scholarly journals Matrix basis for plane and modal waves in a Timoshenko beam

2016 ◽  
Vol 3 (11) ◽  
pp. 160825 ◽  
Author(s):  
Julio Cesar Ruiz Claeyssen ◽  
Daniela de Rosso Tolfo ◽  
Leticia Tonetto
Keyword(s):  
Closed Form ◽  
Timoshenko Beam ◽  
Plane Waves ◽  
Crack Problem ◽  
Wave Interaction ◽  
Beam Model ◽  
High Frequencies ◽  
Transmitted Waves ◽  
The Matrix ◽  
Non Local

Plane waves and modal waves of the Timoshenko beam model are characterized in closed form by introducing robust matrix basis that behave according to the nature of frequency and wave or modal numbers. These new characterizations are given in terms of a finite number of coupling matrices and closed form generating scalar functions. Through Liouville’s technique, these latter are well behaved at critical or static situations. Eigenanalysis is formulated for exponential and modal waves. Modal waves are superposition of four plane waves, but there are plane waves that cannot be modal waves. Reflected and transmitted waves at an interface point are formulated in matrix terms, regardless of having a conservative or a dissipative situation. The matrix representation of modal waves is used in a crack problem for determining the reflected and transmitted matrices. Their euclidean norms are seen to be dominated by certain components at low and high frequencies. The matrix basis technique is also used with a non-local Timoshenko model and with the wave interaction with a boundary. The matrix basis allows to characterize reflected and transmitted waves in spectral and non-spectral form.

scholarly journals Formulation and experimental validation of space-fractional Timoshenko beam model with functionally graded materials effects

2021 ◽  
Author(s):  
Paulina Stempin ◽  
Wojciech Sumelka
Keyword(s):  
Functionally Graded Materials ◽  
Timoshenko Beam ◽  
Functionally Graded ◽  
Beam Model ◽  
Material Parameter Identification ◽  
Graded Materials ◽  
Size Dependent ◽  
Non Local ◽  
Thick Beam ◽  
Euler Bernoulli Beam

AbstractIn this study, the static bending behaviour of a size-dependent thick beam is considered including FGM (Functionally Graded Materials) effects. The presented theory is a further development and extension of the space-fractional (non-local) Euler–Bernoulli beam model (s-FEBB) to space-fractional Timoshenko beam (s-FTB) one by proper taking into account shear deformation. Furthermore, a detailed parametric study on the influence of length scale and order of fractional continua for different boundary conditions demonstrates, how the non-locality affects the static bending response of the s-FTB model. The differences in results between s-FTB and s-FEBB models are shown as well to indicate when shear deformations need to be considered. Finally, material parameter identification and validation based on the bending of SU-8 polymer microbeams confirm the effectiveness of the presented model.

Buckling analysis of double-walled carbon nanotubes embedded in an elastic medium under axial compression using non-local Timoshenko beam theory

2011 ◽  
Vol 225 (2) ◽  
pp. 498-506 ◽  
Author(s):  
M Mohammadimehr ◽  
A R Saidi ◽  
A Ghorbanpour Arani ◽  
A Arefmanesh ◽  
Q Han
Keyword(s):  
Elastic Medium ◽  
Axial Compression ◽  
Timoshenko Beam ◽  
Beam Theory ◽  
Small Scale ◽  
Beam Model ◽  
Critical Buckling Load ◽  
Non Local ◽  
Double Walled Carbon Nanotubes ◽  
Walled Carbon Nanotubes

Using the non-local elasticity theory, Timoshenko beam model is developed to study the elastic buckling of double-walled carbon nanotubes (DWCNTs) embedded in an elastic medium under axial compression. The non-local effects in the normal and transverse shear stress components are considered. The effects of the surrounding elastic medium based on a Winkler model and van der Waals' (vdW) force between the inner and outer nanotubes are taken into account. Considering the small-scale effects, the governing equilibrium equations are derived and the critical buckling loads under axial compression are obtained. The numerical results are reported using the non-local Timoshenko beam theory and compared with those obtained using the non-local Euler—Bernoulli beam theory. The results show that the critical buckling load can be overestimated by the local beam model if the small-scale effect is overlooked for long nanotubes. Furthermore, in order to estimate the non-local critical buckling load of DWCNTs under axial compression, a simplified analysis is carried out and the results are compared with those obtained using molecular mechanics.

The Elastic Field of an Elliptic Inclusion With a Slipping Interface

1986 ◽  
Vol 53 (1) ◽  
pp. 103-107 ◽  
Author(s):  
E. Tsuchida ◽  
T. Mura ◽  
J. Dundurs
Keyword(s):  
Closed Form ◽  
Infinite Series ◽  
Elastic Field ◽  
Elliptic Inclusion ◽  
Numerical Examples ◽  
Elastic Fields ◽  
Bonded Interface ◽  
The Matrix ◽  
Displacement Potentials ◽  
Uniform Expansion

The paper analyzes the elastic fields caused by an elliptic inclusion which undergoes a uniform expansion. The interface between the inclusion and the matrix cannot sustain shear tractions and is free to slip. Papkovich–Neuber displacement potentials are used to solve the problem. In contrast to the perfectly bonded interface, the solution cannot be expressed in closed form and involves infinite series. The results are illustrated by numerical examples.

Wave-Induced Vibrations of an Axial-Loaded Immersed Timoshenko Beam Carrying an Eccentric Tip Mass with Rotary Inertia

2010 ◽  
Vol 54 (01) ◽  
pp. 15-33
Author(s):  
Jong-Shyong Wu ◽  
Chin-Tzu Chen
Keyword(s):  
Closed Form ◽  
Timoshenko Beam ◽  
Forced Vibration ◽  
Natural Frequencies ◽  
Closed Form Solution ◽  
Form Solution ◽  
Rotary Inertia ◽  
Mode Shapes ◽  
Mode Superposition Method ◽  
Tip Mass

Under the specified assumptions for the equation of motion, the closed-form solution for the natural frequencies and associated mode shapes of an immersed "Euler-Bernoulli" beam carrying an eccentric tip mass possessing rotary inertia has been reported in the existing literature. However, this is not true for the immersed "Timoshenko" beam, particularly for the case with effect of axial load considered. Furthermore, the information concerning the forced vibration analysis of the foregoing Timoshenko beam caused by wave excitations is also rare. Therefore, the first purpose of this paper is to present a technique to obtain the closed-form solution for the natural frequencies and associated mode shapes of an axial-loaded immersed "Timoshenko" beam carrying eccentric tip mass with rotary inertia by using the continuous-mass model. The second purpose is to determine the forced vibration responses of the latter resulting from excitations of regular waves by using the mode superposition method incorporated with the last closed-form solution for the natural frequencies and associated mode shapes of the beam. Because the determination of normal mode shapes of the axial-loaded immersed "Timoshenko" beam is one of the main tasks for achieving the second purpose and the existing literature concerned is scarce, the details about the derivation of orthogonality conditions are also presented. Good agreements between the results obtained from the presented technique and those obtained from the existing literature or conventional finite element method (FEM) confirm the reliability of the presented theories and the developed computer programs for this paper.

Closed-form solutions of bending-torsion coupled forced vibrations of a piezoelectric energy harvester under a fluid vortex

2021 ◽  
pp. 1-31
Author(s):  
Xiang Zhao ◽  
Weidong Zhu ◽  
Ying-hui Li
Keyword(s):  
Closed Form ◽  
Torsional Vibration ◽  
Energy Harvester ◽  
High Energy ◽  
Forced Vibrations ◽  
Vibration Energy ◽  
Beam Model ◽  
Vibration Energy Harvester ◽  
Closed Form Solutions ◽  
Piezoelectric Energy

Abstract Vibration energy harvesting problems have strongly developed in recent years. However, many researchers just consider bending vibration models of energy harvesters. As a matter of fact, torsional vibration is also important and cannot be ignored in many cases. In this work, closed-form solutions of bending-torsion coupled forced vibrations of a piezoelectric energy harvester subjected to a fluid vortex are derived. Timoshenko beam model is used for modeling the energy harvester, and the extended Hamilton's principle is used in the modeling process. Since piezoelectric effects in both bending and torsional directions are considered, two kinds of electric coupling effects appear in forced vibration equations, and a new model for the electric circuit equation is developed. Lamb-Oseen vortex model is considered in this study. Both the external aerodynamic force and moment are simple harmonic loads. Three damping coefficients are considered in the present model. Based on Green's function method, closed-form solutions of the piezoelectric energy harvester subjected to the water vortex are derived. Some published results are used to verify the present solutions. It can be concluded through analysis that when torsional vibration is considered, the bandwidth of the high energy area of the voltage becomes large, and the bending-torsion coupled vibration energy harvester can produce more power than a transverse vibration energy harvester.

A new Timoshenko beam model based on modified gradient elasticity: Shearing effect and size effect of micro-beam

2019 ◽  
Vol 223 ◽  
pp. 110946 ◽  
Author(s):  
Bing Zhao ◽  
Jian Chen ◽  
Tao Liu ◽  
Wenhao Song ◽  
Jianren Zhang
Keyword(s):  
Size Effect ◽  
Timoshenko Beam ◽  
Gradient Elasticity ◽  
Beam Model ◽  
Timoshenko Beam Model ◽  
Model Based
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