An improved Mickens’ solution for nonlinear vibrations

Abstract

The Duffing equation and other nonlinear equations of motion of nonlinear vibrations are widely applied in the fields of engineering and science, especially mechanical and electrical engineering. The modified Mickens extended iteration method has been utilized in this article to investigate for further accurate solutions to the Duffing equation and an equation of motion of nonlinear vibrations. The third approximate frequency shows good agreement with the exact result for both oscillators. For simplicity in algebraic calculations, the increasing harmonic terms have been taken to the fifth order. In order to establish the reliability and effectiveness of the proposed approach, the results obtained through this method are compared with those available in the literature. The comparison of the obtained solutions with numerical solutions shows a remarkable accuracy. Also, it becomes evident that the modified Mickens extended iteration method is significantly simpler, more accurate, efficient, and straightforward than other existing methods. The highest percentage error in the third approximate frequency of an equation of motion of nonlinear vibrations is less than 0.0488, and the maximal percentage error in the third approximate frequency of the Duffing oscillator is less than 0.0065. The proposed technique is principally exhibited in nonlinear models where the nonlinear terms are strong, but it can also be broadly applicable to other problems arising in engineering. Mathematics Subject Codes: 34A34, 34C25, 37N15 (AMS Mathematics Subject Classification 2010)

Keywords

• Extended iteration procedure
• Nonlinear vibrations
• Duffing oscillator
• Nonlinearity
• Nonlinear oscillations
• Fourier series