Perturbation Techniques

The classical perturbation technique, as outlined by Etkin,1 proceeds as follows. First, an axis system is defined in relationship to physical quantities, such as the principal body axes or the relative wind velocity. The components of the state parallel to these axes are then identified. A particular steady flight regime is selected

with certain values for the reference components, e. g.,x, i, xr2, xr3, and a perturbed flight with xpi, xp2, Xp-}. The scalar differences

Ax,- = x Pi — xri i — 1,2,3

are the perturbation variables. Because the perturbations are generated by a scalar subtraction, this technique is also called the scalar perturbation method (see Ref. 2). The disadvantage of this technique lies in the fact that all of the formu­lations are tied to one particular coordinate system. A change to other coordinate systems is very difficult to accomplish.

In theoretical work vectors are preferred over components, and perturbations are defined as the vectorial differences between the reference and perturbed vectors. No allusion is made to a particular coordinate system. Because this technique considers the total state variable rather than its components, it is called the total perturbation method. Denoting the state vectors during the reference and perturbed flights as xr and xp, respectively, the total perturbation is defined as

Sx = xp — xr

The total perturbations have the advantage over the scalar perturbations that they hold for any coordinate system. In applications, however, numerical calculations require that vectors be expressed by their components, referred to a particular coordinate system. For instance, the MOI is given in body axes; vehicle acceleration and angular velocity are measured by accelerometers and rate gyros, mounted parallel to the body axes; wind-tunnel measurements are recorded in component form; and the whole framework of aerodynamics is based on force and moment components rather than total values.

To express the total perturbations in components, a transformation matrix must be introduced. In our notation the components of the Sx perturbation, relative to any coordinate system, say ]°, become

[&xfp = [XpfP – [T]DpDr[xr]Dr (7.1)

The subscripts r and p indicate reference and perturbed flights, respectively; [xr]Dr and [xp]Dp are the components as measured during reference and perturbed flights; and [TfPDr is the transformation matrix of the coordinate system associated with the perturbed frame Dp relative to the coordinate system associated with the ref­erence frame Dr.

Every numerical evaluation of equations based on the total perturbation method includes the transformation matrix [T]DpDr. Consequently, the transformation an­gles and their trigonometric functions enter the calculations, increasing the com­plexity of the equations considerably.

Wouldn’t you rather work with a perturbation methodology that combines the general invariance of the total perturbation method for theoretical investigations with the simple component presentation of the scalar perturbation method? We can formulate such a procedure by introducing the rotation tensor RDpDr6l the Dp frame wrt the Dr frame in the following form:

sx — xp — RUpUrxr

The ex perturbation is obtained by first rotating the reference vector xr through RDpDr ancj ^еп subtracting it from the perturbed vector xp. It satisfies our first requirement of invariancy. To show that it reduces to a simple component form, we impose the ]Dp coordinate system and transform the reference vector to the ]Dr system:

[ex]Dp = [.xpfp – [RDpDr]Dp[xr]Dp

= [xpfp – [RDpDr]Up[T]DpDr[xr]Dr = [xpfp – [xrfr

The last equation follows from Eq. (4.6). Note that the transformation matrix of Eq. (7.1) is absent. Because this technique emphasizes the component form of a vector, Eq. (7.2) is referred to as the component perturbation method or alternately as the є perturbations.

When you work with the component perturbation method, the choice of the RDpDr tensor and thus the selection of the frame D is most important. As a general guideline, choose D so that the є perturbation remains small throughout the flight. Especially in atmospheric flight, the selection of D is determined by the require­ment of representing the aerodynamic forces as a function of small perturbations. Then a Taylor-series expansion is possible, and the difficult task of expressing the aerodynamic forces in simple analytical form can be achieved. I propose the des­ignation dynamic frame for D because the dynamic equations of flight mechanics are solved in a coordinate system associated with frame D.

Let us discuss some examples. The dynamic frame of an aircraft is either the body frame В or the stability frame S. In both cases, for small disturbances, the rotation tensors are close to the unit tensor, expressing the fact that the frame Dp has been rotated by small angles from Dr. As will be outlined in more detail in Sec. 7.3, the dynamic frame plays also an important role in the aerodynamic force and moment expansions.

In missile dynamics the situation is similar except that the aeroballistic frame replaces the stability frame. However, for a spinning missile the body frame can­not serve as a dynamic frame because the perturbations of the aerodynamic roll angle can be large. To keep the perturbations small between the wind and dynamic frames, the nonrolling body frame is chosen as dynamic frame. The motions be­tween the body frame and the dynamic frame thus are not explicitly included in the aerodynamic expansion, but rather the derivatives depend on them implicitly. To simplify the notation, I will use the abbreviated form R for RDpDr whenever appropriate. ,

Perturbation techniques enable us to expand the aerodynamic forces in terms ot small variables about the reference flight. Suppose /(x) is the aerodynamic force vector with x representing a state vector. The force during the perturbed flight f(xp) is expressed in view of Eq. (7.2) by

f(xp) — /(ex + Rxr)

Expanding about the reference flight (ex = 0) yields

where 3 //Эх is the Jacobian matrix. The Principle of Material Indifference, famil­iar to us from Sec. 2.1.3, states (see Ref. 3) that the physical process, generating fluid dynamic forces, is independent of spatial attitude. In other words, if xr is rotated through R, the process of functional dependence remains the same. The only difference is that the force has also been rotated through R, i. e.,

Rf{xr) = f{Rxr)

Making use of this fact, Eq. (7.3) becomes

f(Xp) = Rf(Xr) + ЄХ + ■■■ (7.4)

and / behaves like the є perturbations, introduced by Eq. (7.2)

fp = Rfr + sf (7.5)

The component or є perturbations satisfies both requirements of invariancy for theoretical derivations and simple component form for practical calculations. They are a generalization of the classical scalar perturbation method and are particularly well suited to formulate perturbations in a form invariant under time-dependent coordinate transformations.