Abstract
Accurate approximate closed‐form solutions for the cubic‐quintic Duffing oscillator are obtained in terms of elementary functions. To do this, we use the previous results obtained using a cubication method in which the restoring force is expanded in Chebyshev polynomials and the original nonlinear differential equation is approximated by a cubic Duffing equation. Explicit approximate solutions are then expressed as a function of the complete elliptic integral of the first kind and the Jacobi elliptic function cn. Then we obtain other approximate expressions for these solutions, which are expressed in terms of elementary functions. To do this, the relationship between the complete elliptic integral of the first kind and the arithmetic‐geometric mean is used and the rational harmonic balance method is applied to obtain the periodic solution of the original nonlinear oscillator.
Highlights
The cubic-quintic Duffing equation is a differential equation with third- and/or fifth-power nonlinearity 1
The approximate frequency obtained in this paper, combining a cubication procedure based on Chebyshev polynomials expansion of the restoring force, the arithmetic geometric mean for approximating the exact frequency of the cubic Duffing oscillator, and the second-order rational harmonic balance method, is more accurate than the first-order and second-order frequencies obtained using the NHBM and slightly less accurate 0.38% versus 0.23% than the third-order frequency obtained using the NHBM
Using the results obtained by applying a cubication method for the cubic-quintic Duffing oscillator, very simple and accurate expression for the frequency and the solution are obtained for the cubic-quintic Duffing oscillator
Summary
The cubic-quintic Duffing equation is a differential equation with third- and/or fifth-power nonlinearity 1. We use the results previously obtained 4 using the Chebyshev series expansion of the restoring force 5, 6 for a quintic Duffing oscillator Using this approximation, the original nonlinear differential equation is replaced by a cubic Duffing equation, which can be exactly solved. The replacement of the original nonlinear equation by an “approximate cubic Duffing equation” allows us to obtain an approximate frequency-amplitude relation as a function of the complete elliptic integral of the first kind and the solution in terms of the Jacobi elliptic function cn 4. Using these first approximate equations, closed-form expressions for the approximate frequency and the solution in terms of elementary functions are obtained using the arithmetic-geometric mean combined with the second-order rational harmonic balance method
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