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Free vibration analysis of a nonlinear panel coupled with extended cavity using the multi-level residue harmonic balance method
Thin-Walled Structures 98 (2016) 332–336
Contents lists available at ScienceDirect
Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws
Short communication
Free vibration analysis of a nonlinear panel coupled with extended cavity using the multi-level residue harmonic balance method Yiu Yin Lee Department of Architecture and Civil Engineering, City University of Hong Kong, Kowloon Tong, Kowloon, Hong Kong
art ic l e i nf o
a b s t r a c t
Article history: Received 17 February 2015 Received in revised form 13 July 2015 Accepted 7 October 2015
This article addresses the free vibration analysis of a nonlinear panel coupled with extended cavity. In practice, the cavity length of a panel-cavity system is sometimes longer than the panel length. Therefore, this study examines the effect of cavity length on the natural frequency of a nonlinear panel coupled with extended cavity. The multi-level residue harmonic balance method, which was recently developed by the author and his research partners, is used to solve this nonlinear problem. The present harmonic balance solution agrees reasonably well with the results obtained from a previous classical solution and shows that the cavity length is a very important factor that significantly affects the panel vibration and should not be ignored in the modelling process. The natural frequency of a panel-cavity system is very sensitive to the cavity length and decreases significantly when the cavity is longer, due to its larger volume or weaker stiffness. Moreover, when the cavity is very long and its resonant frequencies are close to that of the panel, multiple frequency solutions are obtained. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Nonlinear vibration Harmonic balance method Structural-acoustics Panel cavity system
1. Introduction Over the past few decades, numerous studies on panel-cavity coupling have been published (e.g., Sadri and Younesian [1] and Hui et al. [2]). To the best of the author's knowledge, almost all of these adopted a model in which the cavity length is equal to the panel length, although in practice, it is sometimes longer. Fig. 1 shows two examples – (1) the clamping fixture occupies some spaces and (2) the nine individual panels are mounted on the clamping grids and share one common cavity – that motivated this study on the effect of cavity length. Moreover, studies of this nonlinear structural-acoustic problem are still limited, although many nonlinear panel or linear structural-acoustic problems have been solved (e.g., Younesian et al. [3,4] and Shi et al. [5]). In addition, the author and his research partners [6] recently developed the multi-level residue harmonic balance method to solve nonlinear beam/plate problems. The main advantage of this method is that only one set of nonlinear algebraic equations is generated in the zero-level solution procedure, while higher-level solutions to any desired degree of accuracy can be obtained by solving a set of linear algebraic equations. Table 1 compares the performance of the classical harmonic method and the multi-level residue harmonic balance method in solving cubic nonlinear beam problems. E-mail address: [email protected] http://dx.doi.org/10.1016/j.tws.2015.10.006 0263-8231/& 2015 Elsevier Ltd. All rights reserved.
The number of nonlinear algebraic equations generated in the solution processes in the proposed method is clearly much smaller than that in the classical harmonic method, showing that it is time saving and less complicated.
2. Structural-acoustic formulation Fig. 1 shows a nonlinear panel coupled with extended cavity. The cavity length is longer than the panel length. The acoustic pressure within the cavity induced by the panel in Example I, or in the centre panel in Example II is given by the following homogeneous wave equation (Lee and Ng [7]).
∇2P h −
1 ∂ 2P h = 0, Ca2 ∂t 2
(1)
where Ph is the acoustic pressure induced by the h-th harmonic component of the nonlinear panel vibration and Ca is the speed of sound. It is assumed, in Example II of Fig. 1, that the vibrations of the individual panels are not coupled with each other or that the vibration of one panel does not affect the others. The boundary conditions of the cavity are at x¼ 0 and a
∂P h =0 ∂x
(2a)
Y.Y. Lee / Thin-Walled Structures 98 (2016) 332–336
333
The total panel vibration is the summation of all harmonic components and is given by
wh =
H
∑h = 1.3,5… w h
(4)
According to a previous study (Lee and Ng, 1998) [7] the general multi-acoustic mode solution of Eq. (1) is
P h (x, y , z, t ) U
V
(L
∑u ∑v
=
h uv
(
)
(
h h h sinh μ uv z + Nuv cosh μ uv z
uπ
vπ
))φ
uv (x,
y) T (t ) (5)
uπ
vπ
hω
( a )2 + ( b )2 − ( C )2 ; ϕuv (x, y ) ¼ cos ( a x ) cos ( b y ) is
h where μuv =
a
the acoustic mode; u and v are the acoustic mode numbers; and ω h h is the excitation frequency. L uv and Nuv are coefficients that depend on the boundary conditions at z¼ 0 and z ¼c; U and V are the numbers of acoustic mode numbers used; and T(t) is the time function. By applying the boundary conditions in Eqs. (2c–d) to Eq. (5), h h the unknown modal coefficients, L uv and Nuv can be expressed in h terms of A (t ), and thus the hth harmonic component of the modal acoustic pressure force acting on the panel is given by b
Pch (t ) =
a
∫0 ∫0 P h (x, y, z, t ) ϕ (x, y) dxdy b
a
∫0 ∫0 ϕ (x, y)2dxdy
⇒
Pch (t ) = K hAh (t ) Fig. 1. Examples of a panel coupled with extended cavity.
2 ϕ 2 V ρa (hω) (αuv ) uv α αuv μh
U
K h = − ∑u ∑v
where b
(6)
uv
b
a
h cot h (μuv c)
a
b
;
uv αuv =
a
ϕ uv = ∫ ∫ φuv ϕdxdy ; = ∫ ∫ ϕ2dxdy ; αuv ∫0 ∫0 φuv φuv dxdy ; αuv 0 0 0 0
at y¼ 0 and b
Then, the total modal acoustic pressure force at z¼ c is given by
∂P h =0 ∂y
(2b)
at z ¼0
H
Pch
∑
Pc =
(7)
h = 1,3,5 …
h
∂P =0 ∂z
(2c)
at z ¼c,for
Δa þa'4x 4 Δa, and Δb þb' 4y4 Δb,
∂P h ∂z
= − ρa
∂ 2w h ∂t 2
h
otherwise,
∂P =0 ∂z
In this study, the governing equation of nonlinear panel adopted by Hui et al. [2] is used here and is incorporated with the acoustic force term in equation (7).
ρ
(2d)
d2A +ρωo2 A+βA3 +Pc =0 dt 2 ⇒
where a’' and b' ¼panel width and length; a, b, and c ¼cavity length, width, and depth; and ρa ¼air density. w h is the hth harmonic components of the nonlinear panel displacement at z¼c, which is given in the following form
where ωo=
w h = Ah (t ) ϕ (x, y)
frequency of the panel; β=
(3)
where A (t ) is the modal amplitude while ϕ(x,y) is the mode shape h
(for a simply supported panel, ϕ (x, y ) = sin (
π (y − Δb) π (x − Δa) ) sin ( b‵ )). a‵
ρ
d2A + ρωo2 A + βA3 + dt 2 Eτ 2
π
12ρ (1 − ν 2)
4
H
∑h = 1,3,5... K hAh = 0
(8)
π
(( a ′ )2 +( b ′ )2)= the fundamental linear natural
3 ν2 [( 4 − 4 )(1
cient, γ = 3π E¼ Young's modulus;
Eτ
γ
12 (1 − ν 2) (a ′)4
4
2
= nonlinear stiffness coeffi-
+ r ) + νr ],
r ¼ a’/b’ ¼aspect
ratio;
ν ¼Poisson's ratio; ρ ¼density per unit
Table 1 Comparison between the numbers of algebraic equations generated in the multi-level residue harmonic balance method and classical harmonic balance method for cubic nonlinear beam problem.
Multi-level residue harmonic balance method Classical harmonic balance method
Zero level (one harmonic term)
1st level (two harmonic terms)
2nd level (three harmonic terms)
Nos. of nonlinear algebraic equations
Nos. of linear algebraic equations
Nos. of nonlinear algebraic equations
Nos. of linear algebraic equations
Nos. of nonlinear algebraic equations
Nos. of linear algebraic equations
1
0
1
2
1
3
1
0
2
0
3
0
334
Y.Y. Lee / Thin-Walled Structures 98 (2016) 332–336
thickness; and τ ¼panel thickness.
3. Multi-level residue harmonic balance solution procedure According to [6], the multi-level residue harmonic balance solution form in Eq. (8) is given by
A (t ) ≅ ε0A 0 (t ) + εA1 (t ) + ε2A2 (t ) + …
(9)
where ε is an embedding parameter; and A0 (t ), A1 (t ), and A2 (t ), are the zero, 1st and 2nd level modal amplitude solutions. By inputting equations (9) into (8), consider those terms associated with ε0 and setup the zero level governing equation.
ρA¨ 0 +ρωo2 A 0 +βA 03 +K1A 0 =∆0 (t )
(10)
where A0 (t ) = A0,1 cos (ωt ) , the first and second subscripts in A0,1 ¼the zero level and harmonic number; and ∆0 (t ) is the residue remaining at the zero level. Consider the harmonic balance of cos (ωt ) in equation (10). Then, the following algebraic equation can be obtained
⎛ ⎞ 3 2 ⎜ ρ −ω2+ωo2 + βA 0,1 +K1⎟ A 0,1=0 ⎝ ⎠ 4
(
)
(11)
By solving equation (11), the unknown natural frequency ω and zero level residue Δ0 can be found and expressed in terms of the initial modal amplitude A0,1. Again, by inputting equations (9) into (8), consider those terms associated with ε1 and setup the first level governing equation.
ρA¨ 1 +ρωo2 A1+3βA 02 A1+K1A1,1+K3A1,3 +∆0 (t )=∆1(t )
(12)
where A1 (t ) ¼ A1,1 cos (ωt )+A1,3 cos (3ωt ) ,the first and second subscripts in A1,1 and A1,3 ¼ the 1st level and their harmonic numbers; and ∆1 (t ) is the residue remaining at the 1st level. Consider the harmonic balances of cos (ωt ) and cos (3ωt ) in equation (12) to set up two algebraic equations
∫0
2π / ω
∆1(t ) cos (ωt ) dt = 0;
∫0
2π / ω
∆1(t ) cos (3ωt ) dt = 0
(13a-b)
Note that A0,1 has been found using the zero level solution procedure. Thus, equations (13a-b) are linear equations and contain the two unknowns A1,1 and A1,3. Again, by inputting equations (9) into (8), consider those terms associated with ε2 and setup the second level governing equation.
ρA¨ 2 +ρωo2 A2 +3βA 0 A12 +3βA2 A 02 +K1A2,1+K3A2,3 +K 5A2,5 +∆1(t )=∆2 (t )
(14)
where A2 (t ) ¼ A2,1 sin (ωt )+A2,3 sin (3ωt )+A2,5 sin (5ωt ), ;the first and second subscripts in A2,1, A2,3 and A2,5 ¼the 2nd level and their harmonic numbers; ∆2 (t ) is the residue remaining at the 2nd level. Consider the harmonic balances of cos (ωt ), cos (3ωt ) and cos (5ωt ) in equation (14) to set up three algebraic equations.
∫0
2π / ω
Δ2 (t ) cos (ωt ) dt = 0;
∫0
∫0 ∫0
2π / ω
Δ2 (t ) cos (3ωt ) dt = 0;
2π / ω
Δ2 (t ) cos (5ωt ) dt = 0
(15a-c)
Note that A1,1 and A1,3 have been found using the 1st level solution procedure. Thus, equations (15a-c) are linear equations and contain the three unknowns A2,1, A2,3 and A2,5. Similarly, the pth level governing equations can be setup by inputting equation (9) into (8) and picking up those terms associated with εp, then consider the higher level solution form
Ap (t )=Ap,1 cos (ωt )+Ap,3 cos (3ωt )+Ap,5 cos (5ωt )+…
(16)
Then, consider the harmonic balances of cos (ωt ), cos (3ωt ), cos (5ωt ), … to setup the linear algebraic equations and solve for the unknowns Ap,1, Ap,3, Ap,5 …. Finally, the overall initial amplitude is given by
AI =
∑ p ∑q Ap, q
(17)
where AI ¼overall initial amplitude; p¼ solution level; and q¼ harmonic number
4. Results and discussions In this section, the material properties of the panel in the numerical cases are considered as follows: Young's modulus of the beam E ¼71 109 N/m2; mass density ρ ¼ 2700 kg/m3, Poisson's ratio ν ¼0.3; the panel dimensions a b¼0.3048 m 0.3048 m. In Tables 2a and 2b, Figs. 2 and 3a and b, the panel thickness τ ¼1.2192 mm. In Fig. 4, the panel thickness τ ¼4 1.2192 mm. Tables 2a and 2b show the contributions of the amplitude harmonic components at various solution levels and the natural frequency convergence study for various acoustic modes used, respectively. For smaller initial amplitudes (or smaller frequency ratios), the nonlinearity of the system is also smaller. The zero level solution, which contains the first harmonic component only, is sufficiently accurate. The higher harmonic components in the higher level solutions, which reflect the nonlinearity of the system, are not very significant. For larger initial amplitudes (or higher frequency ratios), the nonlinearity of the system is greater, and thus the contributions of the higher harmonic components are higher. Therefore, the error of the zero level solution, which contains the first harmonic component only, is larger; and higher level approaches, in which more harmonic terms are incorporated, are needed to achieve more accurate results. The four acoustic mode approach and first-level approach can be seen to be adequate to obtain converged and accurate solutions. In Fig. 2, the vibration amplitude ratio, AI/τ, is plotted against the frequency ratio for various cavity depths. The frequency ratios for various cavity depths obtained using the present harmonic balance method agree reasonably well with those obtained from the elliptical integral method in Ref. [2]. When the depth of the cavity is shorter or its volume is smaller, the natural frequencies are higher because its stiffness is stronger. According to the comparison between the slopes of the curves for no cavity and with depth¼0.0508 m, the natural frequency of the panel coupled with the shorter cavity depth is less sensitive to the vibration amplitude, particularly for vibration amplitude ratio40.5. In Fig. 3a, the vibration amplitude ratio, AI/τ, is plotted against the frequency ratio for various cavity lengths. The effect of cavity length is similar to that of cavity depth. When the length of the cavity is longer or its volume is greater, the natural frequencies are lower because its stiffness is weaker. For example, when the cavity length changes from 1 to 1.2, 1.5; and 3 the panel length for a zero amplitude ratio, the natural frequency decreases by 8.5%, 16.2% and 28%, respectively. The zero amplitude ratio implies linear vibration. In Fig. 3b, the frequency ratio is plotted against the Table 2a Amplitude harmonic component contributions at various solution levels, %. Frequency Ratio ω/ωo ¼
Zero level 1st level 2nd level
1.50 99.57 0.43 0.00
1.81 97.72 2.24 0.04
2.29 96.14 3.73 0.12
2.61 95.54 4.30 0.17
Y.Y. Lee / Thin-Walled Structures 98 (2016) 332–336
335
Table 2b Frequency ratio convergence study for various acoustic modes used, ω/ωo. Amplitude ratio AI/τ ¼
1 mode 4 modes 9 modes
0.400 1.5068 1.5027 1.5026
1.108 1.8151 1.8101 1.8099
1.823 2.2965 2.2901 2.2900
2.23 2.6142 2.6069 2.6068
Fig. 2. Vibration amplitude ratio versus frequency ratio for various cavity depths.
cavity length for various cavity depths. The amplitude ratio is zero. There are large differences between the points on the three curves of different cavity depths when the cavity length is equal to the panel length. This implies that when the cavity length is equal to the panel length, the natural frequency is very sensitive to the cavity depth. On the contrary, when the length of the cavity is much longer than that of the panel, the natural frequency is not very sensitive to the cavity depth. There are small differences between the points on the three curves of different cavity depths when the cavity length is equal to 4 the panel length. In Fig. 4, the vibration amplitude ratio, AI/τ, is plotted against the frequency ratio when the cavity length ¼4.25 the panel length and thickness¼4 1.2192 mm. The dotted line represents the result of the one-dimensional cavity or no cavity resonance parallel to the panel length. It can be seen that the first and second cavity resonances significantly affect the natural frequency. As these are within the frequency ratio range considered, there are three natural frequency solutions. The first frequency solution converges to the first cavity resonant frequency, but is lower when the initial amplitude goes to infinity; the second frequency solution is slightly higher than the first cavity resonant frequency when the initial amplitude is small, and converges to the natural frequency solution of no cavity resonance when the initial amplitude is set higher. Then, the second frequency solution converges to the second cavity resonant frequency, but is lower when the initial amplitude goes to infinity. The third frequency solution is almost constant and slightly higher than the second cavity resonant frequency when the initial amplitude is set at less than 1. Then it converges to the natural frequency solution of no cavity resonance when the initial amplitude is set higher. As mentioned above, in practice, the cavity length is usually longer than the panel length (see Fig. 1). Hence, in the modelling process, the cavity length is a very important factor that significantly affects the panel vibration or sound radiation and should not be ignored.
Fig. 3. (a) Vibration amplitude ratio versus frequency ratio for various cavity lengths. (b) Frequency ratio versus cavity length for various cavity depths.
Fig. 4. Vibration amplitude ratio versus frequency length¼ 4.25 panel length and thickness¼ 4 1.2192 mm.
ratio
for
cavity
5. Conclusions The nonlinear structural-acoustic formulation has been introduced for the large amplitude free vibrations of a flexible panel
336
Y.Y. Lee / Thin-Walled Structures 98 (2016) 332–336
coupled with extended cavity, and the multi-level residue harmonic balance method is applied to solve this nonlinear problem. The present harmonic balance solution agrees reasonably well with the results obtained from the elliptic integral method. This study focuses on the effect of cavity length, which was ignored in many previous structural acoustic studies. The results show that the natural frequency of a panel-cavity system is very sensitive to the length of the cavity and may decrease significantly when it is longer, because its volume is greater or its stiffness is weaker. Moreover, when the cavity is very long and the cavity resonant frequencies are close to that of the panel, multiple frequency solutions may be obtained.
Acknowledgements The author would like to express his sincere gratitude and appreciation to Dr. W.Y. Poon's advice and contributions.
References [1] M. Sadri, D. Younesian, Nonlinear free vibration analysis of a plate-cavity system, Thin-Walled Struct. 74 (2014) 191–200. [2] C.K. Hui, Y.Y. Lee, J.N. Reddy, Approximate elliptical integral solution for the large amplitude free vibration of a rectangular single mode plate backed by a multi-acoustic mode cavity, Thin-Walled Struct. 49 (9) (2011) 1191–1194. [3] D. Younesian, M. Sadri, E. Esmailzadeh, Primary and secondary resonance analyses of clamped-clamped micro-beams, Nonlinear Dyn. 76 (4) (2014) 1867–1884. [4] D. Younesian, Z. Saadatnia, H. Askari, Analytical solutions for free oscillations of beams on nonlinear elastic foundations using the variational iteration method, J. Theor. Appl. Mech. 50 (2) (2012) 639–652. [5] S.X. Shi, G.Y. Jin, Z.G. Liu, Vibro-acoustic behaviors of an elastically restrained double-panel structure with an acoustic cavity of arbitrary boundary impedance, Appl. Acoust. 76 (2014) 431–444. [6] A.S.M.Z. Hasan, Y.Y. Lee, A.Y.T. Leung, The multi-level residue harmonic balance solutions of multi-mode nonlinearly vibrating beams on an elastic foundation, J. Vib. Control (2014) 1–18, http://dx.doi.org/10.1177/1077546314562225. [7] Y.Y. Lee, C.F. Ng, Sound insertion loss of stiffened enclosure plates using the finite element method and the classical approach, J. Sound Vib. 217 (2) (1998) 239–260.
Contents lists available at ScienceDirect
Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws
Short communication
Free vibration analysis of a nonlinear panel coupled with extended cavity using the multi-level residue harmonic balance method Yiu Yin Lee Department of Architecture and Civil Engineering, City University of Hong Kong, Kowloon Tong, Kowloon, Hong Kong
art ic l e i nf o
a b s t r a c t
Article history: Received 17 February 2015 Received in revised form 13 July 2015 Accepted 7 October 2015
This article addresses the free vibration analysis of a nonlinear panel coupled with extended cavity. In practice, the cavity length of a panel-cavity system is sometimes longer than the panel length. Therefore, this study examines the effect of cavity length on the natural frequency of a nonlinear panel coupled with extended cavity. The multi-level residue harmonic balance method, which was recently developed by the author and his research partners, is used to solve this nonlinear problem. The present harmonic balance solution agrees reasonably well with the results obtained from a previous classical solution and shows that the cavity length is a very important factor that significantly affects the panel vibration and should not be ignored in the modelling process. The natural frequency of a panel-cavity system is very sensitive to the cavity length and decreases significantly when the cavity is longer, due to its larger volume or weaker stiffness. Moreover, when the cavity is very long and its resonant frequencies are close to that of the panel, multiple frequency solutions are obtained. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Nonlinear vibration Harmonic balance method Structural-acoustics Panel cavity system
1. Introduction Over the past few decades, numerous studies on panel-cavity coupling have been published (e.g., Sadri and Younesian [1] and Hui et al. [2]). To the best of the author's knowledge, almost all of these adopted a model in which the cavity length is equal to the panel length, although in practice, it is sometimes longer. Fig. 1 shows two examples – (1) the clamping fixture occupies some spaces and (2) the nine individual panels are mounted on the clamping grids and share one common cavity – that motivated this study on the effect of cavity length. Moreover, studies of this nonlinear structural-acoustic problem are still limited, although many nonlinear panel or linear structural-acoustic problems have been solved (e.g., Younesian et al. [3,4] and Shi et al. [5]). In addition, the author and his research partners [6] recently developed the multi-level residue harmonic balance method to solve nonlinear beam/plate problems. The main advantage of this method is that only one set of nonlinear algebraic equations is generated in the zero-level solution procedure, while higher-level solutions to any desired degree of accuracy can be obtained by solving a set of linear algebraic equations. Table 1 compares the performance of the classical harmonic method and the multi-level residue harmonic balance method in solving cubic nonlinear beam problems. E-mail address: [email protected] http://dx.doi.org/10.1016/j.tws.2015.10.006 0263-8231/& 2015 Elsevier Ltd. All rights reserved.
The number of nonlinear algebraic equations generated in the solution processes in the proposed method is clearly much smaller than that in the classical harmonic method, showing that it is time saving and less complicated.
2. Structural-acoustic formulation Fig. 1 shows a nonlinear panel coupled with extended cavity. The cavity length is longer than the panel length. The acoustic pressure within the cavity induced by the panel in Example I, or in the centre panel in Example II is given by the following homogeneous wave equation (Lee and Ng [7]).
∇2P h −
1 ∂ 2P h = 0, Ca2 ∂t 2
(1)
where Ph is the acoustic pressure induced by the h-th harmonic component of the nonlinear panel vibration and Ca is the speed of sound. It is assumed, in Example II of Fig. 1, that the vibrations of the individual panels are not coupled with each other or that the vibration of one panel does not affect the others. The boundary conditions of the cavity are at x¼ 0 and a
∂P h =0 ∂x
(2a)
Y.Y. Lee / Thin-Walled Structures 98 (2016) 332–336
333
The total panel vibration is the summation of all harmonic components and is given by
wh =
H
∑h = 1.3,5… w h
(4)
According to a previous study (Lee and Ng, 1998) [7] the general multi-acoustic mode solution of Eq. (1) is
P h (x, y , z, t ) U
V
(L
∑u ∑v
=
h uv
(
)
(
h h h sinh μ uv z + Nuv cosh μ uv z
uπ
vπ
))φ
uv (x,
y) T (t ) (5)
uπ
vπ
hω
( a )2 + ( b )2 − ( C )2 ; ϕuv (x, y ) ¼ cos ( a x ) cos ( b y ) is
h where μuv =
a
the acoustic mode; u and v are the acoustic mode numbers; and ω h h is the excitation frequency. L uv and Nuv are coefficients that depend on the boundary conditions at z¼ 0 and z ¼c; U and V are the numbers of acoustic mode numbers used; and T(t) is the time function. By applying the boundary conditions in Eqs. (2c–d) to Eq. (5), h h the unknown modal coefficients, L uv and Nuv can be expressed in h terms of A (t ), and thus the hth harmonic component of the modal acoustic pressure force acting on the panel is given by b
Pch (t ) =
a
∫0 ∫0 P h (x, y, z, t ) ϕ (x, y) dxdy b
a
∫0 ∫0 ϕ (x, y)2dxdy
⇒
Pch (t ) = K hAh (t ) Fig. 1. Examples of a panel coupled with extended cavity.
2 ϕ 2 V ρa (hω) (αuv ) uv α αuv μh
U
K h = − ∑u ∑v
where b
(6)
uv
b
a
h cot h (μuv c)
a
b
;
uv αuv =
a
ϕ uv = ∫ ∫ φuv ϕdxdy ; = ∫ ∫ ϕ2dxdy ; αuv ∫0 ∫0 φuv φuv dxdy ; αuv 0 0 0 0
at y¼ 0 and b
Then, the total modal acoustic pressure force at z¼ c is given by
∂P h =0 ∂y
(2b)
at z ¼0
H
Pch
∑
Pc =
(7)
h = 1,3,5 …
h
∂P =0 ∂z
(2c)
at z ¼c,for
Δa þa'4x 4 Δa, and Δb þb' 4y4 Δb,
∂P h ∂z
= − ρa
∂ 2w h ∂t 2
h
otherwise,
∂P =0 ∂z
In this study, the governing equation of nonlinear panel adopted by Hui et al. [2] is used here and is incorporated with the acoustic force term in equation (7).
ρ
(2d)
d2A +ρωo2 A+βA3 +Pc =0 dt 2 ⇒
where a’' and b' ¼panel width and length; a, b, and c ¼cavity length, width, and depth; and ρa ¼air density. w h is the hth harmonic components of the nonlinear panel displacement at z¼c, which is given in the following form
where ωo=
w h = Ah (t ) ϕ (x, y)
frequency of the panel; β=
(3)
where A (t ) is the modal amplitude while ϕ(x,y) is the mode shape h
(for a simply supported panel, ϕ (x, y ) = sin (
π (y − Δb) π (x − Δa) ) sin ( b‵ )). a‵
ρ
d2A + ρωo2 A + βA3 + dt 2 Eτ 2
π
12ρ (1 − ν 2)
4
H
∑h = 1,3,5... K hAh = 0
(8)
π
(( a ′ )2 +( b ′ )2)= the fundamental linear natural
3 ν2 [( 4 − 4 )(1
cient, γ = 3π E¼ Young's modulus;
Eτ
γ
12 (1 − ν 2) (a ′)4
4
2
= nonlinear stiffness coeffi-
+ r ) + νr ],
r ¼ a’/b’ ¼aspect
ratio;
ν ¼Poisson's ratio; ρ ¼density per unit
Table 1 Comparison between the numbers of algebraic equations generated in the multi-level residue harmonic balance method and classical harmonic balance method for cubic nonlinear beam problem.
Multi-level residue harmonic balance method Classical harmonic balance method
Zero level (one harmonic term)
1st level (two harmonic terms)
2nd level (three harmonic terms)
Nos. of nonlinear algebraic equations
Nos. of linear algebraic equations
Nos. of nonlinear algebraic equations
Nos. of linear algebraic equations
Nos. of nonlinear algebraic equations
Nos. of linear algebraic equations
1
0
1
2
1
3
1
0
2
0
3
0
334
Y.Y. Lee / Thin-Walled Structures 98 (2016) 332–336
thickness; and τ ¼panel thickness.
3. Multi-level residue harmonic balance solution procedure According to [6], the multi-level residue harmonic balance solution form in Eq. (8) is given by
A (t ) ≅ ε0A 0 (t ) + εA1 (t ) + ε2A2 (t ) + …
(9)
where ε is an embedding parameter; and A0 (t ), A1 (t ), and A2 (t ), are the zero, 1st and 2nd level modal amplitude solutions. By inputting equations (9) into (8), consider those terms associated with ε0 and setup the zero level governing equation.
ρA¨ 0 +ρωo2 A 0 +βA 03 +K1A 0 =∆0 (t )
(10)
where A0 (t ) = A0,1 cos (ωt ) , the first and second subscripts in A0,1 ¼the zero level and harmonic number; and ∆0 (t ) is the residue remaining at the zero level. Consider the harmonic balance of cos (ωt ) in equation (10). Then, the following algebraic equation can be obtained
⎛ ⎞ 3 2 ⎜ ρ −ω2+ωo2 + βA 0,1 +K1⎟ A 0,1=0 ⎝ ⎠ 4
(
)
(11)
By solving equation (11), the unknown natural frequency ω and zero level residue Δ0 can be found and expressed in terms of the initial modal amplitude A0,1. Again, by inputting equations (9) into (8), consider those terms associated with ε1 and setup the first level governing equation.
ρA¨ 1 +ρωo2 A1+3βA 02 A1+K1A1,1+K3A1,3 +∆0 (t )=∆1(t )
(12)
where A1 (t ) ¼ A1,1 cos (ωt )+A1,3 cos (3ωt ) ,the first and second subscripts in A1,1 and A1,3 ¼ the 1st level and their harmonic numbers; and ∆1 (t ) is the residue remaining at the 1st level. Consider the harmonic balances of cos (ωt ) and cos (3ωt ) in equation (12) to set up two algebraic equations
∫0
2π / ω
∆1(t ) cos (ωt ) dt = 0;
∫0
2π / ω
∆1(t ) cos (3ωt ) dt = 0
(13a-b)
Note that A0,1 has been found using the zero level solution procedure. Thus, equations (13a-b) are linear equations and contain the two unknowns A1,1 and A1,3. Again, by inputting equations (9) into (8), consider those terms associated with ε2 and setup the second level governing equation.
ρA¨ 2 +ρωo2 A2 +3βA 0 A12 +3βA2 A 02 +K1A2,1+K3A2,3 +K 5A2,5 +∆1(t )=∆2 (t )
(14)
where A2 (t ) ¼ A2,1 sin (ωt )+A2,3 sin (3ωt )+A2,5 sin (5ωt ), ;the first and second subscripts in A2,1, A2,3 and A2,5 ¼the 2nd level and their harmonic numbers; ∆2 (t ) is the residue remaining at the 2nd level. Consider the harmonic balances of cos (ωt ), cos (3ωt ) and cos (5ωt ) in equation (14) to set up three algebraic equations.
∫0
2π / ω
Δ2 (t ) cos (ωt ) dt = 0;
∫0
∫0 ∫0
2π / ω
Δ2 (t ) cos (3ωt ) dt = 0;
2π / ω
Δ2 (t ) cos (5ωt ) dt = 0
(15a-c)
Note that A1,1 and A1,3 have been found using the 1st level solution procedure. Thus, equations (15a-c) are linear equations and contain the three unknowns A2,1, A2,3 and A2,5. Similarly, the pth level governing equations can be setup by inputting equation (9) into (8) and picking up those terms associated with εp, then consider the higher level solution form
Ap (t )=Ap,1 cos (ωt )+Ap,3 cos (3ωt )+Ap,5 cos (5ωt )+…
(16)
Then, consider the harmonic balances of cos (ωt ), cos (3ωt ), cos (5ωt ), … to setup the linear algebraic equations and solve for the unknowns Ap,1, Ap,3, Ap,5 …. Finally, the overall initial amplitude is given by
AI =
∑ p ∑q Ap, q
(17)
where AI ¼overall initial amplitude; p¼ solution level; and q¼ harmonic number
4. Results and discussions In this section, the material properties of the panel in the numerical cases are considered as follows: Young's modulus of the beam E ¼71 109 N/m2; mass density ρ ¼ 2700 kg/m3, Poisson's ratio ν ¼0.3; the panel dimensions a b¼0.3048 m 0.3048 m. In Tables 2a and 2b, Figs. 2 and 3a and b, the panel thickness τ ¼1.2192 mm. In Fig. 4, the panel thickness τ ¼4 1.2192 mm. Tables 2a and 2b show the contributions of the amplitude harmonic components at various solution levels and the natural frequency convergence study for various acoustic modes used, respectively. For smaller initial amplitudes (or smaller frequency ratios), the nonlinearity of the system is also smaller. The zero level solution, which contains the first harmonic component only, is sufficiently accurate. The higher harmonic components in the higher level solutions, which reflect the nonlinearity of the system, are not very significant. For larger initial amplitudes (or higher frequency ratios), the nonlinearity of the system is greater, and thus the contributions of the higher harmonic components are higher. Therefore, the error of the zero level solution, which contains the first harmonic component only, is larger; and higher level approaches, in which more harmonic terms are incorporated, are needed to achieve more accurate results. The four acoustic mode approach and first-level approach can be seen to be adequate to obtain converged and accurate solutions. In Fig. 2, the vibration amplitude ratio, AI/τ, is plotted against the frequency ratio for various cavity depths. The frequency ratios for various cavity depths obtained using the present harmonic balance method agree reasonably well with those obtained from the elliptical integral method in Ref. [2]. When the depth of the cavity is shorter or its volume is smaller, the natural frequencies are higher because its stiffness is stronger. According to the comparison between the slopes of the curves for no cavity and with depth¼0.0508 m, the natural frequency of the panel coupled with the shorter cavity depth is less sensitive to the vibration amplitude, particularly for vibration amplitude ratio40.5. In Fig. 3a, the vibration amplitude ratio, AI/τ, is plotted against the frequency ratio for various cavity lengths. The effect of cavity length is similar to that of cavity depth. When the length of the cavity is longer or its volume is greater, the natural frequencies are lower because its stiffness is weaker. For example, when the cavity length changes from 1 to 1.2, 1.5; and 3 the panel length for a zero amplitude ratio, the natural frequency decreases by 8.5%, 16.2% and 28%, respectively. The zero amplitude ratio implies linear vibration. In Fig. 3b, the frequency ratio is plotted against the Table 2a Amplitude harmonic component contributions at various solution levels, %. Frequency Ratio ω/ωo ¼
Zero level 1st level 2nd level
1.50 99.57 0.43 0.00
1.81 97.72 2.24 0.04
2.29 96.14 3.73 0.12
2.61 95.54 4.30 0.17
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Table 2b Frequency ratio convergence study for various acoustic modes used, ω/ωo. Amplitude ratio AI/τ ¼
1 mode 4 modes 9 modes
0.400 1.5068 1.5027 1.5026
1.108 1.8151 1.8101 1.8099
1.823 2.2965 2.2901 2.2900
2.23 2.6142 2.6069 2.6068
Fig. 2. Vibration amplitude ratio versus frequency ratio for various cavity depths.
cavity length for various cavity depths. The amplitude ratio is zero. There are large differences between the points on the three curves of different cavity depths when the cavity length is equal to the panel length. This implies that when the cavity length is equal to the panel length, the natural frequency is very sensitive to the cavity depth. On the contrary, when the length of the cavity is much longer than that of the panel, the natural frequency is not very sensitive to the cavity depth. There are small differences between the points on the three curves of different cavity depths when the cavity length is equal to 4 the panel length. In Fig. 4, the vibration amplitude ratio, AI/τ, is plotted against the frequency ratio when the cavity length ¼4.25 the panel length and thickness¼4 1.2192 mm. The dotted line represents the result of the one-dimensional cavity or no cavity resonance parallel to the panel length. It can be seen that the first and second cavity resonances significantly affect the natural frequency. As these are within the frequency ratio range considered, there are three natural frequency solutions. The first frequency solution converges to the first cavity resonant frequency, but is lower when the initial amplitude goes to infinity; the second frequency solution is slightly higher than the first cavity resonant frequency when the initial amplitude is small, and converges to the natural frequency solution of no cavity resonance when the initial amplitude is set higher. Then, the second frequency solution converges to the second cavity resonant frequency, but is lower when the initial amplitude goes to infinity. The third frequency solution is almost constant and slightly higher than the second cavity resonant frequency when the initial amplitude is set at less than 1. Then it converges to the natural frequency solution of no cavity resonance when the initial amplitude is set higher. As mentioned above, in practice, the cavity length is usually longer than the panel length (see Fig. 1). Hence, in the modelling process, the cavity length is a very important factor that significantly affects the panel vibration or sound radiation and should not be ignored.
Fig. 3. (a) Vibration amplitude ratio versus frequency ratio for various cavity lengths. (b) Frequency ratio versus cavity length for various cavity depths.
Fig. 4. Vibration amplitude ratio versus frequency length¼ 4.25 panel length and thickness¼ 4 1.2192 mm.
ratio
for
cavity
5. Conclusions The nonlinear structural-acoustic formulation has been introduced for the large amplitude free vibrations of a flexible panel
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coupled with extended cavity, and the multi-level residue harmonic balance method is applied to solve this nonlinear problem. The present harmonic balance solution agrees reasonably well with the results obtained from the elliptic integral method. This study focuses on the effect of cavity length, which was ignored in many previous structural acoustic studies. The results show that the natural frequency of a panel-cavity system is very sensitive to the length of the cavity and may decrease significantly when it is longer, because its volume is greater or its stiffness is weaker. Moreover, when the cavity is very long and the cavity resonant frequencies are close to that of the panel, multiple frequency solutions may be obtained.
Acknowledgements The author would like to express his sincere gratitude and appreciation to Dr. W.Y. Poon's advice and contributions.
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