Free Vibrations of Timoshenko Beams with Variable Cross Sections: A Lagrangian Approach

2009 ◽  
Author(s):  
N.M. Auciello
Keyword(s):  
Cross Sections ◽  
Free Vibrations ◽  
Lagrangian Approach ◽  
Variable Cross ◽  
Timoshenko Beams

A unified procedure for the vibration analysis of elastically restrained Timoshenko beams with variable cross sections

2020 ◽  
Vol 68 (1) ◽  
pp. 38-47
Author(s):  
Gang Wang ◽  
Wen L. Li ◽  
Wanyou Li ◽  
Zhihua Feng ◽  
Junfang Ni
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Vibration Analysis ◽  
Variable Cross Section ◽  
Variable Cross ◽  
Timoshenko Beams ◽  
Unified Approach ◽  
Cross Section Area ◽  
Section Area ◽  
Elastically Restrained

A generalized analytical method is developed for the vibration analysis of Timoshenko beams with elastically restrained ends. For a beam with any variable cross section along the lengthwise direction, the finite element method is the only unified approach to handle those kinds of problems, since the analytical solutions could not be obtained by the governing equations when the cross section area and the second moment of area changing variably lengthwise. In this article, a unified approach is proposed to study the Timoshenko beam with any variable cross sections. The cross section area and second moment of area of the beam are both expanded into cosine series, which are mathematically capable of representing any variable cross section. The translational displacement and rotation of cross section are expressed in the Fourier series by adding some admissible functions which are used to handle the elastic boundary conditions with more accuracy and high convergence rate. By using Hamilton's principle, the eigenvalues and the coefficients of the Fourier series are both obtained. Some examples are presented to illustrate the excellent accuracy of this method. Analytical solutions of the vibration of the beam are achieved for different combinations of boundary conditions including classical and elastically restrained ones. The derived results can be used as benchmark solutions for testing approximate or numerical methods used for the vibration analysis of Timoshenko beams with any variable cross section.

scholarly journals Analysis of free vibrations of cantilever bars with parabolically variable cross-sections using the Rayleigh’s method

2018 ◽  
Vol 16 (4) ◽  
pp. 5-14
Author(s):  
Olga Szlachetka ◽  
◽  
Marek Chalecki ◽  
Jacek Jaworski, ◽  
◽  
...  
Keyword(s):  
Cross Sections ◽  
Free Vibrations ◽  
Variable Cross

scholarly journals Analytical Modeling of the Mechanical Behavior of MEMS/NEMS-Multilayered Resonators with Variable Cross-Sections for Sensors and Energy Harvesters

IEEE Access ◽  
2021 ◽  
pp. 1-1
Author(s):  
Eustaquio Martinez-Cisneros ◽  
Luis A. Velosa-Moncada ◽  
Ernesto A. Elvira-Hernandez ◽  
Omar I. Nava-Galindo ◽  
Luz Antonio Aguilera-Cortes ◽  
...  
Keyword(s):  
Mechanical Behavior ◽  
Cross Sections ◽  
Analytical Modeling ◽  
Variable Cross ◽  
Energy Harvesters

Linear Thermomechanical Microactuators

1999 ◽  
Author(s):  
Rebecca Cragun ◽  
Larry L. Howell
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Thermal Expansion ◽  
Cross Sections ◽  
Variable Cross ◽  
Element Analysis ◽  
Thermal Actuators ◽  
Output Force ◽  
Analysis Models ◽  
High Output

Abstract Thermomechanical in-plane microactuators (TIMs) have been designed, modeled, fabricated, and tested. TIMs offer an alternative to arrays of smaller thermal actuators to obtain high output forces. The design is easily modified to obtain the desired output force or deflection for specific applications. The operational principle is based on the symmetrical thermal expansion of variable cross sections of the surface micromachined microdevice. Sixteen configurations of TIMs were fabricated of polysilicon. Finite element analysis models were used to predict the deflection and output force for the actuators. Experimental results were also recorded for all sixteen configurations, including deflections and output forces up to 20 micron and 35 dyne.

scholarly journals GEOMETRIC MODELING OF A SHAPE OF PARALLELOGRAM PLATES IN A PROBLEM OF FREE VIBRATIONS USING CONFORMAL RADII

2021 ◽  
pp. 143-159
Author(s):  
A.A. Chernyaev ◽  
Keyword(s):  
Cross Sections ◽  
Geometric Modeling ◽  
Interpolation Method ◽  
Geometric Characteristic ◽  
Free Vibrations ◽  
Theory Of Elasticity ◽  
Constant Thickness ◽  
Basic Frequency ◽  
Plate Vibrations ◽  
The Subject

The paper considers a method of geometric modeling applied when solving basic twodimensional problems of the theory of elasticity and structural mechanics, in particular the applied problems of engineering. The subject of this study is vibrations of thin elastic parallelogram plates of constant thickness. To determine a basic frequency of vibrations, the interpolation method based on the geometric characteristic of the shape of plates (membrane, cross sections of a rod) is proposed. This characteristic represents a ratio of interior and exterior conformal radii of the plate. As is known from the theory of conformal mappings, conformal radii are those obtained by mapping of a plate onto the interior and exterior of a unit disk. The paper presents basic terms, tables, and formulas related to the considered geometric method with a comparative analysis of the curve diagrams obtained using various interpolation formulas. The original computer program is also developed. The main advantage of the proposed method of determining the basic frequency of plate vibrations is a graphic representation of results that allows one to accurately determine the required solution on the graph among the other solutions corresponding to the considered case of parallelogram plates. Although there are many known approximate approaches, which are used to solve the considered problems, only geometric modeling technique based on the conformal radii ratio gives such an opportunity.

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