Steady Reference Flight

I define steady as the nonaccelerated and nonrotating flight and choose the body frame В as the dynamic frame. With шВгІ — 0 (nonrotating reference flight) and therefore lRll — 0, Eqs. (7.18) and (7.19) simplify to

D’eP’r + snBIRBpBrp>Br = Efa + e/, + (E – RBpBr)fgr (7.20)

DlslBJ — єта – I – smt (7.21)

To prepare for the use of the perturbed body coordinates ]Bp, we transform the rotational derivatives to the Bp frame. Let us start with Newton’s equation Eq. (7.20) and use the fact that шВгІ = 0:

D’sp’ = D^sp’g + SlBpIsplB = DBpsplB + ПВрВгєр’в

Substitute the rotational derivative and use the definition of the linear momentum Pb = mvB-

m(DBpevIB + ПВрВгву’в + еїіш RRpRrv’Rr) = efa + eft + (E – RBpBr)fgr (7.22)

Euler’s equation is obtained by a similar transformation

О’еіЦ = DRpeIri + ГBplElBi = DBpslBJ + ГlBpBrElBJ

and substituting it into Eq. (7.21):

DBpElBB! + ГlBpBrElBJ = Em a + Em, (7.23)

Modifying the perturbation e:Irr [Eq. (7.11)] by the definition of the angular mo­mentum Eq. (7.7), and with шВгІ = 0, we obtain

_jB/ /BpI nBpBrjBrl jBp. .BpI nBpBr jBr. .Brl j^P, ,BpBr

SlB tBp – R и lBr = iBpU – K ІВгШ – 1ВрШ

and simplify Eq. (7.23) (with DBpIBR = 0):

l%DBpujBpBr + ПВрВгІврршВрВг — Em a + em, (7.24)

Equations (7.22) and (7.24) are the perturbation equations of steady flight in their invariant form. We select the perturbed body coordinates ]Rp for the component formulation. First we deal with the gravitational term

{[E]Bp – [RBpBr]Bp)[fgr]Bp = ([E]Bp – [RBpBr]Bp)[T]BpI[fgry

= ([T]BpBr ~ [E])[T]Br,[fgr]>

then we express the linear momentum equations in ]Bp coordinates

d єуівЛВр
At

= Wa]Bp + W,}Bp + ([T]BpBr – [E])[T]BrI[fgr}! (7.26) and the angular momentum equation

Г As&pBr~BP

[lZ +№ВрВг]Вр[1вр]P[a>BpBr]Bp = [sma]Bp + [smt]Bp (7.27)

These equations are nonlinear differential equations in the perturbation variables [ev1b]Bp and [aBpBr]Bp. Eq. (7.26) is coupled with Eq. (7.27) through [aBpBr]Bp. In addition, the underlined term of Eq. (7.26) also couples the angular veloc­ity perturbations via the reference velocity. With small perturbation assumptions and therefore neglecting terms of second order, we can linearize the left-hand sides:

= WaBp + w, fp + (iTiBpBr – m)mBr,.uY

[-J BpBr-Bp

KTP =[ema]Bp + [emt]Bp

As you see, the translational equation (7.28) is still coupled with the rotational equa­tion (7.29) through the angular velocity perturbations [a>BpBr]Bp. Equation (7.29) would be uncoupled from Eq. (7.28) were it not for the aerodynamic moment [єта]Вр, which is a function of the linear velocity. Both equations are still nonlin­ear differential equations through their aerodynamic functions.

The perturbation equations for steady flight are the workhorse for linear stabil­ity analysis. They apply equally to aircraft and missiles and have been used as far back as Lanchester, that great British aerodynamicist who introduced the stability derivative. A more intriguing challenge is the modeling of perturbations for un­steady flight. Much of our hard-earned tools will have to be put to use. With them we can study such exotic problems as the stability of cruise missiles in pitch-over dive and the dynamics of agile missile intercepts.