Tolerance modelling of vibrations of periodic three-layered plates with inert core

doi:10.1016/j.compstruct.2015.08.123

Composite Structures

Volume 134, 15 December 2015, Pages 854-861

https://doi.org/10.1016/j.compstruct.2015.08.123 Get rights and content

Abstract

In this note a free vibration analysis of periodic three-layered sandwich structures is performed. Basing on the Kirchhoff’s thin plate theory simplified equations of motion are derived, which are characterised by highly-oscillating, periodic and non-continuous coefficients. In order to obtain a system of equations with constant coefficients, the tolerance averaging technique is used. An application of the proposed tolerance model to analyse free vibration frequencies of a three-layered plate strip is shown – for both lower order frequencies related to its macrostructure and higher order frequencies related to its microstructure. Some comparisons of results of lower frequencies, obtained in the tolerance, the asymptotic and the known homogenised models are presented. Moreover, a certain verification of the proposed model is performed using the Ritz method. It can be observed that the tolerance model can be successfully applied to analyse vibration problems of vast variety of periodic three-layered plates and can significantly improve the optimisation process of such structures.

Introduction

There are many reasons why composite structures are becoming more and more vital for modern engineering. As state of the art technology allows us to combine several different materials into one heterogeneous structure, characterised by physical and mechanical properties, which are unreachable for classic materials, it becomes crucial to develop useful tool for optimising these properties for special engineering purposes.

In this article three-layered ‘sandwich’ composite structures are considered. Investigations on behaviour of such structures have their beginnings in the middle of 20th century and since then the design and optimisation process has been much improved. The analysis of dynamical behaviour of sandwich structures can be found in works of Chonan [1], Oniszczuk [2], [3] and Szcześniak [4], [5], among others. As it became clear that the shape of the core of sandwich structures is of great importance for mechanical properties of the whole structure, many researchers investigate this relation. Hence, many different concepts of periodic (e.g. honeycomb, rectangular, wavy-type, cf. Massimo and Panos [6]) or quasi-periodic (e.g. aluminium or metal foam, cf. Jin-Yih et al. [7], Magnucki and Jasion [8], Grygorowicz et al. [9], Jasion et al. [10]) cores have been presented and the need for a convenient mathematic model of periodic structures has emerged.

Some propositions to describe discrete periodic structures was performed by Brillouin [11], where the vibration analysis of atomic lattice was investigated. Basing on his work several discrete and continuous models of a wave propagation in periodic structures were proposed, e.g. by Mead [12]. A different approach was proposed by Kohn and Vogelius [13], who presented the homogenisation method for periodic plates, which special application was used recently e.g. by Wen-ming et al. [14]. However, governing equations of these methods usually neglect the influence of the microstructure on behaviour of these plates, which in certain engineering cases can prove to be insufficient.

With the development of computers, the finite element method (FEM) become one of the most popular approaches to analyze periodic structures. One should mention the work by Zhi-Jing et al. [15], which shows a vibration analysis of periodic plates using a spectral element method, being a special application of FEM, investigations of Massimo and Panos [6] of wave propagation in sandwich plates with periodic honeycomb core or numerical analysis of vibrations of periodic plates by Yuanwu et al. [16], using the asymptotic homogenisation method. Since the use of FEM for vibration analysis of various periodic structures is much time-consuming, different analytical solutions are proposed.

In this note the analytical solution to a vibration analysis of periodic three-layered structures with an inert core is presented and discussed. Basing on the simplified model, shown by Szcześniak [4], governing equations of motion with coefficients being periodic, non-continuous and highly-oscillating functions are obtained. In order to derive a system of equations with constant coefficients, which take into consideration the effect of the microstructure on the behaviour of the whole structure, the tolerance averaging technique, proposed by Woźniak et al. [17], [18], is applied. Eventually, the obtained solutions are compared to results by the asymptotic and the homogenised models. A certain physical correctness of the proposed model is also shown using the Ritz method.

Section snippets

Modelling foundations

Let Ox₁x₂x₃ be an orthogonal Cartesian coordinate system, t – a time coordinate and $x \equiv (x_{1}, x_{2})$ . The considered structure is assumed to have spans L₁ and L₂ in x₁- and x₂-axis directions, respectively. Hence, its midplane is defined as $Δ \equiv [0, L_{1}] \times [0, L_{2}]$ . By setting $z \equiv x_{3}$ the undeformed plate occupies the region $Λ \equiv {(x, z) : - H (x) / 2 ⩽ z ⩽ H (x) / 2, x \in Δ}$ , where H(x) is a total thickness of the plate.

The outer layers of the considered structure are Kirchhoff’s type thin plates, which are assumed to be symmetric

Basic concepts of the tolerance averaging technique

The tolerance averaging technique was described and developed by Woźniak et al. in a numerous books and publications [17], [18]. Various applications of the method were presented in a series of papers, e.g. by Jędrysiak [19], [20], Jędrysiak and Michalak [21], [22] or Domagalski and Jędrysiak [23]. In the tolerance averaging technique, several introductory concepts are applied, defined below.

Denote a cell at $x \in Λ_{Ω}$ by $Ω (x) \equiv x + Ω$ , and $Λ_{Ω} \equiv Λ \cap \cup_{z \in Ω (x)} Ω (z)$ . The definition of the averaging operation for

The tolerance model

The starting point of the tolerance modelling procedure is the system of Eq. (2). By applying the averaging operator to these equations and transforming them using both the micro–macro decomposition and the tolerance averaging approximations, governing equations with constant coefficients are obtained in the form: $\begin{matrix} \partial_{α β} ({\tilde{B}}_{α β γ δ} \partial_{γ δ} W_{1} + {\tilde{B}}_{1 α β}^{A} v_{1}^{A}) + \tilde{M} {\ddot{W}}_{1} + \tilde{K} (W_{1} - W_{2}) + \underset{̲}{{\tilde{K}}_{1}^{A}} v_{1}^{A} - \underset{̲}{{\tilde{K}}_{2}^{B}} v_{2}^{B} = {\tilde{p}}_{1}, \\ {\tilde{B}}_{1 α β}^{A} \partial_{γ δ} W_{1} + {\tilde{B}}_{1}^{AB} v_{1}^{A} + \underset{̲}{{\tilde{M}}_{1}^{AB}} {\ddot{v}}_{1}^{A} + \underset{̲}{{\tilde{K}}_{1}^{B}} (W_{1} - W_{2}) + \underset{̲}{{\tilde{K}}_{11}^{AB}} v_{1}^{A} - \underset{̲}{{\tilde{K}}_{21}^{AB}} v_{2}^{A} = \underset{̲}{{\tilde{p}}_{1}^{B}}, \\ \partial_{α β} ({\tilde{B}}_{α β γ δ} \partial_{γ δ} W_{2} + {\tilde{B}}_{2 α β}^{A} v_{2}^{A}) + \tilde{M} {\ddot{W}}_{2} + \tilde{K} (W_{2} - W_{1}) + {\tilde{K}}_{2}^{A} \end{matrix}$

The calculation example – free vibrations of plate strips

In this section, the analytical solution to free vibration analysis for periodic three-layered plate strips is presented. In order to evaluate the obtained results, all calculations are performed for: the tolerance, the asymptotic and the homogenised plate model.

Let us consider the structure with geometry presented in Figs. 2 and 3. The plate under consideration can be treated as independent of x₂ coordinate. Hence, it is one-dimensional problem. It is assumed, that both upper and lower plates

Physical correctness of the proposed tolerance model

In this section certain physical correctness of the tolerance model is presented. As higher order vibrations cannot be analysed using neither the asymptotic model nor the homogenised model, the known Ritz method can be applied to derive formulas for higher order frequencies.

Let us consider the plate strip with geometry presented in Fig. 8. The problem of vibrations of the plate under consideration can be treated as independent of x₂ coordinate, hence it is a one-dimensional problem, in which $x \equiv x$

Remarks

In this paper, the tolerance averaging technique is used to calculate the free vibration frequencies of the three-layered composite plate with certain periodic microstructure. Using this modelling procedure the initial system of partial differential equations with highly oscillating, periodic, functional coefficients can be replaced by a system of partial differential equations with constant coefficients, which can be easily solved using methods similar to those used for homogeneous plates.

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Tolerance modelling of vibrations of periodic three-layered plates with inert core

Abstract

Introduction

Section snippets

Modelling foundations

Basic concepts of the tolerance averaging technique

The tolerance model

The calculation example – free vibrations of plate strips

Physical correctness of the proposed tolerance model

Remarks

J Sound Vibr

J Sound Vibr

Thin-Walled Struct

Thin-Walled Struct

J Sound Vibr

Int J Solids Struct

Thin-Walled Struct

Thin-Walled Struct

Thin-Walled Struct

Dynamical behaviour of elastically connected double-beam systems subjected to an impulsive load

Bull JSME

Vibration of elastic sandwich and elastically connected double-plate system under moving loads

Publ Warsaw Univ Technol

Vibration of elastic sandwich and elastically connected double-beam system under moving loads

Publ Warsaw Univ Technol