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where
c (7.30)
CI = 2Uo
b, = 2~r
Taking Laplace transforms, we get
mi Po Aa(s) + (Cyp - mis)Ap(s) - mir (s) = 0
(C.zcr - mis) A(x(s) - mi Po Ap(s) + miq(s) = 0
Cma Aa(s) + (Cmq Cl - Iyls)q(s) - (Ixl - Izi)Por(s) = 0
CnpA,8(s) - (I;I - IAtl)Poq(s) + (Cnrbi - Izis)r (s) = O
(7.31)
(7.32)
(7.33)
(7.34)
(7.35)
INERTIA COUPLING AND SPIN
For a nontrivial solution of these equations, we must have
miPo (Cyp-uiis) O -mi
. (Czu -itiis) -miPo mi 0
Cmu 0 (CmqCJ -IyIS) -(Ixi -Izi)Po
O Cnp -(I;j-I;.i)po (Cnrbi~/;is)
Expanding the determinant, we get the characteristic equation
AiS4 + BiS3 + CIS2 + Dis + Ei - 0
where
Ai = Fyl2tlyll;l .
Bi = -m:t(lyiC,,r bi +I;IC,nqCI) - 17ZllyICICzr,r - ITIiCyplyil;l
Cl = /Ti:pg-Iy I I;l + Cyp [lTl I (Iyl Cnr bi + I; Cmq Cl) + Iy I Iztl Cztr]
633
-0 (7.36)
(7.37)
(7.38)
(7.39)
- mi [mi pg-(lri - lzl)(l;,l - I;1) - Czu(lyICnrbl + IzlCmqCl)
- mi Cmqci Cnrbi + mi Cma,l;l] + tTi:lyl Cnp (7,40)
Di = _ITI21 pg(lyl Cnr bi + I;I Cmq CI) + Cyp [-Cza(lyi Cnrbi + Iz I Cmq CI)
+ m{ pg-(lxl - Izl)(l):l - I;-I) - ITI ICmqCICnr bi + mi Cma I/l]
- mi [CzaCmq CI C,,rbi - mi Cma C,,rbi - (Ix I - /zl)(/;l - I;l )Czcr P~]
- lTZICnp(IT11CmqCI + Czalyl) (7.41)
Ei = mf(lzl - Ixl)(lyl - Ki)P~ - pg [_m21Cmqci Cnrbi + l7~Cnp(lzl - Ixl)
- Cyp(lzl - Irl)(,l;l - I;l)Czcx - I7221Cma(l;l - I;I)]
+ Cyp(Cza Cmq CICnrbi - mi Cmtr Cnr bi) + lTl I Cnp Cza Cmq CI - lTl21Cmct Cntt
(7.42)
Specialcase, Po = O. Itisinteresting to note thatin Eqs. (7.38-7.42),wherever
the roll rate Po appears,it is having only the powers of 2 and 4. For very small
values of roll rates pg N pg N 0. Thus, assuming Po ~ O, Eq. (7.36) reduces to
0
(Czcr - mis)
Cma
0
(Cyp - ITl IS)
0
0
Cnp
O
mi
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