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.

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Dynamics of rotating machines

Online estimation of the frequency of a sinusoidal signal is a classical problem in systems theory that has many practical applications. In this paper the authors provide a solution to the problem of ensuring a globally convergent estimation. More specifically, they propose a new adaptive notch filter whose dynamic equations exhibit the following remarkable features: 1) all signals are globally bounded and the estimated frequency is asymptotically correct for all initial conditions and all frequency values; 2) the authors obtain a simple tuning procedure for the estimator design parameters, which trades-off the adaptation tracking capabilities with noise sensitivity, ensuring (exponential) stability of the desired orbit; and 3) transient performance is considerably enhanced, even for small or large frequencies, as witnessed by extensive simulations. To reveal some of the stability-instability mechanisms of the existing algorithms and motivate our modifications the authors make appeal to a novel nonlinear (state-dependent) time scaling. The main advantage of working in the new time scale is that they remove the coupling between the parameter update law and the filter itself, decomposing the system into a feedback form where the required modifications to ensure stability become apparent. Even though they limit their attention here to the simplest case of a single constant frequency without noise the algorithm is able to track time-varying frequencies, preserve local stability in the presence of multiple sinusoids, and is robust with respect to noise.

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A globally convergent frequency estimator

Unbalance of magnetically levitated rotors causes undesirable synchronous vibrations which may lead to saturation of the magnetic actuator. To avoid this problem we propose a generalized narrow-band notch filter which is inserted into the multivariable feedback without destabilizing the closed loop. The parameters of the generalized notch filter strongly depend on the inverse sensitivity matrix evaluated at rotational speed. This matrix is easily measured a priori and stored in a look-up table. It is shown that a decentralized notch is feasible for weakly gyroscopic rotors. The proposed notch filter approach has advantages in terms of run-time complexity and analytical verification of closed-loop stability. Results from the implementation of the proposed unbalance compensation in industrial magnetic bearing systems are included.

Unbalance compensation using generalized notch filters in the multivariable feedback of magnetic bearings

Nonlinear differential-geometric techniques are proposed for the design of a feedback controller for three-phase voltage-source pulsewidth modulation (PWM) AC/DC boost converters under a cascade control structure. The input-output linearizability of the system modeled in d-q synchronous reference frames is examined. The results lead to a decoupled d-q current control scheme. For the internal DC-bus voltage dynamics (zero dynamics), by considering the d-axis current command as control input and using a square transform on DC-bus voltage, it is shown that this remaining system contains an input memoryless nonlinearity of conic sector type and satisfies the conditions for a Lur'e plant. This suggests a new outer-loop control strategy to control the square of the DC output voltage, rather than DC output voltage itself, utilizing a simple proportional plus integral (PI) controller cascaded to the d-axis current loop. Since most results do not consider the control of zero dynamics, the strategy for the control of zero dynamics via cascade control structure may provide valuable insight for the design of input-output linearized systems in which zero dynamics contain some desired control variables. The absolute tracking concept for Lur'e plants is introduced to prove the global tracking capability with zero steady-state error of the voltage loop. The controlled PWM AC/DC converter has the features of global stability, fast (exponential) tracking of DC-bus voltage command with zero steady-state error, asymptotic rejection of load disturbance, robustness against parameter uncertainties and decoupled dynamical responses between d and q current loops. Also, measurement of load current is not required. All these features are confirmed via laboratory experiments on a 1.5 kVA PC-based controlled prototype.

Input-output linearization and zero-dynamics control of three-phase AC/DC voltage-source converters

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.

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Active Balancing and Vibration Control of Rotating Machinery: A Survey

Active magnetic bearings (AMBs) are an example for industrial applications of digital signal processors (DSPs). The complex control functions involved include multichannel control in a delay-time sensitive system. DSPs are well suited for this challenging application. Hardware and software issues are discussed. Unbalance compensation and other additional functions of the controller are presented. An industrial milling spindle with AMBs and DSP control is discussed in detail. A section on AMB research projects with DSP control indicates future trends. >

C. Gahler

Papers

Unbalance compensation using generalized notch filters in the multivariable feedback of magnetic bearings

Application of digital signal processors for industrial magnetic bearings