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

  IJETT-book-cover           
  
© 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.

Reference
[1] David Wagg, Simon Neild, Nonlinear Vibration with Control for Flexible and Adaptive Structures, Second Edition, Springer International Publishing Switzerland. (2015) 292-298.
[2] S. R. R. Pillai and B. Nageswara Rao, On Non-Linear Free Vibrations of Simply Supported Uniform Beams, Journal of Sound and Vibration 159(3) (1992) 527-531.
[3] W. Lestari and S. Hanagud, Non-Linear Vibration of Buckled Beams: Some Exact Solutions, International Journal of Solids and Structures. 38 (2001) 4741-4757.
[4] M.T. Ahmadian, M. Mojahedi and H. Moeenfar, Free Vibration Analysis of a Non-Linear Beam Using Homotopy and Modified Lindstedt-Poincare Methods, Journal of Solid Mechanics. 1 (2009) 29-36.
[5] Jian-She Peng, Yan Liu, and Jie Yang, A Semi-Analytical Method For Non-Linear Vibration of Euler-Bernoulli Beams with General Boundary Conditions, Hindawi Publishing Corporation, Mathematical Problems in Engineering. 591786 (2010) 17.
[6] Ma Xinmou, Chang Liezhen, and Pan Yutian, Accurate Solutions to Non-Linear Vibration of Cantilever Beam via Homotopy Perturbation Method, Procedia Engineering. 15 (2011) 4768-4773.
[7] Hamid M. Sedighi, Kourosh H. Shirazi, A and Noghrehabadi, Application of Recent Powerful Analytical Approaches on the Non-Linear Vibration of Cantilever Beams, International Journal of Nonlinear Sciences and Numerical Simulation. 13(7-8) (2012) 487-494.
[8] A.Barari, H.D. Kaliji, M. Ghadimi, and G. Domairry, Non-Linear Vibration of Euler-Bernoulli Beams, Latin American Journal of Solids and Structures. 8 (2011) 139-148.
[9] Debabrata Das, Prasanta Sahoo, and Kashinath Saha, A Numerical Analysis of Large Amplitude Beam Vibration Under Different Boundary Conditions and Excitation Patterns, Journal of Vibration and Control. 18(12) (2012) 1900–1915.
[10] S.S. Jafari, M.M. Rashidi, and S. Johnson, Analytical Approximation of Non-Linear Vibration of Euler-Bernoulli Beams, Latin American Journal of Solids and Structures. 13 (2016) 1250-1264.
[11] S. Bagheri, A. Nikkar and H. Ghaffarzadeh, Study of Non-Linear Vibration of Euler-Bernoulli Beams Using Approximate Analytical Techniques, Latin American Journal of Solids and Structures. 11 (2014) 157-168.
[12] Iman Pakar and Mahmoud Bayat, Analytical Study on the Non-Linear Vibration of Euler-Bernoulli Beams, Journal of Vibro Engineering. 14(1) (2012).
[13] Mohammadreza Azimi and Saeed Kariman, Periodic Solution for Vibration of Euler-Bernoulli Beam Subjected to Axial Load Using DTM and HA, Journal of Applied Mechanical Engineering. 2(2) (2013) 1000125.
[14] A.A. Motallebi, M. Poorjamshidian1, and J. Sheikhi, Vibration Analysis of a Nonlinear Beam under Axial Force by Homotopy Analysis Method, Journal of Solid Mechanics. 6(3) (2014) 289-298.
[15] O. Barry, Non-Linear Vibration of an Axially Loaded Beam Carrying Rigid Bodies, AIP Advances. 6(12) (2016).
[16] Shahin Mohammadrezazadeh, Ali-Asghar Jafari and Mohammad Saeid Jafari, An Analytical Approach for the Non-Linear Forced Vibration of the Clamped-Clamped Buckled Beam, Journal of Theoretical and Applied Vibration and Acoustics. 3(2) (2017) 127-144.
[17] Hu Ding, Yi Li and Li-Qun Chen, Non-Linear Vibration of a Beam with Asymmetric Elastic Supports, Nonlinear Dynamics. 95 (2019) 2543–2554.
[18] Van - Hieu Dang and Quang- Duy Le, Analysis of Nonlinear Vibration of Euler-Bernoulli Beams Subjected to Compressive Axial Force via the Equivalent Linearization Method with a Weighted Averaging, International Journal of Scientific and Innovative Mathematical Research. 7(1) (2019) 4-13.
[19] Tobias Friis, Marius Tarpø, Evangelos I. Katsanos and Rune Brincker, Equivalent Linear Systems of Nonlinear Systems, Journal of Sound and Vibration. 469 (2020).
[20] Mahmoud Bayat, Iman Pakar, and Paul Ziehl, Nonlinear Vibration of Axially Loaded Railway Track Systems Using Analytical Approach, Journal of Low-Frequency Noise, Vibration and Active Control. 0(0) (2021) 1–11.
[21] A. Rinc´on-Casado, J. Gonz´alez-Carbajal, D. García-Vallejo and J. Domínguez, Analytical and Numerical Study of the Influence of Different Support Types in the Non-Linear Vibrations of Beams, European Journal of Mechanics / A Solids. 85 (2021) 104113.
[22] Ehsan Loghman, Firooz Bakhtiari-Nejad, Ali Kamali E, Mostafa Abbaszadeh and Marco Amabili, Non-Linear Vibration of Fractional Viscoelastic Micro-Beams, International Journal of Non-Linear Mechanics. 137 (2021) 103811.
[23] Omar Outassafte, Ahmed Adri, Yassine El Khouddar, Said Rifai and Rhali Benamar, Geometrically Non-Linear Free In-Plane Vibration of Circular Arch Elastically Restrained Against Rotation at the Two Ends, International Journal of Engineering Trends and Technology. 69(3) (2021) 85-95.
[24] Walter Lacarbonara and Hiroshi Yabuno, Refined Models of Elastic Beams Undergoing Large In-Plane Motions: Theory and Experiment, International Journal of Solids and Structures. 43 (2006) 5066–5084.
[25] H. Hu, A Classical Perturbation Technique Valid for Large Parameters, Journal of Sound and Vibration. 269 (2004) 409–412.
[26] Singiresu S. Rao, Vibration of Continuous Systems, John Wiley & Sons, Inc., Hoboken, New Jersey. (2007) 326-334.
[27] Robert D. Blevins, Formulas for Dynamics, Acoustics and Vibration, First Edition, John Wiley & Sons, Ltd. (2016) 429-434.