20.01.2013 Views

The Art of the Helicopter John Watkinson - Karatunov.net

The Art of the Helicopter John Watkinson - Karatunov.net

The Art of the Helicopter John Watkinson - Karatunov.net

SHOW MORE
SHOW LESS

You also want an ePaper? Increase the reach of your titles

YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.

Introduction to helicopter dynamics 105<br />

Fig. 3.36 (a) In a two-bladed rotor <strong>the</strong> blades flap with a 180 ◦ relationship. <strong>The</strong> fundamental flapping cancels,<br />

whereas <strong>the</strong> second harmonic flapping adds. Two bladed rotors are prone to 2P hop. (b) <strong>The</strong> third harmonic<br />

cancels in a two-bladed rotor.<br />

rotor, or indeed for a rotor with a higher number <strong>of</strong> blades. In Figure 3.37(b) is shown<br />

<strong>the</strong> effect for <strong>the</strong> third flapping harmonic. Here <strong>the</strong> third harmonic waveform is in <strong>the</strong><br />

same phase for each blade and so <strong>the</strong>re will be summation.<br />

Now that <strong>the</strong> general principle is clear, <strong>the</strong> result for any number <strong>of</strong> blades and any<br />

harmonic can readily be established. This is summarized in Figure 3.38(a) where it will<br />

be seen that <strong>the</strong> lowest hopping frequency is numerically equal to <strong>the</strong> number <strong>of</strong> blades<br />

and that <strong>the</strong>se frequencies may be multiplied by integers to find <strong>the</strong> higher hopping<br />

modes.<br />

When in-plane forces are considered, <strong>the</strong> results are quite different. Figure 3.38(b)<br />

shows <strong>the</strong> result <strong>of</strong> various harmonics with different numbers <strong>of</strong> blades. <strong>The</strong>se forces<br />

will be heterodyned by rotor speed so that <strong>the</strong> result on <strong>the</strong> hull <strong>of</strong> a rotor frequency nP<br />

will be vibration at (n − 1)P and (n + 1)P whereas <strong>the</strong> frequency P is absent. Following<br />

<strong>the</strong> principles <strong>of</strong> vector summation explained above, certain harmonics cancel at <strong>the</strong><br />

hub. Note that as with hop forces, <strong>the</strong> lowest frequencies at which rocking forces can<br />

exist are numerically <strong>the</strong> same as <strong>the</strong> number <strong>of</strong> blades, with <strong>the</strong> harmonics obtained by<br />

integer multiplication. <strong>The</strong> difference between Figure 3.38(a) and Figure 3.38(b) is that

Hooray! Your file is uploaded and ready to be published.

Saved successfully!

Ooh no, something went wrong!