In this study, the nonlinear free vibration of conservative two degrees of freedom systems is analyzed using the homotopy analysis method (HAM). The mathematical model of such systems is described by two second-order coupled differential equations with cubic nonlinearities. First, novel approximate analytical solutions for displacements and frequencies are established using HAM. Then, the homotopy Padé technique is applied to accelerate the convergence rate of the solutions. Comparison between the obtained results and those available in the literature shows that the first-order approximation of homotopy Padé technique leads to accurate solutions with a maximum relative error less than 0.068 percent for all the considered cases.
1.
Dimarogonas
, A. D.
, and Haddad
, S.
, 1992, Vibration for Engineers
, Prentice-Hall
, Englewood Cliffs, NJ
.2.
Cveticanin
, L.
, 1998, “Some Particular Solutions Which Describe the Motion of the Rotor
,” J. Sound Vib.
0022-460X, 212
, pp. 173
–178
.3.
Mahmoud
, G. M.
, and Aly
, S. A. H.
, 2000, “On Periodic Solutions of Parametrically Excited Complex Non-Linear Dynamical Systems
,” Physica A
0378-4371, 278
, pp. 390
–404
.4.
Cveticanin
, L.
, 2001, “Analytic Approach for the Solution of the Complex-Valued Strong Non-Linear Differential Equation of Duffing Type
,” Physica A
0378-4371, 297
, pp. 348
–360
.5.
Nayfeh
, A. H.
, and Balachandran
, B.
, 1989, “Modal Interactions in Dynamical and Structural Systems
,” Appl. Mech. Rev.
0003-6900, 42
, pp. 175
–201
.6.
Tse
, F. S.
, Morse
, I. E.
, and Hinkle
, R. T.
, 1978, Mechanical Vibrations: Theory And applications
, Allyn and Bacon Inc.
, Boston
.7.
Ladygina
, Y. V.
, and Manevich
, A. I.
, 1993, “Free Oscillations of a Nonlinear Cubic System With Two Degrees of Freedom and Close Natural Frequencies
,” J. Appl. Math. Mech.
0021-8928, 57
, pp. 257
–266
.8.
Cveticanin
, L.
, 2002, “The Motion of a Two-Mass System With Non-Linear Connection
,” J. Sound Vib.
0022-460X, 252
, pp. 361
–369
.9.
Cveticanin
, L.
, 2001, “Vibrations of a Coupled Two-Degree-of-Freedom System
,” J. Sound Vib.
0022-460X, 247
, pp. 279
–292
.10.
Vakakis
, A. F.
, and Rand
, R. H.
, 2004, “Non-Linear Dynamics of a System of Coupled Oscillators With Essential Stiffness Nonlinearities
,” Int. J. Non-Linear Mech.
0020-7462, 39
, pp. 1079
–1091
.11.
Lai
, S. K.
, and Lim
, C. W.
, 2007, “Nonlinear Vibration of a Two-Mass System With Nonlinear Stiffnesses
,” Nonlinear Dyn.
0924-090X, 49
, pp. 233
–249
.12.
Qaisi
, M. I.
, and Kilani
, A. W.
, 2000, “A Power-Series Solution for a Strongly Non-Linear Two-Degree-of-Freedom System
,” J. Sound Vib.
0022-460X, 233
, pp. 489
–494
.13.
Nayfeh
, A. H.
, and Mook
, D. T.
, 1979, Nonlinear Oscillations
, Wiley
, New York
.14.
Mickens
, R. E.
, 1996, Oscillations in Planar Dynamic Systems
, World Scientific
, Singapore
.15.
Bellman
, R.
, 1970, Methods of Nonlinear Analysis
, Academic
, New York
.16.
Gelb
, A.
, and Vander Velde
, W. E.
, 1968, Multiple Input Describing Functions and Nonlinear System Design
, McGraw-Hill
, New York
.17.
Kaplan
, W.
, 1958, Ordinary Differential Equations
, Addison-Wesley
, Reading, MA
.18.
Qaisi
, M. I.
, 1996, “A Power Series Approach for the Study of Periodic Motion
,” J. Sound Vib.
0022-460X, 196
, pp. 401
–406
.19.
Wu
, B. S.
, Sun
, W. P.
, and Lim
, C. W.
, 2006, “An Analytical Approximate Technique for a Class of Strongly Non-Linear Oscillators
,” Int. J. Non-Linear Mech.
0020-7462, 41
, pp. 766
–774
.20.
Liao
, S. J.
, 1992, “On the Proposed Homotopy Analysis Techniques for Nonlinear Problems and Its Application
,” Ph.D. thesis, Shanghai Jiao Tong University, Shanghai, China.21.
Liao
, S. J.
, 2003, Beyond Perturbation: Introduction to Homotopy Analysis Method
, Chapman and Hall
, London
/CRC
, Boca Raton, FL
.22.
Liao
, S. J.
, and Cheung
, K. F.
, 2003, “Homotopy Analysis of Nonlinear Progressive Waves in Deep Water
,” J. Eng. Math.
0022-0833, 45
, pp. 105
–116
.23.
Liao
, S. J.
, 2008, “Notes on the Homotopy Analysis Method: Some Definitions and Theorems
,” Commun. Nonlinear Sci. Numer. Simul.
1007-5704, 14
(4
), pp. 983
–997
.24.
Pirbodaghi
, T.
, Ahmadian
, M. T.
, and Fesanghary
, M.
, 2009, “On the Homotopy Analysis Method for Non-Linear Vibration of Beams
,” Mech. Res. Commun.
0093-6413, 36
, pp. 143
–148
.25.
Saff
, E. B.
, and Varga
, R. S.
, 1977, Padé and Rational Approximation
, Academic
, New York
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by American Society of Mechanical Engineers
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