# Elliptic averaging methods using the sum of Jacobian elliptic delta and zeta functions as the generating solution

T. Okabe, Takahiro Kondou, J. Ohnishi

Research output: Contribution to journalArticle

16 Citations (Scopus)

### Abstract

The present paper describes an improved version of the elliptic averaging method that provides a highly accurate periodic solution of a non-linear system based on the single-degree-of-freedom Duffing oscillator with a snap-through spring. In the proposed method, the sum of the Jacobian elliptic delta and zeta functions is used as the generating solution of the averaging method. The proposed method can be used to obtain the non-odd-order solution, which includes both even- and odd-order harmonic components. The stability analysis for the approximate solution obtained by the present method is also discussed. The stability of the solution is determined from the characteristic multiplier based on Floquet's theorem. The proposed method is applied to a fundamental oscillator in a non-linear system. The numerical results demonstrate that the proposed method is very effective for analyzing the periodic solution of half-swing mode for systems based on Duffing oscillators with a snap-through spring.

Original language English 159-169 11 International Journal of Non-Linear Mechanics 46 1 https://doi.org/10.1016/j.ijnonlinmec.2010.08.004 Published - Jan 1 2011

### Fingerprint

Averaging Method
Elliptic function
Delta Function
Riemann zeta function
Nonlinear systems
Duffing Oscillator
Periodic Solution
Nonlinear Systems
Multiplier
Stability Analysis
Approximate Solution
Harmonic
Odd
Degree of freedom
Numerical Results
Theorem
Demonstrate

### All Science Journal Classification (ASJC) codes

• Mechanics of Materials
• Mechanical Engineering
• Applied Mathematics

### Cite this

In: International Journal of Non-Linear Mechanics, Vol. 46, No. 1, 01.01.2011, p. 159-169.

Research output: Contribution to journalArticle

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