Tilt-Wing eVTOL Transition Corridor, Stall and Pitch Trim

The implemented model is a longitudinal tandem tilt-wing equilibrium with a prescribed common wing tilt $\beta$, flight-path angle $\gamma$ and along-path acceleration $a$. Positive body pitch $\theta$ is nose-up; tilt is relative to the body, with 90° in hover configuration. The solver finds total thrust $T$, body pitch and normalized pitch-control command $u$ together.

Each wing has a blown and an unblown zone. For each propeller group, the positive propulsive branch of uniform oblique momentum theory is solved:

Here $U_\parallel=V\cos(\theta+\beta-\gamma)$ and $U_\perp=V\sin(\theta+\beta-\gamma)$. Eight scalar Newton updates are applied in parallel across points. The axial limit agrees with May, Milz and Looye (2023), Eq. 37, attached PDF p. 12. Negative axial inflow is excluded.

In a blown zone, the effective velocity is the freestream vector plus $k_s v_i$ along the thrust axis. In an unblown zone it is the freestream alone. Its direction gives local flow angle $\psi$, so $\alpha=\theta+\beta-\psi$ and $q=\rho|\mathbf U|^2/2$. This follows the local-flow/strip-force construction in May et al., Eqs. 26–34, attached pp. 10–11, using two uniform zones per wing. The finite-wing polar is $C_L=a_L(\alpha-\alpha_0+\tau_f\delta)$, $C_D=C_{D0}+k C_L^2$ and $C_m=C_{m0}+C_{m\delta}\delta$, with angles in radians. Lift and drag are rotated from each local flow direction into earth axes.

Pitch control follows a declared allocation: $d=d_{max}u\sin^2\beta$, $T_f=T(1+d)/2$, $T_a=T(1-d)/2$, and $\delta_f=-\delta_a=\delta_{max}u\cos^2\beta$. Thus differential thrust dominates hover and differential elevons dominate cruise. Forces, thrust moments and local $qScC_m$ are summed about the CG; the moment arm calculation uses body coordinates, including wing height. Fuselage drag acts at the CG. Group electrical power is $T_j(U_\parallel+\kappa_i v_i)/\eta$.

A bounded, finite-difference Newton solve enforces $\Sigma F_x=ma\cos\gamma$, $\Sigma F_z=mg+ma\sin\gamma$, and $\Sigma M=0$. Accepted residuals are below $10^{-5}$ after normalization by weight and weight times total wing-arm distance. Feasible samples also satisfy $|C_L|\leq C_{Lmax}$ in every nonzero-area, moving-air zone, $T_j\leq T_{lim}/2$, $P\leq P_{lim}$, $|u|\leq1$, and $|\theta|\leq\theta_{lim}$. Outside stall, the unbounded attached-flow polar is used only to locate a failed constraint; those force and power values are extrapolations, not post-stall predictions.

• Rigid tandem wings with equal mean chord, common tilt, fixed CG, and independently loaded forward/aft propeller groups. Each group has half the total disk area and half the installed thrust capacity. Wing aerodynamic centers and thrust application points coincide at the specified arms and height.
• Wing slope, drag, maximum lift and control derivatives are design inputs, assumed constant with Reynolds number, tilt and thrust. Maximum positive and negative lift magnitudes are assumed equal; elevon deflection shifts the lift curve. Calibrated finite-wing coefficients are required.
• Uniform induced flow over the specified blown fraction; the same wake multiplier and area fraction apply to both wings. The remaining area is explicitly evaluated in free flow. A stalled uncovered strip excludes a point even when its load is small; at exactly zero local speed it carries no force and imposes no stall test.
• Constant density, straight flight path, zero pitch rate and zero angular acceleration. A nonzero acceleration is a local force-balance prescription, not an integrated trajectory.
• Default mass 25.9 kg, area 0.737 m² and CLmax 1.43 come from NASA/TM-2017-219794, attached p. 28. That report's area unit “m3” is a typographical error; area is m². Geometry, disk area, coefficients, limits and the chosen schedule are illustrative engineering assumptions. This is not a GL-10, VZ-2 or XC-142 performance reproduction.

This is a design estimate for the stated tandem configuration and control allocation, not a flight envelope. A feasible cell establishes only the modeled longitudinal force and pitch balance; it establishes neither static nor dynamic stability, handling qualities, reserve control authority or an executable transition.

The corridor uses 13 airspeeds by 19 tilt angles (5° tilt spacing; default speed spacing 2.92 m/s). The operating path uses 41 independently solved samples. This keeps browser work small while retaining the coupled physics, at the cost of missing narrow bands and exact boundary locations. Cells extend halfway toward neighboring samples only for display. The table reports sample endpoints, preserves disjoint bands, and never certifies the interval between samples. Path position is solved independently on each animation frame.

The equilibrium solver uses at most 16 bounded Newton iterations and the inflow solver eight updates. A gray “Unresolved” cell means no admissible converged solution was obtained or axial inflow was negative; it is not proof that every possible trim branch is impossible. Blank selected-point results have that same meaning. Multiple trim branches and alternative control allocations can exist.

Stall and lift/drag parameters need measured or validated vehicle data. The two-zone approximation omits spanwise gradients, wake contraction and skew, swirl, propeller–propeller and wing–wing interference, flow separation detail and dynamic stall. The wake multiplier is an input, not a computed wake geometry. The electrical power expression uses a lumped efficiency allowance and induced-loss factor; it does not replace motor/propeller maps or include a separate fixed profile-power term.

No descent, windmill-brake operation, gusts, ground effect, lateral/directional trim, actuator rates or wing-tilt torque, structural limits, battery sag/thermal limits, rotor tip Mach/blade stall or failure cases are checked. In particular NASA/TM-2017-219643, attached p. 17, identifies tilt-actuator torque as an additional GL-10 corridor boundary. The present model must not be interpreted as including it. Reference cases verify the stall-speed and actuator-disk submodels; no independent aircraft-level transition or pitch-trim validation is claimed. The corridor drawing uses fixed SI axes; inputs, scalar results and numeric table columns retain their declared units for the viewer's metric/US conversion.