Frequency formula for a class of fractal vibration system

Yi Tian

doi:10.31181/rme200103055y

Abstract

Four fractal nonlinear oscillators (The fractal Duffing oscillator, fractal attachment oscillator, fractal Toda oscillator, and a fractal nonlinear oscillator) are successfully established by He’s fractal derivative in a fractal space, and their variational principles are obtained by semi-inverse transform method. The approximate frequency of the four fractal oscillators are found by a simple frequency formula. The results show the frequency formula is a powerful and simple tool to a class of fractal oscillators.

Highlights

The partial differential equations (PDEs) arise in many fields like the condense matter physics, fluid mechanics, economics and management, etc
Vibration is the intrinsic property of a packing system, and so far there is no way to stop the vibration, the frequency-amplitude is the main factor for designing a packing system (Song, 2020)
The two-scale transform method and fractal frequency formulas are adopted to find the approximate frequency of fractal oscillator equation

Summary

Introduction

The partial differential equations (PDEs) arise in many fields like the condense matter physics, fluid mechanics, economics and management, etc. There are many methods for solving nonlinear PDEs, for example, the homotopy perturbation method (Anjum & He, 2020a, 2020b; He, 2003; He & El-Dib, 2020; Yu, et al, 2019), variational iteration method (Anjum & He, 2019; He, 1999), Taylor series method (He, 2019,2020a; He & Ji, 2019a; He, et al, 2020), Exp-function method (He, 2013; He & Wu, 2006), and variational-based methods (He, 2020b,2021; He & Ai, 2020). Vibration is the intrinsic property of a packing system, and so far there is no way to stop the vibration, the frequency-amplitude is the main factor for designing a packing system (Song, 2020). B2 2 and N is 3 / 2 for non-singular oscillators and 0.8 for singular oscillators

Fractal Duffing oscillator

Fractal attachment oscillator

Fractal Toda oscillator

Conclusions