Abstract
This article attains a new formulation of beam vibrations on an elastic foundation with quintic nonlinearity, including exact expressions for the beam curvature. To achieve a proper design of the beam structures, it is essential to realize how the beam vibrates in its transverse mode, which, in turn, yields the natural frequency of the system. In this direction, a powerful analytical method called the parameter expansion method is employed to obtain the exact solution of the frequency-amplitude relationship. It is clearly shown that the first term in series expansions is sufficient to produce a highly accurate approximation of the above-mentioned system. Finally, the accuracy of the present analytic procedure is evaluated through comparisons with numerical calculations.
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H. M. Sedighi, K. H. Shirazi, and A. Noghrehabadi, “Application of Recent Powerful Analytical Approaches on the Non-Linear Vibration of Cantilever Beams,” Int. J. Nonlinear Sci. Numer. Simulat. 13(7/8), 487–494 (2012); DOI: 10.1515/ijnsns-2012-0030.
H. Rafieipour, A. Lotfavar, and M. H. Mansoori, “New Analytical Approach to Nonlinear Behavior Study of Asymmetrically LCBs on Nonlinear Elastic Foundation under Steady Axial and Thermal Loading,” Lat. Amer. J. Solids Structures 9, 531–545 (2012).
A. Barari, H. D. Kaliji, M. Ghadami, and G. Domairry, “Non-Linear Vibration of Euler-Bernoulli Beams,” Lat. Amer. J. Solids Structures 8, 139–148 (2011).
P. D. Cha and J. M. Rinker, “Enforcing Nodes to Suppress Vibration along a Harmonically Forced Damped Euler-Bernoulli Beam,” J. Vibr. Acoust. 134(5), 051010 (2012); DOI: 10.1115/1.4006375.
H. M. Sedighi, K. H. Shirazi, and J. Zare, “An Analytic Solution of Transversal Oscillation of Quintic Nonlinear Beam with Homotopy Analysis Method,” Int. J. Non-Linear Mech. 47, 777–784 (2012); DOI: 10.1016/j.ijnonlinmec.(2012).04.008.
H. M. Sedighi and K. H. Shirazi, “A New Approach to Analytical Solution of Cantilever Beam Vibration with Nonlinear Boundary Condition,” J. Comput. Nonlinear Dynamics. 7, 034502 (2012); DOI: 10.1115/1.4005924.
H. M. Sedighi, K. H. Shirazi, and J. Zare, “Novel Equivalent Function for Deadzone Nonlinearity: Applied to Analytical Solution of Beam Vibration Using He’s Parameter Expanding Method,” Lat. Amer. J. Solids Structures 9, 443–451 (2012).
H. M. Sedighi, K. H. Shirazi, A. Reza, and J. Zare, “Accurate Modeling of Preload Discontinuity in the Analytical Approach of the Nonlinear Free Vibration of Beams,” Proc. Inst. Mech. Eng., Pt. C: J. Mech. Eng. Sci. 226(10), 2474–2484 (2012); DOI: 10.1177/0954406211435196.
S. E. Motaghian, M. Mofid, and P. Alanjari, “Exact Solution to Free Vibration of Beams Partially Supported by an Elastic Foundation,” Scient. Iranica A 18(4), 861–866 (2011).
M. Nikkhah Bahrami, M. Khoshbayani Arani, and N. Rasekh Saleh, “Modified Wave Approach for Calculation of Natural Frequencies and Mode Shapes in Arbitrary Non-Uniform Beams,” Scient. Iranica B 18(5), 1088–1094 (2011).
J. Amani and R. Moeini, “Prediction of Shear Strength of Reinforced Concrete Beams Using Adaptive Neuro-Fuzzy Inference System and Artificial Neural Network,” Scient. Iranica A 19(2), 242–248 (2012); DOI: 10.1016/j.scient.(2012).02.009.
H. M. Sedighi, K. H. Shirazi, A. R. Noghrehabadi, and A. Yildirim, “Asymptotic Investigation of Buckled Beam Nonlinear Vibration,” Iran. J. Sci. Technol. Trans. Mech. Eng. 36(M2), 107–116 (2012).
T. S. Jang, H. S. Baek, and J. K. Paik, “A New Method for the Non-Linear Deflection Analysis of an Infinite Beam Resting on a Non-Linear Elastic Foundation,” Int. J. Non-Linear Mech. 46, 339–346 (2011).
H. Arvin and F. Bakhtiari-Nejad, “Non-Linear Modal Analysis of a Rotating Beam,” Int. J. Non-Linear Mech. 46, 877–897 (2011).
J. Awrejcewicz, A. V. Krysko, V. Soldatov, and V. A. Krysko, “Analysis of the Nonlinear Dynamics of the Timoshenko Flexible Beams Using Wavelets,” J. Comput. Nonlinear Dyn. 7(1), 011005 (2012).
U. Andreaus, L. Placidi, and G. Rega, “Soft Impact Dynamics of a Cantilever Beam: Equivalent SDOF Model Versus Infinite-Dimensional System,” Proc. Inst. Mech. Eng., Pt C: J. Mech. Eng. Sci. 225(10), 2444–2456 (2011); DOI: 10.1177/0954406211414484.
L. F. Campanile, R. Jähne, and H. Hasse, “Exact Analysis of the Bending of Wide Beams by a Modified Elastica Approach,” Proc. Inst. Mech. Eng., Pt C: J. Mech. Eng. Sci. 225(11), 2759–2764 (2011); DOI: 10.1177/0954406211417753.
S. Bagheri, A. Nikkar, and H. Ghaffarzadeh, “Study of Nonlinear Vibration of Euler-Bernoulli Beams by using Analytical Approximate Techniques,” Lat. Amer. J. Solids Struct. 11, 157–168 (2014).
H. M. Sedighi and K. H. Shirazi, “Asymptotic Approach for Nonlinear Vibrating Beams with Saturation Type Boundary Condition,” Proc. Inst. Mech. Eng., Pt C: J. Mech. Eng. Sci. 227(11) 2479–2486 (2013); DOI: 10.1177/0954406213475561.
A. G. Kolpakov, “On the Analysis of a Plate with a Local Shape Perturbation,” Prikl, Mekh. Tekh. Fiz. 53(4), 171–182 (2012) [J. Appl. Mech. Tech. Phys. 53 (4), 616–625 (2012)].
M. K. Yazdi, H. Ahmadian, A. Mirzabeigy, and A. Yildirim, “Dynamic Analysis of Vibrating Systems with Nonlinearities,” Comm. Theoret. Phys. 57(2), 183–187 (2012).
A. Kamali Eigoli and G. R. Vossoughi, “A Periodic Solution for Friction Drive Microrobots Based on the Iteration Perturbation Method,” Scient. Iranica B 18(3), 368–374 (2011).
M. S. Shadloo and A. Kimiaeifar, “Application of Homotopy Perturbation Method to Find an Analytical Solution for Magneto Hydrodynamic Flows of Viscoelastic Fluids in Converging/Diverging Channels,” Proc. Inst. Mech. Eng., Pt C: J. Mech. Eng. Sci. 225, 347–353 (2011).
A. Koochi, A. S. Kazemi, Y. Tadi Beni, et al., “Theoretical Study of the Effect of Cawsimir Attraction of the Pull-in Behavior of Beam-Type NEMS Using Modified Adomian Method,” Physica E, Low-Dimens. Syst. Nanostruct. 43(4), 625–632 (2010).
B. K. Hammad, A. H. Nayfeh, and E. M. Abdel-Rahman, “On the use of the Subharmonic Resonance As a Method for Filtration,” J. Comput. Nonlinear Dyn. 6(4), 041007 (2011); DOI: 10.1115/1.4003031.
A. Hasanov, “Some New Classes of Inverse Coefficient Problems in Non-Linear Mechanics and Computational Material Science,” Int. J. Non-Linear Mech. 46(5), 667–684 (2011).
A. H. Baferani, A. R. Saidi, and E. Jomehzadeh, “An Exact Solution for Free Vibration of thin Functionally Graded Rectangular Plates,” Proc. Inst. Mech. Eng., Pt C: J. Mech. Eng. Sci. 225(3), 526–536 (2011); DOI: 10.1243/09544062JMES2171.
A. Naderi and A. R. Saidi, “Buckling Analysis of Functionally Graded Annular Sector Plates Resting on Elastic Foundations,” Proc. Inst. Mech. Eng., Pt C: J. Mech. Eng. Sci. 225(2), 312–325 (2011).
J. H. He and D. H. Shou, “Application of Parameter-Expanding Method to Strongly Nonlinear Oscillators,” Int. J. Nonlinear Sci. Numer. Simulat. 8, 121–124 (2007).
H. M. Liu, “Approximate Period of Nonlinear Oscillators with Discontinuities by Modified Lindstedt-Poincaré Method,” Chaos, Solitons Fractals 23(2), 577–579 (2005).
L. Xu, “He’s Parameter-Expanding Methods for Strongly Nonlinear Oscillators,” J. Comput. Appl. Math. 207, 148–154 (2007).
L. Xu, “Application of He’s Parameter-Expansion Method to an Oscillation of a Mass Attached to a Stretched Elastic Wire,” Phys. Lett. 368, 259–262 (2007); DOI: 10.1016/j.physleta. 2007.04.004.
H. M. Sedighi and K. H. Shirazi “Vibrations of Micro-Beams Actuated by an Electric Field via Parameter Expansion Method,” Acta Astronaut. 85, 19–24 (2013); DOI: 10.1016/j.actaastro.2012.11.014.
A. N. Kounadis, J. Mallis, and A. Sbarouni, “Postbuckling Analysis of Columns Resting on an Elastic Foundation,” Arch. Appl. Mech. 75, 395–404 (2006); DOI: 10.1007/s00419-005-0434-1.
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Original Russian Text © H.M. Sedighi, K.H. Shirazi.
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Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 55, No. 6, pp. 186–195, November–December, 2014.
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Sedighi, H.M., Shirazi, K.H. Accurate investigation of lateral vibrations of a quintic nonlinear beam on an elastic foundation: Using an exact formulation of the beam curvature. J Appl Mech Tech Phy 55, 1066–1074 (2014). https://doi.org/10.1134/S0021894414060194
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DOI: https://doi.org/10.1134/S0021894414060194