Analytic Formula for the Vibration and Sound Radiation of a Nonlinear Duct

)is paper addresses the vibration and sound radiation of a nonlinear duct. Many related works assume that the boundaries are linearly vibrating (i.e., their vibration amplitudes are small), or that the duct panels are rigid, and their vibrations can thus be neglected. A classic method combined with Vieta’s substitution technique is adopted to develop an analytic formula for computing the nonlinear structural and acoustic responses. )e development of the analytic formula is based on the classical nonlinear thin plate theory and the three-dimensional wave equation. )e main advantage of the analytic formula is that no nonlinear equation solver is required during the solution procedure. )e results obtained from the proposed classic method show reasonable agreement with those from the total harmonic balance method. )e effects of excitation magnitude, panel length, damping, and number of flexible panels on the sound and vibration responses are investigated.


Introduction
Over the past decades, many studies have considered structural-acoustic problems, plate vibration, and solution methods for nonlinear governing equations (e.g., [1][2][3][4][5][6][7][8][9][10][11][12]).Among various structural-acoustic problems, duct noise has been a particular focus for many years.For example, Venkatesham et al. [13] studied the breakout noise from a rectangular duct with compliant walls; they found that the low-frequency breakout noise was important and should not be neglected, and the coupling between the acoustic waves and the structural waves must be considered in the prediction of the transverse transmission loss.Tang and Lin [14] proposed the use of stiff light composite panels for duct noise reduction.eir selling point was that the actuation strategy would enable the creation of composite panels for duct noise control without the need for a traditional heavy structural mass.According to their results, the mass-spring resonance absorption in the case of a stiff thick panel with a thin flexible plate would be more efficient.Tiseo and Concilio [15] conducted a set of simulations for duct noise control.eir proposed feedback structural and acoustical structural control strategies maximized the damping within an acoustic duct and thus improved noise reduction.Jade and Venkatesham [16] implemented intensity-based experimental techniques for measuring breakout noise and validated "equivalent unfolded plate" analytical models.
ey developed an experimental test setup to measure the input and radiated sound power and vibration level and then calculated the transverse transmission loss and radiation efficiency.ey adopted two measurement techniques: the intensity probe method (P-P method) and the microflown technique (P-U method).A numerical model was developed to predict transverse transmission loss and radiation efficiency, and the results were verified by the experimental results.In the aforementioned research works, nonlinear structural vibration, which should be considered for thin panel structures, was not the focus.e number of published articles on nonlinear structural acoustics is quite limited although a few are worth mentioning (e.g., [17][18][19][20]).
In contrast, there are numerous studies on solution methods for nonlinear governing equations (e.g., [21][22][23][24][25][26][27][28]).For example, Jacques et al. [29] adopted the harmonic balance and finite element methods for solving the problem of the nonlinear vibration of a viscoelastic sandwich beam.Chen et al. [30] analyzed the nonlinear steady state vibrations of plane structures.Both the finite element method and incremental harmonic balance method were used.e nonlinear governing equations were solved by the incremental harmonic balance method.Lee et al. [31] adopted the homotopy perturbation method to solve parabolic partial differential equations with constant coefficients for the nonlinear plate problem.Gao et al. [32] adopted the multiple scales method to analyze the nonlinear primary resonance of functionally graded porous cylindrical shells.Hao et al. [33] developed the nonlinear governing equations of motion for the FGM plates using Hamilton's principle and adopted the asymptotic perturbation method to obtain four-dimensional nonlinear averaged equations.When the aforementioned methods were applied to the nonlinear governing equations, coupled nonlinear algebraic equations were generated that needed to be solved by a nonlinear equation solver.
is motivated the current study to adopt the classical method and Vieta's substitution technique [34] and to develop analytic formulations that would not require an equation solver to compute the nonlinear sound and vibration responses of a rectangular duct.Note that, normally a nonlinear problem would generate a set of nonlinear equations.A nonlinear equation solver is needed to solve them to obtain the nonlinear solutions (e.g.[35]).
e effects of excitation magnitude, panel length, damping, and number of flexible panels on sound and vibration responses were also investigated for various panel cases.
Few studies have examined nonlinear structural acoustics, and studies of nonlinear duct sound and vibration are very limited.e modelling technique presented here is suitable for handling more than one flexible panel.Other works (e.g., [36,37]) only considered one flexible panel.e main advantage of the solution method is that the procedure does not require a nonlinear equation solver; rather, the final nonlinear solutions can be expressed in terms of a set of symbolic parameters with various physical meanings.e method is very suitable for fast engineering calculation purposes.If a numerical method is used, the final solutions are numerical and require more computation time.e limitation of this solution method is that it can only be used for a simply supported plate.For other boundary conditions, the nonlinear modal differential equations are more complicated.us, no analytical solutions can be obtained from the proposed method.

Theory and Formulation
Figure 1 shows an open-end duct made of flexible panels.A piston-like excitation is generated at the other end.e acoustic pressure within the rectangular cavity can be obtained from the well-known homogeneous wave equation [2,11,37]: where p Q (x, y, z) is the acoustic pressure at the position of (x, y, z) induced by the Q th panel mode and k is the wave number.e solution form of the acoustic pressure and the acoustic cavity mode can be expressed in the following form: (2) where P QJ is the acoustic pressure amplitude of the J th acoustic mode; J is the general acoustic mode number; J is the number of acoustic mode used; U, V, and W are the acoustic mode numbers for the x, y, and z directions; and a, b, and c are the dimensions of the cavity.Note that b and c are much smaller than a.It is implied that the acoustic resonant frequencies of nonzero V and W modes are much higher than the first several acoustic resonant frequencies of the duct cavity.
Putting equations ( 2) and ( 3) into (1) and using the technique of integration by parts in [37] yield the following equation: where is the wave number of the J th acoustic mode and α JJ �  vol φ J φ J dx dy dy. e subscripts "vol" and "pan" represent the cavity volume and panel surface, respectively.zp Q /zn is the pressure gradient on the panel surface.
At the panel surface, where ρ a is the air density; ω is the excitation frequency; A o and A Q are the displacement amplitudes of the source and panel, respectively; ϕ Q (x, y) is the Q th panel mode shape, which is a double sine function, sin((mπx)/a)sin((nπy)/b) (the panel is simply supported with immovable edges, and hence, the model shape can be written as a double sine function); and m and n are the structural mode numbers.Putting equation ( 5) into (4) yields the following equation: where α QJ �  pan ϕ Q φ J dx dy and α oJ �  sou φ J dx dy and the subscript "sou" represents the vibration source.
In equation ( 6), the first term on the right side is the modal acoustic stiffness term, i.e., modal acoustic stiffness term, If there are 2 or more flexible panels considered, the overall acoustic stiffness is given by overall acoustic stiffness term, 2

Shock and Vibration
where N is the number of panels and the subscript "aco" means acoustics.It is noted that the term on the right side is the acoustic sti ness force term.Note that in equation ( 8), α QJ /α QQ is the modal contribution factor and α QQ According to [38,39], the Airy stress function derivatives can be expressed as follows: e solution forms of w, u, and v are given in [40]: Hence, the governing equation of Airy stress function is given by 1 12 where is the phase velocity of compressional waves; E is Young's modulus; ρ p is the panel density; and t is time.
In this study, the governing equation of nonlinear forced vibration is dimensional and shown as follows.e following dimensional variables are introduced: en, put equations (12b) and (12c) into (12a) to eliminate the two nondimensional variables (i.e., Λ Q and τ).Hence, the governing equation of nonlinear free vibration is given by the following equation: Consider the acoustic sti ness force term in equation ( 8) and a force term which represents uniform harmonic excitation.Equation (13a) can be rewritten as follows:

Shock and Vibration
where κ � dimensionless excitation parameter, , and note that again α QJ /α QQ is the modal contribution factor, which is the same as the one in equation (8).
In equation (13b), the nonlinearity of the structuralacoustic system is represented by the term β Q Λ 3 Q .Physically, the nonlinearity is caused by the large amplitude vibration.In a typical linear vibration (or small vibration) analysis, the axial and transverse vibrations of a panel are independent, whereas they are coupled in a nonlinear vibration analysis.ey are coupled in the nonlinear analysis because in small deflection cases, it is assumed that the direction of the panel's transverse vibration is still perpendicular to the surface; in large deflection cases, the transverse vibration direction is no longer perpendicular to the surface because the surface is subject to large deformation.Mathematically, the nonlinear term in equations (13a) and (13b) results in multiple solutions, whereas there is only one solution in linear cases.In fact, equations (13a) and (13b) are in the form of the well-known Duffing equation.
Consider the approximation of T(t) ≈ sin(ωt), neglect the higher harmonic terms, and then equation (13b) is rewritten in the following form: 3 4 where Note that there is a limitation in the proposed solution form.In highly nonlinear cases, superharmonic and subharmonic responses occur.As the solution form does not include the components of sin(3ωt) or sin(1/3ωt), the results cannot show these phenomena.
For damped vibration, A Q in equation ( 14) is rewritten into the following complex form: en, substituting equation ( 16) in ( 14), we get the following equation: ; and ζ is the damping ratio.Physically, the damping force is assumed to be proportional to the damping ratio × the velocity (i.e., dΛ Q /dt).e damping ratio represents the level of damping relative to the critical damping (i.e., ζ � 1).
According to Vieta's substitution [34], equation ( 17) can be simplified using the substitution of c � λ − (G 2 /3G 3 ): where 18) can be further simplified using the substitutions of λ � B − (ε 1 /3B) and χ � B 3 : e exact solutions of equation ( 19) are easily obtainable without any nonlinear equation solver.Finally, the solutions of λ are given below: where en, for the calculation of sound radiation, the following equations can be used to find the radiation efficiency [41]: where symmetric modes are the modes of m and n set to odd numbers and antisymmetric modes are the modes of m and/ or n set to even numbers.

Numerical Results and Discussion
e material and physical properties adopted in the following numerical cases are as follows: Young's modulus � 7.1 × 10 10 N/m 2 ; Poisson's ratio � 0. e seven mode solutions are normalized as 100.In the (1,1) and (2,1) mode cases, the four acoustic mode solutions achieve an error rate of less than 1% for different excitation levels (κ � 0.05 to 0.5), while in the (3,1) mode case, the six mode solutions achieve an error rate of around 3.5%.Tables 4-6 show the structural mode contributions for various excitation levels: the larger the excitation level is, the higher the contributions of the two higher modes are.In other words, when the excitation is high, more modes are needed.e comparisons between Tables 4-6 show the modal contributions at the corresponding peak frequencies.e contributions of the dominant modes at the corresponding peak frequencies are always higher than 94%.
us, the use of two structural modes is appropriate for obtaining solutions with good accuracy.Figures 2(a) and 2(b) present the comparisons between the first symmetric and first antisymmetric modal amplitude solutions obtained from the proposed method and the classical harmonic balance method (e.g., [20]).e nondimensional excitation parameter value, κ, is 0.5.e damping ratio, ζ, is 0.02.e panel length a is 3 m, and the number of flexible panels is four.Two harmonic components, sin(ωt) and sin(3ωt), are adopted in the classical harmonic balance method.e results obtained from the two methods are generally in good agreement for the first symmetric and first antisymmetric modal amplitude solutions.
e only slight differences are found at the first nonlinear peak values, around the frequencies ω/ω 1 � 0.9 in Figure 2(a) and 1.15 in Figure 2(b).
e differences are caused by the different definitions of the damping terms in the two methods.
ere is a limitation in the proposed solution form.In highly nonlinear cases, superharmonic and subharmonic responses occur.As the solution form does not include the sin(3ωt) or sin(1/3ωt) components, the results cannot show these phenomena.
Figures 3(a)-3(d) show the symmetric and antisymmetric vibration amplitudes and sound powers plotted against the excitation frequency for various excitation magnitudes.
e damping ratio ζ is 0.02; the first two structural modes and seven acoustic modes are adopted; four flexible panels are considered (N � 4); and the panel length a � 3 m.e peak frequencies and peak values of the structural resonances (i.e., the first two peaks in the figures) increase with the excitation magnitude.e peak values of the other resonances (i.e., acoustic resonances) also increase with the excitation magnitude.e peak frequencies remain almost unchanged, and the peaks are generally more linear, except those under the high excitation level κ � 0.5, which are more nonlinear, and their peak frequencies are around ω/ω 1 � 2.3.e comparisons between the vibration amplitude and sound power curves show that the high frequency acoustic peaks are more important in the sound power figures while the low frequency structural peaks are more important in the vibration amplitude figures because the radiation efficiencies are monotonically increasing with the excitation frequency (Figure 3(e)).e higher the excitation frequency, the higher the radiation efficiency.
e comparisons between the curves of symmetric and antisymmetric cases show that the second or higher acoustic peaks  6 Shock and Vibration are much less signi cant in the antisymmetric case, while the rst acoustic peaks are important in both the symmetric and antisymmetric cases.
Figures 4(a)-4(d) show the symmetric and antisymmetric vibration amplitudes and sound powers plotted against the excitation frequencies for various damping ratios.e excitation level κ 0.2; the rst two structural modes and seven acoustic modes are adopted; four exible panels are considered (N 4); and the panel length a 3 m.Obviously, the damping only a ects the peak and trough values.In the o -resonance range, the damping has no signi cant e ect.If the damping ratio is small (ζ 0.005), the two structural peaks and rst acoustic peaks look more nonlinear, the peak values are the highest among the three damping cases, and the trough values (or antiresonant values) are the lowest.If the damping ratio is high (ζ 0.05), (1) the two structural peaks and rst acoustic peaks, which look more nonlinear in the case of ζ 0.005, become more linear; and (2) the peak values are the lowest and the trough values the highest.Figures 5(a)-5(d) show the symmetric and antisymmetric vibration amplitudes and sound powers plotted against the excitation frequencies for various panel lengths.e excitation level κ 0.2; the damping ratio ζ 0.02; the rst two structural modes and seven acoustic modes are adopted; and four exible panels are considered (N 4).Obviously, the panel length (or cavity length) a ects the structural and acoustic resonant frequencies in all vibration amplitude and sound power gures.Generally, the shorter the panel length is, the higher the resonant frequencies and peak values are.Note that, in the case of a 2, some acoustic peaks are outside the frequency range.In Figures 5(c) and 5(d), the resonant frequencies of the acoustic peaks around ω/ω 1 6.4 to 6.6 are not signicantly a ected by the panel length.e rst antisymmetric acoustic peaks are more nonlinear than the rst symmetric acoustic peaks.
is phenomenon can also be observed in Figures 6(c) and 6(d).
Figures 6(a)-6(d) show the symmetric and antisymmetric vibration amplitudes and sound powers plotted against the excitation frequencies for various numbers of exible panels.e excitation level κ 0.2; the damping ratio ζ 0.02; the rst two structural modes and seven acoustic modes are adopted; and the panel length a 3 m.Figures 6(a) and 6(c) show that the number of panels a ects the peak values, peak frequencies, and o -resonance responses in all cases.If only one exible panel is installed, (1) the rst and second symmetric and antisymmetric structural resonant frequencies among the three cases (i.e., N 1, 2, 4) are highest in both the vibration amplitude and the sound power gures; (2) the rst symmetric and antisymmetric structural resonant peak values in the sound power gures are also the highest, while the rst symmetric and antisymmetric structural resonant peak values in the vibration amplitude gures are nearly the same among the three cases; (3) the second symmetric and antisymmetric structural resonant peak values in the vibration amplitude and sound power gures are the lowest among the three cases; and (4) the rst antisymmetric acoustic peaks are more nonlinear (the jump phenomenon can be seen), while the second antisymmetric structural peaks look more linear.
Figures 7(a)-7(d) show the structural-acoustic peak ratios plotted against the dimensionless excitation level for various structural modes.e damping ratio ζ 0.02; the rst seven acoustic modes are adopted; four exible panels are considered (N 4); and the panel length a 2 m. e structural-acoustic peak ratio is de ned as the ratio of the structural peak value to the highest acoustic peak value.In Figure 7(a), the two structural-acoustic peak ratios are monotonically decreasing with the increasing excitation level.is implies that the acoustic peak is more important or signi cant when the excitation level is higher.In Figures 7(b) and 7(c), the structural-acoustic peak ratios of the vibration amplitude are almost constant when the excitation level ranges from κ 0.05 to 0.12.ere is an abrupt   Shock and Vibration jump in each curve at around κ 0.12 to 0.15.e abrupt jumps are due to the jump-up phenomenon, which is well known in forced nonlinear panel vibration [34].en, the structural-acoustic peak ratios of vibration amplitude are monotonically decreasing with the increasing excitation level.e structural-acoustic peak ratio of sound power is

Shock and Vibration
similar to the other ratios, except it is monotonically increasing when the excitation level is from κ 0.05 to 0.12.In Figure 7(d), the two structural-acoustic peak ratios are monotonically increasing for κ 0.05 to 0.055.ere is an abrupt jump around κ 0.06.en, the two ratios are monotonically decreasing until κ 0.17, at which point another abrupt jump can be seen.e abrupt jumps are also due to the jump-up phenomenon in the acoustic resonance.For κ > 0.17, the structural-acoustic peak ratio of vibration amplitude almost constant, while the structural-acoustic peak ratio of sound power is mildly decreasing.

Conclusions
is study investigated the vibration and sound radiation of nonlinear duct panels using the analytic formula based on the classic method combined with Vieta's substitution technique.e main advantage of the analytic formula is that, during the solution procedure, no nonlinear equation solver is required.In other words, the analytic formula is a set of symbolic parameters with various physical meanings.
e results obtained from the proposed classic method show reasonable agreement with those from the total harmonic balance method.e main ndings can be summarized as follows: (1) the damping ratio and excitation level are the two key factors to determine the nonlinearities of the resonant peaks of nonlinear panels.(2) e acoustic peak values in the sound power gures are generally higher than those in the vibration gures.
is implies that the high-frequency components in the sound responses are more signi cant than those in the vibration responses because the sound radiation e ciency is higher when the response frequency is higher.(3) e antisymmetric modal responses, which have been considered in only a few structural acoustic studies, are found to be as important as (or as large as) the symmetric modal responses.e reason is that the rst antisymmetric structural mode is coupled with the rst antisymmetric acoustic mode, which is the most dominant acoustic mode even though it is less important than the rst symmetric mode.(4) e well-known jump phenomenon is found in the peaks of the structural modes coupled with the acoustic modes, whereas it has previously been found only in the resonant peaks of nonlinear structural vibrations.
Data Availability e data used to support the ndings of this study are included within the article.
is a time function ≤ 1; h is the panel thickness; F is the Airy stress function; w, u, and v are the dimensionless displacements along the z, x, and y directions, respectively; v is Poisson's ratio; ζ mx/a and η ny/a are the dimensionless x and y coordinates; r a/b is the panel aspect

Figure 2 :
Figure 2: (a) Comparison of the symmetric and (b) antisymmetric vibration amplitude results from the proposed method and classical harmonic balance method (κ 0.5, a 3 m, b c 0.5 m, ξ 0.02, and N 4).

Figure 3 :Figure 4 :
Figure 3: (a) Symmetric and (c) antisymmetric vibration amplitude versus excitation frequency for various source excitation magnitudes (a 3 m, b c 0.5 m, ξ 0.02, and N 4); (b) symmetric and (d) antisymmetric sound radiation versus excitation frequency for various source excitation magnitudes (a 3 m, b c 0.5 m, ξ 0.02, and N 4); (e) radiation e ciency versus excitation frequency for various structural modes.