Eigenvalue Assignment Problems in Vibration Using Measured Receptances: Passive Modification and Active Control
Matrix methods using mass, damping and stiffness terms are widely used in vibration analysis of structure and provide the basis for active control of vibrations using state-space methods. However modern vibration test procedures provide reliable receptanc
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partment of Engineering, University of Liverpool, United Kingdom † Department of Mechanical Engineering Louisiana State University, Baton Rouge, USA
Abstract Matrix methods using mass, damping and stiffness terms are widely used in vibration analysis of structure and provide the basis for active control of vibrations using state-space methods. However modern vibration test procedures provide reliable receptance data with usually much greater accuracy than is achievable from system matrices, M, C, K, formed from finite elements. In this article procedures are described for pole placement by passive modification and active control using measured receptances. The theoretical basis of the method is described and experimental implementation is explained.
1
Introduction
Receptances were first used in the passive modification of structures by Duncan (1941) and the pole assigment problem was described as a classical inverse problem in active control by Kautsky et al. (1985). In this article the use of measured receptances for pole assignment in passive modification and active control is described. The considerable advantages of this approach are explained. In conventional modelling the dynamic behaviour of a structure is determined from mass, damping and stiffness matrices, M, C, K, usually obtained from finite elements. In theory the receptance matrix, which is measurable experimentally, is the inverse of the dynamic stiffness matrix, H(s) = Z−1 (s)
(1)
Z(s) = s2 M + sC + K
(2)
G. M. L. Gladwell et al. (eds.), Dynamical Inverse Problems: Theory and Application © CISM, Udine 2011
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J.E. Mottershead, M.G. Tehrani and Y.M. Ram
In reality finite element models contain approximations, assumptions and errors not present in measured data. The terms in the receptance matrix, H(s), are dominated by the eigenvalues (or poles) closest to the frequency of excitation, n ϕk ϕTk ϕ∗ ϕ∗T ( + k k∗) (3) H(s) = s − λk s − λk k=1
th
where ϕk denotes the k eigenvector. It may be shown, however, that the dynamic stiffness terms, in Z(s), are dominated by the poles furthest away - usually the higher modes, most affected by discretisation and the least accurate from a finite element model. We see that H(s) may be represented very accurately by a truncated set of modes whereas Z(s) generally may not be. Furthermore,the problem of dynamic-stiffness error cannot be overcome by inverting the accurately measured matrix of receptances. Within the limited frequency range of a vibration test, and when s = iω, the inversion of H(iω) is not able to reproduce Z(iω) because the effect of the high-frequency modes is negligibly small, within the measurement noise (Berman and Flannelly, 1971). The receptance terms may be expressed by the ratio of two polynomials, H(s) =
N(s) d(s)
(4)
where d(λk ) = 0 is the characteristic equation of the original system, with eigenvalues λk , k = 1, 2, . . . , 2n. The ij th term in N(s), provides the characteristic equation of the zeros, nij (ξk ) = 0, k = 1, 2, . . . , 2m, m ≤ n. This means that the vibration response at c
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