http://www-personal.umich.edu/~dsbaero/library/BhatTopObstrSCL2000.pdf

A topological obstruction to continuous global stabilization of rotational motion and the unwinding phenomenon

This book equips the reader to understand every important aspect of the dynamics of rotating machines. Will the vibration be large? What influences machine stability? How can the vibration be reduced? Which sorts of rotor vibration are the worst? The book develops this understanding initially using extremely simple models for each phenomenon, in which (at most) four equations capture the behavior. More detailed models are then developed based on finite element analysis, to enable the accurate simulation of the relevant phenomena for real machines. Analysis software (in MATLAB) is associated with this book, and novices to rotordynamics can expect to make good predictions of critical speeds and rotating mode shapes within days. The book is structured more as a learning guide than as a reference tome and provides readers with more than 100 worked examples and more than 100 problems and solutions.

/pdf/dynamics-of-rotating-machines-499re2kizd.pdf

Dynamics of rotating machines

Vibration suppression of rotating machinery is an important engineering problem. In this paper, a review of the research work performed in real-time active balanc- ing and active vibration control for rotating machinery, as well as the research work on dynamic modeling and analy- sis techniques of rotor systems, is presented. The basic methodology and a brief assessment of major difficulties and future research needs are also provided.

/pdf/active-balancing-and-vibration-control-of-rotating-machinery-3edr2htqn3.pdf

Active Balancing and Vibration Control of Rotating Machinery: A Survey

This paper presents a new approach, called adaptive autocentering, that compensates for transmitted force due to imbalance in an active magnetic bearing system. Under the proposed control law, a rigid rotor achieves rotation about the mass center and principal axis of inertia. The basic principle of this approach is to perform on-line identification of the physical characteristics of rotor imbalance and to use the identification results to tune a stabilizing controller. This approach differs from the usual strategy of adaptive feedforward compensation, which models the effect of imbalance as an external disturbance or measurement noise, and then cancels this effect by generating a synchronous reference signal. Unlike adaptive feedforward compensation, adaptive autocentering control is frequency independent and works under varying rotor speed. Performance of the control algorithm is demonstrated in simulation examples for the case of rigid rotors with static or dynamic imbalance.

/pdf/adaptive-autocentering-control-for-an-active-magnetic-1fhsi3qjwo.pdf

Adaptive autocentering control for an active magnetic bearing supporting a rotor with unknown mass imbalance

From the Publisher:
Nonlinear Control of Engineering Systems is a research monograph on Lyapunov-based control design and analysis techinques for nonlinear systems. The basic idea is to illustrate how recent Lyapunov-based techniques can be utilized to develop control designs for nonlinear systems including: mechanical systems, electrical systems, robotic systems, aerospace systems, and underactuated systems. The introduction presents: (i) motivation for the Lyapunov-based design and analysis techniques, (ii) an overview of the book, (iii) some simple examples as an aid to the less experienced reader. The book presents a complete and detailed theoretical treatment of the stated control objective, and contains a detailed description of the experimental testbed and the results obtained by implementing the developed control algorithms. The experimental results will serve to complement the theoretical content of the book by: (i) demonstrating the feasibility of implementing "complex" nonlinear control algorithms with current computational hardware, (ii) guiding the reader towards adapting these techniques to his/her own needs or for further research, and (iii) bridging the gap between theoretical design and analysis and real-time implementation.

Nonlinear Control of Engineering Systems

We define the sleeping motion and show that it is a solution of the equations of motion of a balanced top. In the general case of a top with known mass imbalance, we derive two families of globally asymptotically stabilizing control laws for the sleeping motion based on Hamilton-Jacobi-Bellman theory with zero dynamics. Two actuation schemes using torque actuators are considered.

Global stabilization of the spinning top with mass imbalance

The objective of this paper is to describe an imbalance compensation scheme for a rigid rotor supported by magnetic bearings that performs on-line identification of rotor imbalance and allows imbalance cancellation under varying speed of rotation. The proposed approach supplements existing magnetic bearing controls which are assumed to achieve elastic suspension of the rotor. By adopting a physical model of imbalance and utilizing measurements of the spin rate, the proposed algorithm allows the computation of the necessary corrective forces regardless of variations in the spin rate. Convergence of the algorithm is analyzed for single-plane balancing, and is supported by simulation in single- and two-plane balancing, as well as by experimental results in single-plane implementation.

Adaptive Virtual Autobalancing for a Rigid Rotor With Unknown Mass Imbalance Supported by Magnetic Bearings

The Serret-Andoyer transformation is a classical method for reducing the free rigid body dynamics, expressed in Eulerian coordinates, to a 2-dimensional Hamiltonian flow. First, we show that this transformation is the computation, in 3-1-3 Eulerian coordinates, of the symplectic (Marsden-Weinstein) reduction associated with the lifted left-action of SO(3) on T*SO(3)—a generalization and extension of Noether's theorem for Hamiltonian systems with symmetry. In fact, we go on to generalize the Serret-Andoyer transformation to the case of Hamiltonian systems on T*SO(3) with left-invariant, hyperregular Hamiltonian functions. Interpretations of the Serret-Andoyer variables, both as Eulerian coordinates and as canonical coordinates of the co-adjoint orbit, are given. Next, we apply the result obtained to the controlled rigid body with momentum wheels. For the class of Hamiltonian controls that preserve the symmetry on T*SO(3), the closed-loop motion of the main body can again be reduced to canonical form. This simplifies the stability proof for relative equilibria , which then amounts to verifying the classical Lagrange-Dirichlet criterion. Additionally, issues regarding numerical integration of closed-loop dynamics are also discussed. Part of this work has been presented in LumBloch:97a.

/pdf/generalized-serret-andoyer-transformation-and-applications-2s9lvc18ax.pdf

Kai-Yew Lum

Papers

Adaptive autocentering control for an active magnetic bearing supporting a rotor with unknown mass imbalance

Global stabilization of the spinning top with mass imbalance

Adaptive Virtual Autobalancing for a Rigid Rotor With Unknown Mass Imbalance Supported by Magnetic Bearings

Generalized Serret-Andoyer transformation and applications for the controlled rigid body

Adaptive virtual autobalancing for a magnetic rotor with unknown mass imbalance Part I: static balancing