Lateral trim

This section describes lateral trim analysis for the example airplane under two scenarios: one-engine inoperative and cross-wind conditions. In these scenarios, the rudder should be able to trim the airplane such that sideslip angle \(\beta=0\). In both cases, maximum rudder applied should not exceed \(\pm20^{\circ}\). One can also use other requirements (such as spin recovery) for verifying/sizing the rudder. Refer to lecture notes, Section 21.6 in Nicolai and Carichner, and Section 16.4.2 in Raymer for further details.

One-engine out

The one-engine out scenario mentions that rudder should be able to produce enough yawing moment to hold the aircraft at \(\beta=0\) at takeoff speed (1.1 times the stall speed) and CG being at most-aft position. Using yawing moment balance about CG, following condition can be derived (equation 21.24, Nicolai and Carichner) for asymmetric power:

\[ C_{n_{\delta_r}} \delta_r (\bar{X}_{ac_v} - \bar{X}_{cg}) = \frac{T+D}{qS_wb} \bar{Y}_p, \]

where \(\delta_r\) is the rudder deflection (rad), \(T\) is the thrust from one engine, \(D\) is the drag from inoperative engine, \(q\) is the dynamic pressure, and \(\bar{Y}_p\) is the distance between the engine and CG, normalized using wing span \(b\). Other terms have been defined earlier. Above equation can be solved for required \(\delta_r\) to trim the airplane in one-engine out scenario.

The thrust for operative engine is computed using takeoff power from engine and 80% propeller efficiency, while accounting for installation losses. The drag from the inoperative engine is estimated using \(D/q\) for feathered propeller described in parasitic drag section, along with drag from idle engine. The rudder control derivative can be computed using (equation 21.26, Nicolai)

\[ C_{n_{\delta_r}} = 0.9 C_{F_{\beta_v}} V_{vt} \tau, \]

where \(C_{F_{\beta_v}}\) is the side-force curve slope for vertical tail (computed earlier), \(\tau\) is the rudder effectiveness (obtained from figure 21.14, Nicolai) and \(V_{vt}\) is the vertical tail volume coefficient.

Below code block computes thrust and drag for one-engine out scenario:

import numpy as np

rho = 0.00237717 # slugs/cu ft, sea-level
wing_loading = 40 # lbs/sq ft
CLmax_TO = 1.8
Vstall = ( wing_loading * 2 / rho / CLmax_TO )**0.5 # ft/s
V = 1.1 * Vstall # ft/s
q = 0.5 * rho * V**2 # lbs/ft^2

print(f"Speed at takeoff: {V:.2f} ft/s")
print(f"Dynamic pressure at takeoff: {q:.2f} lbs/sq ft")

# Propeller area, sq ft
D = 5.9 # ft
Aprop = np.pi * D**2 / 4

# Thrust
n_prop = 0.8 # prop efficiency
P = 298 # hp, takeoff power
T = 550 * P * n_prop / V # lbs (one-engine only)

D = 0.08 * T + 0.1*Aprop*q # drag of inoperative engine
T = 0.92 * T # account for installation effects

print(f"Thrust from one engine: {T:.2f} lbs")
print(f"Drag from inoperative engine: {D:.2f} lbs")

Speed at takeoff: 150.41 ft/s
Dynamic pressure at takeoff: 26.89 lbs/sq ft
Thrust from one engine: 802.02 lbs
Drag from inoperative engine: 143.25 lbs

Below code block computes required \(\delta_r\) for one-engine out scenario:

# Parameters
mac = 4.3 # ft
Xac_v = 0.25 + 17/mac
Xcg = 0.1861
b = 33 # ft
Sw = 134 # sq ft
tau = 0.58 # figure 21.14, Nicolai
V_vt = 0.065
CFbeta_v = 2.7356 # 1/rad
Yp = 6.5/b # distance between engine and CG

Cn_delta_r = 0.9 * CFbeta_v * V_vt * tau # 1/rad

delta_r = (T + D) * Yp / q / Sw / b / Cn_delta_r / ( (Xac_v - Xcg) * mac / b ) * 180 / np.pi

print(f"Required rudder deflection for one-engine out scenario: {delta_r:.1f} degree")

Required rudder deflection for one-engine out scenario: 1.8 degree

Based on the above calculation, the rudder is more than capable of providing directional control during engine out scenario.

Crosswind

The crosswind scenario requries that aircraft should be able to operate in crosswind speed equal to 20% of takeoff speed. Equivalently, this can be written as \(11.5^{\circ}\) of sideslip at takeoff speed. Using moment balance, requried \(\delta_r\) can be computed using (equation 21.25, Nicolai and Carichner)

\[ C_{n_{\beta}} \beta + C_{n_{\delta_r}} \delta_r = 0. \]

Note that \(C_{n_\beta}\) for takeoff conditions is already computed in yaw static stability section. Below code block computes required \(\delta_r\) for crosswind condition:

Cn_beta = 0.1385 # 1/rad, takeoff
beta = 11.5 * np.pi / 180 # rad, required beta

crosswind_speed = 0.2 * V # ft/s

delta_r = - Cn_beta * beta / Cn_delta_r * 180 / np.pi # deg

print(f"Required rudder deflection for max crosswind speed of {crosswind_speed:.0f} ft/s: {delta_r:.1f} degree")

Required rudder deflection for max crosswind speed of 30 ft/s: -17.2 degree

Based on the above calculation, it can be seen that crosswind condition is the driving requirement for rudder sizing since it requires much larger rudder deflection than one-engine out scenario.

This concludes the lateral trim analysis for the example airplane.