Three-dimensional free vibration analysis of levy-type laminated plates using multi-term extended Kantorovich method

Kumari, Poonam; Behera, Susanta

doi:10.1016/j.compositesb.2017.01.057

Cited by 18 publications

(5 citation statements)

References 40 publications

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“…It should be emphasized that the current results are in good agreement with the 3‐D FE except near to the edge. This disagreement at the very clamped support is well‐reported and discussed in literatures [34–36, 45].…”

Section: Numerical Results and Discussionsupporting

confidence: 76%

“…It should be emphasized that the current results are in good agreement with the 3-D FE except near to the edge. This disagreement at the very clamped support is well-reported and discussed in literatures [34][35][36]45]. Sometimes, the axially graded panel with step-wise variation in the material properties is also used in the structural applications [46].…”

Section: F I G U R E 3 Meshed Panel Geometrymentioning

confidence: 71%

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2‐D Analytical solutions for the multisegmented panel subjected to arbitrary boundary condition

Kumar

Kumari

2022

Z Angew Math Mech

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A closed-form bending solution of the multisegmented panels is presented for different support conditions. The panels can have multiple no. of segments along the x-axis with varying material properties. The governing equations are formulated using the extended Kantorovich method (EKM) in the mixed-form, and a set of ordinary differential equations have been obtained. The continuity of displacement and stresses are satisfied at the interface of each segment. The panels can have simply-support, clamp, and free types of support conditions. The panel interface at each segment along the x-axis is assumed to be perfectly bonded. In this work, the stress and deformation variations for four types of panel configuration have been studied. Two segmented panels having aluminum and Graphite-Epoxy with equal and unequal segments are considered. A foursegmented Al/SiC panel having gradual material variation is also analyzed. The effect of ply-angles on the longitudinal variables is also studied for the equal segmented Al/GrEp panel. The deflection and stresses are compared with the finite element solution and found in good agreement with the EKM results. The current development will lead to developing the solution for stepped panels, dissimilar plates, and more complex cases used in aerospace applications.

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Section: Numerical Results and Discussionsupporting

confidence: 76%

Section: F I G U R E 3 Meshed Panel Geometrymentioning

confidence: 71%

2‐D Analytical solutions for the multisegmented panel subjected to arbitrary boundary condition

Kumar

Kumari

2022

Z Angew Math Mech

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“…Shufrin et al (2008) propose a semi-analytical extended Kantorovich method for buckling analysis of symmetric laminated rectangular plates with general boundary conditions. Poonam and Susanta (2017) gives the closed three-dimensional free vibration solution of Levy-type's supported rectangular laminated plates for the first time by using the extended Kantorovich method. In addition, Akhavan et al (2009) derive the governing equation of the dynamic response of rectangular plates that based on the first-order shear deformation theory.…”

Section: Introductionmentioning

confidence: 99%

Unified solution of some problems of rectangular plates with four free edges based on symplectic superposition method

Su,

Bai,

Hai

2023

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PurposeA unified framework for solving the bending, buckling and vibration problems of rectangular thin plates (RTPs) with four free edges (FFFF), including isotropic RTPs, orthotropic rectangular thin plates (ORTPs) and nano-rectangular plates, is established by using the symplectic superposition method (SSM).Design/methodology/approachThe original fourth-order partial differential equation is first rewritten into Hamiltonian system. The class of boundary value problems of the original equation is decomposed into three subproblems, and each subproblem is given the corresponding symplectic eigenvalues and symplectic eigenvectors by using the separation variable method in Hamiltonian system. The symplectic orthogonality and completeness of symplectic eigen-vectors are proved. Then, the symplectic eigenvector expansion method is applied to solve the each subproblem. Then, the symplectic superposition solution of the boundary value problem of the original fourth-order partial differential equation is given through superposing analytical solutions of three foundation plates.FindingsThe bending, vibration and buckling problems of the rectangular nano-plate/isotropic rectangular thin plate/orthotropic rectangular thin plate with FFFF can be solved by the unified symplectic superposition solution respectively.Originality/valueThe symplectic superposition solution obtained is a reference solution to verify the feasibility of other methods. At the same time, it can be used for parameter analysis to deeply understand the mechanical behavior of related RTPs. The advantages of this method are as follows: (1) It provides a systematic framework for solving the boundary value problem of a class of fourth-order partial differential equations. It is expected to solve more complicated boundary value problems of partial differential equations. (2) SSM uses series expansion of symplectic eigenvectors to accurately describe the solution. Moreover, symplectic eigenvectors are orthogonal and directly reflect the orthogonal relationship of vibration modes. (3) The SSM can be carried to bending, buckling and free vibration problems of the same plate with other boundary conditions.

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“…As early as 1960s, the classical Rayleigh-Ritz method, finite difference method, variational method, Galerkin method, series expansion method and integral equation method were introduced by Leissa (1969) in detail. Since then, many numerical methods have appeared one after another, among which the commonly used numerical methods include finite element method (Kanaka Raju and Venkateswara Rao, 1976), Rayleigh-Ritz method using characteristic orthogonal polynomial (Bhat, 1985), a modified method of the classical initial value technique (Al-Khaiat, 1989), extended Kantorovich method (Kumari and Behera, 2017), element-free Galerkin method (Bahmyari and Rahbar-Ranji, 2012) and isogeometric analysis method (Valizadeh et al, 2013). Compared with numerical methods, the research on analytical methods of this kind of problems is much less.…”

Section: Introductionmentioning

confidence: 99%

Analytical free vibration solutions of fully free orthotropic rectangular thin plates on two-parameter elastic foundations by the symplectic superposition method

Bai

2020

Journal of Vibration and Control

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The free vibration of orthotropic rectangular thin plates with four free edges on two-parameter elastic foundations is studied by the symplectic superposition method. Firstly, by analyzing the boundary conditions, the original vibration problem is converted into two sub-vibration problems of the plates slidingly clamped at two opposite edges. Based on slidingly clamped at two opposite edges, the fundamental solutions of these two sub-vibration problems are respectively derived by the separation variable method of the corresponding Hamiltonian system, and then the symplectic superposition solution of the original vibration problem is obtained by superimposing the fundamental solutions of the two sub-problems. Finally, the symplectic superposition solution obtained in this study is verified by calculating the frequencies and mode functions of several concrete rectangular thin plates with four free edges.

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Three-dimensional free vibration analysis of levy-type laminated plates using multi-term extended Kantorovich method

Cited by 18 publications

References 40 publications

2‐D Analytical solutions for the multisegmented panel subjected to arbitrary boundary condition

2‐D Analytical solutions for the multisegmented panel subjected to arbitrary boundary condition

Unified solution of some problems of rectangular plates with four free edges based on symplectic superposition method

Analytical free vibration solutions of fully free orthotropic rectangular thin plates on two-parameter elastic foundations by the symplectic superposition method

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