Accuracy in Non-linear Frequency Estimate of an Euler-Bernoulli Beam with Strong Geometric Non-Linearity using First Order and Second-Order Perturbation Methods

Accuracy in Non-linear Frequency Estimate of an Euler-Bernoulli Beam with Strong Geometric Non-Linearity using First Order and Second-Order Perturbation Methods
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© 2022 by IJETT Journal
Volume-70 Issue-5
Year of Publication : 2022
Authors : Suresh N Shankaranarayana , Chandrashekara C V
DOI :  10.14445/22315381/IJETT-V70I5P231

How to Cite?

Suresh N Shankaranarayana , Chandrashekara C V, "Accuracy in Non-linear Frequency Estimate of an Euler-Bernoulli Beam with Strong Geometric Non-Linearity using First Order and Second-Order Perturbation Methods," International Journal of Engineering Trends and Technology, vol. 70, no. 5, pp. 285-298, 2022. Crossref, https://doi.org/10.14445/22315381/IJETT-V70I5P231

Abstract
The ability to accurately estimate the non-linear frequency and response of general mechanical structures and beams is critical for their dynamic design. In the present study, the non-linear dynamics of the beam system are investigated using first-order and second-order perturbation techniques. The boundary conditions of pinned-pinned, clamped-clamped, and clamped-pinned ends are explored. The results from the two techniques are compared with the frequency from the exact solution and response from Runge-Kutta 4th order solution. The ratio of non-linear frequency to linear frequency is studied and demonstrated that the ratio increases with the increasing initial deflection and decreasing beam thickness. It is demonstrated that the error with the first-order frequency estimate, as compared with the exact solution, increases with the increasing non-linear frequency ratio. In contrast, the error with the second-order technique reduces with the increasing non-linear frequency ratio. For the given system configuration, the absolute error percentage from both techniques appears to cross over around a non-linear frequency ratio of about 1.8. The response from the first and second-order techniques matches closely with the Runge-Kutta solution in moderate and strong non-linear regimes, respectively.

Keywords
Non-linear vibration, Euler-Bernoulli beam, mid-plane stretch, perturbation technique, free vibration.

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