# EVTOL Tilt-Wing Aircraft Design under Uncertainty Using a Multidisciplinary Possibilistic Approach

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. MAPLA

#### 2.2. Initial Design and Sizing of the Propulsion System

- Disk loading: bigger propeller disk is better in hover due to lower disk loading.
- Ground clearances: smaller propellers would result in more ground clearance when the horizontal tail and the wing are in a horizontal position.
- Propeller-wing interaction: larger propellers would have a higher slipstream height, which results in more lift because of the propeller slipstream in the same slipstream speed.
- Propeller-propeller interaction: smaller propellers with lower slipstream heights allow easier placement of the engines such that the horizontal tail propellers do not lose their slipstream. This slipstream loss can lead to a reduction in thrust from the horizontal tail propellers and an increase in noise emissions.
- Blade rotation mechanism: using very small propellers makes it more difficult to alter the pitch of the blades.
- Safety in OEI conditions: using more and smaller engines, a failure of one would result in less effect on controllability in hover and on a loss of thrust.

_{∞}is the speed perpendicular to the propeller disk, factor k corrects for the power losses, equal to 1.2 in this case, and A is the disc area.

^{3}[46]. Consequently, using MAPLA’s electric propulsion module with the total energy and maximum power requirements, the total battery mass for energy storage was estimated to be 776 kg with a required volume of 0.4 m

^{3}.

^{3}.

#### 2.3. PBDO Method Outline

## 3. MDO Framework

#### 3.1. Methods

- Determine the estimated errors
- (I)
- Model each entry based on the horizontal/vertical tails, wing model and battery management
- (II)
- Calculate the aerodynamics of the eVTOL tilt-wing aircraft
- (III)
- Apply the uncertainty analysis
- (IV)
- Use the best-fit PDF curve for each uncertainty

- Select the initial vector and a set of reliability indices
- Run the IDF-based deterministic optimization
- Run the MDF-based PMA reliability assessment at the current reliability index, and change variables based on the sequential method
- Check for convergence based on the current reliability goal
- (I)
- if yes, update starting vector with the current optimum point, select the next reliability goal and return to Step 3
- (II)
- if no, return to Step 3

- Advance to the next target reliability level
- (I)
- retain solution as a new starting vector
- (II)
- return to Step 3

#### 3.2. Design Variables

#### 3.3. Disciplines

#### 3.3.1. Aerodynamics and Stability

- The maximum lift of a twisted wing
- The zero-lift pitching moment of the twisted wing
- The drag of the twisted wing
- The high-lift surfaces
- Lift of the horizontal tail and elevator surfaces
- The drag of the control and high-lift surfaces

#### 3.3.2. Weight and Balance

#### 3.3.3. Handling Quality

#### 3.3.4. Battery System

#### 3.4. Objective Function

- Zero-lift drag contribution of the wing, horizontal tail, and vertical tail.
- Zero-lift drag contribution of the fuselage and nacelles.
- Zero-lift interference drag contribution of the components.
- Drag contribution of the wing and empennage package at the angle of attack.
- Drag contribution of the fuselage and nacelle at the angle of attack.
- Wing-fuselage interference drag contribution at the angle of attack.

#### 3.5. Constraints

#### 3.6. Source of Uncertainty

## 4. Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Roman Symbols | |

A | disk area |

a | acceleration |

b | span |

$\mathrm{C}$ | capacity |

C_{D} | drag coefficient |

C_{L} | lift coefficient |

$\mathrm{c}$ | ratio of the moveable surface chord to fixed surface chord |

${\mathrm{c}}_{\mathrm{r}}$ | root chord |

${\mathrm{c}}_{\mathrm{t}}$ | tip chord |

$\mathrm{D}$ | drag |

$\mathrm{E}$ | energy |

${\mathrm{E}}_{\mathrm{s}\mathrm{p}}$ | specific energy |

${\mathrm{E}}_{\mathrm{v}\mathrm{o}\mathrm{l}}$ | the volumetric energy density |

dt | time step |

F_{x} | X component of the force acting on the vehicle |

F_{y} | Y component of the force acting on the vehicle |

G | generic functions |

h | altitude |

$\mathrm{i}$ | incidence angle |

k | correction factor for the power losses |

L | lift |

l | distance, from the fuselage nose to the mean aerodynamic chord, parallel to X-axis |

${\mathrm{m}}_{\mathrm{b}\mathrm{a}\mathrm{t}}$ | mass of the battery |

${\mathrm{n}}_{\mathrm{c}}$ | the number of cells |

P | power |

T | thrust |

${V}_{\infty}$ | the speed perpendicular to the propeller disk |

${\mathrm{V}}_{\mathrm{n}}$ | the nominal cell voltage |

${\mathrm{V}}_{\mathrm{t}\mathrm{o}\mathrm{t}}$ | the total voltage of batteries |

${\mathrm{v}}_{\mathrm{b}\mathrm{a}\mathrm{t}}$ | the volume of the battery |

Y | the coupling variable |

${z}_{h}$ | distance, from the quarter chord of the horizontal tail mean aerodynamic chord to the X-axis, parallel to Z-body axis, positive down |

Greek Symbols | |

${\mathsf{\theta}}_{\mathrm{T}}$ | the rotated wing angle with respect to the x-axis |

${\mathsf{\varphi}}_{\mathrm{T}\mathrm{E}}$ | trailing edge sweep angle |

$\mathsf{\Lambda}$ | sweep angle |

$\mathsf{\Gamma}$ | dihedral |

$\rho $ | density |

$\eta $ | the propulsive efficiency at the given speed |

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**Figure 7.**The XDSM of the MDO process to optimize the eVTOL tilt-wing aircraft [54].

**Figure 11.**Geometry of the eVTOL tilt-wing aircraft that is optimized by MAPLA and by using the proposed multidisciplinary possibilistic approach based on the airworthiness and performance requirements.

**Figure 12.**Results for the longitudinal static characteristics of the optimized eVTOL tilt-wing aircraft in the cruise flight conditions: (

**a**). Lift coefficient. (

**b**). Drag coefficient. (

**c**). Pitching moment coefficient.

**Figure 13.**Results for the lateral-directional static characteristics of the optimized eVTOL tilt-wing aircraft: (

**a**). Side-force derivatives. (

**b**). Effective dihedral coefficient. (

**c**). Weathercock stability coefficient.

**Figure 14.**Results for the longitudinal dynamic characteristics of the optimized eVTOL tilt-wing aircraft in cruise flight conditions: (

**a**). Lift coefficient due to pitch rate. (

**b**). Lift coefficient due to vertical acceleration. (

**c**). Pitching moment coefficient due to pitch rate. (

**d**). Pitching moment coefficient due to vertical acceleration.

**Figure 17.**Results for the undamped natural frequency in different acceleration sensitivity values for Short Period oscillations.

Variable | Value | Unit |
---|---|---|

Wing span | 10.97 | $\mathrm{m}$ |

Mean Aerodynamic Chord | 1.511 | $\mathrm{M}$ |

Wing Surface Area | 16 | ${\mathrm{m}}^{2}$ |

Aspect Ratio | 7.52 | - |

Mach | 0.25 | - |

Mass | 980 | $\mathrm{k}\mathrm{g}$ |

CG | 10 | % |

Altitude | 0 | m |

**Table 2.**Performance comparison of different optimization methods [43].

Method | ${\mathbf{f}}_{\mathbf{m}\mathbf{i}\mathbf{n}}$ | ${\mathbf{X}}_{1}$ | ${\mathbf{X}}_{2}$ | ${\mathbf{n}}_{\mathbf{e}\mathbf{v}\mathbf{a}\mathbf{l}}$ |
---|---|---|---|---|

Deterministic | 5.1769 | 3.1134 | 2.0636 | 16 |

Single Loop | 6.6198 | 3.4413 | 3.2866 | 16 |

PMA/Sequential | 6.7043 | 3.4506 | 3.2537 | 651 |

PMA/Double Loop | 6.7043 | 3.4506 | 3.2537 | 1004 |

RIA/Double Loop | 6.7257 | 3.4391 | 3.2866 | 1530 |

**Table 3.**Comparison of different methods’ stability [43].

Method | ${\mathbf{f}}_{\mathbf{m}\mathbf{i}\mathbf{n}}$ | ${\mathbf{X}}_{1}$ | ${\mathbf{X}}_{2}$ | ${\mathbf{n}}_{\mathbf{e}\mathbf{v}\mathbf{a}\mathbf{l}}$ |
---|---|---|---|---|

MDF/Deterministic | 115.797 | 1.648 | 3.004 | 68 |

MDF/Single Loop | 119.942 | 1.656 | 2.913 | 1495 |

MDF/Double Loop/PMA | 119.656 | 1.598 | 2.806 | 16,474 |

MDF/Sequential/PMA | 119.683 | 1.621 | 2.843 | 7413 |

IDF/Double Loop/PMA | 119.446 | 1.600 | 2.787 | 23,765 |

Component | Variable | Details | Limits |
---|---|---|---|

Horizontal Tail | ${\mathrm{i}}_{\mathrm{h}}$ | Incidence angle of horizontal tail, deg | 0 to 3 |

b_{h} | Span of horizontal tail, m | 4 to 6 | |

${\mathrm{c}}_{{\mathrm{r}}_{\mathrm{h}}}$ | Root chord of horizontal tail, m | 1 to 1.45 | |

${\mathrm{c}}_{{\mathrm{t}}_{\mathrm{h}}}$ | Tip chord of horizontal tail, m | 0.5 to 1 | |

${{\mathsf{\Lambda}}_{\mathrm{L}\mathrm{E}}}_{\mathrm{h}}$ | Horizontal tail leading edge sweep angle, deg | 10 to 20 | |

l_{h} | Distance, parallel to X-axis, from the the horizontal tail mean aerodynamic chord to the nose of the fuselage, m | 8 to 8.7 | |

z_{h} | Distance, parallel to Z-axis, from the quarter chord of the horizontal tail mean aerodynamic chord to the X-axis, positive down, m | −0.4 to −0.1 | |

c_{elevator} | Elevator to horizontal tail chord ratio | 0.2 to 0.5 | |

Vertical Tail | b_{v} | Span of vertical tail, m | 1.8 to 2.2 |

${\mathrm{c}}_{{\mathrm{r}}_{\mathrm{v}}}$ | Root chord of vertical tail, m | 1.9 to 2.5 | |

${\mathrm{c}}_{{\mathrm{t}}_{\mathrm{v}}}$ | Tip chord of vertical tail, m | 0.8 to 1.4 | |

${\mathsf{\varphi}}_{\mathrm{T}\mathrm{E}}$ | Vertical tail Trailing edge sweep angle, deg | 10 to 20 | |

z_{v} | Perpendicular distance from root chord of vertical-tail to X-axis, positive down, m | −0.35 to −0.15 | |

l_{v} | Distance along X-axis from the leading edge of tip chord of vertical tail to the nose of the fuselage, m | 8.5 to 9.5 | |

c_{rudder} | Rudder to vertical tail chord ratio, m | 0.2 to 0.5 | |

Wing | ${\mathrm{i}}_{\mathrm{w}}$ | Incidence angle of the wing, deg | 2 to 5 |

${\mathsf{\alpha}}_{\mathrm{t}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{t}}$ | Incidence angle of the wing | −2 to −5 | |

b_{w} | Wing span, m | 10 to 12 | |

${\mathrm{c}}_{{\mathrm{r}}_{\mathrm{w}}}$ | Root chord of the wing, m | 1.8 to 2.5 | |

${\mathrm{c}}_{{\mathrm{t}}_{\mathrm{w}}}$ | Tip chord of the wing, m | 0.8 to 1.1 | |

${{\mathsf{\Lambda}}_{\mathrm{L}\mathrm{E}}}_{\mathrm{w}}$ | Leading edge sweep angle of the wing, deg | 2 to 5 | |

${{\mathsf{\Lambda}}_{\mathrm{T}\mathrm{E}}}_{\mathrm{w}}$ | Trailing edge sweep angle of the wing, deg | −6 to −12 | |

l_{w} | Distance, parallel to X-axis, from the leading edge of wing mean aerodynamic chord to the nose of fuselage, m | 2.2 to 4 | |

z_{w} | Distance, parallel to Z-axis, from the quarter chord of the wing mean aerodynamic chord to the X-axis, positive down, m | 0.2 to 0.6 | |

$\mathsf{\Gamma}$ | Dihedral angle, deg | 5 to 10 | |

Battery Placement | ${\mathrm{l}}_{{\mathrm{v}}_{{\mathrm{b}}_{1}}}$ | Battery set 1, distance from nose, m | 0.8 to 1.3 |

${\mathrm{l}}_{{\mathrm{v}}_{{\mathrm{b}}_{2}}}$ | Battery set 2, distance from nose, m | 6.4 to 6.9 | |

${\mathrm{N}}_{{\mathrm{b}}_{1}}$ | Battery set 1, number of batteries | 1 to 24 | |

${\mathrm{N}}_{{\mathrm{b}}_{2}}$ | Battery set 2, number of batteries | 1 to 24 |

Constraint | Description | Limits |
---|---|---|

${\mathbf{W}}_{\mathbf{e}}$ | Empty weight, kg | ${\mathrm{W}}_{\mathrm{e}}$ < 1700 |

${\mathbf{C}}_{{\mathbf{m}}_{\mathsf{\alpha}}}$ | Pitching moment coefficient, a/rad | $-1.5<{\mathrm{C}}_{{\mathrm{m}}_{\mathsf{\alpha}}}<-0.3$ |

${\mathbf{C}}_{{\mathbf{n}}_{\mathsf{\beta}}}$ | Weathercock stability coefficient, 1/rad | $0.1<{\mathrm{C}}_{{\mathrm{n}}_{\mathsf{\beta}}}<0.4$ |

${\mathbf{C}}_{{\mathbf{l}}_{\mathsf{\beta}}}$ | Effective dihedral coefficient, 1/rad | ${\mathrm{C}}_{{\mathrm{l}}_{\mathsf{\beta}}}<0$ |

${\mathbf{C}}_{{\mathbf{m}}_{\mathbf{q}}}$ | Pitching moment coefficient due to pitch rate, 1/rad | $-30<{\mathrm{C}}_{{\mathrm{m}}_{\mathrm{q}}}<-5$ |

${\mathbf{C}}_{{\mathbf{n}}_{\mathbf{r}}}$ | Damping in yaw derivative, 1/rad | $-1<{\mathrm{C}}_{{\mathrm{n}}_{\mathrm{r}}}<-0.1$ |

$\mathbf{C}{\mathbf{G}}_{\mathbf{a}\mathbf{f}\mathbf{t}}$ | Centre of gravity, maximum afterwards, % | <27% |

$\mathbf{C}{\mathbf{G}}_{\mathbf{f}\mathbf{o}\mathbf{r}}$ | Centre of gravity, maximum forwards, % | >10% |

SM | Static margin | $5\mathrm{\%}\le \mathrm{S}\mathrm{M}\le 10\mathrm{\%}$ |

${\mathsf{\delta}}_{{\mathbf{r}}_{\mathbf{m}\mathbf{a}\mathbf{x}}}$ | Max rudder deflection, deg | $\pm 25$ |

${\mathsf{\delta}}_{{\mathbf{e}}_{\mathbf{m}\mathbf{a}\mathbf{x}}}$ | Max elevator deflection, deg | $\pm 25$ |

$\mathbf{H}{\mathbf{Q}}_{\mathbf{P}}$ | Handling quality, phugoid mode | $\mathrm{H}{\mathrm{Q}}_{\mathrm{P}}=1$ |

$\mathbf{H}{\mathbf{Q}}_{\mathbf{S}\mathbf{P}}$ | Handling quality, short period mode | $\mathrm{H}{\mathrm{Q}}_{\mathrm{S}\mathrm{P}}=1$ |

$\mathbf{H}{\mathbf{Q}}_{\mathbf{D}}$ | Handling quality, Dutch roll mode | $\mathrm{H}{\mathrm{Q}}_{\mathrm{D}}=1$ |

$\mathbf{H}{\mathbf{Q}}_{\mathbf{R}}$ | Handling quality, roll mode | $\mathrm{H}{\mathrm{Q}}_{\mathrm{R}}=1$ |

$\mathbf{H}{\mathbf{Q}}_{\mathbf{S}}$ | Handling quality, spiral mode | $\mathrm{H}{\mathrm{Q}}_{\mathrm{S}}=1$ |

Aerodynamic Characteristics | Deviation |
---|---|

$\mathbf{C}{\mathbf{L}}_{\mathsf{\alpha}}$ | $5\mathrm{\%}$ |

$\mathbf{C}{\mathbf{D}}_{\mathsf{\alpha}}$ | $10\mathrm{\%}$ |

$\mathbf{C}{\mathbf{m}}_{\mathsf{\alpha}}$ | $15\mathrm{\%}$ |

$\mathbf{C}{\mathbf{m}}_{\mathbf{q}}$ | $30\mathrm{\%}$ |

$\mathbf{C}{\mathbf{m}}_{\dot{\mathsf{\alpha}}}$ | $30\mathrm{\%}$ |

$\mathbf{C}{\mathbf{L}}_{\mathbf{q}}$ | NA, assumed 20% |

$\mathbf{C}{\mathbf{L}}_{\dot{\mathsf{\alpha}}}$ | NA, assumed 20% |

${\mathbf{C}}_{{\mathbf{n}}_{\mathsf{\beta}}}$ | $20\mathrm{\%}$ |

${\mathbf{C}}_{{\mathbf{l}}_{\mathsf{\beta}}}$ | $5\mathrm{\%}$ |

${\mathbf{C}}_{{\mathbf{l}}_{\mathbf{p}}}$ | NA, assumed 20% |

${\mathbf{C}}_{{\mathbf{n}}_{\mathbf{p}}}$ | NA, assumed 20% |

${\mathbf{C}}_{{\mathbf{l}}_{\mathbf{r}}}$ | $10\mathrm{\%}$ |

${\mathbf{C}}_{{\mathbf{n}}_{\mathbf{r}}}$ | $10\mathrm{\%}$ |

Component | Variable | Description | Result |
---|---|---|---|

Horizontal Tail | ${\mathrm{i}}_{\mathrm{h}}$ | Incidence angle of horizontal tail, deg | 1.92 |

b_{h} | Span of horizontal tail, m | 4.95 | |

${\mathrm{c}}_{{\mathrm{r}}_{\mathrm{h}}}$ | Root chord of horizontal tail, m | 1.29 | |

${\mathrm{c}}_{{\mathrm{t}}_{\mathrm{h}}}$ | Tip chord of horizontal tail, m | 0.82 | |

${{\mathsf{\Lambda}}_{\mathrm{L}\mathrm{E}}}_{\mathrm{h}}$ | Horizontal tail leading edge sweep angle, deg | 12.2 | |

l_{h} | Distance, parallel to X-axis, from the the horizontal tail mean aerodynamic chord to the nose of the fuselage, m | 8.32 | |

z_{h} | Distance, parallel to Z-axis, from the quarter chord of the horizontal tail mean aerodynamic chord to the X-axis, positive down, m | −0.278 | |

c_{elevator} | Elevator to horizontal tail chord ratio | 0.37 | |

Vertical Tail | b_{v} | Span of vertical tail, m | 1.847 |

${\mathrm{c}}_{{\mathrm{r}}_{\mathrm{v}}}$ | Root chord of vertical tail, m | 1.955 | |

${\mathrm{c}}_{{\mathrm{t}}_{\mathrm{v}}}$ | Tip chord of vertical tail, m | 0.874 | |

${\mathsf{\varphi}}_{\mathrm{T}\mathrm{E}}$ | Vertical tail Trailing edge sweep angle, deg | 17.15 | |

z_{v} | Perpendicular distance from root chord of vertical-tail to X-axis, positive down, m | −0.23 | |

l_{v} | Distance along X-axis from the leading edge of tip chord of vertical tail to the nose of the fuselage, m | 9.08 | |

c_{rudder} | Rudder to vertical tail chord ratio, m | 0.418 | |

Wing | ${\mathrm{i}}_{\mathrm{w}}$ | Incidence angle of the wing, deg | 2.74 |

${\mathsf{\alpha}}_{\mathrm{t}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{t}}$ | Incidence angle of the wing | −3.15 | |

b_{w} | Wing span, m | 11.95 | |

${\mathrm{c}}_{{\mathrm{r}}_{\mathrm{w}}}$ | Root chord of the wing, m | 2.143 | |

${\mathrm{c}}_{{\mathrm{t}}_{\mathrm{w}}}$ | Tip chord of the wing, m | 0.9 | |

${{\mathsf{\Lambda}}_{\mathrm{L}\mathrm{E}}}_{\mathrm{w}}$ | Leading edge sweep angle of the wing, deg | 3.2 | |

${{\mathsf{\Lambda}}_{\mathrm{T}\mathrm{E}}}_{\mathrm{w}}$ | Trailing edge sweep angle of the wing, deg | −9.5 | |

l_{w} | Distance, parallel to X-axis, from the leading edge of wing mean aerodynamic chord to the nose of fuselage, m | 2.76 | |

z_{w} | Distance, parallel to Z-axis, from the quarter chord of the wing mean aerodynamic chord to the X-axis, positive down, m | 0.205 | |

$\mathsf{\Gamma}$ | Dihedral angle, deg | 7.5 | |

Battery Placement | ${\mathrm{l}}_{{\mathrm{v}}_{{\mathrm{b}}_{1}}}$ | Battery set 1, distance from nose, m | 0.9 |

${\mathrm{l}}_{{\mathrm{v}}_{{\mathrm{b}}_{2}}}$ | Battery set 2, distance from nose, m | 6.5 | |

${\mathrm{N}}_{{\mathrm{b}}_{1}}$ | Battery set 1, number of batteries | 16 | |

${\mathrm{N}}_{{\mathrm{b}}_{2}}$ | Battery set 2, number of batteries | 8 |

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## Share and Cite

**MDPI and ACS Style**

Rostami, M.; Bardin, J.; Neufeld, D.; Chung, J.
EVTOL Tilt-Wing Aircraft Design under Uncertainty Using a Multidisciplinary Possibilistic Approach. *Aerospace* **2023**, *10*, 718.
https://doi.org/10.3390/aerospace10080718

**AMA Style**

Rostami M, Bardin J, Neufeld D, Chung J.
EVTOL Tilt-Wing Aircraft Design under Uncertainty Using a Multidisciplinary Possibilistic Approach. *Aerospace*. 2023; 10(8):718.
https://doi.org/10.3390/aerospace10080718

**Chicago/Turabian Style**

Rostami, Mohsen, Julian Bardin, Daniel Neufeld, and Joon Chung.
2023. "EVTOL Tilt-Wing Aircraft Design under Uncertainty Using a Multidisciplinary Possibilistic Approach" *Aerospace* 10, no. 8: 718.
https://doi.org/10.3390/aerospace10080718