Linearization of Model Equations

The nonlinear coupled 6DOF EOMs can be linearized using the small disturbance theory [2]. In using this theory, we assume that the excursions about the reference flight condition are small. Thus, we assume small values of the variables u, v, w, p, q, r, f, U. This theory cannot be applied to situations with large change in motion variables. In this approach, all the variables in the 6DOF model are expressed as a sum of a reference value (steady-state value) and a small perturbation, e. g.,

U = u0 + u V = v0 + v W = w0 + w

P = P0 + p Q = q0 + q R = r0 + r

The nonlinear translational accelerations from the full set of 6DOF equations (Equation 3.24) are expressed as

U = —qw + rv — g sin U + ax

V = —ru + pw + g cos U sin f + ay (5.2)

W = —pv + qu + g cos U cos f + az

From Equations 5.1 and 5.2, the u equation can be expressed in terms of perturbed variables as follows:

u = -(q + q0)(w + w0) + (r + r0)(v + v0) — g sin (U + 00) + (ax + ax0) (5.3)

Equation 5.3 can be simplified by neglecting products of the perturbations and using small-angle approximation. Also, assuming wings-level, symmetric reference flight, we have

w0 = v0 = p0 = q0 = r0 = f 0 = C0 = 0

Setting all disturbances in the perturbed U, V, W equations to zero gives the reference flight conditions as

axo = q0w0 — r0v0 + g sin 60

ay0 = r0 щ — p0w0 — g cos U0 sin f 0 (5.4)

aZ0 = p0v0 — q0u0 — g cos &0 cos f 0

U = — gU cos Uo + ax

v = —ru0 + gf cos U0 + ay w = +qu0 — gU sin U0 + az

The linearized implementation in Equation 5.5 is frequently used for

1. Routine batch analysis of large amount of data in cruise flight regimes

2. Analysis of small amplitude maneuvers

3. Where computational efficiency is important and nonlinear effects are minimal