On the discontinuous dynamics of a class of 2-DOF frictional vibration systems with asymmetric elastic constraints

Wen Zhang; Jinjun Fan; Yuanyuan Peng; Wen Zhang; Jinjun Fan; Yuanyuan Peng

doi:10.3934/mmc.2023024

Mathematical Modelling and Control

2023, Volume 3, Issue 4: 278-305. doi: 10.3934/mmc.2023024

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On the discontinuous dynamics of a class of 2-DOF frictional vibration systems with asymmetric elastic constraints

School of Mathematics and Statistics, Shandong Normal University, Ji'nan, 250014, China

Received: 14 December 2022 Revised: 21 February 2023 Accepted: 03 March 2023 Published: 07 December 2023

In this paper, the discontinuous dynamic behavior of a two-degree-of-freedom frictional collision system including intermediate elastic collision and unilateral elastic constraints subjected to periodic excitation is studied by using flow switching theory. In this system, given that the motion of each object might have a velocity that is either greater than or less than zero and each object experiences a periodic excitation force that has negative feedback, because the kinetic and static friction coefficients differ, the flow barrier manifests when the object's speed is zero. Based on the discontinuity or nonsmoothness of the oscillator's motion generated by elastic collision and friction, the motion states of the oscillator in the system are divided into 16 cases and the absolute and relative coordinates are used to define various boundaries and domains in the oscillator motion's phase space. On the basis of this, the G-function and system vector fields are used to propose the oscillator motion's switching rules at the displacement and velocity boundaries. Finally, some dynamic behaviors for the 2-DOF oscillator are demonstrated via numerical simulation of the oscillator's stick, grazing, sliding and periodic motions and the scene of sliding bifurcation. The mechanical system's optimization designs with friction and elastic collision will benefit from this investigation's findings.
- flow switchability,
- discontinuous dynamical system,
- asymmetric elastic constraints,
- grazing motion,
- stick-sliding motion,
- flow barrier
Citation: Wen Zhang, Jinjun Fan, Yuanyuan Peng. On the discontinuous dynamics of a class of 2-DOF frictional vibration systems with asymmetric elastic constraints[J]. Mathematical Modelling and Control, 2023, 3(4): 278-305. doi: 10.3934/mmc.2023024

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Abstract

In this paper, the discontinuous dynamic behavior of a two-degree-of-freedom frictional collision system including intermediate elastic collision and unilateral elastic constraints subjected to periodic excitation is studied by using flow switching theory. In this system, given that the motion of each object might have a velocity that is either greater than or less than zero and each object experiences a periodic excitation force that has negative feedback, because the kinetic and static friction coefficients differ, the flow barrier manifests when the object's speed is zero. Based on the discontinuity or nonsmoothness of the oscillator's motion generated by elastic collision and friction, the motion states of the oscillator in the system are divided into 16 cases and the absolute and relative coordinates are used to define various boundaries and domains in the oscillator motion's phase space. On the basis of this, the G-function and system vector fields are used to propose the oscillator motion's switching rules at the displacement and velocity boundaries. Finally, some dynamic behaviors for the 2-DOF oscillator are demonstrated via numerical simulation of the oscillator's stick, grazing, sliding and periodic motions and the scene of sliding bifurcation. The mechanical system's optimization designs with friction and elastic collision will benefit from this investigation's findings.

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