Dennis O’Connor

Periodic Motions With Impacting Chatter and Stick in a Gear Transmission System

In this paper, an impact model with possible stick between the two gears is proposed for gear transmission systems, which includes the piecewise backlash model and the traditional impact model for the first time. The new model presented in this paper possesses a time-varying boundary for two dynamical systems either to switch or to impact. Such a model can catch impacting chatter and stick phenomena in gear transmission systems. Based on the new model, periodic impacting chatter and stick in a gear transmission system can be investigated. For doing so, switching sets on the time-varying boundaries are introduced to define basic mappings. Mapping structures based on basic mappings are developed for characterizing motions in gear transmission systems, and from such mapping structures, periodic motions with impacting chatter and stick in such a gear transmission system are predicted analytically. Numerical simulations are performed for illustration of periodic motions with impacting chatter and stick phenomena.

Nonlinear Dynamics of a Gear Transmission System: Part II — Periodic Impacting Chatter and Stick

In Part I, the motion mechanism of impacting chatter and stick motion in the gear transmission dynamical system was discussed. This paper focuses on periodic motions relative to the impacting chatter and stick in order to find the origin of noise and vibration in such a gear transmission system. Such periodic motions are predicted analytically through mapping structures, and the corresponding local stability and bifurcation analysis are carried out. The grazing and stick conditions presented in Part I are adopted to determine the existence of periodic motions, which cannot be achieved from the local stability analysis. Numerical simulations are performed to illustrate periodic motions and stick motion criteria. Such an investigation may provide some clues to reduce the noise in gear transmission systems.Copyright

Nonlinear Dynamics of a Gear Transmission System: Part I — Mechanism of Impacting Chatter With Stick

In this paper, an investigation on nonlinear dynamical behaviors of a transmission system with a gear pair is conducted. The transmission system is described through an impact model with possible stick between the two gears. From the theory of discontinuous dynamical systems, the motion mechanism of impacting chatter with stick is investigated. The onset and vanishing conditions of stick motions are developed, and the condition for maintaining the stick motion is achieved as well. The corresponding physics interpretation is given for a better understanding of nonlinear behaviors of gear transmission systems. Furthermore, such an understanding may be very helpful to improve the efficiency of gear transmission systems.Copyright

System Discontinuity and Switchability

In this Chapter, from Luo (2005, 2006, 2012), system discontinuity and switchability at the boundary will be reviewed. The accessible and inaccessible sub-domains will be introduced in discontinuous dynamic systems. On the accessible domains, the corresponding dynamic systems will be defined. The switchability and tangency (grazing) of a flow to the separation boundary between two adjacent accessible domains will be discussed, and the necessary and sufficient conditions for such passability and tangency of the flow to the boundary will be presented. The product of the two dot products of the boundary normal vector and vector fields will be presented, and the corresponding conditions for the flow switchability to the boundary will be discussed.

Dynamic Mechanism of a Train Suspension System

Nonlinear dynamical behaviors of a train suspension system with impacts are investigated. The suspension system is modelled through an impact model with possible stick between a bolster and two wedges. Based on the mapping structures, periodic motions of such a system are described. To understand the global dynamical behaviors of the train suspension system, system parameter maps are developed. Numerical simulations for periodic and chaotic motions are performed from the parameter maps.Copyright

A General Theory for Flow Passability

In this chapter, from Luo (2008a, 2008b, 2009, 2012a, 2012b), a general theory for the passability of a flow to a specific boundary in discontinuous dynamical systems will be reviewed. The concepts of real and imaginary flows will be introduced. The G-functions for discontinuous dynamical systems will be developed to describe the general theory of the passability of a flow to the boundary. Based on the G-function, the passability of a flow from a domain to an adjacent one will be discussed. With the concepts of real and imaginary flows, the full and half sink and source flows to the boundary will be discussed in detail. A flow to the boundary in a discontinuous dynamical system can be passable or non-passable. Thus, all the switching bifurcations between the passable and non-passable flows will be presented.

A Freight Train Suspension System

From O’Connor and Luo (2014), a freight train suspension system will be presented for all possible types of motion. The suspension system includes a wedge system and a bolster system. The suspension system experiences impacts and friction between wedges and bolster. The impacts cause the chatter motions between wedges and bolster, and the friction cause the stick and non-stick motions between wedges and bolster. Due to the wedge effect, the suspension system may become stuck and not move, which causes the suspension to lose function. To discuss such phenomena in the freight train suspension systems, the theory of discontinuous dynamical systems will be used, and the motion mechanism of impacting chatter with stick and stuck will be discussed. The analytical conditions for the onset and vanishing of stick motions between the wedges and bolster will be presented, and the condition for maintaining stick motion will be obtained as well. The analytical conditions for stuck motion will be developed for the onset and vanishing conditions for stuck motion. Analytical prediction of periodic motions relative to impacting chatter with stick and stuck motions in train suspension will be performed through the mapping dynamics. Numerical simulations will be completed for illustration of periodic motions of stick and stuck motions. Finally, from field testing data, the effects of wedge angle on the motions of the suspension will be presented to find a more desirable suspension response for design.

Dynamical System Interaction

From Luo (2009, 2012), the interaction of two dynamical systems will be discussed. The interaction relations of the two dynamical systems will be treated as boundaries in discontinuous dynamical systems, and such boundaries are time-varying. Thus, the boundary and domains for one of the two dynamical systems is controlled by the other one. The mathematical conditions for such interactions of two dynamical systems will be presented through the theory for the switchability and attractivity of edge flows to the specific edges.

Analytical Solutions for Periodic Motions in a Hardening Mathieu-Duffing Oscillator

Analytical solutions for period-m motions in a hardening Mathieu-Duffing oscillator are obtained using the finite Fourier series solutions, and the stability and bifurcation analysis of such periodic motions are completed. To verify the approximate analytical solutions of periodic motions, numerical simulations of the hardening Mathieu-Duffing oscillator are presented. Period-1 asymmetric and period-2 symmetric motions are illustrated.Copyright

COEXISTING ASYMMETRIC AND SYMMETRIC PERIODIC MOTIONS IN THE MATHIEU-DUFFING OSCILLATOR

In this paper, periodic motions in the Mathieu-Duffing oscillator are analytically predicted through the harmonic balance method. The approximate, analytical solutions of periodic motions are achieved, and the corresponding stability analyses of the stable and unstable periodic solutions are completed. Numerical simulations are provided for a complete picture of coexisting periodic motions.Copyright