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system due to rotation of the Earth.
(ao)e: acceleration of the body relative to an Earth-fixed system OxeYeze.
292e x V:,: Coriolis acceleration observedin an Earth-fixed system.
.(dC~Obb/dt)b x rb: apparent acceleration observed in the body-fixed system due
to a tune rate of change of the angular velocity of the body.
CO:bb X co~ X rb: apparent acceleration ofthe particle due to the angular velocity
of the mowng axes system and recorded in the body fixed system.
Force equations. Multiplying both sides of Eq. (4.249) by 8m, mass of the
particle, we obtain
8m d2R_ ~ 8m<2e x S2e X (Re + Ro) +8m(ao)e + 28rriSle x'V:
8m CTt2 :
+8m(dd,),xl,+8mo3lbx(c3{.bxr_t,) . (4.250)
~ 8Fz (4.251)
where 8Ft is the net external force acting on the particle P of mass 8m.
Summing over the entire body,
>:j: 8F, = E8m?2e x C2e X (Re + Ro) + ~- 8m(ao)c +2E 8m:Qe x vo
Let
+ X) 8m(dc~ x b) + ~_ 8m(colb x colb X rb) (4.252)
:8m =m
~- 8Ft -. F
(4.253)
(4.254)
EQUATIONS OF MOTION AND ESTIMATION OF STABILITY DERIVATIVES 365
We note that
1,
r,g = ~- 8mrb
mrcg = ~~: 8mrh
(4.255)
(4.256)
Then
F = m[(2e x (2e X (R, + Ro) + (ao)e +2C2e x V;j + (dd, )b X i:cg
+(D~b X (c7)l.b x lcb,)]
(4.257)
Specialcases. When the center of gravity of the vehicle coincides with the
origin of the body-fxed coordinate system, rcg = O, then Eq. (4.257) assumes the
form
F = m[CZe x ?2e X (Re + Ro) + (ao)e + 2C2e x V:] (4.258)
For the low-altitude flight dynamic problems, IRol << IRel, then
F =m[C2e X ?2e x Re + (ao)e + 2?2e X Ve] (4.259)
Let us expand the vector products in Eq. (4.259) and derive the equations of
motion in the usual Cartesian form. Let A, denote the latitude of the ongin of the
OxeYeZe axes system. Then,
S2e = 2e<2c COS;t- - F-CeS2e sin A (4.260)
Re - -ke Re (4.261)
1V : = le Vor + Je Voy + ke Voz (4.262)
- ZXe + JYe + kze
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