Proceedings of the Institute of Acoustics

Sound radiation from composite plates 1

H Aygun London South Bank University, School of the Built Environment and Architecture, 103 2 Borough Rd, London, SE1 0AA, UK. aygunh@lsbu.ac.uk 3

4

1. Introduction 5

The vibrations of the structures that are made of isotropic and homogeneous materials has been an 6 important subject in the study of noise control, in the aeronautical industry, in the study of fluid-solid 7 interactions and in construction industry for double leaf partitions as building structures. When a sound 8 wave impinges on a structure, a part of acoustic energy is reflected back into source medium, a part is 9 dissipated in the structure and the rest of acoustic energy is transmitted through the structure into the 10 other medium. In building acoustics applications, the reflected acoustic energy builds up a reverberant 11 sound field in the source room that in turn vibrates the walls. The vibrations in the common walls radiate 12 sound directly into the receiver room. The vibrations in the other walls of the source room travel as 13 structure-borne noise to the all walls of the receiving room and radiate sound into the receiver room. 14

Complex structures may be modelled as systems that are made of individual plate like elements. The 15 theory of vibration of porous and non-porous structures is a well-known branch of engineering 16 mechanics. Previous works on classical theory of the plate [1-7] has investigated vibration of isotropic 17 and anisotropic plates for various boundary conditions. The vibration of porous plates can be described 18 using two coupled equations [8], which are based on Biot’s stress-strain relations [9, 10] and which 19 introduce two types of compressional waves (‘fast’ and ‘slow’) and a shear wave. They assumed that 20 the thickness of plate is smaller than the wavelength and that interaction can take place between the 21 slow waves and the bending waves in the plate. They also ignored the amplitude of the fast wave. 22 Galerkin’s variational techniques were applied to porous plates [11-13], taking into account a classical 23 set of trial functions obtained from the linear combination of trigonometric and hyperbolic functions for 24 various boundary conditions. The effects of fluid loading on the vibration of rectangular porous plates 25 and on their radiated sound power was investigated by including an extra term into the equations for 26 the porous plate vibration, corresponding to the additional external force acting on the plate [14]. 27 Previous study on low frequency vibration of porous plates [15] has demonstrated the existence of low 28 frequency absorption coefficient resonance in configurations consisting of clamped poroelastic plates 29 with an air cavity between the plates and a rigid termination. An analytical model that takes into account 30 the effect of perforations and the effect of the flexural vibrations in the plates has been formulated and 31 used to calculate the insertion loss in the absence, and in the presence of air flow [16, 17]. 32

Important progress in predicting acoustic radiation from baffled structures including plates and beams 33 has been made in the last three decades. Many previous studies have focused on the calculation of the 34 radiation efficiency of these structures [18-21] and the radiation of sound from a baffled, rectangular 35 plate with edges elastically restrained against deflection and rotation [22-23]. The models used for the 36 radiation efficiency of plate-like radiators, ranged from very simple ones based on modal average 37 expressions to refined calculations of the radiation impedance matrix with cross-modal coupling have 38 been evaluated, and developed a new approach based on Taylor’s expansion of the Green’s function 39 [24]. Variational method that is based on the Rayleight-Ritz model, can be used to determine the 40 radiation of sound from the plates that are immersed either in a light fluid or in a heavy fluid by reducing 41 the quadruple integral into a double integral using a specific change of variable and by integrating the 42 double integral with a numerical method using Gaussian quadrature formulae [25-28]. 43

Recently, Aygun and McCann [29] has studied composite recycled glass bead panels in order to assess 44 their suitability for civil engineering application, especially in noisy urban environments, either as 45 structural panel components that also offer acoustic insulation or as dedicated noise barriers for outdoor 46 applications. 47

The aim of this paper is to investigate the vibrational and acoustical parameters of thin composite plates 48 that are made of fibreglass, which are used for applications ranging from aerospace and automotive to 49 construction industry. To author’s best knowledge, vibroacoustical properties of composite plate have 50 been reported in this paper for first time. The deflection of composite plate has been predicted at 51 difference locations on the plate using the classical theory of the vibrating plate for simply supported 52 and clamped boundary conditions. Computational simulations have been carried out to determine 53 deformations of the plate in 3D for different frequency ranges for simply supported and clamped 54

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Proceedings of the Institute of Acoustics

boundary conditions. Here, vibroacoustic indicators including the radiation efficiency, the mean square 1 velocity, and the radiated sound power have been computed for a square composite plate. Furthermore, 2 the radiation impedance matrix has been predicted for simply supported boundary condition by using 3 equations for eigenfunctions and Green’s function without interpolation, convergence and without 4 reducing the quadruple integral into a double integral. 5

2. Theory of plate vibration 6

Wave motion in solid structures stores energy in shear as well as in compression. Many different types 7 of waves can be seen in solid structures when a solid is excited by different ways of stressing. The solid 8 structures should have energy storage capability in the form of potential and kinetic energy in order to 9 allow wave propagation through the medium. Potential energy is stored in solid structure parts that have 10 undergone elastic deformation while kinetic energy is stored in any part of the medium that has mass 11 and is in motion. When a flat plate is subjected to a transverse, time dependent force density F ( x , y , t ), 12 the transverse deflection of the plate is governed by the fourth order differential equation. The 13 transverse vibration under free wave conditions stems entirely from inertial loading. A thin, baffled 14 square plate of dimension a x a (along aces x and y , respectively) and uniform thickness h is considered 15 in this study. The plate displacement induced by bending waves is in the direction of z axis and is 16 function of time. The geometry of the plate is shown in Figure 1. 17

18

z

y

Plate

a

Force

x 0

a

19

Figure 1: The geometry of a baffled plate. 20

21

The flexural wave equation for composite thin plate are given below; 22

where s w is the transverse plate deflection,

s w is the second order derivative of the plate deflection, 24

) 1( 12 / 2 3 v Eh D − = is the flexural rigidity, ( ) 2 2 4 ∇ ∇ = ∇ and 2 2 2 2 2 y x ∂ ∂ + ∂ ∂ = ∇ in the system of 25

coordinates ( x, y ) with x and y parallel to the plate sides of length a and b respectively, s ρ is the mass 26

density, E is the Young’s modulus of the plate, and v is the Poisson ratio of the plate. 27

The plate deflection s w for harmonic wave motion is expressed in the form of the beam functions as 28

where mn A is the unknown coefficients to be determined, m, n = 0, 1, 2, 3 ….. ∞ , and n m Y X and are 31

=

the beam functions in x and y direction respectively. 32

The beam functions have been selected to satisfy different boundary conditions at the edges of the 33 plate. An appropriate trigonometric function for vibrating beams has been used for n m Y X and different 34

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Proceedings of the Institute of Acoustics

boundary conditions. For simply supported plates, the beam functions are ), / sin( ) ( a x m x X m π = and 1

) / sin( ) ( b y n y Y n π = which should satisfy the equations of equilibrium. The boundary conditions for 2

2

simply supported edges of the plate are a x x x w w = = = ∂ ∂ = and 0 for ,0 ,0 2

and 3

2

b y y y w w = = = ∂ ∂ = and 0 for ,0 ,0 2

. The shape of each mode of vibration of plate can be determined 4

from Equation (2) by knowing the relative values of mn A and the values of n m Y X and functions. In the 5

static and dynamic analysis, the excitation function F ( x , y ) has been expanded into double infinite sine 6 series of variables x and y for each value of the couple (m, n) by using the equation below; 7

where mn F are the expansion coefficients. 9

The properties of the plate used for numerical analysis are given in the Table 1. 10

Table 1: Properties of thin composite plate 11

Young’s Modulus (Pa)

Loss Factor

Poisson Ratio

Width (m)

Thickness (m)

Density (kg/m 3 )

Length (m)

0.50 0.50 0.0025 1600 7.489 x 10 9 0.03 0.2

The square velocities of thin composite plate for simply supported boundary condition are computed 12 using a MATLAB code at the centre of the plate for 100 Hz, 500 Hz and 1 kHz in Figures 2 and 3. 13 Circular dot shows the location of the force applied to the plate. Dark red contours show the high 14 pressure areas while dark blue contours show the minimum pressure areas. 15

(a) (b) (c)

0.45

0.45

0.45

0.4

0.4

0.4

0.35

0.35

0.35

0.3

0.3

0.3

0.25

0.25

0.25

0.2

0.2

0.2

0.15

0.15

0.15

0.1

0.1

0.1

0.05

0.05

0.05

16

0.1 0.2 0.3 0.4

0.1 0.2 0.3 0.4

0.1 0.2 0.3 0.4

Figure 2: Square velocity (contour) of fibreglass composite simply supported plate at a) 100 Hz, b) 500 17 Hz, and c) 1 kHz respectively. 18

(a) (b) (c)

0.45

0.45

0.45

0.4

0.4

0.4

0.35

0.35

0.35

0.3

0.3

0.3

0.25

0.25

0.25

0.2

0.2

0.2

0.15

0.15

0.15

0.1

0.1

0.1

0.05

0.05

0.05

19

0.1 0.2 0.3 0.4

0.1 0.2 0.3 0.4

0.1 0.2 0.3 0.4

Figure 3: Square velocity (contour) of fibreglass composite clamped plate at a) 100 Hz, b) 500 Hz, and 20 c) 1 kHz respectively. 21

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Proceedings of the Institute of Acoustics

1

3. Acoustic radiation from a composite plate 2

Mean square velocity is the squared velocity normal to the surface of the plate in a given frequency 3 band and it is averaged over time and the surface area of the plate. It expresses the global behaviour 4 of the vibration of the plate. Mean square velocity of thin composite plate is defined as a time-space 5 average of the square vibrational velocity of plate. Mean square velocity of thin composite plate is given 6 by [22]; 7

where ω π 2 = T , a and b are the coordinates of plates in x and y directions, respectively, and S is the 9

surface area of the plate. 10

The total sound field generated by vibrating plates can be expressed in terms of the distribution of sound 11 pressure over a given surface. The radiated sound power expresses the sound energy radiated into the 12 surrounding environment by vibration of the plate. It can be obtained from the integration of the sound 13 intensity over the plate surface. The radiated sound power of thin composite plate is given by equation 14 (5); 15

where ) ,0, , ( t y x P is the surface acoustic pressure which is often called Rayleigh integral, and is given 17 by the Equation (6), 18

where ) 0, , ;0, , ( y x y x G ′ ′ is the Green function and it is given by the Equation (7); 20

where k is the wave number, and R is the distance between points, and is given by the Equation (8); 22

The size and shape of the vibrating plate play an important role to determine the radiation efficiency of 24 the plate. The radiation efficiency of the plate is small at low frequencies because of larger wavelength 25 while the radiation efficiency approaches unity at higher frequencies because of smaller wavelength of 26 sound. The radiation efficiency expresses the ratio of the vibration energy (mean square velocity) 27 transformed into acoustic energy (sound power). The radiation efficiency of the plate does not depend 28 on the amplitude of vibrating structure. It can be obtained using analytical methods. The radiation 29 efficiency of the plate is given by the Equation (9); 30

Where Z is the characteristic impedance of air. 32

If the plate deflection given by Equation (2) is solved, then the mean square velocity of the plate can 33 easily be predicted. A Gaussian quadrature scheme with 20 terms of the Legendre polynomial has been 34 used to expand the deflection of the plate and mean square velocity of the plate with 30 elements in 35 each direction (m, n). The mean square velocity of composite plate is shown in Figure 4 while the 36

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Proceedings of the Institute of Acoustics

predicted real part of the radiated sound power of a composite plate is shown in Figure 5. The radiation 1 efficiency of the plate has been calculated by using the radiated sound power and the mean square 2 velocity of the plate. The radiation efficiency of a square simply supported composite plate is shown in 3 Figure 6. 4

20

10

0

-10

-20

Mean Square Velocity (dB)

-30

-40

-50

-60

-70

0 100 200 300 400 500 600 700 800 900 1000

5

Frequency (Hz)

Figure 4: Mean square velocities of the composite plate at x = 0.25m and y = 0.25m 6

140

120

100

Radiated Sound Power (dB)

80

60

40

20

0 100 200 300 400 500 600 700 800 900 1000

7

Frequency (Hz)

Figure 5: The real part of radiated sound power of the composite plate at x = 0.25m and y = 0.25m 8

-10

-15

-20

-25

-30

Radiation Efficiency (dB)

-35

-40

-45

-50

-55

0 100 200 300 400 500 600 700 800 900 1000

9

Frequency (Hz)

Figure 6: Radiation efficiency of the composite plate. 10

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Proceedings of the Institute of Acoustics

4. Conclusion 1

An investigation of the vibrational and acoustical parameters of thin composite plate that is made of 2 fibreglass has been carried out. Computational simulations have been carried out to determine 3 deformations of the plate in square velocities for different frequency ranges for simply supported and 4 clamped boundary conditions. It has been shown that clamping the plate at four edges delayed first 5 resonance of the plate by 15 Hz and second resonance of the plate by 30 Hz. Vibroacoustic indicators 6 including the radiation efficiency, the mean square velocity, and the radiated sound power have been 7 computed for a square composite plate. 8

5. Acknowledgement 9

This work was supported by London South Bank University Acoustic research Centre. 10

6. References 11

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19) L. D. Pope and R. C. Leibowitz, “Intermodal coupling coefficients for a fluid-loaded rectangular 1 plate,” J. Acoust. Soc. Am.56, 408-414 (1974) 2

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29) H. Aygun and F. McCann, 2020. “Structural and acoustical performance of recycled glass bead 22 panels”. Journal of construction and building materials . Volume 258, 119581. 23

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