ABSTRACT
The aeroballistics topics discussed so far have built up to where the reader has an appreciation for the techniques required to analyze projectile motion to a great degree of accuracy. The culmination of this study was the development of the equations for a six degree-of-freedom (6-DOF) model which accurately describes the motion of a rigid body through air. With a 6-DOF model in hand, the aeroballistician can examine the effects of a given configuration. The word given was italicized for emphasis because the aeroballistician must know the configuration properties before he or she analyzes the projectile. The implications of this are that without other tools to determine what needs to be changed in a design to alter the projectile behavior, one must simply guess at a new configuration, determine the aerodynamic coefficients, and reanalyze. This process can be very inefficient. The solution to this problem is to develop a theory that can be used to quickly determine what must be changed in a projectile to alter its flight behavior, make the changes, and reassess. This will be the topic for the remainder of this section. Linearized theory was (at least in the opinion of the authors) refined to an exceptional
degree by Murphy [1] in 1963. Other authors before and since [2-4] have developed similar theories and a good description of these can be found in McCoy [5]. The reason the theory is called linearized is the fact that the aerodynamic coefficients are assumed to be functions of the angle of attack in a linear sense. In other words,
Fj / Cj0 ( sinat) or Mj / Cj0d( sinat) (9:1)
Here the subscript j indicates any parameter of interest as introduced earlier. There are good points and bad points (as always) with this technique. The good news is that the mathematics become simple enough to determine quantities of interest extremely quickly and find means of changing a projectile’s flight characteristics quickly. The bad news is that the use of linear coefficients prevents us from duplicating some motions that occur frequently enough in projectile flight to warrant the inclusion of their nonlinear brethren-and the math becomes complicated to boot. We will continue the practice of using the definitions of the appropriate vectors and
scalars based on Ref. [5]. The choice is somewhat arbitrary, but for several years now the authors have used this lucid work as a supplementary textbook and it is a matter of convenience. Our coordinate system is defined as in Figure 9.1. The aerodynamic coefficients introduced in the beginning of this chapter were written
for both forces and moments as
Fj ¼ 12 rV 2SCj (9:2)
Mj ¼ 12 rV 2SdCj (9:3)
We have also defined the angular rates of the projectile as
p ¼ Roll (spin) rate (9:4) q ¼ Pitch rate (9:5) r ¼ Yaw rate (9:6)
The projectile angular position with respect to the velocity vector was given by
a ¼ Angle of attack (9:7) b ¼ Angle of sideslip (9:8)
The aerodynamic coefficients are functions of the rates expressed in Equations 9.4 through 9.6 as well as angular positions expressed in Equations 9.7 and 9.8. Additionally, these coefficients are also functions of the time rate of change of a and b which do not normally coincide with q and r. Thus, we can write
Cj ¼ Cj(a,b, _a, _b, p, q, r) (9:9)
With this nomenclature, any coefficient can normally be expressed as a series expansion in the seven variables
Cj ¼ Cj0 þ Cjaaþ Cjbbþ Cj _a _a _ad V
þ Cj _b _b
_bd V
þ Cjp
pd V
þ Cjq
qd V
þ Cjr
rd V
þ
(9:10)
In Equation 9.10, we have included the terms in parentheses to maintain the nondimensional characteristics of the coefficient. We can see that this expansion results in a large number of terms that must be carried. Seldom in aeroballistics do we require terms in this expression beyond second order, but they can be included if data is available. When we discuss linear aeroballistics, we are limiting ourselves to the eight terms displayed in Equation 9.10.
