Formula Sheet · The History of Aircraft Design

Weights and loadings

Weight build-up (Raymer)

\[ \Wo = W_{\text{crew}} + W_{\text{payload}} + \Wf + \We \]

Wing and power loading

\[ \frac{W}{S} = \frac{\Wo}{S}\ \ (\text{kg/m}^2) \] \[ \frac{W}{P} = \frac{\Wo}{P}\ \ (\text{kg/kW}) \]

Thrust and empty weight

\[ \frac{T}{W} = \frac{T}{\Wo g} \] \[ \frac{\We}{\Wo}\ \ \text{(dimensionless)} \]

Multiply \(\WS\) in kg/m² by \(g\) for N/m². \(1\ \text{lb} = 0.4536\ \text{kg}\) (or \(4.448\ \text{N}\)); \(1\ \text{ft} = 0.3048\ \text{m}\); \(1\ \text{ft}^2 = 0.0929\ \text{m}^2\); \(1\ \text{hp} = 0.7457\ \text{kW}\); \(1\ \text{mph} = 1.609\ \text{km/h} = 0.4470\ \text{m/s}\); \(1\ \text{kt} = 1.852\ \text{km/h}\); \(1\ \text{lb/ft}^2 = 4.882\ \text{kg/m}^2\); \(1\ \text{lb/hp} = 0.6083\ \text{kg/kW}\).

Lift, drag and the parabolic drag polar

\[ L = \tfrac12\rho V^2 S\CL, \quad D = \tfrac12\rho V^2 S\CD, \quad \CD = \CDz + K\CL^2,\ \ K = \frac{1}{\pi A e} \]

Maximum \(L/D\)

\[ \LDmax = \frac12\sqrt{\frac{\pi A e}{\CDz}} \] \[ \CL^* = \sqrt{\pi A e\,\CDz},\ \ \CD = 2\CDz \]

Drag area and zero-lift power

\[ f = \CDz S, \quad D_0 = \tfrac12\rho V^2 f \] \[ P_0 = \tfrac12\rho V^3 f \]

\(A = b^2/S\); a biplane with two equal wings: \(A = b^2/(S/2)\). \(e \approx 0.70\) (biplanes) to \(0.85\). Loftin: \(e = 0.70\) for 1914–18, \(0.75\) later.
Induced drag \(\CL^2/(\pi A e)\): Prandtl's lifting-line theory (1918–19), \(e = 1\) for an elliptic lift distribution.

Lilienthal and the Wrights (\(L\) in lb, \(V\) in mph, \(S\) in ft²)

\[ L = k\,V^2 S\,c_l, \qquad \CL = \frac{k}{0.002557}\,c_l \approx 1.28\,c_l \]

Smeaton's coefficient \(k\): traditional 0.005 (50% too high), the Wrights' 0.0033 (1901), modern 0.00326 \(= \tfrac12\rho(1.28)\) in these units.
For a given weight \(V \propto 1/\sqrt{k}\): a design with \(k = 0.005\) needed 23% more speed than expected.

At \(\CLmax\)

\[ V_s = \sqrt{\frac{2}{\rho\CLmax}\,\frac{W}{S}}, \qquad \frac{W}{S} = \tfrac12\rho V_s^2\,\CLmax \]

For fixed \(V_s\) and \(\Wo\): \(S \propto 1/\CLmax\). Flaps raised \(\CLmax\) from about 1.3 to 2 in the 1930s, so wings could shrink.
\(\rho_0 = 1.225\ \text{kg/m}^3\); use \(W/S\) in N/m² and \(V\) in m/s, then \(\times 3.6\) for km/h.

Standard atmosphere, \(h \le 11\ \text{km}\)

\[ T = 288.15 - 0.0065h\ \ \text{(K, m)} \] \[ a = \sqrt{\gamma R T},\ \ M = V/a \]

Simple sweep theory

\[ M_n = M\cos\Lambda \] \[ \Lambda_{\text{cone}} = 90^\circ - \arcsin(1/M) \]

\(\gamma = 1.4\), \(R = 287\ \text{J/(kg·K)}\); \(a_0 = 340.3\ \text{m/s}\); 11–20 km: \(T = 216.65\ \text{K}\), \(a = 295.1\ \text{m/s}\). \(\rho = p/(RT)\).
Thick straight wings: drag rise near \(M = 0.7\). Real swept wings gain less than \(1/\cos\Lambda\). A leading edge swept behind the Mach cone stays subsonic.

Breguet range (jet), and the sizing equation (Raymer)

\[ R = \frac{V}{C}\,\frac{L}{D}\,\ln\frac{W_{i-1}}{W_i}, \qquad \Wo = \frac{W_{\text{crew}} + W_{\text{payload}}}{1 - \Wf/\Wo - \We/\Wo} \]

\(C\) in 1/h with \(V\) in km/h gives \(R\) in km. Typical cruise \(C\): turbojet 0.9, low-bypass turbofan 0.8, high-bypass 0.5 per hour. Propeller: \(V/C \to \eta_p/C_{\text{bhp}}\).
Jets: best range at \(L/D = 0.866\,\LDmax\). Fraction burned \(= 1 - W_1/W_0 = 1 - e^{-RC/(V\,L/D)}\).
Empty-weight trend \(\We/\Wo = A\,\Wo^{\,C}\) (Raymer Table 3.1); fit to Loftin's jet transports: \(A = 1.117\), \(C = -0.0698\) (kg). Solve for \(\Wo\) by iteration.

1799 Cayley: fixed wing, lift and drag · 1891–96 Lilienthal's glides and tables · 1901 Wrights' wind tunnel · 1903 Wright Flyer · 1909 Blériot crosses the Channel
1915 Junkers J 1, all metal · 1918 Fokker D.VII; Prandtl's lifting-line theory · 1927 Lockheed Vega · 1928 NACA cowling · 1933 Boeing 247 · 1935 Douglas DC-3
1939 He 178, first jet · 1947 F-86 and B-47 swept wings; X-1 passes Mach 1 · 1949 Comet · 1952 area rule · 1954 Dash 80 (707)
1969 Boeing 747, high-bypass turbofans; Concorde · 1970s supercritical wings, winglets · 1987 A320 fly-by-wire · 2009 787, composite primary structure

kg is not a force. Use \(W = mg\) in N in the lift equation and in \(\TW\); ratios like \(\We/\Wo\) do not care.
Unconverted data. Loftin and Raymer use lb, ft², hp, mph: convert before dividing.
km/h in a formula. Divide by 3.6 before squaring or cubing a speed.
\(A = b/S\). Aspect ratio is \(b^2/S\) (or \(b/c\) for a rectangular wing).
Fuel burned is not \(\ln\). Burning 30% gives \(\ln(1/0.7) = 0.357\).
Trend constants in the wrong units. \(A\) changes between kg and lb; \(C\) does not.