Non-linear vibration analysis of specially orthotropic tapered micro-plates with arbitrary located crack: A non-classical analytical approach

2021 ◽  
pp. 095440622110197
Author(s):  
Bhupesh K Chandrakar ◽  
NK Jain ◽  
Ankur Gupta
Keyword(s):  
Governing Equation ◽  
Orthotropic Plate ◽  
Multiple Scales ◽  
Plate Theory ◽  
Couple Stress Theory ◽  
Plate Thickness ◽  
Solution Technique ◽  
Classical Plate Theory ◽  
Non Linear ◽  
The Impact

The present work aims to study the non-linear vibrations in a cracked orthotropic tapered micro-plate. Linear and parabolic variation in the plate thickness is assumed in one as well as two directions. The partial crack is located in the centre, and it is continuous; this crack’s location is arbitrary and can be varied within the centre-line. Based on classical plate theory, the equilibrium principle is applied, and the governing equation of tapered orthotropic plate is derived. Additionally, the microstructure’s effect has been included in the governing equation using the non-classical modified couple stress theory. The simplified line spring model is used to consider the impact of partial crack on the plate dynamics and is incorporated using in-plane forces and bending moments. The introduction of Berger’s formulation brings the non-linearity in the model in terms of in-plane forces. Here, Galerkin’s method has been chosen for converting the derived governing equation into time-dependent modal coordinates, which uses an approximate solution technique to solve the non-linear Duffing equation. The crack is considered along the fibres and across the fibres to show the effect of orthotropy. Results are presented for an orthotropic cracked plate with non-uniform thickness. The effects of the variation of taper constants, crack location, crack length, internal material length scale parameter on the fundamental frequency are obtained for two different boundary conditions. The non-linear frequency response curves are plotted to show the effect of non-linearity on the system dynamics using the method of multiple scales, and the contribution of taper constants and crack parameters on non-linearity is shown with bending-hardening and bending-softening phenomenon. It has been found that vibration characteristics are affected by the taper parameters and fibre direction for a cracked orthotropic plate.

A Non-Classical Analytical Approach for Vibration Analysis of Isotropic and Fgm Plate Containing a Star Shaped Crack

2020 ◽  
Vol 36 (4) ◽  
pp. 465-484
Author(s):  
Ankur Gupta ◽  
Shashank Soni ◽  
N. K. Jain
Keyword(s):  
Governing Equation ◽  
Vibration Analysis ◽  
Plate Theory ◽  
Couple Stress Theory ◽  
Modified Couple Stress Theory ◽  
Multiple Cracks ◽  
Arbitrary Orientation ◽  
Response Curves ◽  
Classical Plate Theory ◽  
Orientation Gradient

ABSTRACTA non-classical analytical model for vibration analysis of thin isotropic and FGM plate containing multiple part-through cracks (star shaped) of arbitrary orientation is proposed. A plate containing four concentric cracks of arbitrary orientation in the form of continuous line is considered for analysis. The proposed governing equation is derived based on classical plate theory and modified couple stress theory. Line spring model is modified to accommodate all the crack terms. The application of Berger’s formulation introduces nonlinearities in the governing equation and then the Galerkin’s method is applied for solving final governing equation. Results for fundamental frequencies for different values of crack length, crack orientation, gradient index and material length scale parameters are presented for two different boundary conditions. Furthermore, to study the phenomenon of bending hardening/softening in a cracked plate, the frequency response curves are plotted for the parameters stated above. Based on the outcomes of this study, it can be concluded that stiffness of the plate is severely affected by the presence of multiple cracks and the stiffness goes on decreasing with increase in number of cracks thereby affecting the fundamental frequency.

Water Entry of a Wedge by Hydroelastic Orthotropic Plate Theory

1999 ◽  
Vol 43 (03) ◽  
pp. 180-193 ◽  
Author(s):  
Odd M. Faltinsen
Keyword(s):  
Impact Velocity ◽  
Cross Sections ◽  
Orthotropic Plate ◽  
Plate Theory ◽  
Three Dimensional ◽  
Water Entry ◽  
Natural Period ◽  
Cross Sectional ◽  
Flow Effects ◽  
The Impact

Water entry of a hull with wedge-shaped cross sections is analyzed. The stiffened platings between two transverse girders on each side of the keel are separately modeled. Orthotropic plate theory is used. The effect of structural vibrations on the fluid flow is incorporated by solving the two-dimensional Laplace equation in the cross-sectional fluid domain by a generalized Wagner's theory. The coupling with the plate theory provides three-dimensional flow effects. The theory is validated by comparison with full-scale experiments and drop tests. The importance of global ship accelerations is pointed out. Hydrodynamic and structural error sources are discussed. Systematic studies on the importance of hydroelasticity as a function of deadrise angle and impact velocity are presented. This can be related to the ratio between the wetting time of the structure and the greatest wet natural period of the stiffened plating. This ratio is proportional to the deadrise angle and inversely proportional to the impact velocity. A small ratio-means that hydroelasticity is important and a large ratio means that hydroelasticity is not important.

Determination of vibration of single-layered graphene sheets using nonlocal modified couple stress theory

2021 ◽  
pp. 2250011
Author(s):  
F. Attar ◽  
R. Khordad ◽  
A. Zarifi
Keyword(s):  
Couple Stress ◽  
Plate Theory ◽  
Couple Stress Theory ◽  
Nonlocal Parameter ◽  
Modified Couple Stress Theory ◽  
Length Scale ◽  
Vibration Modes ◽  
Classical Plate Theory ◽  
Stress Theory ◽  
Modified Couple Stress

The free vibration of single-layered graphene sheet (SLGS) has been studied by nonlocal modified couple stress theory (NMCS), analytically. Governing equation of motion for SLGS is obtained via thin plate theory in conjunction with Hamilton’s principle for two cases: (1) using nonlocal parameter only for stress tensor, (2) using nonlocal parameter for both stress and couple stress tensors. Navier’s approach has been used to solve the governing equations for simply supported boundary conditions. It is found that the frequency ratios decrease with increasing nonlocal parameter and also with enhancing vibration modes, but increase with raising length scale parameter. The nonlocal and length scale parameters are more prominent in higher vibration modes. The obtained results have been compared with the previous studies obtained by using classical plate theory, the modified couple stress theory and nonlocal elasticity theory, separately.

Superharmonic and Subharmonic Resonances of Spiral Stiffened Functionally Graded Cylindrical Shells under Harmonic Excitation

2019 ◽  
Vol 19 (10) ◽  
pp. 1950114 ◽  
Author(s):  
Habib Ahmadi ◽  
Kamran Foroutan
Keyword(s):  
Multiple Scales ◽  
Plate Theory ◽  
Cylindrical Shells ◽  
Harmonic Excitation ◽  
Functionally Graded ◽  
Classical Plate Theory ◽  
Geometrical Properties ◽  
Hardening Nonlinearity ◽  
Subharmonic Resonances ◽  
Theory Of Shells

This paper presents the superharmonic and subharmonic resonances of spiral stiffened functionally graded (SSFG) cylindrical shells under harmonic excitation. The stiffeners are considered to be externally or internally added to the shell. Also, it is assumed that the material properties of the stiffeners are continuously graded in the thickness direction. In order to model the stiffeners, the smeared stiffener technique is used. Within the context of the classical plate theory of shells, the von Kármán nonlinear equations are derived for the shell and stiffeners based on Hooke’s law and the relations of stress-strain. Using Galerkin’s method, the equation of motion is discretized. The superharmonic and subharmonic resonances are analyzed by the method of multiple scales. The influence of the material parameters and various geometrical properties on the superharmonic and subharmonic resonances of SSFG cylindrical shells is investigated. Considering these results, the hardening nonlinearity behavior and jump value of cylindrical shell is less and more than others, when the angle of stiffeners is [Formula: see text] and [Formula: see text], respectively.

On Large Deflection of Symmetric Composite Plates under Static Loading

1986 ◽  
Vol 200 (1) ◽  
pp. 13-19 ◽  
Author(s):  
M Gorji
Keyword(s):  
Shear Deformation ◽  
Plate Theory ◽  
Lateral Displacement ◽  
Shear Deformation Theory ◽  
Composite Plates ◽  
Classical Plate Theory ◽  
Maximum Displacement ◽  
Von Karman ◽  
Non Linear ◽  
Von Kármán

The effect of transverse shear deformation on bending of elastic symmetric laminated composite plates undergoing large deformation (in the Von Karman sense) is considered in the present paper. The non-linear terms of the lateral displacement are considered as an additional set of lateral loads acting on the plate. The solution of a Von Karman type plate is therefore reduced to that of an equivalent plate with small displacements. This method offers an alternative technique for obtaining non-linear solutions to plate problems. The solutions of a number of example problems indicate that the non-linear shear deformation theory results, as expected, in higher values of the lateral displacement than the non-linear solutions from the classical plate theory. The difference in the values of the maximum displacement from both solutions, however, remains essentially constant beyond a certain value of the load. It is also noted that the linear and non-linear solutions deviate at a low value of w/h (w = maximum lateral displacement, h = thickness). Consequently, the extent of w/h within which the small deflection theory is applicable to composite plates is much lower than the value of 0.4 typically used for isotropic plates and depends, in general, upon lamination geometry and the degree of anisotropy.

scholarly journals A Non-Linear Dynamic Model of Ionic Polymer-Metal Composite (IPMC) Cantilever Actuator

2019 ◽  
Vol 16 (1) ◽  
pp. 6332-6347
Author(s):  
D. K. Biswal ◽  
D. Bandopadhya ◽  
S. K. Dwivedy
Keyword(s):  
Governing Equation ◽  
Multiple Scales ◽  
Production Method ◽  
Equation Of Motion ◽  
Metal Composite ◽  
Ionic Polymer ◽  
Solvent Loss ◽  
Non Linear Equation ◽  
Non Linear ◽  
Polymer Metal

This work presents development of an effective non-linear mathematical model for dynamic analysis of Ionic polymer-metal composites (IPMCs) cantilever actuators undergoing large bending deformations under AC excitation voltages. As the IPMC actuator experiences dehydration (solvent loss) in open environment, a model has been proposed to calculate the solvent loss due to applied electric potential following Cobb-Douglas production method. D’Alembert’s principle has been used for the derivation of the governing equation of motion of the system. Generalized Galerkin’s method has been followed to reduce the governing equation to the second-order temporal differential equation of motion. Method of multiple scales has been used to solve the non-linear equation of motion of the system and dehydration effect on the vibration response has been demonstrated numerically.

scholarly journals Ritz Method in Vibration Analysis for Embedded Single-Layered Graphene Sheets Subjected to In-Plane Magnetic Field

Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 515
Author(s):  
Olga Mazur ◽  
Jan Awrejcewicz
Keyword(s):  
Magnetic Field ◽  
Boundary Conditions ◽  
Elastic Foundation ◽  
Magnetic Parameter ◽  
Plate Theory ◽  
Couple Stress Theory ◽  
Ritz Method ◽  
Graphene Sheets ◽  
Classical Plate Theory ◽  
Foundation Parameters

Vibrations of single-layered graphene sheets subjected to a longitudinal magnetic field are considered. The Winkler-type and Pasternak-type foundation models are employed to reproduce the surrounding elastic medium. The governing equation is based on the modified couple stress theory and Kirchhoff–Love hypotheses. The effect of the magnetic field is taken into account due to the Lorentz force deriving from Maxwell’s equations. The developed approach is based on applying the Ritz method. The proposed method is tested by a comparison with results from the existing literature. The numerical calculations are performed for different boundary conditions, including the mixed ones. The influence of the material length scale parameter, the elastic foundation parameters, the magnetic parameter and the boundary conditions on vibration frequencies is studied. It is observed that an increase of the magnetic parameter, as well as the elastic foundation parameters, brings results closer to the classical plate theory results. Furthermore, the current study can be applied to the design of microplates and nanoplates and their optimal usage.

Refined hyperbolic shear deformation plate theory

2014 ◽  
Vol 229 (15) ◽  
pp. 2675-2686 ◽  
Author(s):  
K Nareen ◽  
RP Shimpi
Keyword(s):  
Shear Deformation ◽  
Plate Theory ◽  
Plate Thickness ◽  
Three Dimensional ◽  
Theory Of Elasticity ◽  
Governing Equations ◽  
Dimensional Theory ◽  
Classical Plate Theory ◽  
Shear Deformation Plate Theory ◽  
Elasticity Solutions

The paper presents a novel shear deformation plate theory involving only two variables. Taking a cue from exact three-dimensional theory of elasticity solutions for a plate, hyperbolic functions are used for describing displacement variation across plate thickness. The theory involves only two governing equations, which are uncoupled for statics and are only inertially coupled for dynamics. The shear stress free surface conditions are satisfied. No shear correction factor is required. The theory is variationally consistent, has a strong similarity with classical plate theory, and is simple, yet accurate. Illustrative examples for free vibration and for static flexure demonstrate the effectiveness of the theory.

Analytical Solution for Buckling Behavior of FGM Plate Considering Surface Effect Based on General Third-Order Plate Theory and Non-local Theory

2020 ◽  
Vol 20 (11) ◽  
pp. 2050125
Author(s):  
J. R. Zhong
Keyword(s):  
Surface Effect ◽  
Plate Theory ◽  
Plate Thickness ◽  
Surface Elasticity ◽  
Local Theory ◽  
Solution Technique ◽  
Surface Residual Stress ◽  
Third Order ◽  
Stress Surface ◽  
Non Local

In this paper, the buckling characteristic of FGM plate considering the surface effect is studied based on general third-order plate theory and non-local theory. The surface effect of FGM plate is captured by the surface elasticity theory. The Kirchhoff hypothesis is released by employing parabolic variation of transverse shear strains. By using Navier solution technique, analytical solutions of buckling loads of FGM plate with surface effect are given, and detailed parametric studies are presented to show the relationship between surface effects and the plate thickness, power-law index, surface residual stress, surface moduli and non-local parameter. Furthermore, the surface effect on the buckling characteristic of FGM plate is also discussed.

Experimental Investigation of Nonlinear Vibration of a Thin Rectangular Plate

2019 ◽  
Vol 11 (06) ◽  
pp. 1950059 ◽  
Author(s):  
Sohayb Abdulkerim ◽  
Athanasios Dafnis ◽  
Hans-G Riemerdes
Keyword(s):  
Rectangular Plate ◽  
Plate Theory ◽  
Plate Thickness ◽  
Extraction Procedure ◽  
Time History ◽  
Laser Vibrometer ◽  
Classical Plate Theory ◽  
Thin Rectangular Plate ◽  
Sinusoidal Input ◽  
History Of

In this paper, the geometrical nonlinear vibrations of a rectangular plate have been investigated experimentally and numerically. The experiment was conducted on a thin rectangular plate. The plate was excited close to the first fundamental natural frequency. The time history of velocities of the central point has been measured by using a laser vibrometer. While the numerical investigation has been carried using the Finite Element Method (FEM), the numerical results are validated by analytical and experimental results. In order to develop and test the extraction procedure of the bifurcation plot of a dynamical system, a chaotic pendulum has been analyzed. Then, the same successful code has been used again for the experimental dynamics of the investigated plate. The plate has been forced with a sinusoidal input at a gradually stepped and increased amplitude. For every step, the phase portrait is determined, and then processed to extract the bifurcation map. The resulted map has shown successfully the linear range where the classical plate theory is adequate, and the boundary at which the transition to nonlinearity has occurred. The bifurcation has occurred when the lateral amplitude has reached 50% of the plate thickness.

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