Modes of Oscillation in Radiofrequency Paul Traps

We examine the time-dependent dynamics of ion crystals in radiofrequency traps. The problem of stable trapping of general three-dimensional crystals is considered and the validity of the pseudopotential approximation is discussed. We derive analytically the micromotion amplitude of the ions, rigorously proving well-known experimental observations. We use a method of infinite determinants to find the modes which diagonalize the linearized time-dependent dynamical problem. This allows obtaining explicitly the ('Floquet-Lyapunov') transformation to coordinates of decoupled linear oscillators. We demonstrate the utility of the method by analyzing the modes of a small `peculiar' crystal in a linear Paul trap. The calculations can be readily generalized to multispecies ion crystals in general multipole traps, and time-dependent quantum wavefunctions of ion oscillations in such traps can be obtained.Comment: 24 pages, 3 figures, v2 adds citations and small correction

Application of a parameter identification technique to a hingeless helicopter rotor

A mathematical model of a gyro-controlled, three-bladed hingeless helicopter rotor was developed and parameters of the model were estimated using a parameter identification technique. The flapping and feathering degrees of freedom of the blades were modeled. The equations of the model contain time-varying, periodic coefficients due to the forward speed of the rotor. A digital simulation of the analytical model was compared with wind-tunnel measurements to establish the validity of the model. Comparisons of steady-state and transient solutions of the analytical model with the tunnel measurements gave reasonably good matching of gyro angle but less satisfactory matching of hub moment measurements. Further improvements were obtained by use of a parameter identification technique to adjust as many as 10 parameters of the analytical model. The sensitivity of the blade response to small changes in the parameters was also calculated

Spike Oscillations

According to Belinskii, Khalatnikov and Lifshitz (BKL), a generic spacelike singularity is characterized by asymptotic locality: Asymptotically, toward the singularity, each spatial point evolves independently from its neighbors, in an oscillatory manner that is represented by a sequence of Bianchi type I and II vacuum models. Recent investigations support a modified conjecture: The formation of spatial structures (`spikes') breaks asymptotic locality. The complete description of a generic spacelike singularity involves spike oscillations, which are described by sequences of Bianchi type I and certain inhomogeneous vacuum models. In this paper we describe how BKL and spike oscillations arise from concatenations of exact solutions in a Hubble-normalized state space setting, suggesting the existence of hidden symmetries and showing that the results of BKL are part of a greater picture.Comment: 38 pages, 14 figure

Flat systems, equivalence and trajectory generation

Flat systems, an important subclass of nonlinear control systems introduced via differential-algebraic methods, are defined in a differential geometric framework. We utilize the infinite dimensional geometry developed by Vinogradov and coworkers: a control system is a diffiety, or more precisely, an ordinary diffiety, i.e. a smooth infinite-dimensional manifold equipped with a privileged vector field. After recalling the definition of a Lie-Backlund mapping, we say that two systems are equivalent if they are related by a Lie-Backlund isomorphism. Flat systems are those systems which are equivalent to a controllable linear one. The interest of such an abstract setting relies mainly on the fact that the above system equivalence is interpreted in terms of endogenous dynamic feedback. The presentation is as elementary as possible and illustrated by the VTOL aircraft

Nonlinear and sampled data control with application to power systems

Sampled data systems have come into practical importance for a variety of reasons. The earliest of these had primarily to do with economy of design. A more recent surge of interest was due to increase utilization of digital computers as controllers in feedback systems. This thesis contributes some control design for a class of nonlinear system exhibition linear output. The solution of several nonlinear control problems required the cancellation of some intrinsic dynamics (so-called zero dynamics) of the plant under feedback. It results that the so-dened control will ensure stability in closed-loop if and only if the dynamics to cancel are stable. What if those dynamics are unstable? Classical control strategies through inversion might solve the problem while making the closed loop system unstable. This thesis aims to introduce a solution for such a problem. The main idea behind our work is to stabilize the nonminimum phase system in continuous- time and undersampling using zero dynamics concept. The overall work in this thesis is divided into two parts. In Part I, we introduce a feedback control designs for the input-output stabilization and the Disturbance Decoupling problems of Single Input Single Output nonlinear systems. A case study is presented, to illustrate an engineering application of results. Part II illustrates the results obtained based on the Articial Intelligent Systems in power system machines. We note that even though the use of some of the AI techniques such as Fuzzy Logic and Neural Network does not require the computation of the model of the application, but it will still suer from some drawbacks especially regarding the implementation in practical applications. An alternative used approach is to use control techniques such as PID in the approximated linear model. This design is very well known to be used, but it does not take into account the non-linearity of the model. In fact, it seems that control design that is based on nonlinear control provide better performances

A High-Order, Linear Time-Invariant Model for Application to Higher Harmonic Control and Flight Control System Interaction

This research describes a new methodology for the extraction of a high-order, linear time invariant model, which allows the periodicity of the helicopter response to be accurately captured. This model provides the needed level of dynamic fidelity to permit an analysis and optimization of the AFCS and HHC algorithms. The key results of this study indicate that the closed-loop HHC system has little influence on the AFCS or on the vehicle handling qualities, which indicates that the AFCS does not need modification to work with the HHC system. However, the results show that the vibration response to maneuvers must be considered during the HHC design process, and this leads to much higher required HHC loop crossover frequencies. This research also demonstrates that the transient vibration responses during maneuvers can be reduced by optimizing the closed-loop higher harmonic control algorithm using conventional control system analyses

Analytical stability and simulation response study for a coupled two-body system

An analytical stability study and a digital simulation response study of two connected rigid bodies are documented. Relative rotation of the bodies at the connection is allowed, thereby providing a model suitable for studying system stability and response during a soft-dock regime. Provisions are made of a docking port axes alignment torque and a despin torque capability for encountering spinning payloads. Although the stability analysis is based on linearized equations, the digital simulation is based on nonlinear models

Nonlinear Model Reduction to Fractional and Mixed-Mode Spectral Submanifolds

A primary spectral submanifold (SSM) is the unique smoothest nonlinear continuation of a nonresonant spectral subspace $E$ of a dynamical system linearized at a fixed point. Passing from the full nonlinear dynamics to the flow on an attracting primary SSM provides a mathematically precise reduction of the full system dynamics to a very low-dimensional, smooth model in polynomial form. A limitation of this model reduction approach has been, however, that the spectral subspace yielding the SSM must be spanned by eigenvectors of the same stability type. A further limitation has been that in some problems, the nonlinear behavior of interest may be far away from the smoothest nonlinear continuation of the invariant subspace $E$. Here we remove both of these limitations by constructing a significantly extended class of SSMs that also contains invariant manifolds with mixed internal stability types and of lower smoothness class arising from fractional powers in their parametrization. We show on examples how fractional and mixed-mode SSMs extend the power of data-driven SSM reduction to transitions in shear flows, dynamic buckling of beams and periodically forced nonlinear oscillatory systems. More generally, our results reveal the general function library that should be used beyond integer-powered polynomials in fitting nonlinear reduced-order models to data.Comment: To appear in Chao

Control of an over-actuated nanopositioning system by means of control allocation

This Master’s Thesis is devoted to the analysis and design of a control structure for the nanopositioning system LAU based on the dynamic control allocation technique. The objective is to control the vertical displacement with nanometer precision under a control effort distribution criterion among the actuator set. In this case, the pneumatic actuator is used as a passive gravity compensator while the voice coil motor generates the transient forces. The analysis of the system characteristics allows defining the design criterion for the control allocation. In this direction, the proposed dynamic control allocation stage considers a frequency distribution of the control effort. The lower frequency components are assigned to the pneumatic actuator while the higher frequencies are handled by the voice coil drive. The significant actuator dynamics are compensated through a Kalman filter approach. The position controller is based on a feedback linearization framework with a disturbance observer for enhanced robustness. The experimental validation demonstrates the feasibility of the proposed technique.Diese Masterarbeit widmet sich der Analyse und dem Entwurf einer Regelungsstruktur für das Nanopositioniersystem LAU. Dabei werden Methoden untersucht, welche das notwendige Stellsignal auf zwei Aktoren aufteilen. Ziel ist es, die vertikale Verschiebung des LAU mit Nanometerpräzision zu regeln. In diesem Fall wird der pneumatische Aktor als passiver Schwerkraftkompensator verwendet, während die elktromagnetische Tauchspule die transienten Kräfte erzeugt. Die Analyse der Eigenschaften des LAUSystems ermöglicht die Definition der Entwurfskriterien zur Aufteilung der Stellgröße. In dieser Richtung berücksichtigt die vorgeschlagene dynamische Methode eine Aufteilung der Stellgröße bezüglich der Frequenzanteile. Die niederfrequenten Komponenten werden dem pneumatischen Aktor zugeordnet. Dem elektromagnetische Aktor werden die verbliebenen hochfrequenten Anteile zugeordnet. Die signifikanten Effekte der Aktordynamik in Bezug auf die Bewegungsdynamik werden durch einen Kalman- Filteransatz kompensiert. Nichtlineare Streckenanteile werden basierend auf dem Modell und einem Störbeobachter kompensiert, sodass der verbleibende Anteil des Positionsreglers mit linearen Methoden entworfen werden kann. Die experimentelle Validierung zeigt die Effektivität des untersuchten Konzeptes.Tesi

Least-Squares, Continuous Sensitivity Analysis for Nonlinear Fluid-Structure Interaction

A least-squares, continuous sensitivity analysis method is developed for transient aeroelastic gust response problems to support computationally efficient analysis and optimization of aeroelastic design problems. A key distinction between the local and total derivative forms of the sensitivity system is introduced. The continuous sensitivity equations and sensitivity boundary conditions are derived in local derivative form which is shown to be superior for several applications. The analysis and sensitivity problems are both posed in a first-order form which is amenable to a solution using the least-squares finite element method. Several example and validation problems are presented and solved, including elasticity, fluid, and fluid-structure interaction problems. Significant contributions of the research include the first sensitivity analysis of nonlinear transient gust response, a local derivative formulation for shape variation that requires parameterizing only the boundary, and statement of sufficient conditions for using nonlinear black box software to solve the sensitivity equations. Promising paths for future investigation are presented and discussed