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where
F = le F.r,e + Je Fy,e + ke Fz.e (4.273)
The above system ofequations describes the motion ofa point mass with the only
assumption that the orbital motion of the Earth around the sun is ignored. For the
majority of the fiight dynamics problems, this assumption is usually satisfactory.
However, for interplanetary motions like a mission to the planet Mars or other
distant planets, the orbital motion of the Earth has to be considered. The interested
reader may refer to a standard text on astrodynamics for this purpose.6
4.3.3 Force Equations for Aircraft Motion
In the study of aircraft flight dynamics,it is usual to ignore the rotation of the
Earth about its own axis.ln other words, an Earth-fixed axes system such as the
navigational system XeYeZe is assumed to bc an inertial frame ofreference.ln doing
so, weignore the terms S2<x :Qe X Re and 2(2e x Voe.ln other words, we assume
eob. b = C:Oeb,b and ( Vo)t - ( Vo)e. The error introduced by these assumptions can be
estimated as follows.
Consider an aircraft moving at a flight velocity of 1000 mph-(1466.37 ft/s) at sea
level (speed of sound r. 1100 ft/s). Let the flight path be contained in the equator
(A = 0). This flight velocity corresponds to a flight Mach number of approximately
1.34. With A - 0 in Eq. (4.268), we have
92e x ?2e x Re = k.Ref22
IQe X g2e X Rel = ReS2~
With Re ~ 2.097 x 10J ft and
SZe = 24~60 (2;7;)
- 7.2722 * 10-s rad/s
(4.274)
(4.275)
(4.276)
(4.277)
EQUATIONS OF MOTION AND ESTIMATION OF STABILITY DERIVATIVES 367
we obtain
IS2e x ~e X Rel - 2.097 x 107a.2722 *10-5)2 (4.278)
- 0.11089 ft/SZ
Similarly, from Eq. (4.269), with A - 0,
2S2e x Ve = 292e(- Je Voz + ke Voy)
21S2e x Vel ~ 2S2e Vo
N 2(7.2722 *10-5)1466.37
(4.279)
(4,280)
(4.281)
(4.282)
~ 0.2133 ft/S2 (4.283)
Thus, the errors caused by ignoring the Earth's rotation about its own axis are
relatively small for typical aircraft motions at subsonic and supersonic speeds.
However, these errors can become significant if the 11ight velocity increases.
With these assumptions, Eq. (4.259) reduces to
F = m (ao)e (4.284)
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