This paper studies the vibrational behavior and far-field sound radiation of a submerged stiffened conical shell at low frequencies. The solution for the dynamic response of the conical shell is presented in the form of a power series. A smeared approach is used to model the ring stiffeners. Fluid loading is taken into account by dividing the conical shell into narrow strips which are considered to be local cylindrical shells. The far-field sound pressure is solved by the Element Radiation Superposition Method. Excitations in two directions are considered to simulate the loading on the surface of the conical shell. These excitations are applied along the generator and normal to the surface of the conical shell. The contributions from the individual circumferential modes on the structural responses of the conical shell are studied. The effects of the external fluid loading and stiffeners are discussed. The results from the analytical models are validated by numerical results from a fully coupled finite element/boundary element model.

The reduction of noise emitted by maritime platforms is an important topic in naval research. A truncated cone is used to support the aft-shaft-bearing of the submarine propeller. The structural and acoustic responses of a conical shell are not as widely reported in literature as in the case of cylindrical shells. This is due to the increased mathematical complexity associated with the effect of the variation of the radius along the length of the cone on the elastic waves. Much of the earlier work on the equations of motion for conical shells and forms of solution with different boundary conditions has been summarized by Leissa [

Many researchers have investigated approximate methods to solve the acoustic radiation from vibrating elastic structures. The boundary integral equation formulation is a popular method. A good review on the evolution of boundary element techniques was given by Chien et al. [

In this work, the structural and acoustic responses of a stiffened conical shell submerged in a heavy fluid are presented. Section

For a conical shell, the coordinate system

Coordinate system and displacements for a thin conical shell.

The parameters of the stiffeners.

When the fluid loading on the surface of the stiffened conical shell is considered, the equations of motion can be expressed as

The fluid loading effects on the conical shell are taken into account using an approximate method by dividing the conical shell into

The external pressure

The dynamic response of the conical shell for each circumferential mode number

Free end:

clamped end:

shear-diaphragm (SD) end:

Two excitation cases as shown in Figure

Excitation along the generator and normal to the surface at the smaller end of the conical shell.

In this section, the theory for the boundary element method is used to demonstrate that the far-field acoustic pressure of the shell can be solved by superposing the pressure of the individual elements. An analytical solution of the acoustic pressure radiated by a cylindrical piston in a cylindrical baffle is used to obtain the pressure of conical elements.

According to the Helmholtz boundary integral equation, the acoustic pressure radiated by the arbitrary surface can be expressed as [

By discretizing the surface into

The conical shell is divided into

According to Wang [

The cylindrical baffle and piston.

The radiated pressure from each element of the conical shell can be approximated by the analytical solution of the radiated pressure of a piston in a cylindrical baffle, as shown in Figure

The structural and acoustic responses for the conical shell under harmonic excitation are presented using the following parameters:

A finite element model is developed in ANSYS to validate the results obtained from analytical model presented here. The conical shell model and the ring stiffeners are built using Shell 63 elements and Beam 188 elements, respectively. Fluid 30 and Fluid 130 elements are used to simulate the external fluid and the absorbing boundaries around the fluid domain. ANSYS is used to obtain the structural responses. SYSNOISE is then used to calculate the far-field pressure using the direct boundary element method.

The validity of smeared approach for the stiffened conical shell and the convergence of dividing the conical shell are discussed in what follows. A model of the stiffened conical shell

Range of validity for the stiffeners smeared over the conical shell.

Comparison with FEA results when the space or size of stiffeners changes. (a) Results when the space between stiffeners

As discussed in Section

Comparison for different divisions for the conical shell.

The validity and convergence of the Element Radiation Superposition Method (ERSM) for solving the far-field acoustic pressure of a conical shell are discussed in what follows. Each conical segment with the length of

Validity and convergence of solving far-field pressure.

The structural and acoustic responses of the conical shell due to the excitation shown in Figure

Structural and acoustic responses due to

Structural and acoustic responses due to

The contributions of the independent circumferential modes with mode order

The effect of the fluid loading on the structural and acoustic responses for the two excitation cases is presented for the conical shell in air in Figures

Comparison of conical shells in water and in air, under

Comparison of conical shells in water and in air, under

The effect of the ring stiffeners on the structural and acoustic responses of the conical shell responses is presented in Figures

Comparison of conical shells with and without stiffeners, under

Comparison of conical shells with and without stiffeners, under

An analytical method to predict the structural and acoustic responses of a stiffened conical shell immersed in fluid has been presented. The solution for the dynamic response of a conical shell was presented in the form of a power series. Fluid loading on the shell was taken into account by dividing the shell into narrow strips and stiffeners were modeled using a smeared approach. The Element Radiation Superposition Method was used to solve the far-field pressure of a conical shell. Good agreement between the analytical and FEM/BEM results was observed. Two cases of excitation by considering forces acting in different directions at the smaller end of the conical shell were examined, in which the forces acted along the generator and normal to the surface of the shell. In both cases,

The forces and moments for a conical shell are given by [

The authors declare that there is no conflict of interests regarding the publication of this paper.

All the work in this paper obtains great support from the National Natural Science Foundation of China (51179071) and the Fundamental Research Funds for the Central Universities, HUST: 2012QN056.