Nonlinear fluid-structure interaction of an elastic panel in an acoustically excited two-dimensional inviscid compressible fluid Z. Aginsky and O. Gottlieb Citation: Phys. Fluids 25, 076104 (2013); doi: 10.1063/1.4813814 View online: http://dx.doi.org/10.1063/1.4813814 View Table of Contents: http://pof.aip.org/resource/1/PHFLE6/v25/i7 Published by the AIP Publishing LLC.
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PHYSICS OF FLUIDS 25, 076104 (2013)
Nonlinear fluid-structure interaction of an elastic panel in an acoustically excited two-dimensional inviscid compressible fluid Z. Aginsky and O. Gottlieba) Department of Mechanical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel (Received 7 October 2012; accepted 27 June 2013; published online 22 July 2013)
The focus of this paper is on the asymptotic investigation of the nonlinear fluidstructure interaction of an acoustically excited clamped panel immersed in an inviscid compressible fluid. A multiple-scales analysis of the corresponding two-dimensional unsteady potential flow initial-boundary-value-problem is employed to investigate both primary resonance and a 3:1 internal resonance between the panel fifth and ninth modes. Validation of the asymptotic structural response and the fluid pressure shows good agreement with numerical solution of a weakly nonlinear panel in a quadratic Euler field. The results shed light on the intricate acoustic interaction bifurcation structure which exhibits coexisting bi-stable periodic solutions, and quasiperiodic response reflecting spatially periodic modal energy transfer for both panel and fluid. This behavior is found to occur for panel excitation by finite level acoustic pressure waves that can be a crucial factor for design of high integrity structural systems required for aviation or space where light structures are exposed to intensive acoustic pressure C 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4813814] fluctuations.
I. INTRODUCTION
Acoustic fluid-structure interaction appears in several length scales, from large civil structures,1 to small biological2 and microelectomechanical systems structures.3 Mechanical structures can be exposed to significant acoustic noise in various operating conditions. Finite acoustic pressure fluctuations on aircraft panels can lead to failure due to acoustic fatigue requiring frequent maintenance and inspection.4–8 Structural oscillations induced by incident acoustic waves, result in reflected, transmitted and scattered waves which contribute to the panel loading. The traditional analysis of structural response and its coupled acoustically excited fluid field is usually made by the assumptions of linear vibrations and small amplitude potential flow, respectively. However, experimental analysis reveals distinct nonlinear behavior4, 9 of panels with either geometric or material nonlinearities that exhibit displacement on the order of panel thickness.10–13 In a benchmark experiment at NASA (1992), Maestrello et al.9 placed an aluminum plate in a compressible fluid field with both low and high sound pressure levels (SPL). The aluminum plate was clamped and a normal incidence plane wave was generated by an acoustic driver mounted at the end of an exponential horn. At low SPL the plate motion was periodic. However, at a high SPL the measured plate motion was chaotic-like. They also noted that a decrease of acoustic input by two orders of magnitude (20 dB) did not alter the wide-banded spectral signature of the non-stationary solution. The acoustic pressure radiation was also measured near and away from the center of the plate and revealed similar behavior. Quasiperiodic and chaotic dynamics have been found in several investigations of externally excited panels. However, the majority of these investigations do not consistently incorporate acoustical radiation or acoustic excitation. Examples of numerical investigations of panel dynamics in air a) Author to whom correspondence should be addressed. Electronic mail:
[email protected]
1070-6631/2013/25(7)/076104/35/$30.00
25, 076104-1
C 2013 AIP Publishing LLC
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without fluid coupling include chaotic snap-through buckling of clamped beams,14, 15 and chaotic oscillations in circular plates with free-edges.16 Discrepancies between numerical analysis and experimental measurements were documented for imperfect free-edge circular17 and rectangular plates.18, 19 A small number of numerical investigations have incorporated nonlinear acoustic-panel interaction which account for both the geometric and fluid nonlinearities. Frendi et al.11 investigated the coupling between a vibrating plate and acoustic radiation. They employed the equations of a nonlinear beam to describe the structural dynamics and the acoustic fluid was described using the two-dimensional nonlinear Euler equations. The coupling between the acoustic field and the plate was obtained by substitution of the acoustic pressure on the plate and consequent substitution of the velocity of the plate back to the acoustic field equations. Their results revealed that for low-level of acoustic forcing on the plate the coupling is weak and therefore not needed in order to obtain an accurate prediction of the plate response. However, for higher amplitude forcing, the acoustic coupling was found to be important as it significantly increased the damping on the structure. Recently, Aginsky and Gottlieb20 employed a loose coupling approach with grid adaptation to numerically couple between a quadratic Euler fluid field and an acoustically excited viscoelastic panel with a cubic nonlinearity. Simulations reveal an intricate bifurcation structure near the fifth-mode panel resonance that included coexisting symmetric and asymmetric periodic solutions. They also established that emergence of a nonstationary chaotic like, spatio-temporal solutions were an outcome of orbital stability loss of a fundamental ultrasubharmonic response and that distinct sum and difference frequency combination resonances culminated with quasi-periodic panel dynamics. Examples of asymptotic investigations without fluid-structure interaction include a hingedclamped beam,21 a simply supported rectangular plate,22, 23 and a nearly square plate.24 Nagai et al.25 compared experimental and analytical results of chaotic vibrations of a shallow cylindrical shell-panel with simply supported edges subjected to the gravity and periodic acceleration, and found partial agreement. Recent investigations of fluid-structure interactions involving elastic beams presented the frequency response of clamped-clamped and cantilever beams immersed in compressible fluids26 and the hydrodynamics loads generated by oscillating flexible cylinders due to viscosity and radiation of acoustic waves.27, 28 However, these investigations did not consistently model the fluid-structure coupling and did not take into account nonlinear behavior such as coexisting solutions and energy transfer between modes. Ginsberg29, 30 and Nayfeh31, 32 investigated the acoustic wave propagation where the fluid of the acoustic field assumed to be an inviscid, irrotational perfect gas modeled by a nonlinear unsteady potential equation. Nayfeh31 compared several perturbation methods such as strained coordinates, method of characteristics, renormalization, and multiple-scales for the nonlinear hyperbolic wave analysis and showed the generality and robustness of the multiple-scales method. Geer and Pope33 investigated the sound generation by vibrating bodies. They used an inviscid, ideal gas fluid model to describe the acoustic fluid medium and applied the multiple-scales method to the model equations. However, the structural fluctuations were only modeled by boundary conditions. Therefore, the structural dynamics and the coupling to the fluid, which included nonlinear effects, were omitted. They compared the asymptotic solution to a numerical solution which was obtained using the MacCormack predictor-corrector technique, and found a good agreement for a wide range of acoustic Mach numbers and presented a growing deviation of the asymptotic solution from the numerical solution with increasing acoustic Mach number. Nayfeh34 derived an analytical solution for nonlinear propagation of waves induced by plate vibrations using the method of renormalization for a lossless medium and employed multiple-scale asymptotics for a thermo-viscous fluid. However, these solutions did not incorporate elastic coupling consistently as the plate was modeled only as boundary conditions of the fluid field. Ginsberg29, 35 and Nayfeh and Kelly36 investigated the primary resonance solution of a harmonically excited, nonlinear plate with periodic boundary conditions and an acoustic field. These models had several sources of nonlinearity for the fluid which included nonlinear convection, the pressure-density relation, and the interface condition of the plate-fluid. Another source of nonlinearity was the geometric plate bending induced tension. Sorokin and his colleagues studied nonlinear fluid-structure interaction of elastic structures in heavy fluid loading conditions, i.e., water as an acoustic medium.37–39 Their analysis consisted of an infinitely long nonlinear elastic plate without
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intermediate supports and with periodically fixed hinges which are driven by a lateral force. The fluid-structure interaction was modeled via a contact acoustic pressure using the nonlinear Bernoulli equation and the coupling was modeled using a nonlinear continuity condition. However, the acoustic medium was modeled by a linear acoustic equation. They used multiple-scale asymptotics to obtain an approximate solution for the system which revealed that the response governed by the structural nonlinearity for weak near resonance excitation is governed by the fluid nonlinearity for the case of hard sub-harmonic excitation. Similar results was found by Chang and Liu40 which investigated a nonlinear elastic plate subjected to heavy fluid loading in a magnetic field. Motivated by the above noted discrepancies between experiments and theoretical models that do not incorporate any fluid coupling, and the limited analysis of consistent near-resonance fluidstructure interaction, the focus of this paper is on the asymptotic investigation of the nonlinear fluidstructure interaction of an acoustically excited clamped panel immersed in an inviscid compressible fluid. A multiple-scales analysis of the corresponding two-dimensional unsteady potential flow initial-boundary-value-problem is employed to investigate primary resonance and a 3:1 internal resonance between the panel fifth and ninth modes. Validation of the asymptotic structural response and the fluid pressure is done via a coupled numerical solution for a weakly nonlinear panel in a quadratic Euler field. We close with a discussion of the bi-stable and quasiperiodic pressure fields that emerge from the structure that is subject to harmonic acoustic excitation. II. PROBLEM FORMULATION
The acoustic fluid-structure interaction (A-FSI) problem considered consists of a clamped elastic panel immersed in a compressible fluid that is excited by an incident acoustic pressure plane wave that is normal to the panel surface. The interaction between the incident wave and the plate results in reflected, transmitted, and scattered waves. The system domain is described in Fig. 1. This problem of A-FSI includes several sources of nonlinearity:11, 30 (1) (2) (3) (4)
The transverse displacement of the panel due to geometric induced tension. Wave steepening between the near and far fields. The pressure variation is not much smaller than the ambient pressure. The interface kinematic continuity condition.
A. The fluid field
The most general model for an acoustic fluid is based on the Navier-Stokes equations which account for compressibility, viscosity, and rotationally. In cases where low frequencies are avoided,
FIG. 1. Definition sketch of the domain.
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gravitational effects are negligible.41 In cases where an incident plane wave excites a panel and there is no flow parallel to the structure, the viscosity can be neglected.42 A simpler fluid model can be obtained when the length of acoustic wave is much smaller than the surrounding pressure in a stationary ideal fluid. In this case, the fluid can be assumed irrotational and described by a set of unsteady nonlinear potential flow equations.41 The fluid field is defined by a compressible, inviscid, and ideal gas that is excited by an acoustic incident plane wave, which is normal to the structure surface. In this case the use of nonlinear unsteady potential flow equations29, 30, 32, 36 is suitable. The following equations describe the mass transport, momentum transport, and the thermodynamic state: ∂ρ + ∇ · ρ V = 0, (1) ∂t ρ
∂ V + V · ∇ V ∂t
= −∇ p,
(2)
γ p ρ −1 = , p0 ρ0
(3)
where V is the velocity field components vector, γ is the gas specific heat ratio (γ = 1.401 in air at 25 ◦ C), ρ 0 is the undisturbed density, and p0 is the undisturbed pressure. Following Ginsberg,29 Nayfeh and Kelly,36 and Anderson,43 we make use of Eqs. (1)–(3), with an irrotational fluid assumption (∇ × V = 0), and V = ∇φ to obtain the nonlinear potential and pressure equations truncated to cubic order: φtt − c2 [φx x + φzz ] = − (γ − 1) φt (φx x + φzz ) − 2φx φxt − 2φz φzt ,
p p0
(γ γ−1)
(γ − 1) =1− c2
1 1 2 2 φt + (φx ) + (φz ) , 2 2
(4)
(5)
where c is the speed of sound in the undisturbed fluid. B. The elastic field
We use the two-dimensional beam equation to describe the nonlinear elastic field for a thin ˜ t˜)):11–14 panel (W˜ (x, ˜ ρ p h W˜ t˜t˜ + D W˜ x˜ x˜ x˜ x˜ − N˜ W˜ x˜ x˜ + κ˜ 1 W˜ x˜ x˜ x˜ x˜ t˜ + κ˜ 2 W˜ t˜ = p,
(6)
where h is the panel thickness (we assume h < L / 100) and ρ p is the panel density, the bending stiffness is D = Eh3 / 12(1 − ν 2 ), E is the Young’s modulus, υ is the Poisson ratio, κ 1 is an internal damping coefficient deduced from a Kelvin-Voigt model,44 and κ 2 is the linear equivalent damping coefficient which is proportional to the panel velocity. The linear and nonlinear membrane stiffness and the nonlinear internal damping terms in Eq. (6) are derived in Appendix A: Eh N˜ = N˜ 0 + 2L
x 0 +L
x0
∂ W˜ ∂ x˜
2
κ ˜ Eh dx + 1 DL
x 0 +L
x0
∂ W˜ ∂x
∂ 2 W˜ ∂ t˜∂ x˜
d x,
(7)
where N˜ 0 is a measure of panel pretension. Motived by the benchmark experiment of Maestrello et al.,9 the influence of pretension is not considered in this investigation. The panel is clamped on both sides and is excited from its stable equilibrium. The boundary conditions for clamped panel are
W˜ x0 , t˜ = W˜ x˜ x0 , t˜ = W˜ x0 + L , t˜ = W˜ x˜ x0 + L , t˜ = 0, (8)
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and the initial conditions are ˜ 0) = 0, W˜ (x,
˜ 0) = 0. W˜ t˜ (x,
(9)
C. Flow-structure coupling
The fluid-structure coupling is obtained via modified boundary conditions on the flexible panel or adjacent rigid wall and the fluid field. We assume that the fluid velocity on the panel surface is equal to the panel velocity: Wt (t, x) = φz (t, x, 0) − Wx (t, x) φx (t, x, 0) .
(10)
The acoustic pressure load on the panel is the pressure difference between the sides. The pressures on the incident side and on the other side are P˜ + = P˜∞ + P˜ i + P˜ r + P˜ s , (11) P˜ − = P˜∞ + P˜ t , where P˜∞ is the ambient pressure, and P˜ i , P˜ r , P˜ s , P˜ t are the incident, reflected, scattered, and transmitted pressure fluctuations, respectively. We assume that the fluid on both sides of the panel has the same properties. The scattered radiated pressure and the transmitted radiated pressure are both induced by the panel vibration. Therefore, one field is anti-symmetric to the other with respect to the panel. On the surface of the panel, we thus define the transmitted and scattered pressures:11 P˜ t = − P˜ s .
(12)
The input pressure can be defined as the sum of the incident and the reflected waves: P˜ i + P˜ r =
m
˜n ,
˜ n sin ω˜ n t˜ +
(13)
n=1
where ˜ n is the external forcing pressure amplitudes, ω˜ n = 2π f˜n is the angular frequency, m denotes ˜ n is the phase. Consequently, the acoustic driving the number of single frequency sources, and pressure results in ˜ P˜ = P˜ + − P˜ − = 2 P˜ t − ,
(14)
where we assume zero phases:
˜ =
m
˜ n sin 2π f˜n t˜ .
(15)
n=1
D. Non-dimensional equations
We rescale the two-dimensional (2D) initial-boundary-value problem (IBVP) described in Secs. II A–II C, by the beam length (L) and the elastic time scale (T) to yield the following non-dimensional parameters:
L 4ρ p h 1 − υ 2 x˜ z˜ W˜ t˜ x= , z= , W = , t= , T = , L L L T EI v˜ ρ˜ p˜ u˜ , v= , ρ= , p= − 1, u= L0 L0 ρ0 p0 where the panel moment of inertia is I = h3 /12, and 0 = 1/T. We substitute the non-dimensional parameters into Eqs. (4) and (5) and obtain the following non-dimensional equation for the potential: φtt −
1 [φx x + φzz ] = − (γ − 1) φt (φx x + φzz ) − 2φx φxt − 2φz φzt , C2
(16)
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where C = L0 /c. We neglect the third order terms due to their small contribution.42 The pressure is calculated via the following equation:29, 36
(γ −1) 1 1 2 2 2 γ (1 + p) (17) = 1 − (γ − 1) C φt + (φx ) + (φz ) . 2 2 The nondimensional panel equation is ⎤ ⎡ 1 1 Wtt + Wx x x x − Wx x ⎣ψ Wx2 d x + δˆ3 Wx Wt x d x ⎦ + δˆ1 Wx x x xt + δˆ2 Wt
= χˆ
0
0
ˆ n sin (nt) − p| y=W ,
(18)
n
where
L 2 κ˜ 1 1 − υ 2 κ˜ 2 1 − υ 2 L 4 2 p0 L 3 1 − υ 2 κ˜ 1 Eh 2 ˆ ˆ , δ2 = , δ3 = , ψ = 6 1−υ , , χˆ = EIT EIT DLT h EI (19) and
ˆ n sin (nt) are the multiple frequency inputs. We apply a single source (n = 1) in this δˆ1 =
n
investigation. The non-dimensional coupling is obtained via a compatibility condition describe by the panel velocity:23, 37 Wt (t, x) = φz (t, x, 0) − Wx (t, x) φx (t, x, 0) .
(20)
The nondimensional boundary conditions for the clamped panel are W (0, t) = Wx (0, t) = W (L , t) = Wx (L , t) = 0,
(21)
and the initial conditions are W (x, 0) = 0,
Wt (x, 0) = 0.
(22)
III. ASYMPTOTIC ANALYSIS A. Primary resonance
Investigation of the weakly nonlinear IVBP is carried out using an asymptotic multiple-scales analysis applied directly to the coupled elastic structure and acoustic fluid fields. We investigate primary resonance in three configurations: (i)
Infinitely long beam periodically simply supported, where the distance L between supports is depicted in Fig. 2, and where the beam zero displacement boundary condition is W (x, t) = 0,
(ii) (iii)
x = 0, ±1, ±2, ....
(23)
Infinitely long beam, periodically clamped, where the distance between supports L is depicted in Fig. 3. Single beam, clamped, where distance between supports is L as presented in Fig. 4.
FIG. 2. Infinitely long beam with periodically simplify supports configuration.
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FIG. 3. Infinitely long beam with periodically clamped supports configuration.
1. The multiple-scales solution
We apply the asymptotic multiple-scales method due to its generality and robustness,31, 34 and assume a small panel displacement, a small potential function and a small pressure fluctuation, where ε is a small parameter: W (x, T0 , T1 , T2 ) = εW1 (x, T0 , T1 , T2 ) + ε2 W2 (x, T0 , T1 , T2 ) + ε3 W3 (x, T0 , T1 , T2 ) + ..., φ (x, y, T0 , T1 , T2 ) = εφ1 (x, y, T0 , T1 , T2 ) + ε2 φ2 (x, y, T0 , T1 , T2 ) + ...,
(24)
p (x, y, T0 , T1 , T2 ) = εp1 (x, y, T0 , T1 , T2 ) + ε2 p2 (x, y, T0 , T1 , T2 ) + ..., with the following time scales: T0 fast, T1 slow, and T2 is even slower: Tn = εn t,
n = 0, 1, 2....
(25)
The derivatives are ∂ = D0 + ε D1 + ε2 D2 + O(ε3 ) , ∂t
Dj =
∂2 = D0 2 + 2ε D0 D1 + ε2 D1 2 + 2D0 D2 + O(ε3 ) , ∂t 2
∂ , ∂ Tj
(26)
Dj2 =
∂2 . ∂ Tj 2
(27)
Due to the restrictive assumption of small amplitude panel dynamics where the order of the displacement magnitude is the panel thickness, we assume small acoustic excitation. In order to ensure that the secular interaction of the system occurs at the third order, we also assume both small internal and viscous damping to yield the following parameters: ε2 δ1 = δˆ1 ,
ε2 δ2 = δˆ2 ,
ε2 δ3 = δˆ3 ,
ε2 χ = χ, ˆ
ε n = ˆ n .
(28)
Thus, we obtain the following elastic field equation: ⎡ ⎤ 1 1 Wtt + Wx x x x − ψ Wx x Wx2 d x + ε2 ⎣δ1 Wx x x xt + δ2 Wt − δ3 Wx x Wx Wt x d x ⎦ = ε2 χ ε
0
n sin (nt) − p| y=W .
n
FIG. 4. Finitely clamped-clamped beam configuration.
0
(29)
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The solution of Eq. (29) depends on the type of boundary condition. In the following solutions we use a general set of boundary conditions and excite the beam near its primary resonance where we define the following detuning: ε2 σ = − ωn ,
(30)
where ωn is the beam frequency calculated from the dispersion relation. Substitution of Eqs. (24)– (30) into Eqs. (16)–(20) yields the following recursive set of equations: Order (ε1 ): D0 2 W1 + W1x x x x = 0,
(31)
1 [φ1x x + φ1zz ] = 0, C2
(32)
D0 W1 (x, t) − φ1z (x, 0, t) = 0,
(33)
p1 + γ C 2 D0 φ1 = 0.
(34)
D0 2 W2 + W2x x x x = −2D1 D0 W1 ,
(35)
D 0 2 φ1 −
2
Order (ε ):
D 0 2 φ2 −
1 [φ2x x + φ2zz ] = −2D1 D0 φ1 − (γ − 1) D0 φ1 (φ1x x + φ1zz ) − 2φ1x D0 φ1x −2φ1z D0 φ1z , C2 (36) D0 W2 (x, t) − φ2z (x, 0, t) = −W1x (x, t) φ1x (x, 0, t) − D1 W1 (x, t) ,
(37)
p1 2 1 1 2 2 2 p 2 + γ C D 0 φ2 = − γ C D1 φ1 + (φ1x ) + (φ1z ) . 2γ 2 2
(38)
2
Order (ε3 ) (only the beam equation): D0 2 W3 + W3x x x x = −D1 2 W1 − 2D1 D0 W2 − 2D2 D0 W1 − 2D1 D0 W2 − δ1 D0 W1x x x x − δ2 D0 W1 1 + ψ W1x x
W1x 2 d x + χ ee t − 2χ p1 | y=0 .
(39)
0
Note that Eq. (39) includes both linear internal Kelvin-Voigt and equivalent viscous damping terms and the acoustic excitation terms. However, the cubic Kelvin-Voigt term in Eq. (29) is of higher order due to the small displacement assumption of Eq. (24). Solutions of different boundary conditions of Eqs. (31), (35), and (39) are presented in Sec. III A 3. We note that the boundary conditions of Eqs. (21) and (23) are trivial in all the orders. The solution of the first order system (Eqs. (31)–(34)) is (40) W1 = ζ (x) A (T1 , T2 ) eiωn T0 + A¯ (T1 , T2 ) e−iωn T0 , where ζ (x) is the basis function of the beam that depends on the type of boundary condition. We substitute Eq. (40) into Eq. (33) and obtain a first order potential equation for an outgoing wave: ωn φ1 = (41) X (x) B1 eikz z Q (T1 , T2 ) eiωn T0 + Q¯ (T1 , T2 ) e−iωn T0 , kz
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where kz 2 = zn 2 (C2 zn 4 − 1) and zn 2 = ωn . For an outgoing wave to exist, we need to satisfy the condition of L0 ωn > c.36 We use the matching condition in Eq. (33) and obtain the following relations: B1 = 1,
X (x) = ζ (x) ,
Q (T1 , T2 ) = A (T1 , T2 ) ,
Q¯ (T1 , T2 ) = − A¯ (T1 , T2 ) .
(42)
Recall that the requirement for an outgoing wave eliminates the need for an additional boundary condition. The pressure of the first order is obtained from Eq. (34):
p1 = −Im
iγ ωn 2 2 C ζ (x) eikz z A (T1 , T2 ) eiωn T0 + A¯ (T1 , T2 ) e−iωn T0 . kz
(43)
The solution of the second order system (Eqs. (35)–(38)) is obtained by substitution of Eqs. (30), (40), and (43) into Eq. (35): (44) D0 2 W2 + W2x x x x = −2iωn ζ D1 A (T1 , T2 ) eiωn T0 − A¯ (T1 , T2 ) e−iωn T0 . We note the existence of a secular term in the right-hand side (RHS) of Eq. (44). In order to eliminate the secular term we set: D1 A (T1 , T2 ) = 0.
(45)
Thus, A = A(T2 ) and the second order displacement is identically zero: W2 = 0.
(46)
The second order potential equation is obtained by substituting Eqs. (40) and (45) into Eqs. (36) and (37): D 0 2 φ2 −
1 iωn ikz z ¯ −iωn T0 − D1 Aeiωn T0 + D1 Ae ζe [φ2x x + φ2zz ] = −2 2 C kz
−iωn
3
2 2 (γ − 1) 2 ζ + ζ ζx x − (γ + 1) ζ e2ikz z A2 e2iωn T0 − A¯ 2 e−2iωn T0 . 2 x 2 kz ky
(47)
We note that the secular term in the RHS of Eq. (47) is eliminated since D1 A = 0 from the previous order. Eq. (47) without the secular term is D 0 2 φ2 −
1 [φ2x x + φ2zz ] = C2
−iωn 3
2 2 (γ − 1) 2 (γ e2ikz z A2 e2iωn T0 − A¯ 2 e−2iωn T0 , ζ + ζ ζ − + 1) ζ xx 2 x 2 kz kz
(48)
and the compatibility condition is φ2z (x, 0, t) =
ωn ζx 2 2 2iωn T0 A e − A¯ 2 e−2iωn T0 . kz
(49)
Solution of Eqs. (48) and (49) is φ2 = −i
z n 2 2 2 2i(ωn T0 +kz z) A e ζ − A¯ 2 e−2i(ωn T −kz z)0 . 2 x 2k z
(50)
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The solutions for the potential function and the pressure are based on the first and second order terms. We use the third order beam equation to obtain the evolution equation which describes the slowly varying amplitude A(T2 ). We substitute Eqs. (30), (40), (43), and (47) into Eq. (39) and obtain the following equation: D0 2 W3 + W3x x x x = ⎡ ¯ xx = ⎣−iωn ζ (2D2 +) A+3ψ A2 Aζ
⎤
1 ζx d x+ χ e 2
iσ T2 ⎦ iωn T0
e
1 +ψζx x
0
ζx 2 d x A3 e3iωn T0 +CC, 0
(51) where = δ1
ζx x x x 2χ γ z n 2 C 2 = δ1 z n 4 + δ2 + δ3 + δ2 + ζ kz
(52)
and δ 3 = 2χ γ zn 2 C2 /kz . We eliminate the secular terms in Eq. (51) and obtain a complex evolution equation: 1 ¯ xx − iωn ζ (2D2 + ) A + 3ψ A Aζ
ζx 2 d x + χ eiσ T2 = 0.
2
(53)
0
The third order beam equation without secular terms is 1 D0 W3 + W3x x x x = ψζx x
ζx 2 d x A3 e3iωn T0 + CC,
2
(54)
0
where CC denotes complex conjugate. The solution of Eq. (54) is ψ W3 = − ζ x x 8
1
ζx 2 d x A3 e3iωn T0 − eiωn T0 + C.C.
(55)
0
We multiply Eq. (53) by the linear modal shape function ζ (x) and integrate the result over the domain to obtain the following equation: ⎡ ⎤ 1 1 ¯ x x ζx 2 d x + χ eiσ T2 ⎦d x = 0. ζ ⎣−iωn ζ (2D2 + ) A + 3ψ A2 Aζ (56) 0
0
Solution of Eq. (56) yields a complex evolution equation: − iωn 1 (2D2 + ) A + 3ψ2 ψ A2 A¯ + 3 χ eiσ T2 = 0,
(57)
where 1 1 =
1
0
1 ζx x (x) ζ (x)
ζ (x) d x, 2 = 2
0
1 ζx (x) d xd x, 3 =
ζ (x) d x.
2
0
(58)
0
We use a polar representation, A(T2 ) = a(T2 )e /2 and define the following phase: iθ
(T2 ) = σ T2 − θ (T2 ) .
(59)
We collect the real and the imaginary parts to obtain the following autonomous polar system and note that this slow varying evolution equation has a standard Duffing like form:45 a˙ = − (K 21 + K 22 + K 23 ) a − K 3 sin () , ˙ = σ − K 1 a 2 − K 3 cos () , a
(60)
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FIG. 5. Damping tendencies as a function of the mode number. K21 -circles (black), K22 -triangles (blue), and K23 -squares (red).
where 32 ψ 1 , K 21 = δ1 z n 4 , 2 81 z n 2 1 3 χ = δ3 , K3 = . 2 1 z n 2
K1 = K 23
K 22 =
1 δ2 , 2
(61)
We note three types of linear damping mechanisms: K21 relates to the internal damping coefficient deduced from the structural Kelvin-Voigt model, K22 relates to the linear equivalent fluid damping, and K23 is damping due to acoustic radiation. Fig. 5 depicts three different tendencies for the linear damping coefficients as a function of the exciting acoustic frequency. The results presented are for a clamped-clamped beam with the following parameters: L/h = 100, ε2 δ 1 = 2.823.10−5 , ε2 δ 2 = 2.437, υ = 0.33, E = 66.69.103 MPa, ρ p = 2700 Kg/m3 , γ = 1.4, c = 340 m/s. K21 , K22 , and K23 described by (black) circles, (blue) triangles, and (red) squares, respectively. We note that K22 remains constant throughout the frequency range and that K21 grows as the frequency is increased. However, K23 which consists of damping due to acoustic radiation decreases with increasing frequency. A similar tendency where the fluid damping reduces as the exciting frequency increases was found for a linear cantilever beam vibrating in a Stokes flow regime.46 This linear damping behavior reveals that for a different range of frequencies a different damping mechanism may be significant as depicted in Fig. 5, where the dominant damping mechanism for the first mode is due to acoustic radiation whereas the internal damping from a Kelvin-Voigt model is dominant for the third mode. Furthermore, recall that an outgoing wave will exist only if the following condition is satisfied:36 EI 1 2
ωn > 1. (62) 2 2 c2 L ρph 1 − υ We note that as the ratio of the left-hand side in Eq. (62) approaches the value of unity, the magnitude of the acoustic drag increases dramatically. The left parenthesis in Eq. (62) includes the structural properties, the center parenthesis the fluid parameter, whereas the term in the right parentheses is determined by the beam dispersion equation which depends on the type of boundary condition.
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2. Frequency response and orbital stability
By squaring and adding the two equations in Eq. (60) and multiplying by ε2 we obtain the steady state amplitude response as a function of the detuning from the excitation frequency: k1 1 (εa)2 ± =1− ωn ωn ωn
˜ 2 2
k3 − k˜21 + k˜22 , εa
(63)
where
1 − υ2 k˜21 = κ˜ 1 ωn 2 + κ˜ 2 L 4 , 2E I T
2 p0 L 3 1 − υ 2 γ z n 2 L0 2 ˜k22 = , E I kz c
p L 3 1 − υ 2 ˜ n ˜k3 = 3 0 . 1 E I z n 2
(64)
We explore the stability of the dynamic response by deriving the Jocobian matrix of Eq. (60) at equilibrium. Note that all the parameters in Eq. (63) are positive. Thus, at steady state, the lower and the upper branches of the frequency response are stable, whereas the middle branch is unstable.45 Fig. 6 depicts the frequency response obtained from Eq. (63), for a clamped beam with the following parameters: L = 0.3048 m, L/h = 100, ε2 δ 1 = 0, ε2 δ 2 = 0, υ = 0.33, E = 69.35 GPa, ρ p = 2700 Kg/m3 , γ = 1.4, c = 340 m/s, = 0.0125. We note that Fig. 6 portrays the frequency response of a standard hardening Duffing equation,45 where the solid and dashed lines correspond to stable and unstable solutions, respectively. We note three bifurcation regions. Regions I and III include a unique solution, whereas region II includes coexisting bi-stable solutions separated by an unstable solution. The transitions between regions I-II and II-III are defined by two saddle-node bifurcation points.51
FIG. 6. Frequency response of the beam first mode. The solid and dashed lines depict stable and unstable solution, respectively.
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3. The influence of boundary conditions a. Case of an infinitely long periodically simply supported beam. The linear shape function for a simply supported periodic boundary condition as described in Eq. (23) is
ζ (x) = sin (z n X ) ,
z n = nπ,
(65)
where X = x − x0 and x0 is the coordinate of the left support of the beam. For simplicity we will present the case of x0 = 0. We use Eq. (54) to obtain the third order beam solution and the evolution equations (63) and (64). The third order displacement solution is identical to Nayfeh and Kelly:36 W3 = −
ψn 4 π 4 sin(nπ x)a 3 e3iωn T0 − eiωn T0 + C.C. 64
(66)
We substitute Eq. (65) into Eqs. (40)–(43) and obtain the displacement, the potential, and the pressure first order solutions. We substitute Eq. (65) into Eq. (50) and the solution of the second order potential is φ2 =
−in 4 π 4 [(1 + cos(2π nx))] A2 e2i(ωn T0 +kz z) − A¯ 2 e−2i(ωn T0 +kz z) 2 4k z
(67)
and the pressure of the second order is obtained by substitution of Eqs. (41), (43), and (67) into Eq. (38):
6 6 2n π γ 2 p2 = Im C [(1 + cos(2π nx))] A2 e2i(ωn T0 +kz z) + A¯ 2 e−2i(ωn T0 +kz z) + 2 4k z 1 p1 2 1 (φ1x )2 + (φ1z )2 . (68) − γ C2 + 2γ 2 2 Recall that the total pressure is the sum of the first and the second order contributions. b. Case of an infinitely long periodically clamped beam. We assume the existence of a pretension which ensures that the panel equilibrium configuration is asymptotically stable due to the existence of positive linear structural damping. For the case of clamped-clamped beam, the boundary conditions are as in Eq. (21). We use the linear case, without fluid loading, structural damping and external forces, and obtain the following shape function:13
ζ (x) = cosh (zn X) − cos (zn X) −
cosh (zn ) − cos (zn ) [ sinh (zn X) − sin (zn X)]. sinh (zn ) − sin (zn )
(69)
The frequency, ωn = zn 2 , can be calculated numerically from the beam dispersion relation: cosh (zn ) cos (zn ) − 1 = 0.
(70)
We substitute Eq. (69) into Eqs. (40)–(43) and obtain the first order solution. Substitution of Eq. (69) into Eq. (50) and into Eq. (38) yields a second order potential and a second order pressure. We substitute Eq. (69) into Eq. (55) and obtain the third order solution for the beam displacement. The second order pressure term can be obtained by substitution of Eq. (69) into Eqs. (38), (43), and (50). We note that the derivative in the x direction in the potential expression, and the potential is zero for x = 0 and x = 1. Therefore, there is no flow on these vertical lines and the pressure field takes the form of long separate corridors opposite to the beam. c. Case of a finitely clamped-clamped beam. For the case of a single span panel described in Fig. 4, the contact pressure on the beam surface is described by Eq. (43). However, the pressure field is a function of distance:42, 47, 48 ⎧ ⎫ ⎨ iγ ω 2 1 ζ ⎬ n (71) p1 (x, z, t) = −Im C 2 √ eikd d x A (T1 , T2 ) eiωn T0 + A¯ (T1 , T2 ) e−iωn T0 , ⎩ kz ⎭ d 0
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FIG. 7. Frequency response of the beam first mode, with and without acoustic radiation by dashed line and solid line respectively, = 0.004.
where d = |r − rs |, r is the position vector of the observation point, rs is the position vector of the elemental surface having normal velocity, and where k = ωn /C. Fig. 7 depicts the frequency response obtained from Eq. (63) for the beam mid-span with and without acoustic radiation denoted by dashed and solid lines, respectively for the following parameters: L = 0.3048 m, L/h = 100, ε2 δ 1 = 0, ε2 δ 2 = 2.3896, υ = 0.33, E = 69.35.103 MPa, ρ p = 2700 Kg/m3 , γ = 1.4, c = 340 m/s, the excitation amplitude is = 0.004, and we note that the significant fluid loads are due to acoustic radiation. B. Internal resonance
Due to the cubic nature of the beam nonlinearity we seek ratios between two natural frequencies which are close to three (ωn /ωm ≈ 3) which correspond to conditions for a 3:1 internal resonance. We note that the fifth mode frequency is close to the excitation frequency in the Maestrello experiment,9, 11 and that the ratio between the fifth (ωn = 298.59) and the ninth (ωm = 890.72) modal frequencies is ω9 /ω5 ≈ 2.98. 1. Multiple-scales solution
We assume that the contributions of all the other modes are non-resonant and as such smaller. Thus, we update the assumption of Eq. (24): εW j1 (T0 , T1 , T2 ) + ε2 W j2 (T0 , T1 , T2 ) + ε3 W j3 (T0 , T1 , T2 ) + ..... W j (T0 , T1 , T2 ) = +ε2
j=5,9
εW j1 (T0 , T1 , T2 ) + ε2 W j2 (T0 , T1 , T2 ) + ε3 W j3 (T0 , T1 , T2 ) + ......
(72)
j=1−8 J =5
We explore the internal resonance between the fifth and the ninth modes using Eqs. (27)–(38). The solution of the first order is (n = 5, m = 9): W1 = (73) ζ j (x) A j (T1 , T2 ) eiω j T0 + A¯ j (T1 , T2 ) e−iω j T0 , j=n,m
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where ζ n (x) and ζ m (x) are the n and m modal shape functions that depend on the type of boundary conditions. We substitute Eq. (73) into Eq. (32) and obtain the first order potential solution for an outgoing wave: φ1 =
ωj ζ j (x) eikz j z A j (T1 , T2 ) eiω j T0 − A¯ j (T1 , T2 ) e−iω j T0 , k j=n,m z j
(74)
where
kz j 2 = z j 2 C 2 z j 4 − 1 , z j 2 = ω j .
(75)
We note that the first order potential includes two frequency terms, and that the contact pressure at first order is ⎧ ⎫ ⎨ iγ ω 2 ⎬ j p1 (x, z, t) = −Im C 2 ζ j (x) eikz j z A j (T1 , T2 ) eiω j T0 + A¯ j (T1 , T2 ) e−iω j T0 . ⎩ ⎭ kz j j=n,m
(76) For the case of a single element, described in Fig. 4, the pressure field is48 ⎧ ⎫ ⎨ iγ ω 2 1 ζ ⎬ j eikz d x A j (T1 , T2 ) eiω j T0 + A¯ j (T1 , T2 ) e−iω j T0 , p1 (x, z, t) = −Im C2 ⎩ ⎭ kz j d j=n,m 0
(77) where d = |r − rs |, r is the position vector of the observation point, rs is the position vector of the elemental surface having normal velocity, and where k = ωn /C. The second order beam solution is obtained by substituting of Eq. (73) into Eq. (35): D0 2 W2 + W2x x x x = −2i
ω j ζ j D1 A j (T1 , T2 ) eiω j T0 − A¯ j (T1 , T2 ) e−iω j T0 .
(78)
j=n,m
In order to eliminate the secular term in the RHS of Eq. (78), we set the slow time derivatives to zero: D1 An (T1 , T2 ) = 0,
D1 Am (T1 , T2 ) = 0.
(79)
Thus, the amplitudes An and Am depend only on the slow time (T2 ) and the beam second order solution is zero: W2 = 0.
(80)
The second order potential equation is obtained by substituting Eqs. (73), (74), and (80) into Eqs. (36) and (37): ! 1 2 −2iωn T0 2ik zn z 2 2iωn T0 ¯ A + + φ e e − A e = ϑ [φ ] 2x x 2zz 11 n n C2 ! 2 +ϑ12 e2ikzm z Am 2 e2iωm T0 − A¯m e−2iωm T0 + ϑ2 An Am ei(kzn +kzm )z ei(ωn +ωm )T0 − −ϑ2 A¯ n A¯ m ei(kzn +kzm )z e−i(ωn +ωm )T0 + ϑ3 An A¯ m + ϑ4 A¯ m An ei(kzn +kzm )z ei(ωn −ωm )T0 − − ϑ3 A¯ n Am + ϑ4 Am A¯ n ei(kzn +kzm )z e−i(ωn −ωm )T0 , D 0 2 φ2 −
(81)
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and the compatibility condition is
! ! 2 2 φ2z (x, 0, t) = o11 An 2 e2iωn T0 + A¯n e−2iωn T0 + o12 Am 2 e2iωm T0 + A¯m e−2iωm T0 +
+Am An o2 ei(ωm +ωn )T0 − o2 A¯ m A¯ n e−i(ωm +ωn )T0 + o3 A¯ m An − o4 An A¯ m ei(ωn −ωm )T0 +
+ −o3 Am A¯ n + o4 A¯ n Am e−i(ωn −ωm )T0 , (82)
where the coefficients of Eqs. (81) and (82) are in Appendix B. We substitute Eqs. (73), (76), and (80) into Eq. (39) and obtain the third order beam equation in Appendix C. We define two detuning parameters that describe the nearness of the nth natural frequency (ωn ) to three times that of the mth natural frequency (3ωm ) corresponding to the condition for a 3:1 internal resonance, and the nearness of the external excitation frequency () to the mth resonance frequency (ωm) : ε2 σ1 = ωn − 3ωm ,
(83)
ε2 σ2 = − ωm .
(84)
We obtain the following third order beam equation where we have omitted the non-secular terms (NST): D0 2 W3 + W3x x x x = γ ωm + −iωm Am 2ζm D2 + δ1 ζmx x x x + δ2 ζm + 2χC 2 ζm k zm ⎞ ⎛ 1 1 1 3ψ A¯ m Am 2 ζmx x ζmx 2 d x + 2ψ An A¯ n Am ⎝ζmx x ζnx 2 d x + 2ζnx x ζmx ζnx d x ⎠ + 0
⎛ ψ A¯m An ⎝ζnx x 2
1
1 ζmx 2 + 2ζmx x
0
⎞
0
⎤
0
ζmx ζnx d x ⎠ eiσ1 T2 + χ eiσ2 T2 ⎦ eiωm T0 +
0
γ ωn + −iωn An 2ζn D2 + δ1 ζnx x x x + δ2 ζn + 2χC 2 ζn k zn ⎞ ⎛ 1 1 1 3ψ An 2 A¯ n ζnx x ζnx 2 d x + 2ψ Am A¯ m An ⎝ζnx x ζmx 2 d x + 2ζmx x ζmx ζnx d x ⎠ + 0
1 ψ Am 3 ζmx x
⎤
0
0
ζmx 2 d xe−iσ1 T2 ⎦ eiωn T0 + N .S.T. + C.C.
(85)
0
Elimination of secular terms in Eq. (85) yields a pair of complex evolution equations. We multiply them by ζ j and integrate over the domain to yield: 2 −iωm Am 1 (2D2 + m1 + m2 ) + Sm1 A¯ m Am 2 + Smn2 An A¯ n Am + A¯m An S3 eiσ1 T2 +
+Fm χ eiσ2 T2 = 0, − iωn An 2 (2D2 + n1 + n2 ) + Sn1 An 2 A¯ n + Snm2 Am A¯ m An + S4 Am 3 e−iσ1 T2 = 0,
(86) (87)
where the coefficients of Eqs. (86) and (87) are in Appendix D (Eq. (D1)). We make use of a polar representation which enables elimination of the non autonomous terms in Eqs. (86) and (87):49 1 1 ( p1 − iq1 ) eiν1 T2 , An = ( p2 − iq2 ) eiν2 T2 , 2 2 ν1 = σ2 , ν2 = 3σ2 − σ1 . Am =
(88)
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We collect real and the imaginary parts to yield the following slowly varying set of equations:
81 ωm p˙ 1 = −81 ωm q1 ν1 − 41 ωm (m1 + m2 ) p1 + Sm1 q1 p1 2 + q1 2 +
+Smn2 q1 p2 2 + q2 2 + S3 q2 p1 2 − q2 q1 2 − 2 p2 p1 q1 ,
81 ωm q˙1 = +81 ωm p1 ν1 − 41 ωm (m1 + m2 ) q1 − Sm1 p1 p1 2 + q1 2 −
−Smn2 p1 p2 2 + q2 2 − S3 p2 p1 2 − p2 q1 2 + 2q2 p1 q1 + 8 χ Fm ,
82 ωn p˙ 2 = −82 ωn q2 ν2 − 42 ωn (n1 + n2 ) p2 + Sn1 q2 p2 2 + q2 2 +
+Snm2 q2 p1 2 + q1 2 + S4 3 p1 2 q1 − q1 3 ,
82 ωn q˙2 = +82 ωn p2 ν2 − 42 ωn (n1 + n2 ) q2 − Sn1 p2 p2 2 + q2 2 −
−Snm2 p2 p1 2 + q1 2 + S4 3 p1 q1 2 − p1 3 .
(89)
The slowly varying evolution equations without acoustic radiation and the Kelvin-Voigt damping terms are identical to those of Ref. 49. We multiply Eq. (89) by ε3 and obtain the small amplitude response as a function of the excitation frequency in Appendix E. At steady state the time derivatives are zero and we obtain the following set of algebraic equations for the system frequency response:
ωm − 41 ωm (K m1 + K m2 ) εp1 + Sm1 εq1 ε2 p1 2 + q1 2 + −81 ωm εq1 1 −
+ Smn2 εq1 ε2 p2 2 + q2 2 − S3 ε3 q2 q1 2 − q2 p1 2 + 2 p2 p1 q1 = 0,
ωm 81 ωm εp1 1 − − 41 ωm (K m1 + K m2 ) εq1 − Sm1 εp1 ε2 p1 2 + q1 2 −
− Smn2 εp1 ε2 p2 2 + q2 2 − S3 ε3 p2 p1 2 − p2 q1 2 + 2q2 p1 q1 + 8Fm F = 0,
ωn − 42 ωn (K n1 + K n2 ) εp2 + Sn1 εq2 ε2 p2 2 + q2 2 + −82 ωn εq2 3 −
+ Snm2 εq2 ε2 p1 2 + q1 2 + S4 ε3 3 p1 2 q1 − q1 3 = 0,
(90)
ωn 82 ωn εp2 3 − − 42 ωn (K n1 + K n2 ) εq2 − Sn1 εp2 ε2 p2 2 + q2 2 −
− Snm2 εp2 ε2 p1 2 + q1 2 + S4 ε3 3 p1 q1 2 − p1 3 = 0, where
1 − υ2 p0 L 3 1 − υ 2 2 γ ω j p0 L 3 1 − υ 2 4 2 K j1 = κ˜ 1 ω j + κ˜ 2 L , K j2 = 2 C ,F =
¯ n . EIT EI kz j EI (91) We solve the algebraic Eq. (90) numerically where fixed points in the slowly varying evolution equations correspond to periodic orbits in the original IBVP. The stability of the fixed points solutions is explored using the characteristic polynomial of the Jacobian metrics (λ4 + c1 λ3 + c2 λ2
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FIG. 8. Frequency response of a beam mid span fifth mode (top), ninth mode (center), and the combined response (bottom), triangles-saddle nodes, solid squares-Hopf point, = 0.1.
+ c3 λ + c4 = 0). The solution is asymptotically stable for positive coefficients ci > 0 (i = 1–4) and a positive third Hurwiz determinant 3 = [c3 (c1 c2 − c3 ) − c1 2 c4 ] > 0. We note that coexisting solutions will be determined between saddle node bifurcation points (c4 = 0), and that periodic slowly varying envelopes will be determined between Hopf bifurcation points [3 = 0 and H = (c3 /c1 )0.5 > 0], where the latter indicate to quasiperiodic solutions of the original IBVP. 2. Frequency response and orbital stability
Fig. 8 depicts the response amplitudes of the fifth (top) mode |εa1 | = (p1 2 + q1 2 )0.5 , ninth (center) mode |εa2 | = (p2 2 + q2 2 )0.5 , and the combined (bottom) panel amplitude, |W| = ((εa1 )2 + (εa2 )2 )0.5 , respectively. Fig. 9 depicts the ninth mode amplitude response in a vertical axis logarithmic scale. The panel parameters are: L = 0.3048 m, E = 58.76 GPa, L/h = 300, ε2 δ 1 = 0, ε2 δ 2 = 0.3621, υ = 0.33, ρ p = 2700 Kg/m3 , γ = 1.4, c = 340 m/s and an excitation pressure of = 0.1. Solid and dashed lines correspond to stable and unstable solutions, respectively, and saddle nodes are marked by triangles. We estimate the numerical error by substitution of the solutions of p1 ,q1 ,p2 ,q2 back into Eq. (E1) and calculate the values of (p1 )t , (q1 )t , (p2 )t , and (q2 )t . The values of these time derivatives at steady state are zero, thus we define a numerical error as: err = max{|(p1 )t |, |(q1 )t |, |(p2 )t |, |(q2 )t |}, which was found to be smaller than 10−9 . We note six regions in the frequency response bifurcation diagram depicted in Fig. 8. A single solution is obtained in region I (/ω5 < 1.0532), region III (1.12495 < /ω5 < 1.16432), region IV
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FIG. 9. Frequency response of a beam mid span ninth mode in a vertical axis logarithmic scale, triangles – saddle nodes, solid squares – Hopf point, = 0.1.
(1.16432 < /ω5 < 1.17261), and region V (1.17261 < /ω5 < 1.3155). Three coexisting solutions are found in region II (1.0532 < /ω5 < 1.12495) and region VI (/ω5 > 1.3155). Region IV is also delineated by saddle node (SN5 ). However, we note that the location of this saddle node is beyond the detuning assumption. The pattern of this frequency response is similar to that of a 3:1 internal resonance between the first and second mode of a hinged-clamped beam,21 with the exception of the position of the Hopf points (found in Ref. 21 to overlap region II). We note that the ninth mode amplitude response is almost negligible in the entire region except for the narrow region II. We explore the behavior of Eq. (90) in region IV (1.16432 < /ω5 < 1.17261) by numerically solving Eq. (E1). Fig. 10 depicts the projections of the phase portraits q1 (q2 ) at /ω5 = 1.169 (left), the corresponding power spectra (center) and Poincare’ map (right). The Poincare’ map is generated by sampling the (q1 ,q2 ) points each time the orbit crosses the p1 surface for a constant value of positive qj . This simple periodic solution was the only one found in this interval. Thus, the narrow region between the Hopf points corresponds to quasiperiodic solutions for both panel response and scattered fluid pressure. Fig. 11 depicts the panel quasiperiodic response for /ω5 = 1.165 which is obtained from Eqs. (83), (84), (88), and (72) and the numerical result of Eq. (E1). The panel time series, power spectra (where the frequency is normalized by f5 = ω5 /2π ), phase plane, and Poincare’ map50 are presented top left, top right, bottom left, and bottom right, respectively. We note that the quasiperiodic response consists of amplitude modulation in the time series, a torus in the Poincare’ map, and side bands close to the fifth and ninth modes in power spectra. We explore the influence of the acoustic excitation pressure magnitude on the size of region IV numerically. Fig. 12 depicts the positions of the Hopf points (3 = 0), which bound the region, by solid lines. We note that the positions of the Hopf points shift to the right and away from each other with excitation pressure increasing. The behavior of Eq. (90) in the area between the lines was explored in detail to yield solutions similar to those in Fig. 10.
FIG. 10. Projections of the phase portraits between the Hopf points at /ω1 = 1.169 (left), the corresponding power spectra (center), and Poincare’ map (right).
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FIG. 11. Quasiperiodic panel response ( = 0.1, /ω5 = 1.165): time series (top left), power spectra (top right), phase plane (bottom left), and Poincare’ map (bottom right).
IV. NUMERICAL VALIDATION A. Primary resonance
We validate the case of primary resonance by comparison of the asymptotic solution with numerical integration of a nonlinear 2D clamped panel in an inviscid and compressible Euler fluid field.20 We note that the fluid-structure interaction in an Euler flow field includes the case of a
FIG. 12. Hopf points positions vs. the acoustic excitation pressure magnitude.
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potential flow field. The flow field equations are solved using a second order accurate Beam and Warming finite differencing scheme. We note that the conservative Euler equations do not provide any natural dissipation mechanism and that the selected numerical scheme is weakly unstable. Therefore, following Ref. 51, we incorporate a necessary artificial dissipation model to eliminate spurious behavior. The dissipation model includes both second and fourth order terms that are required to damp both low and high frequency growth, and to control nonlinear instabilities. The boundary conditions of the rigid wall and flexible panel are non-penetration conditions where the normal velocity is zero and the normal gradient sum of the vertical momentum and the pressure is zero. In order to ensure that the acoustic scattered wave will leave the computational domain, non-reflecting boundary conditions are employed on the free edges. The panel equation is solved using an explicit finite difference method, where the nonlinear stiffness terms are calculated using the composite Simpson’s rule of integration. The fluid-structure coupling is obtained by a first order loose coupling approach, with a small time step, where the fluid and structure variables are updated alternatively by the separate fluid (CFD) and structural (CSD) solvers that exchange boundary information with grid adaptation. The numerical results are rescaled using the panel length (L) and acoustic time scale (L/a∞ ) resulting in nondimensional coordinates, displacement and time: x˜ = x / L, z˜ = z / L, W˜ = W / L, t˜ = a∞ t / L, and the corresponding velocities, pressure and ˜ v˜ = u, v / a∞ , P˜ = P / ρ∞ a∞ 2 , and ρ˜ = ρ / ρ ∞ . For convenience we present the density are: u, results in nondimensional form without the “∼” above the variable or parameter. We note that the numerical solver is explained in detail in Ref. 20. We validate the asymptotic results close to the first mode with the following beam parameters: L/h = 100, ε2 δ 1 = 0, ε2 δ 2 = 0, υ = 0.33, E = 69.357.103 MPa, ρ p = 2700 Kg/m3 , γ = 1.4, c = 340 m/s, the excitation amplitude is = 0.012. Fig. 13 depicts the comparison between the asymptotic and the numerical results which are described by circles. We note the good agreement close to primary resonance; however, as the magnitude of the mid-span amplitude increases (or as the excitation frequency is far from linear resonance), the difference between the analytical and the numerical solutions increases (within 5.8%). Fig. 14 depicts the phase plane of the coexisting solutions described in Fig. 13 for /ω1 = 1.2, where stable periodic solutions of large and small amplitude are described by solid lines, and the unstable solution is described by a dashed line which separates the domains of attraction of the stable solutions.
FIG. 13. Frequency response of the beam first mode immerses in air ( = 0.012) by solid line and numerical results by circles.
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FIG. 14. Phase plane of coexisting solutions, stable solutions by solid lines and unstable solution by dashed line (/ω1 = 1.2, = 0.012).
Fig. 15 depicts a comparison between the asymptotic and numerical maximum pressure values (/ω1 = 1.145, = 0.02), where the analytical values and numerical results are described by solid line and circles, respectively. We note the good agreement (∼2.8%) in the far field which corresponds to agreement between the beam numerical and asymptotic amplitude solutions (∼3%). This agreement deteriorates close to the panel (∼9.8% at z = 0.1).
FIG. 15. Comparison of the maximum pressure values, analytical values by solid line and numerical results by circles (/ω1 = 1.145, = 0.02).
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FIG. 16. Frequency response of the beam combined response, |W|, immerses in air ( = 0.1). Stable solutions by solid lines, unstable solutions by dashed lines, triangles-saddle nodes, solid squares-Hopf point. Periodic and quasiperiodic solution numerical results circles and star respectively.
B. Internal resonance
We validate the asymptotic solutions corresponding to the 3:1 internal resonance between the fifth and ninth symmetric modes as described in Sec. III B 2. Fig. 16 depicts the frequency response of the maximal panel mid-span response, |W|, for a pressure excitation of = 0.1. The solid and dashed lines depict stable and unstable asymptotic solutions, respectively. The circles and single star depict periodic and quasiperiodic numerical results, respectively. The numerical solutions include the contributions of both lower and higher modes. In order to compare between the asymptotic and the numerical solutions, we include the contribution of the additional modes (1–4, and 6–8), by computation of the multiple modal shape function values at x = 0.25. We note that the validation was done in the frequency range where the non-resonant modal contributions are smaller than the resonant mode contribution. The contribution of the ninth mode is small in most of the frequency range of Fig. 16 (except for the middle branch in region II), and we thus make use the following relation: ⎛ ⎜ |W ( f )| = ⎜ ⎝W ( f )numrical −
8 j=1 j=5
⎞ ( ⎟ φ j (x = 0.25) a j ( f )⎟ φ5 (x = 0.25) . ⎠
(92)
The amplitudes, aj , are calculated via Eq. (63). We note the good agreement close to ω5 between the analytic solution and the numerical solution (within 3.4%). The numerical solution amplitudes are found to be slightly larger than the asymptotic amplitudes. This behavior corresponds to the reduction in the fluid loading as the frequency increases (both in Figs. 16 and 13), whereas the asymptotic analysis incorporates a detuning from the 5th mode resonance. Numerically obtained quasiperiodic solutions are found in the frequency range 1.1324 < / ω5 < 1.1368. Fig. 17 depicts the quasiperiodic time series (top left), power spectra where the frequency axis is normalized by f5 = ω5 /2π (top right), phase plane (bottom left), and Poincare’ map (bottom right) for x = 4.75, for /ω5 = 1.136. A zoom of the time series in Fig. 17 is shown in Fig. 18. We note the amplitude modulation consists of a torus in the Poincare’ map (defined by 236 points
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FIG. 17. Quasiperiodic mid-span panel response ( = 0.1, /ω5 = 1.136, x = 4.75): time series (top left), power spectra (top right), phase plane (bottom left), and Poincare’ map (bottom right).
sampled each forcing frequency). The power spectra frequency content includes the fifth mode resonance frequency, f5 , and the ninth mode resonance frequency, f9 . We note the good agreement between the maximum numerical mid-span response, |W|, and the predicted asymptotic solution in Fig. 16 where star depicts the quasiperiodic numerical solution. This quasiperiodic solution was obtained for a frequency which is about 3.2% lower than the frequency predicted analytically in Sec. III B 2. Furthermore, we note that the numerical solution described in Fig. 17 has the same pattern as the analytical solution in Fig. 11. The discrepancy in the location of the quasiperiodic solution (i.e., region IV) may be due to the differences between the asymptotic and numerical solutions as the numerical solver includes accurate contributions from all panel modes and incorporates additional artificial dissipation that is not modeled. We note that the ninth mode amplitude, for the quasiperiodic solutions is significantly smaller than the fifth mode amplitude. Fig. 19 depicts a numerical panel span spatial form within a single quasiperiodic modulation for t = 1265.18, 1265.34, 1265.499, 1265.659, 1265.818 (top to bottom), respectively. The temporally sampled spatial span exhibits a clear fifth mode periodic structure. Fig. 20 depicts a comparison between the asymptotic and the numerical maximum pressure values, where the analytical and numerical results are described by a solid line and circles, respectively.
FIG. 18. Quasiperiodic mid-span panel response ( = 0.1, /ω5 = 1.136, x = 4.75): zoom-in of time series.
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FIG. 19. Span Structure of the quasiperiodic response ( = 0.1, /ω5 = 1.136) when t = 1265.18, 1265.34, 1265.499, 1265.659, 1265.818 top to bottom, respectively.
We note a good agreement (∼2.5%) between the asymptotic and numerical maximal pressures in the far field which correspond to the agreement between the asymptotic and the numerical for the panel amplitude (∼2.8%). This agreement deteriorates with decreasing distance to the beam (∼6.6% at z = 0.1).
FIG. 20. Comparison of the maximum pressure values, analytical values by solid line and numerical results by circles (/ω5 = 1.136, = 0.1).
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V. THE PRESSURE FIELD A. Surface pressure
The fluid load due to acoustic radiation (surface pressure) at first order is obtained by Eq. (34), for z = 0:
p1 = −Im
iγ ωn 2 2 C ζ (x) A (T1 , T2 ) eiωn T0 + A¯ (T1 , T2 ) e−iωn T0 . kz
(93)
As described in Sec. III A, we make use of this surface pressure due to acoustic radiation to obtain the frequency response curve of Eq. (63). We note that coexistence of bi-stable solutions is governed by a positive root in Eq. (63), and for k˜21 = 0 we obtain the relation k˜3 / k˜22 > εa, which can be written as follows: γ L 2 0 2
˜ 3 1 ) > zn . εa c 3 L 2 0 2 − 1
(94)
We note that Eq. (94) includes the gas coefficients (left parentheses), the boundary condition type (middle parentheses), and the panel parameters (right parentheses). ˜ which is required Fig. 21 depicts the beam displacement vs. the minimal exciting pressure ( ) for coexisting solutions. The dashed-dotted (black), dotted (blue), and solid (red) lines depict values in air (γ = 1.4, c = 340 m/s), in CO2 (γ = 1.304, c = 258 m/s)30 and in H2 (γ = 1.41, c = 1269.5 m/s),47 respectively. We note that as the ratio of the gas parameters γ /c decreases, the minimum exciting pressure which is required for coexisting solutions is decreased.
B. Scattered pressure field
1. Periodic response
The numerically obtained instantaneous scattered pressure field near primary resonance is depicted in Fig. 22. The panel parameters are: L/h = 100, ε2 δ 1 = 0, ε2 δ 2 = 0, υ = 0.33, E = 69.357.103 MPa, ρ p = 2700 Kg/m3 , γ = 1.4, c = 340 m/s, /ω1 = 1.145, and = 0.02. We note the general form of a simple wave. Fig. 23 depicts the mid-span displacement time series, power
FIG. 21. The displacement vs. minimum exciting pressure required for coexisting solutions, dashed-dotted (black) in air, dotted (blue) in CO2 , and solid (red) in H2 .
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FIG. 22. Numerical result of the instantaneous pressure field (P(x,z) = P – P∞ ) for /ω1 = 1.145 and = 0.02.
spectra 9 where the frequency axis is normalized by f1 = ω1 /2π and a combined phase plane and Poincare’ map (where a single the Poincare’ point is marked by a circle). The corresponding pressure field response is described in Fig. 24 which depicts the time series, power spectra and combined phase plane and Poincare’ map of the mid-plane surface pressure (x = 5, z = 0) and the pressures at the locations of (x = 5, z = 4) and (x = 5, z = 8), top to bottom (x = 0.5 and x = 5 are the mid-plane x-coordinate in the analytical and the numerical analysis respectively). We note that the pressure amplitude decreases with distance from panel and high harmonics in the surface pressure are eliminated with increasing distance from the panel. These higher harmonics are not modeled by the asymptotic solution. However, the contents of the higher harmonics are much smaller than the contents of the governing primary resonance frequency f1 . A comparison of numerical (left) and asymptotic (right) results for the instantaneous pressure field corresponding to periodic response of the internal resonance between the fifth and ninth modes are shown in Fig. 25 (with beam parameters as in Sec. III B 2). We note that both analytical and numerical pressure fields yield a similar spatial pattern of a simple wave scattered from single frequency excitation (/ω5 = 1.1, = 0.1). The temporal response of the corresponding pressure field is shown in Fig. 26 which depicts the time series, power spectra 9 where the frequency axis is normalized by f5 = ω5 /2π , and combined phase plane and Poincare’ map, of the mid-plane surface pressure (z = 0) and the pressures at the near (z = 0.1) and far (z = 8) field locations, respectively. We note that the pressure amplitude decreases with distance from the panel and that the higher harmonics in the surface pressure decrease with increasing distance from the panel. However, the
FIG. 23. Time series, power spectra and combined phase plane and Poincare’ map of the numerical results of the mid-plane displacement for /ω1 = 1.145 and = 0.02.
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FIG. 24. Time series, power spectra, and combined phase plane and Poincare’ map of the numerical results of the mid plane surface pressure (x = 5,z = 0), the pressures at the locations of (x = 5, z = 4) and (x = 5, z = 8), top to bottom respectively for /ω1 = 1.145 and = 0.02.
higher harmonic contents are much smaller than the content of the ω5 and ω9 obtained from the asymptotic solution. 2. Quasiperiodic response
The corresponding pressure field response of the quasiperiodic response is portrayed in Fig. 27, which depicts the time series, power spectra and combined phase plane and Poincare’ map of the
FIG. 25. Numerical (left) and analytical (right) result of the instantaneous pressure field (P(x,z) = P – P∞ ) for /ω5 = 1.1 and = 0.1.
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FIG. 26. Time series, power spectra, and combined phase plane and Poincare’ map of the numerical results of the mid plane surface pressure (x = 4.75, z = 0), the pressures at the locations of (x = 4.75, z = 0.1) and (x = 4.75, z = 8), top to bottom respectively for /ω5 = 1.1 and = 0.1.
mid-plane surface pressure (z = 0) and the near field (z = 0.1) and far field (z = 8), respectively. We note the beating in the pressure amplitude close to the panel surface consists of a small torus similar to the Poincare’ map in Fig. 17. The modulation in pressure amplitude and the higher harmonic content decrease with increasing distance from the panel. The far field response behavior, (Fig. 27 bottom), is similar to the behavior of the far field pressure of Eq. (77), where only the panel response frequencies, shown by Fig. 17 (top right), are scattered and that the pressure amplitude and modulation amplitude decrease with increasing distance from the panel.
VI. CLOSING REMARKS
An asymptotic multiple-scales analysis of an acoustically excited clamped two-dimensional panel immersed in an inviscid compressible fluid was employed to investigate both primary and a 3:1 internal resonance between the panel fifth and ninth modes. Validation of the asymptotic structural response and the fluid pressure was performed by comparison with a numerical solution of the geometrically nonlinear panel in a quadratic Euler field. The acoustic loading on the panel was found to be governed by three different mechanisms which include the fluid parameters, the type of boundary condition and the panel parameters. Furthermore, the acoustic damping was found to reduce with increasing panel spatial modes. This behavior reveals that the dominant damping mechanism may change with different modes, and demonstrates the significance of non-constant acoustic drag loading. The latter is especially important for conditions of energy transfer between spatial modes such as in internal or combination resonance conditions. The results shed light on the intricate acoustic interaction bifurcation structure which exhibits coexisting bi-stable periodic solutions, and quasiperiodic response reflecting spatially periodic modal
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FIG. 27. Time series, power spectra, and combined phase plane and Poincare’ map of the numerical results of the mid plane surface pressure (x = 4.75, z = 0), the pressures at the locations of (x = 4.75, z = 0.1) and (x = 4.75, z = 8), top to bottom respectively for /ω5 = 1.136 and = 0.1 (quasiperiodic).
energy transfer for both panel and fluid. This behavior is found to occur for panel excitation by finite level acoustic pressure waves that can be a crucial factor for design of high integrity structural systems required for aviation or space where light structures are exposed to intensive acoustic pressure fluctuations.
ACKNOWLEDGMENTS
This research was supported in part by the Israel Science Foundation (1475/09) founded by the Israel Academy of Science, and by the 2011 Umbrella Cooperation Program on Modeling and Simulation between RWTH Aachen University, Forschungszentrum Julich, and the Technion. O.G. sincerely thanks C. H. M. Jenkins for many illuminating discussions on spatiotemporal instabilities in acoustically excited panels and membranes.
APPENDIX A: DERIVATION OF THE NONLINEAR VISCOELASTIC PANEL EQUATION OF MOTION
Following Leamy and Gottlieb52 we consider a planar weakly nonlinear pre-tensioned (N0 ) viscoelastic string (of length L) augmented by linear Euler-Bernoulli bending53 which incorporates a Kelvin-Voigt constitutive relationship where the stress (σ ) is a linear function of strain (˜ε) and strain rate (ε˙˜ ):44, 53 σ = E˜ε + F ε˙˜ ,
(A1)
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where E is Youngs modulus and F is a viscoelastic damping parameter. Thus, the equations of motion for the beam-string are54 1 2 ρAutt − N0 ux + EA ux + wx + FA (ut x + wx wtx ) = 0, 2 x (A2) 1 2 ρAwtt − N0 wx + EAwx ux + wx + FAwx (utx + wx wt x ) − (EIwx x x + FIt x x x ) = Qw , 2 x
where ρ, A, and I are the material density, cross-sectional area, and moment of inertia of the beamstring, respectively. u(x,t) and w(x,t) are the longitudinal and transverse components of the planar elastic field respectively. The generalized force component Qw is due to external acoustic excitation and linear equivalent damping (e.g., Eq. (6)). We assume that N0 EA and note that the longitudinal natural frequency is much higher than the transverse natural frequency. Thus, the longitudinal velocity and inertia can be neglected13 to yield a simple spatial relationship between the transverse and longitudinal derivatives. Incorporating fixed longitudinal boundary conditions (u(0, t) = u(L, t) = 0) enables integration of the resulting relationship to yield: w2 ux = c1 (t) − x , 2L
1 c1 = 2L
L w2x dx, 0
1 c1t = L
L wx wtx dx.
(A3)
0
Thus, the resulting initial-boundary-value problem for the pre-tensioned beam-string consists of an integro-differential equation for the transverse mode: ⎡ EA ρAwtt − wx x ⎣N0 + 2L
L
FA w2x dx + L
0
L
⎤ wx wt x dx⎦ + EIwx x x x + FIwt x x x x = Qw .
(A4)
0
APPENDIX B: COEFFICIENTS OF THE SECOND ORDER 2D POTENTIAL EQUATION
The coefficients of Eq. (81) are 2 ω ωn 3 n ϑ11 = − (γ − 1) i ζn ζx xn e2ikzn z − ωn 3 ζn 2 e2ikzn z − 2i ζxn + ωn 3 ζn 2 , k zn k zn 2 2 3 ωm ωm 2ik zm z 3 2 2ik zm z 3 2 − 2i ϑ12 = − (γ − 1) i ζm ζx xm e − ωm ζm e ζxm + ωm ζm , k zm k zm 2 2 k zm k zn ωn ωm ωm 2 ωn 2 2 ζn ζx xm + ζm ζx xn − ωn ωm ζn ζm − ωm ωm ζm ζn − ϑ2 = − (γ − 1) i k zn k zm k zm k zn k zn k zm 2 ωn ωm ωm 2 ωn −2i (B1) ζxn ζxm + ζxn ζxm + ωn 2 ωm ζn ζm + ωm 2 ωn ζn ζm , k zn k zm k zn k zm ωn 2 ωm k zm ωn 2 ωm ϑ3 = − (γ − 1) i − ζn ζx xm + ωn 2 ωm ζn ζm − 2i − ζxn ζxm + ωn 2 ωm ζn ζm , k zn k zm k zn k zn k zm 2 2 ωm ωn k zn ωm ωn ϑ4 = − (γ − 1) i ζm ζx xn − ωm 2 ωm ζm ζn − 2i ζxn ζxm − ωm 2 ωn ζn ζm . k zm k zn k zm k zn k zm
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The coefficients of Eq. (82) are ωn ζxn 2 , k zn ωm o12 = − ζxm 2 , k zm ωn ωm , o2 = − ζxm ζxn + ζxn ζxm k zn k zm ωn o3 = −ζxm ζxn , k zn ωm o4 = −ζxn ζxm . k zm
o11 = −
(B2)
APPENDIX C: THIRD ORDER PANEL EQUATION
The beam third order equation: D0 2 W3 + W3x x x x = χ eiT0 − [2iωn ζn D2 An + δ1 iωn ζx x x xn An + δ2 iωn ζn An ] eiωn T0 − − [2iωm ζm D2 Am + δ1 iωm ζx x x xm Am + δ2 iωm ζm Am ] eiωm T0 + iγ ωn 2 iγ ωm 2 L0 2 ζn 2χ An eiωn T0 + ζm Am eiωm T0 + c k zn k zm ⎧⎡ ⎤ 1 1 ⎨ 2 ψ ⎣ An 3 ζx xn ζxn d xe3iωn T0 + Am 3 ζx xm ζxm 2 d xe3iωm T0 ⎦ + ⎩ 0
⎡
⎣2Am A¯ m An ζx xn
0
1 ζxm 2 d x + 3An 2 A¯ n ζx xn 0
⎡ ⎣2An A¯ n Am ζx xm ⎡ ⎣ An 2 Am ζx xm ⎡ ⎣ An 2 A¯ m ζx xm ⎡ ⎣ A¯m 2 An ζx xn ⎡ ⎣ Am 2 An ζx xn
1 ζxn 2 d x + 4Am A¯ m An ζx xm 0
1 ζxn 2 d x + 3 A¯ m Am 2 ζx xm
0
ζxn 2 d x + 2Am An 2 ζx xn 0
0
1
1 ζxn 2 d x + 2 A¯ m An 2 ζx xn
0
2
ζxm 2 + 2 A¯m An ζx xm 0
ζxm ζxn d x ⎦ eiωm T0 +
0
⎤
⎤ ζxm ζxn d x ⎦ ei(2ωn −ωm )T0 ⎤
ζxm ζxn d x ⎦ ei(ωn −2ωm )T0 +
0
1
1 ζxm 2 d x + 2Am 2 An ζx xm
0
ζxm 2 d x + 4Am An A¯ n ζx xn
⎤
ζxm ζxn d x ⎦ ei(2ωn +ωm )T0 +
0
1
ζxm ζxn d x ⎦ eiωn T0 + 1
0
1
⎤
0
1
1
1
1
0
⎤ ζxm ζxn d x ⎦ ei(2ωm +ωn )T0
⎫ ⎬ ⎭
+ C.C.
(C1)
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APPENDIX D: COEFFICIENTS OF THE COMPLEX EVOLUTION EQUATIONS
The coefficients of Eqs. (86) and (87) are
γ ωj , j1 = δ1 ω j 2 + δ2 , j2 = 2χC 2 kz j 1 j =
ζ j 2 d x, 0
1 S j1 = 3ψ
1 ζ j ζ jxx dx
0
1 Si j2 = 2ψ 0
1 S3 = ψ
0
⎛ ζ j ⎝ζi x x ⎛
ζm ⎝ζnx x
0
1
1 ζ j x 2 d x + 2ζ j x x
0 1
⎞ ζi x ζ j x d x ⎠d x,
0
1
ζmx 2 + 2ζmx x 0
1 S4 = ψ
ζ j x 2 d x,
(D1)
⎞
ζmx ζnx d x ⎠d x,
0
1 ζn ζmx x
0
ζmx 2 d xd x, 0
1 Fj =
ζ j d x. 0
APPENDIX E: EVOLUTION EQUATIONS INDEPENDENT OF ε
The amplitude response as a function of the excitation frequency, which is independent in ε and obtained from Eq. (89):
ωm − 41 ωm (K m1 + K m2 ) εp1 + Sm1 εq1 ε2 p1 2 + q1 2 + 81 ωm ε ( p1 )t = −81 ωm εq1 1 −
+ Smn2 εq1 ε2 p2 2 + q2 2 + S3 ε3 q2 p1 2 − q2 q1 2 − 2 p2 p1 q1 ,
ωm − 41 ωm (K m1 + K m2 ) εq1 − Sm1 εp1 ε2 p1 2 + q1 2 − 81 ωm ε (q1 )t = 81 ωm εp1 1 −
2
2 2 3 − Smn2 εp1 ε p2 + q2 − S3 ε p2 p1 2 − p2 q1 2 + 2q2 p1 q1 + 8Fm F, (E1)
ωn − 42 ωn (K n1 + K n2 ) εp2 + Sn1 εq2 ε2 p2 2 + q2 2 + 82 ωn ε ( p2 )t = −82 ωn εq2 3 −
2
2 2 3 + Snm2 εq2 ε p1 + q1 + S4 ε 3 p1 2 q1 − q1 3 ,
ωn − 42 ωn (K n1 + K n2 ) εq2 − Sn1 εp2 ε2 p2 2 + q2 2 − 82 ωn ε (q2 )t = 82 ωn εp2 3 −
2
2 2 − Snm2 εp2 ε p1 + q1 + S4 ε3 3 p1 q1 2 − p1 3 .
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Z. Aginsky and O. Gottlieb
Phys. Fluids 25, 076104 (2013)
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