Subcritical bifurcation in a self-excited single-degree-of-freedom system with velocity weakening–strengthening friction law: analytical results and comparison with experiments

Nonlinear Dynamics, Sep 2017

The dynamical behavior of a single-degree-of-freedom system that experiences friction-induced vibrations is studied with particular interest on the possibility of the so-called hard effect of a subcritical Hopf bifurcation, using a velocity weakening–strengthening friction law. The bifurcation diagram of the system is numerically evaluated using as bifurcation parameter the velocity of the belt. Analytical results are provided using standard linear stability analysis and nonlinear stability analysis to large perturbations. The former permits to identify the lowest belt velocity \(({v_\mathrm{lw}})\) at which the full sliding solution is stable, the latter allows to estimate a priori the highest belt velocity at which large amplitude stick–slip vibrations exist. Together the two boundaries \([v_\mathrm{lw}, v_\mathrm{up}] \) define the range where two equilibrium solutions coexist, i.e., a stable full sliding solution and a stable stick–slip limit cycle. The model is used to fit recent experimental observations.

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Subcritical bifurcation in a self-excited single-degree-of-freedom system with velocity weakening–strengthening friction law: analytical results and comparison with experiments

Nonlinear Dyn Subcritical bifurcation in a self-excited single-degree-of-freedom system with velocity weakening-strengthening friction law: analytical results and comparison with experiments A. Papangelo 0 1 2 M. Ciavarella 0 1 2 N. Hoffmann 0 1 2 0 N. Hoffmann Imperial College London , Exhibition Road, London SW7 2AZ , UK 1 M. Ciavarella Department of Mechanical Engineering, Center of Excellence in Computational Mechanics, Politecnico di BARI , Viale Gentile 182, 70126 Bari , Italy 2 A. Papangelo ( The dynamical behavior of a single-degreeof-freedom system that experiences friction-induced vibrations is studied with particular interest on the possibility of the so-called hard effect of a subcritical Hopf bifurcation, using a velocity weakening-strengthening friction law. The bifurcation diagram of the system is numerically evaluated using as bifurcation parameter the velocity of the belt. Analytical results are provided using standard linear stability analysis and nonlinear stability analysis to large perturbations. The former permits to identify the lowest belt velocity (vlw) at which the full sliding solution is stable, the latter allows to estimate a priori the highest belt velocity at which large amplitude stick-slip vibrations exist. Together the two boundaries [vlw, vup] define the range where two equilibrium solutions coexist, i.e., a stable full sliding solution and a stable stick-slip limit cycle. The model is used to fit recent experimental observations. Mass-on-moving-belt model; Exponential decaying; Weakening-strengthening friction law; Bistable equilibrium; Subcritical bifurcation 1 Introduction Subcritical as well as supercritical Hopf bifurcations are often encountered in different engineering applications, e.g., aeroelastic response of airfoils with structural nonlinearities [ 1, 2 ], dynamics of ball joints [3], brake squeal [ 4 ]. Engineers are generally more concerned about subcritical (hard) bifurcations as a small perturbation around the equilibrium position can lead the system to large amplitude vibration states, which the structure may not tolerate [ 5 ]. A number of authors have studied the “Mass-on-moving-Belt” model (“MB model” in the following), Tondl [ 6 ], Hetzler et al. [ 7 ], Hetzler [ 8 ], Won and Chung [ 9 ], Nayfeh and Mook [ 10 ], Mitropolskii and Van Dao [ 11 ], Popp [ 12 ], Popp et al. [ 13 ], Hinrichs et al. [ 14 ], Andreaus and Casini [ 15 ], Awrejcewicz and Holicke [ 16 ], Awrejcewicz et al. [ 17 ], which present various types of analysis of a mass-on-belt system with various kinds of friction laws, and provide in some cases, analytical expressions for the change between stick–slip and pure-slip oscillations. Many authors have attempted to use fast vibrations which in some respects seems to transform classical Coulomb friction into viscous-like damping ([ 5, 18 ]). Most often, supercritical bifurcations are found, namely where the system undergoes a smooth transition to a limit cycle (generally involving stick– slip) when the control parameter is varied. In [ 19 ] Hoffmann studied the effect of LuGre type friction law [ 20 ] on the stability of the classical MB model. It was shown that rate-dependent effects act against the destabilizing effect of the velocity decaying friction characteristic. The reader is referred to the review by Awrejcewicz and Olejnik [ 21 ] where the dynamical behavior of different lumped mechanical systems (see also [ 22 ]) with various friction laws has been investigated. Hetzler et al. [ 7 ] (see also [ 23 ]) studied the dynamic behavior of the MB model using different friction characteristics, (exponential and polynomial decaying). They assumed a weakly nonlinear behavior and used a first-order averaging method to find approximate solutions. It was shown that the exponential decaying leads to subcritical Hopf bifurcation while, using a cubic polynomial friction law, the dynamical behavior (subcritical/supercritical) depends on the friction law parameters [ 7 ]. Also in [ 8 ] Hetzler showed that adding a Coulomb frictional damping to the self-excited MB model leads to an “imperfect” Hopf bifurcation scenario where it does not make sense to ask for stability of the steady state but rather one should seek for stability to a certain level of perturbation. Recently, Papangelo et al. [ 24 ] have found localized vibration states in a self-excited chain of mechanical oscillators weakly elastically coupled, which lead to the so-called snaking bifurcations in the bifurcation diagram. A key feature of the system was that, if isolated from the structure, each nonlinear oscillator experiences a subcritical Hopf bifurcation in a certain range of the control parameter (yielding bistability1). However, Papangelo et al. [ 24 ], adopted a polynomial nonlinearity quite remote from a real friction law. Here, perhaps with an eye to the classical Stribeck curve, for the MB model we propose an exponen (...truncated)


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A. Papangelo, M. Ciavarella, N. Hoffmann. Subcritical bifurcation in a self-excited single-degree-of-freedom system with velocity weakening–strengthening friction law: analytical results and comparison with experiments, Nonlinear Dynamics, 2017, pp. 2037-2046, Volume 90, Issue 3, DOI: 10.1007/s11071-017-3779-4