Frequency formula for a class of fractal vibration system

ABSTRACT

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 frequency formulation for oscillator (1) was proposed as (He, et al., 2021a where A is the amplitude, it can be approximated calculated as 3 / 2 for non-singular oscillators and 0.8 for singular oscillators.

Variational principle of fractal Duffing oscillator
In a fractal space, Eq. ( 5) can be described by He's fractal derivative as follows The variational principle (He, 2020c) of Eq. ( 7) can be given by semi-inverse transform method as follows

Fractal frequency formula
Using the two-scale transform method (He & Ji, 2019b) to Eq. ( 7) and assume

Fractal attachment oscillator
Consider the following attachment oscillator (Ren, et al  In (Ren,et al., 2019), the explicit form of the frequency formula of attachment oscillator is not given.

Fractal Toda oscillator
Consider the following Toda oscillator (He, et al., 2021c)  (1 ) lim () The variational principle of Eq. ( 27) can be given by semi-inverse transform method as follows Adopt the frequency formula (3), and the approximate frequency can be easily obtained as follows In (He, et al., 2021c), the frequency formula (34) is not given for Toda oscillator.

Conclusions
In this paper, four nonlinear oscillators are described by He's fractal derivative in a fractal space, and their variational principle are successfully established via semi-inverse transform method.The two-scale transform method and fractal frequency formulas are adopted to find the approximate frequency of fractal oscillator equation.The examples show the frequency formula is simple and effective.
space, Eq. (25) can be described by He's fractal derivative as follows )

Variational principle of a fractal nonlinear oscillator
Using the two-scale transform method to Eq. (37) and assume