scholarly journals Frequency formula for a class of fractal vibration system

2022 ◽  
Vol 3 (1) ◽  
pp. 55-61
Author(s):  
Yi Tian ◽  
Keyword(s):  
Variational Principles ◽  
Duffing Oscillator ◽  
Nonlinear Oscillators ◽  
Vibration System ◽  
Transform Method ◽  
Inverse Transform ◽  
Simple Tool ◽  
Fractal Derivative ◽  
Fractal Space ◽  
Approximate Frequency

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.

scholarly journals A powerful and simple frequency formula to nonlinear fractal oscillators

2020 ◽  
pp. 146134842094783
Author(s):  
Kang-Le Wang ◽  
Chun-Fu Wei
Keyword(s):  
Variational Principle ◽  
Conservation Laws ◽  
Nonlinear Oscillator ◽  
Transform Method ◽  
Inverse Transform ◽  
Simple Tool ◽  
Fractal Derivative ◽  
Fractal Space ◽  
Approximate Frequency

In this work, a fractal nonlinear oscillator is successfully established by fractal derivative in a fractal space, and its variational principle is obtained by semi-inverse transform method. The variational principle can provide conservation laws in an energy form. The approximate frequency of the fractal oscillator is found by a simple fractal frequency formula. An example shows the fractal frequency formula is a powerful and simple tool to fractal oscillators.

scholarly journals He’s frequency formula to fractal undamped Duffing equation

2021 ◽  
pp. 146134842199260
Author(s):  
Guang-Qing Feng
Keyword(s):  
Variational Principle ◽  
Nonlinear Oscillation ◽  
Duffing Equation ◽  
Transform Method ◽  
Simple Method ◽  
Interesting Topic ◽  
Numerical Result ◽  
Fractal Derivative ◽  
Fractal Space ◽  
Approximate Frequency

Nonlinear oscillation is an increasingly important and extremely interesting topic in engineering. This article completely reviews a simple method proposed by Ji-Huan He and successfully establishes a fractal undamped Duffing equation through the two-scale fractal derivative in a fractal space. Its variational principle is established, and the two-scale transform method and the fractal frequency formula are adopted to find the approximate frequency of the fractal oscillator. The numerical result shows that He’s frequency formula is a unique tool for the fractal equations.

scholarly journals Analytical solution for non-linear local fractional Bratu-type equation in a fractal space

2020 ◽  
Vol 24 (6 Part B) ◽  
pp. 3941-3947
Author(s):  
Shao-Wen Yao ◽  
Wen-Jie Li ◽  
Kang-Le Wang
Keyword(s):  
Analytical Solution ◽  
Fractional Derivative ◽  
Variational Formulation ◽  
Decomposition Method ◽  
Approximate Analytical Solution ◽  
Type Equation ◽  
Transform Method ◽  
Inverse Transform ◽  
Non Linear ◽  
Fractal Space

In this paper, the non-linear local fractional Bratu-type equation is described by the local fractional derivative in a fractal space, and its variational formulation is successfully established according to semi-inverse transform method. Finally, we find the approximate analytical solution of the local fractional Bratu-type equation by using Adomina decomposition method.

scholarly journals He’s fractal calculus and its application to fractal KdV equation

2021 ◽  
pp. 100-100
Author(s):  
Xue-Si Ma ◽  
Li-Na Zhang
Keyword(s):  
Analytical Solution ◽  
Iteration Method ◽  
Approximate Analytical Solution ◽  
Kdv Equation ◽  
Variational Iteration Method ◽  
Natural Phenomena ◽  
Transform Method ◽  
Fractal Derivative ◽  
Fractal Calculus ◽  
Fractal Space

He?s fractal calculus is a powerful and effective tool to dealing with natural phenomena in a fractal space. In this paper, we study the fractal KdV equation with He?s fractal derivative. We first adopt the two-scale transform method to convert the fractal KdV equation into its traditional partner in acontinuous space. Finally, we successfully use He?s variational iteration method (HVIM) to obtain its approximate analytical solution.

A NOVEL VARIATIONAL PERSPECTIVE TO FRACTAL WAVE EQUATIONS WITH VARIABLE COEFFICIENTS

Fractals ◽  
2021 ◽  
Author(s):  
XUE-FENG HAN ◽  
KANG-LE WANG
Keyword(s):  
Wave Equations ◽  
Variational Principles ◽  
Variable Coefficients ◽  
Transform Method ◽  
Numerical Examples ◽  
Fractal Models ◽  
Inverse Transform ◽  
Fractal Calculus ◽  
Different Types ◽  
Wave Models

This paper aims at establishing two different types of wave models with unsmooth boundaries by the fractal calculus, and their fractal variational principles are successfully designed by employing the fractal semi-inverse transform method. A new approximate technology is proposed to solve the two fractal models based on the variational principle and fractal two-scale transform method. Finally, two numerical examples show that the proposed method is efficient and accurate, which can be extended to solve different types of fractal models.

scholarly journals Parameters Approach Applied on Nonlinear Oscillators

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Najeeb Alam Khan ◽  
Fatima Riaz ◽  
Nadeem Alam Khan
Keyword(s):  
Natural Frequency ◽  
Initial Conditions ◽  
Duffing Oscillator ◽  
Nonlinear Oscillators ◽  
Solution Procedure ◽  
Power Exponent ◽  
Restoring Force ◽  
Arbitrary Power ◽  
Collocation Points ◽  
Approximate Frequency

We applied an approach to obtain the natural frequency of the generalized Duffing oscillatoru¨+u+α3u3+α5u5+α7u7+⋯+αnun=0and a nonlinear oscillator with a restoring force which is the function of a noninteger power exponent of deflectionu¨+αu|u|n−1=0. This approach is based on involved parameters, initial conditions, and collocation points. For any arbitrary power ofn, the approximate frequency analysis is carried out between the natural frequency and amplitude. The solution procedure is simple, and the results obtained are valid for the whole solution domain.

Dynamics research of Fangzhu’s nanoscale surface

2021 ◽  
pp. 146134842110527
Author(s):  
Pinxia Wu ◽  
Weiwei Ling ◽  
Xiumei Li ◽  
Xichun He ◽  
Liangjin Xie
Keyword(s):  
Energy Balance ◽  
Analytical Solutions ◽  
Numerical Experiments ◽  
Fractal Model ◽  
Balance Method ◽  
Transform Method ◽  
Energy Balance Method ◽  
Collection Rate ◽  
Fractal Derivative ◽  
Approximate Analytical Solutions

In this paper, we mainly focus on a fractal model of Fangzhu’s nanoscale surface for water collection which is established through He’s fractal derivative. Based on the fractal two-scale transform method, the approximate analytical solutions are obtained by the energy balance method and He’s frequency–amplitude formulation method with average residuals. Some specific numerical experiments of the model show that these two methods are simple and effective and can be adopted to other nonlinear fractal oscillators. In addition, these properties of the obtained solution reveal how to enhance the collection rate of Fangzhu by adjusting the smoothness of its surfaces.

scholarly journals Variational principle for nonlinear fractional wave equation in a fractal space

2021 ◽  
pp. 18-18
Author(s):  
Shao-Wen Yao
Keyword(s):  
Wave Equation ◽  
Variational Principle ◽  
Inverse Method ◽  
Scale Method ◽  
Fractional Wave Equation ◽  
Fractal Derivative ◽  
Fractal Space

The fractal derivative is adopted to describe the nonlinear fractional wave equation in a fractal space. A variational principle is successfully established by the semi-inverse method. The two-scale method and He?s exp-function are usedto solve the equation, and a good result is obtained.

A boundary division guiding synchrosqueezed wave packet transform method for rolling bearing fault diagnosis

2021 ◽  
Author(s):  
Bin Pang ◽  
Heng Zhang ◽  
Zhenduo Sun ◽  
Xiaoli Yan ◽  
Chunhua Li ◽  
...  
Keyword(s):  
Fault Diagnosis ◽  
Wave Packet ◽  
Rolling Bearing ◽  
Scale Space ◽  
Transform Method ◽  
Bearing Fault ◽  
Bearing Fault Diagnosis ◽  
Adaptive Parameter ◽  
Inverse Transform ◽  
Band Limited

Abstract Synchrosqueezed wave packet transform (SSWPT) can effectively reconstruct the band-limited components of the signal by inputting the specific reconstructed boundaries and it provides an alternative bearing fault diagnosis method. However, the selection of reconstructed boundaries can significantly affect the fault feature extraction performance of SSWPT. Accordingly, this paper presents a boundary division guiding SSWPT (BD-SSWPT) method. In this method, an adaptive boundary division method is developed to effectively determine the reconstructed boundaries of SSWPT. Firstly, the marginal spectrum of SSWPT, more robust to noise than the Fourier spectrum, is defined for the scale-space division to obtain the initial boundaries. Secondly, the inverse transform of SSWPT is conducted based on the initial boundaries to obtain the initial reconstructed components. Thirdly, a boundary redefinition scheme, composed of clustering and combination, is conducted to redefine the boundaries. Finally, the potential components are extracted by the inverse transform of SSWPT based on the redefined boundaries. The validity of BD-SSWPT is verified by simulated and experimental analysis, and the superiority of BD-SSWPT is highlighted through comparison with singular spectrum decomposition (SSD) and an adaptive parameter optimized variational mode decomposition (AVMD). The results demonstrate that BD-SSWPT identifies more significant fault features and has higher computational efficiency than SSD and AVMD.

Sign in / Sign up

Export Citation Format

Share Document