CHAPTER 7
slope of curve formed by yawing moment coefficient Cn versus angle of sideslip ß, rad-l (deg-l)
yawing damping derivative relative to angle of sideslip rate ß, rad-] (deg-]) slope of curve formed by yaw moment coefficient Cn versus effective control- surface deflection for yaw
(deg-l)
aerodynamic drag force vector, N magnitude of aerodynamic drag force vector D, N
aerodynamic reference length of body, m
elevation angle measured from projec- tion of line-of-sight vector Rl on xyr plane to the vector Rp rad (deg) resultant aerodynamic force vector, N
gravitational force vector, including effects of earth rotation, N
total instantaneous thrust force vector, N
components of aerodynamic force vec- tor FA expressed in the body coordinate system, N
component of gravitational force vec- tor Fg expressed in the body coordinate system, N
components of gravitational force vec- tor Fg expressed in the earth coordinate system, N
magnitude of total instantaneous thrust force vector FP, N
reference thrust force, N
components of thrust vector FP expressd in the body coordinate sys- tem, N
acceleration due to gravity, m/sz acceleration due to gravity at earth sur- face (nominally 9.8 m/s2), m/s2
altitude above mean sea level, m specific impulse of propellant, N-s/kg moments of inertia (diagonal elements of inertia matrix when products of iner- tia are zero), kg-m2
constant in induced drag coefficient, dimensionless
aerodynamic lift force vector, N components of total moment vector M expressed in body coordinate system (roll, pitch, and yaw, respectively), N•m components of aerodynamic moment
7-2
vector MA expressed in body coordinate system (roll, pitch, and yaw, respec- tively), N•m
components of propulsion moment vec- tor Mp expressed in body coordinate system (roll, pitch, and yaw, respec- tively), N•m
magnitude of aerodynamic lift force vector L, N
distance from center of mass to nozzle, m
total moment vector acting on a body, N•m
aerodynamic moment vector, N•m miss distance vector at time of closest approach, directed from missile to tar- get, m
thrust (propulsion) moment vector, N-m instantaneous mass of missile, kg mass at time of launch, kg
aerodynamic normal force vector, N magnitude of aerodynamic normal force vector N, N
load factor in units of g, dimensionless specified maximum load factor in units of g to be applied during maneuver, di- mensionless
position vector of missile, m position vector of target, m initial position vector of target, m eriod of target weave maneuver, s kill probability, dimensionless
components of angular rate vector w ex- pressed in body coordinate system (roll, pitch, and yaw, respectively), rad/s deg/s)
components of angular acceleration expressed in body coordinate system (roll, pitch, and yaw respectively). rad/
s2 (deg/s2)
ambient atmospheric pressure, Pa reference ambient pressure, Pa dynamic pressure parameter, Pa
range vector from missile center of mass to target center of mass, m line-of-sight vector from missile to tar- get expressed in earth coordinates, m line-of-sight vector from target to mis- sile expressed in target coordinate sys- tem, m
radius of the earth, m
MIL-HDBK-1211(MI) components of line-of-sight vector R,
expressed in target coordinate system, m
aerodynamic reference area, mz
transformation matrix from earth to body coordinates, dimensionless transformation matrix from earth to tar- get coordinates, dimensionless
simulated time, s
indicates that the associated variable is calculated at the current calculation time
indicates that the associated variable is calculated at the previous calculation time
time of closest approach,s
time since initiation of the maneuver,s unit vector in direction of lateral-accel- eration-command vector Ac, dimension- less
unit vector in direction of missile cen- terline axis, dimensionless
unit vector in direction of velocity of missile center of mass VM, dimension- less
unit vector in direction of relative veloc- ity vector VT/M, dimensionless
components of absolute linear velocity vector VM expressed in body coordinate system, m/s
components of linear acceleration expressed in body Coordinate system, m/s2
absolute linear velocity vector of a body, m/s
acceleration vector of missile center of mass, m/s
absolute velocity vector of missile cen- ter of mass (equivalent to the vector V for a general body), m/s
velocity vector of target center of mass, m/s
velocity vector of the center of mass of the target relative to the center of mass of the missile, m/s
speed of a body, speed of air relative to a body, magnitude of velocity vector V, In/s
magnitude of velocity vector of the cen- ter of mass of the missile VM, m/s magnitude of velocity vector of target
7-3
center of mass VT, m/s
magnitude of velocity vector of the cen- ter of mass of the target relative to the center of mass of the missile VT/M, m/s weight vector, N
weight, N
instantaneous distance from nose to center of mass, m
distance from missile nose to the refer- ence moment station, m
angle of attack rad (deg) total angle of attack rad (deg)
angle of sideslip (angle of attack in yaw plane), rad (deg)
angle measured from xb-axis to projec- tion of thrust vector FP on xbyb-plane, rad (deg)
angle measured from projection of thrust vector Fpon xbyb-plane to the thrust vector FP, rad (deg)
computation time step, s
general, or effective, angular deflection of control surface relative to body, rad (deg)
effective control-surf’ deflection causing pitching moment, rad (deg) effective control-surface deflection causing rolling moment, rad (deg) effective control-surface deflection causing yawing moment, rad (deg) damping ratio of a second-order system, dimensionless
Euler angle rotation in elevation (pitch angle), rad (deg)
atmospheric density, kg/m3
time constant (time to achieve 63% of a step command in a fit-order system), s Euler angle rotation in roll (roll angle), rad (deg)
target aircraft bank angle (Euler roll angle of the target coordinate system relative to the earth coordinate system), rad (deg)
Euler angle rotation in azimuth (heading angle), rad (deg)
rates of change of Euler angles in head- ing, pitch, and roll, respectively, rad/s (deg/s)
angular rate vector of rotating reference frame relative to inertial frame, rad/s
angular rate vector of target flight path, rad/s (deg/s)
undamped natural frequency of a second-order system, rad/s (deg/s)
7-l INTRODUCTION
Missile and target motions are calculated in a simulation by means of the equations of motion given in Chapter 4 by using values of the various forces acting on the vehicle.
Methods of determining the values of the gravitational, aerodynamic, and propulsive forces for substitution into these equations of motion are given in Chapters 4,5, and 6.
The equations of motion apply to any flying vehicle includ- ing the missile and the target. Integration of the differential equations of motion yields the translational and rotational velocity and position histories of a vehicle throughout the simulated flight. The equations are usually simplified for calculating target motion and may also be simplified for cal- culating missile motion, depending on the objectives of the simulation.
Vehicle translational and rotational equations of motion are usually solved in the body coordinate system if the sim- ulation has five or six degrees of freedom. In simulations with three degrees of freedom, rotational motion is not cal- culated explicitly, and the translational equations of motion are most conveniently solved in the earth reference frame.
With three degrees of freedom, missile angular motion about the pitch axis is calculated implicitly for use in calcu- lating the angle of attack. The use of a second-order transfer function is a convenient method of incorporating realistic missile angular response characteristics into a three-degree- of-freedom simulation. Simulations that calculate pitch and yaw rotational motion implicitly as opposed to including them in the basic equations of motion have been called pseudo-five-degree-of-freedom simulations.
In simulating the motion of the target, the objective is to provide the means to study missile flight response to target flight characteristics. These target flight characteristics may extend from straight and level fright to complex evasive maneuvers. Target position relative to the missile is used in a simulation to calculate missile guidance system responses;
target angular attitude relative to the missile is used to cal- culate target signature, which depends on the relative aspect.