Figure 5-3. Zero-Lift Drag Coefficient
Figure 5-4. Drag Polar
namic moment on a symmetrical configuration is generated by the resultant aerodynamic force acting with a lever arm relative to the axis of rotation. This moment usually is approximated by the normal force acting at the center of pressure with a lever arm equal to the distance between the center of mass and the center of pressure. For asymmetrical configurations, e.g., a missile with deflected control sur- faces, an additional moment can be caused by forces that are equal in magnitude but opposite in direction—therefore not included in the resultant force—and whose lines of action are not colinear.
Calculations of aerodynamic moments are based on moment coefficients derived from wind tunnel tests. Fig. 5- 6 shows a typical set of moment coefficient curves for vari- ous control-surface deflection angles. The indicated moment reference station is the point on the missile, some- times called the aerostation, about which the moments were measured in the wind tunnel. Moment coefficient curves vary with Mach number therefore, tabular data based on figures similar to Fig. 5-6 are usually input to a flight simu- lation for several different Mach numbers. The moment coefficients illustrated in Fig. 5-6 represent only the static component of the total instantaneous moment coefficient. In addition, the dynamic damping components, discussed in par. 5-3, are supplied in the form of the dynamic derivatives The moments in Eqs. 4-46 are about the instantaneous center of mass of the missile. As previously stated however, moment coefficients derived from analytical calculations or from wind tunnel data are specified with respect to. some reference moment station. The reference moment station is often located at the center of mass of the missile after motor burnout but before motor burnout the missile center of mass changes position with time because of the mass redis- tribution that occurs when propellant burns and is expelled.
Therefore, it is necessary to correct the moment coefficient
to make it relative to the instantaneous center of mass.
Equations for calculating this correction for the pitching and yawing moment coefficients
dynamic components are
where
Figure 5-6. Moment Coefficient Versus Angle of Attack (Ref.9) 5-10
MIL-HDBK-1211(MI) r= yaw component of angular rate vector w
expressed in body coordinate system, rad/s (deg/s)
V = speed of body, speed of air relative to body, magnitude of velocity vector V, m/s
xcm~ = instantaneous distance from missile nose to center of mass, m
xref = distance from missile nose to reference moment station, m.
The Instantaneous location of the center of mass xcm depends on the shape and burning characteristics of the pro- pellant grain. This parameter usually is given as a function of time for a given missile and can be input to the simula- tion in tabular form or as an equation that approximates the experimental or analytically estimated data. The damping terms in Eq. 5-12 are not corrected from the reference moment station to the instantaneous center of mass.
Although this constitutes an approximation, these terms are usually neither critical enough nor defined accurately enough to justify correction.
At large angles of attack the curve of aerodynamic pitch- ing and yawing moment coefficients as a function of angle of attack may be very nonlinear. This nonlinearity relates to the fact that large angles of attack can cause the missile tail to move out of the downwash field of the forward compo- nents of the missile, cause a large increase in the tail effi- ciency, and cause a large increase in the slopes of the pitching and yawing moment curves (Ref. 8).
The pitching and yawing moment characteristics are essentially the same in missiles with cruciform symmetry, i.e., Cm = Cn. The aerodynamic moment coefficients Cm
and Cn may be input to a missile flight simulation in the form of extensive tables with interpolation among entries for angle of attack Mach number, control-surface deflection angle, and possibly altitude (which affects Reynolds num- ber). The manner in which the aerodynamic moment coeffi- cients are included in a particular simulation depends on the computing equipment and on the level of detail required by the simulation users to simulate angular motion. A less cum- bersome, but also less detailed, approach is to input rela- tively smaller tables of the pitching moment derivatives as functions of Mach number. The aerodynamic moment coefficient
then given by
where
in the pitch direction is
pitching moment coefficient about reference moment station, dimensionless
slope of curve formed by pitch moment coeffi- cient Cm versus angle of attack a, rad-] (deg-l)
slope of curve formed by pitch moment coeffi-
cient Cm versus control-surface deflection rad-l (deg-1)
angle of attack in pitch plane, rad (deg) angle of effective control-surface deflection in
the pitch direction, rad (deg).
The assumption that the variations of Cm with respect to a and P are linear can often be justified and thus permit the simplified approach given by Eq. 5-13. For example, the Cm
curves are usually relatively linear at angles of attack near the trim angle of attach i.e., on the axis where Cm = O in Fig. 5-6, and deviations of the missile from the trim condi- tion are likely to & small at least for some missiles (Ref. 9).
Eq. 5-13 can be made nonlinear to match wind tunnel data better ‘by the addition of terms in powers of α and
Similarly, the aerodynamic moment coefficient in the
where
= yaw moment coefficient about reference moment station, dimensionless
slope of curve formed by yaw moment coeffi- cient Cn versus angle of sideslip , rad-1 (deg-1)
= slope of curve formed by yaw moment coeffi- cient Cn versus effective control-surface deflec- tion rad-l (deg-1)
= angle of sideslip (angle of attack in yaw plane), rad (deg)
= angle of effective control-surface deflection in the yaw direction, rad (deg).
Many missiles are designed to prevent the aerodynamic roll moment LA, or at least to reduce it to a minimum. For example, the autopilot of a missile containing a roll-rate gyro can sense missile roll rate and issue control commands to null it. Some missiles have control tabs called rollerons that use gyoscopic moments produced by missile roll to adjust the tab automatically in a direction control the roll.
Other missiles are designed to roll continuously. For slender missiles it has been estimated that if the missile roll rate is less than about 20 revolutions per second, the gyroscopic effects are small enough to be neglected (Ref. 10). In this case the terms involving the roll rate p in Eqs. 4-46 can be eliminated. If the roll rate is not considered to be negligible, it is calculated by integrating Eqs. 4-46. These equations involve the roll moment 1., the aerodynamic contribution of whichh is calculated from the roll moment coefficient C1 by using Eq. 5-6. The roll moment coefficient is a function of Mach number, control-surface defection angle, roll rate, and missile speed. One method of calculating the roll moment coefficient is to use two derivatives-
The first derivative accounts for the effective control-sur-
5-11
damping produced by the roll rate p. The expression for is
where
aerodynamic roll moment coefficient about center of mass, dimensionless
roll damping derivative (Subpar. 5-2.2.4), rad-1 (deg-’)
= slope of curve formed by roll moment coeffi- cient Cl versus effective control-surface deflec-
aerodynamic reference length of body, m missile roll rate, rate (deg/s)
speed of a body, speed of air relative to body, magnitude of velocity vector V, m/s
effective control-surface deflection angle corre- sponding to roll, rad (deg).