Active vibration control of rotating machines

Second order matrix equations arise in the description of real dynamical systems. Traditional modal control approaches utilise the eigenvectors of the undamped system to diagonalise the system matrices. Any remaining off-diagonal terms in the modal damping matrix are discarded. A regrettable automatic consequence of this action is the destruction of any notion of the skew-symmetry in the damping. The methods presented in this thesis use the `Lancaster Augmented Matrices' (LAMs) allowing state space representations of the second order systems. `Structure preserving transformations' (SPTs) are used to manipulate the system matrices whilst preserving the structure within the LAMs. Utilisation of the SPTs permits the diagonalisation of the system mass, damping and stiffness matrices for non-classically damped systems. Thus a modal control method is presented in this thesis which exploits this diagonalisation. The method introduces independent modal control in which a separate modal controller is designed in modal space for each individual mode or pair of modes. The modal displacements and velocities for the diagonalised systems are extracted from the physical quantities using first order SPT-based filters. Similarly the first order filters are used to translate the modal force into the physical domain. Derivation of the SPT-filters is presented together with a method by which one exploits the non-uniqueness of the diagonalising filters such that initially unstable filters are stabilised. In the context of active control of rotating machines, standard optimal controller methods enable a trade-off to be made between (weighted) mean-square vibrations and (weighted) mean-square control forces, or in the case of a machines controlled using magnetic bearings the currents injected into the magnetic bearings. One shortcoming of such controllers for magnetic bearings is that no concern is devoted to the voltages required. In practice, the voltage available imposes a strict limitation on the maximum possible rate of change of control force (force slew rate). This thesis presents a method which removes the aforementioned existing shortcomings of traditional optimal control. Case studies of realistic rotor systems are presented to illustrate the modal control and control force rate penalisation methods. The system damping matrices of the case studies contain skew-symmetric components due to gyroscopic forces typical of rotating machines. The SPT-based modal control method is used to decouple the non-classically damped equations of motion into n single degree of freedom systems. Optimal modal controllers are designed independently in the modal space such that the modal state, modal forces and modal force rates are weighted as required. The SPT-based modal control method is shown to yield superior results to the conventional notion of independent modal space control according to reasonable assessment

Active vibration control of rotating machines

Second order matrix equations arise in the description of real dynamical systems. Traditional modal control approaches utilise the eigenvectors of the undamped system to diagonalise the system matrices. Any remaining off-diagonal terms in the modal damping matrix are discarded. A regrettable automatic consequence of this action is the destruction of any notion of the skew-symmetry in the damping. The methods presented in this thesis use the `Lancaster Augmented Matrices' (LAMs) allowing state space representations of the second order systems. `Structure preserving transformations' (SPTs) are used to manipulate the system matrices whilst preserving the structure within the LAMs. Utilisation of the SPTs permits the diagonalisation of the system mass, damping and stiffness matrices for non-classically damped systems. Thus a modal control method is presented in this thesis which exploits this diagonalisation. The method introduces independent modal control in which a separate modal controller is designed in modal space for each individual mode or pair of modes. The modal displacements and velocities for the diagonalised systems are extracted from the physical quantities using first order SPT-based filters. Similarly the first order filters are used to translate the modal force into the physical domain. Derivation of the SPT-filters is presented together with a method by which one exploits the non-uniqueness of the diagonalising filters such that initially unstable filters are stabilised. In the context of active control of rotating machines, standard optimal controller methods enable a trade-off to be made between (weighted) mean-square vibrations and (weighted) mean-square control forces, or in the case of a machines controlled using magnetic bearings the currents injected into the magnetic bearings. One shortcoming of such controllers for magnetic bearings is that no concern is devoted to the voltages required. In practice, the voltage available imposes a strict limitation on the maximum possible rate of change of control force (force slew rate). This thesis presents a method which removes the aforementioned existing shortcomings of traditional optimal control. Case studies of realistic rotor systems are presented to illustrate the modal control and control force rate penalisation methods. The system damping matrices of the case studies contain skew-symmetric components due to gyroscopic forces typical of rotating machines. The SPT-based modal control method is used to decouple the non-classically damped equations of motion into n single degree of freedom systems. Optimal modal controllers are designed independently in the modal space such that the modal state, modal forces and modal force rates are weighted as required. The SPT-based modal control method is shown to yield superior results to the conventional notion of independent modal space control according to reasonable assessment