Modelling wave propagation in two-dimensional structures using finite element analysis

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Abstract

A method is described by which the dispersion relations for a two-dimensional structural component can be predicted from a finite element (FE) model. The structure is homogeneous in two dimensions but the properties might vary through the thickness. This wave/finite element (WFE) method involves post-processing the mass and stiffness matrices, found using conventional FE methods, of a segment of the structure. This is typically a 4-noded, rectangular segment, although other elements can be used. Periodicity conditions are applied to relate the nodal degrees of freedom and forces. The wavenumbers—real, imaginary or complex—and the frequencies then follow from various resulting eigenproblems. The form of the eigenproblem depends on the nature of the solution sought and may be a linear, quadratic, polynomial or transcendental eigenproblem. Numerical issues are discussed. Examples of a thin plate, an asymmetric laminated plate and a laminated foam-cored sandwich panel are presented. For the last two examples, developing an analytical model is a formidable task at best. The method is seen to give accurate predictions at very little computational cost. Furthermore, since the element matrices are typically found using a commercial FE package, the meshing capabilities and the wealth of existing element libraries can be exploited.

Introduction

The propagation of waves in a homogeneous structure is of interest for many applications, especially at higher frequencies. Examples include the transmission of structure-borne sound, shock response and statistical energy analysis. Knowledge of the dispersion relations, group velocity, reflection and transmission characteristics, etc., enables predictions to be made of disturbance propagation, energy transport and so on.

In simple cases, analytical expressions for the dispersion equation can be found (e.g. Refs. [1], [2]). Examples include one-dimensional structures such as rods and thin beams and two-dimensional structures such as thin plates. For more complicated structures or at higher frequencies the analysis becomes more difficult or even impossible, and the dispersion equation is usually transcendental. Finding all the real, imaginary and complex solutions can be difficult and numerical solutions might be sought. Perhaps the underlying assumptions and approximations break down—for example, for a plate, Kirchoff [1], [2], Mindlin [1], [2], [3] or Rayleigh-Lamb [1], [4] theories might be required to accurately model the behaviour as frequency increases. On the other hand, the properties of the cross-section of a homogeneous solid might be complicated. Examples include layered media and laminated, fibre-reinforced, composite constructions. The latter might be modelled as single equivalent uniform plates but at higher frequencies first (or higher-order) shear deformation theories [5], [6] might be necessary. The equations of motion then become very complicated at best.

Thus in such cases numerical approaches are potentially of benefit. In this paper a finite element (FE)-based approach for the analysis of wave propagation in homogeneous, two-dimensional structures is described. The properties can vary in an arbitrary manner through the thickness of the structure. In this wave/finite element (WFE) method the mass and stiffness matrices of a conventional FE model of a very small segment (typically rectangular) of the structure are post-processed by applying periodicity conditions for the propagation of a time-harmonic disturbance through the structure. The approach is similar to that of Abdel-Rahman [7] for the FE analysis (FEA) of periodic structures, except that in the present case the spatial periodicity is arbitrary. Application of the periodicity conditions results in various eigenproblems whose solutions yield the dispersion relations. Since conventional FEA is used, the full power of existing FE packages and their extensive element libraries can be utilised.

For one-dimensional structures there have been applications of the WFE method for free [8] and forced vibration [9], to rail structures [10] (and, in Ref. [11], using periodic structure theory for a track section), laminate plates [8], thin-walled structures [12] and fluid-filled [13], [14] pipes. Mencik and Ichchou [15] applied the method to calculate wave transmission through a joint. There have been various applications of FEA to spatially periodic one-dimensional structures. These include the earlier works of Orris and Petyt [16], [17], Abdel-Rahman [7] and others reviewed by Mead in Ref. [18]. Specific applications include railway tracks [11], truss beams [19] and stiffened cylinders [20], [21]. The general approach is in contrast to the spectral finite element (SFE) method for one-dimensional waveguides (e.g. Refs. [22], [23], [24]) in which new elements, with a space-harmonic displacement along the axis of the waveguide, must be derived on a case-by-case basis. Other authors have applied periodic structure theory and FE to two-dimensional structures, for example Ruzzene et al. [25], who considered cellular cored structures. Duhamel [26] presents a similar approach for forced vibration of a two-dimensional structure. In Ref. [26], he enforces a harmonic motion of the form exp(−iky) in one direction, so that the equation of motion reduces to that for a one-dimensional structure and is subsequently solved using the WFE methods in Refs. [8], [9]. The Green function is then found by evaluating an integral over the wavenumber k. This is a similar approach to “two-dimensional” spectral element methods (e.g. Ref. [27]), where a harmonic dependence in one dimension is imposed, so that a two-dimensional structure reduces to an ensemble of one-dimensional waveguides.

This paper extends the WFE approach to two-dimensional homogeneous structures. In Section 2 the formulation is described. Periodic structure theory is employed, although the periodicity is arbitrary. A single segment of the structure is analysed using conventional FE methods. This is typically a rectangular segment, meshed through the structure's thickness with rectangular elements with only corner nodes. Elements with other shapes or with interior or mid-side nodes can also be used. Periodicity conditions are applied to develop eigenproblems of various forms—linear, polynomial or transcendental—whose solutions yield the dispersion relations, group velocity and so on. The eigenvalue problems and various numerical issues are discussed in Section 3. Section 4 contains numerical examples of thin isotropic and orthotropic plates, for which analytical solutions are available, and laminated composite-reinforced plates for which an analytical solution is not available. Further details and examples can be found in Ref. [28].

Section snippets

The wave/finite element formulation

Consider a solid which is homogeneous in both the x and y directions, but whose properties may vary through its thickness in the z-direction. An example is the laminate shown in Fig. 1(a), in which each layer is uniform. Under the passage of a time-harmonic wave of frequency ω any response variable w(x, y, z, t) varies asw(x,y,z,t)=W(z)ei(ωt-kxx-kyy)where kx=kcos θ and ky=ksin θ are the components of the wavenumber k in the x and y directions and where W(z) is some function through the thickness

Forms of the eigenproblem

The eigenproblems of Eqs. (9), (11) involve the three parameters λx, λy and ω, and take various forms depending on the physical nature of the solution being sought. As will be seen later, some eigensolutions are artifacts of the spatial discretisation of the structure and are not representative of wave motion in the continuum.

First, the matrices K and M, and C and K′ if damping is present, are real and symmetric in the absence of gyroscopic terms. Consequently so, too, is the DSM D. By taking

Numerical examples

In this section various numerical examples are presented to illustrate the application of the WFE method to plates. Further examples of orthotropic, thick and laminated plates and more detailed discussion can be found in Ref. [28].

Concluding remarks

In this paper a wave/finite element (WFE) method for the analysis of wave motion in two-dimensional structures was described. The structures are homogeneous in two dimensions but the properties might vary through the thickness. The method involves post-processing the mass and stiffness matrices of a segment of the structure, produced using conventional FE methods. The size of the FE model is very small. Emphasis was placed on a 4-noded, rectangular segment, although other element types can be

Acknowledgements

Part of this work was carried out while the first author was an Erskine Visiting Fellow at the University of Canterbury, New Zealand. Part of this work was carried out while the second author held a Fellowship funded by the European Doctorate in Sound and Vibration Studies. The authors are grateful for the support provided.

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