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Cartesian Equations
We begin with Newton’s second law, as expressed by Eq. (5.9), and expand the right-hand side to include the aerodynamic and propulsive forces fap and the weight mg:
mD‘vlB = fap + mg (8.1)
On the left side is the rotational derivative relative to the inertial frame /, operating on the velocity vB of vehicle c. m В wrt the inertial frame. This inertial velocity [vBV, expressed in inertial coordinates, is suitable as state variable, but not for formulating the aerodynamic forces. It is the movement of the vehicle relative to the atmosphere that determines the air loads. Because the atmosphere is attached to the Earth (no wind assumption), the aerodynamics depends on the geographic velocity vf of the vehicle wrt Earth E. The relationship between inertial and geographic velocities will be derived first.
The position of the inertial reference frame 1 is oriented in the solar ecliptic, and one of its points I is collocated with the center of the Earth. The Earth frame E, fixed with the geoid, rotates with the angular velocity шЕІ. By definition, the inertial velocity is v B = DlsBi, where sbi is the location of the vehicle’s c. m. wrt point I. To introduce the geographic velocity, we change the reference frame to E:
D! sB[ = DesBi + EIeiSbi (8-2)
first right-hand term
De$bi = De$be + DeSei = De$be = vf
where DesEi is zero because sei is constant in the Earth frame. Substituting into Eq. (8.2), we obtain the relationship between the inertial and geographic velocities:
vb ~ vb + ПЕ! вв1 (8.3)
The difference between the absolute values is approximately 465 m/s at the equator.
Now we are prepared to coordinate Eq. (8.1) for computer programming. The aerodynamic and propulsive forces are expressed in flight-path coordinates ]v, whereas the gravitational acceleration is given in geographic axes ]G. Because the state variables are to be calculated in inertial coordinates, we introduce the transformation matrices [T]GV and T’G and write the component form
= [T f[T }VG[fa, p]v + m[gf) (8.4)
These are the first three differential equations to be solved for the inertial velocity components Wgl1. The second set consists of the inertial position coordinates
= K]’ (8.5)
Don’t forget the initial conditions. You need to specify the velocity and position vectors at launch.
Two transformation matrices must be programmed. The geographic wrt the inertial coordinates TM is composed of the TMs [T]G1 = [T аЕ[Тш, provided by Eqs. (3.13) and (3.12), respectively, whereas [T]VG is given by Eq. (3.25).
For aircraft and tactical missiles you can simplify your simulation by substituting Earth as inertial frame. In Eqs. (8.4) and (8.5) you replace frame / and point I by frame E and point E. The distinction between inertial and geographic velocity disappears and the geographic coordinate system is replaced by the local-level system L:
= [T)VL[fa, P]v + m[g]L (8.6)
= [vEBf (8.7)
Only one TM [T]VL is required and is given by Eq. (3.29). These equations of motion are quite useful for simple near-Earth trajectory work.
The Cartesian formulation is the easiest to implement among the three options. It only suffers from a lack of intuitiveness. Who can compose the inertial velocity [Ug]7 and position [лщ К components into a mental picture? Because of this deficiency, the polar equations are sometimes preferred. They formulate the equations of motion in terms of the geographic velocity v f, which is much easier to visualize.
