119 lines
6.1 KiB
Markdown
119 lines
6.1 KiB
Markdown
---
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title: Weapon Systems Midterm Equation Sheet
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author: Aidan Sharpe
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geometry: margin=1in
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---
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# Aerodynamics
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#### Mach number:
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$M = \frac{v_\text{missile}}{v_\text{sound}}$
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#### Dynamic pressure:
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$Q = \frac{\rho}{2}v^2 \approx 0.7 P M^2$, where $\rho$ is ambient density, $v$ is the missile velocity, $P$ is the ambient pressure, and $M$ is the Mach number.
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#### Pressure waves:
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$\mu = \arcsin\left(\frac{1}{\text{M}}\right)$, the angle of a supersonic shock wave above the direction of motion, where $M$ is the Mach number.
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#### Force coefficients:
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$C_N = \frac{N}{Q S_\text{Ref}} \approx C_{N_\alpha} \alpha$, 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.
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#### Moment coefficients:
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$C_m = \frac{m}{Q S_\text{Ref} L_\text{Ref}}$, where $m$ is the pitching moment, and $L_\text{Ref}$ is the reference length.
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#### Induced drag:
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$C_{D_I} = C_N \alpha = C_{N_\alpha} \alpha^2 = \frac{1}{C_{N_\alpha}} \left(\frac{n_z W}{Q S_\text{Ref}}\right)^2$, where $n_z$ is the maneuver acceleration, and $W$ is the missile weight.
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#### Maneuver G's:
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$n_z = \frac{N}{W} = C_{N_\alpha} \frac{\alpha Q S_\text{Ref}}{W}$, where $N$ is the normal force, and $W$ is the missile weight.
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#### Lift:
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$L = N\cos(\alpha) - A\sin(\alpha) \approx N$, where $N$ is the normal force, and $A$ is axial drag.
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#### Drag:
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$D = A\cos(\alpha) + N\sin(\alpha) \approx A + N\alpha$, where $A$ is axial drag, and $N$ is normal force.
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#### Static margin:
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$SM = CG - CP$, 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.
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#### Body fineness ratio:
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$BFR = \frac{l}{d}$, where $l$ is the missile length, and $d$ is the missile diameter. Typically between 5 and 25.
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#### Nose fineness ratio:
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$NFR = \frac{l}{d}$, where $l$ is the nose length, and $d$ is the maximum nose diameter. Typically between 2 and 4.
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# Rocket Propulsion
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#### Rocket thrust:
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$F = \frac{\dot{W} v_e}{g} + (P_e + P_a)A_e = \frac{P_0 A^*}{C^*}v_e + (P_e + P_a)A_e$, 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.
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#### Mass flow rate:
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$\dot{m} = \frac{\dot{W}}{g}$
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#### Weight flow rate:
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$\dot{W} = g\frac{P_0 A^*}{C^*}$, where $P_0$ is the chamber pressure, $A^*$ is the throat area, and $C^*$ is the characteristic velocity of burned propellants.
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#### Characteristic velocity:
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$C^* = \frac{223}{K}\sqrt{\frac{T_0}{m}}$, where $m$ is molecular weight, $K$ is a function of the specific heat ratio, and $T_0$ is the flame temperature.
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#### Specific impulse:
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$I_{sp} = \frac{F}{\dot{W}}$
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#### Exit velocity:
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$v_e = g I_{sp}$
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#### Ideal burnout velocity:
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$V_{BO_I} = g I_{sp} \ln\left(\frac{W_L}{W_{BO}}\right)$, where $W_L$ is the weight of the vehicle at launch, and $W_{BO}$ is the weight of the vehicle at burnout.
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#### Realistic burnout velocity:
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$V_{BO} = V_{BO_I} - g \sin(\bar{\gamma})T_{BO}$, where $\bar{\gamma}$ is the average flight path angle, and $T_{BO}$ is the time at burnout.
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#### Rocket velocity:
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$v(t) = v_0 + v_e \ln\left(\frac{m_0}{m(t)}\right) - g\sin(\bar{\gamma})t$, where $v_0$ is the initial velocity, $m_0$ is the initial mass, and $m(t)$ is the mass at time $t$.
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# Weapon Control Systems
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#### Total engagement time:
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$TET = \frac{ROF - R_\text{min}}{v_t}$, 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.
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#### Duration of first shot:
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$TOF_1 = \frac{ROF}{v_m + v_t}$, where $v_m$ is the velocity of the missile, and $v_t$ is the velocity of the target.
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#### Depth of fire:
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$DOF = \frac{TET - TOF_1}{T_H} + 1$, where $T_H$ is the homing time.
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#### Time between launches:
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$\Delta T_L = T_H\left(1 + \frac{v_t}{v_m}\right)$, where $T_H$ is homing time, $v_t$ is the velocity of the target, and $v_m$ is the velocity of the missile.
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#### Total launching time:
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$N_L = N \Delta T_L$
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#### Time to go:
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$TGO = \frac{|\vec{R}_{TM}|}{\cos(\theta_m)|\bar{v}_m| + \cos(\theta_t)|v_t}$, 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).
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#### Predicted intercept point:
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$\overrightarrow{PIP} = \vec{R}_T + TGO \vec{v_t}$, where $R_T$ is the current vector from the illuminator to the target.
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#### Power density at the missile seeker:
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$PDMS = \frac{P_T G_T \sigma_{RCS}}{L_{IL} (4\pi)^2 R_T^2 R_{TM}^2}$, 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.
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# Trajectory Design
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#### Midcourse heading error:
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$\varepsilon = \arccos\left(\frac{\vec{R}_{TGO} \cdot \vec{v}_{M}}{|\vec{R}_{TGO}| |\vec{v}_{M}|}\right)$
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#### Terminal guidance heading error:
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$\varepsilon = \arccos\left(\frac{\vec{R}_{TM} \cdot \vec{v}_{TM}}{|\vec{R}_{TM}| |\vec{v}_{TM}|}\right)$
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#### Thrust energy optimization:
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$E_\text{thrust} = E_\text{preburnout drag} + \int\limits_{s_\text{burnout}}^{s_\text{final}} \text{Drag} ds + E_\text{grain} + \frac{1}{2}\frac{W_M}{g}v_\text{final}^2 + W_M(h_\text{final} - h_\text{initial})$, 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.
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#### Optimal dynamic pressure:
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$Q_\text{opt} = \frac{W_M}{s_\text{Ref}}\sqrt{C_A C_{N_\alpha}}$
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#### Cruise altitude:
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$h_\text{opt} \approx 2.3 \times 10^4 \ln\left(\frac{W_M}{s_\text{Ref} M^2 \sqrt{C_A C_{N_\alpha}}}\right)$, where $W_M$ is the weight of the empty missile and $M$ is the Mach number.
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#### Optimal turn:
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$R_\text{opt} = \frac{v^2}{n_z g}$
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