Trans RINA, Vol 152, Part A4, Intl J Maritime Eng, Oct-Dec 2010
step pulse of unknown but high amplitude and short duration in a time domain of infinite extent. The actual shape of these pulses is not critical [9], and can be represented by rectangular ones enabling the Fourier pressure spectrum to be drawn. This is seen to be almost flat over a frequency range including several harmonics of bpf. The Fourier
frequency spectrum for such a
pressure pulse is well known and has the form sinx/x, where x = ωδτ, For the limiting case when δτ → 0 this function, known as the Dirac or delta function, takes the value of 1, or, in other words, the frequency spectrum is flat for all ω. For small finite pulse-time increments of δτ representative of blade passage times the Fourier frequency spectrum plotted against frequency has a very slowly drooping characteristic and vibration is just as likely to occur at all frequencies. In practice, however, vibration can be expected to occur when local resonances in the plating and inboard structure are excited.
For a solitary pressure pulse of 100μs duration, i.e.
midway between the values mentioned previously for the oscillating hydrofoil and the CVG experiments, the frequency spectrum remains effectively flat up to the 5th harmonic of blade rate and beyond.
Hull pressure pulses for the cavitating pictures in [6] are not provided, but the author gives a short dissertation on the subject concluding that to gain an understanding of cavitation rather more than a Fourier-based curve fitting algorithm is necessary. This view is contrary to that given in [9], for example, for simple mechanical shocks to which collapsing cloud cavities can be likened when the period of the pressure pulse, in the lower tens of microseconds, is small compared with the natural period of the ship’s stern structure of the order of 106 times this.
8.
HIGHER HARMONIC PRESSURE LEVELS
In the search of further evidence in support of the conclusion that it was the collapse of
the propeller
cavitating tip vortices that caused the broadband vibration on the ship, it is informative to review the distribution of the higher harmonic pressure
levels
measured on the model compared with their corresponding bpf levels. An inspection of these rms pressures
obtained for the two contending model
propeller designs at the ship’s main operating conditions, of which there were three, led to the following observations.
In nearly a quarter of the results the 2nd harmonic rms pressure levels were greater than the corresponding bpf levels, whilst in nearly a third of the results the 3rd harmonic pressure levels were greater than the 2nd but not necessarily greater than the corresponding bpf levels. Fig. 2 shows two fairly extreme examples of these results in which both 2nd and 3rd rms harmonic pressure levels were greater than the bpf levels. These harmonic pressure distributions are unusual and resulted presumably from
the shock pressures created by the collapsing cavitating vortices.
After the publication of [8] some scepticism may have existed concerning the experimental finding that higher harmonic pressure levels could exceed the blade rate values, but this was dispelled by the publication of [10], in which the author’s showed theoretically it was possible.
Figure. 2. Normalised Harmonic Pressures
An engineering means of gauging the activity of the higher harmonic pressures is to calculate the quadrature sums of the harmonic pressures for each transducer and compare these values with those obtained from tests on a similar ship model that did not
suffer a significant
propeller/cavitation induced vibration problem. This has been done using the following expression. With n = 5 this
led to maximum values close to 2 for many
transducers positions on the model of the as-built ship compared with a similar figure of 1 for the acceptable vessel.
p →n 1 2 9. CONCLUSIONS
• Experiments on the underwater form of a model high- speed twin screw displacement
ship in a large
cavitation tunnel produced evidence, both visual and measured, that led to a plausible explanation for the source of the broadband vibration on the ship.
A-178 ©2010: The Royal Institution of Naval Architects 1[] ∑ pn 2 1/ 2
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