Computer models that simulate the motion of missiles and those that simulate the motion of airplanes are based on the same physical principles: application of propulsive, aerody- namic, and gravitational forces and the responses of the re- spective airframes to those forces. In general, however, missile flight simulations are much more concerned with the details of the missile flight than with the motion of the target aircraft. Inmost missile flight simulations, the motion of the target is precalculated and its time sequence is input as a ta- ble; or very general algorithms (e.g., straight lines, circular arcs, and sinusoids; or simplified responses to maneuver commands) are employed to calculate the target flight path.
Therefore, although the discussion that follows is equally ap- plicable to the simulation of missiles and aircraft, the empha- sis is on missiles.
A mathematical model of the motions of the missile and target is based on Newton’s second law. At each instant of time a force acting on a rigid body (missile or target) results in an instantaneous acceleration of the center of the mass of
the body. The acceleration is directly proportional to the force; the proportionality constant is the reciprocal of the mass of the body. If the force vector passes through the cen- ter of mass of the body, a pure translation results. If the force vector does not pass through the center of mass, a combina- tion of transition and rotation results. The instantaneous ro- tational acceleration of the body is proportional to the moment of the force acting about an axis through the center of mass. In this case the proportionality constant is the recip- rocal of the moment of inertia of the body about that axis.
These concepts are expressed mathematically as the familiar quations
and where
Specific requirements of the laws in the form shown are that the mass be constant and that the accelerations be calculated with respect to an absolute reference frame fixed in inertial space, i.e., a reference frame rigidly associated with the fixed stars (those heavenly bodies that do not show any apprecia- ble change in&ii relative position from century to century) (Ref. 9). It is often convenient to use reference frames that move relative to inertial space; in which case it is necessary to modify the equations to account for the motion of the ref- erence frame. These modifications and the method of han- dling the variable mass of the missile are presented in Chapter 4.
Three basic types of forces act on a missile and are includ- ed in almost all flight simulations-the forces of gravity, propulsion, and aerodynamics. In addition, the gyroscopic moments of internal rotors (or the rotating airframe itself) are sometimes included in simulations; these are discussed in Chapter 4.
3-3.2.1 Gravitational Force
According to Newton’s law of gravitation, every particle in the universe attracts every other particle with a force that varies directly as the product of the two masses and inversely as the square of the distance between them. This gravitation- al mass attraction is directed along the line connecting the masses. For systems of particles, such as a missile and the earth, the resultant gravitational mass attraction is the vector sum of the forces on individual particles. Although the non- spherical mass distribution of the earth affects the magnitude and direction of the resultant attractive force on bodies such
as missiles, the nonspherical components are usually small enough to be neglected in surface-to-air missile applications.
For missiles that operate at altitudes between sea level and 30,000 m, the change in the radial distance affects the grav- itational mass attraction by less than 1%; however, the cor- rection for changes in altitude is so simple that it is usually applied in flight simulations.
The force of gravity observed in a rotating earth reference frame is the vector sum of the force due to gravitational mass attraction force and a “pseudoforce” called centrifugal force (Refs, 10 and 11). Centrifugal force on a body in a rotating frame is called a pseudoforce because it does not exist under Newton’s law in a nonrotating inertial tie of reference.
Since there is no centrifugal force at the poles, the observed gravitational force is equal to the gravitational mass attrac- tion force, and it decreases at lower latitudes to a minimum at the equator. This variation of the observed gravitational force with latitude has only small significance in missile flight simulation because the maximum variation is only about O.5%. If the simulation is to match actual flight-test re- sults, however, the effect of centrifugal force at the latitude of the test range is usually considered.
Since the effects of earth curvature, angular rate, and di- rection of the gravity vector have an insignificant impact on the dynamics of the missile during its relatively short flight, it is possible to assume a Cartesian coordinate system fixed to the surface of a nonrotating earth. Usually (as defined in subpar. 3-3.3) the x- and y-axes of this earth coordinate sys- tem define a plane tangent to the earth at the simulated launch point. Assumptions about the gravitational vector in this coordinate system allow simplifications to be made in the equations of motion. The direction of the gravitational force vector is assumed to be perpendicular to the plane tan- gent to the surface of the earth. All gravity vectors are as- sumed to be parallel rather than to converge toward the center of the earth and the magnitude of the gravity vector contains the centrifugal correction.
3-3.2.2 Propulsive Force
The force of propulsion (thrust) applied to the missile usu- ally is supplied by a rocket motor. The resulting thrust vector usually is designed to pass through the center of mass of the missile so as not to contribute unwanted rotational moments.
Provisions are made in some simulations to study the effects of small thrust misalignment errors (Ref. 7). In simulating missiles that use thrust-vector control, the direction of the thrust vector is responsive to missile control commands.
For applications that use solid propellant rocket motors, the magnitude of the missile thrust is independent of all pa- rameters that change during the flight except time and atmo- spheric pressure. Thus the magnitude of the thrust is supplied to the simulation born a table of thrust as a function of time at a specified reference pressure. At each time advancement
within the simulation, the appropriate value of thrust is se- lected from the table, Interpolation is used when the simulat- ed time falls between the times tabulated. As the simulated missile changes altitude and, therefore, ambient atmospheric pressure, the magnitude of the thrust is corrected by an algo- rithm in the simulation that accounts for the difference be- tween the reference pressure and the pressure at the current altitude. If the effects of rocket propellant grain temperature on thrust are simulated, they are usually included in the thrust table and require no special algorithms within the sim- ulation.
For missiles that use other types of propulsion, simulation of the thrust maybe more complicated. For example, the ef- fects of missile speed and ambient atmospheric air condi- tions must be included in the simulation of the thrust of ram jets or air-augmented rockets.
3-3.2.3 Aerodynamic Force
The magnitudes of aerodynamic forces and moments on a missile of given configuration are a function of the Mach number at which the missile travels and the ambient atmos- pheric pressure or, equivalently, speed and atmospheric density. The methods of dimensional analysis (Refs. 12 and 13) show that the aerodynamic forces FA and moments MA are functionally related to these parameters as expressed by
where
The force and moment coefficients CF and CM, respectively, are functions of Mach number MN and vehicle configura- tion, which includes any control-surface deflections. The dy- namic pressure parameter Q is defined as
or the equivalent form
3-6
where
These equations are evaluated within the simulation at each computational time step. The term CFS represents the aerodynamic force per unit of dynamic pressure, and CMSd represents the aerodynamic moment per unit of dynamic pressure parameter. Reference area S and reference length d are related to missile size; they are constants for any given missile. A complete statement of an aerodynamic coefficient for use as data includes the value of the coefficient plus the reference area and reference length, where applicable, on which the coefficient is based. For surface-to-air missiles the reference area is usually the cross-sectional area of the mis- sile body, and the reference length is usually the diameter of the missile body; however, any other representative area- planform area, wing area, surface area-or length-body length or wing mean aerodynamic chord—may be used. The reference area on which aerodynamic force coefficient data are based must always be specified; otherwise, the data are incomplete and unusable. Likewise, the reference length, in addition to the reference area, for any aerodynamic moment coefficient data must be specified. Care must be taken to en- sure that the reference area and reference length used in the simulation are consistent with those on which the aerody- namic data are based. The dependence of the aerodynamic forces and moments on the aerodynamic shape of the missile is described by the coefficients CF and CM At subsonic speeds the coefficients are relatively constant with Mach number, but at transonic and supersonic speeds they are strongly influenced by Mach number. These coefficients are estimated or derived from wind tunnel or flight tests and are supplied to the simulation in the form of tables as functions of Mach number and control-surface deflection. The mo- ment coefficient CM also depends on the location of the cen- ter of mass of the missile, and this dependency must be taken into amount in the simulation.
Aerodynamic coefficients also depend on the Reynolds number, which represents the ratio of inertial forces to vis- cous forces in the fluid flow under consideration. This de- pendency is relatively weak within the range of Reynolds numbers experienced by most surface-to+tir missiles and can sometimes be neglected. However, since the Reynolds num-
ber varies with altitude, as well as with missile speed and size for simulations of missiles that reach high altitude, it may be necessary to supply tables of aerodynamic coeffi- cients also as functions of altitude.
3-3.2.4 Airframe Response
A missile or an airplane, considered a rigid body in space, is a dynamic system in six degrees of freedom. Its motion in space is defined by six components of velocity, i.e., three translational and three rotational. Simplifications are some- times made in missile flight simulations by approximating—
or neglecting altogether-the degree of freedom that repre- sents missile roll; this results in a five-degree-of-freedown model. Simulations that are further simplified by approxi- mating all three rotational degrees of freedom but that retain the three translational degrees of freedom, are three-degree- of-freedom models.
In simulations with five or six degrees of freedom, the fins are deflected at each computational time step in response to commands from the autopilot. Aerodynamic moments are calculated on the basis of the fin deflections, and solution of the rotational equations of motion yields the missile angle of attack.
In simulations with three degrees of freedom, the differ- ence born six-degree-of-freedom models is that the simulat- ed missile directly assumes an angle of attack corresponding to the lateral acceleration commanded by the guidance mod- el. The calculations of fin deflections and aerodynamic mo- ments are bypassed thus the transient behavior of the missile in developing an angle of attack does not Meet all the de- tailed nonlinear response characteristics that can be included when aerodynamic moments are calculated by using tabular aerodynamic moment data. For many applications in which missile transient response characteristics are known or can be assumed, sufficient simulation fidelity is obtained by em- ploying a transfer function in place of a detailed simulation of the aerodynamic response. The commanded angle of at- tack is the input to the transfer function, and the achieved an- gle of attack is the output. For example, employing a transfer function that corresponds to a second-order dynamic system permits adjusting the time required for the simulated missile to respond to commands and the amount the achieved angle of attack overshoots the commanded angle of attack. (These parameters are important to missile miss distance.) By ad- justing these parameters, the missile response characteristics are calibrated to match flight-test data or the results of more sophisticated simulations.
*It is common usage, almost universal, to call Q the "dynamic pressure". This usage is strictly correct only in the subsonic flow region. In the transonic and supersonic flow regions, the actual measurable dynamic pressure is equal to the dynamic pressure parameter multiplied by a compressibility factor however, the parameter is commonly referred to as dynamic pressure.