Nonlinear steady state vibrations of beams made of the fractional Zener material using an exponential version of the harmonic balance method

Lewandowski, Roman

doi:10.1007/s11012-022-01576-8

Cited by 8 publications

(1 citation statement)

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“…They found that the DLFT-HB is asymptotically equivalent to the LCP-HB. Lewandowski [55] proposed an exponential version of the HB method to investigate the steady state vibration of geometrically nonlinear problems and found that the method works better than the classical HB method. Taghipour et al [56] applied the AFTHB method to investigate multi-harmonic forces and multiharmonic responses of nonlinear structures with strong nonlinearities.…”

Section: Introductionmentioning

confidence: 99%

Improved harmonic balance method for analyzing asymmetric restoring force functions in nonlinear vibration of mechanical systems

Mohammadian,

Ismail

2024

Phys. Scr.

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The non-linear differential equation governing some mechanical systems, such as the non-linear vibrations of FG beams resting on a nonlinear foundation, behaves similarly to the oscillations of a non-linear oscillator with an asymmetric restoring force function that includes both odd and non-odd non-linear terms. The objective of this research is to obtain higher-order approximate analytical solutions for such problems by introducing an enhanced harmonic balance method. By building upon the original problem, two new symmetric systems with odd non-linear terms are introduced, and their higher-order approximate analytical solutions are derived using a novel approach. In proposed method, the restoring force function is represented by its Fourier series expansion. Unlike previous papers, linearizing the equation or taking the first derivative of the Fourier series in the subsequent iteration is unnecessary. However, to enhance the solution’s accuracy, the remaining error from each iteration is utilized in the next one. Finally, by combining the results from the two introduced systems, the analytical period and corresponding periodic solution of the original problem can be obtained. This method is applied to the governing differential equation of FG beams resting on non-linear foundations, a physical non-natural oscillator, and conservative Toda oscillator. The key advantages of this approach are its simplicity and its ability to provide highly precise solutions for both small and large amplitudes in a single iteration.

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Section: Introductionmentioning

confidence: 99%