# TRANSFORMATION OF INERTIAS

The inertia matrix I connects angular momentum with angular velocity [see (4.3,4) and (4.3,5)] via

h = ltd

and hence belongs to the class of matrices covered by (A.4,26). It follows that for two sets of body axes, denoted FBl and Fb 2 connected by the transformation L12, the inertias in frame Fb2 can be obtained from those in FBl by

I2 = L2,I, L12 (В.12,1)

If the two frames are two sets of body axes such that xBl is rotated about ySl through angle £ to bring it to xB?, then (see Appendix A.4)

(B.12,2)

The inertias in frame FB2, denoted by an asterisk, are then obtained from those in FBl, with the usual assumption of symmetry about the xz plane, by the relations

/* = lx cos2 £ + L sin2 £ + 1^ sin 2£

/* = lx sin2 £ + lz cos2 £ – Izx sin 2£ (B.12,3)

= Wx – lz) sin 2£ + /«(sin2 £ – cos2 0

TRANSFORMATION OF STABILITY DERIVATIVES

All of the stability derivatives with respect to linear and angular velocities and velocity derivatives can be expressed as sums of expressions of the form of (A.4,23). That is, with the usual assumptions about separation of longitudinal and lateral motion, we can write

AX |
‘xu |
0 |
Xw~ |
Au |
"0 |
xq |
0" |
V |
"0 |
0 |
0 ‘ |
Дм |
|||

AT |
= |
0 |
0 |
V |
+ |
0 |
<7 |
+ |
0 |
0 |
0 |
V |
|||

A Z. |
_ZH |
0 |
_ A w _ |
. 0 |
Z, |
0 _ |
_ r |
_0 |
0 |
_ Avv_ |

(B.12,4)

A L ‘ |
" 0 |
Lv |
0 ‘ |
A и |
Л |
0 |
к |
~P~ |
"0 |
0 |
0 ‘ |
Ай |
|||

AM |
= |
Mu |
0 |
Mw |
V |
+ |
0 |
Mq |
0 |
q |
+ |
0 |
0 |
0 |
V |

AN _ |
_ 0 |
N0 |
0 . |
_ A w_ |
Л |
0 |
Nr. |
r |
.0 |
0 |

(B.12,5) |

Each of the six matrices of derivatives above transforms according to the rule (A.4,26). When L is given by (B.12,2) we have the transformation from an initial set of body axes (unprimed) to a second set (primed) as follows:

Longitudinal

(Xu)’ = Xu cos3 f – (Xw + ZJ sin £cos f + Zw sin2 £ (Xwy = Xw cos2 £ + (Xu – ZJ sin f cos £ – Z„ sin2 £ (*,)’ = X„ cos £ – Zq sin f (*«)’ = Z* sin2 £ (1)

(X*)’ = – Z* sin £cos І (1)

(zu)’ = zu cos2 f – (Zw – XJ sin £cos ^ Xw sin2 £ (ZJ’ = Zw cos2 £ + (Z„ + XJ sin £ cos £ + XH sin2 f (ZJ’ = Zq cos £ + Xq sin £

(ZJ’ = – Z* sin £ cos £ (1)

(ZJ’ = Zlv cos2 £

(M„)’ = Mu cos ij – Mw sin £

(Mw)’ — Mw cos £; + Mu sin £

(Mqy = Mq

(МйУ = —M# sin £ (1)

(MJ’= M* cos £ (1)

Lateral

0J’ =

(Ур)’ = Ур cos £ – Уг sin £

(Yr)’ = Уг cos £ + Yp sin £

(Lp)’ = Lp cos I – Nv sin £

(Lp)’ = Lp cos2 £ – (Lr + AJ sin £ cos £ + 7Vr sin2 £ (L,.)’ = Lr cos2 £ — (Nr — Lp) sin £cos £ — Np sin2 £ (yVp)’ = Nv cos ^ + Lp sin £

(Np)’ = /Vp cos2 £ – (Nr — Lp) sin £ cos £, — L, sin2 £ (AJ’ = iVr cos2 % + (Lr + Np) sin £ cos £ + Lp sin2 £

(1) For consistency of assumptions, the derivatives with respect to и and (X*)’ are usually ignored.