此篇論文在於探討非線性邊界樑之動靜態問題。本文使用移位函數求解非線性邊界問題。對於靜態非線性邊界樑問題，只需做移位函數法即可將系統求解出；而動態非線性樑問題，將使用線性模態配合內積法，將系統簡化，之後再使用Adomian Decomposition Method做求解。並用疊代法探討其影響系統模態、頻率、振幅之關係。本文提出之方法，可適用於強非線性系統，且可分析非線性參數對系統之影響。

This study discusses the static and dynamic analysis of beam with nonlinear boundary conditions. Shifting function is used to solve nonlinear boundary problem. We just only use the shifting function to solve the static of beam problems of nonlinear boundary condition. Without any assumption, we can solve the dynamic of beam problems of nonlinear boundary condition with Linear modes method, Inner product method and Adomian decomposition method. One can be found the relation between the amplitude、mode and frequency of the system. For strong nonlinear system, one can obtain approximate analysis solutions with good precision, and can investigate the influence of nonlinear parameters on system by the present method.

[1]K. Wolf, O. Gottlieb, “Nonlinear dynamics of a noncontacting atomic force microscope cantilever actuated a piezoelectric layer,”Journal of Applied Physical, Vol. 91, No. 7, pp. 4701-4709, 2001.
[2]R. F. Fing, S. C. Huang, “Dynamic modeling and vibration analysis of the atomic force microscope,” Transactions of ASME, Journal of Vibration and Acoustics, Vol. 123, pp. 502-509, 2001.
[3]Sasaki, N., Tsukada, M., Tamura, R., Abe, K., Sato, N., “Dynamics of the cantilever in noncontact atomic force microscopy,”Applied physics A Material, Vol. 66, pp. 287-291, 1998.
[4]林昆蔚,“非接觸式原子力顯微鏡探針振動分析,”國立成功大學機械工程學系碩士論文,2003.
[5]陳炳勳,“原子力顯微鏡V型微懸臂樑的分析,”國立成功大學機械工程學系碩士論文,2004.
[6]J. A. Turner,“Non-linear vibrations of a beam with
cantilever-Hertzian contact boundary conditions,”Journal of Sound and Vibration, Vol. 275, pp. 177-191, 2004.
[7]U. Rabe, E. Kester, W. Arnold, “Probing linear and nonlinear Tip-Sample interaction forces by atomic force acoustic microscope,”Surface and Interface Analysis, Vol. 27, pp. 386-391, 1999.
[8]T. F. Ma, J. D. Silva, “Iterative solutions for a beam equation with nonlinear boundary conditions of third order,”Applied Mathematics and Computation, Vol. 159, pp. 11-18, 2003.
[9]S. M. Lin, S. Y. Lee, K. W. Lin, “Exact solutions of AFM scanning probes subjected to tip-sample forces,” Journal of mechanics of materials and structures, Vol. 2, pp. 897-910, 2007.
[10]G. V. Rao, N. R. Naidu, “Free vibration and stability behavior of uniform beams and columns with nonlinear elastic end rotational restraints,” Journal of sound and vibration, Vol. 176, pp. 130-135,1994.
[11]W. K. Lee and M. H. Teo, “Two-mode interaction of a beam with a nonlinear boundary condition,” Transactions of ASME, Journal of vibration and Acoustics, Vol. 121, pp. 84-88, 1999.
[12]M. Tabaddor, “Influence of nonlinear boundary conditions on the single-mode response of a cantilever beam,”International Journal of Solids and Structures, Vol. 37, pp. 4915-4931, 2000.
[13]李建賢,“樑的非線性靜態分析,”國立成功大學機械工程學系碩士論文, 2007.
[14]P. B. Beda,“On Deformation of an Euler-Bernolli Beam Under Terminal Force and Couple,” CMES 4, pp. 231-238, 2003.
[15]Y. H. Kuo and S. Y. Lee, “Defection of nonuniform beams resting on a nonlinear elastic foundation,” Computer and Structures, Vol. 51, No.5, pp. 513-519, 1994.

[16]W. C. Xie, H. P. Lee and S. P. Lim, “Normal modes of a non-linear clampled- clampled beam,” Journal of Sound and Vibration, Vol. 250, pp. 339-349, 2002.
[17]Touze, C. Thomas, O. Chaiqne, A. “Hardening/softening behaviour in non-linear oscillations of structural systems using non-linear normal modes,” Journal of Sound and Vibration, Vol. 273,pp. 77-101, 2004.
[18]M. I. Qaisi, “Non-linear normal modes of a continuous system,”Journal of Sound and Vibration, Vol. 209, pp. 561-569, 1998.
[19]M. I. Qaisi, “Normal modes of a continuous system with quadratic and cubic non-linearities,” Journal of Sound and Vibration, Vol. 265, pp. 329-335, 2003.
[20]M. H. Kargarnovin, D. Younessian, D. J. Thompson, C. J. C. Jones,“Response of beams on nonlinear viscoelastic foundations to harmonic moving loads,” Computers and Structures, Vol. 83, pp.1865-1877, 2005.
[21]J. H. He, “A review on some new recently developed nonlinear analytical techniques,” International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 1, pp. 51-70, 2000.
[22]S. Y. Lee, S. M. Lin, “Dynamic analysis of nonuniform beams with time-dependent elastic boundary conditions,” ASME J. Applied Mechanics, Vol. 63, pp. 474-478, 1996.
[23]S. Y. Lee, W. R. Wang, T. Y. Chen, “A general approach on the mechanical analysis of non-uniform beams with non-homogeneous elastic boundary conditions,” ASME J. Vibration and Acoustics Vol.120, pp. 164-169, 1998.
[24]S. Y. Lee, Y. H. Kuo, “Exact solutions for the analysis of general elastically restrained non-uniform beams,” ASME Journal of Applied Mechanics, Vol. 59, pp. S205-S212, 1992.
[25]S. Y. Lee, S. M. Lin, “Analysis of non-uniform Timoshenko beams with non-homogeneous elastic boundary conditions,”Int. J. Sound and Vibration, Vol. 217, No. 3, pp. 223-238, 1998.
[26]A. M. Wazwaz,“The modified decomposition method for analytic treatment of differential equations,” Applied Mathematics and Computation, Vol. 173, pp. 165-176, 2006.
[27]X. G. Luo,“A two-step Adomian decomposition method,” Applied Mathematics and Computation, Vol. 170, pp. 570-583, 2005.
[28]J. Biazar, A. R. Amirtaimoori, “An analytic approximation to the solution of heat equation by Adomian decomposition method and restrictions of the method,” Applied Mathematics and Computation, Vol. 171, pp. 738-745, 2005.
[29]M. M. Hosseini, “Adomian decomposition method with Chebyshev polynomials,” Applied Mathematic and Computation, Vol. 175, pp.1685-1693, 2006.
[30]邱青煌, “Adomian分解法應用在非線性散熱片之熱傳及熱應力之分析,”國立成功大學機械工程學系碩士論文, 2001.

[31]D. Kaya, I. E. Inan, “A convergence analysis of the ADM and an application,” Applied Mathematics and Computation, Vol. 161, pp.1015-1026, 2005.
[32]M. M. Hosseini, H. Nasabzadeh, “on the convergence of Adomian decomposition method,” Applied Mathematics and Computation, Vol.182, pp. 536-543, 2006
[33]Ali H. Nayfeh, “Perturbation Methods,” Wiley Classics Library Edition Published, 2000.
[34]W. Lestari, S. Hanagud, “Nonlinear vibration of buckled beams:some exact solutions,” International Journal of Solids and Structures, Vol. 38, pp. 4741-4757, 2001.
[35]P. Hagedorn, “Non-linear oscillations,” Published in the United States by Oxford University Press, New York, Second edition 1988.