Improvement of the Semi-Analytical Method, for Determining the Geometrically Non-Linear Response of Thin Straight Structures: Part Ii—first and Second Non-Linear Mode Shapes of Fully Clamped Rectangular Plates

Kadiri, M. El; Benamar, Rhali

doi:10.1006/jsvi.2002.5026

Cited by 21 publications

(9 citation statements)

References 18 publications

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“…In order to obtain explicit analytical solutions for the first nonlinear axisymmetric mode shape of the FG circular plates, which are expected to be very useful in engineering applications and in further analytical developments, the improved version of the semi-analytical model developed by El Kadiri, Benamar, and White 28 and El Kadiri and Benamar 29 for beams and rectangular plates has been developed and adapted for the FG circular plate, which are shown to be in a good agreement with the iterative method.…”

Section: Discussionmentioning

confidence: 99%

Geometrically Nonlinear Free Axisymmetric Vibrations Analysis of Thin Circular Functionally Graded Plates Using Iterative and Explicit Analytical Solution

Kaak¹,

Bikri²,

Benamar³

2016

IJAV

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This paper deals with nonlinear free axisymmetric vibrations of functionally graded (FG) thin circular plates whose properties vary in thickness. The inhomogeneity of the plate is characterized by a power law variation of the Young's modulus and mass density of the material along the thickness direction, whereas Poisson's ratio is assumed to be constant. The theoretical model is based on Hamilton's principle and spectral analysis using a basis of admissible Bessel's functions to yield the frequencies of the circular plates under clamped boundary conditions on the basis of the classical plate theory. The large vibration amplitudes problem, reduced to a set of nonlinear algebraic equations, is solved numerically. The nonlinear to linear frequency ratios are presented for various values of the volume fraction index n showing a hardening type nonlinearity. The distribution of the radial bending stresses associated to the nonlinear mode shape is also given for various vibration amplitudes and compared with those predicted by the linear theory. Then, explicit analytical solutions are presented, based on the semi-analytical model previously developed by El Kadiri et al. for beams and rectangular plates. This model allows direct and easy calculation for the first nonlinear axisymmetric mode shape with its associated nonlinear frequencies and nonlinear bending stresses of FG circular plates, which are expected to be very useful in engineering applications and in further analytical developments. An excellent agreement is found with the results obtained by the iterative method.

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Section: Discussionmentioning

confidence: 99%

Geometrically Nonlinear Free Axisymmetric Vibrations Analysis of Thin Circular Functionally Graded Plates Using Iterative and Explicit Analytical Solution

Kaak¹,

Bikri²,

Benamar³

2016

IJAV

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“…As mentioned in [9], the above system permits one to obtain explicitly the basic function contributions ε 2 , ε 3, …, ε 18 of the second and higher order functions, corresponding to a given value of the assigned first basic function contribution a 1 if * ir k , for r i ≠ , is assumed to be negligible compared to * rr k , and direct solution was found as follows:…”

Section: General Formulations and Analysismentioning

confidence: 99%

A Semi-Analytical Approach for the Geometrically Nonlinear Analysis of Skew Plates

Abdelali

Kadiri

Benamar

2014

AMM

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The present work concerns the nonlinear dynamic behaviour of fully clamped skew plates at large vibration amplitudes. A model based on Hamilton’s principle and spectral analysis has been used to study the large amplitude free vibration problem, reducing the non linear problem to solution of a set of non-linear algebraic equations. Two methods of solution have been adopted, the first method uses an improved version of the Newton-Raphson method, and the second leads to explicit analytical expressions for the higher mode contribution coefficients to the first non-linear mode shape of the skew plate examined. The amplitude dependent fundamental mode shape and the associated non-linear frequencies have been obtained by the two methods and a good convergence has been found. It was found that the non-linear frequencies increase with increasing the amplitude of vibration, which corresponds to the hardening type effect, expected in similar cases, due to the membrane forces induced by the large vibration amplitudes. The non-linear mode exhibits a higher bending stress near to the clamps at large deflections, compared with that predicted by linear theory. Numerical details are presented and the comparison made between the results obtained and previous ones available in the literature shows a satisfactory agreement. Tables of numerical results are given, corresponding to the linear and nonlinear cases for various values of the skew angle θ and various values of the vibration amplitude. These results, similar to those previously published for other plates, are expected to be useful to designers in the need of accurate estimates of the non-linear frequencies, the non linear strains and stresses induced by large vibration amplitudes of skew plates.

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“…Considering now the nonlinear algebraic system (24), it has to be solved for determination of the nonlinear frequency response function of the 2-dof systems treated. This system is solved here using the explicit procedure (First formulation) presented for the first time in (El Kadiri et al, 2002b).…”

Section: The Nonlinear Solutionmentioning

confidence: 99%

“…The objective of the present work is the treatment of the transverse steady-state periodic forced response of 2-dof systems with cubic nonlinearities, using the method developed in references (Eddanguir et al, 2009;Beidouri et al, 2008;El Kadiri et al, 2002b;Benamar et al, 1993) and applied to the investigation of nonlinear vibration of different structures, of various types, geometries, and material characteristics.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear transverse steady-state periodic forced vibration of 2-dof discrete systems with cubic nonlinearities

Eddanguir

Beidouri

Benamar

2011

TECM

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A method based on Hamilton’s principle and spectral analysis has been applied recently to nonlinear transverse vibrations of discrete systems with cubic nonlinearities, leading to calculation of the nonlinear free modes of transverse vibration and their associated nonlinear frequencies. The objective of the present work was the extension of this method to the nonlinear forced transverse steady-state periodic response of 2-dof system leading to nonlinear frequency response function in the neighbourhood of the two modes

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Improvement of the Semi-Analytical Method, for Determining the Geometrically Non-Linear Response of Thin Straight Structures: Part Ii—first and Second Non-Linear Mode Shapes of Fully Clamped Rectangular Plates

Cited by 21 publications

References 18 publications

Geometrically Nonlinear Free Axisymmetric Vibrations Analysis of Thin Circular Functionally Graded Plates Using Iterative and Explicit Analytical Solution

Geometrically Nonlinear Free Axisymmetric Vibrations Analysis of Thin Circular Functionally Graded Plates Using Iterative and Explicit Analytical Solution

A Semi-Analytical Approach for the Geometrically Nonlinear Analysis of Skew Plates

Nonlinear transverse steady-state periodic forced vibration of 2-dof discrete systems with cubic nonlinearities

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