203 research outputs found
Sensitivity of stability charts with respect to modal parameter uncertainties for turning operations
On the stability of high-speed milling with spindle speed variation
Spindle speed variation is a well-known technique to suppress regenerative machine tool vibrations, but it is usually considered to be effective only for low spindle speeds. In this paper, the effect of spindle speed variation is analyzed in the high-speed domain for spindle speeds corresponding to the first flip (period doubling) and to the first Hopf lobes. The optimal amplitudes and frequencies of the speed modulations are computed using the semidiscre- tization method. It is shown that period doubling chatter can effectively be suppressed by spindle speed variation, although, the technique is not effective for the quasiperiodic chatter above the Hopf lobe. The results are verified by cutting tests. Some special cases are also discussed where the practical behavior of the system differs from the predicted one in some ways. For these cases, it is pointed out that the concept of stability is understood on the scale of the principal period of the systemâthat is, the speed modulation period for variable spindle speed machining and the tooth passing period for constant spindle speed machining
Bifurcation analysis of delay-induced resonances of the El-Nino Southern Oscillation
Models of global climate phenomena of low to intermediate complexity are very
useful for providing an understanding at a conceptual level. An important
aspect of such models is the presence of a number of feedback loops that
feature considerable delay times, usually due to the time it takes to transport
energy (for example, in the form of hot/cold air or water) around the globe. In
this paper we demonstrate how one can perform a bifurcation analysis of the
behaviour of a periodically-forced system with delay in dependence on key
parameters. As an example we consider the El-Nino Southern Oscillation (ENSO),
which is a sea surface temperature oscillation on a multi-year scale in the
basin of the Pacific Ocean. One can think of ENSO as being generated by an
interplay between two feedback effects, one positive and one negative, which
act only after some delay that is determined by the speed of transport of
sea-surface temperature anomalies across the Pacific. We perform here a case
study of a simple delayed-feedback oscillator model for ENSO (introduced by
Tziperman et al, J. Climate 11 (1998)), which is parametrically forced by
annual variation. More specifically, we use numerical bifurcation analysis
tools to explore directly regions of delay-induced resonances and other
stability boundaries in this delay-differential equation model for ENSO.Comment: as accepted for Proc Roy Soc A, 20 pages, 7 figure
Criticality of Hopf bifurcation in state-dependent delay model of turning processes
International audienc
Finding periodic orbits in state-dependent delay differential equations as roots of algebraic equations
In this paper we prove that periodic boundary-value problems (BVPs) for delay
differential equations are locally equivalent to finite-dimensional algebraic
systems of equations. We rely only on regularity assumptions that follow those
of the review by Hartung et al. (2006). Thus, the equivalence result can be
applied to differential equations with state-dependent delays (SD-DDEs),
transferring many results of bifurcation theory for periodic orbits to this
class of systems. We demonstrate this by using the equivalence to give an
elementary proof of the Hopf bifurcation theorem for differential equations
with state-dependent delays. This is an alternative and extension to the
original Hopf bifurcation theorem for SD-DDEs by Eichmann (2006).Comment: minor revision, correcting mistakes in formulation of Lemma 2.3 and
A.5 (which are also present in the Journal paper): center of neighborhood
must be in , which is the case for the main theore
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 of Simple Balancing Models with State Dependent Switching Control
Time-delayed control in a balancing problem may be a nonsmooth function for a
variety of reasons. In this paper we study a simple model of the control of an
inverted pendulum by either a connected movable cart or an applied torque for
which the control is turned off when the pendulum is located within certain
regions of phase space. Without applying a small angle approximation for
deviations about the vertical position, we see structurally stable periodic
orbits which may be attracting or repelling. Due to the nonsmooth nature of the
control, these periodic orbits are born in various discontinuity-induced
bifurcations. Also we show that a coincidence of switching events can produce
complicated periodic and aperiodic solutions.Comment: 36 pages, 12 figure
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