Effect of the Moving Force on the Radiated Sound. from Rectangular Plates with Elastic Support

Transcription

1 Adv. Studies Theor. Phys., Vol. 3, 9, no. 9, Effect of the Moving Force on the Radiated Sound fro Rectangular Plates with Elastic Support Yufeng Qiao* and Qiai Huang School of Mechanical of Science & Engineering Huazhong University of Science & Technology Wuhan 4374, China Corresponding Author. Tel: E-ail Address: Astract The effect of the oving force on the radiated sound fro the rectangular plates with elastic support is revealed in this paper. An analytical ethod in conjugation with nuerical ethod is also developed to calculate the sound radiated fro the plates. According to this ethod, the radiated sound is calculated for rectangular plate with different elastic support oundary conditions. By coparing these calculation results, the effect of the elastic support on the radiated sound is also studied. Moreover, the effect of the plate ode density is included in the research results. Keywords: Radiated Sound, Rectangular Plates, Elastic Support, Moving Force 1. Introduction Rectangular and rectangular-like plates with elastic-support are frequently encountered in echanical product design. They often play the role of the ain noise source of the product. Thus, control of the radiated noise fro the plates ecoes an iportant issue. Moving force is an ordinary for of force conditions in design. Therefore, it is necessary to otain the effect of the oving force on the radiated sound fro the rectangular plate with elastic support. Since no analytical ethod exists to solve the forced response of the rectangular plate with elastic support, we have to adopt approxiate solutions in conjunction with nuerical ethods. Different techniques have een developed to solve the prole in the past. These techniques include the spline technique [1], Galerkin s ethod [], the finite strip ethod [3], the Ritz ethod [4] and the Rayleigh-Ritz ethod [5]. In fact, the force applied on the plate aye a oving force. Therefore, it is also worth value to study the effect of the oving force on the radiated sound. Gadeyan and Oni [6] studied the dynaic response of the rectangular plate excited y oving force. Takaatake [7] presented a siplified

2 334 Yufeng Qiao and Qiai Huang analytical ethod for the rectangular plate sujected to oving force to exaine the effect of the additional ass due to oving force. The purpose of this paper is to investigate the effect of the oving force on the radiated sound fro the rectangular plate with elastic support. By coparing the oundary condition applied at different sides of the plates, the effect of the oving force and the elastic oundary on the radiated sound fro the plate is studied.. Sound Radiation Governing Equations Consider an elastically supported rectangular plate, which is shown in Fig.1, excited y a point force. The governing equation for the transverse displaceent w( x, y, t ) is given y [8, 9] 4 w( x, y, t) w( x, y, t) D w( x, y, t) + cd + ρh = f () t. (1) t t D= Eh 3 /1 1 μ, E is the Young s odulus, Here D is the flexural rigidity defined y ( ) h is the plate thickness, μ is the Poisson s ratio, c d is the daping coefficient of the rectangular plate, f ( t ) is the force applied on the plate, t is tie, ρ is the ass density, 4 = = + + x x y y 4 4 and is the Laplace operator. Figure.1 Rectangular plate with elastic support and coordinate syste For a haronic excitation of circular frequency ω, we set j t w x, y, t = W x, y e ω, () ( ) ( ) F( t) j t Fe ω W ( x, y ) can e written as the superposition of adequate shape functions =. (3), (4) = 1 n= 1 (, ) = Φ (, ) W x y W x y

3 Effect of the oving force on the radiated sound 335 where Φ ( x, y) is the odal shape functions, Consider a haronic point force f ( ) respectively, as shown in Fig.. The oving force f ( ) W is the displaceent coefficient. t that oves on the plate at speed of v x and v y, t can e expressed as () ( ) δ ( ) δ ( ) f t = F t x x vxt y y vyt, (5) where F () t is the agnitude of the oving force, δ ( x) and δ ( y) are the Dirac Delta x, y is the initial location that the oving force applies on the plate surface. functions, ( ) Figure. Rectangular plate excited y a haronic oving force Using the orthogonal property and assuing proportional daping in the forced virating syste, sustitute Eq. (-5) into Eq. (1), we otain the noralized equation of transverse displaceent W && + ξωw & + ωw = F, (6) where ξ is the generalized daping factor defined y ξ = cd /ρhω. (7) ω is the generalized natural frequency defined y 1 a 4 ω = ( x, y) ( x, y) dydx Φ Φ, (8) F is the noralized force defined y y z ( x, y ) a 1 F = f () t Φ ( x + vxt, y + vyt), (9) v y f ( t ) v x is the generalized ass of the rectangular plate defined y a = ρhφ (, ) x y dydx, (1) For a lightly daped syste ( ξ < 1), we solve Eq. (4) in tie doain y Duhael integral [1]. Then the displaceent coefficient W is given y W = exp ξ ω t a sinϖ t+ cosϖ t ( )( ) 1 t + F exp ξω ( t τ) sinϖ ( t τ) dτ ϖ, (11) where ϖ is the generalized natural frequency of daped free viration defined y x

4 336 Yufeng Qiao and Qiai Huang ϖ = ω 1 ξ, (1) the paraeters a and can e otained fro the initial transverse displaceent w( x, y,) and the initial velocity w& ( x, y,) a = a (,,) + ξω (,,) Φ (, ) ϖ Φ ( x, y) dydx w & x y w x y x y dydx, (13) a = a a (,,) Φ (, ) Φ ( x, y) dydx w x y x y dydx. (14) Consider a case of a haronic point force oves at speed of v x along with the x direction fro the initial location ( x, y ), the oving force f ( t ) can e written, in ters of the ass of the oving force defined y, as f( x, y, t) = f w μ f v w w x + v x +, (15) x xt t where g is the acceleration due to gravity, μ f is the ass ratio e the ratio of the ass due to the oving force to the rectangular plate defined y μ / f = f g. So, the noralized force is expressed as μ Φ * f Φ Φ F = F vx W + vx W & +ΦW && = 1 n= 1 x x. (16) Truncate the indices and n after u and n u ters, respectively, Eq. (6) is expressed in atrix for as ([ I ] + [ M] ){ Q && } + ( Cd + [ C] ){ Q & } + ([ Kd] + [ K] ){ Q} = { F}, (17) where [ I ] is the identity atrix which results fro noralization y the generalized ass of the plate, the vector { Q } is coposed y the displaceent coefficient y { } [ M ] is the generalized ass atrix defined y W defined Q = W, L, W n, L, W,, u L W u unu, (18) T

5 Effect of the oving force on the radiated sound 337 [ C d ] and [ ] Φ Φ Φ Φ Φ Φ Φ 11 L L L L L L L L L L Φ Φ Φ Φ Φ Φ Φ L L L 1 n u 1n u 1n u 1nu [ M ] = μ L L L L L L L F Φ Φ Φ Φ Φ Φ Φ L L L u1 u1 u1 u1 L L L L L L L ΦΦ u u 11 ΦΦ u u 1n Φ u Φ u u u1 Φ u u L L L u u u u u u u u C are the generalized daping atrices defined y 11 1nu 11 u1 11 unu 1nu 11 1nu 1nu u1 1nu unu u1 11 u1 1nu u1 u1 unu, (19) ξω L ξ1 n ω u 1nu, () [ Cd ] = L ξ u1ω u1 L ξω u u u u Φ11Φ Φ 11 11Φ 1n Φ11Φ u 1 Φ11Φ u un u L L L L L L L L L L Φ1 n Φ u 11 Φ1 n Φ u 1n Φ u 1n Φ u u1 Φ1 n Φ u unu L L L 1 n u 1n u 1n u 1nu, (1) C = vxμ L L L L L L L F Φ u1φ11 Φ u1φ1n Φ u u1φu1 Φ u1φunu L L L u1 u1 u1 u1 L L L L L L L ΦΦ u u 11 ΦΦ u u 1n Φ u Φ u u u1 ΦΦ u u u u L L L u u u u u u u u K are the generalized stiffness atrices defined y [ ] [ K d ] and [ ]

6 338 Yufeng Qiao and Qiai Huang ω 11 L ω 1nu, () [ Kd ] = L ω u 1 L ω u u Φ11Φ Φ11Φ 11 1n Φ11Φ u 1 Φ11Φ u un u L L L L L L L L L L Φ1 n Φ u 11 Φ1 n Φ u 1n Φ u 1n Φ u u1 Φ1 n Φ u unu L L L 1n u 1n u 1n u 1nu, (3) [ K] = μfv L L L L L L L x Φ u1φ11 Φ u1φ1n Φ u u1φu1 Φ u1φunu L L L u1 u1 u1 u1 L L L L L L L Φun Φ u 11 Φ un Φ u 1n Φ u un Φ u u1 Φ un Φ u unu L L L u u u u u u u u F is the generalized force vector defined y { } { } F = F, L, Fn, L, F,, u L F u unu, (4) Φij Φij Φ ij =Φ ij vt x, y, Φ ij =, Φ ij =. x x Φ x, y of the rectangular plate is given y [9] Φ ( x, y) = X ( x) Yn ( y), (5) Y y are taken to e the ode shapes of the associated un-daped where ( ) The ode shape function ( ) where ( ) X x and ( ) n eas, having the sae oundary conditions as the plate in x and y directions, respectively. Consider a un-daped ea of length l, the governing equation is given y [9] 4 w( x, y) w( x, t) EI + ρ 4 = (6) x t where EI is the flexural rigidity defined y EI = ( 1 μ ) D, ρ is the ass per unit w x, t = X x sin ωt, we otain length. Suppose ( ) ( ) ( ) ( ) X x = A sinα x+ A cosα x+ A sinhα x+ A coshα x, (7) where the paraeters A 1, respectively, α is the eigenvalue of the A, A 3 and A 4 are deterined y oundary conditions th ode defined y T

7 Effect of the oving force on the radiated sound 339 α = ρ ω EI, (8) S / th is positive integer, S is the ea section area, ω is the ode eigenfrequency. For a rectangular plate with elastic support as shown in Fig.1, the oundary conditions is given y (1) at edge y = () at edge y = (3) at edge x = (4) at edge x = a w w w R D + μ, = y y x 3 3 w w Tw + D + D 3 ( μ ) =, y x y w w w R + D + μ, = y y x 3 3 w w Tw D + D 3 ( μ ) =, y x y w w w R1 D + μ, = x x y 3 3 w w Tw 1 + D + D 3 ( μ ) =, x x y w w w Ra + D + μ, = x x y (3) 3 3 w w Tw a D + D 3 ( μ ) =, x x y where the paraeters R, T, R, T, R 1, T 1, R a and T a are stiffness of the rotational springs and the translational springs at every side, respectively. X x as a strip eleent of the For the ea odal shape function in Eq. (5), we treat ( ) plate. So, the flexural rigidity EI of the ea can e replaced y D ( 1 μ ) Eq. (9) and (3), we otain the following hoogenous linear equation [11] (9) (3) (31). According to u R u R A 3 3 T T 1 A RR u RR u = u cosu+ R sin sin cos cosh sinh sinh cosh A3 T u u u+ RT u u u+ RT u u u+ RT u usin u+ R cos cos sin sinh cosh cosh sinh A4 R u u u RR u u u+ RR u u u RR u 3 where RT = Ta / EI, 3 4 RT = Ta / EI, RR = Ra / EI, RR = Ra / EI, 4 ρ ω /, (33) u = S a EI. According to Eq. (33), the eigenfunction is given y u 8 ( 1 cosucosh u) + u 7 ( RR + RR)( sin ucosh u+ cosusinh u) + u 6 ( R ) 5 RRRsin usinh u + u ( RT + RT)( cosusinh u sin ucosh u) +

8 34 Yufeng Qiao and Qiai Huang ( ) ( T R + T R) 1+ cos cosh + ( T R + T R) cos cosh + R R( T + T)( cos sinh + sin cosh ) + ( T Tsin sinh ) + ( )( ) ( ) 4 u R R R R u u R R R R u u u 3 R R R R u u u u u R R u u u R R R R u u u u R R R R u u Solving Eq. (34), we otain the positive eigenvalues u 1, u, L u. Since the natural frequency of ea can e expressed as 4 ω = u EI / ρ Sa, (35) T T R + R sin cosh cos sinh + T T R R 1 cos cosh =. (34) the odal shape function in x direction X ( x ) can e otained according to Eq. (7). Siilarly, we can suppose Y y = B sin β y+ B cos β y+ B sinh β y+ B cosh β y, (36) ( ) n n n n n n n n n where the paraeters B 1n, respectively, n B n, B 3n and β is the eigenvalue of the B 4n are deterined y oundary conditions, th n ode, n is positive integer. Using the ethod Y y in y direction. entioned aove, we otain the odal shape function n ( ) Once we otain the ode shape function, the transverse displaceent w( x, y, t ) of the rectangular plate can e otain y solving Eq. (17). Especially, for the case of low ass ratio, μ F can e ignored. Thus, the transverse displaceent of the rectangular plate is further siplified to fφ (, ) t x y w( x, y, t) = exp ξω ( t τ) Φ ( vxτ, y ) sin ϖ ( t τ) dτ o = 1 n= 1 ϖ. (37) For a rectangular plate in an infinite affle as shown in Fig.3, the radiated sound pressure fro the plate can e otained y evaluating the Rayleigh surface integral where each eleent area on the plate is treated as a siple point sound source and its contriution is added with an appreciate tie delay. The sound pressure at oservation location (,, ) P x y z is given y [1] P P P jkr jωρ e p ( Q) = w( P) ds π &, (38) S r where ρ is the air density, S is the surface area of the plate, r is the distance etween the oservation location and the infinitesial eleent on the plate.

9 Effect of the oving force on the radiated sound 341 Figure.3 Coordinate syste for evaluation of structural sound pressure The structural sound intensity at the oservation location Q is [1] 1 * I ( Q) = Re p( Q) v ( Q), (39) v Q is the velocity of acoustic ediu at the location Q, * denotes coplex where ( ) conjugate. The sound power radiated into the sei-infinite space aove the plate is given y ' W = I Q ds, (4) ' S ' where S is an aritrary surface which covers area S. According to Eq. (38) and (39), Eq. (4) takes the for jkr ωρ je * ' W = Re w( P) v ( Q) dsds ' 4π S S & r. (41) * Since { J} ( J J ) ( ) Re = + /, where J is a coplex function, we note that jkr kr kr je * 1 je * * je Re w& ( P) v ( Q) = w& ( P) v ( Q) + w& ( P) v( Q) r r r 1 sinθ + icosθ * * sinθ icosθ = w( P) v ( Q) w ( P) v( Q) & + & r r. (4) ' When we suppose S to S, due to the reciprocity relationship etween r 1 and r (Fig.3), ( ) w( Q) v Q = &. Eq. (4) is siplified as ( kr) jkr je * sin * Re w & ( P) w ( Q) = w( P) w ( Q) r & & r &. (43) Then Eq. (41) is given y ωρ a a sin ( kr ) * ' ' W = w( P) w ( Q) dxdy dxdy 4π & r &. (44) When the plate is divided into N eleents, Eq. (44) is approxiated y a finite series

10 34 Yufeng Qiao and Qiai Huang ωρ * sin kr &( r ) & ( r ) ( Δ ), (45) N N W w ri w rj S 4π i = 1 j = 1 r where r i and r j are the position vectors of the center point of two eleents. Figure.4 Discretization of the Rectangular Plate Surface When we define the velocity atrix as (,,,, T w& = w& 1 L w& i L w& N), where T denotes transposition, W is expressed as H W = w& Zw&, (46) where H denotes coplex conjugate and transposition, Z is the resistance atrix defined y sin kr1, sin kr1, N 1 L kr1, kr 1, N sin kr,1 sin kr, N ωρ 1 S L Z = kr,1 kr, N. (47) 4π Nc M M O M sin krn,1 sin krn, L 1 krn,1 krn, where c is the sound velocity in air. In free sound field, according to [1], the radiated sound pressure level L p is given y W 4 L = p 1lg 1lg S 1lg W ρc, (48) 1 where W is reference sound power and defined y 1 Watt. 3. Nuerical Results and Discussions 11 3 Consider a flat rectangular steel plate ( E =.1 1 Pa, ρ = 78 kg /, μ =.8) of size.5.3. When the teperature is o 3 C, the properties of air are ρ = 1.1 kg / and c = 343 / s. The daping coefficients otained fro experients [15, 16] adopted for

11 Effect of the oving force on the radiated sound 343 the nuerical coputations are cd = ρhω/1π, where h is the plate thickness. Choose the location (.5,.15,.5) as the oservation location and 5-1Hz as the frequency range adopted in this nuerical calcualtion. The stiffness of all the springs is 5 Newton /. (a) C-ES-C-ES () ES-C-ES-C (c) ES-ES-ES-ES Figure.5 Rectangular plates with elastic support oundary conditions, where C denotes the claped support, ES denotes the elastic support Consider a case of low ass ratio, then the ass ratio μ f can e ignored. We set the jωt agnitude of the haronic oving force is F ( t) = e Newton. When the force oves at a speed of vx =.5 / s fro the initial location (,.1) along with the line y =.1, the radiated sound pressure level fro the plates at the oservation point is shown in Fig.6.

12 344 Yufeng Qiao and Qiai Huang Figure.6 Radiated sound pressure level fro the rectangular plates in a case of low ass ratio: (a) the sound pressure level radiated fro the ES-C-ES-C plate, () the sound pressure level radiated fro the C-ES-C-ES plate, (c) the sound pressure level radiated fro the ES-ES-ES-ES plate. Coparing the curve of the sound-pressure-level versus frequency, the radiated sound pressure level fro the ES-ES-ES-ES plate (Fig.6 c) is flatter than fro the ES-C-ES-C plate (Fig.6 a) and the C-ES-C-ES plate (Fig.6 ). Especially, when the frequency is higher than 4Hz, the fluctuation of the radiated sound pressure level fro the ES-ES-ES-ES plate (Fig.6 c) is quite sall. Notice that, the value of the radiated sound pressure level fro the ES-ES-ES-ES plate (Fig.6 c) is higher in the frequency range of 4-1Hz. When the oundary condition at the end of the plate is changed fro claped-support to elastic-support (Fig.6 a, c), the frequencies, which ake the radiated sound pressure level fro the ES-C-ES-C plate to axiu or iniu value, does not appear. Moreover, when the frequency is lower than Hz, the radiated sound pressure level fro the ES-ES- ES-ES plate is higher than fro the ES-C-ES-C plate. Changing the oundary condition at the side of the plate fro claped-support to elastic-support (Fig.6, c), when the frequency is in the range of -4Hz, the radiated sound pressure level fro the C-ES-C- ES plate is quite lower than fro the ES-ES-ES-ES plate. Notice that, coparing the radiated sound pressure level fro the ES-C-ES-C plate and the C-ES-C-ES plate (Fig.6 a, ), the curve of the forer is also flatter than the later. We find that the elastic support has great effect on the radiation sound. More elastic-support in the oundary condition of the plate edge ake the curve of the sound-pressure-level versus frequency flatter. Because the plate thickness variation will cause the ode density variation, for investigating the effect of the ode density on the odulated radiated sound roughness fro the plate, for further investigating the effect of the elastic support on the radiation sound, we consider the plate thickness is variale in the range of The radiated sound pressure level fro the rectangular plates is shown in Fig.7.

13 Effect of the oving force on the radiated sound 345 Figure.7 Sound pressure level radiated fro the rectangular plates when the plate thickness is variale in the case of low ass ratio: (a) the sound pressure level radiated fro the ES- C-ES-C plate () the sound pressure level radiated fro the C-ES-C-ES plate, (c) the sound pressure level radiated fro the ES-ES-ES-ES plates When the plate thickness increases, the structural ode density ecoes sparse [17]. With the plate thickness increasing, although the elastic support has less effect on the radiated sound pressure level fro the plate, there are still soe differences. When changing the oundary condition at the end of the plate fro claped-support to elasticsupport, the frequency range, which ake the radiated sound pressure level fro the ES- ES-ES-ES plate higher than fro the ES-C-ES-C plate (Fig.7 a, c) in low frequency range, ecoes wider with the plate thickness increasing. Siilarly, when change the oundary condition at the side of the plate fro claped-support to elastic-support, the frequency

14 346 Yufeng Qiao and Qiai Huang range, which ake the radiated sound pressure level fro the ES-ES-ES-ES plate higher than fro the C-ES-C-ES plate (Fig.7, c) in iddle frequency range, also ecoes wider with the plate thickness increasing. Coparing Fig.6 and Fig.7, we find the sae conclusion with the frequency increasing. Whatever the plate ode density is, ore elastic-support oundary condition akes the fluctuation of the radiated sound pressure level fro the plate less. However, there are less frequencies, which ake the radiated sound pressure level fro the plate to axiu value, appears with the plate thickness increasing. When in a case of high ass ratio, the effect of the oving force speed should e taken into account. For investigating the effect of the oving force on the radiated sound in the case of high ass ratio, we choose two different oving force speeds. Suppose the agnitude of the oving force is f = 1e jωt newton, when the force oves at low speed of vx =.5 / s fro the initial location (,.1) along with the line y =.1, the radiated sound pressure level fro the rectangular plates is shown in Fig.8. Figure.8 Sound pressure level radiated fro the rectangular plates in the case of high ass

15 Effect of the oving force on the radiated sound 347 ratio case when the force oves at a low speed: (a) the sound pressure level radiated fro the ES-C-ES-C plate, () the sound pressure level radiated fro the C-ES-C-ES plate, (c) the sound pressure level radiated fro the ES-ES-ES-ES plate In the case of high ass ratio, coparing Fig.6 and Fig.8, the curve of the sound-pressurelevel versus frequency is quite different fro the one in the case of low ass ratio. Fewer frequencies ake the sound pressure level to axiu value in the case of high ass ratio whatever the oundary condition is. However, the fluctuation of the radiated sound pressure level for the plate is larger than in the case of low ass ratio. Notice that, ore the edge of the plate is elastically supported, the ass ratio has fewer effect on the sound radiated fro the plate. The effect of the elastic support on the radiated sound fro the plate is also different fro in the case of low ass ratio. There is no difference for the effect of elastic support on the sound radiated fro the ES-ES-ES-ES plate (Fig.8 c). However, when the oundary condition at the end of the plate is changed fro elasticsupport to claped-support (Fig.8 a), the curve of the sound-pressure-level versus frequency is quite flatter in low frequency range than the one in the case of low ass ratio. Siilarly, when changing the oundary condition at the side of the plate fro elasticsupport to claped-support, siilar phenoenon is oserved. In order to investigate the effect of the speed of the oving force, we use the sae oving force chose in the case of high ass ratio. When the force oves at high speed of vx = 5 / s fro the initial location (,.1) along with the line y =.1, the radiated sound pressure level fro the rectangular plates is shown in Fig.9.

16 348 Yufeng Qiao and Qiai Huang Figure.9 Sound pressure level radiated fro the rectangular plates in the case of high ass ratio when the force oves at high speed: (a) the sound pressure level radiated fro the ES-C-ES-C plate, () the sound pressure level radiated fro the C-ES-C-ES plate, (c) the sound pressure level radiated fro the ES-ES-ES-ES plate. Coparing Fig.8 and Fig.9, we find the effect of the high speed of the oving force on the radiated sound fro the plates. In the low frequency range of -1Hz, the sound pressure level radiated fro the ES-C-ES-C plate and the ES-ES-ES-ES plate (Fig.9 a, c) is higher than in the case of low speed of the oving force. However, the sound pressure level radiated fro the C-ES-C-ES plate (Fig.9 ) is lower than in the case of low speed in the sae frequency range. Notice that, although the curve of the sound-pressure-level versus frequency is quite siilar with the one in the case of low speed when the frequency is higher than 1Hz, the radiated sound pressure level fro the plate is always higher than the one on the case of low speed of the oving force. There is also soe difference of the effect of the elastic-support on the sound radiated fro the plate. When oundary condition at the side of the plate is elastic-support (Fig.9 a, c), the sound pressure level has less fluctuation in the low frequency range. On the other hand, when the oundary condition at the end of the plate is elastic-support (Fig.9, c), the sound pressure level has less fluctuation in the high frequency range. Notice that, in the low frequency range, ore frequencies ake the radiated sound pressure level fro the plates to axiu value whatever the oundary condition is. For further investigating the effect of the speed of the oving force on the sound radiated fro the plate, we suppose the plate thickness is variale in the range of The radiated sound pressure level fro the rectangular plates at low speed and high speed, respectively, is shown in Fig.1.

17 Effect of the oving force on the radiated sound 349

18 35 Yufeng Qiao and Qiai Huang Figure.1 Radiated sound pressure level fro the rectangular plates in the case of high ass ratio when the plate thickness is variale: (a1) the radiated sound pressure level fro the ES-C-ES-C plate when the force oves at low speed of vx =.5 / s, (a) the radiated sound pressure level fro the ES-C-ES-C plate when the force oves at high speed of vx = 5 / s, (1) the radiated sound pressure level fro the C-ES-C-ES plate when the force oves at low speed of vx =.5 / s, () the radiated sound pressure level fro the C-ES-C- ES plate when the force oves at high speed of vx = 5 / s, (c1) the radiated sound pressure level fro the ES-ES-ES-ES plate when the force oves at low speed of vx =.5 / s, (c) the radiated sound pressure level fro the ES-ES-ES-ES plate when the force oves at high speed of v = 5 / s. x With the plate thickness increasing, although the speed of the oving force has less effect on the radiated sound fro the plates, there are still soe differences. In the low frequency range, the value difference etween the radiated sound pressure level fro the ES-C-ES-C plate and the ES-ES-ES-ES plate (Fig.1 a, c) in the case of high speed and in the low speed (Fig.1 a1, c1) ecoes fewer with the plate thickness increasing. Siilarly, the value difference etween the radiated sound pressure level fro the C-ES-C-ES plate in the high-speed case (Fig.1 ) and in the low-speed case (Fig.1 1) has sae trend with the plate thickness increasing. Coparing Fig.8, Fig.9 and Fig.1, we find that, with the plate thickness increasing, the elastic-support always has great effect on the radiated sound fro the plate whatever the speed of the oving force is. When the oundary condition at the side of the plate is the elastic-support, the curve of the sound-pressure-level versus frequency is flatter in low frequency range, which ecoes wider with the plate thickness increasing. Siilar phenoenon is also oserved in the iddle frequency range when the oundary condition at the end of the plate is elastic-support with the plate thickness increasing. 4. Conclusions Using developed analytical ethod in conjunction with nuerical ethod, the effect of the oving force and the elastic-support on the radiated sound fro the rectangular plate is studied in this paper y coparing the different agnitude and oving speed of the force applied on the plate. Based on the Rayleigh integral, the radiated sound pressure level fro

19 Effect of the oving force on the radiated sound 351 the ES-C-ES-C plate, the C-ES-C-ES plate and the ES-ES-ES-ES plate are calculated when different forces apply on the plates, respectively. According to the calculation results, the effects of the elastic-support and the oving force on sound radiated fro the rectangular plate are revealed, respectively. In the case of low ass ratio, through calculation results of the radiated sound pressure level fro the rectangular plates with elastic support, we get a conclusion that the curve of the sound-pressure-level versus frequency ecoes flat when ore edge of the plate is elastically supported. The sound pressure level in the case of high ass ratio is also verified this conclusion. In order to investigate the effect of the oving force, we choose two cases of low oving speed and high oving speed, respectively. The calculation results show that the speed of the oving force can change the sound radiated fro the plate with elastic support to ecoe flat in special frequency range. The results also show that the plate ode density on the radiated sound fro the plates. References [1] S.R. Soni, R.K. Sankara. Viration of non-unifor rectangular plates: a spline technique ethod of solution. Journal of Sound and Viration 1974, 35(1): [] C. Filipich, P.A.A. Laura, R.D. Santos. A note on viration of rectangular plates. Journal of Sound and Viration 1977, 5(3): [3] Mukherjee, M. Mukhopadhyay. Finite eleent for flexural viration analysis of plates having various shapes and varying rigidities. Coputational Structure 1986, 3(6): [4] R.H. Gutierrez, P.A.A. Laura, R.O. Grossi. Viration of rectangular plates of ilinearly varying thickness and with oundary conditions. Journal of Sound and Viration 1981, 75:33-38 [5] Y.K. Cheung, D. Zhou. The free viration of rectangular plates using a new set of ea function with the Rayleigh-Ritz ethod. Journal of Sound and Viration 1999, 3(5): 73-7 [6] J.A. Gaadeyan, S.T. Oni. Dynaic ehavior of eas and rectangular plates under oving loads. Journal of Sound and Viration 1995, 18(5): [7] H. Takaatake. Dynaic analysis of rectangular plates with stepped thickness sujected to oving loads additional ass. Journal of Sound and Viration 1998, 13(5): [8] T. Takahagi, M. Nakai, Y. Taai. Near Field Sound Radiation Fro Siply Supported Rectangular Plate. Journal of Sound and Viration, 1995, 185(3): [9] A.W. Leissa. Viration of Plates. Washington DC: NASA SP [1] T.P. Chang, H.C. Chang. Viration and uckling analysis of rectangular plates with nonlinear elastic end restraints against rotation. International Journal of Solids Structure 1997, 34(18): [11] W.T. Thoson. Theory of viration with applications. New Jersey: Prentice Hall. 197 [1] L. Rayleigh. Theory of Sound (nd Edition). New York: Dover Pulications [13] S. Schedin, C. Laourge, A. Chaigne. Transient sound fields fro ipacted plates: coparison etween nuerical siulations and experients. Journal of Sound and Viration, 1999, 1(3):

20 35 Yufeng Qiao and Qiai Huang [14] C.C. Sung, J.T. Jan. The response of and sound power radiated y a claped rectangular plate. Journal of Sound and Viration 1997, 7(3): [15] T. Takahagi, M. Nakai, Y. Taai. Near Field Sound Radiation Fro Siply Supported Rectangular Plate. Journal of Sound and Viration, 1995, 185(3): [16] Yuh-Chun Hu, Shyh-Chin Huang. The frequency response and daping effect of threelayer thin shell. Coputers and Structures,, 76(5): [17] G. Xie, D.J. Thopson, C.J.C. Jones. Mode count and odal density of structural systes: relationships with oundary conditions. Journal of Sound and Viration, 4, 7(3-5): Received: Feruary, 7