Static yaw stability derivative#
This section computes yaw static stability derivative \(C_{n_{\beta}}\) for the example airplane. The thrust effects are neglected for this analysis. Using equation 16.42 in Raymer, yaw static stability derivative can be written as:
The first, second, and third term represent yawing moment contribution from wing, fuselage, and vertical tail, respectively. Following sub-sections compute these component-level derivatives and estimates \(C_{n_{\beta}}\) for the airplane.
Wing#
The wing contribution towards yawing moment can be computed as (equation 16.44, Raymer):
where \(A\) is the wing aspect ratio and \(C_L\) is the lift coefficient. Note that sweep effects are not included in the above equation since example airplane has zero quarter-chord sweep. Below code block computes \(C_{n_{\beta_w}}\) at three different flight conditions:
import numpy as np
# Parameters
A = 8
mac = 4.3 # ft
CL_landing = 2.2
CL_takeoff = 1.8
CL_cruise = 0.38
Cn_beta_wing_landing = CL_landing**2/np.pi/A/4
Cn_beta_wing_takeoff = CL_takeoff**2/np.pi/A/4
Cn_beta_wing_cruise = CL_cruise**2/np.pi/A/4
print(f"Yaw static stability derivative for wing (1/rad):")
print(f"landing: {Cn_beta_wing_landing:.4f}")
print(f"takeoff: {Cn_beta_wing_takeoff:.4f}")
print(f"cruise: {Cn_beta_wing_cruise:.4f}")
Yaw static stability derivative for wing (1/rad):
landing: 0.0481
takeoff: 0.0322
cruise: 0.0014
Fuselage#
The fuselage contribution towards yawing moment can be computed as (equation 16.50, Raymer):
where \(V\) is the volume of the fuselage, \(b\) is the wing span, \(S_w\) is the wing reference area, \(D_f\) is the height of the fuselage, and \(W_f\) is the width of the fuselage. The \(V\) for the example airplane is computed using OpenVSP model. Below block computes \(C_{n_{\beta_{fus}}}\):
# Parameters
b = 33 # ft
Sw = 134 # sq ft
V = 417 # cu ft, vsp model
Df = 5.75 # ft
Wf = 5 # ft
Cn_beta_fus = -1.3 * V / Sw / b * Df / Wf
print(f"Yaw static stability derivative for fuselage: {Cn_beta_fus:.4f} 1/rad")
Yaw static stability derivative for fuselage: -0.1410 1/rad
Tail#
The tail contribution towards yawing moment can be computed as (equation 16.39, Raymer)
where \(C_{F_{\beta_v}}\) is the side-force curve slope for vertical tail (similar to lift curve slope for horizontal tail) and is computed using the lift curve slope equation described in aerodynamics section. The \(S_v\) and \(S_w\) are vertical tail area and wing reference area, respectively. The \(\bar{X}_{ac_v}\) denotes normalized x-location of the aerodynamic center of vertical tail. Note that \(\bar{X}_{cg}\) should be most aft CG since it yields smallest moment arm.
NOTE: The \(\bar{X}\) quantities in the above expression are normalized using wing span \(b\)
The \(\eta_v\) and \(\partial \beta_v/\partial \beta\) are analogous to \(\eta_h\) and \(\partial \alpha_h/\partial \alpha\), respectively, and their product is computed using
where \(S_{vs}\) is the vertical tail area if it were extended till fuselage centerline, \(Z_{wf}\) is the vertical distance of the wing above the fuselage centerline, and \(A_{wing}\) is the wing aspect ratio. The \(S_{vs}\) is assumed to be same as \(S_v\) since fuselage centerline is close to vertical tail attachment location. Below code block computes \(C_{n_{\beta_v}}\):
# Parameters
A_vt = 1.55 * 1.47 # factor of 1.55 accounts for fusealge and HT
Sv = 17 # sq ft
Svs = 17 # sq ft
Zwf = -2.54 # ft
Xcg = 0.1861
Xac_v = 0.25 + 17/mac # vt momemt arm is 17 ft
# VT force curve slope calculation
M = 0.3
sweep_vt = 30*np.pi/180 # deg, quarter chord
beta = (1 - M**2)**0.5
eta = 1.0
numerator = 2*np.pi*A_vt
denominator = 2 + (4 + (A_vt*beta/eta)**2 * (1 + (np.tan(sweep_vt)/beta)**2) )**0.5
CFbeta_v = numerator / denominator # 1/rad
print(f"Side force curve slope for VT: {CFbeta_v:.4f} 1/rad")
factor = 0.724 + 3.06 * Svs / Sw - 0.4 * Zwf / Df + 0.009 * A
Cn_beta_vt = CFbeta_v * factor * Sv / Sw * (Xac_v - Xcg) * mac / b
print(f"Yaw static stability derivative for vertical tail at cruise: {Cn_beta_vt:.4f} 1/rad")
Side force curve slope for VT: 2.7356 1/rad
Yaw static stability derivative for vertical tail at cruise: 0.2472 1/rad
Final \(C_{n_\beta}\)#
The final \(C_{n_\beta}\) is just the sum of all three components. The contributions from wing for yaw stability are usually small unless flying at high anlge of attack. Below code computes \(C_{n_\beta}\) for the example airplane at three flight conditions:
Cn_beta_landing = Cn_beta_wing_landing + Cn_beta_fus + Cn_beta_vt
Cn_beta_takeoff = Cn_beta_wing_takeoff + Cn_beta_fus + Cn_beta_vt
Cn_beta_cruise = Cn_beta_wing_cruise + Cn_beta_fus + Cn_beta_vt
print(f"Yaw static stability derivative for the airplane (1/rad):")
print(f"landing: {Cn_beta_landing:.4f}")
print(f"takeoff: {Cn_beta_takeoff:.4f}")
print(f"cruise: {Cn_beta_cruise:.4f}")
Yaw static stability derivative for the airplane (1/rad):
landing: 0.1544
takeoff: 0.1385
cruise: 0.1077
For all three conditions, \(C_{n_\beta}\) is positive which implies that airplane will be statically stable in yaw direction. Also, the \(C_{n_\beta}\) values computed above are close to the historical data reported in Figure 16.20 in Raymer. This concludes yaw static stability estimation, next section describes roll static stability.