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Research Papers: Modeling

Fractional Dynamics: A Statistical Perspective

[+] Author and Article Information
J. A. Machado

Department of Electrotechnical Engineering,  Institute of Engineering of Porto, Rua Dr. Antonio Bernardino de Almeida, 4200-072 Porto, Portugaljtm@isep.ipp.pt

Alexandra Galhano

Department of Electrotechnical Engineering,  Institute of Engineering of Porto, Rua Dr. Antonio Bernardino de Almeida, 4200-072 Porto, Portugalamf@isep.ipp.pt

J. Comput. Nonlinear Dynam 3(2), 021201 (Jan 25, 2008) (5 pages) doi:10.1115/1.2833481 History: Received June 04, 2007; Revised August 10, 2007; Published January 25, 2008

Fractional calculus is a mathematical paradigm that has been increasingly adopted to describe the dynamics of systems with hereditary characteristics, or that reflect an average of a large population of microelements. In this line of thought, this article analyzes the statistical dynamics of a system composed of a large number of micromechanical masses with backlash and impacts. We conclude that, while individual dynamics of each element has an integer-order nature, the global dynamics reveal the existence of both integer and fractional dynamics.

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Copyright © 2008 by American Society of Mechanical Engineers
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Figures

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Figure 6

Parameters {mk−1,αk}, k=1,2, versus M1 for M2=1.0kg, h=10−2m, n=103 microelements, Tw=1.0s, dt=5.0×10−6s

Grahic Jump Location
Figure 7

Parameters {mk−1,αk}, k=1,2, versus M2 for M1=1.0kg, h=10−2m, n=103 microelements, Tw=1.0s, dt=5.0×10−6s

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Figure 1

Mechanical system; (a) multicomponent global structure and (b) elemental component, with two masses M1 and M2 with backlash h, subjected to impacts under the action of force f

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Figure 3

Transfer functions H1(jω), H2(jω) and approximations {m1−1,α1}≈{1.18,−1.02} and {m2−1,α2}≈{0.71,−1.01}, for one simulation with Tw=10.0s, Fmax=10.0N, M1=M2=1.0kg, h=10−2m, dt=5.0×10−6s

Grahic Jump Location
Figure 4

Parameters of the average transfer functions versus ε for Fmax={10,20,30,40,50}, with n=103 microelements, Tw=1.0s, M1=M2=1.0kg, h=10−2m, dt=5.0×10−6s

Grahic Jump Location
Figure 5

Parameters {mk−1,αk}, k=1,2, versus h, for n=103 microelements, Tw=1.0s, M1=M2=1.0kg, h=10−2m, dt=5.0×10−6s

Grahic Jump Location
Figure 2

Time history of ẋ1 and ẋ2 for one simulation with Tw=10.0s, Fmax=10.0N, M1=M2=1.0kg, h=10−2m, dt=5.0×10−6s

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