The Solution of a Spectral Task on Variable Section Compressed Beams Vibrations by Numerical Methods

2019 ◽  
Vol 974 ◽  
pp. 704-710
Author(s):  
Khusen P. Kulterbaev ◽  
Lyalusya A. Baragunova ◽  
Maryana M. Shogenova ◽  
Maryana A. Shardanova
Keyword(s):  
Boundary Conditions ◽  
Free Vibrations ◽  
Hyperbolic Type ◽  
Fundamental Function ◽  
Algebraic Problem ◽  
Negative Function ◽  
Main Equation ◽  
Variable Section ◽  
Adequate Accuracy ◽  
Null Values

Free flexural free vibrations of variable section are considered. The vibrations mathematical model represents the boundary value problem consisting of the hyperbolic type and boundary conditions main equation. By means of separation method of variables the task at the beginning comes to homogeneous differential equation of the fourth order for fundamental function with the corresponding boundary conditions. The grid area of an argument change and fundamental function in it are applied. That leads to an algebraic problem of eigenvalues. Multimodal non-negative function which null values match its eigenvalues is designed. The finite differences methods and coordinate descent in combination with the specified function sections graphic visualization at a small amount of descents with an adequate accuracy for eigenvalues practice are given. The known ways to define fundamental functions are applied.

scholarly journals Vibration of rotating circular cylindrical shells with distributed springs

2021 ◽  
Vol 37 ◽  
pp. 346-358
Author(s):  
Fuchun Yang ◽  
Xiaofeng Jiang ◽  
Fuxin Du
Keyword(s):  
Boundary Conditions ◽  
Galerkin Method ◽  
Shell Theory ◽  
Rotation Speed ◽  
Cylindrical Shells ◽  
Free Vibrations ◽  
Governing Equations ◽  
Natural Response ◽  
Circular Cylindrical Shells ◽  
The Galerkin Method

Abstract Free vibrations of rotating cylindrical shells with distributed springs were studied. Based on the Flügge shell theory, the governing equations of rotating cylindrical shells with distributed springs were derived under typical boundary conditions. Multicomponent modal functions were used to satisfy the distributed springs around the circumference. The natural responses were analyzed using the Galerkin method. The effects of parameters, rotation speed, stiffness, and ratios of thickness/radius and length/radius, on natural response were also examined.

Natural vibrations of micro beams with nonrigid supports

2016 ◽  
Vol 23 (19) ◽  
pp. 3233-3246 ◽  
Author(s):  
Diana V Bambill ◽  
Graciela I Guerrero ◽  
Daniel H Felix
Keyword(s):  
Boundary Conditions ◽  
Couple Stress Theory ◽  
Continuum Theory ◽  
Modified Couple Stress Theory ◽  
Free Vibrations ◽  
Small Scale ◽  
Stress Theory ◽  
Edge Conditions ◽  
Micro Systems ◽  
Micro Structures

The present study aims to provide some new information for the design of micro systems. It deals with free vibrations of Bernoulli–Euler micro beams with nonrigid supports. The study is based on the formulation of the modified couple stress theory. This theory is a nonclassical continuum theory that allows one to capture the small-scale size effects in the vibrational behavior of micro structures. More realistic boundary conditions are represented with elastic edge conditions. The effect of Poisson’s ratio on the micro beam characteristics is also analyzed. The present results revealed that the characterization of real boundary conditions is much more important for micro beams than for macro beams, and this is an assessment that cannot be ignored.

A Nonlinear Shear Deformable Nanoplate Model Including Surface Effects for Large Amplitude Vibrations of Rectangular Nanoplates with Various Boundary Conditions

2015 ◽  
Vol 07 (05) ◽  
pp. 1550076 ◽  
Author(s):  
Reza Ansari ◽  
Mostafa Faghih Shojaei ◽  
Vahid Mohammadi ◽  
Raheb Gholami ◽  
Mohammad Ali Darabi
Keyword(s):  
Boundary Conditions ◽  
Surface Stress ◽  
Equations Of Motion ◽  
Continuation Method ◽  
Free Vibrations ◽  
Model Parameters ◽  
Geometrically Nonlinear ◽  
Residual Surface Stress ◽  
Various Boundary Conditions ◽  
Shear Deformable

In this paper, a geometrically nonlinear first-order shear deformable nanoplate model is developed to investigate the size-dependent geometrically nonlinear free vibrations of rectangular nanoplates considering surface stress effects. For this purpose, according to the Gurtin–Murdoch elasticity theory and Hamilton's principle, the governing equations of motion and associated boundary conditions of nanoplates are derived first. Afterwards, the set of obtained nonlinear equations is discretized using the generalized differential quadrature (GDQ) method and then solved by a numerical Galerkin scheme and pseudo arc-length continuation method. Finally, the effects of important model parameters including surface elastic modulus, residual surface stress, surface density, thickness and boundary conditions on the vibration characteristics of rectangular nanoplates are thoroughly investigated. It is found that with the increase of the thickness, nanoplates can experience different vibrational behavior depending on the type of boundary conditions.

scholarly journals In-Plane free Vibration Analysis of an Annular Disk with Point Elastic Support

2011 ◽  
Vol 18 (4) ◽  
pp. 627-640 ◽  
Author(s):  
S. Bashmal ◽  
R. Bhat ◽  
S. Rakheja
Keyword(s):  
Boundary Conditions ◽  
Orthogonal Polynomials ◽  
Ritz Method ◽  
Free Vibration Analysis ◽  
Free Vibrations ◽  
Outer Edge ◽  
Mode Shapes ◽  
Annular Disk ◽  
Different Boundary Conditions ◽  
Boundary Characteristic

In-plane free vibrations of an elastic and isotropic annular disk with elastic constraints at the inner and outer boundaries, which are applied either along the entire periphery of the disk or at a point are investigated. The boundary characteristic orthogonal polynomials are employed in the Rayleigh-Ritz method to obtain the frequency parameters and the associated mode shapes. Boundary characteristic orthogonal polynomials are generated for the free boundary conditions of the disk while artificial springs are used to account for different boundary conditions. The frequency parameters for different boundary conditions of the outer edge are evaluated and compared with those available in the published studies and computed from a finite element model. The computed mode shapes are presented for a disk clamped at the inner edge and point supported at the outer edge to illustrate the free in-plane vibration behavior of the disk. Results show that addition of point clamped support causes some of the higher modes to split into two different frequencies with different mode shapes.

scholarly journals Linear flexural natural frequencies and stability analysis of spinning Rayleigh beams: application to clamped-clamped beams.

2018 ◽  
Vol 241 ◽  
pp. 01002
Author(s):  
Mohamed Amine Aouadi ◽  
Faouzi Lakrad
Keyword(s):  
Stability Analysis ◽  
Boundary Conditions ◽  
Linear Models ◽  
Natural Frequencies ◽  
Free Vibrations ◽  
Analytical Computation ◽  
Rayleigh Beam ◽  
Clamped Beams

In the present paper 3D bending linear free vibrations of spinning Rayleigh beams are investigated. Four linear models, that differ in the linearization process, are studied. A focus on analytical computation of natural frequencies for a broad range of boundary conditions is highlighted. Then, the conditions of occurrence of divergence and flutter instabilities are determined. Finally, a case study consisting of a clamped-clamped Rayleigh beam is studied. It is found that the free vibrations destabilization process depends on the used linearization approach.

Vibrations of Double-Walled Carbon Nanotubes With Different Boundary Conditions Between Inner and Outer Tubes

2008 ◽  
Vol 75 (2) ◽  
Author(s):  
Kai-Yu Xu ◽  
Elias C. Aifantis ◽  
Yong-Hua Yan
Keyword(s):  
Carbon Nanotubes ◽  
Boundary Conditions ◽  
Resonant Frequency ◽  
Free Vibrations ◽  
Original Method ◽  
Reduced Form ◽  
Vibrational Modes ◽  
Resonant Frequencies ◽  
Different Boundary Conditions ◽  
Double Walled Carbon Nanotubes

Free vibrations of a double-walled carbon nanotube (DWNT) are studied. The inner and outer carbon nanotubes are modeled as two individual elastic beams interacting each other by van der Waals forces. An original method is proposed to calculate the first seven order resonant frequencies and relative vibrational modes. Detailed results are demonstrated for DWNTs according to the different boundary conditions between inner and outer tubes, such as fixed-free, cantilever-free, fixed-simple and fixed-fixed (reduced form) supported ends. Our results indicate that there is a special invariable frequency for a DWNT that is not affected by different combinations of boundary conditions. All vibrational modes of the DWNT must be coaxial when the resonant frequency is smaller than this frequency. Some noncoaxial vibrations will occur when their resonant frequencies exceed the frequency. Especially, the first noncoaxial resonant frequency is still invariable for all different boundary conditions. A change of resonant frequency for various lengths of DWNTs is discussed in detail. In addition, our model predicts a new coaxial-noncoaxial vibrational mode in fixed-simple supports for inner and outer tubes of a DWNT.

An accurate solution method for the vibration analysis of Timoshenko beams with general elastic supports

2014 ◽  
Vol 229 (13) ◽  
pp. 2327-2340 ◽  
Author(s):  
Dongyan Shi ◽  
Qingshan Wang ◽  
Xianjie Shi ◽  
Fuzhan Pang
Keyword(s):  
Boundary Conditions ◽  
Series Expansion ◽  
Solution Method ◽  
Free Vibrations ◽  
Accurate Solution ◽  
Fourier Series Expansion ◽  
Timoshenko Beams ◽  
End Points ◽  
Solution Algorithms ◽  
Wide Range

In this investigation, an accurate solution method is presented for the free vibrations of Timoshenko beams with general elastic restraints at the end points, a class of problems which are rarely attempted in the literatures. Unlike in most existing studies where solutions are often developed for a particular type of boundary conditions, the current method can be generally applied to a wide range of boundary conditions with no need of modifying solution algorithms and procedures. Under the current framework, the displacement and rotation functions are generally sought, regardless of boundary conditions, as an improved trigonometric series in which several supplementary functions introduced to remove the potential discontinuities with the displacement components and its derivatives at the end points and accelerated the series expansion. Mathematically, the current Fourier series expansion is an exact solution for a class of problems with the Timoshenko beam such that both the governing equations and the boundary conditions simultaneously satisfy any specified degree of accuracy. The effectiveness and reliability of the presented solution are demonstrated by comparing the present results with those results published in literatures and finite element method data, and numerous new results for beams with elastic boundary restraints is presented, which may serve as benchmark solution for future researches.

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