4 R.J. Allemang and A.W. Phillips −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 0 50 100 150 200 250 Real Imag State Vector (DOF) Fig. 1.1 Eighth order, pole-weighted vector (state vector) example When comparing modal (base) vectors, at either the short or the long dimension, a pole weighted vector can be constructed independent of the original algorithm used to estimate the poles and modal (base) vectors. For a given order k of the pole weighted vector, the modal (base) vector and the associated pole can be used to formulate the pole weighted vector as follows: f gr D 8 ˆ ˆ ˆ ˆ ˆ< ˆ ˆ ˆ ˆ ˆ: k r f gr : : : 2 r f gr 1 r f gr 0 r f gr 9 > > > > >= > > > > >;r f gr D 8 ˆ ˆ ˆ ˆ ˆ< ˆ ˆ ˆ ˆ ˆ: zk r f gr : : : z2 r f gr z1 r f gr z0 r f gr 9 > > > > >= > > > > >;r (1.3) While the above formulation (on the left) is possible, this form would be dominated by the high order terms if actual frequency units are utilized. Generalized frequency concepts (frequency normalization or Z domain mapping) are normally used to minimize this issue by using the Z domain form (zr ) of the complex modal frequency (œr) as shown above (on the right). The Z domain form of the complex natural frequency is developed as follows: zr De . r= max/ (1.4) zmr De m . r= max/ (1.5) In the above equations, max can be chosen as needed to cause the positive and negative roots to wrap around the unit circle in the Z domain without overlapping (aliasing). Normally, max is taken to be five percent larger than the largest frequency (absolute value of the complex frequency) identified in the roots of the matrix coefficient polynomial. Figures 1.1 and 1.2 are graphical representations of the pole weighted vector (state vector) defined in Eq. 1.3. In this example, the modal (base) vector (at the bottom of Fig. 1.1) is a real-valued normal mode that looks like one period of a
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