Abstract
This work analyzes the stochastic response of a multiphysics system with stick-slip oscillations. The system is composed of two interacting subsystems, a mechanical with Coulomb friction and an electromagnetic (a DC motor). An imposed source voltage in the DC motor stochastically excites the system. This excitation, combined with the dry-friction, induces in the mechanical subsystem stochastic stick-slip oscillations. The resulting motion of the mechanical subsystem can be characterized by a random sequence of two qualitatively different and alternate modes, the stick- and slip-modes, with a non-smooth transition between them. The duration of each stick-mode is uncertain and depends on electromagnetic and mechanical parameters and variables, specially the position of the mechanical subsystem during the stick-mode. Duration and position are dependent random variables and must be jointly analyzed. The objective of this paper is to characterize and quantify this stochastic dependence, a novelty in the literature. The high amount of data required to perform the analysis and to construct joint histograms puts the problem into the class of big data problems.
Similar content being viewed by others
References
L. Anh, Dynamics of mechanical systems with Coulomb friction, vol. 1 (Springer, Berlin, 2002)
Q. Cao, A. Léger, A Smooth and Discontinuous Oscillator: Theory, Methodology and Applications, in Springer Tracts in Mechanical Engineering, vol. 1. (Springer, Berlin, 2017)
P. Glendinning, M. Jeffrey, An introduction to piecewise smooth dynamics, vol. 1 (Birkhäuser, Switzerland, 2019)
M. Jeffrey, Hidden dynamics: the mathematics of switches, decisions and other discontinuous behaviour, vol. 1 (Springer, Switzerland, 2018)
R. Lima, R. Sampaio, Stick-mode duration of a dry-friction oscillator with an uncertain model. J. Sound Vib. 353, 259–271 (2015)
J. Awrejcewicz, P. Olejnik, Stick-slip dynamics of a two-degree-of-freedom system. Int. J. Bifurc. Chaos 13(4), 843–861 (2003)
U. Galvanetto, S. Bishop, Dynamics of a simple damped oscillator under going stick-slip vibrations. Mechanica 34, 337–347 (1999)
R. Leine, D. Van Campen, A. Kraker, L. Van den Steen, Stick-slip vibrations induced by alternate friction models. Nonlinear Dyn. 16(1), 45–54 (2019)
A. Luo, B. Gegg, Stick and non-stick periodic motions in periodically forced oscillators with dry friction. J. Sound Vib. 291(1–2), 132–168 (2006)
P. Olejnik, J. Awrejcewicz, Application of Hénon method in numerical estimation of the stick-slip transitions existing in Filippov-type discontinuous dynamical systems with dry friction. Nonlinear Dyn. 73(1), 723–736 (2013)
M. Jeffrey, Modeling with nonsmooth dynamics, in Frontiers in Applied Dynamical Systems: Reviews and Tutorials, vol. 7. (Springer, Switzerland, 2020)
M. Bengisu, A. Akay, Stick-slip oscillations: dynamics of friction and surface roughness. J. Acoust. Soc. Am. 105(1), 194–205 (1999)
Y. Braiman, F. Family, H. Hentschel, Nonlinear friction in the periodic stick-slip motion of coupled oscillators. Phys. Rev. B 55(8), 5491–5504 (1997)
M. Jeffrey, The ghosts of departed quantities in switches and transitions. SIAM Rev. 60(1), 116–136 (2017)
A. Léger, E. Pratt, Qualitative Analysis of Nonsmooth Dynamics: A Simple Discrete System with Unilateral Contact and Coulomb Friction (Elsevier, ISTE Press, Great Britain, 2016)
E. Berger, Friction modeling for dynamic system simulation. Appl. Mech. Rev. 55(6), 535–577 (2002)
A. Fidlin, Nonlinear Oscillations in Mechanical Engineering (Springer, EUA, 2006)
B. Vande Vandre, D. Van Campen, A. De Kraker, An approximate analysis of dry-friction-induced stick-slip vibrations by a smoothing procedure. Nonlinear Dyn. 19, 157–169 (1999)
J. Awrejcewicz, M. Fečkan, P. Olejnik, On continuous approximation of discontinuous systems. Nonlinear Anal. 62(7), 1317–1331 (2005)
J. Awrejcewicz, P. Olejnik, Friction pair modeling by a 2-DOF system: numerical and experimental investigations. Int. J. Bifurc. Chaos 15(6), 1931–1944 (2005)
J. Awrejcewicz, L. Dzyubak, C. Grebogi, Estimation of chaotic and regular (stick-slip and slip-slip) oscillations exhibited by coupled oscillators with dry friction. Nonlinear Dyn. 42(2), 383–394 (2005)
J. Awrejcewicz, Y. Pyryev, Chaos prediction in the duffing type system with friction using Melnikov’s functions. Nonlinear Anal.: Real World Appl. 7(1), 12–24 (2006)
J. Awrejcewicz, Y. Pyryev, Occurrence of stick-slip phenomenon. J. Theor. Appl. Mech. 45(1), 33–40 (2007)
A. Barakat, R. Lima, R. Sampaio, P. Hagedorn, Bimodal parametric excitation of a micro-ring gyroscope. Proc. Appl. Math. Mech. 20(1), e202000153 (2021)
J.-F. Deü, W. Larbi, R. Ohayon, R. Sampaio, Piezoelectric shunt vibration damping of structural-acoustic systems: finite element formulation and reduced-order model. J. Vib. Acoust. 136(3), 031007 (2014)
W. Larbi, Numerical modeling of sound and vibration reduction using viscoelastic materials and shunted piezoelectric patches. Comput. Struct. 232, 105822 (2020)
R. Lima, C. Soize, R. Sampaio, Robust design optimization with an uncertain model of a nonlinear vibro-impact electro-mechanical system. Commun. Nonlinear Sci. Numer. Simul. 23, 263–273 (2015)
R. Lima, R. Sampaio, Two parametric excited nonlinear systems due to electromechanical coupling. J. Braz. Soc. Mech. Sci. Eng. 38, 931–943 (2016)
R. Lima, R. Sampaio, P. Hagedorn, J.-F. Deü, Comments on the paper ’On nonlinear dynamics behavior of an electro-mechanical pendulum excited by a nonideal motor and a chaos control taking into account parametric errors’ published in this Journal. J. Braz. Soc. Mech. Sci. Eng. 41, 552 (2019)
W. Manhães, R. Sampaio, R. Lima, P. Hagedorn, Two coupling mechanisms compared by their Lagrangians, in: Proceeding of the XVIII International Symposium on Dynamic Problems of Mechanics (DINAME 2019), Búzios, Brazil (2019), pp. 1–4
W. Manhães, R. Sampaio, R. Lima, P. Hagedorn, J.-F. Deü, Lagrangians for electromechanical systems. Mec. Comput. XXXVI 42, 1911–1934 (2018)
R. Lima, R. Sampaio, Electromechanical system with a stochastic friction field. Mec. Comput. XXXVII 18, 667–677 (2019)
A. Wijata, K. Polczyński, J. Awrejcewicz, Theoretical and numerical analysis of regular one-side oscillations in a single pendulum system driven by a magnetic field. Mech. Syst. Signal Process. 150, 107229 (2021)
K. Polczyński, S. Skurativskyi, M. Bednarek, J. Awrejcewicz, Nonlinear oscillations of coupled pendulums subjected to an external magnetic stimulus. Mech. Syst. Signal Process. 154, 107560 (2021)
R. Lima, R. Sampaio, Stick-slip oscillations in a multiphysics system. Nonlinear Dyn. 100, 2215–2224 (2020)
R. Lima, R. Sampaio, Stick-slip oscillations in a stochastic multiphysics system, in: Proceedings of the 5th International Symposium on Uncertainty Quantification and Stochastic Modeling (Uncertainties 2020), Rouen, France, pp. 3–17 (2020)
R. Lima, R. Sampaio, Construction of a statistical model for the dynamics of a base-driven stick-slip oscillator. Mech. Syst. Signal Process. 91, 157–166 (2017)
R. Lima, R. Sampaio, Parametric analysis of the statistical model of the stick-slip process. J. Sound Vib. 397, 141–151 (2017)
T. Hlalele, S. Du, Analysis of power transmission line uncertainties: status review. J. Electr. Electr. Syst. 5(3), 1–5 (2016)
G. Sivanagaraju, S. Chakrabarti, S. Srivastava, Uncertainty in transmission line parameters: estimation and impact on line current differential protection. IEEE Trans. Instrum. Meas. 63(6), 1496–1504 (2014)
R. Lima, R. Sampaio, What is uncertainty quantification? J. Braz. Soc. Mech. Sci. Eng. 40, 155 (2018)
R. Lima, R. Sampaio, Uncertainty quantification and cumulative distribution function: how are they related?, Bayesian Inference and Maximum Entropy Methods in Science and Engineering.Springer Proceedings in Mathematics & Statistics (2017), (253–260)
M. Márquez, I. Boussaada, H. Mounier, S.-I. Niculescu, Analysis and control of oilwell drilling vibrations: a time-delay systems approach, in Advances in Industrial Control (Springer, Switzerland, 2015)
D. Sivia, J. Skilling, Data Analysis: A Bayesian Tutorial, 2nd edn. (Oxford University Press, New York, 2006)
M. Cartmell, Introduction to Linear, Parametric and Nonlinear Vibrations, vol. 260 (Springer, 1990)
M. Dantas, R. Sampaio, R. Lima, Asymptotically stable periodic orbits of a coupled electromechanical system. Nonlinear Dyn. 78, 29–35 (2014)
M. Dantas, R. Sampaio, R. Lima, Existence and asymptotic stability of periodic orbits for a class of electromechanical systems: a perturbation theory approach. Z. Angew. Math. Phys. 67, 2 (2016)
M. Dantas, R. Sampaio, R. Lima, Phase bifurcations in an electromechanical system. IUTAM Procedia 1(19), 193–200 (2016)
E. Souza de Cursi, R. Sampaio, Uncertainty Quantification and Stochastic Modeling with Matlab (Elsevier, ISTE Press, 2015)
R. Lima, R. Sampaio, Modelagem Estocástica e Geração de Amostras de Variáveis e Vetores Aleatórios, Vol. 70 of Notas de Matemática Aplicada, SBMAC (2012), http://www.sbmac.org.br/arquivos/notas/livro_70.pdf
Acknowledgements
The authors acknowledge the support given by FAPERJ, CNPq and CAPES.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lima, R., Sampaio, R. Random stick-slip oscillations in a multiphysics system. Eur. Phys. J. Plus 136, 879 (2021). https://doi.org/10.1140/epjp/s13360-021-01860-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjp/s13360-021-01860-8