Linearized Harmonic Balancing Approach for Accurate Solutions to the Dynamically Shifted Oscillator

Abstract

The analytical approximate technique developed by Wu et al for conservative oscillators with odd nonlinearity is used to construct approximate frequency-amplitude relations and periodic solutions to the dynamically shifted oscillator. This nonlinear oscillator is described by an equation of motion which includes a linear restoring force and an anti-symmetric, constant force which is a nonlinear force depending only upon the sign of the displacement. By combining Newton's method with the method of harmonic balance, analytical approximations to the oscillation frequency and periodic solutions are constructed for this oscillator and the approximate periods obtained are valid for the complete range of oscillation amplitudes. Excellent agreement of the approximate frequencies and periodic solutions with the exact ones are demonstrated and discussed and the results reveal that this technique is very effective and convenient for solving this class of conservative nonlinear oscillatory systems.

Cubication of conservative nonlinear oscillators

“…Substituting equations ( 17) and ( 18) into equation (19) we can write the approximate frequency as a function of the first two terms of the Chebyshev series expansion of the nonlinear function f (x) as follows:…”

“…The exact frequency € ω e ( A) for this oscillator is given as follows [8,18,19] € ω e ( A) = ω 0 1− 2 π sin…”

“…The nonlinear differential equation for this case can be written as follows [18,19]: with initial conditions given in equation ( 2). The parameter x 0 is the dynamic shift and it can be positive or negative, but in the last situation A > 2 |x 0 | [8,18].…”

“…In figure 5 we compare the nonlinear function f (x) ω 2 0 = x + x 0 sgn(x) with its cubic approximation obtained using the Chebyshev series expansion, f Ch (x) ω 2 0 = 1 + 8x 0 πA x − 16x 0 3πA 3 x 3 . The exact frequency ω e (A) for this oscillator is given as follows [8,18,19]:…”

The Variational Approach Coupled with an Ancient Chinese Mathematical Method to the Relativistic Oscillator

“…with initial conditions ( ) ( ) 0 0 , 0 = ′ = u A u . Recently many analytical methods were proposed to solve various nonlinear oscillators, such as the parameter-expansion method [1][2][3], the energy balance method [4,5], the harmonic balance method [6,7,8], the homotopy perturbation method [9], He's amplitude-frequency formulation [10,11], a complete review on analytical approach to nonlinear oscillators was given in Ref. [12].…”