A modified harmonic balance method to obtain higher-order approximations to strongly nonlinear oscillators

Abstract

We propose a new method, namely, the modified harmonic balance method. This paper also analyses and offers the high-order approximate periodic solutions to the strongly nonlinear oscillator with cubic and harmonic restoring force. The existing harmonic balance method cannot be applied directly to such kind of nonlinear oscillators in the presence of forcing term. It is possible if we rewrite the original form of the nonlinear oscillators. If we do so, the results are valid only for small values of amplitude of the oscillation. Moreover, after applying the existing harmonic balance method, a set of complicated higher-order nonlinear algebraic equations are obtained. Analytical investigation

Global residue harmonic balance method for strongly nonlinear oscillator with cubic and harmonic restoring force

“…1 A modified energy balance method (MEBM) was proposed for obtaining the higher-order approximations. 34 We remark that MEBM may perform less efficient than HBM in some special cases. El-Dib and Matoog pointed out that the period solution of ( 1) is available when the coefficients of (1) satisfy some constrained condition.…”

Analytical study of the vibrating double-sided quintic nonlinear nano-torsional actuator using higher-order Hamiltonian approach

“…Getting a relation between different parameters from equations ( 20), (22), and ( 23) is not easy manually. Using Mathematica software program, the second-order algebraic equation set will be ready to solve to get the natural frequency ω and values of a and b, so the second-order approximation solution is given by equation (19) after knowing the values of the variables, some of the results are listed in Table 1 for the Van der Waals force and Table 2 for the Casimir effect.…”

“…2, compact support radial basis functions by Chen et al, 3 variational iteration techniques, [4][5][6] the greedy algorithm, 7 local RBF method, 8 semi inverse method, 9,10 iteration perturbation method, 11 variational principle, [12][13][14][15][16] Taylor series method 17 and local methods such as the local Kansa method and local method of approximate particular solutions, 18 He's frequency formulation, 19 and many more. [20][21][22][23][24][25][26][27][28][29] Nonlinear oscillators have been widely utilized to represent numerous physical systems particularly in applied sciences and engineering including the vibrations of plates and beams, the vibrations induced on different structures by fluid flow, the large amplitude oscillations of centrifugal governor systems, the free vibration of a restrained uniform beam undergoing large amplitudes of oscillation and carrying intermediate lumped mass, the oscillations of magneto-elastic mechanical systems or pendulum-like systems, the oscillations of pendulum-like systems or propagation of a short electromagnetic pulse in a nonlinear medium, and so on (see, for example, Refs. [30][31][32][33][34].…”

Accurate analytical solution of the circular sector oscillation by the modified harmonic balance method

“…Recently, several methods have been introduced and developed to obtain approximate solutions for (NDEs) due to their complexity and the difficulty of solving them through traditional perturbation techniques. For example, variational iteration method, 5 homotopy perturbation method, 6 max-min approach, [7][8][9] global residue harmonic balance method (GRHBM) for obtaining higher-order approximate solutions, [10][11][12] modified homotopy perturbation method, [13][14][15] energy balance method, 16,17 Hamiltonian approach, [18][19][20] iteration perturbation technique, 21 coupled homotopy-variational approach, [22][23][24] frequency-amplitude formulation, 25,26 multiple scales technique, 27 parameter expansion method, 28 averaging method, 29 iteration method, 30 and Laplace variational iteration method. 31 The harmonic balance method (HBM) is one of the main techniques for obtaining approximate analytical solutions to NDEs describing oscillatory systems.…”

“…31 The harmonic balance method (HBM) is one of the main techniques for obtaining approximate analytical solutions to NDEs describing oscillatory systems. 4,6,[32][33][34] In recent decades, some researchers have studied the behavior of the circular sector oscillator generally modeled using NDEs. For example, Shaban et al, 13 investigated the numerical behavior of the nonlinear system using the modified homotopy perturbation method (HBM).…”