Response of a class of mechanical oscillators described by a novel system of differential-algebraic equations. (English) Zbl 1389.34045

The paper studies vibration of one-degree-of-freedom lumped parameter systems whose constituents are described through novel implicit constitutive relations. In the classical approach constitutive expressions are provided for the force in terms of appropriate kinematical variables (displacement and velocity) which, when substituted into the balance of linear momentum, lead to a single governing ordinary differential equation for the system as a whole. However, many important physical problems cannot specify the forces explicitly in terms of the kinematic variables, but the kinematic parameters are functions of the force. There is an implicit relation between the forces acting on the system and appropriate kinematical variables. In that case the mathematical model of the vibration system is a differential-algebraic system which has to be solved simultaneously. Standard techniques such Poincaré sections, bifurcation diagrams and Lyapunov exponents are used for study. Four examples are investigated: oscillator with a spring with non-standard cubic constitutive relation, inverse Duffing oscillator, oscillator with rigid-elastic spring with linear dashpot and Bingham fluid which does not flow unless a certain threshold in the force is reached in the dashpot. The obtained results give interesting new physical insight in the problems.


34A09 Implicit ordinary differential equations, differential-algebraic equations
34C28 Complex behavior and chaotic systems of ordinary differential equations
70H45 Constrained dynamics, Dirac’s theory of constraints
34C23 Bifurcation theory for ordinary differential equations
34D08 Characteristic and Lyapunov exponents of ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations


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[1] J.-P. Aubin, A. Cellina: Differential Inclusions. Set-Valued Maps and Viability Theory. Grundlehren der Mathematischen Wissenschaften 264, Springer, Berlin, 1984. · Zbl 0538.34007 · doi:10.1007/978-3-642-69512-4
[2] Buliček, M.; Gwiazda, P.; Malek, J.; Rajagopal, K. R.; Świerczewska-Gwiazda, A.; Robinson, J. C. (ed.); etal., On flows of fluids described by an implicit constitutive equation characterized by a maximal monotone graph. Mathematical Aspects of Fluid Mechanics, No. 402, 23-51 (2012), Cambridge · Zbl 1296.35137
[3] M. Buliček, P. Gwiazda, J. Malek, A. Świerczewska-Gwiazda: On unsteady flows of implicitly constituted incompressible fluids. SIAM J. Math. Anal. 44 (2012), 2756-2801. · Zbl 1256.35074 · doi:10.1137/110830289
[4] M. Buliček, J. Malek, K. R. Rajagopal: On Kelvin-Voigt model and its generalizations. Evol. Equ. Control Theory (electronic only) 1 (2012), 17-42. · Zbl 1371.74067 · doi:10.3934/eect.2012.1.17
[5] J.-F. Colombeau: New Generalized Functions and Multiplication of Distributions. North-Holland Mathematics Studies 84, North-Holland Publishing, Amsterdam, 1984. · Zbl 0532.46019
[6] J.-F. Colombeau: Multiplication of Distributions. A Tool in Mathematics, Numerical Engineering and Theoretical Physics. Lecture Notes in Mathematics 1532, Springer, Berlin, 1992. · Zbl 0815.35002
[7] K. Deimling: Multivalued Differential Equations. De Gruyter Series in Nonlinear Analysis and Applications 1, Walter de Gruyter, Berlin, 1992. · Zbl 0760.34002 · doi:10.1515/9783110874228
[8] J.-P. Eckmann, S. Oliffson Kamphorst, D. Ruelle: Recurrence plots of dynamical systems. Europhys. Lett. 4 (1987), 973-977.
[9] A. F. Filippov: Classical solutions of differential equations with multi-valued right-hand side. SIAM J. Control 5 (1967), 609-621. · Zbl 0238.34010 · doi:10.1137/0305040
[10] A. F. Filippov: Differential Equations with Discontinuous Righthand Sides (F.M. Arscott, ed.). Mathematics and Its Applications: Soviet Series 18, Kluwer Academic Publishers, Dordrecht, 1988. · Zbl 0664.34001 · doi:10.1007/978-94-015-7793-9
[11] D. T. Kaplan, L. Glass: Direct test for determinism in a time series. Phys. Rev. Lett. 68 (1992), 427-430. · doi:10.1103/PhysRevLett.68.427
[12] O. Kopaček, V. Karas, J. Kovař, Z. Stuchlik: Transition from regular to chaotic circulation in magnetized coronae near compact objects. Astrophys. J. 722 (2010), 1240-1259. · doi:10.1088/0004-637X/722/2/1240
[13] N. Marwan, M. C. Romano, M. Thiel, J. Kurths: Recurrence plots for the analysis of complex systems. Phys. Rep. 438 (2007), 237-329. · doi:10.1016/j.physrep.2006.11.001
[14] L. Meirovitch: Elements of Vibration Analysis. McGraw-Hill Book Company, Düsseldorf, 1975. · Zbl 0359.70039
[15] E. Ott: Chaos in Dynamical Systems. Cambridge University Press, Cambridge, 2002. · Zbl 1006.37001 · doi:10.1017/CBO9780511803260
[16] D. PraŽak: Remarks on the uniqueness of second order ODEs. Appl. Math., Praha 56 (2011), 161-172. · Zbl 1224.34008
[17] D. PraŽak, K.R. Rajagopal: Mechanical oscillators described by a system of differential-algebraic equations. Appl. Math., Praha 57 (2012), 129-142. · Zbl 1249.34017
[18] K. R. Rajagopal: A generalized framework for studying the vibrations of lumped parameter systems. Mech. Res. Commun. 37 (2010), 463-466. · Zbl 1272.74297 · doi:10.1016/j.mechrescom.2010.05.010
[19] E. E. Rosinger: Generalized Solutions of Nonlinear Partial Differential Equations. North-Holland Mathematics Studies 146, North-Holland Publishing, Amsterdam, 1987. · Zbl 0635.46033
[20] E. E. Rosinger: Nonlinear Partial Differential Equations. An Algebraic View of Generalized Solutions. North-Holland Mathematics Studies 164, North-Holland, Amsterdam, 1990. · Zbl 0717.35001
[21] O. Semerak, P. Sukova: Free motion around black holes with discs or rings: between integrability and chaos-II. Mon. Not. R. Astron. Soc. 425 (2012), 2455-2476. · doi:10.1111/j.1365-2966.2012.21630.x
[22] D. E. Stewart: Rigid-body dynamics with friction and impact. SIAM Rev. 42 (2000), 3-39. · Zbl 0962.70010 · doi:10.1137/S0036144599360110
[23] P. Sukova: Chaotic geodesic motion around a black hole and disc. J. Phys. Conf. Ser. 314 (2011), Article ID 012087. · Zbl 1371.74067
[24] P. Sukova, O. Semerak: Recurrence of geodesics in a black-hole-disc field. AIP Conference Proceedings 1458 (2012), 523-526. · doi:10.1063/1.4734475
[25] P. Sukova, O. Semerak: Free motion around black holes with discs or rings: between integrability and chaos-III. Mon. Not. R. Astron. Soc. 436 (2013), 978-996. · doi:10.1093/mnras/stt1587
[26] Y. Ueda: Randomly transitional phenomena in the system governed by Duffing’s equation. J. Stat. Phys. 20 (1979), 181-196. · doi:10.1007/BF01011512
[27] K. Zaki, S. Noah, K. R. Rajagopal, A. R. Srinivasa: Effect of nonlinear stiffness on the motion of a flexible pendulum. Nonlinear Dyn. 27 (2002), 1-18. · Zbl 1050.70014 · doi:10.1023/A:1017512317443
[28] http://www.recurrence-plot.tk/
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