5. EXPERIMENTAL CHECK ON MATHEMATICAL MODEL
least expensive,
THE
A complete experimental verification would be rather difficult, or at
perhaps needing an
inertial navigation system on board the vessel to give simultaneous measurement of v, p, r and , as well as rudder angle and forward speed. However a partial verification has been attempted, by measuring and recording:
a) rudder angle δ, with a potentiometer; b) heading , with a directional gyroscope ( = r); c) forward speed u0, approximately, by generating a pulse for every revolution of the chainwheel (= 5.556 propeller revolutions).
.
Two vessels were taken over a deliberately sinuous route at speeds varying between about 1.9 and 2.5m/s, and samples of δ and were recorded every 20ms. After reading them back, both signals were digitally low-pass filtered to a bandwidth of about 5Hz, and the signal was differentiated to give r. Figure 8 shows typical traces of δ, and r for the vessel to which the Table 1 data refer.
deliberate course changes, while the higher frequencies are the response to measurement noise (or
disturbances), as described above. The “noise” in δ and r is surprisingly large, a good deal more than in the simulation.
by the need to steer a weaving course. It may have been enhanced psychologically
The low-frequency variations represent the possibly
This gain is proportional to speed, as one might expect, so to allow for other speeds one could write it as:
r = 0.695(u0/2.2) δ or r/u0 = 0.316 m-1 δ (15)
One would not expect r to respond to changes in δ instantaneously, and an estimate of the theoretical delay can be got by putting s = j in the transfer-function, then expanding it in powers of and ignoring all but the first power. This gives, at 2.2m/s:
r = 0.695(1 j0.083) δ (16) In other words, r is expected to lag by 83ms. In fact
the experimental -samples were taken 10ms after the associated -samples, so the traces should theoretically show a 73ms delay.
It is clearly not possible to see any evidence of a δ-to-r time shift in the traces of Figure 8. Even if one makes an expanded plot of a short section, it still does not seem possible to make any meaningful measurement of it. The yaw-rate does seem to respond almost instantaneously to the rudder.
But it is possible to measure the gain. The speed varied somewhat throughout the run, so it was split into 10- second sections. For each section average speed and the average value of dr/d were calculated, and Figure 9 shows a scatter plot of the points. They may be seen to be not inconsistent with the slope of 0.316m-1 deduced in equation 15.
Figure 8: Rudder angle, heading and yaw-rate from an experimental record. Yaw-rate is offset for clarity
Equation 12 gives the theoretical transfer-function from rudder angle to yaw-rate for the vessel of Table 1 and Figure 4. The relationship between δ and r will not be affected by the feedbacks or sensor noise (though it would be by external
disturbances).
The transfer-
function is too complicated to attempt a complete comparison with noisy experimental results, but we can compare their low-frequency behaviour. Putting s = 0 in the transfer-function gives the zero-frequency gain at 2.2m/s as:
r = 0.695s-1 δ (14)
Figure 9: Comparison of experimental and theoretical values of rudder gain
Similar tests were done with the vessel in Figure 3, but gave dr/d consistently below the computed value (on average about 82% of it). The reason is thought to be that “Daring” has a deep sharply vee’d bow. When this bow is dragged sideways through the water by the front
©2007: Royal Institution of Naval Architects
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