Hamiltonian-Based Frequency-Amplitude Formulation for Nonlinear Oscillators

Abstract

Complex mechanical systems usually include nonlinear interactions between their components which can be modeled by nonlinear equations describing the sophisticated motion of the system. In order to interpret the nonlinear dynamics of these systems, it is necessary to compute more precisely their nonlinear frequencies. The nonlinear vibration process of a conservative oscillator always follows the law of energy conservation. A variational formulation is constructed and its Hamiltonian invariant is obtained. This paper suggests a Hamiltonian-based formulation to quickly determine the frequency property of the nonlinear oscillator. An example is given to explicate the solution process.

Hybrid rayleigh–van der pol–duffing oscillator: Stability analysis and controller

“…15 The inherent pull-in instability of MEMS systems can be completely overcome by the fractal vibration theory. [16][17][18] The homotopy perturbation method (HPM) 19 and the Hamitonian approach 20 are two main analytical tools for nonlinear vibration systems. The combination of the Laplace transforms, Lagrange multiplier, fractional complex transforms, and Mohand transform with HPM was employed to find approximate solutions for nonlinear partial differential equations.…”

Application of a Heuristic Method to Solve Nonlinear Oscillators with Irrational Forces

“…It is worth mentioning that many of the modern techniques for solving oscillatory problems have been established by J. H. He. If the reader wishes to go deeper into the study, origin and fundamentals of collocation methods, energy balance methods, variational methods and Hamiltonian techniques we recommend references [24,25,26,27,28,29,30,31].…”

Exact solutions of Hirota–Maccari system forced by multiplicative noise in the Itô sense

“…The analytical solutions of nonlinear differential equations which describe mathematical models are not easy to obtain. For NLPDEs, there are many appropriate numerical and analytical methods in the literature, developed by authors such as the exp(Àf(ς))-expansion method, 1 the Jacobi elliptic function method, 2 the variational iteration method, 3,4 the ðG 0 =GÞ-expansion method, 5,6 the homotopy perturbation method, 7 the modified homotopy perturbation method, 8 the Riccati-Bernoulli sub-ODE method, 9 perturbation method, 10,11 He's frequency formulation, 12,13 the sine-cosine method, [14][15][16] the Hirota's method, 17 the tanhsech method, 18,19 etc. While there are some methods to find the analytical solutions of fractional partial differential equations such as the fractal variational principle, [20][21][22][23] the Lie group analysis method, 24,25 and the perturbation method.…”