Chapter 5 Robust Performance Tailoring with Tuning - SSL - MIT

energy is distributed almost evenly among the first (ACS), third and eleventh (second
axial) modes. The ACS mode and second axial mode contribute a small percentage
to the output in the nominal design, but account for 38% and 26% of the energy,
respectively, in the PT design. Comparing the natural frequencies, it is found that
the PT design frequencies are at least a factor of two lower than those of the nominal
design. The lower frequencies make sense considering the fact that the cross-sectional
areas of the beam have decreased greatly from the nominal system (0.10 m) to the
PT design (0.03 m). The cross-section diameters map directly to area and inertia
(Equations 2.31 and 2.30) which in turn map to the stiffness matrix (Equation 2.2).
Lowering the cross-sectional diameters lowers the global stiffness matrix of the system
and, in turn, the natural frequencies.
The final step in understanding the differences between the two systems is to ex-
amine the mode shapes of the critical modes, plotted in Figure 2-8. The ACS mode
changes between the two systems (Figure 2-8(a)) partly due to the change in inertia,
which in turn effects the modal frequency. The shape of this mode (Figure 2-8(a))
is similar in the nominal and PT design but its frequency is much lower in the tai-
lored case (PT). There is a large difference in percent energy contribution from this
mode between the two systems (from 14% to 38.5%), but the absolute RMS values
accumulated in the motion are of similar magnitude (59 and 39µm).
The major improvement in performance is due to the tailoring of the second
bending mode (Figure 2-8(b)). Notice that in the nominal design the two outer nodal
points, or points of zero deflection, are located slightly away from the endpoints of
the array towards the center. However, in the PT design, these nodal points are right
at the ends of the array, where the collectors are located. Since the collector motions
feature prominently in the OPD equation (Equation 2.9), tailoring the mode shape
such that the nodal points are located at the collectors significantly reduces the OPD
output in this mode, and, as a result, the total RMS OPD.
The second and fourth modes (Figures 2-8(c) and 2-8(d)) are account for zero
energy in the nominal case, but a finite (although small) amount of energy in the PT
design. In the nominal design the system is perfectly symmetric so these modes are
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