Advertisement

Nonlinear Dynamics

, Volume 88, Issue 4, pp 2435–2455 | Cite as

Dual input–output pairs for modeling hysteresis inspired by mem-models

  • Jin-Song Pei
  • François Gay-Balmaz
  • Joseph P. Wright
  • Michael D. Todd
  • Sami F. Masri
Original Paper
  • 243 Downloads

Abstract

We call attention to a dual-pair concept for modeling hysteresis involving instantaneous switching: Specifically, there are two input–output pairs for each hysteresis model under one specific input, namely a differential pair and an integral pair. Currently in engineering mechanics, only one pair is being recognized and utilized, not the other. Whereas this dual-pair concept is inherent in the differential and algebraic forms of memristors and memcapacitors, the concept has not been carried over to memristive system theory, nor to memcapacitive system theory. We show that the “zero-crossing” feature in memristors, memcapacitors, and memristive/memcapacitive models (i.e., the “mem-models”) is also a feature of the differential pairs of well-known non-mem-models, examples of which are Ramberg–Osgood, Bouc–Wen, bilinear hysteresis, and classical Preisach. The dual-pair concept thus connects mem-models and non-mem-models, thereby facilitating the modeling of hysteresis, and raising a set of scientific questions for further studies that might not otherwise come to awareness.

Keywords

Hysteresis Memcapacitive system Piecewise smooth system Hybrid dynamical system Classical Preisach model Mullins effect 

List of symbols

\(\dot{x}\)

Velocity

x

Displacement

a

Absement, the first time integral of displacement, x

\(\dot{r}\)

The first time derivative of r

r

Resisting force or characteristic force of an element

p

General momentum, the first time integral of r

\(\rho \)

The first time integral of p

t

Time

\(\mathbf {y}\)

State variables, see Tables 1, 2, 3 and 4

w

Internal state variable

u

Driving force

G

See Tables 1 and 3

F

See Tables 2 and 4

\(\mathbf {g}\)

See Tables 1 and 3

\(\mathbf {f}\)

See Tables 2 and 4

Notes

Acknowledgements

The first author would like to acknowledge the Vice President for Research of the University of Oklahoma for the support of the Faculty Investment Program.

References

  1. 1.
    Bernardo, M.D., Budd, C.J., Champneys, A.R., Kowalczyk, P.: Piecewise-smooth Dynamical Systems Theory and Applications, Applied Mathematical Sciences vol. 163. Springer (2008)Google Scholar
  2. 2.
    Bouc, R.: Modèle mathématique d’hystérésis. Acustica 21, 16–25 (1971)MATHGoogle Scholar
  3. 3.
  4. 4.
    Caughey, T.K.: Random excitation of a system with bilinear hysteresis. J. Appl. Mech. 27, 649–652 (1960a)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Caughey, T.K.: Sinusoidal excitation of a system with bilinear hysteresis. J. Appl. Mech. 27, 640–643 (1960b)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chua, L.O.: Memrister-the missing circuit element. IEEE Trans. Circuit Theory CT 18(5), 507–519 (1971)CrossRefGoogle Scholar
  7. 7.
    Chua, L.O.: Resistance switching memories are memristors. Appl. Phys. A Mater. Sci. Process. 102, 765–783 (2011)CrossRefMATHGoogle Scholar
  8. 8.
    Chua, L.O., Kang, S.M.: Memristive devices and systems. In: Proceedings of the IEEE, vol. 64, pp. 209–223 (1976)Google Scholar
  9. 9.
    Di Ventra, M., Pershin, Y.V., Chua, L.O.: Circuit elements with memory: Memristors, memcapacitors, and meminductors. In: Proceedings of IEEE, vol. 97, pp. 1717–1724 (2009)Google Scholar
  10. 10.
    Diani, J., Fayolle, B., Gilormini, P.: A review on the mullins effect. Eur. Polymer J. 45, 601–612 (2009)CrossRefGoogle Scholar
  11. 11.
    Ismail, M., Ikhouane, F., Rodellar, J.: The hysteresis bouc-wen model, a survey. Arch. Comput. Methods Eng. 16, 161–188 (2009)CrossRefMATHGoogle Scholar
  12. 12.
    Janičić, V., Ilić, V.: Preisach model gui version 1.0. http://www.mathworks.com/matlabcentral/fileexchange/47763-hysteresis-in-ferromagnetic-gui (2014)
  13. 13.
    Jeltsema, D., Scherpen, J.M.A.: Multidomain modeling of nonlinear networks and systems: energy- and power-based perspectives. IEEE Control Syst. Mag. 29(4), 28–59 (2009)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Jennings, P.C.: Periodic response of a general yielding structure. J. Eng. Mech. Division Proc. Am.Soc. Civ. Eng. 90(EM2), 131–166 (1964)Google Scholar
  15. 15.
    Kádár, G., Szabó, Z., Vértesy, G.: Product preisach model parameters from measured magnetization curves. Physica B 343, 137–141 (2004)CrossRefGoogle Scholar
  16. 16.
    Karnopp, D.C., Margolis, D.L., Rosenberg, R.C.: System Dynamics: Modeling, Simulation, and Control of Mechatronics Systems, 5th edn. Wileys, Hoboken (2012)Google Scholar
  17. 17.
    Krasnosel’skiǐ, M.A., Pokrovskiǐ, A.V.: Systems with Hysteresis. Springer, Berlin (1983)MATHGoogle Scholar
  18. 18.
    LeVeque, R.J.: Wave propagation software, computational science, and reproducible research. In: Proceedings of the International Congress of Mathematicians, pp. 1227–1253. European Mathematical Society, Madrid (2006)Google Scholar
  19. 19.
    Mayergoyz, I.: Mathematical Models of Hysteresis and Their Applications. Elsevier Series in Electromagnetism. Elsevier Science Ltd, Oxford (2003)Google Scholar
  20. 20.
    Oster, G.F., Auslander, D.M.: The memristor: a new bond graph element. ASME J. Dyn. Syst. Meas. Control 94(3), 249–252 (1973)CrossRefGoogle Scholar
  21. 21.
    Paynter, H.M.: Analysis and Design of Engineering Systems: Class Notes for M.I.T. Course 2.751. M.I.T. Press, Cambridge (1961)Google Scholar
  22. 22.
    Paytner, H.M.: The gestation and birth of bond graphs. http://www.me.utexas.edu/~longoria/paynter/hmp/Bondgraphs.html (2000)
  23. 23.
    Pei, J.S., Wright, J.P., Gay-Balmaz, F., Todd, M.D.: Choices in state variables for using state-event location to model piecewise restoring force, to be submitted for review for publication in Nonlinear Dynamics (2017)Google Scholar
  24. 24.
    Pei, J.S., Wright, J.P., Todd, M.D., Masri, S.F., Gay-Balmaz, F.: Understanding memristors and memcapacitors for engineering mechanical applications. Nonlinear Dyn. 80(1), 457–489 (2015)CrossRefGoogle Scholar
  25. 25.
    Scruggs, J.T., Gavin, H.P.: The Control Handbook. 2nd edn. (three volume set), 3526 pp. CRC Press, Boca Raton (2010)Google Scholar
  26. 26.
    Shampine, L.F., Gladwell, I., Brankin, R.W.: Reliable solution of special event location problems for odes. ACM Trans. Math. Softw. 17(1), 11–25 (1991)Google Scholar
  27. 27.
    Sivaselvan, M.V., Reinhorn, A.M.: Hysteretic models for deteriorating inelastic structures. ASCE J. Eng. Mech. 126(6), 633–640 (2000)CrossRefGoogle Scholar
  28. 28.
    Szabó, Z.: Preisach functions leading to closed form permeability. Physica B 372, 61–67 (2006)CrossRefGoogle Scholar
  29. 29.
    Szabó, Z., Tugyi, I., Kádár, G., Füzi, J.: Identification procedures for scalar preisach model. Physica B 343, 142–147 (2004)CrossRefGoogle Scholar
  30. 30.
    The Mathworks Inc: Matlab. http://www.mathworks.com/ (2014)
  31. 31.
    Wen, Y.K.: Equivalent linearization for hysteretic systems under random excitation. ASME J. Appl. Mech. 47, 150–154 (1980)CrossRefMATHGoogle Scholar
  32. 32.
    Wright, J.P., Pei, J.S.: Solving dynamical systems involving piecewise restoring force using state event location. ASCE J. Eng. Mech. 138(8), 997–1020 (2012)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  • Jin-Song Pei
    • 1
  • François Gay-Balmaz
    • 2
  • Joseph P. Wright
    • 3
  • Michael D. Todd
    • 4
  • Sami F. Masri
    • 5
  1. 1.School of Civil Engineering and Environmental ScienceUniversity of OklahomaNormanUSA
  2. 2.CNRS LMDEcole Normale Supérieure de ParisParisFrance
  3. 3.Division of Applied ScienceWeidlinger Associates Inc.New YorkUSA
  4. 4.Department of Structural EngineeringUniversity of California, San DiegoLa JollaUSA
  5. 5.Sonny Astani Department of Civil and Environmental EngineeringUniversity of Southern CaliforniaLos AngelesUSA

Personalised recommendations