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The Equations Collected
The kinematical and dynamical equations derived in the foregoing are collected below for convenience. The assumption that Cxz is a plane of symmetry is used, so that Ixv = Iyz = 0, and (4.6,2) are added to (4.5,9) to give (4.7,2).
’Note that the inertias of the rotors are also included in Is.
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X — mg sin в = т(йЕ + qwE — rvE) |
(a) |
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Y + mg cos в sin ф = m(vE + ruE — pwE) |
(b) |
(4.7,1) |
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Z + mg cos в cos ф = m(wE + pvE — quE) |
(c) |
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L = IJ> – Izxr + qr(L – Iy) – Lxpq + qh’z – rh’y |
(a) |
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M = Iyq + rp(Ix – Iz) + IJp2 – r2) + rh’x – ph’z |
(b) |
(4.7,2) |
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N = Izr- Izxp + pq(Iy – lx) + Izxqr + ph’y – qh’x |
(c) |
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p = ф — ф sin в |
(a) |
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q = в cos ф + ф cos в sin ф |
(b) |
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r = ф cos в cos ф — 0 sin ф |
(c) |
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ф = p + (q sin ф + r cos ф) tan в |
(d) |
(4.7,3) |
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в = q cos ф — r sin ф |
(e) |
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ф = (</ sin ф + r cos ф) sec в |
(f) |
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xE = uE cos в cos ф + u£(sin ф sin в cos ф — cos ф sin Ф) + w£(cos ф sin в cos ф + sin ф sin Ф) |
(a) |
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yE = uE cos в sin ф + u£(sin ф sin в sin ф + COS Ф COS Ф) |
(4.7,4) |
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+ w£(cos ф sin в sin ф — sin Ф COS Ф) |
(b) |
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ZE = —UE sin в + vE sin ф cos в + wE COS Ф cos в |
(c) |
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uE = u + Wx |
(a) |
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Vе = V + Wy |
(b) |
•(4.7,5) |
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wE = w + Wz |
(c) |