a × 2b must satisfy the boundary conditions at the edges and also satisfy the plate equation of dynamic equilibrium. Due to excitation, the plate is assumed to be vibrating with simple harmonic motion in one of its natural modes. The transverse displacement of a point (x, y) on the surface of the plate at any time t corresponding to the (m, n)th mode of vibration is given by
(1)
Wmn = Wren O(x) O ( y ) e TM
where the plate dimensions and the coordinate system are shown in Fig, 1. The acoustic pressure distribution in the far field is given by Wallace. 18
p - --Kcp~Wei(O,t+KR)I a l b 2ztR
-,
-h
O(x)O(y)
.'z~,,',
x e-'(~.+~)
dxdy
(2)
where, 1 = 2aK cos e sin 0 and s -- 2bK sin e sin 0. The displacement functions satisfying the conditions at the edges of the plate are given by I
O(x) =
cOSym ~aa
+ RmcoshTm
form---2,4,6,..,
and
(3) -a_
8
Md. K. Rahman, A. Ertas
z
R
% y / ~
7"
Fig. 1. Rectangular flat plate with coordinate system.
O(x)
(;a) +
= sin 7~m
'(;a)
R m sinh ~m
form=3,5,7,..,
and - a < x _ < a
(4)
where sin (75~) Rm - sinh (75~)
(5)
and ~m are the roots of the equation (6)
tan(2 ) +tanh ( 2 ) =0 and sin ( ~ - )
(7)
m
sinh ( ~ ~-) and ~'m are the roots of the equation tan(2 ) - t a n h ( ~ ) O(y)
(8)
=0
= sin (n - 1)n(y + b) 2b f o r n = 2 , 3 , 4 . . . . and
-b
(9)
Acoustic power radiation from a rectangular plate
9
As the displacement functions are dependent upon the odd-even characteristic of m, it is necessary to consider the even and odd values of m separately. Case I: Even values o f m
Substituting the appropriate displacement functions 0 (x) and 0(y) into eqn (2) and integrating, the acoustic pressure distribution, P, becomes
P_ 8pcKogWabei(~t+ KR) Rrc2(n - 1) I?msin ( ~ ) c ° s
l (2)-
'c°s ( ~ )
Rm
sin (~)1
(~2 _ /2)
"q- (~2 _}_/2)
(lO)
[Ymsinh (Z~) c°s ( ~ ) + lc°sh ( ~ ) f
x]
COS(s/2)
i sin 0/2)
sin ( ~ ) ] }
}
where cos (s/2) is used when n is an even integer and i sin (s/2) is used when is an odd integer. The far-field acoustic intensity, I, is given by I = IPI2 (11)
n
pc
Substituting P from eqn (10) yields I=
32pcK2o92W2a2b2 { lr4R2( n _ 1)2
1
[
(Z_~)
(7 2 - 12 ) Pm sin
cos
_ sin ( ~ ) ] + R(72m + 12) [~,msinh ( 2 ) -'c°s (~2) +lc°sh ( ~ )
2
sin (~)1 } x
(~)
cos ( / )
cos(42) }2 sin(s/2)
[s2/r~a(n_l)2_l]
(12)
Md. K. Rahman, A. Ertas
10 where
l = 2 a K c o s ~ sin 0 = (2a0) cos c¢sin O)/c
Cv/3Pm (1 - v2)
(13)
cos ~ sin 0
s = 2 b K s i n ~ sin 0 = (2b0) sin a sin O)/c _
(14)
Ra Rt2f~2v/-~ s i n ~ s i n 0 Cv/3Pm (1 - v2)
and the frequency is given by W a r b u r t o n ~ as
0)2 ~.
Eh 2g2 f2re4 a 4 12(1 - v2)pm
(15)
where
it 2 : G.4~+ (a/b) 4 G~4 + 2 (a/b) 2 [vH,: Hv .
.
.
.
+ (1 - v) Jx Jr]
(16)
The total average p o w e r radiated from one side o f the baffled plate is then given by Pw =
0 J0
pc
R 2 sin 0d 0d c~ =
o Jo
IR 2 sin 0d 0d c~
(17)
or
128pcK20)2W2a2b2 f~/2 [~/2 {
Pw =
5 1-~ --lc°s ( Z ~ )
sin ( ~ ) ]
J0
30
1 (])2 _ / 2 )
{
sin
cos
(72 q- ,2)
COS(s/2) +/cosh
(7_~)
7m sin
x
[s2/rr2~n -- ~ 2 _ 1]
}2 xsin0d0dc~
Defining KRa=-
2~
co
2 --2f-b
a
and
0) c
- acoustic wave n u m b e r
Rt---
h
a
(19)
(20)
11
Acoustic power radiation f r o m a rectangular plate
Substituting eqns (19) and (20) into eqn (18), yields 2 4 4 n/2 f=/2 8pW 2 2 4r E 2g2 RaRtTt [
ew :
777
1
,o
Jo
[~'msin (~)c°s (~)- 'c°s (~~) sin (~)]
Rm
+ (~ + t2) [Tmsinh ( ~ ) c o s f x ~
[
(~)+/cosh
(~)
sin ( ~ ) ] } 2
COS(s/2) "]2 sinO/2) } x sin 0d 0d~ [$2/7~2 ( n - 1)2 -- 1]
(21)
It is clear from eqn (21) that the total average power radiated from one side of the plate does not depend upon the distance R of the receiving point from the center of the plate. Only the acoustic intensity varies inversely with the square of R, as seen in eqn (12).
Case II: Odd values of m Substituting the appropriate displacement functions 0 (x) and 0(y) into eqn (2) and integrating, the acoustic pressure distribution, P, becomes
P _ 8pcKcoWab ei(O)t+KR)i R~ 2 (n - 1) '
+ (7~ + / 2 )
( X 'J
[
lsinh
cos(s/2) !sin(s/Z) [S2//~2 (?t - 1)2 - 11
cos
- 7"cosh
l
--
sin
} (22)
where cos(s/2) is used when n is an even integer and i sin(s/2) is used when n is an odd integer. The far-field acoustic intensity,/, is
12
Md. K. Rahman, A. Ertas
I=
1
32pcK20)2W2a2b2
~4R2(n~
-/sin
-cosh
°.
/sinh + ,,2R~ (~'m -}- /2)
cos
(, ?)sin
[7'mcos ( _ ~ ) sin ( ~ )
{(7~- t2)
(~)3}2
cos (23)
sinO/2)
[s2/rc2~n-~2 - l ]
x
where l, s, 0)2, and 2~ are given by eqns (13), (14), (15), and (16), respectively. The total average power radiated from one side of the baffled plate becomes Pw
2 44[ 8pW 2rE 22 g RaRtg
2f. 2
95Z(;-- STd-1
'~
+ ( ~ 2 +/2)
lsinh
cos
cos(,/2) }2 sinO/2)
×
[s2/~z2(n- 1) 2 -
1]
- 7mc°sh
x sinOdOd~
co ( )l --
(24)
RESULTS AND DISCUSSION The total average power radiation from one side of a rectangular flat plate vibrating in its natural modes is found under clamped simply supported edge conditions. The appropriate beam functions which define the deformation mode in two perpendicular directions and satisfy the conditions at the edges of the vibrating plate under consideration have been combined for defining the deformation mode of the respective plate under vibration. In this analysis, the boundary conditions considered are two opposing ends fixed and the other two ends simply supported. In fact, the beam functions satisfy the boundary conditions for plates with fixed or freely supported edges, but are only approximate for free edges. The input variables for any rectangular flat plate with uniform thickness are the aspect ratios (b/a) and the thickness ratios (h/a). The results are found for three values of the aspect ratios and three values of the thickness ratios.
Acoustic power radiation,[?orn a rectangular plate
13
Three values of the plate parameters considered are 0-5, 1.0, and 2.0 for the aspect ratio and 0.001, 0.002, and 0.004 for thickness ratio. The following values of other parameters required in numerical calculations are used: density of the surrounding medium, p = 1.21 kg/m3; velocity of sound in the surrounding medium, c = 341 m/s; density of the plate material, Pm = 7700"0kg/m3; modulus of elasticity of the plate material, E = 206GPa; amplitude of vibration, W = 0-0001 m; and Poisson's ratio, v = 0.4. The values of acoustic power radiation are obtained for various plate parameters and mode orders in the x and y directions. Figures 2, 3, and 6-9 show the total average acoustic power radiated from one side of the baffled plate, plotted against the number of nodal lines in the y direction for a fixed mode order in the x direction. In Figs 4 and 5, the total average acoustic power radiation is plotted against the number of nodal lines in the x direction for fixed mode orders in the y direction. The influence of different aspect ratios, thickness ratios, and mode orders are also compared in these figures. In order to offer a plausible explanation of the nature of the curves in Figures 2-5, it is necessary to refer to Figs 10-13. In Figs 10 and 11 it is seen that, with the increase of mode order, n, the frequency factor increases. Figures 12 and 13 show the same trend of 2r with increasing m, the mode order in the x direction. From the definition of the frequency factor 2f by Warburton, ~ it is observed that the frequency of vibration is directly proportional to the frequency factor 2r. Therefore, it can be concluded that the nature of the variation of frequency of the plates will be the same as the frequency factor, because they have the same trend. Hence, it can be
10 6
A I'-10 4
•
m=3
AR=O.50 10 3 O m 10 2 1o 1 rr nr LJJ
lO o lo . I
O rl 10 -2 10 -3
n
(NO.
OF
NODAL
LINE,
IN
Y-DIRECTION)
Fig. 2. Power radiation for low values o f m.
Md. K. Rahman, A. Ertas
14 10 5
m=20 A k,,, Iv Z
•
m=22
•
m=21
~L
AR__--1..o0
/f~,
10 4 ,
0 m I-.< nn"
10 3 -
LU
0
O.
10 2 0
10
20
30
40
n (NO. OF NODAL LINE$1N Y-DIRECTION)
Fig. 3. Power radiation for high values of m.
concluded that, with the increase in value of m or n or both, the frequency of vibration would increase. Figures 2 and 3 show the acoustic power radiation plotted against nodal lines in the y direction, while Figs 4 and 5 show the same in the x direction for constant values of the aspect ratio and thickness ratio. In Fig. 2, it is seen that the acoustic power radiation increases somewhat monotonically at the lower range of n and fluctuates at higher values of n. This is more obvious from Fig. 3. The variations of power radiation follow the same trend with variation of m as is shown in 10 4
~" I.,<
10 3
~. Z
10 2
0 I-<
10 1
~n"
10 0
rr [,,U
10 " 1
o
O.
10 - 2
10 - 3
,
10
T
~o
30
m( NO. O F N O D A L LINE, IN X-DIRECTION) Fig. 4. Power radiation for low values of n.
4O
Acoustic power radiation from a rectangular plate
15
10 6 --'-¢P--10 5
a
TR=O.O01
/
TR--'0.O02
< AR=I.00
10 4
0
n-
n-
/
\ / 'k)= /
= 10 3 10 2
¢IJ
l0 1
0
10 0 10 -1 10 - 2 0
10
20
30
40
m (NO. OF N O D A L L I N E , IN X-DIRECTION)
Fig. 5. Power radiation for high values of n. Figs 4 and 5. These can be explained by the fact that the power radiation should increase with the increase of the frequency of vibration which, in its turn, increases with the increase in value of m and n. At lower values of m and n, the rate of increase of frequency is higher. Therefore, the power radiation increases rapidly at the lower range of mode orders. However, the power radiation does not show a significant rise with increasing values of m and n at the higher range. The graphs show a waviness after certain values of m and n. At the higher range of mode orders, the values of m and n have less influence over the acoustic power radiation. The waviness of the graphs after certain values of mode orders are due to the interference of the waves from the x and y directions. At the higher mode orders, the wave forms from the two directions interfere frequently. The acoustic power radiation fluctuates due to this interference. When the two crests from the two directions superimpose, the total average acoustic power radiation increases. On the other hand, when the crests from one direction superimpose on the troughs from another, the total average acoustic power radiation decreases, as depicted by the roots of the graphs. The frequency of this waviness is dependent upon the interferences of the waves. The number of interferences increases with the increase in the mode order. Thus, the frequency of the waviness has a lower value at the low range of mode order and increases with rise in mode orders. It is observed in Figs 3 and 5 that the rate of increase in the acoustic power radiation approaches zero at very high mode numbers. The variation of the acoustic power radiation can be explained with the help of the variation of acoustic pressure in the surrounding medium due to the
16
Md. K. Rahman, A. Ertas
vibration of the plate. There are various factors which affect the acoustic power radiation, among which the frequency of vibration, the effective radiating surface of the plate, and the effective acoustic pressure are the major three factors among them. It is already known that the frequency of vibration increases significantly with increase in mode order at the lower range, but at the high range, the rate of increase of frequency of vibration falls. At low mode order, the effective radiating surface is larger because, in this case, the points of zero displacement in the plate are less. But, with increasing mode order, the regions of this zero displacement increases, so that the effective radiating surface decreases. Again, due to the vibration of the plate, waveforms are generated in the surrounding medium, resulting in its alternate compression and rarefaction. At very high mode orders, the plate wave number becomes very high. Therefore, alternate crests and roots of the plate wave come closer to one another. At any instant of time, a particular crest produces a compression in the surrounding medium while the neighboring root produces a rarefaction. These alternate compressions and rarefactions cause the acoustic energy to travel through the surrounding medium. When the plate wave number becomes very high, the neighboring roots and crests are so close to each other that the rarefaction produced by the root partially neutralizes the compression produced by the crest. The effective pressure variation decreases. Thus, the intensity of absolute pressure variation is high at the lower mode orders and low at the higher mode orders. Hence, in the low range of mode order, the acoustic power radiation from the plate rapidly rises owing to the positive contributions of the frequency of vibration, effective pressure variation and effective radiating surface of the plate. But at high mode order, the effective acoustic pressure has no effect on power variation. The effect of frequency of vibration becomes insignificant at high range of mode order. Also, the effective radiating surface of the plate implies negative effect on power variation. Therefore, the absolute amount of acoustic power radiation does not show any further increase with mode number. Rather, it attains a stable magnitude in this case. Figures 6 and 7 represent the variation of acoustic power radiation with variation of the thickness ratio of the plate. It is seen from these figures that the acoustic power radiation increases when the thickness ratio is increased. This is because more power will be needed to excite a plate if its thickness is increased. Figure 6 shows the variation of power radiation with thickness ratio for low values of nodal lines in the x direction and Fig. 7 for high values of the same nodal line. In Figs 8 and 9, the effect of aspect ratio of the plate over the power radiation is shown. Figure 8 shows the variation of average power with
Acoustic power radiation from a rectangular plate
17
10 5. n=20 p. ,<
s
n=22
;
n=21
AR=I.00
z
_~
10 4
0 D-CJ ~Z
n" 10 3 UJ
0
10 2 10 n
20
30
40
(NO. OF NODAL LINESIN Y-DIRECTION)
Fig. 6. Power radiation for different values of thickness ratio and low values of m. 10 6 A
I-I-v
z O
----.-¢I----
TR=0.001
.I
TR=0.002
•
TR=O.O04
/1~
A.=l.oo
10 $
/
\
p
<
,< rr 10 4 rr UJ
0
a. 10 3
10 2
!
0
10
•
i
20
i
30
40
n (NO. OF N O D A L L I N E , IN Y - D I R E C T I O N )
Fig. 7. Power radiation for different values of thickness ratio and high value of m. aspect ratio at a lower value of m and Fig. 9 for higher values of rn. It can be seen from Fig. 8 that the total average acoustic power radiation increases with decrease in the values of the aspect ratio, but this is not true for higher values of m. It is found that the total acoustic power radiation increases with increase in the aspect ratio at higher values o f m and at the lower range ofn. This is shown in Fig. 9. The trend is gradually reversed with increase in
Md. K. Rahman, A. Ertas
18 10 6
p~ 104 v ~
103
~ <
102
AR-O.5 AR=I,00 AR=2.00
• •
10 S
A
TR=O.001
p /
~ /
I~ 101 ill O 10 o Q. 10 " 1 10 - 2
+ 10
0
r 20
, 30
40
n (NO. OF NODAL LIN~IN Y-DIRECTION)
Fig. 8. Power radiation for different values of aspect ratio and low value of m. 10 5 . •- - " - - o - - -
pI--
AR=0.5
4,,
AR=I.00
•
AR-2.00
/
TR=O.001
[
10 4
z
0 I--
t~ 10 3 UJ
0
0..
10 2 0
10
20
30
40
n (NO. OF N O D A L LINESlN Y-DIRECTION)
Fig. 9. Power radiation for different values of aspect ratio and high value of m. the value of n. These variations of acoustic power radiation can be explained with the help of Figs 10-13. These figures show the variation of dimensionless frequency factor with the variation of aspect ratio for different values of m and n. It is observed that, at all mode orders, the dimensionless frequency factor increases with decrease in aspect ratio. As the dimensionless frequency factor increases, the frequency of vibration also increases, which in turn increases the total acoustic power. Therefore,
Acoustic power radiation from a rectangular plate
19
10 8 AR=0.S0
0 I-. M.
10 7
8
AR=I.00
¢
AR=2.00
m=4 10 6
f= u.
10 5
~ 10 4 ~ 10 3 10 2
i
',
12
1
22
i
32
n (NO. OF NODAL LINE, IN Y-DIRECTION)
Fig. 10. Dimensionless frequency factor for different values of aspect ratio and low value of re.
10 8 ~ <
AR=0.S0
----e----
AR=1.00
-
AR=2.00
= m z 10 7
10S4 0
40
n (NO. OF NODAL LINE$1N Y-DIRECTION)
Fig. 11. Dimensionless frequency factor for different values of aspect ratio and high value of re.
average acoustic power radiation increases as the aspect ratio is decreased. At lower mode orders, the a m o u n t of power radiation mainly depends upon the effective radiating surface of the plate. The influence of frequency over the power radiation is less dominant in this case. Effective radiating
20
Md. K. Rahman, A. Ertas 10 7 1 ~ :
106
AR=0.50
8
AR=I.00
•
AR=2.00
~== '°s 1 u,. 1041 ; ,021 101| 0
!
i
i
10
20
30
40
m (NO. OF NODAL LINE$1N X-DIRECTION)
Fig. 12. Dimensionless frequency factor for different values of aspect ratio and low value ofn. 10 7
10 6
== ~: 10 5 M.
-~ 10 4
n=20 -.---.-o---- AR=O.S0 4
AR=I.00
•
AR=2.00
10 3 10
20
30
40
m (NO. OF NODAL LINI~IN X-DIRECTION)
Fig. 13. Dimensionlessfrequencyfactor for differentvalues of aspect ratio and high value ofn.
surface is more in the case of a plate with lower aspect ratios at low values of mode order in the x direction. For this reason, at low mode numbers in the x direction, the plate with a lower aspect ratio radiates more power. This is shown in Fig. 8. At high values o f m in the lower range o f n, the plate
Acoustic power radiation from a rectangular plate
21
with higher aspect ratio radiates higher amounts of power, as shown in Fig. 9. This is because, at these mode orders, the effective radiating surface is greater in the case of a plate with higher aspect ratio, as the longer side of the plate is divided into fewer waves. The effective radiating surface decreases with increasing mode order but the frequency of vibration increases. Here the increase in the frequency of vibration far exceeds the reduction in the effective radiating surface. Thus, the frequency of vibration begins to play a dominating role over the acoustic power radiation from the plate. From Fig. 11, it is seen that the increase in the frequency of vibration with increase in the mode numbers is more in the case of a plate with an aspect ratio of 0.50. The amount of acoustic power radiation shows a steeper rise in this case, as shown in Fig. 8. The amount of power radiation from the plates with the other two aspect ratios is smaller.
CONCLUSIONS The purpose of this study was to investigate the noise-generating characteristics of a rectangular flat plate subjected to mixed boundary conditions. The results indicate that: 1. The average acoustic power radiated from one side of a baffled plate vibrating in its natural modes increases with increase in mode order. In the lower range of mode orders, this increasing tendency is high. At the higher range of mode order, the acoustic power radiation is in a stable state with increase in mode order. After a certain mode number, the acoustic power radiation from the plates begin to show a wavy nature. 2. The average power radiated from a baffled plate increases with increase in thickness ratio. 3. With increase in aspect ratio, the average power radiated from a baffled plate decreases at low values of m. For high values of m in the low range of n, the average power radiation increases with increasing aspect ratio. However, this tendency gradually reverses at the higher range of m and n values.
REFERENCES !. Warburton, G. B., The vibration of rectangular plates. Proc. Inst. Mech. Eng., 168 (1954) 371--84. 2. Young, D. & Felger, R.P., Tables o f Characteristic Functions Representing Normal Modes o f a Vibrating Beam. The University of Texas, Publication No.
22
3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.
Md. K. Rahman, A. Ertas
4913, Bureau of Engineering Research, Engineering Research Series, No. 44, 1949. Skudrzyk, E.J., Grandlegan der Akustic. Springer-Verlag, Vienna, Austria, 1954. Skudrzyk, E.J., Sound radiation of a system with a finite or infinite number of resonances. J. Acoustical Soc. of America, 30 (1958) 1152. Feit, D., Sound radiation from orthotropic plates. J. Acoustical Soc. of America, 47 (1970) 388-9. Lowenthal, S. & Tournois, P., Radiation impedance of membranes and plates: Their acoustic coupling with the propagating medium. J. Acoustical Soc. of America, 35 (1963) 1423-8. Thompson, W. Jr & Rattayya, J, V., Acoustic power radiation by an infinite plate excited by a concentrated moment. J. Acoustical Soc. of America, 36, (1964) 1488-90. Tsakonas, S., Chen, C. Y, & Jacobs, W. R., Acoustic radiation of an infinite plate excited by the field of a ship propeller. J. Acoustical Soc. of America, 36 (1964) 1708-17. Maidanik, G., Acoustic radiation from a driven infinite plate backed by a parallel infinite baffle. J. Acoustical Soc. of America, 41--42 (1967) 27-3 t. Magrab, E. B. & Reader, W. T., Farfield radiation from an infinite elastic plate excited by a transient point loading. J. Acoustical Soc. of America, 43--44 (1968) 1623-7. Suzuki, II. & Tichy, J., Sound radiation from an elastically supported circular plate. J. Acoustical Soc. of America, 65 (1979) 106-11. Feit, D., Pressure radiated by a point excited elastic plate. J. Acoustical Soc. of America, 40 (1966) 1494 9. Felt, D. & Saurenman, J. J,, Sound Radiation by Beam Stiffened Plates. Cambridge Acoustical Association, Cambridge, MA, Ref. No. U-356-213, 1971. Gorman, R. M., Vibration and acoustic radiation from line excited ribstiffened damped plates in water. Ph.D thesis, the Pennsylvania State University, Pennsylvania, USA, 1974. Ahmed, A., Acoustics of rectangular flat plates. MSc thesis, BUET, 1988. Mandal, N. K., Acoustics of rectangular flat plates of mixed boundary conditions. MSc Thesis, BUET, Bangladesh University of Engineering and Technology, Bangladesh, 1989. Peng, H. &Keltie, R. F., Sound radiation from finite plates under the action of multiple random point forces. J. Acoustical Soc. of America, 85(1) (1989) 57-67. Wallace, C. E., Radiation resistance of a rectangular panel. J. Acoustical Soc. ~?['America, 51(3) (part 2) (1972) 936-45.
Acoustic Power Radiation from a Rectangular Plate with Mixed Boundary Conditions
M d . K. R a h m a n & A. Ertas Department of Mechanical Engineering, Texas Tech University, Lubbock, Texas, USA
ABSTRACT The total average acoustic power radiation from one side of a baffled rectangular flat plate, vibrating in its natural modes under two opposing ends fixed and the other two ends simply supported boundary conditions, has been investigated. Numerical results on the acoustic power radiation by the plate have been obtained for various mode orders, aspect ratios, and thickness ratios. Expressions for the acoustic power radiation by plates under specific edge restraints have been derived in terms of the far-field acoustic pressure distribution.
NOMENCLATURE Half of the length of the plate b = Half of the width of the plate C = Velocity of sound in air E = Modulus of elasticity of the plate material f = Frequency of vibration of the plate g -- Acceleration due to gravity G x = A function of m, factor of the dimensionless frequency factor a , = A function of n, factor of the dimensionless frequency factor H x = A function of m, factor of the dimensionless frequency factor A function of n, factor of the dimensionless frequency factor 4 = A function of m, factor of the dimensionless frequency factor Jy A function of n, factor of the dimensionless frequency factor a
3
4
Md. K. Rahman, A. Ertas
h K m n P Pw R Ra
= = = = = -= =
Rt
---
r t W w x, y, z 7
= = = -= = = -= = =
O(x) 0(y) 2 2f p
Pm
•
v = R, 0, ~ = {o =
Thickness of plate Acoustic wave number N u m b e r of nodal lines in x direction N u m b e r of nodal lines in y direction Acoustic pressure distribution Total acoustic power radiation Distance of the receiving point from the center of the plate Aspect ratio of the plate, b/a Thickness ratio of the plate, h/a Distance of the receiving point from the elemental area Time Amplitude of vibration Instantaneous displacement of the plate surface Cartesian coordinate system Roots of the equation, tan(7/2) T tanh(7/2) = 0 Displacement function in x direction Displacement function in y direction Wavelength of sound Dimensionless frequency factor Density of the surrounding medium Density of the plate material Poisson's ratio Spherical coordinate system Circular frequency of the plate vibration
INTRODUCTION With the growth of mechanization in modern life, the problem of noise is also growing steadily. The mechanical and electrical equipment which are an integral part of modern living are also the sources of many acoustical problems. Nearly every piece of mechanical equipment in a building is a source of sound and vibration. The size and complexity of machines, increasing road traffic, etc., contribute to the ever-increasing level of noise. The quality of life is being degraded by these noise sources. The problem is already recognized as one of primary importance and demands immediate attention. W o r k on the study of the vibration characteristics of rectangular plates began in the last part of the 19th century when it was realized that a large number of structural systems in civil, mechanical, and aerospace engineer-
Acoustic power radiation from a rectangular plate
5
ing are composed of these plate components. The dynamic behavior of these structures was first studied by Rayleigh. He developed the fundamental equations governing the vibration of plates. Since the middle of the 20th century, noise has been recognized as a source of environmental pollution which ultimately affects human comfort. From that time, research in this field has attracted much attention. The study of acoustic radiation characteristics of flat plates has not yet reached a satisfactory level. The vibration of plates was studied by many investigators in the past. Most of the initiators confined their study to the determination of the frequency of vibrating plates. In 1954, Warburton ~ derived approximate expressions for the frequencies of all the modes of vibration of isotropic plates subjected to any combination of free, simply supported or clamped edges. He applied the Rayleigh method, assuming that the deflections of the plates could be represented by suitable characteristic functions satisfying the boundary conditions. 2 In his analysis, Warburton first developed a set of beam functions satisfying the edge conditions of beams and applied these functions to the vibration of plates. He expressed the frequency in terms of a dimensionless frequency factor, which in turn is a function of the mode shapes of the vibrating plates. Although there has been much analysis of the vibration of solid structures such as rectangular plates, the study of acoustic radiation due to their vibration of flat plates is rather lacking. In 1954, Skudrzyk 3"4 obtained the solution to the radiation of power from an elastic plate for a time-harmonic point force. He used the classical plate theory. The solutions for the acoustic radiated pressure from a point-excited plate using the classical theory was given by Feit 5 and Skudrzyk. 4 The radiation impedance of membranes and plates of finite width, carrying a traveling wave, was calculated by Lowenthal and Tournois. 6 The method used led to a relating simple analytic expression. The problem of acoustic coupling of these strips with the fluid propagating medium was then examined. An expression was derived for the acoustic power radiated by an infinite uniform isotropic plate excited by a time-harmonic moment acting at a point on the plate by T h o m p s o n and Rattayya 7 in 1964. The effect of radiation loading was included. Results were obtained that were valid below the coincidence frequency. Tsakonas et al. 8 determined the acoustic radiation of an infinite plate excited by the field of a ship propeller both by theoretical and numerical methods. In 1967, Maidanik 9 computed the farfield acoustic pressure generated by an infinite plate driven by a point and a line force. The plate considered here was backed by a fluid medium that was terminated by a parallel infinite baffle. He also applied the same methodology to a coated infinite plate. Magrab and Reader 1° obtained
6
Md. K. Rahman, A. Ertas
an expression for the far-field pressure radiated by a thin elastic plate bounded on one side by a fluid medium when the plate was excited by a time-varying point load. In order to discuss the sound radiation from a direct-radiator loudspeaker in the high-frequency region, an elastically supported circular plate in an infinite baffle was used by Suzuki and Tichy. ~ The normal modes of this plate were consistently treated from the free boundary to the simply supported boundary, which were the limiting cases of the elastically supported boundary, by defining a dimensionless parameter representing the ratio of the edge stiffness to the bending stiffness of the plate. To improve the high-frequency prediction of elastic plates, Feit ~2employed the Timoshenko-Miudlin theory for such a prediction. This theory adds shear deformation and rotary inertia to the classical flexural theory. There are two dispersion curves for this plate. The flexural branch (acoustic) has phase and group velocities approaching the Rayleigh velocity of the plate material as the frequency increases. Sound radiation from a beam-reinforced plate, excited by point or line forces, has been investigated by a few authors. Feit and Saurenman ~3 analyzed the acoustic radiation of a point-excited plate reinforced with a beam, but confined their interests to high frequencies. Gorman t4 obtained the solution for a plate reinforced by many beams and excited by a line force parallel to the beam. His solution thus makes the beam's reaction on the plate as purely rotary and transverse impedances without a flexural wave traveling in the beams. Ahmed ~5 and Mandal ~6 studied the acoustic power radiation from one side of a baffled rectangular isotropic plate for different thickness ratio, aspect ratio, and mode orders. The problem of acoustic radiation from baffled finite elastic plates driven by multiple random point forces was analyzed by Peng and Keltie. j7 Expressions for the surface acoustic intensity and radiated sound power were derived neglecting the fluid loading. By assuming the plates to be lightly damped, and neglecting the modal coupling effects, approximate solutions for the acoustic intensity and sound power were obtained. An experimental study was also carried out to measure the surface acoustic intensity patterns and to compare them with the analytical results. As appears from the foregoing survey, the majority of the research works in the field of vibration noise relate to the natural frequency of rectangular plates. Not much work has been done to analyze the acoustic radiation characteristics of rectangular flat plates vibrating in their natural modes. Work on the natural frequency and acoustic radiation of rectangular plates is continuing and has a long way to go before full understanding is achieved.
Acoustic power radiation from a rectangular plate
7
The main objective of this research is to study the noise-generating characteristics of a rectangular flat plate subjected to two opposing ends fixed and the other two ends simply supported boundary conditions.
MATHEMATICAL MODELING In the analysis, the following assumptions have been made: 1. The thickness of the plate is small compared to the dimensions, and the thickness is uniform. 2. The plate is free from applied load. 3. The mid-plane remains neutral during bending. 4. The cross-sectional plane remains plane after deformation. 5. The plate is isotropic, elastic, and homogeneous. 6. Deflections are small in comparison with the thickness of the plate. 7. The normal stresses in the direction transverse to the plate can be neglected. These stresses must be zero at the free surfaces and, provided that the plate is thin, it is reasonable to assume that they are zero at any section. 8. The influence of shear and rotating inertia is neglected. 9. Only transverse displacement is considered. The vibrating rectangular plate of dimension 2a × 2b must satisfy the boundary conditions at the edges and also satisfy the plate equation of dynamic equilibrium. Due to excitation, the plate is assumed to be vibrating with simple harmonic motion in one of its natural modes. The transverse displacement of a point (x, y) on the surface of the plate at any time t corresponding to the (m, n)th mode of vibration is given by
(1)
Wmn = Wren O(x) O ( y ) e TM
where the plate dimensions and the coordinate system are shown in Fig, 1. The acoustic pressure distribution in the far field is given by Wallace. 18
p - --Kcp~Wei(O,t+KR)I a l b 2ztR
-,
-h
O(x)O(y)
.'z~,,',
x e-'(~.+~)
dxdy
(2)
where, 1 = 2aK cos e sin 0 and s -- 2bK sin e sin 0. The displacement functions satisfying the conditions at the edges of the plate are given by I
O(x) =
cOSym ~aa
+ RmcoshTm
form---2,4,6,..,
and
(3) -a_
8
Md. K. Rahman, A. Ertas
z
R
% y / ~
7"
Fig. 1. Rectangular flat plate with coordinate system.
O(x)
(;a) +
= sin 7~m
'(;a)
R m sinh ~m
form=3,5,7,..,
and - a < x _ < a
(4)
where sin (75~) Rm - sinh (75~)
(5)
and ~m are the roots of the equation (6)
tan(2 ) +tanh ( 2 ) =0 and sin ( ~ - )
(7)
m
sinh ( ~ ~-) and ~'m are the roots of the equation tan(2 ) - t a n h ( ~ ) O(y)
(8)
=0
= sin (n - 1)n(y + b) 2b f o r n = 2 , 3 , 4 . . . . and
-b
(9)
Acoustic power radiation from a rectangular plate
9
As the displacement functions are dependent upon the odd-even characteristic of m, it is necessary to consider the even and odd values of m separately. Case I: Even values o f m
Substituting the appropriate displacement functions 0 (x) and 0(y) into eqn (2) and integrating, the acoustic pressure distribution, P, becomes
P_ 8pcKogWabei(~t+ KR) Rrc2(n - 1) I?msin ( ~ ) c ° s
l (2)-
'c°s ( ~ )
Rm
sin (~)1
(~2 _ /2)
"q- (~2 _}_/2)
(lO)
[Ymsinh (Z~) c°s ( ~ ) + lc°sh ( ~ ) f
x]
COS(s/2)
i sin 0/2)
sin ( ~ ) ] }
}
where cos (s/2) is used when n is an even integer and i sin (s/2) is used when is an odd integer. The far-field acoustic intensity, I, is given by I = IPI2 (11)
n
pc
Substituting P from eqn (10) yields I=
32pcK2o92W2a2b2 { lr4R2( n _ 1)2
1
[
(Z_~)
(7 2 - 12 ) Pm sin
cos
_ sin ( ~ ) ] + R(72m + 12) [~,msinh ( 2 ) -'c°s (~2) +lc°sh ( ~ )
2
sin (~)1 } x
(~)
cos ( / )
cos(42) }2 sin(s/2)
[s2/r~a(n_l)2_l]
(12)
Md. K. Rahman, A. Ertas
10 where
l = 2 a K c o s ~ sin 0 = (2a0) cos c¢sin O)/c
Cv/3Pm (1 - v2)
(13)
cos ~ sin 0
s = 2 b K s i n ~ sin 0 = (2b0) sin a sin O)/c _
(14)
Ra Rt2f~2v/-~ s i n ~ s i n 0 Cv/3Pm (1 - v2)
and the frequency is given by W a r b u r t o n ~ as
0)2 ~.
Eh 2g2 f2re4 a 4 12(1 - v2)pm
(15)
where
it 2 : G.4~+ (a/b) 4 G~4 + 2 (a/b) 2 [vH,: Hv .
.
.
.
+ (1 - v) Jx Jr]
(16)
The total average p o w e r radiated from one side o f the baffled plate is then given by Pw =
0 J0
pc
R 2 sin 0d 0d c~ =
o Jo
IR 2 sin 0d 0d c~
(17)
or
128pcK20)2W2a2b2 f~/2 [~/2 {
Pw =
5 1-~ --lc°s ( Z ~ )
sin ( ~ ) ]
J0
30
1 (])2 _ / 2 )
{
sin
cos
(72 q- ,2)
COS(s/2) +/cosh
(7_~)
7m sin
x
[s2/rr2~n -- ~ 2 _ 1]
}2 xsin0d0dc~
Defining KRa=-
2~
co
2 --2f-b
a
and
0) c
- acoustic wave n u m b e r
Rt---
h
a
(19)
(20)
11
Acoustic power radiation f r o m a rectangular plate
Substituting eqns (19) and (20) into eqn (18), yields 2 4 4 n/2 f=/2 8pW 2 2 4r E 2g2 RaRtTt [
ew :
777
1
,o
Jo
[~'msin (~)c°s (~)- 'c°s (~~) sin (~)]
Rm
+ (~ + t2) [Tmsinh ( ~ ) c o s f x ~
[
(~)+/cosh
(~)
sin ( ~ ) ] } 2
COS(s/2) "]2 sinO/2) } x sin 0d 0d~ [$2/7~2 ( n - 1)2 -- 1]
(21)
It is clear from eqn (21) that the total average power radiated from one side of the plate does not depend upon the distance R of the receiving point from the center of the plate. Only the acoustic intensity varies inversely with the square of R, as seen in eqn (12).
Case II: Odd values of m Substituting the appropriate displacement functions 0 (x) and 0(y) into eqn (2) and integrating, the acoustic pressure distribution, P, becomes
P _ 8pcKcoWab ei(O)t+KR)i R~ 2 (n - 1) '
+ (7~ + / 2 )
( X 'J
[
lsinh
cos(s/2) !sin(s/Z) [S2//~2 (?t - 1)2 - 11
cos
- 7"cosh
l
--
sin
} (22)
where cos(s/2) is used when n is an even integer and i sin(s/2) is used when n is an odd integer. The far-field acoustic intensity,/, is
12
Md. K. Rahman, A. Ertas
I=
1
32pcK20)2W2a2b2
~4R2(n~
-/sin
-cosh
°.
/sinh + ,,2R~ (~'m -}- /2)
cos
(, ?)sin
[7'mcos ( _ ~ ) sin ( ~ )
{(7~- t2)
(~)3}2
cos (23)
sinO/2)
[s2/rc2~n-~2 - l ]
x
where l, s, 0)2, and 2~ are given by eqns (13), (14), (15), and (16), respectively. The total average power radiated from one side of the baffled plate becomes Pw
2 44[ 8pW 2rE 22 g RaRtg
2f. 2
95Z(;-- STd-1
'~
+ ( ~ 2 +/2)
lsinh
cos
cos(,/2) }2 sinO/2)
×
[s2/~z2(n- 1) 2 -
1]
- 7mc°sh
x sinOdOd~
co ( )l --
(24)
RESULTS AND DISCUSSION The total average power radiation from one side of a rectangular flat plate vibrating in its natural modes is found under clamped simply supported edge conditions. The appropriate beam functions which define the deformation mode in two perpendicular directions and satisfy the conditions at the edges of the vibrating plate under consideration have been combined for defining the deformation mode of the respective plate under vibration. In this analysis, the boundary conditions considered are two opposing ends fixed and the other two ends simply supported. In fact, the beam functions satisfy the boundary conditions for plates with fixed or freely supported edges, but are only approximate for free edges. The input variables for any rectangular flat plate with uniform thickness are the aspect ratios (b/a) and the thickness ratios (h/a). The results are found for three values of the aspect ratios and three values of the thickness ratios.
Acoustic power radiation,[?orn a rectangular plate
13
Three values of the plate parameters considered are 0-5, 1.0, and 2.0 for the aspect ratio and 0.001, 0.002, and 0.004 for thickness ratio. The following values of other parameters required in numerical calculations are used: density of the surrounding medium, p = 1.21 kg/m3; velocity of sound in the surrounding medium, c = 341 m/s; density of the plate material, Pm = 7700"0kg/m3; modulus of elasticity of the plate material, E = 206GPa; amplitude of vibration, W = 0-0001 m; and Poisson's ratio, v = 0.4. The values of acoustic power radiation are obtained for various plate parameters and mode orders in the x and y directions. Figures 2, 3, and 6-9 show the total average acoustic power radiated from one side of the baffled plate, plotted against the number of nodal lines in the y direction for a fixed mode order in the x direction. In Figs 4 and 5, the total average acoustic power radiation is plotted against the number of nodal lines in the x direction for fixed mode orders in the y direction. The influence of different aspect ratios, thickness ratios, and mode orders are also compared in these figures. In order to offer a plausible explanation of the nature of the curves in Figures 2-5, it is necessary to refer to Figs 10-13. In Figs 10 and 11 it is seen that, with the increase of mode order, n, the frequency factor increases. Figures 12 and 13 show the same trend of 2r with increasing m, the mode order in the x direction. From the definition of the frequency factor 2f by Warburton, ~ it is observed that the frequency of vibration is directly proportional to the frequency factor 2r. Therefore, it can be concluded that the nature of the variation of frequency of the plates will be the same as the frequency factor, because they have the same trend. Hence, it can be
10 6
A I'-10 4
•
m=3
AR=O.50 10 3 O m 10 2 1o 1 rr nr LJJ
lO o lo . I
O rl 10 -2 10 -3
n
(NO.
OF
NODAL
LINE,
IN
Y-DIRECTION)
Fig. 2. Power radiation for low values o f m.
Md. K. Rahman, A. Ertas
14 10 5
m=20 A k,,, Iv Z
•
m=22
•
m=21
~L
AR__--1..o0
/f~,
10 4 ,
0 m I-.< nn"
10 3 -
LU
0
O.
10 2 0
10
20
30
40
n (NO. OF NODAL LINE$1N Y-DIRECTION)
Fig. 3. Power radiation for high values of m.
concluded that, with the increase in value of m or n or both, the frequency of vibration would increase. Figures 2 and 3 show the acoustic power radiation plotted against nodal lines in the y direction, while Figs 4 and 5 show the same in the x direction for constant values of the aspect ratio and thickness ratio. In Fig. 2, it is seen that the acoustic power radiation increases somewhat monotonically at the lower range of n and fluctuates at higher values of n. This is more obvious from Fig. 3. The variations of power radiation follow the same trend with variation of m as is shown in 10 4
~" I.,<
10 3
~. Z
10 2
0 I-<
10 1
~n"
10 0
rr [,,U
10 " 1
o
O.
10 - 2
10 - 3
,
10
T
~o
30
m( NO. O F N O D A L LINE, IN X-DIRECTION) Fig. 4. Power radiation for low values of n.
4O
Acoustic power radiation from a rectangular plate
15
10 6 --'-¢P--10 5
a
TR=O.O01
/
TR--'0.O02
< AR=I.00
10 4
0
n-
n-
/
\ / 'k)= /
= 10 3 10 2
¢IJ
l0 1
0
10 0 10 -1 10 - 2 0
10
20
30
40
m (NO. OF N O D A L L I N E , IN X-DIRECTION)
Fig. 5. Power radiation for high values of n. Figs 4 and 5. These can be explained by the fact that the power radiation should increase with the increase of the frequency of vibration which, in its turn, increases with the increase in value of m and n. At lower values of m and n, the rate of increase of frequency is higher. Therefore, the power radiation increases rapidly at the lower range of mode orders. However, the power radiation does not show a significant rise with increasing values of m and n at the higher range. The graphs show a waviness after certain values of m and n. At the higher range of mode orders, the values of m and n have less influence over the acoustic power radiation. The waviness of the graphs after certain values of mode orders are due to the interference of the waves from the x and y directions. At the higher mode orders, the wave forms from the two directions interfere frequently. The acoustic power radiation fluctuates due to this interference. When the two crests from the two directions superimpose, the total average acoustic power radiation increases. On the other hand, when the crests from one direction superimpose on the troughs from another, the total average acoustic power radiation decreases, as depicted by the roots of the graphs. The frequency of this waviness is dependent upon the interferences of the waves. The number of interferences increases with the increase in the mode order. Thus, the frequency of the waviness has a lower value at the low range of mode order and increases with rise in mode orders. It is observed in Figs 3 and 5 that the rate of increase in the acoustic power radiation approaches zero at very high mode numbers. The variation of the acoustic power radiation can be explained with the help of the variation of acoustic pressure in the surrounding medium due to the
16
Md. K. Rahman, A. Ertas
vibration of the plate. There are various factors which affect the acoustic power radiation, among which the frequency of vibration, the effective radiating surface of the plate, and the effective acoustic pressure are the major three factors among them. It is already known that the frequency of vibration increases significantly with increase in mode order at the lower range, but at the high range, the rate of increase of frequency of vibration falls. At low mode order, the effective radiating surface is larger because, in this case, the points of zero displacement in the plate are less. But, with increasing mode order, the regions of this zero displacement increases, so that the effective radiating surface decreases. Again, due to the vibration of the plate, waveforms are generated in the surrounding medium, resulting in its alternate compression and rarefaction. At very high mode orders, the plate wave number becomes very high. Therefore, alternate crests and roots of the plate wave come closer to one another. At any instant of time, a particular crest produces a compression in the surrounding medium while the neighboring root produces a rarefaction. These alternate compressions and rarefactions cause the acoustic energy to travel through the surrounding medium. When the plate wave number becomes very high, the neighboring roots and crests are so close to each other that the rarefaction produced by the root partially neutralizes the compression produced by the crest. The effective pressure variation decreases. Thus, the intensity of absolute pressure variation is high at the lower mode orders and low at the higher mode orders. Hence, in the low range of mode order, the acoustic power radiation from the plate rapidly rises owing to the positive contributions of the frequency of vibration, effective pressure variation and effective radiating surface of the plate. But at high mode order, the effective acoustic pressure has no effect on power variation. The effect of frequency of vibration becomes insignificant at high range of mode order. Also, the effective radiating surface of the plate implies negative effect on power variation. Therefore, the absolute amount of acoustic power radiation does not show any further increase with mode number. Rather, it attains a stable magnitude in this case. Figures 6 and 7 represent the variation of acoustic power radiation with variation of the thickness ratio of the plate. It is seen from these figures that the acoustic power radiation increases when the thickness ratio is increased. This is because more power will be needed to excite a plate if its thickness is increased. Figure 6 shows the variation of power radiation with thickness ratio for low values of nodal lines in the x direction and Fig. 7 for high values of the same nodal line. In Figs 8 and 9, the effect of aspect ratio of the plate over the power radiation is shown. Figure 8 shows the variation of average power with
Acoustic power radiation from a rectangular plate
17
10 5. n=20 p. ,<
s
n=22
;
n=21
AR=I.00
z
_~
10 4
0 D-CJ ~Z
n" 10 3 UJ
0
10 2 10 n
20
30
40
(NO. OF NODAL LINESIN Y-DIRECTION)
Fig. 6. Power radiation for different values of thickness ratio and low values of m. 10 6 A
I-I-v
z O
----.-¢I----
TR=0.001
.I
TR=0.002
•
TR=O.O04
/1~
A.=l.oo
10 $
/
\
p
<
,< rr 10 4 rr UJ
0
a. 10 3
10 2
!
0
10
•
i
20
i
30
40
n (NO. OF N O D A L L I N E , IN Y - D I R E C T I O N )
Fig. 7. Power radiation for different values of thickness ratio and high value of m. aspect ratio at a lower value of m and Fig. 9 for higher values of rn. It can be seen from Fig. 8 that the total average acoustic power radiation increases with decrease in the values of the aspect ratio, but this is not true for higher values of m. It is found that the total acoustic power radiation increases with increase in the aspect ratio at higher values o f m and at the lower range ofn. This is shown in Fig. 9. The trend is gradually reversed with increase in
Md. K. Rahman, A. Ertas
18 10 6
p~ 104 v ~
103
~ <
102
AR-O.5 AR=I,00 AR=2.00
• •
10 S
A
TR=O.001
p /
~ /
I~ 101 ill O 10 o Q. 10 " 1 10 - 2
+ 10
0
r 20
, 30
40
n (NO. OF NODAL LIN~IN Y-DIRECTION)
Fig. 8. Power radiation for different values of aspect ratio and low value of m. 10 5 . •- - " - - o - - -
pI--
AR=0.5
4,,
AR=I.00
•
AR-2.00
/
TR=O.001
[
10 4
z
0 I--
t~ 10 3 UJ
0
0..
10 2 0
10
20
30
40
n (NO. OF N O D A L LINESlN Y-DIRECTION)
Fig. 9. Power radiation for different values of aspect ratio and high value of m. the value of n. These variations of acoustic power radiation can be explained with the help of Figs 10-13. These figures show the variation of dimensionless frequency factor with the variation of aspect ratio for different values of m and n. It is observed that, at all mode orders, the dimensionless frequency factor increases with decrease in aspect ratio. As the dimensionless frequency factor increases, the frequency of vibration also increases, which in turn increases the total acoustic power. Therefore,
Acoustic power radiation from a rectangular plate
19
10 8 AR=0.S0
0 I-. M.
10 7
8
AR=I.00
¢
AR=2.00
m=4 10 6
f= u.
10 5
~ 10 4 ~ 10 3 10 2
i
',
12
1
22
i
32
n (NO. OF NODAL LINE, IN Y-DIRECTION)
Fig. 10. Dimensionless frequency factor for different values of aspect ratio and low value of re.
10 8 ~ <
AR=0.S0
----e----
AR=1.00
-
AR=2.00
= m z 10 7
10S4 0
40
n (NO. OF NODAL LINE$1N Y-DIRECTION)
Fig. 11. Dimensionless frequency factor for different values of aspect ratio and high value of re.
average acoustic power radiation increases as the aspect ratio is decreased. At lower mode orders, the a m o u n t of power radiation mainly depends upon the effective radiating surface of the plate. The influence of frequency over the power radiation is less dominant in this case. Effective radiating
20
Md. K. Rahman, A. Ertas 10 7 1 ~ :
106
AR=0.50
8
AR=I.00
•
AR=2.00
~== '°s 1 u,. 1041 ; ,021 101| 0
!
i
i
10
20
30
40
m (NO. OF NODAL LINE$1N X-DIRECTION)
Fig. 12. Dimensionless frequency factor for different values of aspect ratio and low value ofn. 10 7
10 6
== ~: 10 5 M.
-~ 10 4
n=20 -.---.-o---- AR=O.S0 4
AR=I.00
•
AR=2.00
10 3 10
20
30
40
m (NO. OF NODAL LINI~IN X-DIRECTION)
Fig. 13. Dimensionlessfrequencyfactor for differentvalues of aspect ratio and high value ofn.
surface is more in the case of a plate with lower aspect ratios at low values of mode order in the x direction. For this reason, at low mode numbers in the x direction, the plate with a lower aspect ratio radiates more power. This is shown in Fig. 8. At high values o f m in the lower range o f n, the plate
Acoustic power radiation from a rectangular plate
21
with higher aspect ratio radiates higher amounts of power, as shown in Fig. 9. This is because, at these mode orders, the effective radiating surface is greater in the case of a plate with higher aspect ratio, as the longer side of the plate is divided into fewer waves. The effective radiating surface decreases with increasing mode order but the frequency of vibration increases. Here the increase in the frequency of vibration far exceeds the reduction in the effective radiating surface. Thus, the frequency of vibration begins to play a dominating role over the acoustic power radiation from the plate. From Fig. 11, it is seen that the increase in the frequency of vibration with increase in the mode numbers is more in the case of a plate with an aspect ratio of 0.50. The amount of acoustic power radiation shows a steeper rise in this case, as shown in Fig. 8. The amount of power radiation from the plates with the other two aspect ratios is smaller.
CONCLUSIONS The purpose of this study was to investigate the noise-generating characteristics of a rectangular flat plate subjected to mixed boundary conditions. The results indicate that: 1. The average acoustic power radiated from one side of a baffled plate vibrating in its natural modes increases with increase in mode order. In the lower range of mode orders, this increasing tendency is high. At the higher range of mode order, the acoustic power radiation is in a stable state with increase in mode order. After a certain mode number, the acoustic power radiation from the plates begin to show a wavy nature. 2. The average power radiated from a baffled plate increases with increase in thickness ratio. 3. With increase in aspect ratio, the average power radiated from a baffled plate decreases at low values of m. For high values of m in the low range of n, the average power radiation increases with increasing aspect ratio. However, this tendency gradually reverses at the higher range of m and n values.
REFERENCES !. Warburton, G. B., The vibration of rectangular plates. Proc. Inst. Mech. Eng., 168 (1954) 371--84. 2. Young, D. & Felger, R.P., Tables o f Characteristic Functions Representing Normal Modes o f a Vibrating Beam. The University of Texas, Publication No.
22
3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.
Md. K. Rahman, A. Ertas
4913, Bureau of Engineering Research, Engineering Research Series, No. 44, 1949. Skudrzyk, E.J., Grandlegan der Akustic. Springer-Verlag, Vienna, Austria, 1954. Skudrzyk, E.J., Sound radiation of a system with a finite or infinite number of resonances. J. Acoustical Soc. of America, 30 (1958) 1152. Feit, D., Sound radiation from orthotropic plates. J. Acoustical Soc. of America, 47 (1970) 388-9. Lowenthal, S. & Tournois, P., Radiation impedance of membranes and plates: Their acoustic coupling with the propagating medium. J. Acoustical Soc. of America, 35 (1963) 1423-8. Thompson, W. Jr & Rattayya, J, V., Acoustic power radiation by an infinite plate excited by a concentrated moment. J. Acoustical Soc. of America, 36, (1964) 1488-90. Tsakonas, S., Chen, C. Y, & Jacobs, W. R., Acoustic radiation of an infinite plate excited by the field of a ship propeller. J. Acoustical Soc. of America, 36 (1964) 1708-17. Maidanik, G., Acoustic radiation from a driven infinite plate backed by a parallel infinite baffle. J. Acoustical Soc. of America, 41--42 (1967) 27-3 t. Magrab, E. B. & Reader, W. T., Farfield radiation from an infinite elastic plate excited by a transient point loading. J. Acoustical Soc. of America, 43--44 (1968) 1623-7. Suzuki, II. & Tichy, J., Sound radiation from an elastically supported circular plate. J. Acoustical Soc. of America, 65 (1979) 106-11. Feit, D., Pressure radiated by a point excited elastic plate. J. Acoustical Soc. of America, 40 (1966) 1494 9. Felt, D. & Saurenman, J. J,, Sound Radiation by Beam Stiffened Plates. Cambridge Acoustical Association, Cambridge, MA, Ref. No. U-356-213, 1971. Gorman, R. M., Vibration and acoustic radiation from line excited ribstiffened damped plates in water. Ph.D thesis, the Pennsylvania State University, Pennsylvania, USA, 1974. Ahmed, A., Acoustics of rectangular flat plates. MSc thesis, BUET, 1988. Mandal, N. K., Acoustics of rectangular flat plates of mixed boundary conditions. MSc Thesis, BUET, Bangladesh University of Engineering and Technology, Bangladesh, 1989. Peng, H. &Keltie, R. F., Sound radiation from finite plates under the action of multiple random point forces. J. Acoustical Soc. of America, 85(1) (1989) 57-67. Wallace, C. E., Radiation resistance of a rectangular panel. J. Acoustical Soc. ~?['America, 51(3) (part 2) (1972) 936-45.