Free vibration of axially loaded Reddy-Bickford beam on elastic soil using the differential transform method

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.

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

International Journal of Computational Materials Science and Engineering ◽

10.1142/s2047684118500203 ◽

2018 ◽

Vol 07 (03) ◽

pp. 1850020 ◽

Cited By ~ 11

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.

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