Linear and Angular Momentum Equations

We use the component perturbation method to formulate the general perturbation equations of atmospheric flight. In this section I derive the perturbed linear and angular momentum equations and follow up with a detailed discussion of the aerodynamic force expansion in the next section.

The linear momentum of the body В with mass m relative to an inertial frame I is given by Eq. (5.3):

Pb = mvB (7.6)

where Vg is the linear velocity of the c. m. В relative to frame I. The angular momentum Ig1 of body В relative to frame / and referred to the c. m. В is defined by the MOI tensor Ig of body В referred to the c. m. В and the angular velocity vector со81:

jBI ___ jB..BI /"7 -74

lB = IgO) (7.7)

Using Eq. (7.2), the following є perturbations of the state vectors are generated:

svb= vbp – RvBr (7-8)

єшв1 = шВрІ – RojBrI (7.9)

and for the linear and angular momenta

£Pb = Pbp ~ яРвг (7-Ю)

Q]BI ___ jBpI DjBrI

ЄІв ~lBP Khr

Generalizing these equations for second-order tensors yields for the MOI tensor

e1bb = 1% – RlfrR (7.12)

and the skew-symmetric form of the angular velocity vector

sflDI = flDpI – RflDr, R (7.13)

Newton’s and Euler’s equation are replicated from Eq. (6.38):

DIpIB=f = fa+f,+fg (7.14)

D’lBJ = m = ma + m, (7.15)

where / represents the forces and m the moments relative to the c. m. B. The subscripts a, t, and g refer to aerodynamics, propulsion, and gravity, respectively. Both equations are valid for the reference and perturbed flights.

To derive the linear momentum equations, let Eq. (7.14) describe the perturbed flight

D/PBp = fap + ftp + fgp

and introduce the є perturbations for each term

D’spl + D‘ (,Rpl) = efa + Rfar + ef, + Rftr + efg + Rfgr (7.16)

Let us modify the second term on the left side by applying the generalized Euler theorem, the chain rule, and the definition of the angular velocity vector Eq. (4.47). With Eq. (7.13) we obtain

D1 є pi + EflDIRp‘Br + RD’plBr

— Rfar + Rftr + Rfgr + efa + sft + £fg (7.17)

The underlined terms are actually Eq. (7.14) applied to the reference flight and rotated through R. They are satisfied identically. The last term can be rewritten using the fact that the gravitational force is the same for the perturbed and reference flights fgp = fgr:

sfg = fgp ~ Rfgr = (E – R)fgr

The perturbation equation of the angular momentum is derived in the same way. Both equations are summarized as follows:

D’spl + e^’R^pI = Bfa + ef, +(E~ RDpDr)fgr (7.18)

D1 eIbJ + £nDIRDpDrlBBr‘ = Etna + Etnt (7.19)

These are the general perturbation equations of atmospheric flight mechanics. No small perturbation assumptions have been made as yet. They are expressed in an invariant form, i. e., they hold for all coordinate systems. Two types of vari­ables appear. The linear and angular momenta of the reference flight p’Br and lRfj. are known as functions of time; and the component perturbations are marked by a preceding e. The latter expressions єр! в and eIbJ represent the unknowns. The
aerodynamic forces and moments will be discussed in Sec. 7.3. Evaluating the per – turbational thrust and gravity forces is straightforward and will not be addressed.

The first terms on the left-hand sides of Eqs. (7.18) and (7.19) are the time rate of change of linear and angular momenta, whereas the second terms account for unsteady reference flights. Both equations are coupled nonlinear differential equations. To help you gain insight into the structure of the perturbation equations, I will derive two special cases: the all-important perturbations about a steady reference flight and the equations for turning reference flight.