Effect of Winkler and Pasternak elastic foundation on the vibration of rotating functionally graded material cylindrical shell

2018 ◽  
Vol 232 (24) ◽  
pp. 4564-4577 ◽  
Author(s):  
Muzamal Hussain ◽  
Muhammad Nawaz Naeem ◽  
Mohammad Reza Isvandzibaei
Keyword(s):  
Cylindrical Shell ◽  
Boundary Conditions ◽  
Elastic Foundation ◽  
Natural Frequencies ◽  
Cylindrical Shells ◽  
Functionally Graded ◽  
Radius Ratio ◽  
Wave Number ◽  
Simply Supported ◽  
Elastic Foundations

In this paper, vibration characteristics of rotating functionally graded cylindrical shell resting on Winkler and Pasternak elastic foundations have been investigated. These shells are fabricated from functionally graded materials. Shell dynamical equations are derived by using the Hamilton variational principle and the Langrangian functional framed from the shell strain and kinetic energy expressions. Elastic foundations, namely Winkler and Pasternak moduli are inducted in the tangential direction of the shell. The rotational motions of the shells are due to the Coriolis and centrifugal acceleration as well as the hoop tension produced in the rotating case. The wave propagation approach in standard eigenvalue form has been employed in order to derive the characteristic frequency equation describing the natural frequencies of vibration in rotating functionally graded cylindrical shell. The complex exponential functions, with the axial modal numbers that depend on the boundary conditions stated at edges of a cylindrical shell, have been used to compute the axial modal dependence. In our new investigation, frequency spectra are obtained for circumferential wave number, length-to-radius ratio, height-to-radius ratio with simply supported–simply supported and clamped–clamped boundary conditions without elastic foundation. Also, the effect of elastic foundation on the rotating cylindrical shells is examined with the simply supported–simply supported edge. To check the validity of the present method, the fundamental natural frequencies of non-rotating isotropic and functionally graded cylindrical shells are compared with the open literature. Also, a comparison is made for infinitely long rotating with the earlier published paper.

Vibrations of rotating cylindrical shells with functionally graded material using wave propagation approach

2018 ◽  
Vol 232 (23) ◽  
pp. 4342-4356 ◽  
Author(s):  
Muzammal Hussain ◽  
M Nawaz Naeem ◽  
Aamir Shahzad ◽  
Mao-Gang He ◽  
Siddra Habib
Keyword(s):  
Wave Propagation ◽  
Natural Frequencies ◽  
Volume Fraction ◽  
Cylindrical Shells ◽  
Functionally Graded ◽  
Wave Number ◽  
Type I ◽  
Rotating Speed ◽  
Simply Supported ◽  
Fundamental Frequencies

Fundamental natural frequencies of rotating functionally graded cylindrical shells have been calculated through the improved wave propagation approach using three different volume fraction laws. The governing shell equations are obtained from Love’s shell approximations using improved rotating terms and the new equations are obtained in standard eigenvalue problem with wave propagation approach and volume fraction laws. The effects of circumferential wave number, rotating speed, length-to-radius, and thickness-to-radius ratios have been computed with various combinations of axial wave numbers and volume fraction law exponent on the fundamental natural frequencies of nonrotating and rotating functionally graded cylindrical shells using wave propagation approach and volume fraction laws with simply supported edge. In this work, variation of material properties of functionally graded materials is controlled by three volume fraction laws. This process creates a variation in the results of shell frequency. MATLAB programming has been used to determine shell frequencies for traveling mode (backward and forward) rotating motions. New estimations show that the rotating forward and backward simply supported fundamental natural frequencies increases with an increase in circumferential wave number, for Type I and Type II of functionally graded cylindrical shells. The presented results of backward and forward simply supported fundamental natural frequencies corresponding to Law I are higher than Laws II and III for Type I and reverse effects are found for Type II, depending on rotating speed. Our investigations show that the decreasing and increasing behaviors are noted for rotating simply supported fundamental natural frequencies with increasing length-to-radius and thickness-to-radius ratios, respectively. It is found that the fundamental frequencies of the forward waves decrease with the increase in the rotating speed, and the fundamental frequencies of the backward waves increase with the increase in the rotating speed. This investigation has been made with three different volume fraction laws of polynomial (Law I), exponential (Law II), and trigonometric (Law III). The presented numerical results of nonrotating isotropic and rotating functionally graded simply supported are in fair agreement with parts of other earlier numerical results.

A semi-analytical three-dimensional free vibration analysis of functionally graded curved panels integrated with piezoelectric layers

2014 ◽  
Vol 21 (4) ◽  
pp. 571-587 ◽  
Author(s):  
Hamid Reza Saeidi Marzangoo ◽  
Mostafa Jalal
Keyword(s):  
Boundary Conditions ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Differential Quadrature Method ◽  
Three Dimensional ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Simply Supported ◽  
Piezoelectric Layers ◽  
Curved Panels

AbstractFree vibration analysis of functionally graded (FG) curved panels integrated with piezoelectric layers under various boundary conditions is studied. A panel with two opposite edges is simply supported, and arbitrary boundary conditions at the other edges are considered. Two different models of material property variations based on the power law distribution in terms of the volume fractions of the constituents and the exponential law distribution of the material properties through the thickness are considered. Based on the three-dimensional theory of elasticity, an approach combining the state space method and the differential quadrature method (DQM) is used. For the simply supported boundary conditions, closed-form solution is given by making use of the Fourier series expansion, and applying the differential quadrature method to the state space formulations along the axial direction, new state equations about state variables at discrete points are obtained for the other cases such as clamped or free-end conditions. Natural frequencies of the hybrid curved panels are presented by solving the eigenfrequency equation, which can be obtained by using edges boundary conditions in this state equation. The results obtained for only FGM shell is verified by comparing the natural frequencies with the results obtained in the literature.

Buckling analysis of functionally graded annular sector plates resting on elastic foundations

2010 ◽  
Vol 225 (2) ◽  
pp. 312-325 ◽  
Author(s):  
A Naderi ◽  
A R Saidi
Keyword(s):  
Boundary Conditions ◽  
Elastic Foundation ◽  
Buckling Analysis ◽  
Functionally Graded ◽  
Buckling Load ◽  
Critical Buckling Load ◽  
Elastic Foundations ◽  
Annular Sector ◽  
Sector Plate ◽  
Annular Sector Plate

In this study, an analytical solution for the buckling of a functionally graded annular sector plate resting on an elastic foundation is presented. The buckling analysis of the functionally graded annular sector plate is investigated for two typical, Winkler and Pasternak, elastic foundations. The equilibrium and stability equations are derived according to the Kirchhoff's plate theory using the energy method. In order to decouple the highly coupled stability equations, two new functions are introduced. The decoupled equations are solved analytically for a plate having simply supported boundary conditions on two radial edges. Satisfying the boundary conditions on the circular edges of the plate yields an eigenvalue problem for finding the critical buckling load. Extensive results pertaining to critical buckling load are presented and the effects of boundary conditions, volume fraction, annularity, plate thickness, and elastic foundation are studied.

Effect of Porosity on Flexural Vibration of CNT-Reinforced Cylindrical Shells in Thermal Environment Using GDQM

2018 ◽  
Vol 18 (10) ◽  
pp. 1850123 ◽  
Author(s):  
Hamed Safarpour ◽  
Kianoosh Mohammadi ◽  
Majid Ghadiri ◽  
Mohammad M. Barooti
Keyword(s):  
Cylindrical Shell ◽  
Boundary Conditions ◽  
Thermal Loading ◽  
Thermal Environment ◽  
Differential Quadrature Method ◽  
Cylindrical Shells ◽  
Flexural Vibration ◽  
Distribution Patterns ◽  
Functionally Graded ◽  
Equivalent Material

This article investigates the flexural vibration of temperature-dependent and carbon nanotube-reinforced (CNTR) cylindrical shells made of functionally graded (FG) porous materials under various kinds of thermal loadings. The equivalent material properties of the cylindrical shell of concern are estimated using the rule of mixture. Both the cases of uniform distribution (UD) and FG distribution patterns of reinforcements are considered. Thermo-mechanical properties of the cylindrical shell are supposed to vary through the thickness and are estimated using the modified power-law rule, by which the porosities with even and uneven types are approximated. As the porosities occur inside the FG materials during the manufacturing process, it is necessary to consider their impact on the vibration behavior of shells. The present study is featured by consideration of different types of porosities in various CNT reinforcements under various boundary conditions in a single model. The governing equations and boundary conditions are developed using Hamilton's principle and solved by the generalized differential quadrature method. The accuracy of the present results is verified by comparison with existing ones and those by Navier's method. The results show that the length to radius ratio and temperature, as well as CNT reinforcement, porosity, thermal loading, and boundary conditions, play an important role on the natural frequency of the cylindrical shell of concern in thermal environment.

scholarly journals Modeling the Size Effect on the Mechanical Behavior of Functionally Graded Curved Micro/Nanobeam

2018 ◽  
Vol 1 (2) ◽  
Author(s):  
Seyyed Amirhosein Hosseini1 ◽  
O. Rahmani2
Keyword(s):  
Size Effect ◽  
Boundary Conditions ◽  
Natural Frequencies ◽  
Power Index ◽  
Nonlocal Parameter ◽  
Functionally Graded ◽  
Opening Angle ◽  
Simply Supported ◽  
Fg Nanobeam ◽  
Good Agreement

The size effect on the free vibration and bending of a curved FG micro/nanobeam is studied in this paper. Using the Hamilton principle the differential equations and boundary conditions is derived for a nonlocal Euler-Bernoulli curved micro/nanobeam.  The material properties vary through radius direction. Using the Navier approach an analytical solution for simply supported boundary conditions is obtained where the power index law of FGM, the curved micro/nanobeam opening angle, the effect of aspect ratio and nonlocal parameter on natural frequencies and the radial and tangential displacements were analyzed. It is concluded that increasing the curved micro/nanobeam opening angle results in decreasing and increasing the frequencies and displacements, respectively. To validate the natural frequencies of curved nanobeam, when the radius of it approaches to infinity, is compared with a straight FG nanobeam and showed a good agreement.

scholarly journals Three-Dimensional Vibration Analysis of a Functionally Graded Sandwich Rectangular Plate Resting on an Elastic Foundation Using a Semi-Analytical Method

Materials ◽  
2019 ◽  
Vol 12 (20) ◽  
pp. 3401 ◽  
Author(s):  
Cui ◽  
Zhou ◽  
Ye ◽  
Gaidai ◽  
Li ◽  
...  
Keyword(s):  
Boundary Conditions ◽  
Rectangular Plate ◽  
Elastic Foundation ◽  
Power Law ◽  
Analytical Method ◽  
Sandwich Plate ◽  
Natural Frequencies ◽  
Three Dimensional ◽  
Functionally Graded ◽  
Sandwich Plates

The three-dimensional vibration of a functionally graded sandwich rectangular plate on an elastic foundation with normal boundary conditions was analyzed using a semi-analytical method based on three-dimensional elasticity theory. The material properties of the sandwich plate varied with thickness according to the power law distribution. Two types of functionally graded material (FGM) sandwich plates were investigated in this paper: one with a homogeneous core and FGM facesheets, and another with homogeneous panels and an FGM core. Various displacements of the plates were created using an improved Fourier series consisting of a standard Fourier cosine series along with a certain number of closed-form auxiliary functions satisfying the essential boundary conditions. The vibration behavior of the FGM sandwich plate, including the natural frequencies and mode shapes, was obtained using the Ritz method. The effectiveness and accuracy of the suggested technique were fully verified by comparing the natural frequencies of sandwich plates with results from investigations of other functionally graded sandwich rectangular plates in the literature. A parametric study, including elastic parameters, foundation parameters, power law exponents, and layer thickness ratios, was performed, and some new results are presented.

scholarly journals Vibration of FG Shell Rested on Winkler Foundation

2020 ◽  
Author(s):  
Abdellatif Selmi
Keyword(s):  
Cylindrical Shell ◽  
Elastic Foundation ◽  
Functionally Graded ◽  
Wave Number ◽  
Winkler Foundation ◽  
Shell Material ◽  
Graded Materials ◽  
Maximum Value ◽  
Graded Material ◽  
Point To Point

Abstract Love’s first a pproximation theory is employed with the combination of Winkler term for the vibration of functionally graded cylindrical shell. MATLAB software is utilized for the vibration of functionally graded cylindrical shell with elastic foundation of Winkler and t he results are verified with the open literature. For isotropic materials, the physical properties are same everywhere where the laminated and functionally graded materials, they vary from point to point. Here the shell material has been taken as functiona lly graded material. The influence of the elastic foundation, wave number, length - and height - to - radius ratios is investigated with different boundary conditions. The frequencies of length - to - radius and height - to - radius ratio are counter part of each other . The frequency first increases and gain maximum value in the midway of the shell length and then lowers down for the variations of wave number. It is found that due to inducting the elastic foundation of Winkler, the frequencies increases.

scholarly journals Frequency Analysis for Functionally Graded Material Cylindrical Shells: A Significant Case Study

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Rabia Anwar ◽  
Madiha Ghamkhar ◽  
Muhammad Imran Khan ◽  
Rabia Safdar ◽  
Muhammad Zafar Iqbal ◽  
...  
Keyword(s):  
Functionally Graded Materials ◽  
Natural Frequencies ◽  
Constitutive Relations ◽  
Cylindrical Shells ◽  
Frequency Equation ◽  
Functionally Graded ◽  
Graded Materials ◽  
Simply Supported ◽  
Love’S Shell Theory ◽  
Graded Material

Cylindrical shells play an important role for the construction of functionally graded materials (FGMs). Functionally graded materials are valuable in order to develop durable materials. They are made of two or more materials such as nickel, stainless steel, zirconia, and alumina. They are extremely beneficial for the manufacturing of structural elements. Functionally graded materials are broadly used in several fields such as chemistry, biomedicine, optics, and electronics. In the present research, vibrations of natural frequencies are investigated for different layered cylindrical shells, those constructed from FGMs. The behavior of shell vibration is based on different parameters of geometrical material. The problem of the shell is expressed from the constitutive relations of strain and stress with displacement, as well as it is adopted from Love’s shell theory. Vibrations of natural frequencies (NFs) are calculated for simply supported-simply supported (SS-SS) and clamped-free (C-F) edge conditions. The Rayleigh–Ritz technique is employed to obtain the shell frequency equation. The shell equation is solved by MATLAB software.

Dynamic Responses of Functionally Graded Sandwich Beams Resting on Elastic Foundation Under Harmonic Moving Loads

2018 ◽  
Vol 18 (09) ◽  
pp. 1850112 ◽  
Author(s):  
Wachirawit Songsuwan ◽  
Monsak Pimsarn ◽  
Nuttawit Wattanasakulpong
Keyword(s):  
Boundary Conditions ◽  
Elastic Foundation ◽  
Natural Frequencies ◽  
Moving Load ◽  
Beam Theory ◽  
Equation Of Motion ◽  
Functionally Graded ◽  
Dynamic Responses ◽  
Harmonic Load ◽  
Sandwich Beams

This paper investigates the free vibration and dynamic response of functionally graded sandwich beams resting on an elastic foundation under the action of a moving harmonic load. The governing equation of motion of the beam, which includes the effects of shear deformation and rotary inertia based on the Timoshenko beam theory, is derived from Lagrange’s equations. The Ritz and Newmark methods are employed to solve the equation of motion for the free and forced vibration responses of the beam with different boundary conditions. The results are presented in both tabular and graphical forms to show the effects of layer thickness ratios, boundary conditions, length to height ratios, spring constants, etc. on natural frequencies and dynamic deflections of the beam. It was found that increasing the spring constant of the elastic foundation leads to considerable increase in natural frequencies of the beam; while the same is not true for the dynamic deflection. Additionally, very large dynamic deflection occurs for the beam in resonance under the harmonic moving load.

scholarly journals Dynamic Stability of Bi-Directional Functionally Graded Porous Cylindrical Shells Embedded in an Elastic Foundation

2020 ◽  
Vol 10 (4) ◽  
pp. 1345 ◽  
Author(s):  
Farshid Allahkarami ◽  
Hasan Tohidi ◽  
Rossana Dimitri ◽  
Francesco Tornabene
Keyword(s):  
Boundary Conditions ◽  
Dynamic Stability ◽  
Elastic Foundation ◽  
Dynamic Instability ◽  
Cylindrical Shells ◽  
Shear Deformation Theory ◽  
Functionally Graded ◽  
Practical Applications ◽  
Energy Devices ◽  
Various Boundary Conditions

This paper investigates the dynamic buckling of bi-directional (BD) functionally graded (FG) porous cylindrical shells for various boundary conditions, where the FG material is modeled by means of power law functions with even and uneven porosity distributions of ceramic and metal phases. The third-order shear deformation theory (TSDT) is adopted to derive the governing equations of the problem via the Hamilton’s principle. The generalized differential quadrature (GDQ) method is applied together with the Bolotin scheme as numerical strategy to solve the problem, and to draw the dynamic instability region (DIR) of the structure. A large parametric study examines the effect of different boundary conditions at the extremities of the cylindrical shell, as well as the sensitivity of the dynamic stability to different thickness-to-radius ratios, length-to-radius ratios, transverse and longitudinal power indexes, porosity volume fractions, and elastic foundation constants. Based on results, the dynamic stability of BD-FG cylindrical shells can be controlled efficiently by selecting appropriate power indexes along the desired directions. Furthermore, the DIR is highly sensitive to the porosity distribution and to the extent of transverse and longitudinal power indexes. The numerical results could be of great interest for many practical applications, as civil, mechanical or aerospace engineering, as well as for energy devices or biomedical systems.

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