Large amplitude free flexural vibrations of laminated composite skew plates

https://doi.org/10.1016/j.ijnonlinmec.2004.04.003Get rights and content

Abstract

Here, the large amplitude free flexural vibration behaviors of thin laminated composite skew plates are investigated using finite element approach. The formulation includes the effects of shear deformation, in-plane and rotary inertia. The geometric non-linearity based on von Karman's assumptions is introduced. The non-linear governing equations obtained employing Lagrange's equations of motion are solved using the direct iteration technique. The variation of non-linear frequency ratios with amplitudes is brought out considering different parameters such as skew angle, number of layers, fiber orientation, boundary condition and aspect ratio. The influence of higher vibration modes on the non-linear dynamic behavior of laminated skew plates is also highlighted. The present study reveals the redistribution of vibrating mode shape at certain amplitude of vibration depending on geometric and lamination parameters of the plate. Also, the degree of hardening behavior increases with the skew angle and its rate of change depends on the level of amplitude of vibration.

Introduction

The increased utilization of thin-walled structural components with relatively low flexural rigidity in the design of space vehicles and their vibration characteristics at large amplitudes in response to the conditions they are subjected to, have attracted the attention of many researchers in recent years. These studies are reviewed and are well documented by Leissa [1], [2], Bert [3], Sathyamoorthy [4], Chia [5], and Kapania and Raciti [6]. It is observed from the existing literature that the large amplitude flexural vibration of rectangular and circular plates has received considerable attention of the researchers. However, limited work has been focused on the geometrically non-linear free vibration analysis of plates other than rectangular/circular plates.

The plates with non-rectangular plan-forms like skew plates find wide application in the aerospace industry. Though a huge body of literature exists on the linear free vibration of isotropic and single layer orthotropic skew plates, such analysis concerning the laminated composite skew plates has received relatively little attention, despite the increasing use of such components in aircraft [7]. The notable recent contributions pertaining to linear vibration behavior of laminated composite skew plates are dealt with [8], [9], [10], [11], [12], [13], [14], [15], [16]. Iyengar and Umaretiya [8] have studied a hybrid laminated skew plate using Galerkin's method. Malhotra et al. [9] have carried out the vibration of rhombic orthotropic plates using parallelogrammic orthotropic plate finite element whereas, Krishnan and Deshpande [10] have examined the free vibration of thin cantilever skew laminates employing finite element method, based on discrete Kirchhoff theory. Kapania and Singhvi [11], and Anlas and Goker [12] have analytically solved the problem using Ritz method. Wang [13] has studied using B-spline Rayleigh–Ritz method, while Han and Dickinson [14] have applied a Ritz Hierarchical finite element in analyzing the laminated skew plates. A numerical approach with assumed solution considering Green function is introduced in the work of Hosokawa et al. [15]. Reddy and Palaninathan [16] have dealt with the problem using high precision plate bending finite element. An attempt is also made to study the vibration characteristics of skew sandwich plates with laminated facings by Makhecha et al. [17] and Wang et al. [18]. It may be concluded that most of these work are carried out based on classical plate theory. Further, the non-linear free vibration of skew plates is treated sparsely in the literature.

Nowinski [19] and most recently Ray et al. [20] have analytically studied the non-linear vibration behavior of isotropic skew plates. Sathyamoorthy and Pandalai [21] have analyzed single layer orthotropic skew plates for movable in-plane edges and Berger approximation for immovable in-plane edge conditions. Prathap and Varadan [22], and Sathyamoorthy and Chia [23] have investigated large amplitude vibration of anisotropic skew plates using the Galerkin method on the basis of a single term assumed vibration mode. All these investigations introduce assumed mode while solving the problem. It is known that one-term approximation for the free vibration mode is insufficient for rectangular as well as skew plates, and such results lose accuracy with increase in aspect ratio and skew angles. It may be further observed that the geometrically non-linear free vibration of higher modes of structures including rectangular seems to be scarce in the literature [24]. Also, to the best of the authors’ knowledge, the work on the non-linear free flexural vibrations of laminated anisotropic composite skew plates is not commonly yet available in the literature.

For accurate solution, the assumed mode shape in the analytical approach should have more terms that lead to unwieldy algebraic and more numerical work. Numerical methods such as finite element procedure is preferable over the analytical methods, while studying the large amplitude dynamic analysis of structures, as there is no need for an a priori assumption of the mode shapes, and the solution itself predict the mode shapes [25]. Such analysis for skew plates appears to be lacking in the literature.

In the present paper, a four-noded shear flexible quadrilateral high precision plate bending element developed recently [26], [27] is extended to analyze the large amplitude vibrations of thin laminated composite skew plates. As the element is free from locking phenomenon, all the energy terms are evaluated using full numerical integration scheme. The formulation includes in-plane and rotary inertia effects. Furthermore, the present study makes use of incremental matrices to represent the geometrical non-linearity. The non-linear governing equations obtained here are solved using direct iteration technique where the linear mode shape is taken as the starting vector. The amplitude–frequency relationships are predicted. While calculating the results, necessary convergence criteria [28] are satisfied for the displacement vector and frequency value. The formulation developed here is validated with the available analytical results. A detailed parametric study, to bring out the influences of the skew angle, lay-up, aspect ratio, and boundary conditions on the non-linear free vibration behavior of laminated skew plates has been carried out. Further, the significance of non-linear free vibration of higher modes is investigated in the present analysis.

Section snippets

Formulation

Fig. 1 shows the rectangular Cartesian co-ordinate system along with the associated covariant base vectors (g1,g2,g3) and contravariant base vectors (1g,2g,3g) for the skew plate having a and b as the length and width, and ψ as the skew angle. (g1,g2,g3) and (1g,2g,3g) are related to the rectangular Cartesian unit base vectors (e1,e2,e3) byg1=e1,g2=(sinψ)e1+(cosψ)e2,g3=e3and1g=e1−(tanψ)e2,2g=(secψ)e2,3g=e3.

The covariant components of the displacement vector u (u=u11g+u22g+u33g) of a shear

Solution procedure

The vibration problem is solved using eigenvalue formulation. To solve the non-linear eigenvalue problems, an iterative procedure is used. Firstly, the eigenvector (mode shape) is obtained from the linear vibration analysis, neglecting the non-linear stiffness matrix in Eq. (12) and then normalized. Next, the normalized vector is amplified/scaled up so that the maximum displacement is equal to the desired amplitude, say w/h=0.4 (w is the maximum lateral displacement, h is the thickness of the

Results and discussion

The study, here, has been focused on the large amplitude free flexural vibration characteristics of thin laminated composite skew plates. The material properties, unless specified otherwise, used in the present analysis areEL/ET=40.0,GLT/ET=0.6,GTT/ET=0.5,νLT=0.25,where E, G and ν are Young's modulus, shear modulus and Poisson's ratio. Subscripts L and T represent the longitudinal and transverse directions respectively with respect to the fibers. All the layers are of equal thickness. The

Conclusions

Large amplitude free vibration of composite skew plate has been investigated using a four-noded shear flexible quadrilateral high precision plate bending element. The present formulation accounts for moderately finite deformation of the plate. Numerical studies are conducted here to examine the effect of skew angle, fiber angle orientation, aspect ratio, and boundary conditions on the large-amplitude frequency and the mode shapes of composite skew plate. Some general observations are made as

References (33)

Cited by (0)

View full text