Nonlinear vibro-acoustic behavior of cylindrical shell under primary resonances

doi:10.1016/j.ijnonlinmec.2021.103682

International Journal of Non-Linear Mechanics

Volume 130, April 2021, 103682

https://doi.org/10.1016/j.ijnonlinmec.2021.103682 Get rights and content

Highlights

•
Vibro-acoustic behavior of cylindrical shell under primary resonances is studied.
•
Donnell’s nonlinear shallow shell theory is used to model the cylindrical shell.
•
Jump phenomenon can be seen in transmission loss curve for the case ( $Ω \approx ω_{1}$ ).
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The detuning parameter is the bifurcation parameters.
•
Neimark–Sacker, flip, and period-3 bifurcations occurred during shell vibration.

Abstract

Nonlinear vibro-acoustic behavior of cylindrical shell excited by an oblique incident plane sound wave under primary resonances is analytically examined in this paper. Donnell’s nonlinear shallow shell theory is used to model the cylindrical shell. Coupled nonlinear differential equations of the system are analytically derived using Galerkin’s approach. The Multiple Scales Method is hence, employed to solve the corresponding nonlinear equations. Then, the effects of crucial design parameters including incident sound wave amplitude, damping ratio and thickness of the shell on the characteristics of the sound transmission loss are studied for different resonance cases. In addition, the effect of detuning parameter on the bifurcation and behavior of the limit cycle under primary resonance is examined. The results show that the detuning parameter is a bifurcation parameter and Neimark–Sacker, flip, and period-3 bifurcations occur when this parameter is varied. Also, according to the results, by getting away from the resonance frequencies, excitation level incorporates no significant effect on the transmission loss of the shell.

Introduction

Vibro-acoustic analysis of cylindrical shells has extensively been investigated in the recent years because of their variety of engineering applications. Several cases may be addressed in aircraft, mechanical, marine and automotive engineering. Different aspects of vibro-acoustic behavior of cylindrical shell have already been addressed in the literature. Noise transmission through and radiation from the cylindrical shells were studied by many researchers. Daneshjou et al. [1] calculated the transmission loss of a homogeneous isotropic thick-walled cylindrical shell due to an oblique plane wave. Oliazadeh and Farshidianfar [2] used exact analytical approach to investigate sound transmission losses in double- and triple-walled thin cylindrical shells with varying air gap sizes. A theoretical model was developed by Zhang et al. [3] to investigate the effect of perforation on the sound transmission through a double-walled cylindrical thin shell excited by a plane wave. An analytical model based on statistical energy analysis theory was developed and experimentally validated by Oliazadeh et al. [4] to examine sound transmission through a thin-walled circular cylindrical shell. A precise transfer matrix method was developed by Wang et al. [5] to calculate the sound radiation of submerged double-walled cylindrical shell with uniformly distributed annular plates and arbitrary boundary conditions. The wave and finite element (WFE) method was developed by Kingan et al. [6] for analyzing sound transmission through, and radiation from an infinitely long cylindrical structure filled and surrounded by fluids.

Sound transmission has also been investigated through sandwich cylinders. Among these works, Magniez et al. [7] used the first-order shear deformation theory to investigate the transmission loss of an infinite multilayer cylinder composed of two orthotropic thin skins separated by an isotropic polymer core. Using the three-dimensional theory of elasticity, the acoustic wave transmission across an imperfect-bonding double-walled sandwich cylinder with FGM-core was studied by Talebitooti et al. [8].

Laminated composite cylindrical shell structures have widely been used particularly in aerospace and marine industries due to their high strength to weight ratio. Several studies have been carried out to investigate transmission loss through these structures. For example, Talebitooti et al. [9] used the Third-order Shear Deformation Theory to investigate the transmission loss of the laminated composite cylindrical shell in contrast to plane sound wave. The transmission loss of an arbitrary thick infinite piezo-laminated cylindrical shell filled with and submerged in compressible fluids was analytically estimated by Rabbani et al. [10].

Effects of poroelastic material on sound transmission, particularly in high frequencies, have been an interesting subject of studies in the recent years. An analytical model was presented by Zhou et al. [11], [12] to investigate the effects of external mean flow and exterior turbulent boundary layer on the transmission loss of a double shell lined with poroelastic material. Liu and Hi [13] calculated the transmission loss of random incidence in the diffuse field through double-shell sandwich composite structures with the poroelastic core. Using a mixed “Biot–Shell” analytical model, Magniez et al. [14] calculated sound transmission through orthotropic sandwich cylinders with a poroelastic core. An extended full method presented based on Biot theory with considering the 3-D wave propagation was developed by Talebitooti et al. [15] to investigate the transmission loss of poroelastic cylindrical shell. They [16] also used this method to study the effect of external subsonic flow on sound transmission through poroelastic cylindrical shell. Using non-dominated sorting genetic algorithm, Talebitooti et al. [17] optimized sound transmission through the laminated composite cylindrical shell with sandwiching a layer of porous material as an intermediate layer.

A large number of studies are available on nonlinear vibrations of circular cylindrical shells. Pellicano et al. [18] investigated non-linear vibration of simply supported circular cylindrical shells considering geometric nonlinearities and the effect of viscous structural damping. Using five classical non-linear shell theories, the response of an empty and simply supported circular cylindrical shell subjected to radial harmonic excitation has been computed by Amabili [19]. The nonlinear vibrations of empty or fluid-filled circular cylindrical shells, clamped at both ends and subjected to a radial harmonic force excitation, have been studied by Karagiozis et al. [20]. Using both Donnell’s non-linear theory retaining in-plane displacements and the Sanders–Koiter non-linear theory, Amabili et al. [21] investigated the effect of geometric imperfections on nonlinear stability of circular cylindrical shells conveying incompressible fluid. Sanders–Koiter nonlinear shell theory was used by Kurylov and Amabili [22] to investigate Large-amplitude (geometrically nonlinear) forced vibrations of circular cylindrical shells with different boundary conditions. Alijani and Amabili [23] investigated geometrically nonlinear forced vibrations of water-filled arbitrary laminated circular cylindrical shells using the Amabili–Reddy nonlinear higher-order shear deformation theory. In an experimental–numerical study, Amabili et al. [24] examined nonlinear vibrations of a water-filled circular cylindrical shell subjected to radial harmonic excitation in the spectral neighborhood of the lowest resonances, using a seamless aluminum sample. Semi-analytical solution of nonlinear vibrations of circular cylindrical shell made of carbon nanotube fiber-reinforced composite excited by a radial harmonic force with considering structural damping was studied by Yadav et al. [25].

In most of the literatures reviewed above, although Sound Transmission Loss (STL) across various types of shells has been studied, there has been no investigation on the nonlinear acoustic behavior of the cylindrical shell in contrast to plane sound wave, so far. Therefore, in this paper an analytical approach is developed to study the nonlinear vibro-acoustic behavior of the cylindrical shells based on Donnell’s theory. Then, the frequency–response for the radial deflection of the shell and transmission loss through the cylindrical shell due to an oblique plane sound wave are calculated using the method of multiple scales (MMS). MMS has also been used in other nonlinear vibro-acoustic problems to calculate frequency–response of structure vibration and transmission loss [26], [27], [28].

The rest of the current paper is organized as follows. In Section 2, the vibro-acoustic model of the sound transmission through a cylindrical shell is presented. Then using the Galerkin method based on the boundary conditions of the system, nonlinear differential equations of the system are achieved. In Section 3, the method of multiple scales is utilized to determine the frequency–response of the shell due to sound wave excitation. In Section 4, results of the current model and their discussion and parametric studies are presented. Finally, conclusions are provided in Section 5.

Section snippets

System model

Fig. 1 shows the geometry and coordinate system of a cylindrical shell with thickness $h$ , mean surface radius $R$ and length $L$ considered for the study in this paper. The cylindrical shell is described with the coordinates $(x, r, θ)$ , in which $x$ , $r$ and $θ$ denote the axial, radial and circumferential coordinates, respectively. Also, the displacements in the axial, radial and circumferential directions are denoted by $u (x, θ)$ , $w (x, θ)$ and $v (x, θ)$ , respectively.

Using the Donnell’s nonlinear shallow shell theory,

Solution method

In this section, the method of multiple scales (MMS) is employed to find an analytical solution for the frequency–response of the proposed nonlinear vibro-acoustic model. In this approach, the response can be presented by an expansion which is a function of multiple-independent scales $T_{i}$ . According to this method, the solution of Eqs. (21) to (23) can be represented by an expansion having the form [32] $\begin{matrix} A_{1 n} (T_{0}) = ε u_{11} (T_{0}; T_{1}; T_{2}) + ε^{2} u_{12} (T_{0}; T_{1}; T_{2}) + ε^{3} u_{13} (T_{0}; T_{1}; T_{2}) \\ B_{1 n} (T_{0}) = ε u_{21} (T_{0}; T_{1}; T_{2}) + ε^{2} u_{22} (T_{0}; T_{1}; T_{2}) + ε^{3} u_{23} (T_{0}; T_{1}; T \end{matrix}$

Sound transmission loss

The sound transmission loss is defined as the ratio of the incident sound powers and the transmitted sound powers through unit length of the shell along the axial direction. $T L = 10 {log}_{10} (\frac{W^{I}}{\sum_{n = 0}^{\infty} W_{n}^{T}})$ in which $W^{I}$ is incident sound power and $W_{n}^{T}$ is the $n$ th term of transmitted sound powers corresponding to the $n$ th mode and can be defined as follows [33]: $W^{I} = R \frac{{P_{0}}^{2} c o s (ψ)}{ρ_{1} c_{1}}$ $W_{n}^{T} = \frac{1}{2} R e a l {\int_{0}^{2 π} \int_{0}^{L} P_{T}^{3} \frac{\partial w}{\partial t} d x d θ}$

For case $Ω \approx ω_{1}$ , substituting Eqs. (4), (10) into Eq. (52) yields: $W_{n}^{T} = \frac{1}{2} R e a l {\int_{0}^{1} \int_{0}^{2 π} P_{t}^{n} H_{n}^{1} (K_{3 r} r) cos (n θ) e^{j (Ω t -)}$

Simulations, results and discussion

To validate the model presented in this paper, its predictions of the responses are compared with the numerical results reported by Amabili et al. [30]. In this reference, nonlinear vibration of a simply supported circular cylindrical shell was analyzed. The shell characteristics in this reference are: $L = 0.2$ m, $R = 0.1$ m, $h = 0.247 \times 1 0^{- 3} m$ , $E = 71.02 \times 1 0^{9} Pa$ , $ρ = 2796 \frac{kg}{m^{3}}$ , $υ = 0.31$ . Also, the damping ratio and the amplitude of the external excitation are considered $2 ζ = 0.001$ and $f_{m n} = 0.0012 h^{2} ρ ω_{m n}^{2}$ ,

Conclusion

Vibro-acoustic behavior of cylindrical shell under primary resonances was analytically studied in this paper. Donnell’s nonlinear shallow shell theory was used to derive the partial differential equations of the shell radial motion. Then, the Galerkin method was employed to achieve the coupled nonlinear ordinary differential equations of the system. Also, in order to solve the nonlinear equations of system, Multiple Scales method was used. Closed-form expressions were obtained for the amplitude

CRediT authorship contribution statement

Amir Hossein Orafa: Conceptualization, Methodology, Software, Validation, Formal analysis. Mohammad Mahdi Jalili: Conceptualization, Investigation, Data curation, Writing - original draft. Ali Reza Fotuhi: Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Nonlinear vibro-acoustic behavior of cylindrical shell under primary resonances

Highlights

Abstract

Introduction

Section snippets

System model

Solution method

Sound transmission loss

Simulations, results and discussion

Conclusion

CRediT authorship contribution statement

Declaration of Competing Interest

Int. J. Mech. Sci.

J. Sound Vib.

J. Sound Vib.

J. Sound Vib.

Ocean Eng.

Wave Motion

J. Sound Vib.

Int. J. Mech. Sci.

Compos. Struct.

J. Sound Vib.

J. Sound Vib.

Compos. Struct.

J. Sound Vib.

J. Sound Vib.

J. Sound Vib.

Aerosp. Sci. Technol.