scholarly journals Dynamic Characteristics of Deeply Buried Spherical Biogas Digesters in Viscoelastic Soils

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Hongwei Hou ◽  
Shihu Gao ◽  
Qianqian Guo ◽  
Long Chen ◽  
Bing Wu ◽  
...  
Keyword(s):  
Cyclic Loading ◽  
Fractional Derivative ◽  
Elastic Medium ◽  
Shell Structure ◽  
Fractional Derivatives ◽  
Analytic Solution ◽  
Analytic Solutions ◽  
Dynamic Responses ◽  
Significant Difference ◽  
The Continuum

The harmonic vibration characteristics of a deeply buried spherical methane tank in viscoelastic soil subjected to cyclic loading in the frequency domain are investigated. The dynamic behavior of the soil is described based on the theory of fractional derivatives. By introducing potential functions, the closed-form expressions for the displacement and the stress of the viscoelastic soil surrounding the deeply buried spherical methane tank are obtained. Two die structures are considered: a homogeneous elastic medium and a shell structure. Based on the theory of elastic motion and the Flügge theory, analytic solutions for the dynamic responses of the spherical methane tank in a fractional-derivative viscoelastic soil are derived explicitly. Analytic solution expressions of the undetermined coefficients are determined by using the continuum boundary conditions. The system dynamic responses to the homogeneous elastic medium and the shell structure and the influences of the parameters of the fractional derivative, soil, and die on the dynamic characteristic of the system are compared and analyzed. The results indicate a significant difference between the dynamic responses of the die structures for the two models.

Nonlinear Dynamics Characteristic of Rotor-Bearing System by a Finite Element Model

2009 ◽  
Author(s):  
Chaofeng Li ◽  
Zhaohui Ren ◽  
Xiaopeng Li ◽  
Bangchun Wen
Keyword(s):  
Nonlinear Dynamic ◽  
Discrete Model ◽  
Element Model ◽  
Dynamic Responses ◽  
Runge Kutta Method ◽  
Significant Difference ◽  
Rotor Bearing ◽  
Comparison Of The Results ◽  
Bearing System ◽  
The Continuum

The nonlinear dynamic behavior of a rotor-bearing system is analyzed with its continuum model based on the analysis of the discrete model, with considering some other important influencing factors besides the nonlinear factors of the bearing, such as, the effect of inertia distribution and shear, transverse-torsion, structural geometric parameters of the system, which make the description of the system more embodiment and avoid the casualness of selection of system parameters. The dynamic responses of the continuum system and discrete system in the same unbalance condition are approached by the Runge-Kutta method and Newmark-β method. With the comparison of the results, significant difference about the dynamic characteristics is found with the addition of the considered factors. It is suggested that the substitution of discrete model by the continuum ones can get more accurate and abundant results. Furthermore, these results can provide more accurate verification and reference for the experiment and nonlinear dynamic design of the more complicated rotor system.

Numerical solution of fractionally damped beam by homotopy perturbation method

Open Physics ◽  
2013 ◽  
Vol 11 (6) ◽  
Author(s):  
Diptiranjan Behera ◽  
Snehashish Chakraverty
Keyword(s):  
Fractional Derivative ◽  
Perturbation Method ◽  
Numerical Solution ◽  
Fractional Derivatives ◽  
Homotopy Perturbation Method ◽  
Analytic Solutions ◽  
Step Function ◽  
Homotopy Perturbation ◽  
Derivatives Of ◽  
Appl Math

AbstractThis paper investigates the numerical solution of a viscoelastic continuous beam whose damping behaviours are defined in term of fractional derivatives of arbitrary order. The Homotopy Perturbation Method (HPM) is used to obtain the dynamic response. Unit step function response is considered for the analysis. The obtained results are depicted in various plots. From the results obtained it is interesting to note that by increasing the order of the fractional derivative the beam suffers less oscillation. Similar observations have also been made by keeping the order of the fractional derivative constant and varying the damping ratios. Comparisons are made with the analytic solutions obtained by Zu-feng and Xiao-yan [Appl. Math. Mech. 28, 219 (2007)] to show the effectiveness and validation of this method.

scholarly journals The Approximate Analytic Solution of the Time-Fractional Black-Scholes Equation with a European Option Based on the Katugampola Fractional Derivative

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 214
Author(s):  
Sivaporn Ampun ◽  
Panumart Sawangtong
Keyword(s):  
Fractional Derivative ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Analytic Solution ◽  
Analytic Solutions ◽  
Approximate Analytic Solution ◽  
European Option ◽  
European Option Pricing ◽  
Homotopy Perturbation ◽  
Black Scholes

In the finance market, it is well known that the price change of the underlying fractal transmission system can be modeled with the Black-Scholes equation. This article deals with finding the approximate analytic solutions for the time-fractional Black-Scholes equation with the fractional integral boundary condition for a European option pricing problem in the Katugampola fractional derivative sense. It is well known that the Katugampola fractional derivative generalizes both the Riemann–Liouville fractional derivative and the Hadamard fractional derivative. The technique used to find the approximate analytic solutions of the time-fractional Black-Scholes equation is the generalized Laplace homotopy perturbation method, the combination of the generalized Laplace transform and homotopy perturbation method. The approximate analytic solution for the problem is in the form of the generalized Mittag-Leffler function. This shows that the generalized Laplace homotopy perturbation method is one of the most effective methods to construct approximate analytic solutions of the fractional differential equations. Finally, the approximate analytic solutions of the Riemann–Liouville and Hadamard fractional Black-Scholes equation with the European option are also shown.

On Finite Part Integrals and Hadamard-Type Fractional Derivatives

2018 ◽  
Vol 13 (9) ◽  
Author(s):  
Li Ma ◽  
Changpin Li
Keyword(s):  
Fractional Derivative ◽  
Singular Integral ◽  
Fractional Derivatives ◽  
Continuous Functions ◽  
Finite Part ◽  
Absolutely Continuous Functions ◽  
Absolutely Continuous ◽  
Fundamental Properties ◽  
Logarithmic Series ◽  
The Right

This paper is devoted to investigating the relation between Hadamard-type fractional derivatives and finite part integrals in Hadamard sense; that is to say, the Hadamard-type fractional derivative of a given function can be expressed by the finite part integral of a strongly singular integral, which actually does not exist. Besides, our results also cover some fundamental properties on absolutely continuous functions, and the logarithmic series expansion formulas at the right end point of interval for functions in certain absolutely continuous spaces.

scholarly journals Fractional-Order Colour Image Processing

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 457
Author(s):  
Manuel Henriques ◽  
Duarte Valério ◽  
Paulo Gordo ◽  
Rui Melicio
Keyword(s):  
Image Processing ◽  
Edge Detection ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Grey Scale ◽  
Colour Image ◽  
Colour Image Processing ◽  
Image Processing Algorithms ◽  
Processing Algorithms

Many image processing algorithms make use of derivatives. In such cases, fractional derivatives allow an extra degree of freedom, which can be used to obtain better results in applications such as edge detection. Published literature concentrates on grey-scale images; in this paper, algorithms of six fractional detectors for colour images are implemented, and their performance is illustrated. The algorithms are: Canny, Sobel, Roberts, Laplacian of Gaussian, CRONE, and fractional derivative.

scholarly journals Time-Domain Analysis of Fractional Electrical Circuit Containing Two Ladder Elements

Electronics ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 475
Author(s):  
Ewa Piotrowska ◽  
Krzysztof Rogowski
Keyword(s):  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Electric Circuit ◽  
Theoretical Models ◽  
Time Domain Analysis ◽  
Electrical Circuit ◽  
State Variable ◽  
State Space System ◽  
Derivatives Of ◽  
Different Order

The paper is devoted to the theoretical and experimental analysis of an electric circuit consisting of two elements that are described by fractional derivatives of different orders. These elements are designed and performed as RC ladders with properly selected values of resistances and capacitances. Different orders of differentiation lead to the state-space system model, in which each state variable has a different order of fractional derivative. Solutions for such models are presented for three cases of derivative operators: Classical (first-order differentiation), Caputo definition, and Conformable Fractional Derivative (CFD). Using theoretical models, the step responses of the fractional electrical circuit were computed and compared with the measurements of a real electrical system.

scholarly journals Constant curvature coefficients and exact solutions in fractional gravity and geometric mechanics

Open Physics ◽  
2011 ◽  
Vol 9 (5) ◽  
Author(s):  
Dumitru Baleanu ◽  
Sergiu Vacaru
Keyword(s):  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Geometric Mechanics ◽  
Fractional Dynamics ◽  
Constant Matrix ◽  
Curve Flows ◽  
Physical Interactions ◽  
Nonholonomic Manifolds ◽  
Lagrange Mechanics ◽  
Fractional Gravity

AbstractWe present a study of fractional configurations in gravity theories and Lagrange mechanics. The approach is based on a Caputo fractional derivative which gives zero for actions on constants. We elaborate fractional geometric models of physical interactions and we formulate a method of nonholonomic deformations to other types of fractional derivatives. The main result of this paper consists of a proof that, for corresponding classes of nonholonomic distributions, a large class of physical theories are modelled as nonholonomic manifolds with constant matrix curvature. This allows us to encode the fractional dynamics of interactions and constraints into the geometry of curve flows and solitonic hierarchies.

scholarly journals A Convergent Collocation Approach for Generalized Fractional Integro-Differential Equations Using Jacobi Poly-Fractonomials

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 979
Author(s):  
Sandeep Kumar ◽  
Rajesh K. Pandey ◽  
H. M. Srivastava ◽  
G. N. Singh
Keyword(s):  
Differential Equation ◽  
Fractional Derivative ◽  
Collocation Method ◽  
Caputo Fractional Derivative ◽  
Fractional Derivatives ◽  
Special Cases ◽  
Collocation Approach ◽  
Linear And Nonlinear ◽  
Simulation Results ◽  
Theoretical Results

In this paper, we present a convergent collocation method with which to find the numerical solution of a generalized fractional integro-differential equation (GFIDE). The presented approach is based on the collocation method using Jacobi poly-fractonomials. The GFIDE is defined in terms of the B-operator introduced recently, and it reduces to Caputo fractional derivative and other fractional derivatives in special cases. The convergence and error analysis of the proposed method are also established. Linear and nonlinear cases of the considered GFIDEs are numerically solved and simulation results are presented to validate the theoretical results.

scholarly journals Vibration Equation of Fractional Order Describing Viscoelasticity and Viscous Inertia

Open Physics ◽  
2019 ◽  
Vol 17 (1) ◽  
pp. 850-856 ◽  
Author(s):  
Jun-Sheng Duan ◽  
Yun-Yun Xu
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Harmonic Excitation ◽  
Steady State Response ◽  
Vibration System ◽  
Derivative Operator ◽  
Vibration Equation ◽  
State Response ◽  
Frequency Relation

Abstract The steady state response of a fractional order vibration system subject to harmonic excitation was studied by using the fractional derivative operator ${}_{-\infty} D_t^\beta,$where the order β is a real number satisfying 0 ≤ β ≤ 2. We derived that the fractional derivative contributes to the viscoelasticity if 0 < β < 1, while it contributes to the viscous inertia if 1 < β < 2. Thus the fractional derivative can represent the “spring-pot” element and also the “inerterpot” element proposed in the present article. The viscosity contribution coefficient, elasticity contribution coefficient, inertia contribution coefficient, amplitude-frequency relation, phase-frequency relation, and influence of the order are discussed in detail. The results show that fractional derivatives are applicable for characterizing the viscoelasticity and viscous inertia of materials.

scholarly journals How Many Proximal Screws Are Needed for a Stable Proximal Humerus Fracture Fixation?

2021 ◽  
Vol 12 ◽  
pp. 215145932199274
Author(s):  
Hyojune Kim ◽  
Myung Jin Shin ◽  
Erica Kholinne ◽  
Janghyeon Seo ◽  
Duckwoo Ahn ◽  
...  
Keyword(s):  
Cyclic Loading ◽  
Proximal Humerus Fracture ◽  
Locking Plate ◽  
Proximal Humerus ◽  
Humerus Fracture ◽  
Proximal Fragment ◽  
Maximum Displacement ◽  
Significant Difference ◽  
Load To Failure ◽  
Screw Group

Purpose: This biomechanical study investigates the optimal number of proximal screws for stable fixation of a 2-part proximal humerus fracture model with a locking plate. Methods: Twenty-four proximal humerus fracture models were included in the study. An unstable 2-part fracture was created and fixed by a locking plate. Cyclic loading and load-to-failure tests were used for the following 4 groups based on the number of screws used: 4-screw, 6-screw, 7-screw, and 9-screw groups. Interfragmentary gaps were measured following cyclic loading and compared. Consequently, the load to failure, maximum displacement, stiffness, and mode of failure at failure point were compared. Results: The interfragmentary gaps for the 4-screw, 6-screw, 7-screw, and 9-screw groups were significantly reduced by 0.24 ± 0.09 mm, 0.08 ± 0.06 mm, 0.05 ± 0.01 mm, and 0.03 ± 0.01 mm following 1000 cyclic loading, respectively. The loads to failure were significantly different between the groups with the 7-screw group showing the highest load to failure. The stiffness of the 7-screw group was superior compared with the 6-screw, 9-screw, and 4-screw groups. The maximum displacement before failure showed a significant difference between the comparative groups with the 4-screw group having the lowest value. The 7-screw group had the least structural failure rate (33.3%). Conclusion: At least 7 screws would be optimal for proximal fragment fixation of proximal humerus fractures with medial comminution to minimize secondary varus collapse or fixation failure. Level of Evidence: Basic science study.

Sign in / Sign up

Export Citation Format

Share Document