Effect of skew angle on free vibration responses of sandwich composite plate

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1 Effect of skew angle on free vibration responses of sandwich composite plate 1 Pankaj Katariya, 2* Subrata Kumar Panda, 3 Trupti Ranjan Mahapatra ( 1 Research scholar, Department of Mechanical Engineering, NIT Rourkela, Odisha, India.) ( 2 Assistatnt professor, Department of Mechanical Engineering, NIT Rourkela, Odisha, India.) 3 Assistant professor, School of Mechanical Engineering, KIIT University, Bhubaneswar, Odisha, India (pk.pankajkatariya@gmail.com, * call2subrat@gmail.com, trmahapatrafme@kiit.ac.in ) Abstract: - In the present article, the free vibration behaviour of the shear deformable skew sandwich composite plate is examined numerically by using the finite element technique. The plate model is created utilizing higher-order shear deformable hypothesis to evade the shear correction factor. The convergence and the validation of the model have been studied by measuring various types of case accessible in literature. From the examination study, it can be effectively assumed that the currently suggested and established model is equipped for explaining the skew sandwich structures with satisfactory precision including the geometrical complexities. Hence, the model is stretched out to analyze the impacts of the different geometrical parameters, for example, thickness proportion, skew angle, lamination sequence, and end condition on the vibration behavior of the skew sandwich composite plate. Keywords: Sandwich composite, vibration, skew angle, finite element analysis, HSDT I. Introduction The vibration control in any structure made of established or current material is an essential issue as a creator perspective. It is exceptionally notable that the present day materials, sandwich composite are very best in enterprises because of their properties, for example, low thickness, less weight, high solidness and quality in contrast with the traditional materials. These materials are exceptionally prescribed in cutting edge structures like an air ship and an air-bearer where the skew plates of sandwich composites are ideal in spite of the scientific challenges experience amid their investigation. These skew plates are presented as sub-structures as sideways plates amid the planning of the scaffolds and the swept wings of an air ship. These structures are constantly presented to different sorts of combined loadings in their working tenure. Therefore it starts generating the vibration and because of which the life of the structure or sutural part's reduces and the structure may not work moreover. Henceforth, it is essential to investigate the vibration responses of skew sandwich composite structures and calculate the accurate frequency parameters. The vibration behavior of the sandwich composite configurations has been now contemplated by different scientists in the past by creating or adjusting methods and additionally numerical strategies. A portion of the pertinent reviews on vibration behavior of sandwich composite structures with and without skew are demonstrated here to examine to fill the gap. The free vibration analysis of thin composite skew plates are studied by [1] building up a fournoded quadrilateral plate component utilizing finite element method (FEM) considering von- Karman kind of strain relation. The free vibration and bending responses of the covered delicate center skew sandwich plates are dissected by [2-3] utilizing a C 0 finite element (FE) display in view of the higher-order zigzag theory (HOZT). The free vibration responses of skew sandwich plates made out of an orthotropic center and overlaid facings is considered by embracing p-ritz strategy [4]. The correct arrangements of the higher-order shear

2 deformation theory (HSDT) used to concentrate the free vibration and bending responses of isotropic, orthotropic and rectangular plates [5]. The vibration responses of the thick rectangular laminates with various end conditions are accounted by [6] utilizing the p-ritz technique in view of the main request Reissner-Mindlin plate hypothesis. The free vibration behavior of thinto-reasonably thick composite and sandwich skew plates are investigated by [7-8] by utilizing differential quadrature method (DQM) and moving slightest squares differential quadrature (MLSDQ) technique, separately in the structure of the first-order shear deformation theory (FSDT) by considering the von-karman sort of strain connection. The frequencies are acquired for composite plate utilizing neighborhood and worldwide higher-arrange hypothesis and spoke to by [9-10]. The free vibration conduct of the composite and sandwich plates is examined by [11-12] utilizing the higher-arrange singular layer hypothesis and worldwide nearby HSDT. The free vibration reactions of composite bended shell board under the consolidated stacking researched by [13-15] utilizing Green-Lagrange sort of strain connection in the structure of the HSDT. The vibration conduct of the composite bended shell board under hygro-warm environment exhibited by [16-17] utilizing Green-Lagrange kind of strain connection in the system of the FSDT kinematics. The free vibration reactions of skew overlay and sandwich composite plate/shell structures are gotten utilizing the FSDT [18-20]. In light of the accessible information crevice from the extensive survey, the present article expects to build up the scientific model to dissect the free vibration reaction of skew sandwich composite plate. There are no reviews has been found yet on the free vibration examination of skew sandwich composite plate for the overlay confronting and diverse center utilizing HSDT kinematic model to the best of creators' information. Additionally, few reviews are found on the skew sandwich composite plates are utilizing the FSDT. Notwithstanding that a general scientific model has been created in view of the HSDT mid-plane kinematics for skew sandwich composite plate. The nine-noded isoparametric Lagrangian component with ten degrees of flexibility for every hub is utilized for the discretization of the area for the numerical examination reason. In perspective of that to acquire the sought vibration reactions a homemade PC code is created in MATLAB environment in conjunction with the FEM model and joining and approval conduct have likewise checked. At last, the vibration reactions are figured for various geometrical parameters (the thickness proportions, the skew edges, the cover plans, and the bolster conditions) to demonstrate the significance of the by and by proposed and created higher-order model. II. Mathematical Modelling In the current work, skew sandwich composite plate is embraced as seemed in Figure 1 to get the desirable vibration behaviour. The measurement of the sandwich composite plate is length a, width b, core thickness is hc, thickness of the face sheet hf and whole plate thickness is consider as h. The setup of the skew plate is shown in Figure 2 where the skew edge angle is meant as ϕ. The proposed numerical model for skew sandwich composite plate is created by utilizing the higher-order mid-plane kinematics as [15]: (1) where, u, v and w are the dislodging of any point inside the plate along x, y and z headings, individually. u0, v0 and w0 are the mid-plane uprooting of any point inside the board along x, y and z bearings, individually. x, y and z are the pivot of ordinary to the mid-plane and expansion terms, separately. The capacities u0 *, v0 *, x * and * y are higher-order terms of Taylor arrangement development 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 broadly acknowledged numerical apparatus for various types of the basic issue. A nine noded isoparametric component with ten degrees of flexibility (DOFs) per hub is

3 decided for the discretization reason and the relocation vector 0 anytime on the midsurface is given by: The details can be seen in [15]. Figure 1: Sandwich configuration of plate with laminate facings. Figure 2: Skew plate configuration (3) Presently, the changed nodal co-ordinates in Cartesian arrange framework is characterized utilizing the change network [TS] as: (4) where, [Ts] is change grid and can be seen in [2]. The summed up strain-relocation connection is embraced to present the removal conduct inside the plate as [21]: (5) where,[t] is the thickness arrange and is the mid-plane strain vector. The mid-plane strain vector as far as nodal removal vector can be composed as: (6) where, [B] is item type of differential administrators and shape works in strain terms. The required constitutive connection of any discretionary k th layer of sandwich plate can be given by [21]: (7) where, [Q] k, k and k are the diminished anxiety framework, the anxiety vector and the strain vectors, individually. The kinetic energy equation (T) of the sandwich composite plate can be communicated as: where,, * 0 and * 0 (8) are the thickness, uprooting vector and the first-order differential of the relocation vector regarding time, individually. Presently, the Eq. (2) is utilized as a part of the kinetic energy Eq. (8) for "N" number of the layered plate and the kinetic energy can be revamped as: where, m zk T k zk 1 f f dz is the inertia matrix. (9) The strain energy (U) of skew sandwich plate can be expressed as: T dv (10) U 0.5 v The generalized governing equation can be obtained using Hamilton s principle, and can be expressed as: ( T U) dt 0 (11) Now, the equilibrium equation of motion of the skew sandwich composite plate can be expressed as:

4 2 [ K] [ M] { } 0 (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 discussion The vibration behavior of skew sandwich composite plate is figured numerically utilizing a PC code created in MATLAB environment. The results acquired are contrasted with the accessible distributed writing with check the convergence and additionally the validation of the current established model. The end conditions embraced in the present investigation are same as in [15]. A. Convergence and validation study In this area, the free vibration behavior of skew sandwich composite plate is figured for various situations. The convergence and validation investigation of the present formed scientific model has been completed for the free vibration analysis of skew sandwich composite plate. For the investigation reason, two distinctive overlay condition are considered i.e., (0 /90 /C/90 /0 ) and ([0 /90 ]4/C/[90 /0 ]4), where C speaks to the core of the sandwich and the skew edge is considered as ϕ = 0 and 15 and the aspect ratio is a/b = 2. The end conditions are acquired for all edges basically simply supported (SSSS) plate with thickness of the center material is considered as 0.8h, and thickness ratio (a/h) is taken as 20 utilizing a similar material property and the non-dimensional equation same as in [4]. The reactions got are displayed in Table 1 for both meeting and approval examine and the outcomes are contrasted and the accessible distributed writing. It is seen from the table the present readings are meeting great with the work refinement and validating with distributed writing [4]. It is likewise vital to say that, a (6 6) mesh size is sufficiently adequate to process the vibration analysis of skew sandwich plate and the same is utilized as a part of the further examination. B. Numerical examples In light of the above convergence and validation examination the present generated numerical FE model is further stretched out to concentrate the impact of different geometrical parameters on the free vibration conduct of sandwich plate. Keeping in mind the end goal to demonstrate the impact of the diverse geometrical parameters on the free vibration behavior of skew sandwich composite plate; two vibration problems are tackled and the impacts are basically talked about here. The vibration behavior are studied for skew sandwich composite plate with two distinctive lamination schemes i.e., (0 /90 /C/0 /90 ) and ([0 /90 ]2/C/[0 /90 ]2) for two diverse end conditions i.e., all edges simply supported (SSSS) and all edges clamped (CCCC). The results are acquired for different thickness ratios and skew angles and introduced in Figures 3-4. It is seen from the Figures 3-4, that the non-dimensional frequencies diminish with an expansion in the thickness ratios. It is additionally seen from the results that the recurrence values diminish with an expansion in the skew edges. It is exceptionally notable that the stiffness of the structure changes as the end condition changes and it is seen from the results that the vibrational frequency values diminishes with change in end conditions from CCCC to SSSS. It is extremely outstanding that the lamination scheme conspires likewise influence the frequency and it is seen from the results that the vibrational frequency values increases with an expansion in the overlay facings of sandwich plate. IV. Conclusion The present analysis depicts the free vibration responses of skew sandwich composite plate structure. As an initial step, a general numerical model in light of the HSDT kinematics is created for skew sandwich composite plate in conjunction with the FEM. Notwithstanding that, an isoparametric quadrilateral Lagrangian component with 90 DOFs per component is

5 utilized for the discretisation of the proposed space. It is additionally essential to say that all the nonlinear higher-order terms of the in-plane and out of plane anxiety terms are incorporated into the definition keeping in mind the end goal to accomplish the exact reaction. Non-dimensional fundamental frequency () (0/90/C/0/90) SSSS CCCC a/h = 0 a/h = 0 a/h = 15 a/h = 15 a/h = 45 a/h = Thickness ratio (a/h) Figure 3 Non-dimensional fundamental frequency of square (0 /90 /C/0 /90 ) skew sandwich plate Notwithstanding that, the results got are likewise validated with the accessible distributed writing. At long last, a few cases are investigated by changing the geometrical parameters (thickness ratios, lamination type, skew angle, and end condition) and in light of that few focuses are drawn out: The non-dimensional essential frequencies diminishes with an expansion in the thickness ratios this is a direct result of the reality the thickness ratios increment so the structure turns out to be thin and the stiffness of the plate changes. The skew angle likewise influences the frequency values incredibly and it is noticed that the frequency values diminishes with increment in skew angle. The end conditions and in addition the lamination schemes additionally influence the vibration results essentially. V. References Non-dimensional fundamental frequency () ([0/90 /C/[0/90] 2 ) SSSS CCCC a/h = 0 a/h = 0 a/h = 15 a/h = 15 a/h = 45 a/h = Thickness ratio (a/h) Figure 4 Non-dimensional fundamental frequency of square ([0 /90 ] 2/C/[0 /90 ] 2) skew sandwich plate The variational approach is utilized to obtain the system generalized equations. In addition, a home-made PC code has been produced in MATLAB environment with the assistance of the proposed scientific model and the convergence behavior is checked and it is comprehended that a (6 6) mesh size is sufficiently adequate to found out the results. [1] Singha, M.K., Daripa, R., Nonlinear vibration of symmetrically laminated composite skew plates by finite element method, International Journal of Non-Linear Mechanics, Vol. 42, 2007, pp: [2] Chalak, H.D., Chakrabarti, A., Sheikh, A.H., Iqbal, M.A., C0 FE model based on HOZT for the analysis of laminated soft core skew sandwich plates: Bending and Vibration, Applied Mathematical Modelling, Vol. 38, 2014, pp: [3] Pandey, S., Pradyumna, S., A new C0 higher-order layerwise finite element formulation for the analysis of laminated and sandwich plates, Composite Structures, Vol. 131, 2015, pp: [4] Wang, C.M., Ang, K.K., Yang, L., Free vibration of skew sandwich plates with laminated facings, Journal of Sound and Vibration, Vol. 235, 2000, pp: [5] Reddy, J.N., Phan, P.H., Stability and vibration of isotropic, orthotropic and laminated plates according to a higher order shear deformation theory, Journal of Sound and Vibration, Vol. 89, 1985, pp:

6 [6] Liew, K.M., Solving the vibration of thick symmetric laminates by Reissner-Mindlin plate theory and the p-ritz method, Journal of Sound and Vibration, Vol. 198, 1996, pp: [7] Malekzadeh, P., Differential quadrature large amplitude free vibration analysis of laminated skew plates based on FSDT, Composite Structures, Vol. 83, 2008, pp: [8] Liew, K.M., Huang, Y.Q., Reddy, J.N., Vibration analysis of symmetrically laminated plates based on FSDT using the moving least squares differential quadrature method, Computer Methods in Applied Mechanics and Engineering, Vol. 192, 2003, pp: [9] Wu, C.P., Chen, W.Y., Vibration and stability of laminated plates based on a local higher-order plate theory, Journal of Sound and Vibration, Vol. 177, 1994, pp: [10] Matsunaga, H., Vibration and stability of cross-ply laminated composite plates according to a global higher-order plate theory, Composite Structure, Vol. 48, No. 4, 2000, pp: [11] Cho, K.N., Bert, C.W., Striz, A.G., Free vibration of laminated rectangular plates analyzed by higher-order individual-layer theory, Journal of Sound and Vibration, Vol. 145, 1991, pp: [12] Zhen, W., Wanji, C., Free vibration of laminated composite and sandwich plates using global-local higher-order theory, Journal of Sound and Vibration, Vol. 298, 2006, pp: [13] Panda, S.K., Singh, B.N., Nonlinear free vibration of spherical shell panel using higher order shear deformation theory A finite element approach, International Journal of Pressure Vessels and Piping, Vol. 86, 2009, pp: [14] Mahapatra, T.R., Kar, V.R., Panda, S.K., Nonlinear free vibration analysis of laminated composite doubly curved shell panel in hygrothermal environment, Journal of Sandwich Structures and Materials, Vol. 0 No. 0, 2015, pp: [15] Katariya, P.V., Panda, S.K., Thermal buckling and vibration analysis of laminated composite curved shell panel, Aircraft Engineering and Aerospace Technology: An International Journal, Vol. 88, No. 1, 2016, pp: [16] Naidu, N.V.S., Sinha, P.K., Nonlinear free vibration analysis of laminated composite shells in hygrothermal environments, Composite Structures, Vol. 77, 2007, pp: [17] Naidu, N.V.S., Sinha, P.K., Nonlinear finite element analysis of laminated composite shells in hygrothermal environments, Composite Structures, Vol. 69, 2005, pp: [18] Singh, G., Raju, K.K., Rao, G.V., Iyengar, N.G.R., Non-linear vibrations of simply supported rectangular cross-ply plates, Journal of Sound and Vibration, Vol. 142, No. 2, 1990, pp: [19] Wang, S., Free vibration analysis of skew fibre-reinforced composite laminates based on first-order shear deformation plate theory, Computers & Structures, Vol. 63, 1997, pp: [20] Nguyen-Van, H., Mai-Duy, N., Karunasena, W., Tran-Cong, T., Buckling and vibration analysis of laminated composite plate/shell structures via a smoothed quadrilateral flat shell element with in-plane rotations, Computers and Structures, Vol. 89, 2011, pp: [21] Jones, R.M., Mechanics of composite materials. Taylor and Francis Inc., USA, [22] Cook, R.D., Malkus, D.S., Plesha, M.E., Witt, R.J., Concepts and applications of finite element analysis, John Wiley & Sons Pte. Ltd., Singapore, 2000.

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