Abstract
This article deals with the question, to what extent damping due to nonsmooth Coulomb friction may affect the stability and bifurcation behavior of vibrational systems with self-excitation due to negative effective damping which—for the smooth case—is related to a Hopf bifurcation of the steady state.
Without damping due to Coulomb friction, the stability of the trivial solution is controlled by the effective viscous damping of the system: as the damping becomes negative, the steady state loses stability at a Hopf point. Adding Coulomb friction changes the trivial solution into a set of equilibria, which—for oscillatory systems—is asymptotically stable for all values of effective viscous damping. The Hopf point vanishes and an unstable limit cycle appears which borders the basin of attraction of the equilibrium set. Moreover, the influence of nonlinear damping terms is discussed.
The effect of Coulomb frictional damping may be seen as adding an imperfection to the classical smooth Hopf scenario: as the imperfection vanishes, the behavior of the smooth problem is recovered.
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References
Altintas, Y., Budak, E.: Analytical prediction of stability lobes in milling. CIRP Ann. 44(1), 357–362 (1995)
Beards, C.: Damping in structural joints. Shock Vib. Dig. 11, 35–41 (1979)
Bolotin, V., Herrmann, G.: Nonconservative Problems of the Theory of Elastic Stability, vol. 1991. Pergamon Press, Elmsford (1963)
Boyaci, A., Hetzler, H., Seemann, W., Proppe, C., Wauer, J.: Analytical bifurcation analysis of a rotor supported by floating ring bearings. Nonlinear Dyn. 57(4), 497–507 (2009)
Bronstein, I., Semendjajew, K., Musiol, G., Mühlig, H.: Taschenbuch der Mathematik. Harri Deutsch Verlag (2008)
Deimling, K.: Multivalued Differential Equations. de Gruyter, Berlin (1992)
Den Hartog, J.: Mechanical Vibrations. Dover, New York (1985). (Originally published: McGraw-Hill 1956)
D’Souza, A., Dweib, A.: Self-excited vibrations induced by dry friction, part 2: Stability and limit-cycle analysis. J. Sound Vib. 137(2), 177–190 (1990)
Dweib, A., D’Souza, A.: Self-excited vibrations induced by dry friction, Part 1: Experimental study. J. Sound Vib. 137(2), 163–175 (1990)
Filippov, A.: Differential Equations with Discontinuous Righthand Sides. Springer, Berlin (1988)
Fulcher, L., Scherer, R., Melnykov, A., Gateva, V., Limes, M.: Negative Coulomb damping, limit cycles, and self-oscillation of the vocal folds. Am. J. Phys. 74, 386 (2006)
Gaul, L., Lenz, J.: Nonlinear dynamics of structures assembled by bolted joints. Acta Mech. 125(1), 169–181 (1997)
Genta, G.: Dynamics of Rotating Systems. Springer, Berlin (2005)
Hagedorn, P.: Non-linear Oscillations. Clarendon, Oxford (1988)
Hahn, W.: Stability of Motion. Springer, Berlin (1967)
Hetzler, H.: On moving continua with contacts and sliding friction: Modeling, general properties and examples. Int. J. Solids Struct. 46(13), 2556–2570 (2009)
Hetzler, H., Schwarzer, D., Seemann, W.: Analytical investigation of steady-state stability and Hopf-bifurcations occurring in sliding friction oscillators with application to low-frequency disc brake noise. Commun. Nonlinear Sci. Numer. Simul. 12(1), 83–99 (2007)
Hetzler, H., Schwarzer, D., Seemann, W.: Steady-state stability and bifurcations of friction oscillators due to velocity-dependent friction characteristics. Proc. Inst. Mech. Eng., Proc., Part K, J. Multi-Body Dyn. 221(3), 401–412 (2007)
Hetzler, H., Seemann, W.: Friction induced flutter instability—on modeling and simulation of brake-squeal. Proc. Appl. Math. Mech. 8(1) (2008)
Ibrahim, R.: Friction-induced vibration, chatter, squeal and, chaos; Part I: Mechanics of contact and friction. Appl. Mech. Rev. 47(7), 209–226 (1994)
Insperger, T., Gradišek, J., Kalveram, M., Stépán, G., Winert, K., Govekar, E.: Machine tool chatter and surface location error in milling processes. J. Manuf. Sci. Eng. 128, 913 (2006)
Kauderer, H.: Nichtlineare Mechanik. Springer, Berlin (1958)
LaSalle, J., Lefschetz, S.: Stability by Lyapunov’s Direct Method (with Applications). Academic Press, New York (1961)
Leine, R., Nijmeijer, H.: Dynamics and Bifurcations of Non-smooth Mechanical Systems, vol. 18. Springer, Berlin (2004)
Leine, R., van de Wouw, N.: Stability and Convergence of Mechanical Systems with Unilateral Constraints. Springer, Berlin (2008)
Maidanik, G.: Energy dissipation associated with gas-pumping in structural joints. J. Acoust. Soc. Am. 40, 1064 (1966)
Muszynska, A.: Whirl and whip–rotor/bearing stability problems. J. Sound Vib. 110(3), 443–462 (1986)
Myers, C.: Bifurcation theory applied to oil whirl in plain cylindrical journal bearings. J. Appl. Mech. 51, 244 (1984)
Popp, K.: Nichtlineare Schwingungen mechanischer Strukturen mit Füge-oder Kontaktstellen. Z. Angew. Math. Mech. 74(3), 147–165 (1994)
Tondl, A.: Some Problems of Rotor Dynamics. Publishing House of the Czechoslovak Academy of Sciences (1965)
Tondl, A.: Quenching of self-excited vibrations: effect of dry friction. J. Sound Vib. 45(2), 285–294 (1976)
Yabuno, H., Kunitho, Y., Kashimura, T.: Analysis of the van der Pol system with Coulomb friction using the method of multiple scales. J. Vib. Acoust. 130, 041008 (2008)
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Hetzler, H. On the effect of nonsmooth Coulomb friction on Hopf bifurcations in a 1-DoF oscillator with self-excitation due to negative damping. Nonlinear Dyn 69, 601–614 (2012). https://doi.org/10.1007/s11071-011-0290-1
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DOI: https://doi.org/10.1007/s11071-011-0290-1