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.

X — mg sin в = т(йЕ + qwE — rvE)

(a)

Y + mg cos в sin ф = m(vE + ruE — pwE)

(b)

(4.7,1)

Z + mg cos в cos ф = m(wE + pvE — quE)

(c)

L = IJ> – Izxr + qr(L – Iy) – Lxpq + qh’z – rh’y

(a)

M = Iyq + rp(Ix – Iz) + IJp2 – r2) + rh’x – ph’z

(b)

(4.7,2)

N = Izr- Izxp + pq(Iy – lx) + Izxqr + ph’y – qh’x

(c)

p = ф — ф sin в

(a)

q = в cos ф + ф cos в sin ф

(b)

r = ф cos в cos ф — 0 sin ф

(c)

ф = p + (q sin ф + r cos ф) tan в

(d)

(4.7,3)

в = q cos ф — r sin ф

(e)

ф = (</ sin ф + r cos ф) sec в

(f)

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)

yE = uE cos в sin ф + u£(sin ф sin в sin ф + COS Ф COS Ф)

(4.7,4)

+ w£(cos ф sin в sin ф — sin Ф COS Ф)

(b)

ZE = —UE sin в + vE sin ф cos в + wE COS Ф cos в

(c)

uE = u + Wx

(a)

Vе = V + Wy

(b)

•(4.7,5)

wE = w + Wz

(c)

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