Configurational symmetries

Most aircraft and guided missiles have a planar or cruciform external shape. The planar configuration dominates among aircraft and cruise missiles, while missiles that execute rapid terminal maneuvers have cruciform airframes. Two types of symmetry are, therefore, con­sidered: reflectional and tetragonal (90 deg rotational) symmetries.

To derive the conditions of vanishing derivatives, precise definitions of these symmetries are required. In the case of reflectional symmetry, the existence of a
plane, satisfying certain conditions, is required, whereas tetragonal symmetry calls for an axis with specific characteristics.

In Chapter 2,1 introduced the reflection tensors M and in Chapter 4 the tetragonal symmetry tensor R<x). In body coordinates they have the form

[Mf =	"l 0	1 H – о О О ___ _l	and |^9о Iй =	"l 0 0 0	о” -1
	0	0 1		0 1	0

[/?9o]B, with a determinant of +1, is a proper rotation, whereas [MH is improper because its determinant value is — 1. For an aircraft the displacement vectors ssp, originating from the symmetry plane and extending to the surface, occur in pairs, related by

s’SP = MsSP

and similarly, for a missile, the displacement vectors ssa, reaching from the sym­metry axis to the surface, also occur in pairs related by

s’sa — ^90 ssa

These relationships together with the PMI, already encountered in earlier chapters, lead us to the desired conditions for vanishing derivatives. Noll3 has provided a precise mathematical formulation. Applied to the aerodynamic problem at hand, the PMI asserts that the physical process of generating aerodynamic forces <7,- from the variables zj is independent of spatial attitude. For any rotation tensor Rin, in tensor subscript notation and summation over repeated indices, it states

Rmdn{zj} = di{Rjpzp} (7.48)

Read Eq. (7.48) with me from left to right: the vector valued function dn of the state vector zj, rotated through the rigid rotation Rm, equals the same vector valued function of the state variables rotated through the same tensor Rjp. A functional with the properties expressed by Eq. (7.48) is called an isotropic function. The rotation is allowed to be proper or improper; i. e., its determinant can be plus or minus one.

Let us apply the PMI first to planar vehicles. Suppose Eq. (7.45) describes the aerodynamics of a particular wind-tunnel test result:

У І = di{zj)

Consider a second test under the same conditions, but with flow variables Zj mirrored by the reflection tensor Mjm

У і — di{MjPZp]

The resulting aerodynamics y’ should also be mirrored.

y = Mmyn

Therefore, equating the last two relationships, and with Eq. (7.45), we obtain

Mmd„{zj} = dilMjpZp] (7.49)

just like the PMI, Eq. (7.48) states. But if the external configuration of the test object possesses planar symmetry, the aerodynamics is indistinguishable in the two tests

di{zj} = dt {MjpZp}

and therefore substituting into Eq. (7.49) we obtain the condition for vanishing derivatives

Mindn{Zj} = di{Zj} (7.50)

We expanded both sides in Taylor series. In body coordinates the elements of Mm consist of +1 and -1 terms only. Those derivatives that exhibit different signs because of Min must be zero! If you read my paper, you will see that the derivation is somewhat more complicated. Yet Eq. (7.50), with the abbreviation of Eq. (7.46), leads eventually to the relationship between the derivatives:

j-yhh’-ji. _ j-_____ |^2д+&+і’ + 1 jyhh’"Jk

Rule 1: The aerodynamic derivatives Щ’,г "1к of a vehicle with reflectional

symmetry vanish if the sum Е Д. + к + і + 1 is an odd number.

When the exponent of ( — 1) is odd, a negative sign will appear at the right-hand side of Eq. (7.51). The same derivatives with different signs can only be equal if their values are zero. The subscript і indicates the force or moment components and the superscripts j, j2, ■ ■ ■, jk designate the components of the state vector of the of the kth partial derivative. To convert from the derivatives with physical variables to their subscript notation Т)/‘л"’л, use Table 7.1.

Let us apply Rule 1 to the example, Eq. (7.47): ‘Ejk+k + i + 1 =(1+5 + 11) + 3+4+1 =25. The derivative does not exist; a result you would have predicted if you are an aerodynamicist.

To derive the condition for vanishing aerodynamic derivatives of vehicles with tetragonal symmetry, we make use of the fact that a cruciform vehicle has two

Table 7.1 Association of dependent and independent variables with subscripts and superscripts ij j і 1 u X 2 V Y 3 w Z 4 p L 5 q M 6 r N 7 u 8 V 9 w 10 Sp — 11 Sq 12 Sr —

planes of reflectional symmetry. The two planes are rotated into each other by the tetragonal symmetry tensor Д90, and they intersect at the axis of symmetry. The PMI is applied twice to the two symmetry planes. The first one we carried out already for the reflectional symmetry plane. Therefore, Rule 1 applies also to cruciform vehicles. We derive the second condition by rotating the original experiment through 90 deg and applying the PMI the second time. I will spare you the details. The result is the relationship

(jqq2–qk _ +£+/>+1 Qqxqv-qk

where C is related to the D derivative by simply exchanging every second or third subscript. Thus the rule for vanishing derivatives for cruciform vehicles is stated as follows.

Rule 2: The aerodynamic derivative D’n"Jk of a vehicle with tetragonal

symmetry vanishes if the sum Ед + к + і + 1 is an odd number (Rule 1) or if Yqk + к + p + 1 is an odd number as well.

The relationship of the subscripts between oJiur"Jk and cqpqi"4k is given by Table 7.2.

As a test case, do you expect Nwp$q to exist for an aircraft or a missile? It is the control-coupling derivative of pitch control Sq, contributing to the yawing moment N, in the presence of a vertical velocity component w and roll rate p. For an aircraft we have Nwpgq = T>|411. Applying Rule 1, Ед + к + і + 1 = 3 + 4+11+3 + 6+1 = 28, we get an even number, and therefore the derivative is nonzero. For a missile, with Rule 2, Cf412 = 411, and Eqk + к + p + 1 =

2 + 4+12 + 3+ 5 + 1 =27 is an odd number, and the derivative vanishes. Did you guess correctly?

Let us try another example: Ywsr = D 12 is the yawing force derivative Y caused by rudder control Sr in the presence of down wash w. It survives the test for planar

Table 7.2 Subscript and superscript relationship between the D and C derivatives i, jk P’Qk 1 1 2 3 3 2 4 4 5 6 6 5 7 7 8 9 9 8 10 10 11 •12 12 • 11

vehicles (from Rule l:Ejj + f+i + l = 3 + 12 + 2 + 2+1 = 20), indicating that, for aircraft, the derivative is linearly dependent on the downwash. For missiles, however, with Cf11 = D12 (Rule 2: ‘Eqi +k + p +1 =2+11+2 + 3 + 1 = 19), the derivative does not exist. Physically speaking, the downwash is symmetrical for cruciform configurations. It affects the side force not linearly, which would result in a sign change, but quadratically, as shown by the existence of the derivative Yw28r — з — c|2 1 ^ : Syt + & + г + I = 3 + 3+12 + 3 + 2 + 1 = 24, and Zq^ + k + p+ l= 2 + 2+ ll + 3+ 3 + l= 22.

These rules are quite helpful not only for modeling but also for investigating nonlinear effects. I put them to good use in my dissertation, describing the nonlinear aerodynamic phenomena of Magnus rotors with higher-order derivatives. The real challenge of course is the extraction of these derivatives from wind-tunnel or free – flight tests, which we leave to the expert.

I do not have space here to discuss the physical interpretation of aerodynamic derivatives in any more detail. You will find the linear derivatives explained by Pamadi6 or Etkin.1 For nonlinear phenomena you have to search the specialist literature that applies to your particular modeling problem.