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Free Vibration Analysis of Functionally Graded Beams with General Elastically End Constraints by DTM  [PDF]
Nuttawit Wattanasakulpong, Variddhi Ungbhakorn
World Journal of Mechanics (WJM) , 2012, DOI: 10.4236/wjm.2012.26036
Abstract: The differential transformation method (DTM) is applied to investigate free vibration of functionally graded beams supported by arbitrary boundary conditions, including various types of elastically end constraints. The material properties of functionally graded beams are assumed to obey the power law distribution. The main advantages of this method are known for its excellence in high accuracy with small computational expensiveness. The DTM also provides all natural frequencies and mode shapes without any frequency missing. Fundamental frequencies as well as their higher frequencies and mode shapes are presented. The significant aspects such as boundary conditions, values of translational and rotational spring constants and the material volume fraction index on the natural frequencies and mode shapes are discussed. For elastically end constraints, some available results of special cases for isotropic beams are used to validate the present results. The new frequency results and mode shapes of functionally graded beams resting on elastically end constraints are presented.
非均匀Winkler弹性地基上变厚度矩形板自由振动的DTM求解 Free vibration analysis for rectangular plates with variable thickness resting on a non-uniform Winkler elastic foundation by DTM
Free vibration analysis for rectangular plates with variable thickness resting on a non-uniform Winkler elastic foundation by DTM

- , 2018, DOI: 10.7511/jslx20170217002
Abstract: 针对非均匀Winkler弹性地基上变厚度矩形板的自由振动问题,通过一种有效的数值求解方法——微分变换法(DTM),研究其量纲固有频率特性。已知变厚度矩形板对边为简支边界条件,其他两边的边界条件为简支、固定或自由任意组合。采用DTM将非均匀Winkler弹性地基上变厚度矩形板量纲化的自由振动控制微分方程及其边界条件变换为等价的代数方程,得到含有量纲固有频率的特征方程。数值结果退化为均匀Winker弹性地基上矩形板以及变厚度矩形板的情形,并与已有文献采用的不同求解方法进行比较,结果表明,DTM具有非常高的精度和很强的适用性。最后,在不同边界条件下分析地基变化参数、厚度变化参数和长宽比对矩形板量纲固有频率的影响,并给出了非均匀Winkler弹性地基上对边简支对边固定变厚度矩形板的前六阶振型。
For free vibration problem of rectangular plates with variable thickness resting on a non-uniform foundation and by an effective solving numerical method called differential transformation method (DTM),the dimensionless natural frequency characteristics are investigated.Two opposite edges of plates are assumed to be simply supported and other two edges can be changed into simply supported,camped or free boundary conditions arbitrarily.By using DTM,dimensionless normalized governing differential equation of rectangular plates with variable thickness resting on a non-uniform Winkler elastic foundation and boundary conditions are transformed to the equivalent algebraic equations,which can derive equations of dimensionless natural frequency.The example results are back to cases for uniform Winkler rectangular plates and rectangular plates with variable thickness,which are compared with different methods in present literature.The result shows that DTM have very higher accuracy and stronger applicability.Finally,the influence of the varied foundation parameter,the varied thickness parameter and the aspect ratio on dimensionless natural frequencies are analyzed for different boundary conditions and deriving the first six mode shapes for CSCS plate with variable thickness resting on a non-uniform Winkler elastic foundations.
Thermomechanical buckling oftemperature-dependent FGM beams
Kiani, Y.;Eslami, M.R.;
Latin American Journal of Solids and Structures , 2013, DOI: 10.1590/S1679-78252013000200001
Abstract: buckling of beams made of functionally graded materials (fgm) under thermomechanical loading is analyzed herein. properties of the constituents are considered to be functions of temperature and thickness coordinate. the derivation of the equations is based on the timoshenko beam theory, where the effect of shear is included. it is assumed that the mechanical and thermal nonhomogeneous properties of beam vary smoothly by distribution of the power law index across the thickness of the beam. the equilibrium and stability equations for an fgm beam are derived and the existence of bifurcation buckling is examined. the beam is assumed under three types of thermal loadings; namely, the uniform temperature rise, heat conduction across the thickness, and linear distribution across the thickness. various types of boundary conditions are assumed for the beam with combination of roller, clamped, and simply-supported edges. in each case of boundary conditions and loading, closed form solutions for the critical buckling temperature of the beam is presented. the results are compared with the isotropic homogeneous beams, that are reported in the literature, by reducing the results of the functionally graded beam to the isotropic homogeneous beam.
Vibration Control of FGM Piezoelectric Plate Based on LQR Genetic Search  [PDF]
Kouider Bendine, Rajan L. Wankhade
Open Journal of Civil Engineering (OJCE) , 2016, DOI: 10.4236/ojce.2016.61001
Abstract: Active vibration control of functionally graded material (FGM) plate with integrated piezoelectric layers is studied. In this regard, a finite element model based on the classical plate theory is adopted and extended to the case of FGM plate to obtain a space state equation. Rectangular four node and eight node elements are used for the analysis purpose. The material proprieties of FG plate are assumed to be graded along the thickness direction. In order to control the vibration of the plate, an LQR controller has been designed and developed. The weighing factors are obtained by using genetic algorithm. The proposed results of finite element modeling are verified with the results obtained using ANSYS. Also the validation of methodology is done with comparing the results with that of available in literature and found in well agreement. Further analysis is performed for three sets of power law exponent n = 0, 1 and 100 which gives benchmark results for vibration control of FGM piezoelectric plate.
Vibrations of Three-Layered Cylindrical Shells with FGM Middle Layer Resting on Winkler and Pasternak Foundations  [PDF]
Abdul Ghafar Shah,Aalia Ali,Muhmmad Nawaz Naeem,Shahid Hussain Arshad
Advances in Acoustics and Vibration , 2012, DOI: 10.1155/2012/507031
Abstract: Vibrations of a cylindrical shell composed of three layers of different materials resting on elastic foundations are studied out. This configuration is formed by three layers of material in thickness direction where the inner and outer layers are of isotropic materials and the middle is of functionally graded material. Love shell dynamical equations are considered to describe the vibration problem. The expressions for moduli of the Winkler and Pasternak foundations are combined with the shell dynamical equations. The wave propagation approach is used to solve the present shell problem. A number of comparisons of numerical results are performed to check the validity and accuracy of the present approach. 1. Introduction The circular cylindrical shells have been found in various engineering applications ranging from large civil and mechanical structures to small electrical components for many years. Vibrations of cylindrical shells are the most wide studied area of research because of their simple geometrical shapes. First of all, Love [1] presented the linear thin shell theory established on the Kirchhoff’s hypothesis for plates. All other shell theories have been built on this theory by modifying some physical terms involving therein. Numerical solutions of shell vibration problem started to come out in the thirties of the twentieth century and were presented by Flügge [2]. Sharma [3, 4] gave an analysis of thin circular cylinders. Sharma approximated the axial model dependence by beam functions and they used Rayleigh-Ritz technique to solve the shell problem. Paliwal et al. [5] investigated the vibrations of a thin circular cylindrical shell attached with elastic foundations. Loy et al. [6] investigated the vibrations of functionally graded material cylindrical shells, made up of FG material composed of stainless steel and nickel. The purpose of work was to examine natural frequencies, influence of the constituent volume fractions, and effects of configurations of constituent materials on their frequencies. Pardhan et al. [7] studied the vibrations of functionally graded material (FGM) cylindrical shell structured from stainless steel and zirconia. Zhang et al. [8] studied the vibration frequencies of cylindrical shells with fluid-filled. They used wave propagation approach and they compared the uncoupled frequencies with available results in the literature. Najafizadeh and Isvandzibaei [9] studied the vibrations of thin-walled cylindrical shells with ring supports composed of functionally graded material comprised of stainless steel and nickel. Arshad
FGM环扇形板的面内自由振动分析 In-plane free vibration analysis of FGM annular sector plates
In-plane free vibration analysis of FGM annular sector plates

- , 2018, DOI: 10.7511/jslx20170703001
Abstract: 假定功能梯度材料(FGM)的物性参数沿环扇形板径向按照幂律梯度变化,基于平面线弹性理论,建立了FGM环扇形板面内自由振动的运动控制微分方程。采用二维微分求积法(DQM)对FGM环扇形板面内自由振动的量纲运动控制微分方程进行离散,数值求解了不同边界条件下FGM环扇形板面内自由振动的量纲固有频率,同时也给出了FGM环扇形板扇形角为π/4时有限元商用软件ANSYS的部分计算结果,验证了本文方法的正确性。结果表明,在相应边界条件下,FGM环扇形板的梯度指标、内外半径比以及扇形角对量纲固有频率均有影响,其计算结果和分析方法可供设计和研究参考。
Assuming that the material physical parameters of the annular sector plates follows a power law distribution along the radius and based on plane linear theory of elasticity,the governing differential equations of motion for in-plane free vibration of a functionally graded material (FGM) annular sector plate are established.Then using two-dimensional differential quadrature method (DQM),the dimensionless governing differential equations of motion of in-plane free vibration of FGM annular sector plates are discretized and the dimensionless natural frequencies of the in-plane free vibration of FGM annular sector plates under the corresponding boundary conditions are solved numerically.Some results of dimensionless natural frequencies of FGM annular sector plate with π/4 angle of the sector are computed by the finite element commercial software ANSYS.They show that the current method is effective and accurate.When the plates are under the corresponding boundary conditions,the results indicate that the gradient index of FGM,the internal and external radius ratio and the angle of the sector all affect the dimensionless natural frequencies,the calculation results and the analysis method can be used in design and research.
An extended isogeometric analysis for vibration of cracked FGM plates using higher-order shear deformation theory  [PDF]
Loc V. Tran,Vinh Phu Nguyen,M. Abdel Wahab,H. Nguyen-Xuan
Computer Science , 2014,
Abstract: A novel and effective formulation that combines the eXtended IsoGeometric Approach (XIGA) and Higher-order Shear Deformation Theory (HSDT) is proposed to study the free vibration of cracked Functionally Graded Material (FGM) plates. Herein, the general HSDT model with five unknown variables per node is applied for calculating the stiffness matrix without needing Shear Correction Factor (SCF). In order to model the discontinuous and singular phenomena in the cracked plates, IsoGeometric Analysis (IGA) utilizing the Non-Uniform Rational B-Spline (NURBS) functions is incorporated with enrichment functions through the partition of unity method. NURBS basis functions with their inherent arbitrary high order smoothness permit the C1 requirement of the HSDT model. The material properties of the FGM plates vary continuously through the plate thickness according to an exponent function. The effects of gradient index, crack length, crack location, length to thickness on the natural frequencies and mode shapes of simply supported and clamped FGM plate are studied. Numerical examples are provided to show excellent performance of the proposed method compared with other published solutions in the literature.
Internal Vibration and Synchronization of Four Coupled Self-Excited Elastic Beams  [PDF]
Miguel A. Barron
Open Journal of Applied Sciences (OJAppS) , 2016, DOI: 10.4236/ojapps.2016.68050
Abstract: The vibration behavior and the synchronization between some internal points of four coupled self-excited beams are numerically studied. Coupling through the root of the beams is considered. The transverse displacements of the internal points and the beam tips are monitored, and the power spectra of the resulting time series are employed to determine the oscillation frequencies. The synchronization between beams is analyzed using phase portraits and correlation coefficients. Numerical results show multiple frequencies in the vibration pattern, and complex patterns of synchronization between pairs of beams.
Analytical study on the vibration frequencies of tapered beams
Bayat, Mahmoud;Pakar, Iman;Bayat, Mahdi;
Latin American Journal of Solids and Structures , 2011, DOI: 10.1590/S1679-78252011000200003
Abstract: a vast amount of published work can be found in the field of beam vibrations dealing with analytical and numerical techniques. this paper deals with analysis of the nonlinear free vibrations of beams. the problem considered represents the governing equation of the nonlinear, large amplitude free vibrations of tapered beams. a new implementation of the ancient chinese method called the max-min approach (mma) and homotopy perturbation method (hpm) are presented to obtain natural frequency and corresponding displacement of tapered beams. the effect of vibration amplitude on the non-linear frequency is discussed. in the end to illustrate the effectiveness and convenience of the mma and hpm, the obtained results are compared with the exact ones and shown in graphs and in tables. those approaches are very effective and simple and with only one iteration leads to high accuracy of the solutions. it is predicted that those methods can be found wide application in engineering problems, as indicated in this paper.
Effect of Vertical Vibration on Block Foundation Resting on Homogeneous and Layered Medium
Ankesh Kumar, Dr. Bappaditya Manna, Prof. K. S. Rao
International Journal of Engineering Research , 2014,
Abstract: In the present study, an investigation is carried out to determine the effect of soil–rock and rock–rock foundation systems on dynamic response of block foundations under vertical mode of vibration. The half-space theory is used for the analysis of foundation resting on homogeneous soil and rocks. The finite element program having transmitting boundaries is considered for layered system considering soil–rock and rock–rock combinations. The analysis is carried out in details for soil– rock and weathered rock–rock systems and the different equations are presented for above combinations. The effect of top layer thicknesses, shear wave velocity and eccentric moments are also simulated. The rock–rock systems considered are sandstone, shale and limestone underlain by basalt rock. It is interpreted that as the shear wave velocity ratio increase the natural frequency increases and the peak displacement amplitude decreases.
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