Static Stability and Control

AERE 3550 Flight Dynamics Chapter 2 Nelson

Pitch Static Stability

• Lift Force Model for an Airfoil:

  • The standard model for the lift force coefficient on an airfoil is:         $C_L = C_{L\forall}(\alpha - \alpha_0)$ where $\alpha_0$ represents the zero lift line (ZLL) angle of attack, and $\alpha_0$ is the angle of attack.

• Geometric and Scaled Parameters:

  • Let $x_{cg}$denote the distance from the wing leading edge to the plane center of gravity (cg).

  • Let $x_{ac}$ denote the distance from the wing leading edge to the plane aerodynamic center (ac).

  • The scaled center of gravity h and scaled aerodynamic center $h_{ac}$ (or $h_{n}$) are normalized by the airfoil mean chord length c:         $h = \frac{x_{cg}}{c}$         $h_{ac} = \frac{x_{ac}}{c}$

  • The parameter h represents the distance from the wing leading edge to the aircraft cg.

• Airfoil Pitching Moment Coefficient:

  • Scaled by $Q S c$, the pitching moment coefficient about the center of gravity is given by:         $C_m = C_{mac} + C_L(h - h_{ac}) = C_{mac} + C_{L\alpha}(\alpha - \alpha_0)(h - h_{ac})$

  • The pitch moment derivative with respect to angle of attack is:         $C_{m\alpha} = \frac{dC_m}{d\alpha} = C_{L\alpha}(h - h_{ac})$

  • For the wing alone to act as a stabilizing component of the aircraft, it must satisfy $C_{m\alpha}<0$.

  • Because $C_{L\alpha}>0$ always holds, achieving $C_{m\alpha}<0$ requires $h<h_{ac}$ . In typical wing configurations, this condition does not hold, meaning the isolated wing is inherently a destabilizing component.

• Pitch Stability Cases:

  • Plane Scenario #1: $C_{m\alpha}>0$ . A positive perturbation in the pitch angle $\alpha$ generates a positive pitching moment $C_{m}$ , driving $\alpha$further away from equilibrium. Plane #1 is longitudinally unstable.

  • Plane Scenario #2: $C_{m\alpha}<0$ , but the equilibrium point $\alpha_0$ (where $C_{m}(\alpha_0)=0$) is negative ($\alpha_0<0$). This corresponds to a wing pitched downward, which is non-existent in practical aircraft flight operations.

  • Plane Scenario #3: $C_{m\alpha}<0$ and $\alpha_0>0$. This configuration represents a longitudinally stable, realistic aircraft.

• Pitch Stiffness Definitions:

  • For an aircraft in a longitudinally balanced (equilibrium) condition at angle of attack $\alpha_0\ge0$, consider a disturbance shifting the angle of attack to $\alpha=\alpha_0+\Delta\alpha$

  • Positive Pitch Stiffness (Longitudinal Stability): The non-zero pitching moment $C_{m}\left(\alpha\right)$acts to restore $\alpha$ back toward $\alpha_0$. Mathematically:         $\frac{\partial C_m}{\partial \alpha}\Big|_{\alpha=\alpha_0} < 0$

  • Negative Pitch Stiffness (Longitudinal Instability): The resulting moment drives $\alpha$ further away from $\alpha_0$. Mathematically:         $\frac{\partial C_m}{\partial \alpha}\Big|_{\alpha=\alpha_0} > 0$

  • Zero Pitch Stiffness (Neutral Longitudinal Stability):         $\frac{\partial C_m}{\partial \alpha}\Big|_{\alpha=\alpha_0} = 0$

Component Contributions to Longitudinal Stability

• Wing / Wing-Body Pitch Coefficients:

  • $C_{Lw} = C_{L0w} + C_{L\alpha w}\alpha_w$

  • $C_{mw} = C_{m0w} + C_{m\alpha w}\alpha_w$

  • $C_{m0w} = C_{macw} + C_{L0w}(h - h_{nwb})$

  • $C_{m\alpha w} = C_{L\alpha w}(h - h_{nwb})$

  • For wing-body combinations, subscript $w$ is replaced by $wb$. When only wing-alone or tail-alone lift derivatives are provided, they must be corrected for finite aspect ratio (AR):         $C_{L\alpha wb} = \frac{C_{L\alpha w}}{1 + \frac{C_{L\alpha w}}{\pi AR}}$

• Tail Pitch Coefficients:

  • $C_{Lt} = C_{L\alpha t}\alpha_t = C_{L\alpha t}\left[\alpha_w\left(1 - \frac{d\epsilon}{d\alpha}\right) - i_w - i_t - \epsilon_0\right]$

  • $C_{mt} = C_{m0t} + C_{m\alpha t}\left[\alpha_w\left(1 - \frac{d\epsilon}{d\alpha}\right)\right]$

  • $C_{m0t} = \eta V_H C_{L0t}(i_w + i_t + \epsilon_0)$

  • Horizontal Tail Volume Ratio ($V_{H}$):         $V_H = \frac{l_t S_t}{c S_w}$  where $l_{t}$ is the distance between wing and tail mean aerodynamic centers, $S_{t}$ is tail area, $S_{w}$ is wing area, and c is mean chord length.

  • Tail Efficiency ($\eta$):        

     $\eta = \frac{Q_t}{Q_w} = \frac{0.5\,\rho_t V_t^2}{0.5\,\rho_w V_w^2}$       

     Typically, $0.8<\eta<1.2$.

  • Tail Downwash Angle ($\epsilon$):         $\epsilon = \epsilon_0 + \frac{d\epsilon}{d\alpha}\alpha_w$         where the downwash gradient is:         $\frac{d\epsilon}{d\alpha} = \frac{2 C_{L\alpha w}}{\pi AR_w}$

• Entire Aircraft Coefficients (Wing/Body + Tail):

  • Total Lift Coefficient:         $C_L = C_{Lwb} + C_{Lt}\frac{S_t}{S_w}$         $C_L = C_{L0} + C_{L\alpha}\alpha$         where:         $C_{L0} = C_{L0wb} + \eta C_{L\alpha t}\frac{S_t}{S_w}(i_w + i_t + \epsilon_0)$         $C_{L\alpha} = C_{L\alpha wb} + \eta C_{L\alpha t}\frac{S_t}{S_w}\left(1 - \frac{d\epsilon}{d\alpha}\right)$

  • Total Pitching Moment Coefficient:         $C_m = C_{m0} + C_{m\alpha}\alpha$         where:         $C_{m0} = C_{m0wb} + \eta V_H C_{L0t}(i_w + i_t + \epsilon_0)$         $C_{m\alpha} = C_{L\alpha wb}(h - h_{acwb}) - \eta V_H C_{L\alpha t}\left(1 - \frac{d\epsilon}{d\alpha}\right)$

Design Calculation Example: Sizing the Tail

• Given Aircraft Parameters:

  • Wing/body moment equation: $C_{mwb} = -0.05 - 0.0035\alpha$ (with $\alpha$ in degrees).

  • Wing data: $C_{L0w} = 0.26$, $S_w = 178\,ft^2$, $b_w = 35.9\,ft$, $c = 5\,ft$, $AR_w = 7.3$, $i_w = 2^\circ$, $C_{L\alpha wb} = 0.07\,deg^{-1} = 4.01\,rad^{-1}$, $h = 0.1$.

  • Tail data: $l_t = 14.75\,ft$, $AR_t = 4.85$, $C_{L\alpha t} = 0.073\,deg^{-1} = 4.18\,rad^{-1}$, $\eta = 1.0$, $C_{L0t}(\alpha=0) = 0.26$.

  • Target aircraft moment equation: $C_m = 0.15 - 0.025\alpha$.

• Step-by-Step Solution:

  1. Determine Wing/Body Neutral Point ($h_{nwb}$):         $C_{m\alpha wb} = C_{L\alpha wb}(h - h_{nwb}) \implies -0.0035 = 0.07(0.1 - h_{nwb})$         $0.1 - h_{nwb} = -0.05 \implies h_{nwb} = 0.15$

  2. Calculate Downwash Gradient ($\frac{d\epsilon}{d\alpha}$):         $\frac{d\epsilon}{d\alpha} = \frac{2 C_{L\alpha w}}{\pi AR_w} = \frac{2(0.07)}{\pi (7.3)} \approx 0.0226$

  3. Solve for Horizontal Tail Volume Ratio ($V_H$) and Tail Area ($S_t$):         $C_{m\alpha} = C_{m\alpha wb} - \eta V_H C_{L\alpha t}\left(1 - \frac{d\epsilon}{d\alpha}\right)$         $-0.025 = -0.0035 - (1.0) V_H (0.073)(1 - 0.0226)$         $-0.0215 = -0.07136 V_H \implies V_H = 0.453$         Using $V_H = \frac{l_t S_t}{c S_w}$:         $0.453 = \frac{14.75 S_t}{5(178)} \implies S_t = 27.3\,ft^2$

  4. Solve for Tail Incidence Angle ($i_t$):         $C_{m0} = C_{m0wb} + \eta V_H C_{L\alpha t}(i_w + i_t)$         $0.15 = -0.05 + (1.0)(0.453)(0.073)(2^\circ + i_t)$         $0.20 = 0.03307(2^\circ + i_t) \implies i_t = -2.75^\circ$

Neutral Point and Static Margin

• Stick-Fixed Neutral Point Definition ($h_n$ or $h_{NP}$):

  • The point where $C_{m\alpha} = 0$ defines neutral stability for the complete aircraft:         $h_n = h_{acwb} + \eta V_H \frac{C_{L\alpha t}}{C_{L\alpha wb}}\left(1 - \frac{d\epsilon}{d\alpha}\right)$

  • For the wing/body alone, the neutral point is $h_{acwb}$. The tail adds a positive stability margin (safety cushion):         $\Delta h = \eta V_H \frac{C_{L\alpha t}}{C_{L\alpha wb}}\left(1 - \frac{d\epsilon}{d\alpha}\right)$

  • In the design example above, $h_{acwb} = 0.15$, and the tail contribution is $\Delta h = 0.307$, elevating the overall neutral point to $h_n = 0.457$.

• Static Margin ($K_n$):

  • Defined as the negative distance between the center of gravity and the stick-fixed neutral point:         $K_n = -(h - h_n) = h_n - h$

  • Positive pitch stiffness ($C_{m\alpha} < 0$) is maintained whenever $h < h_n$.

• Alternative Formulation & Experimental Neutral Point Estimation:

  • Let $h_{nt} - h = \frac{l_t}{c}$. Then $V_H = \frac{S_t l_t}{S_w c} = \frac{S_t}{S_w}(h_{nt} - h)$.

  • Expressing moment in terms of lift:         $C_m = C_{macwb} + C_{Lwb}(h - h_{nwb}) - \eta \frac{S_t}{S_w}(h_{nt} - h) C_{Lt}$

  • Differentiating $C_m$ with respect to total lift coefficient $C_L$:         $\frac{dC_m}{dC_L} = h - h_n$         $h_n \approx h - \frac{dC_m}{dC_L}$

  • This equation allows experimental estimation of $h_n$ by measuring the change in pitching moment coefficient resulting from small changes in lift coefficient across test angles of attack.

• Example 2: Rearward CG Limit Calculation:

  • Given: Most rearward CG position limit $x_{cg} = 25\,ft$, $l_t = 55\,ft$, $x_{acwb} = 21\,ft$, mean chord $c = 19.26\,ft$, required static margin $K_n = h_n - h \ge 0.05$.

  • Assumptions: $C_{L\alpha t} = C_{L\alpha wb}$, $\eta = 1.0$, $\frac{\partial\epsilon}{\partial\alpha} = 0.25$.

  • Normalized positions: $h = \frac{25}{19.26} = 1.30$, $h_{nwb} = \frac{21}{19.26} = 1.09$.

  • $V_H = \frac{S_t}{S_w}\left(\frac{55}{19.26}\right) = 2.856\frac{S_t}{S_w}$

  • $h_n = 1.09 + (1.0)\left(2.856\frac{S_t}{S_w}\right)(1)(1 - 0.25) = 1.09 + 2.142\frac{S_t}{S_w}$

  • Enforcing static margin requirement:         $h_n - h = 0.05 \implies \left(1.09 + 2.142\frac{S_t}{S_w}\right) - 1.30 = 0.05$         $2.142\frac{S_t}{S_w} = 0.26 \implies \frac{S_t}{S_w} = 0.131$

• Independence of $C_{m0}$ from CG Location:

  • The zero-lift pitching moment coefficient $C_{m0}$ is independent of the center of gravity position $h$. Algebraic reduction demonstrates that CG shift terms cancel at zero lift.

Longitudinal Control

• Elevator Deflection ($\delta_e$):

  • Downward elevator deflection is defined as positive ($\delta_e > 0$).

  • A positive deflection generates positive lift ($\Delta L > 0$) and a negative pitching moment ($\Delta M < 0$).

• Linearized Control Equations:

  • $C_L = C_{L0} + C_{L\alpha}\alpha + C_{L\delta e}\delta_e$

  • $C_m = C_{m0} + C_{m\alpha}\alpha + C_{m\delta e}\delta_e$

• Example 3: NAVION Airplane Equilibrium and Acceleration Analysis:

  • Aircraft Data: $W = 2750\,lbs$, $S = 184\,ft^2$, $\rho = 0.002377\,slugs/ft^3$, $V = 158\,ft/s$, $C_{L\alpha} = 4.44\,rad^{-1}$, $C_{m\alpha} = -0.683\,rad^{-1}$, $C_{L\delta e} = 0.355\,rad^{-1}$, $C_{m\delta e} = -0.923\,rad^{-1}$.

  • Level Equilibrium Flight ($\delta_e = 0$):         $C_L = \frac{W}{0.5\,\rho V^2 S} = \frac{2750}{0.5(0.002377)(158)^2(184)} = 0.40417$         $\alpha_0 = \frac{C_L}{C_{L\alpha}} = \frac{0.40417}{4.44} = 0.091\,rad = 5.214^\circ$         $C_{m0} = -C_{m\alpha}\alpha_0 = -(-0.683)(0.091) = 0.062$

  • Vertical Acceleration of $0.1g$:         $L = 1.1 W = 3025\,lbs \implies C_L = 0.40417 \times 1.1 = 0.4445$         System matrix equation:         $\begin{bmatrix} C_{L\alpha} & C_{L\delta e} \\ C_{m\alpha} & C_{m\delta e} \end{bmatrix} \begin{bmatrix} \alpha \\ \delta_e \end{bmatrix} = \begin{bmatrix} C_L - C_{L0} \\ -C_{m0} \end{bmatrix}$         $\begin{bmatrix} 4.44 & 0.355 \\ -0.683 & -0.923 \end{bmatrix} \begin{bmatrix} \alpha \\ \delta_e \end{bmatrix} = \begin{bmatrix} 0.4445 \\ -0.062 \end{bmatrix}$         Solving the linear system yields:         $\alpha = 0.1007\,rad = 5.77^\circ$         $\delta_e = -0.0073\,rad = -0.42^\circ$

• Control Derivative Expressions:

  • Flap effectiveness parameter:         $\tau = \frac{d\alpha_t}{d\delta_e}$

  • Elevator lift derivative:         $C_{L\delta e} = \eta \frac{S_t}{S_w} C_{L\alpha t} \tau$

  • Elevator Control Power ($C_{m\delta e}$):         $C_{m\delta e} = -\eta V_H C_{L\alpha t} \tau = -\eta \frac{S_t l_t}{S_w c} C_{L\alpha t} \tau$

  • In terms of scaled CG position $h$:         $C_{m\delta e} = -\eta \frac{S_t}{S_w} C_{L\alpha t} \tau \left[\frac{l_t}{c} - (h - h_{nwb})\right]$

• Example 4: NAVION Elevator Area Determination:

  • Given: $V_H = 0.66$, $\eta = 1.0$, $C_{L\alpha t} = 3.91\,rad^{-1}$, $C_{m\delta e} = -0.923$, tail area $S_t = 43\,ft^2$.

  • $C_{m\delta e} = -\eta V_H C_{L\alpha t} \tau \implies -0.923 = -(1.0)(0.66)(3.91)\tau \implies \tau = 0.358$

  • From empirical flap effectiveness charts, $\tau = 0.358$ corresponds to an area ratio $\frac{S_e}{S_t} \approx 0.17$.

  • Elevator area:         $S_e = 0.17 S_t = 0.17(43) = 7.31\,ft^2$

• Trimmed Flight Conditions and Trimmed Lift Curve:

  • Setting pitching moment to equilibrium ($C_m = 0$):         $\alpha_{trim} = \frac{C_{m0} C_{L\delta e} - C_{Ltrim} C_{m\delta e}}{\det}$         $\delta_{etrim} = \frac{-C_{m0} C_{L\alpha} - C_{Ltrim} C_{m\alpha}}{\det}$         where $\det = C_{L\alpha} C_{m\delta e} - C_{L\delta e} C_{m\alpha}$.

  • Trimmed Lift Curve Equation:         $C_{Ltrim} = \left[C_{L\alpha} - C_{L\delta e}\frac{C_{m\alpha}}{C_{m\delta e}}\right]\alpha_{trim} + C_{L0} - C_{L\delta e}\frac{C_{m0}}{C_{m\delta e}}$

  • Key Features of Trimmed Lift:

    1. Trimmed lift is linear with respect to $\alpha_{trim}$.

    2. The slope of the trimmed lift curve is smaller than the basic untrimmed lift curve slope $C_{L\alpha}$ by a factor of $\frac{C_{L\delta e} C_{m\alpha}}{C_{m\delta e}}$.

    3. The zero-trimmed-lift angle of attack is positive.

    4. The lift at zero trimmed angle of attack ($\alpha_{trim} = 0$) is positive.

• Example 5: Forward CG Limit Analysis:

  • Given aircraft curves with $C_{L0} = 0.03 + 0.08\alpha\,(deg)$, $\frac{dC_m}{dC_L} = -0.15$, $h = 0.25$, $\delta_e$ bounds $-15^\circ \le \delta_e \le 15^\circ$.

  • Fixed-stick neutral point: $h_n = h - \frac{dC_m}{dC_L} = 0.25 - (-0.15) = 0.40$.

  • Control power: $C_{m\delta e} = -0.028\,deg^{-1}$.

  • Maximum lift coefficient at $\alpha = 15^\circ$: $C_{Lmax} = 0.03 + 0.08(15) = 1.23$.

  • Maximum positive elevator moment from $\delta_{emax} = -15^\circ$:         $\Delta C_{mmax} = C_{m\delta e} \delta_{emax} = -0.028(-15) = 0.42$

  • Trim moment balance at landing condition (RED line):         $C_{m0} + C_{m\alpha}\alpha_{max} + C_{m\delta e}\delta_{emax} = 0$         $0.17 + C_{m\alpha}(15) - 0.42 = 0 \implies C_{m\alpha} = -0.039\,deg^{-1}$

  • Forward CG Limit calculation:         $h_{forward} = h_n + \frac{C_{m\alpha}}{C_{L\alpha}} = 0.40 + \frac{-0.039}{0.08} = -0.087$

• Example 6: Stick-Free Neutral Point ($h_n^{\prime}$):

  • Hinge moment parameters: $C_{h0} = 0$, $C_{h\alpha} = -0.003\,deg^{-1}$, $C_{h\delta} = -0.005\,deg^{-1}$, $\tau = 0.55$, $V_H = 0.4$, $\eta = 1.0$.

  • Elevator free factor ($f$):         $f = 1 - \frac{C_{h\alpha}}{C_{h\delta}}\tau$         Using provided aircraft empirical factor $f = 0.88$.

  • Stick-Free Neutral Point Equation:         $h_n^{\prime} = h_n - \eta V_H \frac{C_{L\alpha t}}{C_{L\alpha wb}}\left(1 - \frac{d\epsilon}{d\alpha}\right)f$         $h_n^{\prime} = 0.4 - 0.0256 = 0.3744$

  • The stick-free neutral point ($0.3744$) lies ahead of the stick-fixed neutral point ($0.40$).

Directional (Weathercock) Static Stability

• Sideslip Angle ($\beta$) and Sign Convention:

  • Sideslip $\beta$ is positive when the relative wind comes from the right side of the aircraft nose.

  • Clockwise yawing moments ($N$) are defined as positive.

  • Directional or weathercock stability requires a positive yawing moment derivative:         $C_{n\beta} = \frac{\partial C_n}{\partial \beta} > 0$

• Wing/Fuselage Contribution ($C_{n\beta wb}$):

  • $C_{n\beta wb} = -k_n k_{Rl} \frac{S_{fs} l_f}{S b}$

  • $k_n$: Empirical wing/body interference factor (function of CG position $x_m/l_f$, fuselage height ratio $h_1/h_2$, and maximum depth/width).

  • $k_{Rl}$: Empirical correction factor for fuselage Reynolds number $R_l = \frac{V l_f}{\nu}$.

  • $S_{fs}$: Projected side area of the fuselage.

  • $l_f$: Total length of the fuselage.

  • $S, b$: Wing area and wing span.

• Vertical Tail Contribution ($C_{n\beta v}$):

  • Side force generated at vertical tail:         $Y_v = -C_{L\alpha v} \alpha_v Q_v S_v$

  • Vertical tail effective angle of attack:         $\alpha_v = \beta + \sigma$         where $\sigma$ is the sidewash angle.

  • Side force derivative:         $C_{L\beta v} = C_{L\alpha v}\left(1 + \frac{\partial\sigma}{\partial\beta}\right) > 0$

  • Restoring yawing moment:         $N_v = -Y_v l_v = C_{L\alpha v}(\beta + \sigma) Q_v S_v l_v$

  • Vertical tail stability derivative:         $C_{n\beta v} = \eta_v V_v C_{L\alpha v}\left(1 + \frac{\partial\sigma}{\partial\beta}\right)$         where $V_v = \frac{S_v l_v}{S b}$ is the vertical tail volume ratio, and $\eta_v = \frac{Q_v}{Q}$.

• Empirical Sidewash Gradient Expression:

  • $\left(1 + \frac{\partial\sigma}{\partial\beta}\right)\eta_v = 0.724 + 3.06\frac{S_v / S}{1 + \cos\Lambda_{c/4 w}} + 0.4\frac{z_w}{d} + 0.009 AR_w$

  • $z_w$: Distance parallel to z-axis from wing root quarter-chord to fuselage centerline.

  • $d$: Maximum vertical depth of fuselage.

  • $\Lambda_{c/4 w}$: Sweep angle of wing quarter-chord.

• Rudder Control Derivatives:

  • Rudder angle $\delta_r$ deflection produces yawing moment:         $\Delta C_n = C_{n\delta r} \delta_r = -\eta_v V_v C_{L\alpha v} \tau \delta_r$

  • Rudder Control Effectiveness ($C_{n\delta r}$):         $C_{n\delta r} = -\eta_v V_v C_{L\alpha v} \tau = -\eta_v V_v C_{L\delta r}$

  • Interrelations between directional derivatives:         $C_{n\delta r} = -C_{n\beta v} \frac{\tau}{1 + \frac{\partial\sigma}{\partial\beta}}$         $\frac{\partial\sigma}{\partial\beta} = -1 - \frac{C_{n\beta v}}{C_{n\delta r}}\tau$

• Example 7: NAVION Sidewash Gradient Evaluation:

  • Given derivatives: $C_{l\beta} = -0.074\,rad^{-1}$, $C_{n\beta} = 0.071\,rad^{-1}$ (corrected from sign typo $-0.071$), $C_{l\delta r} = 0.107\,rad^{-1}$, $C_{n\delta r} = -0.072\,rad^{-1}$, $V_v \eta_v = 0.66$.

  • Permissible range for flap effectiveness $\tau$: $0 \le \tau \le 0.8$.

  • Evaluating sidewash gradient:         $\frac{\partial\sigma}{\partial\beta} = -1 - \frac{0.071}{-0.072}\tau = -1 + 0.986\tau$

  • For $0 \le \tau \le 0.8$, the gradient satisfies $\frac{\partial\sigma}{\partial\beta} \le -0.47$.

  • Because $\frac{\partial\sigma}{\partial\beta} < 0$, sidewash reduces $C_{n\beta v}$, exerting a destabilizing effect on weathercock stability.

• Example 8: Vertical Tail Sizing Design Calculation:

  • Target stability: $C_{n\beta} = +0.1\,rad^{-1} = 0.0017\,deg^{-1}$ at speed $V = 150\,m/s$ (sea level, $\nu = 1.46 \times 10^{-5}\,m^2/s$).

  • Dimensions: $S = 21.3\,m^2$, $b = 10.4\,m$, $z_w = 0.4\,m$, $d = 1.6\,m$, $l_f = 13.7\,m$, $x_m = 8\,m$, $w_f = 1.6\,m$, $h_1 = 15.4\,m$, $h_2 = 1.6\,m$, $s = 1.07\,m$, $l_v = 4\,m$, $C_{L\alpha v} = 0.1\,deg^{-1} = 5.73\,rad^{-1}$, $\Lambda_{c/4 w} = 15^\circ = 0.262\,rad$.

  • Fuselage Contribution ($C_{n\beta wb}$):         $\frac{x_m}{l_f} = \frac{8}{13.7} = 0.58 \implies k_n \approx 0.0015$         $R_l = \frac{150(13.7)}{1.46 \times 10^{-5}} = 1.41 \times 10^8 \implies k_{Rl} \approx 2.0$         $C_{n\beta wb} = -(0.0015)(2.0)\frac{15.4(13.7)}{21.3(10.4)} = -0.0029\,deg^{-1} = -0.166\,rad^{-1}$

  • Required Vertical Tail Contribution ($C_{n\beta v}$):         $C_{n\beta v} = C_{n\beta} - C_{n\beta wb} = 0.1 - (-0.166) = 0.266\,rad^{-1}$

  • Sidewash & Area Calculation:         $AR_w = \frac{b^2}{S} = \frac{(10.4)^2}{21.3} = 5.08$         $\left(1 + \frac{\partial\sigma}{\partial\beta}\right)\eta_v = 0.724 + 3.06\frac{S_v / 21.3}{1 + \cos(15^\circ)} + 0.4\frac{0.4}{1.6} + 0.009(5.08) = 0.873 + 0.1437 S_v$         $V_v = \frac{S_v l_v}{S b} = \frac{4 S_v}{21.3(10.4)} = 0.01805 S_v$         Substituting into $C_{n\beta v} = V_v C_{L\alpha v} \left(1 + \frac{\partial\sigma}{\partial\beta}\right)\eta_v$:         $0.266 = (0.01805 S_v)(5.73)(0.873 + 0.1437 S_v)$         $0.266 = 0.1034 S_v (0.873 + 0.1437 S_v) = 0.09026 S_v + 0.01486 S_v^2$         $0.01486 S_v^2 + 0.09026 S_v - 0.266 = 0$         Solving the quadratic equation yields:         $S_v = 3.73\,m^2$

Roll Static Stability

• Definition & Condition:

  • An airplane possesses static roll stability if a restoring rolling moment is developed when disturbed from a wings-level attitude.

  • Condition for static roll stability:         $C_{l\beta} = \frac{\partial C_l}{\partial \beta} < 0$

• Roll-Induced Sideslip Mechanics:

  • Initial velocity components before roll: $[u, v, w] = [V\cos\alpha, 0, V\sin\alpha]$.

  • Velocity vector in body coordinates after pure roll angle $\phi$:         $\begin{bmatrix} u^{\prime} \\ v^{\prime} \\ w^{\prime} \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos\phi & \sin\phi \\ 0 & -\sin\phi & \cos\phi \end{bmatrix} \begin{bmatrix} V\cos\alpha \\ 0 \\ V\sin\alpha \end{bmatrix} = \begin{bmatrix} V\cos\alpha \\ V\sin\alpha\sin\phi \\ V\sin\alpha\cos\phi \end{bmatrix}$

  • Sideslip velocity generated: $v^{\prime} = V\sin\alpha\sin\phi$.

  • Induced sideslip angle:         $\beta = \sin^{-1}\left(\frac{v^{\prime}}{V}\right) \approx \sin\alpha\sin\phi \approx \alpha\phi$

  • Roll moment derivative with respect to bank angle $\phi$:         $C_l = C_{l\beta}\beta = C_{l\beta}\alpha\phi$         $C_{l\phi} = \frac{\partial C_l}{\partial \phi} = C_{l\beta}\alpha$

  • For positive angle of attack ($\alpha > 0$), positive roll stiffness ($C_{l\phi} < 0$) requires $C_{l\beta} < 0$.

• Dihedral Effect ($\Gamma$):

  • Dihedral angle $\Gamma$ produces a change in angle of attack on the lowered wing:         $\Delta\alpha \approx \beta \tan\Gamma \approx \beta\Gamma$

  • The lowered wing experiences increased lift relative to the raised wing, generating a negative restoring rolling moment.

• Aileron Roll Control & Control Power:

  • Incremental rolling moment from an aileron element at span location $y$:         $\Delta L = \Delta(y) \times y = \left[C_{Ll} Q (c\,dy)\right] y$

  • Aileron control power integral across aileron span $y_1$ to $y_2$:         $C_{l\delta a} = \frac{2 C_{L\alpha w} \tau}{S b} \int_{y_1}^{y_2} c y \, dy$

  • For a tapered wing ($c(y) = c_r \left[1 - \frac{2(1-\lambda)}{b}y\right]$):         $C_{l\delta a} = \frac{2 C_{L\alpha w} \tau c_r}{S b} \left[\frac{y_2^2 - y_1^2}{2} - \frac{2(1-\lambda)}{3b}(y_2^3 - y_1^3)\right]$

  • Expressed using aileron center location $\mu_a = \frac{y_1 + y_2}{b}$ and width $\Delta_a = \frac{y_2 - y_1}{b/2}$:         $C_{l\delta a} = \frac{2 C_{L\alpha w} \tau c_r}{S b} \mu_a \Delta_a \left[ 1 - (1-\lambda)\bar{\mu}_a \right]$

  • Optimum Spanwise Aileron Location:

    • For taper ratio $\lambda = 0.5$, maximum control power occurs near the wingtip ($\mu_{a, opt} \approx \frac{b}{2}$).

    • For taper ratio $\lambda = 0.25$, optimum position moves inward ($\mu_{a, opt} \approx \frac{b}{3}$).

• Example 9: Aileron Control Power Calculation:

  • Given: Business aircraft with $S = 21.3\,m^2$, $b = 10.4\,m$, $AR = 5.08$, 2D lift slope $C_{L\alpha\infty} = 0.1\,deg^{-1} = 5.73\,rad^{-1}$, $c_a/c = 0.25 \implies \tau = 0.48$.

  • Span bounds: $y_1 = 3.4\,m$, $y_2 = 4.8\,m \implies \Delta_a = 1.4\,m$, center $\mu_a = 4.1\,m$, root chord $c_r = 2.75\,m$, taper ratio $\lambda = 0.487$.

  • 3D Wing Lift Curve Slope:         $C_{L\alpha w} = \frac{C_{L\alpha\infty}}{1 + \frac{C_{L\alpha\infty}}{\pi AR}} = \frac{5.73}{1 + \frac{5.73}{\pi (5.08)}} = 4.21\,rad^{-1}$

  • Integrating over aileron span yields roll control power:         $C_{l\delta a} = 0.17\,rad^{-1}$

Stability Derivatives and Parameter Sign Summary

• Standard Sign Corrections for Stability Derivatives:

  • $C_{y\beta}$ (Side force derivative due to sideslip): Must be negative ($C_{y\beta} < 0$).

  • $C_{n\beta}$ (Directional stability derivative): Must be positive ($C_{n\beta} > 0$) for weathercock stability.

  • $C_{l\delta a}$ (Aileron roll control power): Positive downward left aileron deflection generates positive roll moment ($C_{l\delta a} > 0$).

  • $C_{n\delta a}$ (Yaw moment due to aileron): Negative value indicates adverse yaw (aircraft yaws opposite to intended roll direction).

• Fundamental Flight Dynamics Relations:

  • Steady Level Flight Equilibrium:         $W = L = Q S C_L \implies C_L = \frac{W}{Q S}$

  • Trim Angle of Attack:         $\alpha_0 = \frac{C_{L0}}{C_{L\alpha}}$

  • Static Margin Equation:         $K_n = h_n - h = -\frac{C_{m\alpha}}{C_{L\alpha}}$