scholarly journals Free Vibration Analysis of Arbitrary-Shaped Plates Based on the Improved Rayleigh–Ritz Method

2019 ◽  
Vol 2019 ◽  
pp. 1-14
Author(s):  
Wenjie Guo ◽  
Qingsong Feng
Keyword(s):  
Free Vibration ◽  
Strain Energy ◽  
Ritz Method ◽  
Free Vibration Analysis ◽  
Rectangular Domain ◽  
Structural Domain ◽  
Engineering Problems ◽  
Elastic Potential Energy ◽  
Practical Engineering ◽  
Admissible Functions

In this investigation, an improved Rayleigh–Ritz method is put forward to analyze the free vibration characteristics of arbitrary-shaped plates for the traditional Rayleigh–Ritz method which is difficult to solve. By expanding the domain of admissible functions out of the structural domain to form a rectangular domain, the admissible functions of arbitrary-shaped plates can be described conveniently by selecting the appropriate admissible functions. Adopting the spring model to simulate the general boundary conditions, the problems of vibration of the arbitrary plate domain can be solved perfectly. Then, a numerical method is introduced to figure out the structure strain energy, kinetic energy, and elastic potential energy of the boundary. Finally, comparing the result with the simulation results and reference examples, the accuracy and convergence of this method are testified. Therefore, an effective new method is proposed for the guidance of the related research and practical engineering problems.

Free Vibration Analysis of Functionally Graded Cylindrical Shells Stiffened by Uniformly and Non-Uniformly Disturbuted Ring Stiffeners

2009 ◽  
Author(s):  
S. A. Moeini ◽  
M. Rahaeifard ◽  
M. T. Ahmadian ◽  
M. R. Movahhedy
Keyword(s):  
Free Vibration ◽  
Functionally Graded Materials ◽  
Vibration Analysis ◽  
Strain Energy ◽  
Ritz Method ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Shear Stresses ◽  
Graded Materials ◽  
Energy Evaluation

Free vibration analysis of a transversely stiffened circular thin hollow cylinder made of functionally graded materials (FGMs) is analytically evaluated. Functionally graded materials are inhomogeneous composites which are usually made from a mixture of metal and ceramic. The gradient compositional variation of the constituents from one surface to the other provides an elegant solution to the problem of high transverse shear stresses induced when two dissimilar materials with large differences in material properties are bonded. In this paper, application of an FGM made of two different materials is investigated by applying Ritz method. While cylinder is assumed to be thin, strain energy evaluation is performed by Sander’s theorem. Stiffeners which are not necessarily in the same uniform shape are treated as discrete elements and can be placed on both sides of the cylinder or concentrate in the middle wall. Bending, stretching and wrapping effects of stiffeners are considered in calculation of strain energy. Evaluation of kinetic energy of stiffeners is performed by taking into account rotary and translational inertia. To apply Ritz method, polynomial functions are used and natural frequencies and mode shapes of ring stiffened thin cylinder are investigated. Results are compared and verified with previous theoretical and experimental studies of stiffened thin cylinders. Comparison indicates a good agreement between results.

Free Vibration Analysis of Rectangular Thin Plate With Multiple Openings Under General Boundary Conditions

2018 ◽  
Author(s):  
Rui Nie ◽  
Tianyun Li ◽  
Xiang Zhu ◽  
Wenjie Guo
Keyword(s):  
Free Vibration ◽  
Thin Plate ◽  
Ritz Method ◽  
Free Vibration Analysis ◽  
Energy Functional ◽  
General Boundary ◽  
Variation Method ◽  
Rectangular Thin Plate ◽  
Functional Variation ◽  
Elastic Potential Energy

The free vibration characteristics of a rectangular thin plate with series of openings are studied based on the Rayleigh-Ritz method in this paper. Firstly, the strain energy and kinetic energy of the plate are calculated utilizing the modified Fourier series. Then, uniformly distributed transitional and rotational springs are applied to deal with general boundary supports, and the elastic potential energy of the springs can be obtained. Furthermore, the plate is divided into several parts according to its amounts of openings and the energy of each parts are calculated separately and the spring stiffness of cut line between two separated parts are the same. Finally, the governing equation of the plate is obtained with the energy functional variation method. The present method is proved to be accurate by comparing the natural frequencies with those calculated by the finite element method. Besides, the influence of the amounts of openings and the area of total openings are discussed.

Theoretical Study on Space Coupling Free Vibration Analysis of Self-Anchored Cable-Stayed Suspension Bridge

2014 ◽  
Vol 501-504 ◽  
pp. 1107-1111
Author(s):  
Miao Feng
Keyword(s):  
Free Vibration ◽  
Strain Energy ◽  
Coupling Effect ◽  
Longitudinal Vibration ◽  
Free Vibration Analysis ◽  
Flexural Vibration ◽  
Suspension Bridge ◽  
Large Displacement ◽  
Generalized Variational Principle ◽  
Suspension Bridges

Based on Large-displacement Non-linear Elastic Generalized Variational Principle, coupling effect of axial and flexural action, shearing strain energy, torsional strain energy of stiffening girder were considered, the large-displacement incomplete generalized potential energy functional of space coupling free vibration of a three-span self-anchored cable-stayed suspension bridge was presented. By constraint variation, fundamental differential equations of vertical flexural vibration, lateral flexural vibration, longitudinal vibration and torisional vibration were formulated, also presented the equations for the main tower with respect to longitudinal and lateral vibration. The linear free vibration differential equation was obtained when the nonlinear items were discarded. This approach provides theoretical basis for analysis of natural vibration character of self-anchored cable-stayed suspension bridges.

scholarly journals Reflections on the Hysteretic Damping Model

2009 ◽  
Vol 16 (5) ◽  
pp. 529-542 ◽  
Author(s):  
Nuno Maia
Keyword(s):  
Free Vibration ◽  
Point Of View ◽  
The Other ◽  
Hysteretic Damping ◽  
Engineering Problems ◽  
Practical Engineering ◽  
The Subject ◽  
The One ◽  
Damping Model ◽  
Free Vibration Response

This paper presents a reflection on a recently proposed solution to the problem of the free vibration response with the constant hysteretic damping model, that has been presented in some conferences in recent years, by the author himself and some of his colleagues. On the one hand, as expected, the subject has been received with natural criticism, mainly due to the well-known non-causal behaviour of the model in free vibration. On the other hand, it was not easy to understand what could be wrong in that proposal, as apparently everything was perfect from a mathematical point of view. The author decided that this subject deserved a more careful and detailed analysis and – in this kind of tutorial paper – the issue seems to have been clarified. It is concluded that the proposed solution involving the constant hysteretic damping corresponds in fact to an equivalent viscously damped model; it is therefore concluded that the application of the constant hysteretic damping to model the free vibration of practical engineering problems should be considered only in the perspective of an equivalent viscously damped model.

scholarly journals Correction to: Fractional strain energy and its application to the free vibration analysis of a plate

2018 ◽  
Vol 25 (6) ◽  
pp. 2239-2239
Author(s):  
Zaher Rahimi ◽  
Siros Shafiei ◽  
Wojciech Sumelka ◽  
Ghader Rezazadeh
Keyword(s):  
Free Vibration ◽  
Vibration Analysis ◽  
Strain Energy ◽  
Free Vibration Analysis

Free vibration analysis of functionally graded double-tapered beam rotating in thermal environment considering geometric nonlinearity, shear deformability, and Coriolis effect

2017 ◽  
Vol 232 (12) ◽  
pp. 2244-2262 ◽  
Author(s):  
Suman Pal ◽  
Debabrata Das
Keyword(s):  
Mathematical Model ◽  
Free Vibration ◽  
Material Properties ◽  
Volume Fraction ◽  
Thermal Environment ◽  
Ritz Method ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Coriolis Effect ◽  
Governing Equations

The present work investigates the free vibration behavior of double-tapered functionally graded beams rotating in thermal environment, using an improved mathematical model. The functional gradation for ceramic–metal compositions, following power-law, is considered to be symmetric with respect to the mid-plane, leading to metal-rich core and ceramic-rich outer surfaces of the beam. The temperature dependence of the material properties are considered using Touloukian model. The nonlinearity in strain–displacement relationships for both the axial and transverse shear strains are considered. Firstly, the governing equations for deformed beam configuration under time-independent centrifugal loading are obtained using minimum total potential energy principle, and the solution is obtained following Ritz method. Then the free vibration problem of the centrifugally deformed beam is formulated employing Lagrange’s principle and considering tangent stiffness of the deformed beam configuration. Coriolis effect is considered in the mathematical model, and the governing equations are transformed to the state-space for obtaining an eigenvalue problem. The results for the first two modes of both chord-wise and flap-wise vibrations are presented in nondimensional plane to show the effects of taperness parameter, root-offset parameter, volume fraction exponent, operating temperature, and functionally graded material composition. The results in comparative form are presented for both temperature-dependent and temperature-independent material properties.

Vibration and Buckling Response of Width Tapered Laminated Composite Beams Using Ritz Method for Rotorcraft Blade

2010 ◽  
Author(s):  
Vijay Kumar Badagi ◽  
Rajamohan Ganesan
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Composite Beam ◽  
Ritz Method ◽  
Compressive Force ◽  
Laminated Composite ◽  
Free Vibration Analysis ◽  
Composite Beams ◽  
Buckling Analysis ◽  
Laminated Composite Beam

In this study, Symmetric cross-ply linear width tapered laminated composite beam is considered. Due to the variety of width tapered composite beams and the complexity of the analysis, no closed-form analytical solution is available at present regarding free vibration response. Therefore in the present work, the Ritz method is used for the free vibration analysis with considering uni-axial compressive and tensile force. The elastic stiffness of the width tapered composite beam is analyzed compared to uniform laminated composite beam. Free vibration which is significant to investigate the dynamic characteristics of the structure using Ritz method with and without effect of axial tensile and compressive force is analyzed. The analysis is based on 1D laminated beam theory. The governing equations are obtained by means of Hamilton’s principle. Tsai-Wu failure analysis is considered to find the tensile and compressive failure force for each ply in the laminate. Buckling analysis is conducted to find the critical buckling force for the laminated composite beam-column subjected to different sets of boundary conditions. Simply supported, Clamped-free, Clamped-Clamped edge boundary conditions are considered. A detailed parametric study is conducted on tapered composite beams made of NCT/301 graphite-epoxy to investigate the effects of the ratio of the width of the thick section to thin section, boundary conditions, effects of axial and compressive force on natural frequency and buckling analysis.

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