Nonlinear Vibrations of Plates in Axial Pulsating Flow

2014 ◽  
Author(s):  
E. Tubaldi ◽  
M. Amabili ◽  
F. Alijani
Keyword(s):  
Flow Velocity ◽  
Nonlinear Vibrations ◽  
Plate Theory ◽  
Fluid Model ◽  
Equations Of Motion ◽  
Transmural Pressure ◽  
Pulsating Flow ◽  
Pulsation Frequency ◽  
Rectangular Plates ◽  
Flow Perturbation

A theoretical approach is presented to study nonlinear vibrations of thin infinitely long rectangular plates subjected to pulsatile axial inviscid flow. The case of plates in axial uniform flow under the action of constant transmural pressure is also addressed for different flow velocities. The equations of motion are obtained based on the von Karman nonlinear plate theory retaining in-plane inertia via Lagrangian approach. The fluid model is based on potential flow theory and the Galerkin method is applied to determine the expression of the flow perturbation potential. The effect of different system parameters such as flow velocity, pulsation amplitude, pulsation frequency, and channel pressurization on the stability of the plate and its geometrically nonlinear response to pulsating flow are fully discussed. In case of zero uniform transmural pressure numerical results show hardening type behavior for the entire flow velocity range when the pulsation frequency is spanned in the neighbourhood of the plate’s fundamental frequency. Conversely, a softening type behavior is presented when a uniform transmural pressure is introduced.

LATEST DEVELOPMENTS ON THE DYNAMIC STABILITY AND NONLINEAR PARAMETRIC RESPONSE OF GEOMETRICALLY IMPERFECT RECTANGULAR PLATES

1998 ◽  
Vol 22 (4B) ◽  
pp. 501-518 ◽  
Author(s):  
G.L. Ostiguy ◽  
L. St-Georges ◽  
S. Sassi
Keyword(s):  
Dynamic Stability ◽  
Asymptotic Method ◽  
Plate Theory ◽  
Lateral Displacement ◽  
Equations Of Motion ◽  
Rectangular Plates ◽  
Rational Analysis ◽  
Geometric Imperfection ◽  
Direct Integration Method ◽  
Parametric Response

The authors present a rational analysis of the effect of initial geometric imperfections on the dynamic stability and nonlinear parametric response of general rectangular plates, the plate theory used in the analysis may described as the dynamic analog of the von Kármán’s large deflection theory and is derived in terms of the stress function, the lateral displacement and the initial geometric imperfection. The governing equations are satisfied using the orthogonality properties of the assumed functions. The temporal response of the system is analyzed using a first-order asymptotic method and various types of resonances are investigated. The temporal equations of motion describing the nonlinear dynamic behaviour of the imperfect plates are also solved using a direct integration method and the results are compared with those obtained by the asymptotic method.

Nonlinear Vibrations of Pressurized Functionally Graded Plates Using Higher-Order Thickness Stretching Theory

2014 ◽  
Author(s):  
F. Alijani ◽  
M. Amabili
Keyword(s):  
Nonlinear Vibrations ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Shear Deformation Theory ◽  
Higher Order ◽  
Functionally Graded ◽  
Numerical Continuation ◽  
Normal Strain ◽  
Rectangular Plates ◽  
Functionally Graded Plates

Nonlinear vibrations of moderately thick functionally graded (FG) rectangular plates are investigated by considering a higher-order shear deformation theory that takes into account the thickness deformation effect. The geometrically nonlinear strain-displacement relationships are derived retaining full non-linear terms in the in-plane and transverse displacements and the three-dimensional constitutive equations are used by removing the assumption of zero transverse normal strain. The plate is assumed to have immovable boundary conditions at the edges. The equations of motion are obtained by using multi-modal energy approach. A code based on pseudo arc-length continuation and collocation scheme is utilized for numerical continuation and bifurcation analysis. Results show that higher-order thickness deformation theories yield a significant accuracy improvement for nonlinear vibrations of highly pressurized functionally graded plates.

scholarly journals Nonlinear Dynamic Response of Functionally Graded Rectangular Plates under Different Internal Resonances

2010 ◽  
Vol 2010 ◽  
pp. 1-12 ◽  
Author(s):  
Y. X. Hao ◽  
W. Zhang ◽  
X. L. Ji
Keyword(s):  
Dynamic Response ◽  
Nonlinear Dynamic ◽  
Internal Resonance ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Functionally Graded ◽  
Rectangular Plates ◽  
Internal Resonances ◽  
Third Order ◽  
Nonlinear Dynamic Response

The nonlinear dynamic response of functionally graded rectangular plates under combined transverse and in-plane excitations is investigated under the conditions of 1 : 1, 1 : 2 and 1 : 3 internal resonance. The material properties are assumed to be temperature-dependent and vary along the thickness direction. The thermal effect due to one-dimensional temperature gradient is included in the analysis. The governing equations of motion for FGM rectangular plates are derived by using Reddy's third-order plate theory and Hamilton's principle. Galerkin's approach is utilized to reduce the governing differential equations to a two-degree-of-freedom nonlinear system including quadratic and cubic nonlinear terms, which are then solved numerically by using 4th-order Runge-Kutta algorithm. The effects of in-plane excitations on the internal resonance relationship and nonlinear dynamic response of FGM plates are studied.

Nonlinear Vibrations of Elastically-Constrained Rotating Disks

1998 ◽  
Vol 120 (2) ◽  
pp. 475-483 ◽  
Author(s):  
L. Yang ◽  
S. G. Hutton
Keyword(s):  
Differential Equations ◽  
Nonlinear Vibrations ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Rotating Disk ◽  
Transverse Displacement ◽  
Rotating Disks ◽  
Thin Plate Theory ◽  
Force Equilibrium

An analysis of nonlinear vibrations of an elastically-constrained rotating disk is developed. The equations of motion, which are two coupled nonlinear partial differential equations corresponding to the transverse force equilibrium and to the deformation compatibility, are first developed by using von Karman thin plate theory. Then the stress function is analytically solved from the compatibility equation by assuming a multi-mode transverse displacement field. Galerkin’s method is applied to transform the force equilibrium equation into a set of coupled nonlinear ordinary differential equations in terms of time functions. Finally, numerical integration is used to solve the time governing equations, and the effects of nonlinearity on the vibrations of a rotating disk are discussed.

Numerical Formulation of Nonlinear Vibrations of Elastically-Constrained Rotating Disks

1995 ◽  
Author(s):  
Longxiang Yang ◽  
Stanley G. Hutton
Keyword(s):  
Differential Equations ◽  
Nonlinear Vibrations ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Rotating Disk ◽  
Transverse Displacement ◽  
Rotating Disks ◽  
Numerical Formulation ◽  
Force Equilibrium

Abstract An analysis of nonlinear vibrations of an elastically-constrained rotating disk is developed. The equations of motion, which are two coupled nonlinear partial differential equations corresponding to the transverse force equilibrium and to the deformation compatibility, are first developed by using von Karman thin plate theory. Then the stress function is analytically solved from the compatibility equation by assuming a multi-mode transverse displacement field. Galerkin’s method is applied to transform the force equilibrium equation into a set of coupled nonlinear ordinary differential equations in terms of time functions. Finally, numerical integration is used to solve the time governing equations, and the effects of nonlinearity on the vibrations of a rotating disk are discussed.

Bifurcation and Chaos of the Rectangular Moderate Thickness Cracked Plates on an Elastic Foundation

2006 ◽  
Vol 324-325 ◽  
pp. 399-402
Author(s):  
Yong Gang Xiao
Keyword(s):  
Elastic Foundation ◽  
Nonlinear Equations ◽  
External Load ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Rectangular Plates ◽  
Bifurcation And Chaos ◽  
Periodic Load ◽  
Hamilton Variational Principle ◽  
Nonlinear Equations Of Motion

Based on Reissner plate theory and using Hamilton variational principle, the nonlinear equations of motion are derived for the moderate thickness rectangular plates with transverse surface penetrating crack on an elastic foundation under the action of periodic load. The suitable expressions of trial functions satisfied all boundary conditions and crack’s continuous conditions are proposed. By using the Galerkin method and the Runge-Kutta integration method, the nonlinear equations are solved. The possible bifurcation and chaos of the system are analyzed under the action of external load. In numerical calculation, the influences of the different location and depth of crack and external load on the bifurcation and chaos of the rectangular moderate thickness plates with freely supported boundary are discussed.

On the Frequency Response of a Spinning Disk With Large Deformations

2010 ◽  
Author(s):  
Ramin M. H. Khorasany ◽  
Stanley G. Hutton
Keyword(s):  
Frequency Response ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Rotating Disk ◽  
Nonlinear Frequency ◽  
Response Characteristics ◽  
Geometrical Nonlinear ◽  
Oscillation Frequencies ◽  
Multi Mode ◽  
Different Levels

In this paper, the effect of geometrical nonlinear terms, caused by a space fixed point force, on the frequencies of oscillations of a rotating disk with clamped-free boundary conditions is investigated. The nonlinear geometrical equations of motion are based on Von Karman plate theory. Using the eigenfunctions of a stationary disk as approximating functions in Galerkin’s method, the equations of motion are transformed into a set of coupled nonlinear Ordinary Differential Equations (ODEs). These equations are then used to find the equilibrium positions of the disk at different discrete blade speeds. At any given speed, the governing equations are linearized about the equilibrium solution of the disk under the application of a space fixed external force. These linearized equations are then used to find the oscillation frequencies of the disk considering the effect of large deformation. Using multi mode approximation and different levels of nonlinearity, the frequency response of the disk considering the effect of geometrical nonlinear terms are studied. It is found that at the linear critical speed, the nonlinear frequency of the corresponding mode is not zero. Results are presented that illustrate the effect of the magnitude of disk displacement upon the frequency response characteristics. It is also found that for each mode, including the effect of the geometrical nonlinear terms due to the applied load causes a separation in the frequency responses of its backward and forward traveling waves when the disk is stationary. This effect is similar to the effect of a space fixed constraint in the linear problem. In order to verify the numerical results, experiments are conducted and the results are presented.

Dynamical Behaviors of Sinusoidal Nanocomposite Plates Subjected to Thermomechanical Loads

2021 ◽  
Author(s):  
Vu Ngoc Viet Hoang ◽  
Dinh Gia Ninh
Keyword(s):  
Thermal Environment ◽  
Plate Theory ◽  
Micromechanical Model ◽  
Functionally Graded ◽  
Geometrical Parameters ◽  
Rectangular Plates ◽  
Sine Function ◽  
Classical Plate Theory ◽  
Wolfram Mathematica ◽  
Number Of Cycles

In this paper, a new plate structure has been found with the change of profile according to the sine function which we temporarily call as the sinusoidal plate. The classical plate theory and Galerkin’s technique have been utilized in estimating the nonlinear vibration behavior of the new non-rectangular plates reinforced by functionally graded (FG) graphene nanoplatelets (GNPs) resting on the Kerr foundation. The FG-GNP plates were assumed to have two horizontal variable edges according to the sine function. Four different configurations of the FG-GNP plates based on the number of cycles of sine function were analyzed. The material characteristics of the GNPs were evaluated in terms of two models called the Halpin–Tsai micromechanical model and the rule of mixtures. First, to verify this method, the natural frequencies of new non-rectangular plates made of metal were compared with those obtained by the Finite Element Method (FEM). Then, the numerical outcomes are validated by comparing with the previous papers for rectangular FGM/GNP plates — a special case of this structure. Furthermore, the impacts of the thermal environment, geometrical parameters, and the elastic foundation on the dynamical responses are scrutinized by the 2D/3D graphical results and coded in Wolfram-Mathematica. The results of this work proved that the introduced approach has the advantages of being fast, having high accuracy, and involving uncomplicated calculation.

scholarly journals Non-Local Buckling Analysis of Functionally Graded Nanoporous Metal Foam Nanoplates

Coatings ◽  
2018 ◽  
Vol 8 (11) ◽  
pp. 389 ◽  
Author(s):  
Yanqing Wang ◽  
Zhiyuan Zhang
Keyword(s):  
Metal Foam ◽  
Plate Theory ◽  
Constitutive Relations ◽  
Equations Of Motion ◽  
Local Buckling ◽  
Functionally Graded ◽  
Small Scale ◽  
Refined Plate Theory ◽  
Nanoporous Metal ◽  
Non Local

In this study, the buckling of functionally graded (FG) nanoporous metal foam nanoplates is investigated by combining the refined plate theory with the non-local elasticity theory. The refined plate theory takes into account transverse shear strains which vary quadratically through the thickness without considering the shear correction factor. Based on Eringen’s non-local differential constitutive relations, the equations of motion are derived from Hamilton’s principle. The analytical solutions for the buckling of FG nanoporous metal foam nanoplates are obtained via Navier’s method. Moreover, the effects of porosity distributions, porosity coefficient, small scale parameter, axial compression ratio, mode number, aspect ratio and length-to-thickness ratio on the buckling loads are discussed. In order to verify the validity of present analysis, the analytical results have been compared with other previous studies.

scholarly journals Comparison of Galerkin, POD and Nonlinear-Normal-Modes Models for Nonlinear Vibrations of Circular Cylindrical Shells

2006 ◽  
Author(s):  
M. Amabili ◽  
C. Touze´ ◽  
O. Thomas
Keyword(s):  
Nonlinear Vibrations ◽  
Circular Cylindrical Shell ◽  
Equations Of Motion ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Harmonic Excitation ◽  
Nonlinear Responses ◽  
Nonlinear Normal ◽  
The Galerkin Method ◽  
Proper Orthogonal

The aim of the present paper is to compare two different methods available to reduce the complicated dynamics exhibited by large amplitude, geometrically nonlinear vibrations of a thin shell. The two methods are: the proper orthogonal decomposition (POD) and an asymptotic approximation of the Nonlinear Normal Modes (NNMs) of the system. The structure used to perform comparisons is a water-filled, simply supported circular cylindrical shell subjected to harmonic excitation in the spectral neighbourhood of the fundamental natural frequency. A reference solution is obtained by discretizing the Partial Differential Equations (PDEs) of motion with a Galerkin expansion containing 16 eigenmodes. The POD model is built by using responses computed with the Galerkin model; the NNM model is built by using the discretized equations of motion obtained with the Galerkin method, and taking into account also the transformation of damping terms. Both the POD and NNMs allow to reduce significantly the dimension of the original Galerkin model. The computed nonlinear responses are compared in order to verify the accuracy and the limits of these two methods. For vibration amplitudes equal to 1.5 times the shell thickness, the two methods give very close results to the original Galerkin model. By increasing the excitation and vibration amplitude, significant differences are observed and discussed.

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