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

- 168 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

- \(\mathbf {y}\)
*w*Internal state variable

*u*Driving force

*G**F*- \(\mathbf {g}\)
- \(\mathbf {f}\)

## 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.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.Bouc, R.: Modèle mathématique d’hystérésis. Acustica
**21**, 16–25 (1971)MATHGoogle Scholar - 3.Canfield, R.: central_diff.m. http://www.mathworks.com/matlabcentral/fileexchange/12-central-diff-m/content/central_diff.m (2015)
- 4.Caughey, T.K.: Random excitation of a system with bilinear hysteresis. J. Appl. Mech.
**27**, 649–652 (1960a)MathSciNetCrossRefGoogle Scholar - 5.Caughey, T.K.: Sinusoidal excitation of a system with bilinear hysteresis. J. Appl. Mech.
**27**, 640–643 (1960b)MathSciNetCrossRefGoogle Scholar - 6.Chua, L.O.: Memrister-the missing circuit element. IEEE Trans. Circuit Theory CT
**18**(5), 507–519 (1971)CrossRefGoogle Scholar - 7.Chua, L.O.: Resistance switching memories are memristors. Appl. Phys. A Mater. Sci. Process.
**102**, 765–783 (2011)CrossRefMATHGoogle Scholar - 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.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.Diani, J., Fayolle, B., Gilormini, P.: A review on the mullins effect. Eur. Polymer J.
**45**, 601–612 (2009)CrossRefGoogle Scholar - 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.Janičić, V., Ilić, V.: Preisach model gui version 1.0. http://www.mathworks.com/matlabcentral/fileexchange/47763-hysteresis-in-ferromagnetic-gui (2014)
- 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.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.Kádár, G., Szabó, Z., Vértesy, G.: Product preisach model parameters from measured magnetization curves. Physica B
**343**, 137–141 (2004)CrossRefGoogle Scholar - 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.Krasnosel’skiǐ, M.A., Pokrovskiǐ, A.V.: Systems with Hysteresis. Springer, Berlin (1983)MATHGoogle Scholar
- 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.Mayergoyz, I.: Mathematical Models of Hysteresis and Their Applications. Elsevier Series in Electromagnetism. Elsevier Science Ltd, Oxford (2003)Google Scholar
- 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.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.Paytner, H.M.: The gestation and birth of bond graphs. http://www.me.utexas.edu/~longoria/paynter/hmp/Bondgraphs.html (2000)
- 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.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.Scruggs, J.T., Gavin, H.P.: The Control Handbook. 2nd edn. (three volume set), 3526 pp. CRC Press, Boca Raton (2010)Google Scholar
- 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.Sivaselvan, M.V., Reinhorn, A.M.: Hysteretic models for deteriorating inelastic structures. ASCE J. Eng. Mech.
**126**(6), 633–640 (2000)CrossRefGoogle Scholar - 28.Szabó, Z.: Preisach functions leading to closed form permeability. Physica B
**372**, 61–67 (2006)CrossRefGoogle Scholar - 29.Szabó, Z., Tugyi, I., Kádár, G., Füzi, J.: Identification procedures for scalar preisach model. Physica B
**343**, 142–147 (2004)CrossRefGoogle Scholar - 30.The Mathworks Inc: Matlab. http://www.mathworks.com/ (2014)
- 31.Wen, Y.K.: Equivalent linearization for hysteretic systems under random excitation. ASME J. Appl. Mech.
**47**, 150–154 (1980)CrossRefMATHGoogle Scholar - 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