Aerodynamic Symmetry of Aircraft and Missiles

The number of aerodynamic derivatives in the Taylor series increases vastly with higher-order terms. Even the linear derivatives add up to 12 x 6 = 72, more than the aerodynamicist would like to deal with. Fortunately, the configurational symmetries of aircraft and missiles reduce the number of nonzero derivatives drastically.

Maple and Synge4 investigated the vanishing of aerodynamic derivatives in the presence of rotational and reflectional symmetries. They considered the dependence of the aerodynamic forces on linear and angular velocities only and employed complex variables to derive the results. The Maple-Synge theory con­tributed to the solution of many nonlinear ballistic problems in the past. However, with the advent of guided missiles the dependency of the aerodynamic forces on unsteady flow effects and control effectiveness has gained in importance.

In my dissertation and later in a paper5 I derived, starting with the Principle of Material Indifference, rules of vanishing derivatives for aircraft and guided missiles. The aerodynamic forces are assumed functions of linear and angular velocities, linear accelerations, and control surface deflections. I will summarize the results with enough detail so that you can apply the rules successfully, but spare you the derivations. For the curious among you, my paper provides the details.

The functional form of Eqs. (7.40) and (7.41) will be used, but subscript notations will be substituted for the dependent and independent variables. The kth-order derivative of the Taylor-series expansion will be formulated in these subscripts. After reviewing the planar and tetragonal symmetry tensors, thought experiments are conducted that engage the Principle of Material Difference in discarding zero derivatives. Rules will be given for vanishing derivatives by adding up sub – and superscripts. For ease of application, two charts are presented that sift out the vanishing derivatives up to second order for missiles and up to third order for aircraft.

7.3.1.1 Taylor-series expansion. We begin with the aerodynamic func­tionals of Eqs. (7.40) and (7.41), select the dynamic coordinate system, and introduce components for the forces, moments, and dependent variables:

(7.42)

The acceleration components require additional comments. The [DAvB]D_deriva­tive must be transferred to the D frame before it can be expressed as [vABD = [й, i), tii] components

[Davab]° = [Ddva]D + [£2“]dK]D = [vAB]° +

The additional term [£2ZM]D[t;g]£> is absorbed in the [t^J and (coBA ] dependencies. Now we introduce the subscripted independent variables

Zj = {u, v, to, p, q, г, й, v, tii, 8p, 8q, 8r}, 7 = 1,2,…, 12 (7.43)

The two velocity components v and to, if expressed in body coordinates, can also be viewed as angle of attack а = arctan(io/и) and side-slip angle fi = arcsin(v/y/u2 + v2 4- to2). The variables 8p,8q,8r represent the missile
controls—roll, pitch, and yaw—or the aircraft effectors—aileron, elevator and rudder. The dependent variables are abbreviated by

yi = {X, Y,Z, L,M, N}; i = 1,2,…, 6 (7.44)

With these abbreviations Eq. (7.42) can be summarized as

УІ = di(Zj); і = 1,2,…. 6; j = 1, 2,…, 12 (7.45)

The aerodynamic functional is expanded into a Taylor series in terms of the 12 state variable components zj, relative to the reference state zj. The Taylor expansion is mathematically justified if the partial derivatives in the expansion are continuous and the expansion variables A Zj = Zj – Zj are small. For aircraft and missiles the aerodynamic forces are continuous functions of their states for most flight maneuvers. However, unsteady effects, such as vortex shedding, can introduce discontinuities that cannot be presented accurately by this method. In subscript notation the Taylor series assumes the form

The partial derivatives, evaluated at the reference flight conditions, are the aerody­namic derivatives. The kih derivative is а к + 1 order tensor and is abbreviated by

It is a function of the implicit variables M and Re. As an example, the third-order rolling moment derivative with і = 4, j = 1, 72 = 5, 73 = 11 becomes, by correlating the subscripts with Eqs. (7.43) and (7.44),

84 ■ ■ ■ = ~-Э-_Е_ = L (7.47)

dz.1dz.5dzn dudqd(8q) Uqbq

This is the rolling moment derivative caused by the forward velocity component u, the pitch rate q, and the pitch control deflection Sq.