Application to flexible laboratory structure
5.3 Identification of modal model for the structure
5.3.6 Modal model of structure
The final step in the construction of a modal model for the system is to combine the modal identification results from the tests for each actuator into a 3-input/3- output model of the system. This can be accomplished by appending together three modal models, where each of the modal models is identified from the measured response of the structure to a single actuator in turn. Standard model reduction techniques could be used to reduce this system to the 9 structural (i.e., 18 states) of the structure, although the neat arrangement of the modal system matrices as
Table 5.1 "Pre-move" modal property data identified using input VCA 1.
Mode Wr (r ¢lr) cp~r) ¢¥) '1/J~r)
(Hz) %
FLX 1 0.872 0.878 -0.168 -0.623 0.764 0.041 FLX 1 0.893 1.327 0.778 -0.607 -0.159 -0.137 TOR 1 1.590 0.884 0.573 0.605 0.552 -0.144 FLX 2 2.250 2.845 0.708 -0.683 -0.181 0.303 FLX 2 2.292 3.001 -0.258 0.913 -0.316 0.145 TOR2 3.858 5.248 -0.299 -0.709 -0.639 0.070 FLX 3 3.598 1.212 0.676 -0.533 -0.508 0.102 FLX 3 3.613 1.800 -0.024 0.733 0.680 0.325 TOR3 5.993 1.590 0.587 0.588 0.556 -0.078
Table 5.2 "Pre-move" modal property data identified using input VCA 2.
Mode Wr (r ¢lr) cp~r) cp~r) '1/J~r) (Hz) %
FLX 1 0.869 0.669 -0.120 -0.622 0.774 -0.088 FLX 1 0.900 1.193 0.802 -0.531 -0.274 -0.087 TOR 1 1.586 0.780 0.595 0.592 0.544 0.174 FLX 2 2.257 3.148 0.672 -0.731 -0.120 0.303 FLX 2 2.303 2.786 -0.723 0.133 0.678 0.165 TOR2 3.763 3.924 -0.587 -0.809 0.020 -0.080 FLX 3 3.625 2.252 -0.327 -0.663 -0.674 0.615 FLX 3 3.630 2.313 -0.031 0.750 0.661 0.368 TOR3 5.977 1.700 0.585 0.565 0.582 0.086
Table 5.3 "Pre-move" modal property data identified using input VCA 3.
Mode Wr (r </Ar) </J~) </J~r) 1/J~r) (Hz) %
FLX 1 0.871 0.625 -0.260 -0.842 0.472 0.420 FLX 1 0.871 0.657 0.237 0.925 -0.300 0.334 TOR 1 1.587 0.765 0.593 0.614 0.521 0.145 FLX 2 2.294 2.164 0.755 -0.585 -0.296 0.099 FLX 2 2.302 2.097 0.026 0.695 -0.718 0.209 TOR2 3.626 2.309 -0.309 -0.688 -0.657 0.642 FLX 3 3.731 2.426 -0.254 -0.009 -0.967 -0.130 FLX 3 3.632 2.458 -0.006 0.751 0.661 0.379 TOR3 5.990 1.764 0.582 0.656 0.480 0.083
Table 5.4 "Post-move" modal property data identified using input VCA 1.
Mode Wr (r </J~r) </J~r) </J~) 1/J~r) (Hz) %
FLX 1 0.877 0.867 -0.926 0.034 0.375 -0.215 FLX 1 0.876 0.753 0.633 0.443 -0.634 -0.233 TOR 1 1.634 0.671 0.576 0.555 0.600 -0.146 FLX 2 2.199 2.785 -0.006 0.758 -0.653 0.199 FLX 2 2.300 2.121 -0.907 0.229 0.354 0.094 TOR2 3.673 2.646 0.589 0.436 0.681 0.546 FLX 3 3.733 4.232 -0.596 -0.245 -0.765 0.390 FLX 3 3.893 1.885 0.511 -0.604 0.611 0.128 TOR3 6.334 1.027 0.547 0.572 0.612 -0.077
Table 5.5 "Post-move" modal property data identified using input VCA 2.
Mode Wr (r ¢lr) ¢~r) ¢~r) 'lj;~r)
(Hz) %
FLX 1 0.879 0.806 -0.047 -0.679 0.733 0.181 FLX 1 0.884 0.370 -0.583 -0.782 -0.220 0.076 TOR1 1.633 0.640 0.573 0.568 0.591 0.137 FLX 2 2.204 2.836 0.151 0.681 -0.716 0.212 FLX 2 2.299 2.244 0.877 -0.468 -0.113 0.074 TOR2 3.661 1.644 0.801 0.586 0.127 -0.372 FLX 3 3.661 1.613 0.787 0.340 -0.515 0.191 FLX 3 3.924 1.570 0.319 0.270 -0.908 -0.072 TOR3 6.330 1.023 0.542 0.567 0.621 0.080
Table 5.6 "Post-move" modal property data identified using input VCA 3.
Mode Wr (r ¢ir) ¢~r) ¢~r) 'lj;~r)
(Hz) %
FLX 1 0.881 0.496 0.792 0.165 0.589 0.052 FLX 1 0.882 0.458 0.549 -0.441 -0.710 0.132 TOR1 1.632 0.565 0.569 0.564 0.598 0.185 FLX 2 2.278 2.133 0.704 0.469 -0.533 -0.329 FLX 2 2.281 1.968 0.065 0.872 -0.485 0.235 TOR2 3.659 1.464 0.469 0.366 0.804 -0.373 FLX3 3.658 1.496 0.247 -0.093 0.965 0.157 FLX3 3.952 1.355 0.985 -0.022 -0.171 0.082 TOR3 6.326 1.052 0.553 0.569 0.609 0.079
sub-components of the state matrices is lost in the reduction. For the modal model that is constructed in this section, no model reduction is performed, and the model that is used is created by appending the modal model that predicts the acceleration response to input at VCA 1 to the modal model created using VCA 2 as input, yielding a 36-state system. For the experiments discussed in this chapter, VCA 1 is used exclusively to provide the system excitation, and VCA 2 is used exclusively for control purposes. Note that the simplest method would be to excite all three actuators at once, in order to simultaneously identify the participation factors for each mode. However, this was not performed prior to the relocation of the structure, so this method could not be used (at least for the pre-move model).
The modal equations for the identified model can be used for control design purposes. Consider the modal system excited by the pth input VCA and controlled by the qth VCA,
(5.2)
where A= diag (wt,W2, ... ,wg) is the diagonal matrix of modal frequencies,
e
=diag (2(twl, 2(2w2, ... , 2(gwg) is the linear viscous damping matrix, and '1/Jp and '1/Jq are the pth and qth columns of the modal participation factor matrix, which has one column for each VCA input. The input ground motion, which is translated into a force from the pth VCA that acts on the structure, is given by w9(t) E IR, and the feedback control force acting through the qth VCA is given by u(t) E IR. The input ground motion used for the controller design is a Kanai-Tajimi linear filter (Clough and Penzien 1975) with natural frequency of 50 rad/sec and a damping ratio of 50%, similar to the one described in Section 4.3.
The state equation for the uncontrolled modal model based on the rth VCA
input is
(5.3)
where Yr(t) E 1R3 represents the measured acceleration outputs and zr(t) E 1R6 are the performance variables for VCA input r. The matrices A, 8, and \1! from (5.2) are subscripted with an r as well to indicate the modal model identified using excitation from a particular actuator. The matrix of partial modeshape factors is given by
<I> r E 1R9 x 3. For the experimental studies performed throughout this chapter, the 1st
VCA provides the system excitation, and the 2nd VCA is used for feedback control.
The 3rd VCA is not used.
The modal models that are identified for the two VCA inputs are used to model the response of the laboratory structure for controller analysis and design. Similar to the benchmark structure application (Chapter 4), a high-fidelity SIMULINK (1994b) model is used to accurately describe the laboratory system. The first ten seconds of the simulated acceleration response of the system to the identification input from Section 5.3.2, which is acting through VCA 1, is shown in Figure 5.3, and the SIMULINK (1994b) block diagram is pictured in Figure 5.4. The fast Fourier transforms (FFTs) of the simulated and measured response are shown in Figure 5.5, where the simulated response (solid line) and measured response (dashed line) nearly coincide for most of the modes of the structure (except for the first flexural mode).
For the first flexural mode, the peak of model response is much smaller than the one found from FFT of the response data. This could result from several factors.
The first is that the area under this peak is small, so the resulting best-fit model from MODE-ID may not be very sensitive to this first flexural mode peak. In addition, significant ambient vibrations could exist that the structure's response
to this "unmeasured" input is comparable to the forced response in this frequency range, which would cause noise in the modal properties that are identified for this mode.
The final state equations that are used for control design are
{5.4)
where y(t) E IR3 represents the measured outputs as before, v(t) E IR3 is the measure- ment noise and modeling error term, and z(t) E IR7 the performance outputs, which are the relative displacements and velocities at the measured locations as well as the actuator command signal, u(t) E IR. Note that the control effort is weighted by
"fu, and the measurement noise/modeling error v(t) is scaled by Vrms· Furthermore, the measured outputs for the controller design have no feed-through of the input or control force, as is standard in controller design.