本文旨在利用Adomian修正分解法（Adomian modified decomposition method）推導各式柱之振動問題，包含無受力、軸向力和風力的均勻、非均勻柱體之振動問題，並且試改變其參數探討不同參數對自然頻率之影響，而後利用數值計算軟體求解出數值解。文中首先介紹Adomian分解法的基本定義、性質和推導方式，並以柱之振動問題配合各種不同之邊界條件與其他文獻數值解比較，說明Adomian分解法在推導數值解的方便性與正確性。
研究結果顯示，無因次自然頻率收斂情況以邊界條件彈簧端-自由端收斂最為快速，且無因次自然頻率會隨著錐度比和彈簧模數增加而增加、移動彈簧模數對無因次自然頻率影響大於旋轉彈簧模數。若討論含有外力之強迫振動問題，則結果顯示無因次自然頻率附近會有共振現象發生。

In this paper, we use the Adomian modified decomposition method to derive the vibration equation of all kinds of columns, including the vibration problem of uniform and non-uniform columns with zero external force, axial force, and wind force. We first introduce the basic definition, the properties, the derivation of Adomian decomposition method, and discuss the application of Adomian decomposition method in various fields and problems. We solve the vibration problem of a column with different boundary conditions, and evaluate the efficiency and accuracy of the Adomian decomposition method by comparing the numerical solutions obtained with those found in previous literature. The original linear or nonlinear differential equation is transformed by the Adomian modified decomposition method, and few assumptions are required to achieve fast convergence and accuracy. This removes the need to carry out complex derivations, and shortens processing time, while still achieving results with satisfactory hit rate compared with previous literature.

The results of this study show that the dimensionless natural frequency convergence is the most rapid with boundary conditions of spring end - free end. And the dimensionless natural frequency increases as the taper ratio and the modulus of spring increases. The effect of the moving spring modulus on the dimensionless natural frequency is greater than that of the rotary spring. Resonance occurs when the vibration forcing frequency is at or very close to dimensionless natural frequency of the system by analyzing the dynamic behavior of forced vibrations.

[1] LaBryer, P.J. Attar, “A harmonic balance approach for large-scale problems in nonlinear structural dynamics,” Computers & Structures, Vol.88, pp.1002–1014, 2010.
[2] Mahmoud, A.A., Awadalla, R., Nassar, M.M., “Free vibration of non-uniform column using DQM,” Mechanics Research Communications, Vol.38, pp.443–448, 2011.
[3] Uzny, S., Osadnik, M., “An Influence of the Longitudinal Spring Element on Free Vibrations of the Partially Tensioned Column,” Procedia Engineering, Vol.177, pp.135–140, 2017.
[4] Ruocco, E., Wang, C.M., Zhang, H., Challamel, N., “An approximate model for optimizing Bernoulli columns against buckling,” Engineering Structures, Vol.141, pp.316–327, 2017.
[5] Kraav, T., Lellep, J., “Elastic Stability of Uniform and Hollow Columns,” Procedia Engineering, Vol.172, pp.570–577, 2017.
[6] Demirdag, O., Yesilce, Y., “Solution of free vibration equation of elastically supported Timoshenko columns with a tip mass by differential transform method,” Advances in Engineering Software, Vol.42, pp.860–867, 2011.
[7] Taha, M., Essam, M., “Stability behavior and free vibration of tapered columns with elastic end restraints using the DQM method,” Ain Shams Engineering Journal, Vol.4, pp.515–521, 2013.
[8] Ruocco, E., Zhang, H., Wang, C.M., “Hencky bar-chain model for buckling analysis of non-uniform columns,” Structures, Vol.6, pp.73–84, 2016.
[9] Abassy, T.A., El-Tawil, M.A., Saleh, H.K., “The solution of Burgers’ and good Boussinesq equations using ADM–Padé technique,” Chaos, Solitons & Fractals, Vol.32, pp.1008–1026, 2007.
[10] Leung, A.Y.T., Yang, H.X., Chen, J.Y., “Parametric bifurcation of a viscoelastic column subject to axial harmonic force and time-delayed control,” Computers & Structures, Vol.136, pp.45–55, 2014.
[11] Lenci,S., Mazzilli,C.E.N., “Asynchronous free oscillations of linear mechanical systems: A general appraisal and a digression on a column with a follower force,” International Journal of Non-Linear Mechanics, 2017.
[12] Chernin, L., Vilnay, M., Shufrin, I., “Blast dynamics of beam-columns via analytical approach,” International Journal of Mechanical Sciences, Vol.106, pp.331–345, 2016.
[13] Dunne, J.F., Hayward, P., “A split-frequency harmonic balance method for nonlinear oscillators with multi-harmonic forcing,” Journal of Sound and Vibration, Vol.295, pp.939–963, 2006.
[14] Kuczera, K., “Linear force constants in harmonic force fields,” Journal of Molecular Structure, Vol.117, pp.11–18, 1984.
[15] Ritto, T.G., Escalante, M.R., Sampaio, R., Rosales, M.B., “Drill-string horizontal dynamics with uncertainty on the frictional force,” Journal of Sound and Vibration, Vol.332, pp.145–153, 2013.
[16] Mailybaev, A.A., Seyranian, A.P., “Stabilization of statically unstable columns by axial vibration of arbitrary frequency,” Journal of Sound and Vibration, Vol.328, pp.203–212, 2009.
[17] Hsu, J.C., “Application of the Adomian Modified Decomposition Method to the Free Vibrations of Beams,” Dissertation for Doctor of Philosophy, Department of Mechanical Engineering, National Cheng-Kung University, January 2009.
[18] Tsai, P.Y., “Applications of the Hybrid Laplace Adomian Decomposition Method to Nonlinear Physical Systems,” Dissertation for Doctor of Philosophy, Department of Mechanical Engineering, National Cheng-Kung University, January 2010.
[19] Hwang, Y.S., “Application of The Differential Transformation Method to Vibration Problems,” Dissertation for Master, Department of Mechanical Engineering, National Cheng-Kung University, July 2006.
[20] 邱政田, 機械振動學概論, 五南圖書出版公司, 2005.
[21] Adomian, G., “Solving Frontier Problems of Physics: The Decomposition Method,” Kluwer Academic Publishers, 1994.
[22] Dimarogonas, A.D., “Vibration for Engineers,” 2nd ed., Prentice-Hall, Inc., 1996.
[23] Meirovitch, L., “Fundamentals of Vibrations,” McGraw-Hall, International Edition, 2001.
[24] Thomson, W. T., “Theory of Vibration with Applications,” 2nd ed., 1981.
[25] Weaver, W., Timoshenko, S.P., and Young, D.H., “Vibration Problems in Engineering,” 5th ed., John Wiley & Sons, Inc., 1990.
[26] Adomian, G., and Rach, R., “Modified decomposition solution of
nonlinear partial differential equations,” Computers and Mathematics withApplications 23, pp.17-23, 1992.
[27] Adomian, G., and Rach, R., “Nonlinear transformation of series,” Applied Mathematics Letters 4, pp.69-71, 1991.
[28] Chen, C. N., “Vibration of prismatic beam on an elastic foundation by the differential quadrature element method,” Computers & Structures 77, pp.1-9,2000.
[29] Grant, D. A., “The effect of rotary inertia and shear deformation on the frequency and normal mode equations of uniform beams carrying a concentrated mass,” Journal of Sound and Vibration 57(3), pp.357-365, 1978.
[30] Thambiratnam, D., and Zhuge, Y., “Free vibration analysis of beams on elastic foundation,” Computers & Structures 60(6), pp.971-980, 1996.