Static roll stability derivative

This section computes roll static stability derivative \(C_{l_{\beta}}\) for the example airplane. The thrust effects are neglected for this analysis. Using equation 16.43 in Raymer, roll static stability derivative can be written as

\[ C_{l_{\beta}} = C_{l_{\beta_w}} + C_{l_{\beta_v}}. \]

The first and second term represent rolling moment contribution from wing and vertical tail, respectively. The fuselage contribution towards rolling is usually based on wing-fuselage intersection which is already included in \(C_{l_{\beta_w}}\). Following sub-sections compute these component-level derivatives and estimates \(C_{l_{\beta}}\) for the example airplane.

Wing

The wing contribution towards rolling moment can be computed as (equation 16.47, Raymer):

\[ C_{l_{\beta_w}} = \left( \frac{C_{l_{\beta_w}}}{C_L} \right) C_L + (C_{l_\beta})_\Gamma + C_{l_{\beta_{wf}}} \]

The first term represents rolling moment derivative for a swept, tapered wing with no geometric dihedral. The second term accounts for the geometric dihedral while the last term is for wing-fuselage interaction.

The \(C_{l_{\beta_w}}/C_L\) is computed as \(-0.02 \text{ rad}^{-1}\) using Figure 16.21 in Raymer based on the aspect ratio and taper ratio of the wing. The \((C_{l_\beta})_\Gamma\) is computed using (equation 16.45, Raymer)

\[ (C_{l_\beta})_\Gamma = -\frac{C_{L_{\alpha}} \Gamma}{4} \left[ \frac{2(1 + 2\lambda)}{3(1 + \lambda)} \right], \]

where \(\Gamma\) is the geometric dihedral angle (in radians), \(C_{L_{\alpha}}\) is the lift-curve slope and \(\lambda\) is the taper ratio. The wing-fuselage interaction is computed using (equation 16.46, Raymer)

\[ C_{l_{\beta_{wf}}} = -1.2 \frac{\sqrt{A} Z_{wf} (D_f + W_f)}{b^2}, \]

where \(Z_{wf}\) is the vertical distance of the wing above the fuselage centerline, \(D_f\) is the height of the fuselage, \(W_f\) is the width of the fuselage, and \(A\) is the wing aspect ratio.

Below block computes \(C_{l_{\beta_w}}\) at three different flight conditions:

import numpy as np

# Parameters
A = 8
b = 33 # ft
taper_ratio = 0.4
dihedral = 5 * np.pi / 180 # rad
Df = 5.75 # ft
Wf = 5 # ft
Zwf = -2.54 # ft
CL_alpha = 5.0 # 1/rad

# Lift coefficient
CL_cruise = 0.38
CL_takeoff = 1.8
CL_landing = 2.2

# rolling moment derivative
Cl_beta_w_CL = -0.02 # 1/rad
Cl_beta_gamma = - CL_alpha * dihedral / 4 * ( 2 * (1 + 2*taper_ratio) / 3 / (1 + taper_ratio) ) # 1/rad
Cl_beta_wf = -1.2 * A**0.5 * Zwf * (Df + Wf) / b**2

Cl_beta_w_cruise = Cl_beta_w_CL * CL_cruise + Cl_beta_gamma + Cl_beta_wf
Cl_beta_w_takeoff = Cl_beta_w_CL * CL_takeoff + Cl_beta_gamma + Cl_beta_wf
Cl_beta_w_landing = Cl_beta_w_CL * CL_landing + Cl_beta_gamma + Cl_beta_wf

print(f"Roll static stability derivative for wing (1/rad):")
print(f"cruise: {Cl_beta_w_cruise:.4f}")
print(f"takeoff: {Cl_beta_w_takeoff:.4f}")
print(f"landing: {Cl_beta_w_landing:.4f}")

Roll static stability derivative for wing (1/rad):
cruise: -0.0160
takeoff: -0.0444
landing: -0.0524

Tail

The tail contribution towards rolling moment can be computed as (equation 16.41, Raymer)

\[ C_{l_{\beta_v}} = - C_{F_{\beta_v}} \frac{\partial \beta_v}{\partial \beta} \eta_v \frac{S_v}{S_w} \bar{Z}_v. \]

Some of the terms in the above equation are defined and computed in yaw static stability derivative section. The \(\bar{Z}_v\) represents the vertical distance of the tail aerodynamic center from the CG, normalized using wing span \(b\). Below code block computes \(C_{l_{\beta_v}}\):

# Parameters
Zv = 2.5 / b
CFbeta_v = 2.7356
factor = 1.3609
Sv = 17 # sq ft
Sw = 134 # sq ft

Cl_beta_vt = - CFbeta_v * factor * Sv / Sw * Zv

print(f"Roll static stability derivative for vertical tail: {Cl_beta_vt:.4f} 1/rad")

Roll static stability derivative for vertical tail: -0.0358 1/rad

Final \(C_{l_\beta}\)

The final \(C_{l_\beta}\) is sum of the individual components. Below code computes \(C_{l_\beta}\) for the example airplane at three flight conditions:

Cl_beta_landing = Cl_beta_w_landing + Cl_beta_vt
Cl_beta_takeoff = Cl_beta_w_takeoff + Cl_beta_vt
Cl_beta_cruise = Cl_beta_w_cruise + Cl_beta_vt

print(f"Roll static stability derivative for the airplane (1/rad):")
print(f"cruise: {Cl_beta_cruise:.4f}")
print(f"takeoff: {Cl_beta_takeoff:.4f}")
print(f"landing: {Cl_beta_landing:.4f}")

Roll static stability derivative for the airplane (1/rad):
cruise: -0.0518
takeoff: -0.0802
landing: -0.0882

For all three conditions, \(C_{l_\beta}\) is negative which implies that airplane will be statically stable in roll direction. This concludes roll static stability estimation, next section describes lateral direction trim analysis.