Elsevier

Computers & Structures

Volume 70, Issue 3, February 1999, Pages 363-376
Computers & Structures

Vibration and damping analysis of fluid filled orthotropic cylindrical shells with constrained viscoelastic damping

https://doi.org/10.1016/S0045-7949(98)00192-8Get rights and content

Abstract

The finite element code is developed based on the displacement field proposed by Wilkins et al. to find out the natural frequencies and loss factors of fluid filled cylindrical shells with a constrained viscoelastic layer in between two facings made of composite material. The fluid effect on the natural frequencies and loss factors are taken care of by the added mass of the fluid on the structure. Effects of the fluid height, ratio of core to facing thickness and facing fibre orientation on the natural frequencies and loss factors of cylindrical shells are presented.

Introduction

Composite materials are now widely in use due to their light weight and high specific strength. The use of viscoelastic material as a core material in shells of revolution is an effective way of damping the vibration and noise in the structures subjected to dynamic loading. Free vibration analysis of orthotropic conical shell has been studied by Wilkins et al.[1]. Ramesh and Ganesan2, 3, 4 analysed the vibration and damping characteristics of isotropic and orthotropic cylindrical shells. Many papers are available on vibration and damping analysis of cylindrical and conical shells, but only a few papers are available on spherical shells. Akihiko Okazaki et al.[5] analysed the damping characteristics of spherical shells.

Analysis of free vibration characteristics of fluid filled shells with viscoelastic damping can be used to improve the performances of liquid propellant rockets, large capacity oil storage tanks against earthquake loading etc. Analysis of fluid filled cylindrical shells with fluid, but without viscoelastic damping can be found in Refs.6, 7, 8, 9, 10, 11. Addition of a damping layer in fluid filled shells may be useful in the dynamic characteristics under different types of loading such underwater explosion and earthquake loading.

To the authors’ knowledge no literature is available for the analysis of the general shell of revolution made of orthotropic facings and also no literature is available on fluid filled shells with viscoelastic damping. General shell finite element formulation can be used for the analysis of cylindrical, conical, spherical, paraboloidal and any shells of revolution. Hence, the aim of this paper is to develop a general shell finite element code for a viscoelastic shell based on the displacement field proposed by Wilkins et al.[1] and apply the same for the orthotropic cylindrical shell with and without fluid. In the present study, damping in the fluid filled shell is compared with damping of shells without fluid.

Section snippets

Finite element formulation—structure

A three noded isoparametric element is used for the structure. A multilayered general shell element for the structure is shown in Fig. 1. The displacement field used in the analysis is proposed by Wilkins et al.[1] for sandwich conical shells with orthotropic facings and a honeycomb core. The effects of shear deformation are accounted for in the facings.

The displacements u, v, w are along with s, θ and z co-ordinate directions, respectively. The above displacements are defined in terms of

Finite element formulation—fluid domain

An eight noded isoparametric element is used for the fluid domain. The fluid element used is shown in Fig. 2.

From Zienkiewicz and Newton[8], the wave equation of the fluid is given by[H]{p}−[G]ω2{p}−ω2[S]{ui}=0.The fluid pressure variation {p} due to vibration was assumed to be{p}=[NF]{pi}[NF]=[N']cosmθ [N']=[N1N2…N8]N1=0.25(1−ξ)(1−η)(−ξ−η−1),N2=0.50(1−ξ2)(1−η),N3=0.25(1+ξ)(1−η)(ξ−η−1),N4=0.50(1+ξ)(1−η2),N5=0.25(1+ξ)(1+η)(ξ+η−1),N6=0.50(1−ξ2)(1+η),N7=0.25(1−ξ)(1+η)(−ξ+η−1),N8=0.50(1−ξ)(1−η2),

Analysis

For the analysis, the facing is assumed to be made of Kevlar–epoxy with the material properties E1=76.0×109 N/m2, E2=5.5×109 N/m2, E3=E2, G12=G13=G23=5.5×109 N/m2, Poisson’s ratio=0.34 and ρf=1460 kg/m3. The material damping of the facings is not included in the analysis. The core is assumed to be made of PVC and its material properties are Ec=(2.3×107, 0.782×107) N/m2, vc=0.34 and ρc=1340 kg/m3. It is assumed that these properties do not vary with frequency or temperature. The structure is

Validation

In order to validate the element, comparisons are made with the results presented by Ramesh and Ganesan[4] for the viscoelastically damped structure without fluid and comparisons are made with the results presented by Han and Liu[7] for the fluid filled cylindrical shell.

Table 1 shows the comparison of the non-dimensional frequencies and loss factors of the empty shell having the R/t ratio of 100 and tc/tf ratio of 1 made of graphite–epoxy material as described in Ref.[4]. The fibre orientation

Variation of frequency and loss factor with circumferential mode number

Fig. 3 shows the variation of natural frequency and loss factor for a tall shell made of Kevlar–epoxy material with axial fibre orientation (fibre angle 0°). The figure gives the values for empty, half-filled and fully filled shells for various values of tc/tf (core to facing thickness ratio). It is seen from Fig. 3 for the empty tall shell, with the increase in the tc/tf ratio, that the frequency falls, for the circumferential mode number, to less than four. In contrast, with an increase in

Conclusion

The vibration and damping characteristics of fluid filled orthotropic cylindrical shells have been studied by using the finite element method. With various core to facing thickness ratios, various fluid heights and for 0° and 90° fibre orientation. In lower circumferential modes, as the core to facing thickness ratio increases, frequency decreases. In contrast, at higher modes frequency increases as the core to facing thickness increases. Decreasing–increasing trends of frequency with

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