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Papers by Seamus Garvey
International Journal of Environmental Studies, 2014
ABSTRACT
2015 18th International Conference on Electrical Machines and Systems (ICEMS), 2015
The most common way of solving the quadratic eigenvalue problem (QEP) (λ 2 M +λD +K)x = 0 is to c... more The most common way of solving the quadratic eigenvalue problem (QEP) (λ 2 M +λD +K)x = 0 is to convert it into a linear problem (λX + Y )z = 0 of twice the dimension and solve the linear problem by the QZ algorithm or a Krylov method. In doing so, it is important to understand the influence of the linearization process on the accuracy and stability of the computed solution. We discuss these issues for three particular linearizations: the standard companion linearization and two linearizations that preserve symmetry in the problem. For illustration we employ a model QEP describing the motion of a beam simply supported at both ends and damped at the midpoint. We show that the above linearizations lead to poor numerical results for the beam problem, but that a two-parameter scaling proposed by Fan, Lin and Van Dooren cures the instabilities. We also show that half of the eigenvalues of the beam QEP are pure imaginary and are eigenvalues of the undamped problem. Our analysis makes use of recently developed theory explaining the sensitivity and stability of linearizations, the main conclusions of which are summarized. As well as arguing that scaling should routinely be used, we give guidance on how to choose a linearization and illustrate the practical value of condition numbers and backward errors.
Given a pair of distinct eigenvalues (λ 1 , λ 2 ) of an n×n quadratic matrix polynomial Q(λ) with... more Given a pair of distinct eigenvalues (λ 1 , λ 2 ) of an n×n quadratic matrix polynomial Q(λ) with nonsingular leading coefficient and their corresponding eigenvectors, we show how to transform Q(λ) into a quadratic of the form Q d (λ)
When the dynamics of any general second order system are cast in a state-space format, the initia... more When the dynamics of any general second order system are cast in a state-space format, the initial choice of the state-vector usually comprises one partition representing system displacements and another representing system velocities. Coordinate transformations can be defined which result in more general definitions of the state-vector. This paper discusses the general case of coordinate transformations of state-space representations for second order systems. It identifies one extremely important subset of such coordinate transformations -namely the set of structure-preserving transformations for second order systems -and it highlights the importance of these. It shows that one particular structure-preserving transformation results in a new system characterised by real diagonal matrices and presents a forceful case that this structure-preserving transformation should be considered to be the fundamental definition for the characteristic behaviour of general second order systems -in preference to the eigenvalue-eigenvector solutions conventionally accepted.
Static (Guyan) reduction is one of the kingpins upon which most large-scale Finite-Element dynami... more Static (Guyan) reduction is one of the kingpins upon which most large-scale Finite-Element dynamic analyses depend. Using Guyan reduction, it is possible to discard some degrees of freedom from the model during the elementmerge process and thereby prevent the system matrices becoming excessively large. The reduction works extremely well for undamped or very lightly damped structures. For structures in which the damping is higher, it is obviously inappropriate because the reduction transformation developed is completely oblivious to any damping terms. Possible solutions to this problem have been proposed but all have notable shortcomings. This paper proposes that a powerful reduction method can exist for general linear second-order systems which can be considered to be the logical extension of Guyan reduction for systems having significant damping terms. The method rests on the use of elementary structure-preserving coordinate transformations for second-order systems.
Second-order matrix equations arise in the description of real dynamical systems. Traditional mod... more 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. A regrettable consequence of this approach is the discarding of residual off-diagonal terms in the modal damping matrix. This has particular importance for systems containing skew-symmetry in the damping matrix which is entirely discarded in the modal damping matrix. In this paper, a method to utilise modal control using the decoupled second-order matrix equations involving non-classical damping is proposed. An example of modal control successfully applied to a rotating system is presented in which the system damping matrix contains skew-symmetric components. r
Archives of Electrical Engineering, 2015
Journal of Energy Storage, 2015
Proceedings of 1995 American Control Conference Acc 95, Jun 21, 1995
Volume 5: 19th Biennial Conference on Mechanical Vibration and Noise, Parts A, B, and C, 2003
ABSTRACT
International Journal of Environmental Studies, 2014
ABSTRACT
2015 18th International Conference on Electrical Machines and Systems (ICEMS), 2015
The most common way of solving the quadratic eigenvalue problem (QEP) (λ 2 M +λD +K)x = 0 is to c... more The most common way of solving the quadratic eigenvalue problem (QEP) (λ 2 M +λD +K)x = 0 is to convert it into a linear problem (λX + Y )z = 0 of twice the dimension and solve the linear problem by the QZ algorithm or a Krylov method. In doing so, it is important to understand the influence of the linearization process on the accuracy and stability of the computed solution. We discuss these issues for three particular linearizations: the standard companion linearization and two linearizations that preserve symmetry in the problem. For illustration we employ a model QEP describing the motion of a beam simply supported at both ends and damped at the midpoint. We show that the above linearizations lead to poor numerical results for the beam problem, but that a two-parameter scaling proposed by Fan, Lin and Van Dooren cures the instabilities. We also show that half of the eigenvalues of the beam QEP are pure imaginary and are eigenvalues of the undamped problem. Our analysis makes use of recently developed theory explaining the sensitivity and stability of linearizations, the main conclusions of which are summarized. As well as arguing that scaling should routinely be used, we give guidance on how to choose a linearization and illustrate the practical value of condition numbers and backward errors.
Given a pair of distinct eigenvalues (λ 1 , λ 2 ) of an n×n quadratic matrix polynomial Q(λ) with... more Given a pair of distinct eigenvalues (λ 1 , λ 2 ) of an n×n quadratic matrix polynomial Q(λ) with nonsingular leading coefficient and their corresponding eigenvectors, we show how to transform Q(λ) into a quadratic of the form Q d (λ)
When the dynamics of any general second order system are cast in a state-space format, the initia... more When the dynamics of any general second order system are cast in a state-space format, the initial choice of the state-vector usually comprises one partition representing system displacements and another representing system velocities. Coordinate transformations can be defined which result in more general definitions of the state-vector. This paper discusses the general case of coordinate transformations of state-space representations for second order systems. It identifies one extremely important subset of such coordinate transformations -namely the set of structure-preserving transformations for second order systems -and it highlights the importance of these. It shows that one particular structure-preserving transformation results in a new system characterised by real diagonal matrices and presents a forceful case that this structure-preserving transformation should be considered to be the fundamental definition for the characteristic behaviour of general second order systems -in preference to the eigenvalue-eigenvector solutions conventionally accepted.
Static (Guyan) reduction is one of the kingpins upon which most large-scale Finite-Element dynami... more Static (Guyan) reduction is one of the kingpins upon which most large-scale Finite-Element dynamic analyses depend. Using Guyan reduction, it is possible to discard some degrees of freedom from the model during the elementmerge process and thereby prevent the system matrices becoming excessively large. The reduction works extremely well for undamped or very lightly damped structures. For structures in which the damping is higher, it is obviously inappropriate because the reduction transformation developed is completely oblivious to any damping terms. Possible solutions to this problem have been proposed but all have notable shortcomings. This paper proposes that a powerful reduction method can exist for general linear second-order systems which can be considered to be the logical extension of Guyan reduction for systems having significant damping terms. The method rests on the use of elementary structure-preserving coordinate transformations for second-order systems.
Second-order matrix equations arise in the description of real dynamical systems. Traditional mod... more 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. A regrettable consequence of this approach is the discarding of residual off-diagonal terms in the modal damping matrix. This has particular importance for systems containing skew-symmetry in the damping matrix which is entirely discarded in the modal damping matrix. In this paper, a method to utilise modal control using the decoupled second-order matrix equations involving non-classical damping is proposed. An example of modal control successfully applied to a rotating system is presented in which the system damping matrix contains skew-symmetric components. r
Archives of Electrical Engineering, 2015
Journal of Energy Storage, 2015
Proceedings of 1995 American Control Conference Acc 95, Jun 21, 1995
Volume 5: 19th Biennial Conference on Mechanical Vibration and Noise, Parts A, B, and C, 2003
ABSTRACT