scholarly journals Non-Linear Forced Vibration of Fully Clamped FGM Skew Plates using Homogenization Technique

2019 ◽  
Vol 286 ◽  
pp. 03002
Author(s):  
H. Moulay Abdelali ◽  
R. Benamar
Keyword(s):  
Forced Vibration ◽  
Functionally Graded ◽  
Mode Shapes ◽  
Skew Angle ◽  
Linear Mode ◽  
Homogenization Technique ◽  
Skew Plate ◽  
Non Linear ◽  
Excitation Force ◽  
The Right

The present work concerns the geometrically non-linear forced vibration of fully clamped functionally graded skew plates (FGSP). The theoretical model based on Hamilton’s principle and spectral analysis previously applied to obtain the non-linear mode shapes of thin straight structures is used. A homogenization technique is developed to reduce the FGSP problem under consideration to that of an equivalent isotropic homogeneous skew plate. Results are given for the linear and non-linear fundamental frequency of fully clamped FGSP, considering different parameters, such as the skew angle, the excitation force level. The results show a non linearity of the hardening type with a shift to the right of the bent non linear frequency response function, in the neighbourhood of the fundamental mode.

Geometrically Non-Linear Free Vibration of Fully Clamped FGM Skew Plates Using Homogenization Technique

2015 ◽  
Vol 1105 ◽  
pp. 370-380
Author(s):  
Moulay Abdelali Hanane ◽  
Khalid El Bikri ◽  
Benamar Rhali
Keyword(s):  
Free Vibration ◽  
Functionally Graded ◽  
Mode Shapes ◽  
Power Law Distribution ◽  
Skew Angle ◽  
Linear Mode ◽  
Homogenization Technique ◽  
Skew Plate ◽  
Non Linear ◽  
Good Agreement

The present work concerns the geometrically non-linear free vibration of fully clamped functionally graded skew plates (FGSP). The theoretical model based on Hamilton’s principle and spectral analysis is used. A homogenization technique has been developed to reduce the FGSP problem under consideration to that of an isotropic homogeneous skew plate. The material properties of the skew plate examined herein are assumed to be graded in the thickness direction of the plate according to the power-law distribution in terms of volume fractions of the constituents. Results are given for the linear and non-linear fundamental frequency considering different parameters. The non-linear mode shapes exhibit a maximum value in the bending stress at the centre of the plate. It is found also that the non-linear frequencies increase with increasing the amplitude of vibration and increasing the skew angle, which corresponds to the hardening type effect. A good agreement is found with published results.

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

2014 ◽  
Vol 704 ◽  
pp. 118-130
Author(s):  
Hanane Moulay Abdelali ◽  
Mounia El Kadiri ◽  
Rhali Benamar
Keyword(s):  
Mode Shape ◽  
Algebraic Equations ◽  
Skew Angle ◽  
Geometrically Nonlinear Analysis ◽  
Good Convergence ◽  
Linear Mode ◽  
Skew Plate ◽  
Linear Algebraic Equations ◽  
Non Linear ◽  
Analytical Expressions

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.

Large Amplitude Free Vibration Analysis of Functionally Graded Annular Plates

2014 ◽  
Vol 971-973 ◽  
pp. 548-564 ◽  
Author(s):  
Boutahar Lhoucine ◽  
Khalid El Bikri ◽  
Benamar Rhali
Keyword(s):  
Free Vibration ◽  
Material Properties ◽  
Effective Properties ◽  
Plate Thickness ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Mode Shapes ◽  
Graphical Form ◽  
Wide Range ◽  
Non Linear

The geometrically non-linear axisymmetric free vibration of functionally graded annular plate (FGAP) having both edges clamped is analyzed in this paper. The material properties of the constituents are assumed to be temperature-independent and the effective properties of FGAP are graded in thickness direction according to a simple power law function in terms of the volume fractions. Based on the classical Plate theory and von Karman type non-linear strain-displacement relationships, the nonlinear governing equations of motion are derived using Hamilton’s principle. The problem is solved by a numerical iterative procedure in order to obtain more accurate results for vibration amplitudes up to twice the plate thickness. The numerical results are given for the first two axisymmetric non-linear mode shapes, for a wide range of vibration amplitudes and they are presented either in a tabular or in a graphical form, to show the significant effects that the large vibration amplitudes and the variation in material properties have on the non-linear frequencies and the associated bending stresses of the FGAP.

scholarly journals Natural Frequency and Mode Shapes of Exponential Tapered AFG Beams on Elastic Foundation

2016 ◽  
Vol 9 ◽  
pp. 9-25 ◽  
Author(s):  
Hareram Lohar ◽  
Anirban Mitra ◽  
Sarmila Sahoo
Keyword(s):  
Elastic Foundation ◽  
Static Problem ◽  
Functionally Graded ◽  
Mode Shapes ◽  
Dimensionless Frequency ◽  
Energy Principle ◽  
Various Boundary Conditions ◽  
Non Linear ◽  
Beams On Elastic Foundation ◽  
Minimum Potential Energy Principle

A displacement based semi-analytical method is utilized to study non-linear free vibration and mode shapes of an exponential tapered axially functionally graded (AFG) beam resting on an elastic foundation. In the present study geometric nonlinearity induced through large displacement is taken care of by non-linear strain-displacement relations. The beam is considered to be slender to neglect the rotary inertia and shear deformation effects. In the present paper at first the static problem is solved through an iterative scheme using a relaxation parameter and later on the subsequent dynamic analysis is carried out as a standard eigen value problem. Energy principles are used for the formulation of both the problems. The static problem is solved by using minimum potential energy principle whereas in case of dynamic problem Hamilton’s principle is employed. The free vibrational frequencies are tabulated for exponential taper profile subject to various boundary conditions and foundation stiffness. The dynamic behaviour of the system is presented in the form of backbone curves in dimensionless frequency-amplitude plane and in some particular case the mode shape results are furnished.

scholarly journals Multimode Analysis of the Dynamics and Integrity of Electrically Actuated MEMS Resonators

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Serge Bruno Yamgoué ◽  
Alain Juvenal Tchiegang
Keyword(s):  
Time Integration ◽  
Free Vibrations ◽  
Galerkin Projection ◽  
Mode Shapes ◽  
Galerkin Procedure ◽  
Linear Mode ◽  
Mems Resonators ◽  
Numerical Time Integration ◽  
The Right ◽  
Subharmonic Resonances

We present a theoretical investigation of the dynamic behavior of a microelectromechanical system (in brief, MEMS) device modelled as a clamped-clamped microbeam subjected to electrostatic and electrodynamic actuation. We use the Galerkin projection technique to reduce the partial integro-differential equation governing the dynamics of the microbeam to a system of coupled ordinary differential equations which describe the interactions of the linear mode shapes of the microbeam. Analytical solutions are derived and their stability is studied for the simplest reduced-order model which takes into account only the first linear mode in the Galerkin procedure. We further investigate the influence of the first few higher modes on the Galerkin procedure, and hence its convergence, by analysing the boundaries between pull-in and pull-in-free vibrations domains in the space of actuation parameters. These are determined for the various multimode combinations using direct numerical time integration. Our results show that unsafe domains form V-like shapes for actuation frequencies close to the superharmonic, fundamental, and subharmonic resonances. They also reveal that the single first-mode reduced model usually considered underestimates the left branches and overestimates the right branches of these boundaries.

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