Kirk Green

Investigation of a multi-ball, automatic dynamic balancing mechanism for eccentric rotors

This paper concerns an analytical and experimental investigation into the dynamics of an automatic dynamic balancer (ADB) designed to quench vibration in eccentric rotors. This fundamentally nonlinear device incorporates several balancing masses that are free to rotate in a circumferentially mounted ball race. An earlier study into the steady state and transient response of the device with two balls is extended to the case of an arbitrary number of balls. Using bifurcation analysis allied to numerical simulation of a fully nonlinear model, the question is addressed of whether increasing the number of balls is advantageous. It is found that it is never possible to perfectly balance the device at rotation speeds comparable with or below the first natural, bending frequency of the rotor. When considering practical implementation of the device, a modification is suggested where individual balls are contained in separate arcs of the ball race, with rigid partitions separating each arc. Simulation results for a partitioned ADB are compared with those from an experimental rig. Close qualitative and quantitative match is found between the theory and the experiment, confirming that for sub-resonant rotation speeds, the ADB at best makes no difference to the imbalance, and can make things substantially worse. Further related configurations worthy of experimental and numerical investigation are proposed.

A two-parameter study of the locking region of a semiconductor laser subject to phase-conjugate feedback

We present a detailed bifurcation analysis of a single-mode semiconductor laser subject to phase-conjugate feedback, a system described by a delay differential equation. Codimension-one bifurcation curves of equilibria and periodic orbits and curves of certain connecting orbits are presented near the lasers locking region in the two-dimensional parameter plane of feedback strength and pump current. We identify several codimension-two bifurcations, including a double-Hopf point, Belyakov points, and a T-point bifurcation, and we show how they organize the dynamics. This study is the first example of a two-parameter bifurcation study, including bifurcations of periodic and connecting orbits, of a delay system. It was made possible by new numerical continuation tools, implemented in the package DDE-BIFTOOL, and showcases their usefulness for the study of delay systems arising in applications.

Computing unstable manifolds of periodic orbits in delay differential equations

We present the first algorithm for computing unstable manifolds of saddle-type periodic orbits with one unstable Floquet multiplier in systems of autonomous delay differential equations (DDEs) with one fixed delay. Specifically, we grow the one-dimensional unstable manifold Wu (q) of an associated saddle fixed point q of a Poincare map defined by a suitable Poincare section Σ. Starting close to q along the linear approximation to Wu(q) given by the associated eigenfunction, our algorithm grows the manifold as a sequence of points, where the distance between points is governed by the curvature of the one-dimensional intersection curve Wu(q) ∩ Σ of Wu (q) with Σ. Our algorithm makes it possible to study global bifurcations in DDEs. We illustrate this with the break-up of an invariant torus and a subsequent crisis bifurcation to chaos in a DDE model of a semiconductor laser with phase-conjugate feedback.

Bistability and torus break-up in a semiconductor laser with phase-conjugate feedback

Abstract We present a detailed study of a route to chaos via quasiperiodicity on a torus in a semiconductor laser with phase-conjugate feedback. Highlighting the use of new tools that go far beyond mere simulation, we compute bifurcation diagrams and unstable manifolds of saddle periodic orbits. In this way, we show how a torus breaks up with a final sudden onset of chaos in a crisis bifurcation. We also identify regions of bistability between periodic solutions and other attractors in the system.

Bifurcation Analysis of a Spatially Extended Laser with Optical Feedback

Vertical cavity surface-emitting lasers (VCSELs) are a new type of semiconductor laser characterized by the spatial extent of their disk-shaped output apertures. As a result, a VCSEL supports several optical modes (patterns of light) transverse to the direction of light propagation. When any laser is coupled to other optical elements, there is unavoidable optical feedback via reflecting surfaces, which influences the stability of the laser output. For a VCSEL, the question is how the transverse optical modes interact dynamically in the presence of optical feedback and how this affects stability of the system. In this paper, we start from a PDE description of the VCSEL. We proceed by using an expansion in suitable eigenfunctions to resolve the spatial dependence. In the presence of optical feedback we obtain a model in the form of a system of delay differential equations (DDEs). As we show with the example of a VCSEL that supports two transverse modes, the spatially expanded DDE model is small enough to al...

Bifurcation analysis of frequency locking in a semiconductor laser with phase-conjugate feedback

We present a detailed study of the external-cavity modes (ECMs) of a semiconductor laser with phase-conjugate feedback. Mathematically, lasers with feedback are modeled by delay differential equations (DDEs) with an infinite-dimensional phase space. We employ new numerical bifurcation tools for DDEs to continue steady states and periodic orbits, irrespective of their stability. In this way, we show that the periodic orbits corresponding to the ECMs are connected to the steady state solution associated with the locking range of the laser. We also identify symmetric and nonsymmetric homoclinic orbits and hysteresis in the system.

Analytical theory of external cavity modes of a semiconductor laser with phase conjugate feedback

The rate equations describing a laser with phase conjugate feedback are analyzed in the case of non-zero detuning. For low feedback rates and detuning, the stability diagram of the steady state is similar to the laser subject to injection. A stable steady state may loose its stability through a Hopf bifurcation exhibiting a frequency close to the relaxation oscillation frequency of the solitary laser. We also construct time-periodic pulsating intensity solutions exhibiting frequencies close to an integer multiple of the external cavity frequency. These solutions have been found numerically for the zero detuning case and play an important role in the bifurcation diagram.

Bifurcation analysis of a multi-transverse-mode VCSEL

We study the behaviour of a multi-transverse-mode vertical-cavity surface-emitting laser subject to optical feedback in which the optical modes are coupled through the external round-trip. Starting from a delayed partial differential equation description of the spatial optical mode profiles and the carrier diffusion, we first use eigenfunction expansion techniques to resolve the spatial dependence. The resulting system of delay differential equations is then amenable to a full nonlinear bifurcation analysis by means of numerical continuation techniques. As illustration, we present bifurcation diagrams of a two-mode VCSEL in the plane of feedback strength versus feedback phase. In this way, we identify a number of changes in the structure and bifurcations of the VCSELs dynamics. In particular, we find coexisting stable steady state solutions, which bifurcate to stable in-phase and anti-phase periodic solutions with vastly differing frequencies. We show how these periodic solutions give rise to quasiperiodic and chaotic laser dynamics.

Structured Pseudospectra and Random Eigenvalues Problems in Vibrating Systems

This paper introduces the concept of pseudospectra as a generalized tool for uncertainty quantification and propagation in structural dynamics. Different types of pseudospectra of matrices and matrix polynomials are explained. Particular emphasis is given to structured pseudospectra for matrix polynomials, which offer a deterministic way of dealing with uncertainties for structural dynamic systems. The pseudospectra analysis is compared with the results from Monte Carlo simulations of uncertain discrete systems. Two illustrative example problems, one with probabilistic uncertainty with various types of statistical distributions and the other with interval type of uncertainty, are studied in detail. Excellent agreement is found between the pseudospectra results and Monte Carlo simulation results.

STRUCTURED PSEUDOSPECTRA FOR MATRIX FUNCTIONS: THEORY AND APPLICATION

Abstract We introduce structured pseudospectra for analytic matrix functions and derive computable formulae. The results are applied to the sensitivity analysis of the eigenvalues of a time-delay system arising from laser physics.