Accurate analytical solutions to nonlinear oscillators by means of the Hamiltonian approach

Abstract

Communicated by J. CashThe purpose of this paper is to apply the Hamiltonian approach to nonlinear oscillators. The Hamiltonian approach is applied to derive highly accurate analytical expressions for periodic solutions or for approximate formulas of frequency. A conservative oscillator always admits a Hamiltonian invariant, H, which stays unchanged during oscillation. This property is used to obtain approximate frequency-amplitude relationship of a nonlinear oscillator with high accuracy. A trial solution is selected with unknown parameters. Next, the Ritz-He method is used to obtain the unknown parameters. This will yield the approximate analytical solution of the nonlinear ordinary differential equations.In contrast with the traditional methods, the proposed method does not require any small parameter in the equation.

Nonlinear free and forced vibrations of curved single walled carbon nanotube on a Pasternak elastic foundation

“…This approach is a kind of energy method with a vast application in conservative oscillatory systems [18][19][20][21][22][23][24].…”

Extension of the Hamiltonian approach with general initial conditions

“…Since that time, a significant number of papers, where the suggested method is extended and applied, have been published. In the papers of Akbarzade and Kargar (2011a) and Akbarzade and Kargar (2011), the Hamiltonian method is applied for obtaining accurate analytical solutions to nonlinear oscillators. Using this method, He et al (2010) and Bayat et al (2014) obtained the solution for the Duffing-harmonic equation, and Cveticanin et al (2010a derived solutions for the generalized nonlinear oscillator with a fractional power.…”

Approximate Periodic Solution for the Nonlinear Helmholtz-Duffing Oscillator via Analytical Approaches

“…A lot of researchers have worked in this field and have proposed a lot of methods for demonstrating the dynamics responses of these systems [1][2][3][4]. They have developed this field of science and have analyzed the responses of the nonlinear vibration problems such as Duffing oscillators [5][6][7][8][9][10], nonlinear dynamics of a particle on a rotating parabola [11], nonlinear oscillators with discontinuity [12], oscillators with noninteger order nonlinear connection [13], the plasma physics equation [14], and van der Pol oscillator [15,16]. The Helmholtz-Duffing equation is a nonlinear problem with the quadratic and cubic nonlinear terms.…”