IJRMET Vol. 7, IssuE 1, NoV 2016 - ApRIl 2017

w w w . i j r m e t . c o m InternatIonal Journal of research In MechanIcal engIneerIng & technology 21

Issn : 2249-5762 (online) | Issn : 2249-5770 (Print)

Effect of Skew Angle on Free Vibration Responses of Sandwich Composite Plate

1Pankaj Katariya, 2Dr. Subrata Kumar Panda, 3Trupti Ranjan Mahapatra 1,2Dept. of Mechanical Engineering, NIT Rourkela, Odisha, India

3School of Mechanical Engineering, KIIT University, Bhubaneswar, Odisha, India

AbstractIn the present article, the free vibration responses of the shear deformable skew sandwich composite plate is investigated numerically with the help of finite element method. The skew sandwich plate model is developed using higher-order shear deformable theory to avoid shear correction factor. The convergence and the validity of the model have been checked by solving different kinds of example available in published literature. From the comparison study, it can be easily understood that the presently proposed model is capable of solving the skew sandwich structures with adequate accuracy including the geometrical complexities. Subsequently, the model is extended to examine the effects of the various geometrical parameters such as thickness ratio, skew angle, lamination scheme and support condition on the vibration responses of the skew sandwich composite plate.

KeywordsSandwich Composite, Vibration, Skew Angle, Finite Element Analysis, HSDT

I. IntroductionThe vibration control in any structure made of classical or modern material is an important issue as a designer point of view. It is very well known that the modern materials, sandwich composite are highly preferable in industries due to their properties such as low density, less weight, high stiffness and strength in comparison to the conventional materials. These materials are highly recommended in modern structures like an aircraft and an air-carrier where the skew plates of sandwich composites are preferable despite of the mathematical difficulties encounter during their analysis. These skew plates are introduced as sub-structures in the form of oblique plates during the designing of the bridges and the swept wings of an aircraft. These structures are always exposed to various kinds of combined loadings though out their service period. This in turn results the generation of vibration and due to which the life of the structure or sutural member’s decreases and the structure may fail also. Hence, it is important to study the vibration behaviour of skew sandwich composite structures and predict the actual frequency parameters. The vibration responses of the sandwich composite structures have been already studied by various researchers in past by developing or modifying theories based on the analytical and/or numerical methods. Some of the relevant studies on vibration responses of sandwich composite structures with and without skew are presented here to discuss to fill the gap. The free vibration responses of thin composite skew plates are investigated by [1] developing a four-noded quadrilateral plate element using Finite Element Method (FEM) considering von-Karman type of strain relation. The free vibration and bending behaviour of the laminated soft core skew sandwich plates are analysed by [2-3] using a C0 Finite Element (FE) model based on the Higher-order Zigzag Theory (HOZT). The free vibration behaviour of skew sandwich plates composed of an orthotropic core and laminated facings is studied by adopting p-Ritz method

[4]. The exact solutions of the Higher-order Shear Deformation Theory (HSDT) used to study the free vibration and buckling behaviour of isotropic, orthotropic and laminated rectangular plates [5]. The vibration behaviour of the thick rectangular laminates with different support conditions are reported by [6] employing the p-Ritz method based on the first-order Reissner-Mindlin plate theory. The free vibration responses of thin-to-moderately thick composite and sandwich skew plates are analysed by [7-8] by using Differential Quadrature Method (DQM) and Moving Least Squares Differential Quadrature (MLSDQ) method, respectively in the framework of the first-order shear deformation theory (FSDT) by considering the von-Karman type of strain relation. The frequencies are obtained for composite plate using local and global higher-order theory and represented by [9-10]. The free vibration behaviour of the composite and sandwich plates is investigated by [11-12] using the higher-order individual-layer theory and global-local HSDT. The free vibration responses of composite curved shell panel under the combined loading investigated by [13-15] using Green-Lagrange type of strain relation in the framework of the HSDT. The vibration behaviour of the composite curved shell panel under hygro-thermal environment presented by [16-17] using Green-Lagrange type of strain relation in the framework of the FSDT kinematics. The free vibration responses of skew laminate and sandwich composite plate/shell structures are obtained using the FSDT [18-20].

Based on the available knowledge gap from the comprehensive review, the present article aims to develop the mathematical model to analyse the free vibration response of skew sandwich composite plate. There are no studies has been found yet on the free vibration analysis of skew sandwich composite plate for the laminate facing and different core using HSDT kinematic model to the best of authors’ knowledge. Also, few studies are found on the skew sandwich composite plates are using the FSDT. In addition to that a general mathematical model has been developed based on the HSDT mid-plane kinematics for skew sandwich composite plate. The nine-noded isoparametric Lagrangian element with ten degrees of freedom per node is employed for the discretisation of the domain for the numerical analysis purpose. In view of that to obtain the desired vibration responses a home-made computer code is developed in MATLAB environment in conjunction with the FEM model and convergence and validation behaviour have also checked. Finally, the vibration responses are computed for different geometrical parameters (the thickness ratios, the skew angles, the lamination schemes, and the support conditions) to show the importance of the presently proposed and developed higher-order mathematical model.

II. Mathematical ModellingIn the present work, skew sandwich composite plate is adopted as shown in Figure 1 to obtain the desired vibration responses. The dimension of the sandwich composite plate is length a, width b, thickness of the core hc, thickness of the face sheet hf and total

IJRMET Vol. 7, IssuE 1, NoV 2016 - ApRIl 2017 Issn : 2249-5762 (online) | Issn : 2249-5770 (Print)

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thickness is consider as h. The configuration of the skew plate is represented in fig. 2 where the skew angle is denoted as ϕ. The proposed mathematical model for skew sandwich composite plate is developed by using the higher-order mid-plane kinematics as [15]:

(1)

where, u, v and w are the displacement of any point within the plate along x, y and z directions, respectively. u0, v0 and w0 are the mid-plane displacement of any point within the panel along x, y and z directions, respectively. θx, θy and θz are the rotation of normal to the mid-plane and extension terms, respectively. The functions and are higher-order terms of Taylor series expansion in the mid-plane of the plate.

Now, the Eq. (1) can also be written in matrix form as:

(2)

The details of individual terms of Eq. (2) can be seen in [15].The FEM is widely accepted numerical tool for different kinds of the structural problem. A nine noded isoparametric element with ten Degrees Of Freedom (DOFs) per node is chosen for the discretization purpose and the displacement vector at any point on the mid-surface is given by:

(3)

The details can be seen in [15].

Fig. 1: Sandwich Configuration of Plate With Laminate Facings

Fig. 2: Skew Plate Configuration

Now, the transformed nodal co-ordinates in Cartesian coordinate system are defined using the transformation matrix [TS] as:

(4)

where, [Ts] is transformation matrix and can be seen in [2].The generalized strain-displacement relation is adopted to introduce the displacement behaviour within the plate as [21]:

(5)where, [T] is the thickness coordinate and is the mid-plane strain vector.The mid-plane strain vector in terms of nodal displacement vector can be written as:

(6)

where, [B] is product form of differential operators and shape functions in strain terms.The desired constitutive relation of any arbitrary kth layer of sandwich plate can be given by [21]:

(7)where, [Q]k, and are the reduced stress matrix, the stress vector and the strain vectors, respectively.The kinetic energy expression (T) of the sandwich composite plate can be expressed as:

(8)

where, , and are the density, displacement vector and the first-order differential of the displacement vector with respect to time, respectively.Now, the Eq. (2) is used in the kinetic energy Eq. (8) for ‘N’ number of the layered plate and the kinetic energy can be rewritten as:

(9)

where, is the inertia matrix.The strain energy (U) of skew sandwich composite plate can be expressed as:

IJRMET Vol. 7, IssuE 1, NoV 2016 - ApRIl 2017

w w w . i j r m e t . c o m InternatIonal Journal of research In MechanIcal engIneerIng & technology 23

Issn : 2249-5762 (online) | Issn : 2249-5770 (Print)

(10)

The generalized governing equation can be obtained using Hamilton’s principle, and can be expressed as:

(11)

Now, the equilibrium equation of motion of the skew sandwich composite plate can be expressed as:

2[ ] [ ] 0K M −Ω ∆ = (12)

where, Ω is the eigenvalue (frequency) of the free vibrated skew sandwich composite plate and Δ is the corresponding eigenvector (mode shapes).

III. Results and DiscussionThe vibration responses of skew sandwich composite plate are computed numerically using a computer code developed in MATLAB environment. The responses obtained are compared with the available published literature to check the efficacy as well as the validity of the presently developed model. The support conditions adopted in the present analysis are same as in [15].

A. Convergence and Validation StudyIn this section, the free vibration responses of skew sandwich composite plate are computed for different conditions. The convergence and validation study of the present developed mathematical model has been carried out for the free vibration responses of skew sandwich composite plate. For the analysis purpose, two different lamination schemes are considered i.e., (0°/90°/C/90°/0°) and ([0°/90°]4/C/ [90°/0°]4), where C represents the core of the sandwich as well the skew angle is considered as ϕ = 0° and 15° and the aspect ratio is a/b = 2. The responses are obtained for all edges simply-supported (SSSS) plate with thickness of the core material is considered as 0.8h, and thickness ratio (a/h) is taken as 20 using the same material property and the non-dimensional formula same as in [4]. The responses obtained are presented in Table 1 for both convergence and validation study and the results are compared with the available published literature.

Table 1: Convergence and Validation Study of Simply-Supported Skew Sandwich Composite Plate

No. of elements[0°/90°/C/90°/0°] [0°/90°]4/C/[90°/0°]4

Ω(2×2) 13.6733 14.2940 13.3182 14.1565(3×3) 13.1020 13.1842 12.7506 13.1871(4×4) 12.9488 12.6786 12.5965 12.7214(5×5) 12.8943 12.4243 12.5412 12.4845(6×6) 12.8713 12.2782 12.5178 12.3485(7×7) 12.8605 12.1863 12.5088 12.1015

Ref. [4] 12.063 12.767 12.489 13.225

It is observed from the table the present results are converging well with the mesh refinement and validated with published literature [4]. It is also important to mention that, a (6×6) mesh size is sufficient enough to compute the vibration responses of skew sandwich plate and the same is used in the further analysis.

B. Numerical ExamplesBased on the above convergence and validation study the present developed numerical FE model is further extended to study the effect of various geometrical parameters on the free vibration behaviour of sandwich plate. In order to show the effect of the different geometrical parameters on the free vibration responses of skew sandwich composite plate; two vibration problems are solved and the effects are briefly discussed here. The vibration responses are obtained for skew sandwich composite plate with two different lamination schemes i.e., (0°/90°/C/0°/90°) and ([0°/90°]2/C/[0°/90°]2) for two different support conditions i.e., all edges simply-supported (SSSS) and all edges clamped (CCCC). The responses are obtained for various thickness ratios and skew angles and presented in figures 3-4. It is noticed from the Figures 3-4, that the non-dimensional fundamental frequency decreases with an increase in the thickness ratios. It is also observed from the figures that the frequency values decreases with an increase in the skew angles. It is very well known that the stiffness of the structure changes as the support condition changes and it is observed from the responses that the frequency values decreases with change in support conditions from CCCC to SSSS. It is very well known that the lamination schemes also affect the frequency and it is observed from the figures that the frequency values increases with an increase in the laminate facings of sandwich plate.

Fig. 3: Non-dimensional Fundamental Frequency of Square (0°/90°/C/0°/90°) Skew Sandwich Plate

Fig. 4: Non-dimensional Fundamental Frequency of Square ([0°/90°]2/C/[0°/90°]2) Skew Sandwich Plate

IJRMET Vol. 7, IssuE 1, NoV 2016 - ApRIl 2017 Issn : 2249-5762 (online) | Issn : 2249-5770 (Print)

w w w . i j r m e t . c o m 24 InternatIonal Journal of research In MechanIcal engIneerIng & technology

IV. ConclusionThe present study deals with the free vibration behaviour of skew sandwich composite plate structure. As a first step, a general mathematical model based on the HSDT kinematics is developed for skew sandwich composite plate in conjunction with the FEM. In addition to that, an isoparametric quadrilateral Lagrangian element with 90 DOFs per element is employed for the discretisation of the proposed domain. It is also important to mention that all the nonlinear higher-order terms of the in-plane and out of plane stress and strain terms are included in the formulation in order to achieve the accurate response. The variational approach is used to derive the system governing equations. Further, a home-made computer code has been developed in MATLAB environment with the help of the proposed mathematical model and the convergence behaviour is checked and it is understood that a (6×6) mesh size is sufficient enough to compute the responses. In addition to that, the responses obtained are also validated with the available published literature. Finally, some examples are solved by varying the geometrical parameters (thickness ratio, lamination scheme, skew angle, and support condition) and based on that few points are drawn out:

The non-dimensional fundamental frequencies decreases with • an increase in the thickness ratios this is because of the fact the thickness ratios increase so the structure becomes thin and the stiffness of the plate changes. The skew angle also affects the frequency values greatly • and it is observed that the frequency values decreases with increase in skew angle. The support conditions as well as the lamination schemes • also affect the vibration responses significantly.

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