NATURAL FREQUENCIES OF THICK ARBITRARY QUADRILATERAL PLATES USING THE pb–2 RITZ METHOD

doi:10.1006/jsvi.1996.0489

Journal of Sound and Vibration

Volume 196, Issue 4, 3 October 1996, Pages 371-385

https://doi.org/10.1006/jsvi.1996.0489 Get rights and content

Abstract

This paper presents a free vibration analysis of thick arbitrary quadrilateral plates based on the Mindlin shear deformation theory. The recently developed pb-2 Ritz method has been employed to compute the solution. The method features the versatile the pb-2 Ritz functions defined by the product of a two-dimensional polynomial (p-2) and a basic function (b) to approximate the deflections and rotations. The arbitrary thick quadrilateral plate is first mapped onto a basic square plate. The basic function is then formed from the product of boundary expressions of the basic square plate, each raised to an appropriate power to satisfy the various geometric boundary conditions of the actual plate. Stiffness and mass matrices are numerically integrated over the domain of the basis plate by using Guassian quadrature. New sets of natural frequency parameters for quadrilateral Mindlin plates with different combinations of boundary conditions and various thickness-to width ratios are presented. Wherever possible, the present results are verified by comparison with existing analytical and experimental values from open literature. The results from the pb-2 Ritz method are also directly compared with those determined by using the finite element method. The effect of the Poisson ratio on the natural frequencies of a thick cantilever quadrilateral plate is also investigated.

References (0)

Cited by (50)

An investigation on the stochastic thermal vibration behaviors for laminated combined composite plate systems
2023, Mechanical Systems and Signal Processing
The present work aims to study the stationary and nonstationary stochastic thermal vibration properties of the laminated combined plate structures using meshless method. The combined system structure consists of several rigidly connected laminated rectangular plates, where the thermal effect of each subplate is considered according to the thermo-elastic theory. The Hamilton’s principle combined with the first-order shear deformation theory (FSDT) is adopted for generating the vibration formulas of the divided plates. The boundary and rigid compatibility conditions of the combined plates are imposed by massless springs with variable stiffness. The displacement components of the subplate system are approximated by 2D Chebyshev meshfree shape functions and the stochastic acceleration excitation is taken into account by introducing the pseudo excitation method (PEM). The modal behaviors and the stochastic dynamic solutions of laminated combined plates subjected to thermal loads are then successfully obtained. A considerable number of computational examples on the free vibration and random responses of the combined laminated plates are carried out to verify the convergence, accuracy and efficiency of the presented meshless model. Finally, systematic parametric studies are conducted to investigate the effects of thermal factor, acceleration excitations and boundary constraints on the free vibration and random dynamic characteristics of the laminated combined plates.
New insights into the nonlinear stability of nanocomposite cylindrical panels under aero-thermal loads
2023, Composite Structures
In the present study, thermal-aerodynamic postbuckling and thermal flutter behaviors of functionally graded (FG) graphene nanoplatelets (GPLs) reinforced composite (GPLRC) cylindrical panels are theoretically investigated. Homogenization techniques such as the modified Halpin-Tsai model and Voigt’s rule are applied to calculate resultant material properties. The derived model of the cylindrical panel is based on assumptions of the first-order shear deformation theory (FSDT) and von Kármán geometrical nonlinearity. The aerodynamic load induced by the supersonic airflow is determined by the first-order piston theory, whilst the thermal load is based on the quasi-steady thermal stress theory. The element-free improved moving least-square Ritz (IMLS-Ritz) method is applied to develop the nonlinear equations solved with the aid of the Newton-Raphson method. The accuracy and effectiveness of the established numerical model are verified by comparisons with the results from the literature. Effects of key parameters such as aerodynamic pressure, temperature rise, GPL distributions and weight fraction, and geometric properties of cylindrical panels are carried out to comprehensively examine their aerothermoelastic responses. Finally, new insights into the stability of graphene platelet reinforced panels under aerodynamic and thermal loadings are presented and deeply discussed.
Modeling and experimental study on free vibration of plates with curved edges
2021, Journal of Sound and Vibration
This paper reports a modeling and experimental study on the free vibration characteristics of plates with curved edges. The model is established by the first-order shear deformation theory (FSDT) and Chebyshev differential quadrature method (CDQM). The one-to-one coordinate transformation technique is introduced into the CDQM to map the plate with curved edges into a square plate. The admissible displacement functions of the square plate are expanded by two-dimensional Chebyshev polynomials and discretized by Gauss-Lobatto sampling points. The boundary conditions are applied to the plate according to the projection matrix method. Experimental studies of six aluminum plates with different shapes are carried out to investigate the vibration characteristics and verify the validity of the proposed CDQM. Furthermore, the results of the present CDQM are also compared with those of the finite element method (FEM) and existing numerical approaches to examine its efficiency and accuracy. The results show that the current CDQM can rapidly and accurately compute the vibration characteristics of plates with curved edges under different boundary conditions.
Flutter analysis of laminated composite quadrilateral plates reinforced with graphene nanoplatelets using the element-free IMLS-Ritz method
2020, Aerospace Science and Technology
This paper investigates the flutter characteristics of graphene nanoplatelet (GPL) reinforced laminated composite quadrilateral plates using the element-free IMLS-Ritz method. The modified Halpin-Tsai model and rule of mixture are employed to predict the effective material properties including Young's modulus, mass density and Poisson's ratio. The energy functions of GPL reinforced composite (GPLRC) quadrilateral plates are obtained by the first-order shear deformation theory (FSDT) and first order piston theory. Based on the IMLS-Ritz approximation, the discrete dynamic equation of GPLRC quadrilateral plates is derived. The accuracy of the IMLS-Ritz results is examined by comparing the natural frequencies with those obtained from published values. A comprehensive parametric study is carried out, with a particular focus on the effects of weight fraction, distribution pattern, total number of layers, geometry and size of GPL reinforcements and geometric parameters of quadrilateral plates on the flutter boundary of GPLRC quadrilateral plates.
Vibration analysis of arbitrary straight-sided quadrilateral plates using a simple first-order shear deformation theory
2018, Results in Physics
This paper studies the free vibration characteristics of arbitrary straight-sided quadrilateral plates using a simple first-order shear deformation theory (SFSDT), which contains only four unknown displacement components. The arbitrary straight-sided quadrilateral plate is mapped into a unit square plate uniformly, and accordingly, the problem can be solved directly by the existing vibration modeling method of rectangular plate. The admissible functions of displacements are generally expressed as superposition of the periodic functions based on the improved Fourier series method. All the series expansion coefficients can be determined by the Rayleigh-Ritz procedure. Combined with the artificial virtual spring technology, the present method could be used to analyze the vibration characteristics of quadrilateral plates under arbitrary boundary conditions. Convergence and accuracy of the present method are checked out through some numerical examples of plate with rectangular, skew, trapezoidal and general quadrilateral shapes, and various boundary conditions. In addition, some new results and new conclusions have been given as the benchmark for future research.
Geometrically nonlinear analysis of laminated composite quadrilateral plates reinforced with graphene nanoplatelets using the element-free IMLS-Ritz method
2018, Composites Part B: Engineering
Citation Excerpt :
Arbitrarily quadrilateral plate as an important structural component used in engineering has also received lots of research attention. The linear vibration of arbitrarily quadrilateral plates was investigated by pb-2 Ritz method [31], DQ method [32,33], and singular convolution method [34]. Recently, the element free improved moving least squares (IMLS) Ritz method [35,36] has been successfully developed and used to study the free vibration of CNT reinforced composite square plates resting on elastic foundations or with elastically restrained edges [37–43].
This paper investigates the nonlinear bending of graphene nanoplatelet (GPL) reinforced laminated composite quadrilateral plates using the element-free IMLS-Ritz method. The effective material properties including Young's modulus, mass density and Poisson's ratio are determined by the modified Halpin-Tsai model and rule of mixture. The first-order shear deformation theory (FSDT) and the IMLS-Ritz approximation are employed to obtain the discrete nonlinear governing equation of quadrilateral plates with large deformation. The Newton-Raphson method is used to solve the nonlinear equation. The accuracy of the IMLS-Ritz results is examined by comparing with the published values. A comprehensive parametric study is carried out, with a particular focus on the effects of geometric parameters of quadrilateral plates and GPLs distribution pattern, weight fraction, total number of layers, and geometry and size of GPLs on the nondimensional deflection of GPLs reinforced laminated composite quadrilateral plate.

View all citing articles on Scopus

View full text

Regular articleNATURAL FREQUENCIES OF THICK ARBITRARY QUADRILATERAL PLATES USING THE pb–2 RITZ METHOD

Abstract

Regular article
NATURAL FREQUENCIES OF THICK ARBITRARY QUADRILATERAL PLATES USING THE pb–2 RITZ METHOD