Stability implications of delay distribution for first-order and second-order systems

Kiss, G., & Krauskopf, B. (2009). Stability implications of delay distribution for first-order and second-order systems. Early version, also known as pre-print Link to publication record in Explore Bristol Research PDF-documen

Realizability of the normal form for the triple-zero nilpotency in a class of delayed nonlinear oscillators

The effects of delayed feedback terms on nonlinear oscillators has been extensively studied, and have important applications in many areas of science and engineering. We study a particular class of second-order delay-differential equations near a point of triple-zero nilpotent bifurcation. Using center manifold and normal form reduction, we show that the three-dimensional nonlinear normal form for the triple-zero bifurcation can be fully realized at any given order for appropriate choices of nonlinearities in the original delay-differential equation.Comment: arXiv admin note: text overlap with arXiv:math/050539

On the bistable zone of milling processes

LRR-RLK genes identified in Physcomitrella patens and Selaginella moellendorffii. (XLS 4566 kb

Regenerative chatter in self-interrupted plunge grinding

This research is supported by National Natural Science Foundation of China under Grant Nos.11572224 and 11502048, and Fundamental Research Funds for the Central Universities under Grant No. ZYGX2015KYQD033. We would like to thank Dr. Pankaj Wahi for an initial discussion during YY’s stay in Aberdeen.Peer reviewedPublisher PD

On the bistable zone of milling processes

A modal-based model of milling machine tools subjected to time-periodic nonlinear cutting forces is introduced. The model describes the phenomenon of bistability for certain cutting parameters. In engineering, these parameter domains are referred to as unsafe zones, where steady-state milling may switch to chatter for certain perturbations. In mathematical terms, these are the parameter domains where the periodic solution of the corresponding nonlinear, time-periodic delay differential equation is linearly stable, but its domain of attraction is limited due to the existence of an unstable quasi-periodic solution emerging from a secondary Hopf bifurcation. A semi-numerical method is presented to identify the borders of these bistable zones by tracking the motion of the milling tool edges as they might leave the surface of the workpiece during the cutting operation. This requires the tracking of unstable quasi-periodic solutions and the checking of their grazing to a time-periodic switching surface in the infinite-dimensional phase space. As the parameters of the linear structural behaviour of the tool/machine tool system can be obtained by means of standard modal testing, the developed numerical algorithm provides efficient support for the design of milling processes with quick estimates of those parameter domains where chatter can still appear in spite of setting the parameters into linearly stable domains

Control of Drilling Vibrations: A Time-Delay System-Based Approach

International audienceThe main purpose of this study is the control of both axial and torsional vibrations occurring along a rotary oilwell drilling system. This work completes a previous author's paper [Boussaada I., Mounier H., Niculescu S-I., Cela A. 2012] which presents the description of the qualitative dynamical response of a rotary drilling system with a drag bit, using a model that takes into consideration the axial and the torsional vibration modes of the bit. The studied model, based on the interface bit-rock, contains a couple of wave equations with boundary conditions consisting of the angular speed and the axial speed at the top additionally to the angular and axial acceleration at the bit whose contain a realistic frictional torque. Our analysis is based on the center manifold theorem and normal forms theory whose allow us to simplify the model. By this way we design two control laws allowing to suppress the undesired vibrations guaranteeing a regular drilling process. The obtained results are numerically illustrated

Instability regimes and self-excited vibrations in deep drilling systems

peer reviewe

State-dependent distributed-delay model of orthogonal cutting

In this paper we present a model of turning operations with state-dependent distributed time delay. We apply the theory of regenerative machine tool chat- ter and describe the dynamics of the tool-workpiece sys- tem during cutting by delay-diferential equations. We model the cutting-force as the resultant of a force sys- tem distributed along the rake face of the tool, which results in a short distributed delay in the governing equation superimposed on the large regenerative de- lay. According to the literature on stress distribution along the rake face, the length of the chip-tool inter- face, where the distributed cutting-force system is act- ing, is function of the chip thickness, which depends on the vibrations of the tool-workpiece system due to the regenerative efect. Therefore, the additional short de- lay is state-dependent. It is shown that involving state- dependent delay in the model does not afect linear sta- bility properties, but does afect the nonlinear dynamics of the cutting process. Namely, the sense of the Hopf bi- furcation along the stability boundaries may turn from sub- to supercritical at certain spindle speed regions

Dynamics, Nonlinear Instabilities, and Control of Drill-strings

Drill strings are flexible, slender structures, which are many kilometers long. They are used to transmit the rotary motion to the drill bit in the process of drilling a borehole. Due to the flexibility of the drill string and nonlinear interactions between the drill bit and rock, these systems often experience severe vibrations, and these vibrations may cause excessive wear of the drill bit and equipment damage. The aim of this dissertation effort is to further the understanding of the underlying mechanism leading to the undesired vibratory motions of drill strings, as well as to develop a viable control strategy that is applicable for mitigation of harmful vibrations. A reduced-order drill-string model with coupled axial and torsional dynamics is constructed. Nonlinear effects associated with dry friction, loss of contact, and the state-dependent delay, which all arise from cutting mechanics are considered. For the sake of analyses, a non-dimensionalized form of the governing equations is provided. Next, in order to study the local stability of the drill-string system, a linear system associated with the state-dependent delay is derived. The stability analysis of this linearized system is carried out analytically by using the D-subdivision scheme. The obtained results are illustrated in the terms of stability crossing curves, which are presented in the plane of non-dimensional rotation speed and non-dimensional cutting depth; non-dimensional rotation speed and cutting coefficient, respectively. For the nonlinear analysis, a numerical continuation method is developed and used to follow periodic orbits of systems with friction, loss of contact, and state-dependent delay. Bifurcation diagrams are constructed to capture the possible routes from either a nominal stable operational state or a stable limit-cycle motion without stick-slip to a limit-cycle motion with stick-slip. It is shown that the system can experience subcritical Hopf bifurcations of equilibrium solutions and cyclic fold bifurcations. Furthermore, with the preceding work, an observer-based on controller design is proposed by using a continuous pole placement method for time delay systems. The effectiveness of the controller in suppressing stick-slip behavior is shown through simulations. The primary contributions of this dissertation are summarized as follows: i) analytical determination of the stable operational region; ii) revelation of the routes to torsional stick-slip vibrations; and iii) construction of a feasible control scheme to mitigate the destructive vibrations caused by complex nonlinear effects