Sound absorption of a quadratic and cubic nonlinearly vibrating curved panel absorber

Applied Mathematical Modelling 36 (2012) 5574–5588

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Sound absorption of a quadratic and cubic nonlinearly vibrating curved panel absorber Y.Y. Lee a,⇑, J.L. Huang b, C.K. Hui a, C.F. Ng c a

Department of Building and Construction, City University of Hong Kong, Kowloon Tong, Kowloon, Hong Kong Department of Applied Mechanics and Engineering, Sun Yat-sen University, Guangzhou 510275, PR China c Department of Civil and Structural Engineering, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong b

a r t i c l e

i n f o

Article history: Received 31 May 2011 Received in revised form 28 December 2011 Accepted 4 January 2012 Available online 12 January 2012 Keywords: Nonlinear structural vibration Panel absorber Structural acoustics

a b s t r a c t This study investigated the sound absorption of a nonlinearly vibrating curved panel backed by a cavity. Very few studies on similar nonlinear structural-acoustic problems have been conducted to date, although there have been many on nonlinear plate or linear structural-acoustic problems. A curved panel is considered because the overall absorption bandwidth can be designed by appropriately adjusting the panel curvature, which is a key factor in controlling the structural resonant frequencies and absorption peaks. The theoretical formulation is developed based on the assumptions of quadratic and cubic nonlinear structural vibrations, and the linear acoustic pressure induced within the cavity. An approach based on the numerical integration method is developed to solve the nonlinear governing equation of the structural-acoustic problem. In the parametric study, the panel displacement amplitude converges with an increasing number of modes. The effects of excitation level, cavity depth, and damping factor are also examined. The quadratic and cubic nonlinearities and their effects on the sound absorption are also investigated. An experiment was conducted. The theoretical and experimental observations correspond reasonably with each other. Ó 2012 Elsevier Inc. All rights reserved.

1. Introduction The dynamics of nonlinear structures have been widely investigated in various studies (e.g. [1–3]). Some researchers have worked on curved and flat beams/panels that are subject to sinusoidal excitation. For example, Emam and Nayfeh [4], Afaneh and Ibrahim [5], and Tseng and Dugundji [6] adopted the two-mode or multimode approach to study the effects of higher modes on the behavior of period doubling and nonlinear vibration. Emam and Nayfeh pointed out that the single-mode approach leads to significant errors in nonlinear dynamic behavior. Yamaki et al. [7,8] studied the dynamic responses of a clamped beam and the nonlinear modal couplings. Abou-Rayan et al. [9] performed a theoretical study of the nonlinear single-mode responses of a simply supported curved beam to parametric excitation, and Feng and Hu [10] presented their nonlinear beam work using the multimode approach. According to these studies, a multimode approach is necessary in analyzing the structural acoustic problem discussed in this paper. In fact, very few studies have considered the nonlinear vibration effect of a panel backed by a cavity, although many have considered linear structural-acoustics, which are of considerable interest to many researchers. The structural acoustic problem of a panel backed by a cavity has also been the focus of many researchers. For example, Lee et al. [11,12], Lyon [13], Pretlove [14], Oldham and Hillarby [15], and Puri et al. [16]

⇑ Corresponding author. E-mail address: [email protected] (Y.Y. Lee). 0307-904X/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2012.01.006

Y.Y. Lee et al. / Applied Mathematical Modelling 36 (2012) 5574–5588

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have all adopted linear approaches. In practice, a panel backed by a cavity (i.e. a panel absorber or a drum) is usually made of light-weight thin metal or elastic membrane, but very few studies have considered the effects of nonlinear vibration on sound absorption. In fact, considerable research has been conducted into the nonlinear structural stiffness in a vibration isolation/absorption system (e.g. [17,18]). This paper was inspired by the concept of nonlinear vibration isolation; thus, the theoretical and experimental work described here investigates the effects of nonlinear vibration on the sound absorption of a curved panel absorber. In practice, acoustic panel absorbers are usually made of thin panels so that they can be easily excited to vibrate nonlinearly. The results in the paper can be a piece of useful reference for panel absorber design. 2. Theory 2.1. Governing differential equation Fig. 1 shows an initially curved panel or a shallow shell backed by a cavity that is simply supported at two opposite sides and free at the other two and subject to transverse harmonic acoustic excitation. In the theoretical model, it is assumed that the flexural bending along the width is neglected. Note that the experimental plate configuration in [23] is similar to that in this study. The experimental resonant frequencies of the plate bending modes neglected in this study fell in other frequency ranges not concerned, and their modal amplitudes were found not very large. Thus, the structure is simplified and considered as a beam-like panel. The governing differential equation of a curved beam that is subject to a harmonic excitation [19] is given by: 0000

_ þ EIw ¼ € þ Xw qAw

EA 00  00 Þ ðw þ w L

Z

L 0

 2 1  0 w0 þ ðw0 Þdx þ FðtÞ: w 2

ð1aÞ

Consider the acoustic pressure within the cavity acting on the panel. Eq. (1a) is modified and given by: 0000

€ þ Xw _ þ EIw ¼ qAw

EA 00  00 Þ ðw þ w L

Z

L 0

  1  0 w0 þ ðw0 Þ2 dx  PðtÞ þ FðtÞ; w 2

ð1bÞ

where w = the transverse displacement caused by the panel bending; _ w € = the first and second derivatives of the transverse displacement with respect to time t; w; w0 , w00 , w000 = the first, second and fourth derivatives of the transverse displacement with respect to the spatial variable x;  = the initial center deflection of the panel; w E = Young’s modulus of the panel; q = the material density of the panel; X = the damping coefficient of the panel; L = panel length; F(t) = (Ag(sin(xt) = external sound pressure; P(t) = internal sound pressure at z = c; note that the initial center panel deflection is much smaller than c and therefore the cavity depth is assumed constant and equal to c; A = B  h = the cross-section area of the panel; B = panel width; c = cavity depth; h = panel thickness; x = excitation frequency; j = dimensionless excitation parameter; and g = gravitational acceleration = 9.81 m/s2.

External sound excitation Curved panel

z Acoustic pressure induced by the panel motion

x

y L

Fig. 1. A curved panel backed by a rectangular cavity.

c

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Y.Y. Lee et al. / Applied Mathematical Modelling 36 (2012) 5574–5588

Consider that the transverse displacement is expressed in terms of panel mode shapes:

wðx; tÞ ¼

N X

qi ðtÞ/i ðxÞ;

ð2Þ

i

where qi = the modal amplitude of the ith mode; /i = the ith structural mode shape, which is normalized so that the maximum value of each mode shape is equal to 1 (e.g. /13 ðx; yÞ ¼ sinðpLxÞ sinð3pB yÞ); i = the structural mode number; and N = the number of structural modes considered.  = qou1, where qo is the transverse displacement at the center. Hence, the The initial deflection shape is assumed to be w residual can be defined by substituting Eq. (2) into (1b), as follows:

D ¼ qA

N X

€i /i þ qAn q

N X

i

xi q_ i /i þ EI

i

N EA X  qi /00i þ qo /001 L i

!

N X

0000

qi /i

i N X

qo qi

Z

L

0

i¼1

/01 /0i dx

! Z L N X N 1X 0 0 þ qq / / dx þ PðtÞ  FðtÞ; 2 i¼1 j¼1 i j 0 i j

ð3Þ

0000

where /0i ; /00i ; /i are the first, second and fourth derivatives of the ith mode shape, and i and j are the mode numbers. nxi = X, where n = modal damping coefficient. Eq. (3) contains both quadratic and cubic nonlinear terms. Using the Galerkin approach [20], the weighted residual in Eq. (3) is set to zero. Multiplying um by each term on the righthand side of Eq. (3) and taking the integration over the beam length give:

qA

N X

€i a0;0 q i;m þ qAn

i

N X

xi q_ i a0;0 i;m þ EI

N X

i

qi a4;0 i;m

i

" # N N X N N X N X N X EA X 1 1;1 2;0 1X 1;1 2;0 2 1;1 2;0 1;1 2;0  a1;i a1;m qo qi þ ða1;i aj;m þ ai;j a1;m Þqo qi qj þ ai;j ak;m qi qj qk þ Pm ðtÞ þ F m ðtÞ ¼ 0; L 2 2 i i i j j k ð4Þ RL RL RL R L 00 R L 0000 2;0 4;0 where Pm ¼ 0 /m Pdx is the modal sound pressure; F m ¼ 0 /m Fdx; a0;0 i;m ¼ 0 /i /m dx; ai;m ¼ 0 /i /m dx; ai;m ¼ 0 /i /m dx; R L 0 0 a1;1 i;m ¼ 0 /i /m dx; and i, m and k are the structural mode numbers. The internal acoustic pressure acting on the panel is derived in this subsection. The acoustic pressure within the rectangular cavity induced by the nonlinear panel vibration is given by the following homogeneous wave equation [21].

r2 Cðx; y; z; tÞ 

1 @ 2 Cðx; y; z; tÞ ¼ 0; @t 2 C 2a

ð5Þ

where Cðx; y; z; tÞ is the acoustic pressure within the cavity, Ca is the speed of sound, and C at z = c is equal to P (i.e. Cðx; y; c; tÞ ¼ Pðx; y; tÞ). The boundary conditions of the rectangular cavity are two opposite sides open and two opposite sides closed. That is,

@C ¼ 0 at x ¼ 0 and L; @x

ð6aÞ

C ¼ 0 for open end at y ¼ 0 and B;

ð6bÞ

@C ¼ 0 at z ¼ 0; and @z

ð6cÞ

@C ¼ qa w at z ¼ c: @z

ð6dÞ

where qa = air density. By applying the boundary conditions in Eqs. (6a) and (6b), the solution for Eq. (5) can be expressed [21] as:

Cðx; y; z; tÞ ¼

U X ðLv sinhðv zÞ þ K v coshðv z ÞÞ/v ðx; yÞ;

v

ð7Þ

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where v = the acoustic mode number; U = the number of acoustic modes used; Lv and Kv = functions that depend on the boundary condition at z = 0 and the displacement w(t) at z = c; P uv ðx; yÞ ¼ the acoustic modeðe:g: u23 ðx; yÞ ¼ cosð2pL xÞ sin x2 w ¼ Ni qi /i ð3pB yÞ; and rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2p2 3p2  x 2 þ B  Ca uv = a function that depends on the acoustic mode number and excitation frequency (e.g. l23 ¼ L According to [22] which is about the large amplitude free vibration of a simply-supported panel backed by a cavity, the higher harmonic components of the internal acoustic pressure CðtÞ are very small. The acoustic part in the structural-acoustic formulation is linear, which is the same as that in this study. Considering that the higher harmonic components are negP € can be rewritten as x2 w ¼ Ni qi /i in Eq. (6d). Then, by applying the boundary conditions in Eqs. (6a)–(6d) to Eq. ligible, w (7) to eliminate Lv and K v , the pressure at z = c is given by:

Pðx; y; tÞ ¼ qa x2

N X U X cothðcv cÞ ci;v uv ðx; yÞqi ðtÞ; i

v

v v ;v

lv

ð8Þ

where

vv ;v ¼

Z

Z

B

0

L

0

u2v dxdy and ci;v ¼

Z 0

B

Z 0

L

/i um dxdy:

By substituting Eq. (8) into (4), the differential equation can be solved using the Runge–Kutta time domain numerical integration [23], and the root-mean square modal amplitude |qi| can be obtained. The overall root-mean square displacement amplitude is defined by:

qall

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N uX 2 ¼t jqi j :

ð9Þ

i

Note that in Refs. [24,25,4,26,6], the researchers expressed the governing beam/plate equations in terms of the modal coordinates using classical approaches. The modal equations were then solved by multiple scales, harmonic balance or numerical integration methods, etc. The advantages were that the nonlinear couplings of the modes could be directly studied from the spectra of the modal responses, and that the number of modal governing equations was relatively small (normally, not more than six). Thus, the computational effort for solving the modal equation was not an obstacle. The disadvantages were that when the number of modes was larger than six, the procedures for deriving the nonlinear modal governing equations were quite tedious, especially for plates. Moreover, these classical approaches are only suitable for cases with specific boundary conditions. For finite element method, some studies of nonlinear plate vibration (such as those of Han and Petyt [27] and Locke [28]) expressed the governing equations in terms of the physical coordinates. Green and Killey [29] solved the finite element matrix equation to obtain the time domain vibration responses using a numerical integration scheme. The finite element method, in conjunction with the numerical integration scheme, could be used for complex structures, but the computational time was long because of the large matrix size, and the modal responses could not be obtained directly. In this study, the number of modes is not large, and thus the direct integration method is adopted. 2.2. Sound absorption In the previous section, the numerical integration method is used to derive the modal response. In the resonant system of a panel absorber, the impedance contains the mass, stiffness, and damping elements in series. The normalized acoustic impedance of the ith panel mode can be given by:

Z i ¼ Ri þ JIi ;

ð10Þ

pffiffiffiffiffiffiffi where Ri and Ii are the real and imaginary parts of the impedance; J ¼ 1. According to [24], the real part is given by

Rp;mn ¼ qnxo ;

ð11aÞ

where Ii is the imaginary part that represents the mass and stiffness reactance and is given by

Ii ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðjZ i j2  R2i Þ;

where jZ i j ¼ q

jF i j _

a C a jqi j

ð11bÞ

= the normalized modal magnitude of the ith panel mode impedance, and jF i j and jq_ i j are the modal root-

mean square magnitudes of the external sound pressure and the velocity of the ith panel mode, which can be obtained using the numerical integration to solve Eq. (4). Thus, the real and imaginary parts of the impedance and modal displacements can be obtained and substituted into the following equations to obtain the absorption. According to [12], the overall impedance and absorption coefficient are given in Eqs. (12a) and (12b).

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Y.Y. Lee et al. / Applied Mathematical Modelling 36 (2012) 5574–5588

Z ¼

¼ X

N X 1 Zi i

!1 ð12aÞ

;

 4ReðZÞ : 2   2 ð1 þ ReðZÞÞ þ ðImðZÞÞ

ð12bÞ

3. Theoretical results Using the Runge–Kutta time domain numerical integration, the overall root-mean square displacement amplitude in Eq. (9) can be obtained; the sound absorption in Eq. (12b) can then be calculated. The material properties are as follows: Young’s modulus = 7.1  1010 N/m2, Poisson’s ratio = 0.3, damping ratio = 0.02, and mass density = 2700 kg/m3. The curved panel size

qall/h normalized overall displacement

0.6

(a)

0.5

0.4

0.3

0.2

0.1

0 0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

ω/ω1 Normalized excitation frequency 0.6

(b)

Sound absorption

0.5

0.4

0.3

0.2

0.1

0 0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

ω/ω1 Normalized excitation frequency Fig. 2. The overall displacement and sound absorption against the normalized excitation frequency (excitation parameter = 5.33; damping ratio n = 0.02; cavity depth c = 200 mm).

Table 1a Structural modal contributions at the normalized excitation frequency = 1 (four acoustic modes used).

Four structural modes Three structural modes Two structural modes One structural mode

1st mode

2nd mode

3rd mode

4th mode

95.63% 95.63% 95.64% 100%

4.36% 4.36% 4.36% N/A

0.01% 0.01% N/A N/A

0.00% N/A N/A N/A

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Y.Y. Lee et al. / Applied Mathematical Modelling 36 (2012) 5574–5588 Table 1b Structural modal contributions at the normalized excitation frequency = 1.5 (four acoustic modes used).

Four structural modes Three structural modes Two structural modes One structural mode

1st mode

2nd mode

3rd mode

4th mode

9.96% 9.96% 10.00% N/A

89.67% 89.67% 90.00% 100%

0.37% 0.37% N/A N/A

0.00% N/A N/A N/A

Table 1c Convergence study of normalized center displacement amplitude for various acoustic modes used (four structural modes used).

Nine acoustic modes used Four acoustic modes used One acoustic mode used

Normalized excitation frequency = 1

Normalized excitation frequency = 1.5

1.000 1.000 1.044

1.000 1.000 1.000

qall/h normalized overall displacement

0.6

(a)

0.5

0.4

κ = 5.33 0.3

κ = 1.69

0.2

κ = 0.533

0.1

0 0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

ω/ω1 Normalized excitation frequency

κ = 0.533

(b)

0.7

κ = 1.69

Sound absorption

0.6

κ = 5.33

0.5 0.4 0.3 0.2 0.1 0 0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

ω/ω1 Normalized excitation frequency Fig. 3. The overall displacement and sound absorption against the normalized excitation frequency for various excitation parameters (damping ratio n = 0.02; cavity depth c = 200 mm).

is 350 mm  350 mm  1 mm, and the initial center deflection is 2.4 mm. The external pressure excitation is zero for L P x > 0.4L, which is uniformly distributed over the panel surface, and its dimensionless magnitude is k = 5.33 for 0.4L P x P 0. The first four acoustic modes and first four structural modes are used in the convergence study. Fig. 2a shows

Y.Y. Lee et al. / Applied Mathematical Modelling 36 (2012) 5574–5588

qall/h normalized overall displacement

5580 0.6

(a)

0.5

ξ =0.01 0.4

0.3

0.2

0.1

0 0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

ω /ω1 Normalized excitation frequency

qall/h normalized overall displacement

0.6

(b) 0.5

ξ =0.02

0.4

0.3

0.2

0.1

0 0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

ω/ω1 Normalized excitation frequency qall/h normalized overall displacement

0.6

(c) 0.5

ξ =0.04

0.4

0.3

0.2

0.1

0 0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

ω/ω1 Normalized excitation frequency Fig. 4. The overall displacement against the normalized excitation frequency for various damping ratios (excitation parameter = 5.33; cavity depth c = 200 mm).

the overall dimensionless vibration amplitude plotted against the normalized excitation frequency. The cavity depth is 200 mm. The four structural resonant frequencies are 117.9 Hz, 151.9 Hz, 341.7 Hz, and 607.4 Hz, and the four acoustic resonant frequencies are 485.7 Hz, 1086.1 Hz, 1457.1 Hz, and 1751.3 Hz. Tables 1a and 1b shows the contributions of the structural modes to the overall response at the normalized excitation frequencies (1 and 1.5, respectively). The third and fourth modes contribute less than 0.5% in all cases. In other words, the solutions with the first two modes can achieve an error rate of less than 0.5%. Table 1c presents the convergence study of normalized center displacement amplitude for various acoustic modes used and normalized excitations (four structural modes used). The displacement amplitudes of nine acoustic modes are normalized as 1. The differences between the solutions of 4 and 9 acoustic modes used are almost zero. In

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Y.Y. Lee et al. / Applied Mathematical Modelling 36 (2012) 5574–5588 0.9

(a)

0.8

Sound absorption

0.7

ξ =0.01

0.6 0.5 0.4 0.3 0.2 0.1 0 0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

ω/ω1 Normalized excitation frequency

0.9 0.8

(b)

Sound absorption

0.7 0.6

ξ =0.02

0.5 0.4 0.3 0.2 0.1 0 0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

ω/ω1 Normalized excitation frequency 0.9 0.8

(c)

Sound absorption

0.7 0.6

ξ =0.04

0.5 0.4 0.3 0.2 0.1 0 0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

ω/ω1 Normalized excitation frequency Fig. 5. The sound absorption against the normalized excitation frequency for various damping ratios (excitation parameter = 5.33; cavity depth c = 200 mm).

other words, the effect of the higher modes on the structural displacement can be ignored. Besides, it can be seen from Tables 1a to 1c that the solutions with the two structural modes and four acoustic modes can achieve an error rate of less than 0.5%. Fig. 2b shows the sound absorption, which corresponds to Fig. 2a and plotted against the normalized excitation frequencies. The well-known ‘‘jump phenomenon’’ can be seen around the two resonant bandwidths. The first and second peaks are inclined towards left and right, respectively, because the quadratic and cubic nonlinearities induce the softening and hardening effects on the panel. Comparison between Fig. 3a and b shows the normalized root mean square displacements and sound absorptions for various external pressure magnitudes. The absorption peaks due to the nonlinear resonance are

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Y.Y. Lee et al. / Applied Mathematical Modelling 36 (2012) 5574–5588

qall/h normalized overall displacement

0.6

(a) Cavity depth = 200mm

0.5

Cavity depth = 10mm 0.4

No Cavity

0.3

0.2

0.1

0 0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

ω/ω1 Normalized excitation frequency 0.6

Cavity depth = 200mm

(b)

Cavity depth = 10mm

Sound absorption

0.5

No Cavity

0.4

0.3

0.2

0.1

0 0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

ω/ω1 Normalized excitation frequency Fig. 6. The overall displacement and sound absorption against the normalized excitation frequency for various cavity depths (excitation parameter = 5.33; damping ratio n = 0.02).

significantly wider than the typical linear peaks. The overall resonant bandwidth becomes wider by setting the resonant frequencies closer. A larger the excitation level results in a higher the degree of nonlinearity that can be seen at the absorption peak. The displacement peak values in the case of low excitation level are much smaller than those in high excitation level, but the absorption peak values are higher. In these three cases, the second absorption peaks due to the second modes are higher than the first ones. Comparison between Figs. 4 and 5 shows the normalized root-mean square displacements and sound absorptions plotted against the excitation frequency for various damping factors. A higher the damping factor results in lower displacement peak value, but a higher absorption peak value is found. Comparison between Fig. 6a and b shows the normalized root-mean square displacements and sound absorptions plotted against the excitation frequency for various cavity depths, and the displacements and sound absorptions are almost identical in these cases. In general, cavity depth does not play an important role because panel stiffness is much higher than cavity stiffness. Comparison between Figs. 7 and 8 shows the normalized root-mean square displacements and sound absorptions plotted against the excitation frequency for various structural resonant frequencies. The thickness, width and length of the panels in the three cases are the same (only the initial center deflections are different). In Fig. 7, the displacement peaks in these cases are tuned to 0.35–0.5 by appropriately setting the excitation parameter. In Fig. 8b, the sound absorption peak is much lower and almost negligible, although the displacement peaks in Fig. 8a are very high. It can be seen that if the resonant frequency of a panel mode is higher, the absorption peak value due to that mode is higher, and a high vibration peak does not imply a high absorption peak.

4. Experimental results The theoretical results show the effect of nonlinear vibration on the sound absorption of a panel absorber. Therefore, experiments were conducted to verify these results. However, due to the difficulty of setting up powerful excitation

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Y.Y. Lee et al. / Applied Mathematical Modelling 36 (2012) 5574–5588

qall/h normalized overall displacement

0.6

(a) 0.5

ω1 = 25.1 × 2π radian/s ω2 = 37.9 × 2π radian/s

0.4

qo = 0.5mm κ = 0.0533

0.3

0.2

0.1

0 0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

1.8

2

1.8

2

ω/ω1 Normalized excitation frequency

qall/h normalized overall displacement

0.6

(b)

0.5

0.4

0.3

ω1 = 67.8 × 2π radian/s ω2 = 75.9 × 2π radian/s

0.2

qo = 1.4mm κ = 0.533

0.1

0 0.4

0.6

0.8

1

1.2

1.4

1.6

ω/ω1 Normalized excitation frequency

qall/h normalized overall displacement

0.6

(c)

0.5

ω1 = 117.8 × 2π radian/s ω2 = 151.9 × 2π radian/s

0.4

qo = 2.4mm κ = 5.33

0.3

0.2

0.1

0 0.4

0.6

0.8

1

1.2

1.4

1.6

ω/ω1 Normalized excitation frequency Fig. 7. The overall displacement against the normalized excitation frequency for various structural resonant frequencies (damping ratio n = 0.02; no cavity).

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Y.Y. Lee et al. / Applied Mathematical Modelling 36 (2012) 5574–5588 0.6

(a)

Sound absorption

0.5

0.4

ω1 = 25.1 × 2π radian/s ω2 = 37.9 × 2π radian/s

0.3

qo = 0.5mm

0.2

0.1

0 0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

1.8

2

ω/ω1 Normalized excitation frequency 0.6

(b) Sound absorption

0.5

0.4

ω1 = 67.8 × 2π radian/s ω2 = 75.9 × 2π radian/s

0.3

qo = 1.4mm 0.2

0.1

0 0.4

0.6

0.8

1

1.2

1.4

1.6

ω/ω1 Normalized excitation frequency 0.6

(c)

Sound absorption

0.5

0.4

ω1 = 117.8 × 2π radian/s ω2 = 151.9 × 2π radian/s

0.3

qo = 2.4mm

0.2

0.1

0 0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

ω/ω1 Normalized excitation frequency Fig. 8. The sound absorption against the normalized excitation frequency for various structural resonant frequencies (excitation parameter = 5.33; damping ratio n = 0.02; no cavity).

equipment in the tube, only slightly nonlinear vibration was measured. So far, no previous work on experimental nonlinear panel absorber has been done. In fact, the authors tried different ways to conduct the nonlinear experiment. The result obtained in this paper is by far the best. A concrete tube with interior measurements of 38 cm  38 cm  3 m was used in this experiment (Fig. 9). A loudspeaker with input by a function generator and amplifier for producing pure tone sinusoidal wave was placed at one end of the tube. This arrangement generated standing waves inside the tube. A microphone was placed inside the tube to monitor the sound

5585

Y.Y. Lee et al. / Applied Mathematical Modelling 36 (2012) 5574–5588 Loudspeaker amplifier

Sound level meter

Clamping fixture

Signal generator

Carriage

Loudspeaker

(380mm × 380mm × 3m)

Accelerometer amplifier

Curved panel (350mm × 350mm × 0.5mm) Signal analyzer

Fig. 9. The experiment setup.

0.6

(a)

Relative displacement

0.5 0.4 0.3 0.2 0.1 0.0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x/L 1

(b)

Relative displacement

0.8 0.6 0.4 0.2 0 -0.2 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.4 -0.6 -0.8 -1

x/L Fig. 10. The measured structural mode shapes.

pressure, and the pure tone sinusoidal signal was kept constant. In this experiment, an aluminum curved panel (thickness, 0.5 mm; initial center deflection, 0.7 mm; length, 350 mm; and width, 350 mm) was mounted to cover an opening at the side wall. The beam was not perfectly simply supported (it was a case of no cavity backing the panel) and the sound pressure within the duct is assumed to be uniformly acting over the panel surface (note that the wavelength of the high cut-off frequency (80 Hz)  the panel length or width). The dynamic responses of the panel were measured using accelerometers placed at the center. The first two symmetric mode shapes measured are shown in Fig. 10a and b (their resonant frequencies are 37.5 Hz and 77 Hz). This experiment only considered the first two symmetric modes because the sound excitation was nearly uniform and thus no significant antisymmetric responses were measured. Fig. 11a and b shows experimental

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Normalized overall displacement

0.8 0.7 Theory

0.6 0.5

Measurement

0.4 0.3 0.2 0.1 0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

ω/ω1 Normalized excitation frequency Fig. 11. The measured and theoretical normalized overall displacements.

1.0 Modal contribution of nd the 2 mode

0.9

Modal Contribution

0.8 0.7 Modal contribution of st the 1 mode

0.6 0.5 0.4

Numerical Integration

0.3 0.2

Approximate formula eq. (14)

0.1 0.0 0.0

0.5

1.0

1.5

2.0

2.5

3. 0

3.5

ω/ω1 Normalized excitation frequency Fig. 12. The modal contributions obtained from the numerical integration and approximate formula.

Sound absorption

0.20

0.15

Theory

Measurement

0.10

0.05

0.00 0.0

0.5

1.0

1.5

2.0

2.5

ω/ω1 Normalized excitation frequency Fig. 13. The measured and theoretical sound absorptions.

3.0

3.5

Y.Y. Lee et al. / Applied Mathematical Modelling 36 (2012) 5574–5588

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root-mean square displacement amplitudes, which reasonably agree with the predictions calculated according to the experimental panel configuration (the excitation parameter is set K = 0.042 in Fig. 11; the damping ratio is set n = 0.01). The two calculated resonant frequencies are 33.9 Hz and 85.4 Hz. The difference between the measured and calculated resonant frequencies is caused by the imperfectly simply supported edges or partially clamped and partially simply supported edges. Fig. 11 shows that the peak at the first resonant frequencies are slightly inclined towards left (the ‘‘jump phenomenon’’ can be seen), while the peaks at the second resonant frequencies are more linear (i.e. no hardening effect seen in the theoretical result). The loudspeaker was not powerful enough to induce highly nonlinear panel vibrations. The experimental second peak is wider than the theoretical one (which looks sharper). This may be caused by the inaccurate damping ratio and resonant frequency, which affect the shape and value of the peak. The external excitation level of K = 0.042 is equivalent to the 98 dB root-mean square sound pressure level. The conversion formula is given by:

pffiffiffi!

SPL ¼ 20 log

qhgk= 2 Pref

ð13Þ

;

where Pref = 2  105 Pa = reference sound pressure. In Fig. 12, the modal contributions of the case in Fig. 11 obtained from the numerical integration well agree with the crosses and are calculated using the following approximate equation: F 2m ðx2 x2m Þ2 þðnxxm Þ2

dm ðxÞ ¼ P

F 2i N i ðx2 x2 Þ2 þðnxx Þ2 i i

;

ð14Þ

where m is the modal contribution of the mth mode; n is the damping ratio; x is the ith resonant frequency; Fm and Fi are already defined in Eqs. (4) and (11b). Note that of course, the modal contribution of each mode can obtained from the numerical direction method. Eq. (14) is just an approximate equation for quick calculation of modal contribution for this problem. The comparison in Fig. 12 shows that Eq. (14) is accurate enough for calculation of modal contribution. The empirical modal responses can be obtained by multiplying the overall vibration amplitude by the modal contribution in Eq. (14). The variable Fm in Eq. (14) is calculated using the measured mode shapes. The imaginary part of each modal impedance and corresponding sound absorption is then calculated using Eqs. (12a) and (12b). Fig. 13 shows the empirical and theoretical sound absorptions of the case in Fig. 11. Both empirical and theoretical absorption peaks due to the second modes are much higher than those due to the first ones, although their vibration peaks are lower. Further, both empirical and theoretical absorption peaks due to the first modes are very flat. A higher resonant frequency of a panel mode results in higher absorption peak value due to this mode, and a high vibration peak does not imply a high absorption peak. 5. Conclusions This paper has presented a multimode formulation for the nonlinear vibrations of a curved panel backed by a cavity. The sound absorption is obtained by solving the multimode differential equations using the numerical integration method. The convergence study shows the number of acoustic and structural modes needed for an accurate result. The effects of excitation level, cavity depth, and damping factor have been investigated. The quadratic and cubic nonlinearities and their effects on sound absorption have also been investigated. An experiment was conducted. The theoretical and experimental observations correspond reasonably with each other. The main findings include the following: (1) the well-known ‘‘jump phenomenon’’ can be seen in the sound absorption curves; (2) the absorption peaks due to nonlinear resonances are wider than those due to linear resonances because of the wider resonant bandwidth (but the peaks values could be lower); and (3) a higher resonant frequency of a panel mode results in a higher absorption peak value due to this mode, and a high vibration peak does not imply a high absorption peak. Furthermore, it is recommended that the further work on this topic should consider ‘‘micro-perforation’’, which is a kind of acoustic damping treatment to widen the absorption bandwidth. The combination of the nonlinear resonant vibration and micro-perforation effects is expected to further enhance and optimize the absorption performance of a panel absorber. Acknowledgments The research reported in this paper was fully supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China [9041356 (CityU116408)]. References [1] M. Baghani, R.A. Jafari-Talookolaei, H. Salarieh, Large amplitudes free vibrations and post-buckling analysis of unsymmetrically laminated composite beams on nonlinear elastic foundation, Appl. Math. Model. 35 (1) (2011) 130–138. [2] Y.G. Wang, J.L. Shi, X.Z. Wang, Large amplitude vibration of heated corrugated circular plates with shallow sinusoidal corrugations, Appl. Math. Model. 33 (9) (2009) 3523–3532. [3] O. Civalek, Nonlinear analysis of thin rectangular plates on Winkler–Pasternak elastic foundations by DSC–HDQ methods, Appl. Math. Model. 31 (3) (2007) 606–624.

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