Stick-slip vibrations of a self-excited SD oscillator with Coulomb friction
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ORIGINAL PAPER
Stick-slip vibrations of a self-excited SD oscillator with Coulomb friction Zhixin Li · Qingjie Cao · Zairan Nie
Received: 24 April 2020 / Accepted: 8 October 2020 © Springer Nature B.V. 2020
Abstract In this paper, the stick-slip vibrations of an archetypal self-excited smooth and discontinuous (SD) oscillator are investigated. The mathematical model of the self-excited SD oscillator is established by employing Coulomb’s law to formulate the friction between the surfaces of the mass and the moving belt. Complex dynamical behaviors are demonstrated by equilibrium analysis including stability analysis and supercritical pitchfork bifurcations of the system. Closed-form solutions for both stick-slip motions and pure slip motions of the system can be derived and utilized to examine the influence of the belt speed on the steady-state of the system by using Hamilton function. The evolution of sliding regions and the collision of the trajectories with the sliding region are presented for the forced selfexcited system resorting to the numerical simulations. The results obtained here offer an opportunity for us to understand the conversion mechanism between the stick and the slip motions for the friction systems with geometric nonlinearity in engineering. Keywords Self-excited SD oscillator · Stick-slip vibration · Coulomb friction · Sliding region · Geometry nonlinearity Z. Li (B) · Z. Nie School of Mathematics and Physics, Hebei University of Engineering, Handan 056038, People’s Republic of China e-mail: [email protected] Q. Cao School of Astronautics, Harbin Institute of Technology, Harbin 150001, People’s Republic of China
1 Introduction Friction is a very complicated dynamics phenomenon and plays a crucial role in various engineering systems, such as mechanical engineering, civil engineering and seismology [1–3]. Moreover, it is a typically nonsmooth factor that may induce self-excited vibration in extensive engineering systems, such as brake systems [4], bearing systems [5], rotating drilling systems [6], rotor rubbing systems [7], and so on. A great number of comprehensive models have been established, such as Coulomb friction model, Dahl model, Sribeck friction, Karnopp model and LuGre model [8,9]. Among them, the Coulomb friction model is the simplest one and has been extensively adopted. Friction induced self-excited vibration, also known as stick-slip vibration, often cause some undesired effects observed in everyday life as well as engineering applications, including noise of a squeaking door, squeaky chalk on a blackboard, brake squeal, rattling joints of a robot, chattering machine tools and others. The self-excited vibrations between contact interfaces may lead to wear and damage of a machinery or a failure of a mechanical system. Scholars have done a variety of theoretical and experimental research on the related problems caused by the stick-slip vibration [10– 12]. Many researchers have attempted to understand stick-slip vibrations by using a simple oscillator excited by a moving belt
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