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Research Papers

Energy Considerations for Mechanical Fractional-Order Elements

[+] Author and Article Information
Carl F. Lorenzo

NASA Glenn Research Center,
Cleveland, OH 44135
e-mail: Carl.F.Lorenzo@nasa.gov

Tom T. Hartley

University of Akron,
Akron, OH 44325-3904
e-mail: thartley@uakron.edu

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received July 30, 2013; final manuscript received September 23, 2013; published online October 13, 2014. Assoc. Editor: J. A. Tenreiro Machado.

J. Comput. Nonlinear Dynam 10(1), 011014 (Oct 13, 2014) (7 pages) Paper No: CND-13-1192; doi: 10.1115/1.4025772 History: Received July 30, 2013; Revised September 23, 2013

This paper considers the energy aspects of fractional-order elements defined by the equation: force is proportional to the fractional-order derivative of displacement, with order varying from zero to two. In contrast to the typically conservative assumption of classical physics that leads to the potential and kinetic energy expressions, a number of important nonconservative differences are exposed. Firstly, the considerations must be time-based rather than displacement or momentum based variables. Time based equations for energy behavior of fractional elements are presented and example applications are considered. The effect of fractional order on the energy input and energy return of these systems is shown. Importantly, it is shown that the history, or initialization, has a significant effect on energy response. Finally, compact expressions for the work or energy, are developed.

Copyright © 2015 by ASME
Topics: Displacement , Physics
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References

Figures

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Fig. 1

Linear fractional element

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Fig. 2

Effect of order λ on work and displacement for a unit step in force. λ = 0.0 to 2.0 in steps of 0.20, kλ = 1.0.

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Fig. 3

Effect of order λ on displacement for a ramp in force. λ = 0.0 to 1.0 in steps of 0.20, kλ = 1.0.

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Fig. 4

Effect of order λ on work for a ramp in force. λ = 0.0 to 1.0 in steps of 0.20, kλ = 1.0.

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Fig. 5

Effect of order λ on displacement for a ramp in force. λ = 1.0 to 2.0 in steps of 0.20, kλ = 1.0.

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Fig. 6

Effect of order λ on work for a ramp in force. λ = 1.0 to 2.0 in steps of 0.20, kλ = 1.0.

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Fig. 13

Effect of order, λ, on displacement for split-triangular pulse in force. λ = 1.0 to 2.0 in steps of 0.20, kλ = 1.0.

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Fig. 12

Relaxation displacement for a triangular pulse in force. λ = 0 to 1.0 in steps of 0.10, kλ = 1.0.

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Fig. 11

Return energy efficiency, η, versus order λ for a triangular pulse in force. λ = 0 to 1.0 in steps of 0.025, kλ = 1.0.

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Fig. 10

Maximum and returned energy for a triangular pulse in force. λ = 0 to 1.0 in steps of 0.025, kλ = 1.0.

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Fig. 9

Force versus displacement for a triangular pulse in force. λ = 0 to 1.0 in steps of 0.10, kλ = 1.0.

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Fig. 8

Effect of order λ on work for a triangular pulse in force. λ = 0 to 1.0 in steps of 0.10, kλ = 1.0.

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Fig. 7

Effect of order λ on displacement for a triangular pulse in force. λ = 0 to 1.0 in steps of 0.10, kλ = 1.0.

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Fig. 15

Return energy efficiency, η, versus order λ for a split-triangular pulse in force. λ = 1.0 to 2.0 in steps of 0.25, kλ = 1.0.

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Fig. 14

Effect of order λ on work for a split-triangular pulse in force. λ = 1.0 to 2.0 in steps of 0.20, kλ = 1.0.

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Fig. 16

Effect of initialization on displacement for λ of 0.10 and 0.80 with initialization starting at a = 0, and 1.0. kλ = 1.0. Dashed lines λ = 0.1; solid lines λ = 0.8.

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Fig. 17

Effect of initialization on work for λ of 0.10 and 0.80 with initialization starting at a = 0, and 1.0. kλ = 1.0. Dashed lines λ = 0.1; solid lines λ = 0.8.

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