Öz
In the view of memory effect of hysteresis, this work aims to interpret hysteresis nonlinearities in terms of Riemann-Liouville fractional derivative which is a singular operator with memory and hereditary properties. For this purpose, Duhem hysteresis, a model defined by a first order differential equation, is considered and adapted to a fractional order differential equation. Since the fractional order Duhem hysteresis cannot be solved by an analytical scheme, Grünwald-Letnikov approximation is used to obtain numerical solutions. Thus, the effect of fractional order derivative to Duhem hysteresis is demonstrated with graphics obtained by this approximation and plotting using MATLAB. As a result, it is observed that the fractional order model exhibits hysteresis behavior for the orders that are smaller than 1.
Anahtar Kelimeler
Duhem hysteresis Riemann-Liouville fractional derivative fractional order differential equations Grünwald-Letnikov approximation
Kaynakça
- Krasnosel'skii, M.A., and Pokrovskii, A.V., Systems with hysteresis, Springer Verlag, (1989). Mayergoyz, I.D., Mathematical models of hysteresis, Springer Verlag, Berlin, (1991). Macki, J.W., Nistri, P., and Zecca, P., Mathematical models for hysteresis, Siam Review, 35, 1, 9-123, (1993). Visintin, A., Differential models of hysteresis, Springer, Berlin. (1994). Duhem, P., Die dauernden aenderungen und die thermodynamik, Zeitschrift fur Physikalische Chemie, 22, 543-589, (1897). Bagley, R.L., and Torvik, P.J., On the fractional calculus model of viscoelastic behavior, Journal of Rheology, 30, 1, 133—155, (1986). Padovan, J., and Sawicki, J. T., Diophantine type fractional derivative representation of structural hysteresis, Computational Mechanics, 19, 335-340, (1997). Machado, J.A., Analysis and design of fractional order digital control systems, Systems Analysis Modelling Simulation, 27, 107-122, 1997. Darwish, M.A., and El-Bary, A.A., Existence of fractional integral equation with hysteresis, Applied Mathematics and Computation, 176, 684-687, (2006). Schafer, I., and Kruger, K., Modeling of coils using fractional derivatives, Journal of Magnetism and Magnetic Materials, 307, 91-98, (2006). Duarte, F. and Machado, J.A., Fractional dynamics in the describing function analysis of nonlinear friction, Proceedings of the 2nd IFAC Workshop on Fractional Differentiation and Its Applications, Porto, Portugal, 39, 11, 218-223, (2006). Deng, W. and Lü, J., Generating multi-directional multi-scroll chaotic attractors via a Fractional Differential Hysteresis system, Physics Letters A, 369, 438-443, (2007). Duarte, F. and Machado, J.A., Describing function of two masses with backlash, Nonlinear Dynamics, 56, 409-413, (2009). Özdemir N. and İskender, B.B., Fractional order control of fractional diffusion systems subject to input hysteresis, Journal of Computational and Nonlinear Dynamics, ASME, 5, 2, 021002 (6 pages), (2010). İskender, B. B., Özdemir, N. and N., Karaoglan, A.D., Parameter optimization of fractional order PI^λ D^μ controller using response surface methodology, Discontinuity and Complexity in Nonlinear Physical Systems, Series: Nonlinear Systems and Complexity, Vol. 6, Machado, J. A. T, Baleanu, D., Luo, A.C.J. (Eds.), Chapter 5, ISBN 978-3-319-01411-1, (2014). Spanos, P.D., Di Matteo, A. and Pirotta, A., Steady-state dynamic response of various hysteretic systems endowed with fractional derivative elements, Nonlinear Dynamics98,4,3113-3124, (2019). Guyomar, D., Ducharne, B. and Sebald, G., Dynamical hysteresis model of ferroelectric ceramics under electric field using fractional derivatives, Journal of Physics D: Applied Physics, 40, 6048-6054, (2007). Zhu, Z. and Zhou, X., A novel fractional order model for dynamic hysteresis of piezoeletrically actuated fast tool servo, Materials, 5, 2465-2485. (2012). Zhu, Z., To, S., Li, Y., Zhu, W-L. and Bian, L., External force estimation of a piezo-actuated compliant mechanism based on a fractional order hysteresis model, Mechanical Systems and Signal Processing, 110, 296-306, (2018). Ding, C., Cao, J. and Chen, Y.Q., Fractional-order model and experimental verification for broadband hysteresis in piezoelectric actuators, Nonlinear Dynamics, 98,3143-3153, (2019). Caputo, M., and Carcione, J.M., Hysteresis cycles and fatigue criteria using anelastic models based on fractional derivatives, Rheologica Acta, 50, 107-115, (2011). Caputo, M. and Fabrizio, M., On the notion of fractional derivatvie and applications to the hysteresis phenomena, Meccanica, 52, 3043-3052, (2017). Naser, M.F.M. and Ikhouane, F., Consistency of the Duhem model with hysteresis, Mathematical Problems in Engineering, 2013, 586130, (16 pages), (2013). Colemann, B.D., and Hodgdon, M.L., A constitutive relation for rate-independent hysteresis in ferromagnetically soft materials, International Journal of Engineering Science, 24, 897-919, (1986). Colemann, B.D. and Hodgdon, M.L., On a class of constitutive relations for ferromagnetic hysteresis, Archive for Rational Mechanics and Analysis, 99, 375-396, (1987). Oldham, K. B. and Spanier, J., The fractional calculus, Academic Press, New York, (1974). Miller, K.S. and Ross B., An introduction to the fractional calculus and fractional differential equations, Wiley, New York. (1993). Podlubny I., Fractional differential equations, Academic Press, San Diego, (1999). Hodgdon, M.L., Applications of a theory of ferromagnetic hysteresis, IEEE Transactions on Magnetics, 24, 218-221, (1988). Su, C.Y., Stepanenko, Y., Svoboda, J. and Leung, T.P., Robust and adaptive control of a class of nonlinear systems with unknown backlash-like hysteresis, IEEE Transactions on Automatic Control, 45, 2427-2432, (2000).
Öz
Bu çalışma, histerisisin hafıza etkisini göz önüne alarak, doğrusal olmayan histeresis davranışının hafıza ve kalıtım özelliğine sahip tekil olmayan Riemann-Liouville kesirli türevi açısından yorumlamayı amaçlamaktadır. Bunun için, birinci mertebeden diferansiyel denklem ile tanımlanan bir model olan Duhem histeresis göz önüne alınmış ve kesirli mertebeden bir diferansiyel denkleme uyarlanmıştır. Kesirli mertebeden Duhem histeresis analitik bir yöntem ile çözülemeyeceğinden, nümerik çözümleri elde etmek için Grünwald-Letnikov yaklaşımı kullanılmıştır. Böylece, kesirli mertebeden türevin modele etkisi bu yaklaşım göre elde edilen ve MATLAB kullanılarak çizdirilen grafikler ile gösterilmiştir. Sonuç olarak, kesirli mertebeden modelin 1 den küçük mertebeler için histerisis etkisi gösterdiği gözlemlenmiştir.
Anahtar Kelimeler
Duhem histerisis Riemann-Liouville kesirli türevi kesirli mertebeden diferansiyel denklemler Grünwald-Letnikov yaklaşımı
Kaynakça
- Krasnosel'skii, M.A., and Pokrovskii, A.V., Systems with hysteresis, Springer Verlag, (1989). Mayergoyz, I.D., Mathematical models of hysteresis, Springer Verlag, Berlin, (1991). Macki, J.W., Nistri, P., and Zecca, P., Mathematical models for hysteresis, Siam Review, 35, 1, 9-123, (1993). Visintin, A., Differential models of hysteresis, Springer, Berlin. (1994). Duhem, P., Die dauernden aenderungen und die thermodynamik, Zeitschrift fur Physikalische Chemie, 22, 543-589, (1897). Bagley, R.L., and Torvik, P.J., On the fractional calculus model of viscoelastic behavior, Journal of Rheology, 30, 1, 133—155, (1986). Padovan, J., and Sawicki, J. T., Diophantine type fractional derivative representation of structural hysteresis, Computational Mechanics, 19, 335-340, (1997). Machado, J.A., Analysis and design of fractional order digital control systems, Systems Analysis Modelling Simulation, 27, 107-122, 1997. Darwish, M.A., and El-Bary, A.A., Existence of fractional integral equation with hysteresis, Applied Mathematics and Computation, 176, 684-687, (2006). Schafer, I., and Kruger, K., Modeling of coils using fractional derivatives, Journal of Magnetism and Magnetic Materials, 307, 91-98, (2006). Duarte, F. and Machado, J.A., Fractional dynamics in the describing function analysis of nonlinear friction, Proceedings of the 2nd IFAC Workshop on Fractional Differentiation and Its Applications, Porto, Portugal, 39, 11, 218-223, (2006). Deng, W. and Lü, J., Generating multi-directional multi-scroll chaotic attractors via a Fractional Differential Hysteresis system, Physics Letters A, 369, 438-443, (2007). Duarte, F. and Machado, J.A., Describing function of two masses with backlash, Nonlinear Dynamics, 56, 409-413, (2009). Özdemir N. and İskender, B.B., Fractional order control of fractional diffusion systems subject to input hysteresis, Journal of Computational and Nonlinear Dynamics, ASME, 5, 2, 021002 (6 pages), (2010). İskender, B. B., Özdemir, N. and N., Karaoglan, A.D., Parameter optimization of fractional order PI^λ D^μ controller using response surface methodology, Discontinuity and Complexity in Nonlinear Physical Systems, Series: Nonlinear Systems and Complexity, Vol. 6, Machado, J. A. T, Baleanu, D., Luo, A.C.J. (Eds.), Chapter 5, ISBN 978-3-319-01411-1, (2014). Spanos, P.D., Di Matteo, A. and Pirotta, A., Steady-state dynamic response of various hysteretic systems endowed with fractional derivative elements, Nonlinear Dynamics98,4,3113-3124, (2019). Guyomar, D., Ducharne, B. and Sebald, G., Dynamical hysteresis model of ferroelectric ceramics under electric field using fractional derivatives, Journal of Physics D: Applied Physics, 40, 6048-6054, (2007). Zhu, Z. and Zhou, X., A novel fractional order model for dynamic hysteresis of piezoeletrically actuated fast tool servo, Materials, 5, 2465-2485. (2012). Zhu, Z., To, S., Li, Y., Zhu, W-L. and Bian, L., External force estimation of a piezo-actuated compliant mechanism based on a fractional order hysteresis model, Mechanical Systems and Signal Processing, 110, 296-306, (2018). Ding, C., Cao, J. and Chen, Y.Q., Fractional-order model and experimental verification for broadband hysteresis in piezoelectric actuators, Nonlinear Dynamics, 98,3143-3153, (2019). Caputo, M., and Carcione, J.M., Hysteresis cycles and fatigue criteria using anelastic models based on fractional derivatives, Rheologica Acta, 50, 107-115, (2011). Caputo, M. and Fabrizio, M., On the notion of fractional derivatvie and applications to the hysteresis phenomena, Meccanica, 52, 3043-3052, (2017). Naser, M.F.M. and Ikhouane, F., Consistency of the Duhem model with hysteresis, Mathematical Problems in Engineering, 2013, 586130, (16 pages), (2013). Colemann, B.D., and Hodgdon, M.L., A constitutive relation for rate-independent hysteresis in ferromagnetically soft materials, International Journal of Engineering Science, 24, 897-919, (1986). Colemann, B.D. and Hodgdon, M.L., On a class of constitutive relations for ferromagnetic hysteresis, Archive for Rational Mechanics and Analysis, 99, 375-396, (1987). Oldham, K. B. and Spanier, J., The fractional calculus, Academic Press, New York, (1974). Miller, K.S. and Ross B., An introduction to the fractional calculus and fractional differential equations, Wiley, New York. (1993). Podlubny I., Fractional differential equations, Academic Press, San Diego, (1999). Hodgdon, M.L., Applications of a theory of ferromagnetic hysteresis, IEEE Transactions on Magnetics, 24, 218-221, (1988). Su, C.Y., Stepanenko, Y., Svoboda, J. and Leung, T.P., Robust and adaptive control of a class of nonlinear systems with unknown backlash-like hysteresis, IEEE Transactions on Automatic Control, 45, 2427-2432, (2000).
Ayrıntılar
| Birincil Dil | İngilizce |
|---|---|
| Bölüm | Araştırma Makalesi |
| Yazarlar | |
| Yayımlanma Tarihi | 10 Ocak 2020 |
| Gönderilme Tarihi | 23 Ağustos 2019 |
| Yayımlandığı Sayı | Yıl 2020 Cilt: 22 Sayı: 1 |