Iranian Journal of Mechanical Engineering Transactions of ISME

Iranian Journal of Mechanical Engineering Transactions of ISME

Nonlinear vibration of beam on elastic foundation

Authors
Mousa Rezaee 1
Masoud Minaei 2
1 Professor, Department of Mechanical Engineering, Tabriz University
2 PhD. Student, Department of Mechanical Engineering, Tabriz University
Abstract
The large amplitude free vibration behaviour of an Euler-Bernoulli beam with immovable ends subjected to the thermal loads is investigated. Applying the Hamilton’s principle, the beam governing equation of motion is derived. By implementing the Galerkin’s method, the ordinary nonlinear differential equation is derived and because of the large coefficient of the nonlinear term, the Homotopy Perturbation Method (HPM) is used to solve the governing nonlinear equation and the accuracy of the mentioned method is investigated. The results are validated by comparing them with those available in the literature. Moreover, the effects of the system parameters including foundation coefficients, thermal load and vibrational modes on the system nonlinear vibration behaviour are investigated.
Keywords
Euler-Bernoulli Beam
Large amplitude vibration
Homotopy Perturbation Method
Nonlinear elastic foundation
Thermal load

Subjects

[1] He, J.H., "Homotopy Perturbation Technique", Computer Methods in Applied Mechanical Engineering, Vol. 178, pp. 257-262, (1999).
 
[2] He, J.H., "A Coupling Method of Homotopy Technique and Perturbation Technique for Nonlinear Problems", International Journal of Nonlinear Mechanics, Vol. 35, pp. 37-43, (1999).
 
 [3] He, J.H., "A Variational Iteration Method to Duffing Equation (in Chinese)", Chinese Journal of Computational Physics, Vol. 16, pp. 121-133, (1999).
 
[4] He, J.H., "An Elementary Introduction to the Homotopy Perturbation Method", Computers and Mathematics with Applications, Vol. 57, pp. 410-412, (2009).
 
[5] Nayfeh, A.H., and Mook, D.T., "Nonlinear Oscillations", Wiley Interscience, New York, (1979).
 
[6] Nayfeh, A.H., "Introduction to Perturbation Techniques", Wiley Interscience, New York, (1981).
 
[7] Liao, S.J., and Chwang, A.T., "Application of Homotopy Analysis Method in Nonlinear Oscillations", ASME Journal of Applied Mechanics, Vol. 65, pp. 914-922, (1998).
 
[8] He, J.H., "New Interpretation of Homotopy Perturbation Method", International Journal of Modern Physics B, Vol. 20, pp. 2561-2568, (2006).
 
[9] He, J.H., "Homotopy Perturbation Method with an Auxiliary Term", Abstract and Applied Analysis, Vol. 2012, pp. 1-7, (2012).
 
[10] He, J.H., "Comment on He’s Frequency Formulation for Nonlinear Oscillators", European Journal of Physics, Vol. 29, pp. 19-22, (2008).
 
[11] He, J.H., "Preliminary Rreport on the Energy Balance for Nonlinear Oscillations", Mechanics Research Communications, Vol. 29, pp. 107-111, (2002).
[12] Ganji, D.D., Gorji, M., Soleimani, S., and Esmaeilpour, M., "Solution of Nonlinear Cubic-quantic Duffing Oscillators using He’s Energy Balance Method", Journal of Zhejiang University: Science A, Vol. 10, pp. 1263-1268, (2009).
 
[13] Mickens, R.E., "Mathematical and Numerical Study of the Duffing-harmonic Oscillator", Journal of Sound and Vibration, Vol. 244, pp. 563-567, (2001).
 
[14] Yazdi, M.K., Mirzabeigy, A., and Abdollahi, H., "Nonlinear Oscillators with Non-polynomial and Discontinuous Elastic Restoring Forces", Nonlinear Science Letters A, Vol. 3, pp. 48-53, (2012).
 
[15] He, J.H., "Max-min Approach to Nonlinear Oscillators", International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 9, pp. 207-210, (2008).
 
[16] Yazdi, M.K., Ahmadian, H., Mirzabeigy, A., and Yildirim, A., "Dynamic Analysis of Vibrating Systems with Nonlinearities", Communications in Theoretical Physics, Vol. 57, pp. 183-187, (2012).
 
[17] He, J.H., "Some Asymptotic Methods for Strongly Nonlinear Equations", International Journal of Modern Physics B, Vol. 20, pp. 1141-1199, (2006).
 
[18] Noor, M.A., and Khan, W.A., "New Iterative Methods for Solving Nonlinear Equation by using Homotopy Perturbation Method", Applied Mathematics and Computation, Vol. 219, pp. 3565-3574, (2012).
 
 [19] Momani, S., Erjaee, G.H., and Alnasr, M.H., "The Modified Homotopy Perturbation Method for Solving Strongly Nonlinear Oscillators", Computers and Mathematics with Applications, Vol. 58, pp. 2209-2220, (2009).
 
[20] Hu, M., and Wang, L., "Periodic Solutions for Some Strongly Nonlinear Oscillators by He’s Energy Balance Method", World Academy of Science Engineering and Technology, Vol. 68, pp. 620-622, (2012).
 
[21] Ramezani, A., Alasty, A., and Akbari, J., "Effects of Rotary Inertia and Shear Deformation on Nonlinear Free Vibration of Microbeams", ASME Journal of Vibration and Acoustics, Vol. 128, pp. 611-615, (2006).
 
[22] Pirbodaghi, T., Ahmadian, M.T., and Fesanghary, M., "On the Homotopy Analysis Method for Nonlinear Vibration of Beams", Mechanics Research Communications, Vol. 36, pp. 143-148, (2009).
 
[23] Sedighi, H.M., Shirazi, K.H., and Zare, J., "An Analytic Solution of Transversal Oscillation of Quantic Nonlinear Beam with Homotopy Analysis Method", International Journal of Nonlinear Mechanics, Vol. 47, pp. 777-784, (2012).
 
[24] Sarma, B.S., Vardan, T.K., and Parathap, H., "On Various Formulation of Large Amplitude Free Vibration of Beams", Journal of Computers and Structures Vol. 29, pp. 959-966, (1988).
 
[25] Poorjamshidan, M., Sheikhi, J., Mahjoobmoghadas, S., and Norouzinia, M., "An Analytic Solution of Transversal Vibrations and Frequency Response of Quantic Nonlinear Beam", Modares Mechanical Engineering, Vol. 13, pp. 1-9, (2014).
 
[26] Wang, L., Ni, Q., Li, M., and Qian, Q., "The Thermal Efect on Vibration and Instability of Carbon Nanotubes Conveying Fluid", Physica E, Vol. 40, pp. 3179-3182, (2008).
 
Iranian Journal of Mechanical Engineering Transactions of ISME
Volume 18, Issue 1 - Serial Number 42
System Dynamics and Solid Mechanics
Spring 2016
Pages 41-60

  • Receive Date 13 April 2015
  • Revise Date 12 June 2015
  • Accept Date 13 July 2015