Uwe Prells

Isospectral Vibrating Systems, Part 2: Structure Preserving Transformations

The study of inverse problems for n × n-systems of the form L(λ) ≔ Mλ2 + Dλ + K is continued. In this paper it is assumed that one vibrating system is specified and the objective is to generate isospectral families of systems, i.e., systems which reproduce precisely the eigenvalues of the given system together with their multiplicities. Two central ideas are developed and used, namely, standard triples of matrices, and structure preserving transformations.

Partial derivatives of repeated eigenvalues and their eigenvectors

The analysis of inverse problems in linear modeling often require the sensitivities of the eigenvalues and eigenvectors. The calculation of these sensitivities is mathematically related to the corresponding partial derivatives, which do not exist for any parameterization. Inasmuch as eigenvalues and eigenvectors are coupled by the constitutional equation of the general eigenvalue problem, their derivatives are coupled, too. Conditions on the parameterization are derived and formulated as theorems, which ensure the existence of the partial derivatives of the eigenvalues and eigenvectors with respect to these parameters. The application of the theorems is demonstrated by examples.

A Relationship Between Defective Systems and Unit-Rank Modification of Classical Damping

A common assumption within the mathematical modeling of vibrating elastomechanical system is that the damping matrix can be diagonalized by the modal matrix of the undamped model. These damping models are sometimes called classical or proportional. Moreover it is well known that in case of a repeated eigenvalue of multiplicity m, there may not exist a full sub-basis of m linearly independent eigenvectors. These systems are generally termed defective. This technical brief addresses a relation between a unit-rank modification of a classical damping matrix and defective systems. It is demonstrated that if a rankone modification of the damping matrix leads to a repeated eigenvalue, which is not an eigenvalue of the unmodified system, then the modified system is defective. Therefore defective systems are much more common in mechanical systems with general viscous damping than previously thought, and this conclusion should provide strong motivation for more detailed study of defective systems.

Estimating turbogenerator foundation parameters

Abstract Turbogenerators in power stations are often placed on foundation structures that are flexible over the running range of the machine and can therefore contribute to its dynamics. Established methods of obtaining structural models for these foundations, such as the finite element method or modal testing, have proved unsuccessful because of complexity or cost. Another method of foundation system identification, using the unbalance excitation applied by the rotor itself during maintenance run-downs, has previously been proposed but has not yet been experimentally verified. In this paper the necessary theory is developed and certain issues critical to the success of the estimation are examined. The method is tested in both simulation and experiment using a two-bearing rotor rig and good fits between model and measurement are obtained. The predictive capacity of the estimated models when the system is excited with a different unbalance is not as good, and it is surmised that this may be due among other things to inaccurate bearing models.

Use of geometric algebra: compound matrices and the determinant of the sum of two matrices

In this paper we demonstrate the capabilities of geometric algebra by the derivation of a formula for the determinant of the sum of two matrices in which both matrices are separated in the sense that the resulting expression consists of a sum of traces of products of their compound matrices. For the derivation we introduce a vector of Grassmann elements associated with an arbitrary square matrix, we recall the concept of compound matrices and summarize some of their properties. This paper introduces a new derivation and interpretation of the relationship between p–forms and the pth compound matrix, and it demonstrates the use of geometric algebra, which has the potential to be applied to a wide range of problems.

Calculating derivatives of repeated and nonrepeated eigenvalues without explicit use of eigenvectors

The analysis of inverse problems in parametric model updating often require the sensitivities of eigenvalues. The calculation of these sensitivities is mathematically related to the derivatives of the eigenvalues with respect to the model parameters. A common method to calculate these derivatives is the Nelson method, which requires the eigenvectors. The method introduced in this paper is derived from the characteristic equation of the underlying general eigenvalue problem and allows the derivatives of eigenvalues with respect to the model parameters to be calculated without explicit use of the eigenvectors. The method is extended for the case of repeated eigenvalues, which leads to restrictions on the parameterization. For repeated eigenvalues of multiplicity two, these restrictions are formulated expicitly. Applications and limitations of the method are demonstrated by examples.

Compound Matrices and Pfaffians: A Representation of Geometric Algebra

We consider the Clifford algebra Cl n (F) where the field F is the real R or the complex numbers C. It is well known that an m-form x 1/\⋯/\ x m can be represented by the mth compound matrix of the n-by-m matrix X := [x 1, ⋯, x m ] ∈ F n×m relative to the basis {θ 1, θ 2, ⋯, θ 2n } 2254; 1, e 1, ⋯, e 1Λ⋯Λe n } of the underlying Grassmann algebra G n (F). Since the Clifford product ● is related to the Grassmann product A via x ● y = x Λ y + x T y, x, y ∈ F n , the question of a corresponding representation of the Clifford product x 1 ·⋯· x m arises in a natural way. We will show that the Clifford product of an odd (even) number of vectors corresponds to a linear combination of forms of odd (even) grade where the coefficients of these linear combinations are Pfaffians of certain matrices which can be understood as the skew symmetric counterpart of the corresponding Gramians. Based on this representation we calculate the mth Clifford power \( \underline x ^m :{\rm{ = }}\overbrace {x \bullet \cdots \bullet x}^m \) of a vector x ∈ F n which enables the extension of an analytical function f : F → F to their corresponding Clifford function f:F n → Cl n(F).

Decoupling the Equations of Isospectral Flow

Three (m × n) matrices {K, D, M} represent a second-order system in the form (K + Dλ+ Mλ2). If m = n, system eigenvalues are defined as the values of λ for which det(K + Dλ+ Mλ2) = 0. If {K, D, M} are continuous functions of a real scalar parameter, σ, eigenvalues and dimensions of the associated eigenspaces remain constant if and only if the rates of change of {K, D, M} obey certain ODEs called the isospectral flow equations. The integration of these matrix differential equations is of interest here. This paper explains the motivation behind this work in terms of vibrating systems and it reports two related hypotheses concerning how the solutions to these equations may be decoupled. Work underway towards proving and using these hypotheses is presented. No existing known solutions allow this decoupling in general.

STRUCTURE PRESERVING TRANSFORMATIONS AND ISOSPECTRAL FLOWS FOR SECOND ORDER SYSTEMS

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. The regular λ-matrix λ2 M + λD + K with M,D,K ∊ R n×n defines a second-order system. A one-parameter trajectory of such a system {M(t),D(t),K(t)} is an isospectral flow (or more correctly an equivalence flow) if the eigenvalues and the dimensions of the associated eigenspaces are the same for all parameter values t ∊ R . This paper presents the general form for real isospectral flows of real-valued second order systems.Copyright

The Role of Clifford Algebra in Structure-Preserving Transformations for Second-Order Systems

Second-order dynamic systems described by the equation \( Kx + C\dot x + M\ddot x = F \) (where the dots indicate differentiation with respect to time) are of immense importance in engineering. The system matrices, K, C, M, are real (N × N) matrices and the vectors of displacement and force x, F each contain N functions of time. For many of the analyses performed on these systems, a generalised eigenvalue problem involving two (2N × 2N) matrices is set up and solved. It is common that most or all of the resulting eigenvalue-eigenvector pairs are complex. The numerical methods currently used for solving this generalised eigenvalue problem (GEP) do not take full advantage of its very particular structure. In particular, they do not provide any way to capitalise on the symmetry very often present in K, C, M [1]. Moreover, the structure in this problem results in constraints on the eigensolutions which make it possible to store those solutions more compactly but these constraints are generally ignored. There is compelling evidence that a more natural approach is possible. The role of Clifford Algebra in this more natural approach is examined.