By M.A. Laughton, M.G. Say
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19) is a direct consequence of the diagonal dominance of We and Wo' and the fact that the eigenvalues of the grammian product are invariant. The profiles of grammians and system matrix A in modal coordinates are drawn in Fig. 1. , We == E(xx T ). Thus, the diagonal dominance results in the following conclusion that under white noise or impulse, excitation modes are almost independent, or almost orthogonal. 1. 0417 Indeed, the grammians are diagonally dominant. 2922, Wo2 Controllability and Observability System matrix A D Values: • 39 Grammians D zero, small, large.
This is done not only for the sake of analytical simplicity, but also because it is the most favorable way to measure or estimate it. This is the way, for example, to estimate the material damping in the finite-element analysis of large flexible structures. The resulting damping is a proportional one. In another approach, a damping matrix proportional either to the mass, or to the stiffness matrix, or to both, is introduced. This technique produces proportional damping as well. c: -400 a. 1. Transfer function G of the simple system (solid line) and its modes G J (dashed line), G 2 (dash-dot line), and G3 (dotted line).
For example, the number of degrees of freedom of the finite-element model is very large. Therefore the nodal state representation is seldom used in engineering practice. 12). 28) In this representation xI is the vector of the modal displacements, and x2 is the vector of modal velocities. , from a finite-element model), the damping matrix is commonly evaluated in the modal coordinates. Usually, the damping estimation is more accurate in modal coordinates. 4. 737 22 Chapter 2 Presented above, state-space representation IS not a modal state representation (although it was obtained using modal displacements, qm)' The modal state-space representation has a triple (Am' Bm, Cm) characterized by the block-diagonal matrix, Am' and the related input and output matrices Am = diag( Ami), i = 1, 2, ...