When an aircraft is flown through defended airspace, the pilot may perform evasive maneuvers to make it more diffi- cult for defensive gunfire or missiles to intercept his air- craft. If the pilot is aware that he is being engaged by a
particular type of missile, he may perform evasive maneu- vers prescribed for use against that particular type of mis- sile. To be most effective, the timing and direction-or directions of compound maneuvers-may be important.
The magnitudes of the accelerations of evasive maneuvers are particularly important. When a pilot is not aware of a specific engagement by defensive fire, he may perform a more or less continuous series of maneuvers, called jinking, while flying through known defended regions. Other exam- ples of target maneuvers that might be included in a missile flight simulation are terrain-following and terrain-avoidance flight paths or map-of-the-earth fright paths flown by heli- copters for concealment.
Usually, the fidelity required to model the target flight path is insufficient to warrant the use of sophisticated numerical integration techniques for solving the equations of motion. The improved Euler method (Chapter 1O) is commonly used to update target position and velocity from one calculation time to the next (Ref. 3). By employing this”
method, the target position is updated by using
The target acceleration vector AT for substitution into Eqs.
7-22 and 7-23 is calculated by using
The method of calculating the flight path angular rate vector varies depending on the type of target maneuver, The fright path angular rate vector for controlling target maneuvers can be input as a constant or as a tabular function of time, or it can be calculated within the simulation. Equations for cal- culating the angular rate vector for target turns and jinking 7-14
fright paths in horizontal planes are given in the paragraphs that follow. Equations for maneuver components in the ver- tical plane or more complicated flight paths are beyond the scope of this handbook.
To calculate the angular rate vector that will produce the desired maneuver, a mathematical relationship between that vector and the desired maneuver is needed. Before describ- ing this relationship, a parameter called load factor, com- monly used to describe the magnitudes of vehicle maneuvers, is discussed.
7-4.2.1 Load Factor
When applied to coordinated aircraft maneuvers, no side- slip, the load factor is equal to the ratio of the lift to the weight of the aircraft (Ref. 4):
Since W = mg, the maneuver load factor ng is equivalently expressed as the ratio of the lift acceleration (the component of acceleration caused by the lift force )to the accelera- tion due to gravity g:
Thus the maneuver load factor is the lift acceleration expressed in units of the acceleration due to gravity (called g’s). A 2-g maneuver has a lift acceleration equal to twice the acceleration due to gravity. In straight and level flight the lift must equal the weight to give a load factor of 1 g.
The magnitudes of target maneuvers are usually specified in terms of the load factor; however, the actual acceleration of a vehicle needed by a flight simulation must also include the 1-g downward acceleration of gravity.
7-4.2.2 Horizontal TurnS
In a horizontal turn, bank angle is a function of only the load factor. Given the load factor, the bank angle can be calculated by using
Fig. 7-l(A) shows an airplane performing a coordinated, horizontal turn with a load factor of 2 g. The bank angle for a 2-g, horizontal turn is 60 deg regardless of speed. The ver- tical component of the lift vector L is exactly equal and opposite to the airplane weight; otherwise, the airplane would not remain in the horizontal plane. The horizontal component of the lift vector L produces a lateral accelera- tion that causes the flight path to turn. The 2-g load factor vector is directed along the lift vector, as shown in Fig. 7- 1(B). The gravitational component is 1 g directed vertically downward. The vector sum of these two accelerations-due
Figure 7-1. Forces and Accelerations in Horizontal, 2-g Turn
to lift and gravity—is the total acceleration vector, which has a magnitude of directed horizontally toward the cen- ter of the turn.
For horizontal, constant-speed, coordinated turns, the magnitude of the total acceleration of the aircraft can be cal- culated for any given load factor by using
A positive value of ATC produces right-hand turns; a nega- tive value produces left-hand turns.
If changes in speed are desired during the turn, the mag- nitude of target velocity VT is varied accordingly.
7-4.2.3 Weaves in Horizontal Plane
Although pilots employ different types of jinking maneu- vers, one that is commonly employed in simulations is a simple weaving flight path in a horizontal plane.
7-4.2.3.1 Cosine Weave
The weaving flight path can be modeled as a cosine curve by calculating the maneuver acceleration as a function of time by using
If horizontal, coordinated maneuvers are assumed, the max-
imum maneuver acceleration for substitution into Eq. 7-30 is calculated by using
The maximum load factor and the period of the maneuver are set by inputs to the simulation. The time in. in Eq. 7-30 is calculated as the difference between current simulation time t and the time when the maneuver was initiated. The maneuver initiation time may be input, or it can be calcu- lated within the simulation as a function of the engagement, such as the time when the missile reaches a specified range from the target.
At each computation time the instantaneous maneuver acceleration ATC is substituted into Eq. 7-29 to determine the instantaneous angular rate vector wT of the flight path, which in turn is substituted into Eq. 7-24 to yield the instan- taneous target acceleration vector AT The target accelera- tion vector AT is then substituted into Eqs. 7-22 and 7-23 to give the position and velocity of the target at the end of the current computation interval.
7-4.2.3.2 Circular-Arc Weave
A jinking flight path that employs the maximum load fac- tor a greater percentage of the time is similar to the cosine weave except each alternating segment of the fright path is a circular arc rather than a cosine curve. The maneuver accel- eration for the circular arc weave is calculated at each com- putation time by using
where sgn [ ] indicates the algebraic sign.{+ or -) of the argument dimensionless. The maximum maneuver acceler- ation for substitution into Eq. %32 is calculated using Eq. 7- 31
As it stands, Eq. 7-32 introduces discontinuities in the target acceleration at each switch in flight-path direction, i.e., at each change in sign. In reality, to switch from a maneuver toward, for example, the left to one toward the right, the airplane must roll from a left-bank angle to a right- bank angle, which takes a finite time. Such unrealistic dis- continuities are not permissible in missile flight simulations that calculate miss distance because the miss distance could be significantly affected by them. One method of remedying this problem is to pass ATc through a first-order transfer function (low-pass filter) before using it in Eq. 7-29. In a digital simulation, this transfer function is given by (Ref. 3)
7-16
The variable ATaCh is then employed in Eq. 7-29 in place of ATC. The time constant is selected and input by the user of the simulation to give a realistic representation of the time it takes the target to switch maneuver directions.
7-4.2.4 Roll Attitude
The roll attitude (bank angle) of the target is often required in a simulation to determine the attitude of the tar- get reference frame. When the instantaneous load factor for a coordinated maneuver in a horizontal plane is known, Eq.
7-27 can be used to calculate the roll angle. When the load factor varies, e.g., when the maneuver acceleration is calcu- lated using Eq. 7-30 or Eq. 7-33, however, a convenient method used to calculate the instantaneous roll angle, with- out having to calculate the instantaneous load factor, is given by
If the first-order transfer function is not used, e.g., when the cosine weave is simulated, ATach in Eq. 7-34 is equal to Arc.