Nonlinear Statical and Dynamical Models of Fractional Derivative Viscoelastic Body

2005 ◽  
Author(s):  
H. Nasuno ◽  
N. Shimizu
Keyword(s):  
Fractional Derivative ◽  
Power Function ◽  
Exponential Function ◽  
Theoretical Consideration ◽  
Viscoelastic Body ◽  
Nonlinear Behavior ◽  
Dynamical Models ◽  
Dynamical Behaviors ◽  
Viscoelastic Coefficient ◽  
Sinusoidal Excitation

The authors have been conducting experiments on the investigation of nonlinear quasi-statical and dynamical behaviors of a viscoelastic body described by the fractional derivative law. Pre-stress due to pre-displacement induces high damping performance during sinusoidal excitation. To understand this behavior, nonlinear statical and dynamical models are investigated by theoretical consideration. The authors establish and propose appropriate models to describe the nonlinear behavior of the fractional derivative viscoelastic body. The nonlinearity of the viscoelastic coefficient for quasi-statical compressive displacement may be described by the power function with respect to pre-displacement and the nonlinearity of the viscoelastic coefficient for sinusoidal excitation may be described by the exponential function with respect to pre-displacement. Some discussions on the values of the viscoelastic coefficients.

Using the generalized Adams-Bashforth-Moulton method for obtaining the numerical solution of some variable-order fractional dynamical models

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Mohamed M. Khader
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Numerical Solution ◽  
Numerical Results ◽  
Convergence Order ◽  
Variable Order ◽  
Numerical Treatment ◽  
Dynamical Models ◽  
Fractional Modeling ◽  
Dynamics Problems

AbstractThis paper is devoted to introduce a numerical treatment using the generalized Adams-Bashforth-Moulton method for some of the variable-order fractional modeling dynamics problems, such as Riccati and Logistic differential equations. The fractional derivative is described in Caputo variable-order fractional sense. The obtained numerical results of the proposed models show the simplicity and efficiency of the proposed method. Moreover, the convergence order of the method is also estimated numerically.

IDENTIFICATION OF VISCOELASTIC MODEL OF FOUR-WIRE SUSPENSION SYSTEM IN OPTICAL PICKUP ACTUATOR

2017 ◽  
Vol 41 (5) ◽  
pp. 731-744
Author(s):  
Ren J. Chang ◽  
Zheng Y. Liu
Keyword(s):  
Transfer Function ◽  
Fractional Derivative ◽  
Dynamic Model ◽  
Classical Model ◽  
Viscoelastic Model ◽  
Optical Pickup ◽  
Static Tests ◽  
Wire Suspension ◽  
Suspension Structure ◽  
Sinusoidal Excitation

A novel viscoelastic model of four-wire suspension structure with damping gel in an optical pickup actuator was identified and validated. A two-stage method was developed for the identification of inertia, damping, and spring parameters in the dynamic model. The inertia and spring parameters were identified from static tests. With the identified parameters in the dynamic model, the damping parameters were identified through sinusoidal excitation tests. The accuracy of utilizing fractional derivative to model the damping of polymer damper was validated by carrying out error analysis. The fractional transfer function with voltage input was identified and compared with the transfer function of classical model.

Global Dynamics of a Parasite-Host Model with Nonlinear Incidence Rate

2015 ◽  
Vol 25 (08) ◽  
pp. 1550102 ◽  
Author(s):  
Yilei Tang
Keyword(s):  
Incidence Rate ◽  
Global Attractor ◽  
Exponential Function ◽  
Limit Cycles ◽  
Hopf Bifurcations ◽  
Global Dynamics ◽  
Nonlinear Incidence Rate ◽  
Nonlinear Incidence ◽  
Dynamical Behaviors ◽  
Fractional Function

The paper is concerned with the effect of a nonlinear incidence rate Sp Iq on dynamical behaviors of a parasite-host model. It is shown that the global attractor of the parasite-host model is an equilibrium if q = 1, which is similar to that of the parasite-host model with a nonlinear incidence rate of the fractional function [Formula: see text]. However, when q is greater than one, more positive equilibria appear and limit cycles arise from Hopf bifurcations at the positive equilibria for the model with the incidence rate Sp Iq. It reveals that the nonlinear incidence rate of the exponential function Sp Iq for generic p and q can lead to more complicated and richer dynamics than the bilinear incidence rate or the fractional incidence rate for this model.

The Wind-Sand Flux Structure of Grassland on the Northern Foot of Yinshan Mountain

2014 ◽  
Vol 668-669 ◽  
pp. 1530-1537
Author(s):  
Hong Tao Jiang ◽  
Chun Rong Guo ◽  
Chun Xing Hai ◽  
Shan Shan Sun ◽  
Yun Hu Xie ◽  
...  
Keyword(s):  
Vertical Distribution ◽  
Power Function ◽  
Exponential Function ◽  
Vertical Direction ◽  
Power Functions ◽  
Height Range ◽  
Sand Flux ◽  
Soil Flux ◽  
The Vertical Distribution ◽  
Better Than

Sand samplers were layed out in the grassland located in the northern foot of Yinshan Mountain for collecting soil flux samples from 0 to 1.5m height above the surface from Mar., 1, 2008 to Feb., 29, 2009.Exponential and Power functions were both used for describing vertical distribution of sand flux in the grassland, the results indicated that determination coefficient of Power function varied from 0.898 to 0.992 while 0.432 to 0.661 for exponential function. Power function is better than exponential function in describing the vertical distribution of both annual and seasonal soil flux, summer excluded. Annual cumulative percentage of each height was determined indirectly according to the power function mentioned above, the result indicated that up to 2m height,15-25% of soil flux concentrated with in 10cm above the surface,25-35% of soil flux concentrated within 20cm above the surface,30-40% of soil flux concentrated within 30 cm above the surface, 43-54% of soil flux concentrated within 50 cm above the surface,85-90% of soil flux concentrated within 150 cm above the surface, respectively. No significant differences of soil flux structures in spring, autumn, winter and in the whole year were found. The research on wind erosion of grassland in the vertical direction more dispersed, in the height range of sediment accumulated percentage was lower than that of the previous research.

The Method to Determine the Expression for Fatigue Curves

2007 ◽  
Vol 353-358 ◽  
pp. 331-334
Author(s):  
Hong Liang Yi ◽  
Ming Tu Ma ◽  
Zhi Gang Li ◽  
Hao Zhang
Keyword(s):  
Power Function ◽  
Exponential Function ◽  
Constant Term ◽  
Empirical Expressions ◽  
Fatigue Data ◽  
Function Expression ◽  
Two Parameter

There are three common empirical expressions used for the fatigue curves, which are power function, exponential function and three-parameter power function expression, respectively. The mathematical difference between the former two and the latter is whether there exists the constant term S0 in the equations. The S0 can be calculated to determine whether the two-paprameter expression or three-parameter expression should be used. If the two-parameter expression should be used, the power function and exponential function expressions can be compared to determine which one is the optimum one. Finally, the method has been validated by several groups of fatigue data.

LIMITS OF FRACTIONAL DERIVATIVES AND COMPOSITIONS OF ANALYTIC FUNCTIONS

2016 ◽  
Vol 103 (1) ◽  
pp. 104-115 ◽  
Author(s):  
THOMAS H. MACGREGOR ◽  
MICHAEL P. STERNER
Keyword(s):  
Fractional Derivative ◽  
Unit Disk ◽  
Power Function ◽  
Complex Plane ◽  
Analytic Functions ◽  
Fractional Derivatives ◽  
Hadamard Product ◽  
Open Unit ◽  
Open Unit Disk

Suppose that the function $f$ is analytic in the open unit disk $\unicode[STIX]{x1D6E5}$ in the complex plane. For each $\unicode[STIX]{x1D6FC}>0$ a function $f^{[\unicode[STIX]{x1D6FC}]}$ is defined as the Hadamard product of $f$ with a certain power function. The function $f^{[\unicode[STIX]{x1D6FC}]}$ compares with the fractional derivative of $f$ of order $\unicode[STIX]{x1D6FC}$. Suppose that $f^{[\unicode[STIX]{x1D6FC}]}$ has a limit at some point $z_{0}$ on the boundary of $\unicode[STIX]{x1D6E5}$. Then $w_{0}=\lim _{z\rightarrow z_{0}}f(z)$ exists. Suppose that $\unicode[STIX]{x1D6F7}$ is analytic in $f(\unicode[STIX]{x1D6E5})$ and at $w_{0}$. We show that if $g=\unicode[STIX]{x1D6F7}(f)$ then $g^{[\unicode[STIX]{x1D6FC}]}$ has a limit at $z_{0}$.

scholarly journals Stability of solutions for generalized fractional differential problems by applying significant inequality estimates

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mohammed D. Kassim ◽  
Thabet Abdeljawad ◽  
Saeed M. Ali ◽  
Mohammed S. Abdo
Keyword(s):  
Fractional Derivative ◽  
Power Function ◽  
Sufficient Conditions ◽  
Research Paper ◽  
Stability Of Solutions ◽  
Initial Value ◽  
Regularization Techniques ◽  
Fractional Differential ◽  
The Stability ◽  
Generalized Fractional Derivative

AbstractIn this research paper, we intend to study the stability of solutions of some nonlinear initial value fractional differential problems. These equations are studied within the generalized fractional derivative of various orders. In order to study the solutions’ decay to zero as a power function, we establish sufficient conditions on the nonlinear terms. To this end, some versions of inequalities are combined and generalized via the so-called Bihari inequality. Moreover, we employ some properties of the generalized fractional derivative and appropriate regularization techniques. Finally, the paper involves examples to affirm the validity of the results.

scholarly journals Series Representation of Power Function

2018 ◽  
Author(s):  
Petro Kolosov
Keyword(s):  
Numerical Methods ◽  
Power Function ◽  
Exponential Function ◽  
Finite Differences ◽  
Series Representation ◽  
Discrete Case ◽  
Natural Numbers ◽  
Binomial Theorem ◽  
Binomial Expansion

In this paper described numerical expansion of natural-valued power function xn, in point x = x0 where n, x0 - natural numbers. Apply- ing numerical methods, that is calculus of finite differences, namely, discrete case of Binomial expansion is reached. Received results were compared with solutions according to Newton’s Binomial theorem and MacMillan Double Bi- nomial sum. Additionally, in section 4 exponential function’s ex representation is shown.

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