Elsevier

Composite Structures

Volume 142, 10 May 2016, Pages 140-154
Composite Structures

An exact spectral dynamic stiffness theory for composite plate-like structures with arbitrary non-uniform elastic supports, mass attachments and coupling constraints

https://doi.org/10.1016/j.compstruct.2016.01.074Get rights and content

Abstract

This paper presents an exact spectral dynamic stiffness (SDS) theory for composite plates and plate assemblies with arbitrary non-uniform elastic supports, mass attachments and elastic coupling constraints. The theory treats the above supports, attachments and constraints in a sufficiently general, but accurate manner, which can be applied to various SDS formulations as well as classical dynamic stiffness formulations for both modal and dynamic response analysis. The methodology is concise but can be easily applied to complex plate-like structures with any arbitrary boundary conditions. It retains all the advantages of a recently developed SDS method which gives exact results with excellent computation efficiency. The results computed by the present theory are validated against published results. In order to demonstrate the practical applicability of the theory, three wide ranging engineering composite structures are investigated. For benchmarking purposes, results computed from the current theory are accurate up to the last figure quoted.

Introduction

Dynamic analysis of plate-like structures with arbitrary non-uniform elastic supports (Fig. 1(a)), mass attachments (Fig. 1(b)) and elastic coupling constraints (Fig. 1(c)) has always been a challenging problem in many engineering areas. Such an analysis is important to avoid resonance or undesirable dynamic response, control acoustic emission and power flow [1], analyse the effects of cracks [2] or damaged boundaries [3] for identifying the location and destructiveness of these cracks and damages, and many others. The applicability of the proposed theory include, but not limited to, buildings, bridges, ships, aeroplanes, space structures, armoured vehicles, automobiles, machines, robots, optical beam pointing systems and so on. Non-uniform elastic supports, mass attachments and elastic coupling constraints are expected to change the dynamic behaviour of a structure significantly. It is timely and pertinent to review briefly the published work focused on the above three general types of non-classical boundary conditions (BC) and/or continuity conditions (CC). Fig. 1 depicts the above three general types of BC and/or CC corresponding to the Kirchhoff plate theory which serve as illustrating examples.

A wide range of methods have been proposed in the literature for free vibration of plates with elastic supports (see Fig. 1(a)) but they are generally limited to uniform distribution [4], [5], [6], [7], [8], [9], [10], [11], [12]. One of the most widely used methods in this endeavour is the Ritz method [4], [5], [6], [7], [8], which is generally limited to uniformly distributed elastic supports due to the admissible functions adopted in the variable separable assumptions. Other methods such as the Fourier series based analytical method [9], [10], the finite strip method [11], [12] also appear to have similar limitations. The treatment of non-uniform elastic supports is obviously much more demanding when using any established method. Understandably, only a few research has attempted such problems [3], [13], [14], [15], [16], [17], [18]. However, most of the published methods have limitations of different natures. Some methods can only be applied to plates with restricted boundary conditions. For example, the Spectral Collocation [3] and the Discrete Singular Convolution [17] methods appear to be limited to non-uniform rotational elastic supports and they have problems with non-uniform translational elastic supports and/or when any free edge is encountered. Moreover, the results computed by most of these methods do not seem to be sufficiently accurate. This is mainly due to the numerical instability which generally occurs and prevents these methods from computing more accurate results by using higher order basis functions. Additionally, most of the existing methods are only applied to single plates [3], [13], [14], [15], [16], [17], [18], [19] and very few are applied to plates with intermediate uniform elastic supports [6], [7]. The application to more complex plate assemblies with non-uniform elastic supports is no doubt a difficult task when using most of the existing analytical methods.

Even though a lot of research on plates with point mass attachments has been reported [7], [20], [21], [22], [23], there are very few publications on line mass attachments (see Fig. 1(b)) and that too are limited to uniformly distributed ones [24], [25]. (Note that point mass attachments can be considered as a special case of non-uniform line mass attachments.) The problem of non-uniformly distributed mass attachments (non-uniform m(ξ) and/or I2(ξ) in Fig. 1(b)) has not been attempted in any great detail, although it admittedly has many applications in engineering. For example, an engine or a missile attached to an aircraft wing can be idealised as non-uniformly distributed mass with non-uniform rotatory inertia. Also, the shear wall structures of multi-story buildings can be modelled as plates placed vertically and subjected to non-uniform line mass and spring attachments representing the dynamic effects of the floors. The only reported research activities on plates subjected to line mass attachments are all restricted to cases with very simple boundary conditions, and furthermore the line mass attachments are considered to be uniformly distributed [24], [25] as opposed to non-uniform distribution. To the best of the authors’ knowledge, there is no published literature for plate vibration with non-uniformly distributed mass attachments.

Solving the vibration problem of plate assemblies with elastic coupling constraints (see Fig. 1(c)) is even more challenging, which has received rather sporadic attention [26]. Du et al. [26] used a certain modified double Fourier series in the Rayleigh–Ritz method which was applied to two coupled plates with uniform elastic coupling constraints (constant Cw and Cϕ). However, the formulation of this type is quite tedious even when applied to a model comprising only two coupled isotropic plates with uniform elastic constraints. Moreover, the results computed by such a method [26] are not really accurate. One of the reasons for this might be the numerical instability, which seems to be a stumbling block for this method when applied to more general cases of complex structures. Any non-uniform elastic coupling constraints (Cw(ξ) and Cϕ(ξ) as in Fig. 1(c)) could lead to even more serious numerical instability problems when using this method.

Against the above background, an exact spectral dynamic stiffness (SDS) theory is developed covering all of the above three general types of non-uniform boundary conditions (BC) and/or continuity conditions (CC) as illustrated in Fig. 1. The development of the current theory is based on a recently developed spectral dynamic stiffness method (SDSM) [27], [28], [29] with significant extensions to deal with the aforementioned three general types of non-uniform BC and/or CC along any of the line nodes of a composite plate assembly. These proposed enhancements have many engineering applications as mentioned earlier. Additionally, the theory inherits all the merits of the recently developed SDSM [27], [28], [29] including the high accuracy, computational efficiency and robustness. For instance, the current method has as much as 100-fold advantage of computational speed over the conventional finite element method. Any required natural frequency within low to high frequency ranges can be computed by the proposed theory with any desired accuracy. The theory is versatile and the composite plate assemblies can of course, be subjected to any arbitrary boundary conditions in the present method.

The SDS matrices for any arbitrarily distributed elastic supports and/or mass attachments and/or elastic coupling constraints are formulated in a concise manner and the theory is capable of handling simple as well as complex structures. Essentially, the development of such SDS matrices originates from expressing both sides of the corresponding constraint equations in terms of a modified Fourier series [30]. The developed SDS matrices of the above non-uniform BC and/or CC are superposed directly onto the corresponding components of the SDS matrix of the composite plate-like structure. The formulation procedure can be regarded as a series-based exact formulation in a sense similar to that of the SDSM [27], [28]. It should be noted that the current SDS theory is completely general so that it can be applied to different SDS formulations based on different governing differential equations (GDE), e.g. [27], [28], [29], [31]. Of course, the theory can be degenerated, as a special case, to be applied to classical dynamic stiffness method (DSM) based on different GDE, e.g., those in conjunction with refined theories like Carrera’s Unified formulation [32], [33], [34], [35], [36], for more details see Section 2.2.3. Moreover, the current theory can be used not only in modal analysis but also in dynamic response analysis, although the focus of this paper is on modal analysis. As the solution technique for modal analysis, an enhanced Wittrick–Williams algorithm [27], [28] is applied to the superposed SDS matrix to compute any required natural frequencies to any desired accuracy. For illustrative purposes, the current theory is applied to a number of engineering problems. It should be noted that all SDSM results presented in this paper (shown in bold) are accurate up to the last figures quoted and therefore, they can serve as benchmark solutions.

The paper is organised as follows. Section 2.1 reviews briefly the general framework and some properties of the spectral dynamic stiffness method (SDSM) developed in [27], [28], [29]. The spectral dynamic stiffness theory for non-uniform elastic supports and mass attachments is then presented in Section 2.2.1, which is followed by that for non-uniform elastic coupling constraints, see Section 2.2.2. The current SDS theory can also be degenerated and applied to the classical DSM, see Section 2.2.3. In Section 3.1, investigations on convergence and computational efficiency are carried out and the SDSM results are validated by published results. From Sections 3.2 to 3.4, three practical problems in different engineering areas are illustrated. Finally, significant conclusions are drawn in Section 4.

Section snippets

Theory

In essence, the purpose of this paper is to substantially extend a recently developed spectral dynamic stiffness method (SDSM) [27], [28], [29] to more general and diversified cases. Therefore, the basic framework and properties of the SDSM are briefly summarised below to provide the necessary background for the SDS development for arbitrary non-uniform elastic supports, mass attachments and elastic coupling constraints.

Results and applications

The theory developed above is implemented in a MATLAB program which computes exact natural frequencies and mode shapes of plates and plates assemblies with arbitrary non-uniformly distributed elastic supports and/or mass attachments and/or elastic coupling constraints. In Section 3.1, the convergence and computational efficiency analyses are performed, and then the current SDS theory is validated by published results from the literature. Next, three practical composite plate-like structures are

Conclusions

An exact spectral dynamic stiffness (SDS) theory has been developed for free vibration analysis of composite plate-like structures with arbitrary non-uniform elastic supports, mass attachments as well as elastic coupling constraints. The principal conclusions are:

  • (i)

    By using the modified Fourier series, the spectral dynamic stiffness (SDS) matrices for any arbitrarily distributed elastic supports, mass attachments and elastic coupling constraints are formulated analytically in exact conformity

Acknowledgements

The authors appreciate the support given by EPSRC, UK through a Grant EP/J007706/1 which made this work possible.

References (43)

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