Elsevier

Journal of Sound and Vibration

Volume 365, 17 March 2016, Pages 157-171
Journal of Sound and Vibration

Modal density of thin composite cylindrical shells

https://doi.org/10.1016/j.jsv.2015.11.030Get rights and content

Abstract

Modal density is an important parameter in Statistical Energy Analysis (SEA) based response estimation. Many space structures use composite cylinders. Modal densities of such structural elements are not reported. In this work an expression for modal density of composite cylindrical shells is derived. Its characteristics and sensitivity to various parameters are discussed. The frequency at which the modal density has a maximum is derived. Modal densities of typical composite cylinders are obtained. It is shown that computing modal density considering an equivalent isotropic cylinder can lead to significant errors.

Introduction

Estimation of responses of structures to high frequency dynamic excitation is normally carried out using a technique called Statistical Energy Analysis (SEA) [1], [2]. One of the important parameters required for SEA based calculations is the number of resonant modes present in a frequency band. To be useful for SEA based calculations, the number of modes has to be estimated using a closed form expression and cannot be through a method like Finite Element method. Therefore expressions for modal densities of several structural forms are derived and are in use [3], [4], [5].

Many spacecraft structures have a central cylinder which is a cylindrical shell made of composite material. There are several studies reported on the modal densities of cylindrical shells. Heckl [6] obtained the modal density of a thin cylinder for the out of plane motion. In his work, natural frequency was obtained as the frequency at which the impedance vanished and the number of modes was estimated from the maximum value of half waves possible below the given frequency. Another work on modal densities of shells is by Bolotin [7] in which the modal densities of thin shells are obtained using wavenumber diagrams and expressed the modal density in terms of elliptical integrals. Szechenyi [8] obtained the modal density of a thin cylinder by measuring the area in the wavenumber plane and proposed empirical relations for its estimation. Maymon [9] presented the modal densities of stringer stiffened shells by adding the modal densities of monocoque cylinder with those of stiffeners.

Modal densities of sandwich cylinders are also reported. Wilkinson [10] derived an expression for modal densities of sandwich cylinders incorporating shear deformation of the core and Erickson [11] modified the expression considering rotary inertia. Ferguson and Clarkson [12] obtained an expression for estimating modal density of paraboloidal structural element. Elliot [13] presented expressions for the modal densities of thin as well as honeycomb sandwich cylindrical shells in the form of integrals which were evaluated numerically.

All the above works are on the isotropic shells or honeycomb sandwich shells with isotropic face sheets and no expressions are reported for composite cylindrical shells.

In this work an expression for modal density of composite cylinder is derived. For deriving the expression for modal density, an expression for the natural frequency is required. A closed form expression for frequency of orthotropic cylindrical shells is reported by Soedel [14]. This expression is based on Donnell’s shallow shell theory where shear deformation and rotary inertia are neglected. First the expression for natural frequency is derived. Expression for the modal density is then derived following an approach similar to those for isotropic shells. To validate the expression, mode count obtained using this expression is compared with the number of modes computed by a Finite Element Model using NASTRAN. Characteristics of modal densities of composite cylinders are discussed. Modal densities of typical composite cylinders are then presented. Modal densities if calculated using the expression for isotropic shells with equivalent isotropic properties are discussed.

Section snippets

Governing differential equations and natural frequency

There are several theories for describing the elastic behavior of the shells. In the present work, Donnell׳s shallow shell theory [14] is considered. The coordinate axes are longitudinal (x), tangential (θ) and radial (r) as shown in Fig. 1. The displacement along the longitudinal direction is ux, along the tangential direction (linear displacement) is uθ and along the radial direction is ur. Let the radius of the shell be ‘a’ and the length be ‘L’.

Mode count and modal density

Average number of modes per unit frequency is called modal density. The response of the system is greatly dependent on the modal density. If the modal density of a structure is larger, its response also will generally be higher.

Cylinder having identical properties in two directions

In many cylinders properties in the two material–property directions are approximately the same. In such cases the above expression for modal density can be cast in a simple form.

Characteristics of modal density

Characteristics of the modal densities are now obtained for specific extreme or asymptotic cases which will also confirm the correctness of the derived expression to some extent. These characteristics are obtained using the approximate expression. Though the approximate expression is not accurate for low values of β, the study can provide valuable insight to the characteristics.

Mode count using equivalent cylinders

In the absence of the expression for computing the modal densities of composite cylinders derived here one could consider an equivalent cylinder but isotropic and use the expression for isotropic shells. The modal densities of the cylinder, properties of which are given in Section 4.4, are computed using Eq. (42) and compared with the results obtained for equivalent cylinders. It is to be mentioned that the cylinder considered is a honeycomb sandwich cylinder with composite face sheets. The

Conclusions

Expressions for estimating mode count and modal density of a thin composite cylinder are derived. The modal density increases with frequency and has a maximum at certain frequency beyond which it converges to the modal density of flat panels. The expression is then simplified for cylinders having equal properties along the two orthogonal directions. An approximate expression is then derived and the conditions under which it can estimate the modal density with a reasonable accuracy are

References (18)

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