Impact of fuel supply impedance and fuel staging on gas turbine ...

Acoustics
2
1.5
|det(S)|
1
0.5
0
4600
4400
4200 4000 3800
ω real
[rad/s]
3600
200
0
ω im
[rad/s]
−200
Figure 3.7: Example <strong>of</strong> a search path with one initial estimate to determine
the minimum <strong>of</strong>|det(S)|→0
advantage <strong>of</strong> this unconstrained non-linear optimization is that it does not
require numerical or analytical gradients. The method requires an initial estimate
<strong>of</strong> the variables, the real <strong>and</strong> imaginary part <strong>of</strong> the frequency (ω real ,ω im ),
<strong>and</strong> calculates the corresponding starting simplex. For each new iteration a
point close to the current simplex is determined. If the function value <strong>of</strong> this
new point is smaller than the value <strong>of</strong> the current simplex it is replaced by
the new one. The search is stopped by reaching a user-defined tolerance level.
Figure 3.7 shows the calculation from one starting point. For a wide frequency
range several initial estimates <strong>and</strong> therefore several calculations have to be
performed. The amount <strong>of</strong> calculations depends on the estimated number <strong>of</strong>
eigenfrequencies <strong>of</strong> the system. Here, care has to be taken in choosing good
initial estimates to ensure that every eigenfrequency can be found by the rou-
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