6.1 KiB
| title | author | geometry |
|---|---|---|
| Weapon Systems Midterm Equation Sheet | Aidan Sharpe | margin=1in |
Aerodynamics
Mach number:
M = \frac{v_\text{missile}}{v_\text{sound}}
Dynamic pressure:
Q = \frac{\rho}{2}v^2 \approx 0.7 P M^2, where \rho is ambient density, v is the missile velocity, P is the ambient pressure, and M is the Mach number.
Pressure waves:
\mu = \arcsin\left(\frac{1}{\text{M}}\right), the angle of a supersonic shock wave above the direction of motion, where M is the Mach number.
Force coefficients:
C_N = \frac{N}{Q S_\text{Ref}} \approx C_{N_\alpha} \alpha, where N is the normal force, S_\text{Ref} is the maximum cross-sectional area (calculate using diameter of the missile), C_{N_\alpha} is a constant, and \alpha is the angle of attack.
Moment coefficients:
C_m = \frac{m}{Q S_\text{Ref} L_\text{Ref}}, where m is the pitching moment, and L_\text{Ref} is the reference length.
Induced drag:
C_{D_I} = C_N \alpha = C_{N_\alpha} \alpha^2 = \frac{1}{C_{N_\alpha}} \left(\frac{n_z W}{Q S_\text{Ref}}\right)^2, where n_z is the maneuver acceleration, and W is the missile weight.
Maneuver G's:
n_z = \frac{N}{W} = C_{N_\alpha} \frac{\alpha Q S_\text{Ref}}{W}, where N is the normal force, and W is the missile weight.
Lift:
L = N\cos(\alpha) - A\sin(\alpha) \approx N, where N is the normal force, and A is axial drag.
Drag:
D = A\cos(\alpha) + N\sin(\alpha) \approx A + N\alpha, where A is axial drag, and N is normal force.
Static margin:
SM = CG - CP, where CG is the center of gravity, and CP is the center of pressure. Unstable when SM > 0 (CG is aft of CP). As a rule of thumb, SM \approx -0.5d, where d is the missile diameter.
Body fineness ratio:
BFR = \frac{l}{d}, where l is the missile length, and d is the missile diameter. Typically between 5 and 25.
Nose fineness ratio:
NFR = \frac{l}{d}, where l is the nose length, and d is the maximum nose diameter. Typically between 2 and 4.
Rocket Propulsion
Rocket thrust:
F = \frac{\dot{W} v_e}{g} + (P_e + P_a)A_e = \frac{P_0 A^*}{C^*}v_e + (P_e + P_a)A_e, where \dot{W} is the propellant weight flow rate, v_e is the exhaust exit velocity, P_e is the exit pressure, P_a is the outside pressure, and A_e is the nozzle exit area.
Mass flow rate:
\dot{m} = \frac{\dot{W}}{g}
Weight flow rate:
\dot{W} = g\frac{P_0 A^*}{C^*}, where P_0 is the chamber pressure, A^* is the throat area, and C^* is the characteristic velocity of burned propellants.
Characteristic velocity:
C^* = \frac{223}{K}\sqrt{\frac{T_0}{m}}, where m is molecular weight, K is a function of the specific heat ratio, and T_0 is the flame temperature.
Specific impulse:
I_{sp} = \frac{F}{\dot{W}}
Exit velocity:
v_e = g I_{sp}
Ideal burnout velocity:
V_{BO_I} = g I_{sp} \ln\left(\frac{W_L}{W_{BO}}\right), where W_L is the weight of the vehicle at launch, and W_{BO} is the weight of the vehicle at burnout.
Realistic burnout velocity:
V_{BO} = V_{BO_I} - g \sin(\bar{\gamma})T_{BO}, where \bar{\gamma} is the average flight path angle, and T_{BO} is the time at burnout.
Rocket velocity:
v(t) = v_0 + v_e \ln\left(\frac{m_0}{m(t)}\right) - g\sin(\bar{\gamma})t, where v_0 is the initial velocity, m_0 is the initial mass, and m(t) is the mass at time t.
Weapon Control Systems
Total engagement time:
TET = \frac{ROF - R_\text{min}}{v_t}, where ROF is the range of open fire, R_\text{min} is the range of the final shot, and v_t is the velocity of the target.
Duration of first shot:
TOF_1 = \frac{ROF}{v_m + v_t}, where v_m is the velocity of the missile, and v_t is the velocity of the target.
Depth of fire:
DOF = \frac{TET - TOF_1}{T_H} + 1, where T_H is the homing time.
Time between launches:
\Delta T_L = T_H\left(1 + \frac{v_t}{v_m}\right), where T_H is homing time, v_t is the velocity of the target, and v_m is the velocity of the missile.
Total launching time:
N_L = N \Delta T_L
Time to go:
TGO = \frac{|\vec{R}_{TM}|}{\cos(\theta_m)|\bar{v}_m| + \cos(\theta_t)|v_t}, where R_{TM} is the vector from the missile to the target, \theta_m is the angle between \bar{v_m} and \vec{R}_{TM}, \bar{v_m} is the average remaining weapon velocity, \theta_t is the angle between \vec{R}_{TM} and v_t, and v_t is the target velocity (assumed constant).
Predicted intercept point:
\overrightarrow{PIP} = \vec{R}_T + TGO \vec{v_t}, where R_T is the current vector from the illuminator to the target.
Power density at the missile seeker:
PDMS = \frac{P_T G_T \sigma_{RCS}}{L_{IL} (4\pi)^2 R_T^2 R_{TM}^2}, where P_T is the illuminator transmit power, G_T is the antenna gain, \sigma_{RCS} is the radar cross section of the target, L_{IL} is the total of the transmit losses of the illuminator, R_T is the distance from the RF source to the target, and R_{TM} is the distance from the target to the missile seeker.
Trajectory Design
Midcourse heading error:
\varepsilon = \arccos\left(\frac{\vec{R}_{TGO} \cdot \vec{v}_{M}}{|\vec{R}_{TGO}| |\vec{v}_{M}|}\right)
Terminal guidance heading error:
\varepsilon = \arccos\left(\frac{\vec{R}_{TM} \cdot \vec{v}_{TM}}{|\vec{R}_{TM}| |\vec{v}_{TM}|}\right)
Thrust energy optimization:
E_\text{thrust} = E_\text{preburnout drag} + \int\limits_{s_\text{burnout}}^{s_\text{final}} \text{Drag} ds + E_\text{grain} + \frac{1}{2}\frac{W_M}{g}v_\text{final}^2 + W_M(h_\text{final} - h_\text{initial}), where s is the incremental path length of the trajectory, v_\text{final} is the final velocity of the interceptor, h is the interceptor altitude, W_M is the weight of the interceptor without fuel.
Optimal dynamic pressure:
Q_\text{opt} = \frac{W_M}{s_\text{Ref}}\sqrt{C_A C_{N_\alpha}}
Cruise altitude:
h_\text{opt} \approx 2.3 \times 10^4 \ln\left(\frac{W_M}{s_\text{Ref} M^2 \sqrt{C_A C_{N_\alpha}}}\right), where W_M is the weight of the empty missile and M is the Mach number.
Optimal turn:
R_\text{opt} = \frac{v^2}{n_z g}