Variational Approach Method for Nonlinear Oscillations of the Motion of a Rigid Rod Rocking Back and Cubic

Abstract

Abstract-This paper deals with Approximate Analytical Solutions to nonlinear oscillations of a conservative, non-natural, single-degreeof-freedom system with odd nonlinearity. By extending the Variational approach proposed by He, we established approximate analytical formulas for the period and periodic solution.To illustrate the applicability and accuracy of the method, two examples are presented: (i) the motion of a rigid rod rocking back and forth on the circular surface without slipping, and (ii) Cubic-Quintic Duffing Oscillators. Comparison of the result which is obtained by this method with the obtained result by the Exact solution reveals that the He's Variational approach is very effective and convenient and can be easily extended to other nonlinear systems and can therefore be found widely applicable in engineering and other sciences.

Periodic Solution for Strongly Nonlinear Vibration Systems by He’s Energy Balance Method

“…To overcome the shortcomings, many new techniques have appeared in open literature [3][4][5][6][7][8][9][10][11][12][13][14], such as non-perturbative methods [3], homotopy perturbation method [4][5][6][7], perturbation techniques [8], Lindstedt-Poincaré method [9,10], parameter-expansion method [11,12] and parameterized perturbation method [13,14]. Recently, some approximate variational methods, including approximate energy method [15][16][17]32], variational iteration method [18][19][20][21][22] and variational approach [23][24][25][26][27][28][29][30][31], to solution, bifurcation, limit cycle and period solutions of nonlinear equations have been given much attention.…”

Solution of the damped cubic–quintic Duffing oscillator by using Jacobi elliptic functions

“…Next, we substitute Eqs. (9) and (10) into Eqs. (5) and (6), to get after some mathematical steps that:…”

“…It is well-known that the derivation of closed-form solutions that describe the dynamical response of these nonconservative oscillators becomes difficult since, in general, the equation of motion does not admit an exact analytic solution in terms of known (tabulated) functions terms [7]. Therefore, many perturbation and numerical techniques have been developed to obtain approximate solutions that describe the dynamical response of ordinary differential equations with cubic and quintic nonlinear terms [8][9][10][11][12][13].…”

Comparison of Energy Balance Period with Exact Period for Arising Nonlinear Oscillator Equations

“…To overcome the D.D. Ganji [2,3], Variational Iteration method (VIM) [4][5][6][7][8][9], Homotopy Perturbation method [10][11][12][13][14][15][16][17][18] and book-keeping parameter perturbation method [19], just to name a few, a review on some developed nonlinear analytical methods elaborately can be found in [20][21][22][23][24][25][26][27][28][29][30][31][32]. In He's Energy Balance method, a variational principle for the nonlinear oscillation is established, then a Hamiltonian is constructed, from which the angular frequency can be readily obtained by collocation method.…”