Free vibration analysis of Euler–Bernoulli nanobeam using differential transform method

2018 ◽  
Vol 07 (03) ◽  
pp. 1850020 ◽  
Author(s):  
Subrat Kumar Jena ◽  
S. Chakraverty
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Numerical Technique ◽  
Equations Of Motion ◽  
Free Vibration Analysis ◽  
Differential Transform Method ◽  
Inverse Transformation ◽  
Algebraic Equations ◽  
Transform Method ◽  
Differential Transform

In this paper, a semi analytical-numerical technique called differential transform method (DTM) is applied to investigate free vibration of nanobeams based on non-local Euler–Bernoulli beam theory. The essential steps of the DTM application include transforming the governing equations of motion into algebraic equations, solving the transformed equations and then applying a process of inverse transformation to obtain accurate mode frequency. All the steps of the DTM are very straightforward, and the application of the DTM to both the equations of motion and the boundary conditions seems to be very involved computationally. Besides all these, the analysis of the convergence of the results shows that DTM solutions converge fast. In this paper, a detailed investigation has been reported and MATLAB code has been developed to analyze the numerical results for different scaling parameters as well as for four types of boundary conditions. Present results are compared with other available results and are found to be in good agreement.

Solution of Free Vibration Equations of Euler-Bernoulli Cracked Beams by Using Differential Transform Method

2011 ◽  
Vol 110-116 ◽  
pp. 4532-4536 ◽  
Author(s):  
K. Torabi ◽  
J. Nafar Dastgerdi ◽  
S. Marzban
Keyword(s):  
Analytical Solution ◽  
Differential Equations ◽  
Free Vibration ◽  
Boundary Conditions ◽  
Differential Transform Method ◽  
Beam Model ◽  
Cracked Beam ◽  
Transform Method ◽  
Simple Method ◽  
Differential Transform

In this paper, free vibration differential equations of cracked beam are solved by using differential transform method (DTM) that is one of the numerical methods for ordinary and partial differential equations. The Euler–Bernoulli beam model is proposed to study the frequency factors for bending vibration of cracked beam with ant symmetric boundary conditions (as one end is clamped and the other is simply supported). The beam is modeled as two segments connected by a rotational spring located at the cracked section. This model promotes discontinuities in both vertical displacement and rotational due to bending. The differential equations for the free bending vibrations are established and then solved individually for each segment with the corresponding boundary conditions and the appropriated compatibility conditions at the cracked section by using DTM and analytical solution. The results show that DTM provides simple method for solving equations and the results obtained by DTM converge to the analytical solution with much more accurate for both shallow and deep cracks. This study demonstrates that the differential transform is a feasible tool for obtaining the analytical form solution of free vibration differential equation of cracked beam with simple expression.

Free vibration analysis of two parallel beams connected together through variable stiffness elastic layer with elastically restrained ends

2016 ◽  
Vol 20 (3) ◽  
pp. 275-287 ◽  
Author(s):  
Alborz Mirzabeigy ◽  
Reza Madoliat ◽  
Mehdi Vahabi
Keyword(s):  
Differential Equations ◽  
Elastic Layer ◽  
Flexural Rigidity ◽  
Free Vibration Analysis ◽  
Variable Stiffness ◽  
Differential Transform Method ◽  
Mode Shapes ◽  
Algebraic Equations ◽  
Transform Method ◽  
Differential Transform

In this study, free transverse vibration of two parallel beams connected together through variable stiffness Winkler-type elastic layer is investigated. Euler–Bernoulli beam hypothesis has been applied and the support is considered to be translational and rotational elastic springs in each ends. Linear and parabolic variation has been considered for connecting layer. The equations of motion have been derived in the form of coupled differential equations with variable coefficients. The differential transform method has been applied to obtain natural frequencies and normalized mode shapes of system. Differential transform method is a semi-analytical approach based on Taylor expansion series which converts differential equations to recursive algebraic equations and does not need domain discretization. The results obtained from differential transform method have been validated with the results reported by well-known references in the case of two parallel beams connected through uniform elastic layer. The effects of variation type and total stiffness of connecting layer, flexural rigidity ratio of beams, and boundary conditions on behavior of system are investigated and discussed in detail.

The Flapwise Bending Free Vibration Analysis of Micro-rotating Timoshenko Beams Using the Differential Transform Method

2017 ◽  
Vol 24 (20) ◽  
pp. 4868-4884 ◽  
Author(s):  
Hadi Arvin
Keyword(s):  
Boundary Conditions ◽  
Vibration Analysis ◽  
Slenderness Ratio ◽  
Equations Of Motion ◽  
Free Vibration Analysis ◽  
Beam Model ◽  
Transform Method ◽  
Differential Transform ◽  
Material Length Scale ◽  
Material Length

The flapwise bending free vibration analysis of isotropic rotating Timoshenko microbeams, including the size effects, is presented in this paper. A nonclassical theory, i.e. the Modified Couple Stress theory, has been employed to include the size effect in the presented formulation. By consideration of the Timoshenko beam assumptions the shear deformation and the rotary inertia effects are taken into account. The Hamilton’s principle is applied to the obtained strain and kinetic energy relations to derive the nonlinear equations of motion and the associated boundary conditions. After nondimensionalization of the equations of motion and the corresponding boundary conditions, the linearized form of the equations of motion and the accompanied boundary conditions are developed. A semi-analytical approach, i.e. the differential transform method, is implemented to achieve the flapping and axial frequencies. The achieved results are validated via comparison with the available results in the literature. The material length scale, shear deformation consideration, rotating speed and the slenderness ratio influences on the natural frequencies are examined. The results demonstrate that the slenderness ratio and the thickness to the material length scale parameter quotient are the dominant indicators in determining the usage of the nonclassical theories against classical theories. On the other hand, the precision in determination of the higher modes frequencies motivates us to implement the Timoshenko beam model instead of the Euler–Bernoulli beam model.

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