Asymptotic analysis for the coupled wavenumbers in an infinite fluid-filled flexible cylindrical shell: The beam mode

Abstract

Using asymptotics, the coupled wavenumbers in an infinite fluid-filled flexible cylindrical shell vibrating in the beam mode (viz. circumferential wave order $n=1$) are studied. Initially, the uncoupled wavenumbers of the acoustic fluid and the cylindrical shell structure are discussed. Simple closed form expressions for the structural wavenumbers (longitudinal, torsional and bending) are derived using asymptotic methods for low- and high-frequencies. It is found that at low frequencies the cylinder in the beam mode behaves like a Timoshenko beam. Next, the coupled dispersion equation of the system is rewritten in the form of the uncoupled dispersion equation of the structure and the acoustic fluid, with an added fluid-loading term involving a parameter $\mu$ due to the coupling. An asymptotic expansion involving $\mu$ is substituted in this equation. Analytical expressions are derived for the coupled wavenumbers (as modifications to the uncoupled wavenumbers) separately for low- and high-frequency ranges and further, within each frequency range, for large and small values of $\mu$. Only the flexural wavenumber, the first rigid duct acoustic cut-on wavenumber and the first pressure-release acoustic cut-on wavenumber are considered. The general trend found is that for small $\mu$, the coupled wavenumbers are close to the in vacuo structural wavenumber and the wavenumbers of the rigid-acoustic duct. With increasing $\mu$, the perturbations increase, until the coupled wavenumbers are better identified as perturbations to the pressure-release wavenumbers. The systematic derivation for the separate cases of small and large $\mu$ gives more insight into the physics and helps to continuously track the wavenumber solutions as the fluid-loading parameter is varied from small to large values. Also, it is found that at any frequency where two wavenumbers intersect in the uncoupled analysis, there is no more an intersection in the coupled case, but a gap is created at that frequency. This method of asymptotics is simple to implement using a symbolic computation package (like Maple).

Introduction

Wave propagation in fluid-filled cylindrical shells is a classical topic of interest to the structural acoustics community. This is evidenced by the large volume of literature on the subject. In contrast to the planar fluid–structure interaction problems, the fluid–shell system poses an additional challenge by having the motion in the three coordinate directions coupled due to the shell curvature. In this context, the efforts of some researchers have been in finding the fluid-shell coupled wavenumbers. They have numerically solved the coupled dispersion equation by varying individual parameters [1], [2], [3], [4] and also discussed the physics related to the coupled wavenumbers. However, this numerical root-finding approach is laborious, and the solutions are not amenable to easy understanding. For example, to understand the influence of the fluid on the structure or vice versa, an analytical expression for the coupled wavenumbers would be very useful. It would then be easy to identify the coupled wavenumbers as a modification to the uncoupled wavenumbers. To this end, the method of asymptotics appears to be a seemingly indispensable approach as it leads to analytical expressions for the coupled wavenumbers, which conveniently are perturbations to the uncoupled wavenumbers.

The essence of asymptotic analysis is to arrive at a solution for a complicated system which in some way is near to a solvable simpler system with known analytical solutions. Using asymptotics, analytical expressions can be found for the complicated system also and these expressions are slightly modified (or perturbed) versions of those of the simpler system. This method is widely prevalent in solving for weakly nonlinear systems [5]. The method has also been widely used in the field of nonlinear acoustics [6].

Defining the fluid-loading effect in the form of a perturbation parameter, asymptotic analysis has been used to analyze structures in contact with infinite acoustic domains for plane [7], [8], [9] and cylindrical [10], [11] geometries. In contrast, for systems with finite acoustic domains, such as a flexible acoustic duct, studies have been mainly of experimental [12], [13], [14] or of numerical [1], [2], [15], [16] nature. Applications of the asymptotic method to flexible acoustic ducts have not come to our notice. Studies using asymptotic methods to find the coupled wavenumbers of structural acoustic systems in two different geometries have been recently carried out by us [17], [18].

In this study, we consider an infinite fluid-filled flexible circular cylindrical shell (see Fig. 1). Our interest is to find the coupled structural acoustic wavenumbers for this system as perturbations to the uncoupled structural and the acoustic wavenumbers. Numerical solutions to this problem have already been presented [1]. Here, we wish to bring more insight into the character of the wavenumber solutions using asymptotics.

As a representative case, we choose to study the beam mode (viz. circumferential wave order $n=1$) in detail in the present article. The higher order circumferential modes ($n>1$) closely resemble this mode (in a qualitative sense) [19]. Further, $n=1$ mode is the lowest order mode in which the vibrations in all three directions (viz. radial, longitudinal and torsional) are coupled. In contrast, for the axisymmetric mode the torsional displacement is uncoupled from the other two. For the axisymmetric mode, we have found analytical expressions for the coupled wavenumbers using asymptotic methods [18].

For the acoustic fluid in a cylindrical duct, simple closed form expressions for the wavenumbers in the $n=1$ mode are well-known. However, the dispersion relation for the cylindrical shell is an unwieldy polynomial equation. Hence, in the first part of the article, simple expressions for the in vacuo shell wavenumbers will be derived for low and high frequencies using asymptotics. In the process of derivation, we will show that at low frequencies the uncoupled shell (equivalently the in vacuo shell) behaves as a Timoshenko beam, whereas at high frequencies the shell resembles a plate of identical thickness [19].

Next, it will be shown that the dispersion relation of the coupled system is related to the uncoupled structural and acoustic system through a fluid-loading parameter. Using this as the perturbation parameter, we will find the coupled wavenumber expressions for small and large fluid-loading. The inherent nature of the asymptotic method provides analytical expressions for the coupled wavenumbers in terms of the uncoupled wavenumber expressions and a correction factor involving the fluid-loading parameter.