Analysis of dynamical behaviors of a 2-DOF friction oscillator with elastic impacts and negative feedbacks
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ORIGINAL PAPER
Analysis of dynamical behaviors of a 2-DOF friction oscillator with elastic impacts and negative feedbacks Min Gao · Jinjun Fan
Received: 13 May 2020 / Accepted: 17 August 2020 © Springer Nature B.V. 2020
Abstract In this paper, the flow switching theory of discontinuous dynamical systems is used as the main tool to study the dynamical behaviors of a two-degreeof-freedom friction oscillator with elastic impacts, where considering that the inequality of static and kinetic friction forces results in the existence of flow barriers at the relative velocity between the mass and conveyor belt tending to zero. The negative feedback is also taken into account, i.e., when the direction of the relative velocity between the mass and conveyor belt changes, the constant force in the periodic excitation force will also change to adapt to the variation in speed. Due to the discontinuity resulted from the friction and non-smoothness resulted from impact, the motion of the two masses can be divided into the following kinds: non-sliding motion (or free motion), sliding motion, non-sliding–stick motion and sliding– stick motion, and the phase space in system is divided into different domains and boundaries, where the vector field is continuous in each region but discontinuous on the velocity boundary. The corresponding normal vectors and G-functions on the separation boundaries are introduced to acquire the switching conditions for all possible motions such as passable motion, sliding motion, grazing motion and stick motion. The mapping theory provides an effective method for the discussion of periodic motion in discontinuous dynamical sysM. Gao · J. Fan (B) School of Mathematics and Statistics, Shandong Normal University, Ji’nan 250014, People’s Republic of China e-mail: [email protected]
tems. The four-dimensional switching sets are defined by the means of the form of direct product of twodimensional switching sets, and the mapping structures are introduced based on the switching sets to describe periodic motions. Eventually, in order to explain more intuitively the above motions in such a friction–impact system, the velocity, displacement, G-functions time histories and the trajectories in phase plane for the masses are illustrated by simulation numerically. Keywords Dry friction · Discontinuous dynamical system · Elastic impact · Flow barrier · Flow switchability · Sliding motion · Grazing motion · Stick motion
1 Introduction The use of mechanical equipment is accompanied by a series of unavoidable problems, such as vibration, friction and noise, which are not conducive to the normal operation of the equipment. The mechanical systems with dry friction and gap couplings among the system components frequently appear in production and life. The vibration and friction generated by the system will reduce the safety and reliability of the whole system, such as machine tool chatter and vehicle brake noise. In addition, the friction and vibration are not always harmful in machinery, for example, the belt drivers, car and
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