# On the Handling Qualities of Two Flying Wing Aircraft Configurations

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Theory of Longitudinal-Lateral Coupling

#### 2.1. Basic Coupled and Decoupled Modes

_{g}and damping ratios ζ

_{g}of four modes g = 1,..., 4. The fundamental issue is whether these four modes can be related to the phugoid ‘1′, short period ‘2′, dutch roll ‘3′ and helical ‘4′ modes, as suggested in Table 1. This identification should be possible when the coupling is weak, that is the coupled modes ${\omega}_{g}$ with g = 1,..., 4, differ little from the decoupled modes ${\overline{\omega}}_{g}$, in the sense ${\left({\omega}_{g}-{\overline{\omega}}_{g}\right)}^{2}<<{\overline{\omega}}_{g}{}^{2}.$

#### 2.2. Weak Coupling and Mode Properties

^{2}is negligible.

_{g}and damping ratio Δζ

_{g}changes due to coupling:

^{+}> λ

^{−}since |Ω| < ω, so that λ

_{−}has the slowest decay. In the case of amplification Equations (21a,b), the response is still oscillatory Equations (18a,b) if ζ

^{2}< 1, but it has exponentially increasing instead of decreasing amplitude with time constant Equation (21b)

^{+}> λ

^{−}> 0 and the fastest growing mode is λ

^{+}, which could be used instead of |ζ| in the time constant Equation (20b) and time to double amplitude Equations (21c,d).

#### 2.3. Calculation of Frequency and Amplification Changes

^{8}in C in Equation (6b) and in the product of A in Equation (3b) by B in Equation (4b), and thus cancels by subtraction. The coefficients in (27) are the products of three modal factors of the coupled characteristic polynomial Equation (6b), and thus are polynomials of degree six in λ, with leading term λ

^{6}, viz.:

## 3. Natural Stability of Flying-Wing Aircraft

#### 3.1. Relevance of Longitudinal-Lateral Coupling

_{ij}has the dimensions of inverse time. There are three cases: (i) if the coupling sub-matrices vanish Equations (7c,d) then the longitudinal and transversal modes are strictly decoupled; (ii) if the coupling submatrices Equations (7c,d) have terms comparable to those of the longitudinal Equation (7a) or lateral Equation (7b) stability matrices Equations (7a,d), then there is strong coupling; (iii) if, when compared with the longitudinal Equation (7a) and lateral Equation (7b) stability matrices of O(1), the coupling submatrices Equations (7c,d) have terms O(ε) which are small but non-negligible ε

^{2}<< 1, then weak coupling results.

#### 3.2. Longitudinal and Lateral Handling Qualities

_{s}< 1. This is checked next noting that: (i) using Equation (35b) the natural frequency is Equation (37a):

_{p}> 0.04 in Table 10 to ensure level 1 HQ in Table 11. The short-period is monotonic in the remaining flight conditions 1b and 1d–1h, with one stable and one unstable mode (Table 7). Since the damping is negative, it follows (Table 10) that HQs do not even meet level 3. The oscillatory cases of short-period mode all have damping (Figure 3) in the range of level 1 HQ (Table 11) for all flight cases A, B, C as indicated in Table 11 and illustrated in Figure 2.

_{d}> 0.02 and oscillation frequency Ω

_{d}> 0.40 rad.s

^{−1}specifying an upper rectangle. Their product Ω

_{d}.ξ

_{d}> 0.008 rad.s

^{−1}may not satisfy the third condition Ω

_{d}.ξ

_{d}> 0.05 rad.s

^{−1}that specifies a hyperbola. The hyperbola Ω

_{d}.ξ

_{d}= 0.05 cuts ξ

_{d}= 0.02 at Ω

_{d}= 0.05/0.02 = 2.5 rad.s

^{−1}and cuts Ω

_{d}= 0.4 rad.s

^{−1}at ξ

_{d}= 0.05/0.4 = 0.125. Thus, the region of level 2 HQs for the dutch roll lies on the right and level 3 HQs on the left of the hyperbola in Figure 7. The hyperbola on Figure 7 is one of the three hyperbolas on Figure 8, namely that which coincides with the hyperbola closest to the axis in Figure 8. The level 1 HQs for the dutch roll in flight conditions B + C impose the same condition on oscillation frequency Ω

_{d}> 0.40 but higher damping ξ

_{d}> 0.08 shifting the rectangle to the right; the third condition is also more stringent Ω

_{d}.ξ

_{d}> 0.15 than for level 2 shifting the second hyperbola upward and to the right in Figure 8. The level 1 HQs for flight condition A are still more stringent shifting the rectangle (Ω

_{d}> 0.40, ξ

_{d}> 0.19) further the right and the third hyperbola Ω

_{d}.ξ

_{d}> 0.35 further upward and to the right in Figure 8. None of the flight conditions lies within the third or second hyperbolas in Figure 7 and thus Level 1 HQs for the dutch roll are not attained. Since for the dutch roll level 1 HQs are not met in any flight condition, and not even level 3 is met for flight conditions 2e, 2f and 2h, all other flight conditions are level 2 or 3. As indicated in the Table 11 and illustrated in Figure 7. The dutch roll HQs are level 3 for flight conditions 1f, 1g, 2a–2d and 2g; the remaining flight conditions 1a–1d and 1g–1j have level 2 HQs for the dutch roll as indicated in Table 11 and illustrated in Figure 7 and Figure 8.

_{s}appearing to the square; (ii) the normal acceleration relates to the lift and is thus specified Equation (39b) by the lift coefficient that is proportional to the lift slope multiplied C

_{Lθ}to the pitch angle relative to the angle of zero lift assumed to be small:

#### 3.3. Manouever Points of Two Kinds

_{1}and x

_{2}of mean aerodynamic chord. The stability matrix depends on the c.g. position, and thus also the damping ratio ζ of all modes. For small c.g. excursions this dependence may be taken to the linear:

_{1}, ζ

_{2}are the dampings at c.g. positions respectively x

_{1}, x

_{2}and k is the slope:

^{+}is used in (47):

## 4. Assessment of BWB 1 and BWB 2 Designs

#### 4.1. The Dutch Roll, Spiral and Roll Modes

#### 4.2. The Phugoid and Short-Period Modes

#### 4.3. Implications for Control System Design

## 5. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

${d}_{\mathit{a}}$ | coefficients of polynomials (28) |

${d}_{gb}$ | coefficients of polynomials (29) |

$k$ | slope of manouever point linear approximation (45c) |

$p$ | x-component of angular velocity (4a) |

$q$ | y-component of angular velocity (1) |

$r$ | z-component of angular velocity (4a) |

$u$ | x component of linear velocity (1) |

$x$ | position of c.g. as percentage of m.a.c. (45a) |

${x}_{*}$ | critical c.g. position for manouever point (46a–c) |

$v$ | y-component of linear velocity (4a) |

$w$ | z-component of linear velocity (1) |

$\ddot{z}$ | vertical acceleration (38) |

A | characteristic polynomial of longitudinal stability sub-matrix (3b) |

B | characteristic polynomial of lateral stability sub-matrix (4b) |

C | characteristic polynomial of complete stability matrix (6a,b) |

$\overline{C}$ | characteristic polynomial of decoupled complete stability matrix (13b) |

${C}_{g}$ | modal factor (10) |

${\overline{C}}_{g}$ | modal factor for decoupled stability matrix (13a) |

${C}_{L\theta}$ | lift coefficient slope (39b) |

${E}_{g}$ | weak coupling coefficient (14c) |

T | time to double amplitude (21c,d) |

X | aircraft state variables (1, 4a) |

X_{i} | coupled flight variables (5) |

${\overline{X}}_{g}$ | decoupled flight variables (15) |

Z_{ij} | stability matrix (2b) |

ε | small quantity (8) |

θ | Euler angle of pitch (1) |

δ_{ab} | identity matrix (3a) |

$\phi $ | Euler angle of bank (4a) |

$\psi $ | Euler angle of sideslip (Table 5) |

${\lambda}^{\pm}$ | eigenvalues (3a) for modes (18a,b; 20a–c) |

$\zeta $ | damping ratio (3b) |

$\overline{\zeta}$ | decoupled damping ratio (9b) |

$\omega $ | natural exact coupled frequency (3b) |

$\overline{\omega}$ | natural decoupled frequency (9a) |

$\Omega $ | oscillation frequency (19d) |

$\tau $ | time constant (21b) |

$\xi $ | amplification ratio ($\xi =-\zeta $) |

$\Delta C$ | difference between the exact coupled $C$ and decoupled $\overline{C}$ complete characteristic polynomial (27) |

$\Delta {C}_{g}$ | difference between the exact coupled ${C}_{g}$ and decoupled ${\overline{C}}_{g}$ modal factor (14a–c) |

$\Delta \omega $ | difference between the exact coupled $\omega $ and decoupled $\overline{\omega}$ natural frequency (9a) |

$\Delta \zeta $ | difference between the exact coupled $\zeta $ and decoupled $\overline{\zeta}$ damping ratio (9b) |

Subscripts | |

p or 1 | phugoid mode |

s or 2 | short period mode |

d or 3 | dutch roll mode |

h or 4 | helical mode |

r or 4^{-} | roll mode |

l or 4^{+} | spiral mode |

Superscripts | |

$\overline{X}$ | decoupled value of $X$ |

Abbreviations | |

c.g. | center of gravity |

m.a.c. | mean aerodynamic chord |

CAP | Control Anticipation Parameter (38) |

BWB | Blended Wing Body |

HQs | handling qualities |

Symbols | |

$\dot{X}$ | time derivative of $X$ |

$\Delta X$ | variation of $X$ |

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**Figure 11.**Manouever points of the second kind for oscillatory modes becoming monotonic modes (convergent or divergent) before zero damping.

Type | Mode | Frequency | Damping |
---|---|---|---|

Longitudinal | Phugoid | ${\omega}_{p}\equiv {\overline{\omega}}_{1}$ | ${\zeta}_{p}={\overline{\zeta}}_{1}$ |

Short period | ${\omega}_{s}\equiv {\overline{\omega}}_{2}$ | ${\zeta}_{s}={\overline{\zeta}}_{2}$ | |

Lateral | Dutch roll | ${\omega}_{d}\equiv {\overline{\omega}}_{3}$ | ${\zeta}_{d}={\overline{\zeta}}_{3}$ |

Helical | ${\omega}_{h}\equiv {\overline{\omega}}_{4}$ | ${\zeta}_{h}={\overline{\zeta}}_{4}$ |

Mode | Natural Frequency | Damping Ratio |
---|---|---|

Decoupled | ${\overline{\omega}}_{g}$ | ${\overline{\zeta}}_{g}$ |

Weakly coupled | ${\overline{\omega}}_{g}+\Delta {\omega}_{g}$ | ${\overline{\zeta}}_{g}+\Delta {\zeta}_{g}$ |

Condition | ${\left(\Delta {\omega}_{g}\right)}^{2}<<{\left({\overline{\omega}}_{g}\right)}^{2}$ | ${\left(\Delta {\zeta}_{g}\right)}^{2}<<{\left({\overline{\zeta}}_{g}\right)}^{2}$ |

Strongly coupled | ${\omega}_{g}$ | ${\zeta}_{g}$ |

Condition | $\Delta {\omega}_{g}\equiv {\omega}_{g}-{\overline{\omega}}_{g}~{\overline{\omega}}_{g}$ | $\Delta {\zeta}_{g}\equiv {\zeta}_{g}-{\overline{\zeta}}_{g}~{\overline{\zeta}}_{g}$ |

Design | Flight Condition | Mass | Speed | Altitude | Flaps | c.g. |
---|---|---|---|---|---|---|

BWB | Case | ×10^{3} kg | kts | ×10^{3} ft | degrees | % mac |

1 | 1a | 550 | 176 | 0 | 15/25 | 25 |

1b | 550 | 176 | 0 | 15/25 | 35 | |

1c | 550 | 200 | 0 | 15/25 | 25 | |

1d | 550 | 200 | 0 | 15/25 | 35 | |

1e | 670 | M = 0.85 | 39 | clean | 35 | |

1f | 670 | M = 0.85 | 39 | clean | 39 | |

1g | 760 | M = 0.85 | 35 | clean | 35 | |

1h | 760 | M = 0.85 | 35 | clean | 39 | |

1i | 700 | 300 | 0 | clean | 35 | |

1j | 700 | M = 0.70 | 30 | clean | 39 | |

2 | 2a | 550 | 176 | 0 | clean | 35 |

2b | 550 | 176 | 0 | clean | 39 | |

2c | 550 | 200 | 0 | clean | 35 | |

2d | 550 | 200 | 0 | clean | 39 | |

2e | 670 | M = 0.85 | 39 | clean | 35 | |

2f | 670 | M = 0.85 | 39 | clean | 39 | |

2g | 760 | M = 0.85 | 35 | clean | 35 | |

2h | 760 | M = 0.85 | 35 | clean | 39 |

**Table 4.**Re-arranged 8x8 stability matrix for case 1a in Table 3.

$\mathit{u}$ [m/s] | $\mathit{w}$ [m/s] | $\mathit{q}$ [rad/s] | $\mathit{\theta}$ [rad] | $\mathit{v}$ [m/s] | $\mathit{p}$ [rad/s] | $\mathit{r}$ [rad/s] | $\mathit{\varphi}$ [rad] | |
---|---|---|---|---|---|---|---|---|

$\dot{\mathit{u}}$[m/s^{2}] | −2.10 × 10^{−4} | 1.51 × 10^{−1} | −8.91 | −9.94 × 10^{−1} | 1.13 × 10^{−7} | 0 | 0 | 0 |

$\dot{w}$[m/s^{2}] | −1.54 × 10^{−1} | −6.55 × 10^{−1} | 8.05 × 10^{1} | 1.10 × 10^{−1} | −1.24 × 10^{−6} | 0 | 0 | 0 |

$\dot{q}$[rad/s^{2}] | 5.98 × 10^{−4} | −7.16 × 10^{−3} | −6.13 × 10^{−1} | 0 | −1.07 × 10^{−9} | −4.93 × 10^{−5} | 4.93 × 10^{−5} | 0 |

$\dot{\theta}$[rad/s] | 0 | 0 | 1.00 | 0 | 0 | 0 | 0 | 0 |

$\dot{v}$[m/s^{2}] | 4.66 × 10^{−17} | 4.66 × 10^{−17} | 0 | 0 | −5.27 × 10^{−2} | 1.11 × 10^{1} | −8.81 × 10^{1} | 9.94 × 10^{−1} |

$\dot{p}$[rad/s^{2}] | −1.34 × 10^{−16} | −1.19 × 10^{−17} | 0 | 0 | −6.68 × 10^{−3} | −9.07 × 10^{−1} | 2.30 × 10^{−1} | 0 |

$\dot{r}$[rad/s^{2}] | −5.06 × 10^{−15} | −2.03 × 10^{−19} | 0 | 0 | 2.68 × 10^{−3} | −1.85 × 10^{−1} | −1.12 × 10^{−1} | 0 |

$\dot{\varphi}$[rad/s] | 0 | 0 | 0 | 0 | 0 | 1.00 | 1.11 × 10^{−1} | 0 |

**Table 5.**Oscillation frequency and damping ratio of natural modes for case 1g in the Table 3.

Type | Mode | Frequency Damping | De-Coupled | Weakly Coupled Approximation | Fully Coupled |
---|---|---|---|---|---|

Longitudinal | Phugoid | $\begin{array}{l}{\Omega}_{p}/{\lambda}_{1}^{-}\\ {\zeta}_{p}/{\lambda}_{1}^{+}\end{array}$ | 0.201769 0.114 | 0.202 0.114 | 0.201968 0.113596 |

Short period | $\begin{array}{l}{\Omega}_{s}/{\lambda}_{2}^{-}\\ {\zeta}_{s}/{\lambda}_{2}^{+}\end{array}$ | /0.124521 /1.7013 | /0.124 /−1.700 | /0.124266 /−1.70135 | |

Lateral | Dutch roll | $\begin{array}{l}{\Omega}_{d}/{\lambda}_{3}^{-}\\ {\zeta}_{d}/{\lambda}_{3}^{+}\end{array}$ | 0.845291 0.0595375 | 0.845291 0.0595 | 0.845291 0.0595375 |

Helical | $\begin{array}{l}{\Omega}_{h}/{\lambda}_{4}^{-}\\ {\zeta}_{h}/{\lambda}_{4}^{+}\end{array}$ | /−4.28162 × 10^{−6} /−1.13662 | /−4.28 × 10^{−6}/−1.137 | /−4.28162 × 10^{−6} /−1.13662 |

**Table 6.**Complete 9 × 9 stability matrix for case 1a in Table 3.

$\mathit{u}$ [m/s] | $\mathit{v}$ [m/s] | $\mathit{w}$ [m/s] | $\mathit{p}$ [rad/s] | $\mathit{q}$ [rad/s] | $\mathit{r}$ [rad/s] | $\mathit{\varphi}$ [rad] | $\mathit{\theta}$ [rad] | $\mathit{\psi}$ [rad] | |
---|---|---|---|---|---|---|---|---|---|

$\dot{u}$ [m/s^{2}] | −2.10 × 10^{−4} | 1.13 × 10^{−7} | 1.51 × 10^{−1} | 0 | −8.91 | 0 | 0 | −9.94 × 10^{−1} | 0 |

$\dot{v}$ [m/s^{2}] | 4.66 × 10^{−17} | −5.27 × 10^{−2} | 4.66 × 10^{−17} | 1.11 × 10^{1} | 0 | −8.81 × 10^{1} | 9.94 × 10^{−1} | 0 | 0 |

$\dot{w}$ [m/s^{2}] | −1.54 × 10^{−1} | −1.24 × 10^{−6} | −6.55 × 10^{−1} | 0 | 8.05 × 10^{1} | 0 | 0 | 1.10 × 10^{−1} | 0 |

$\dot{p}$ [rad/s^{2}] | −1.34 × 10^{−16} | −6.68 × 10^{−3} | −1.19 × 10^{−17} | −9.07 × 10^{−1} | 0 | 2.30 × 10^{−1} | 0 | 0 | 0 |

$\dot{q}$ [rad/s^{2}] | 5.98 × 10^{−4} | −1.07 × 10^{−9} | −7.16 × 10^{−3} | −4.93 × 10^{−5} | −6.13 × 10^{−1} | 4.93 × 10^{−5} | 0 | 0 | 0 |

$\dot{r}$ [rad/s^{2}] | −5.06 × 10^{−15} | 2.68 × 10^{−3} | −2.03 × 10^{−19} | −1.85 × 10^{−1} | 0 | −1.12 × 10^{−1} | 0 | 0 | 0 |

$\dot{\varphi}$ [rad/s] | 0 | 0 | 0 | 1.00 | 0 | 1.11 × 10^{−1} | 0 | 0 | 0 |

$\dot{\theta}$ [rad/s] | 0 | 0 | 0 | 0 | 1.00 | 0 | 0 | 0 | 0 |

$\dot{\psi}$ [rad/s] | 0 | 0 | 0 | 0 | 0 | 1.01 | 0 | 0 | 0 |

**Table 7.**Eigenvalues ${\lambda}^{\pm}$ of natural modes involving the damping ratio $\zeta $, natural $\omega $ and oscillation $\Omega $ frequencies.

Stability Mode | Longitudinal | Lateral | |||
---|---|---|---|---|---|

Phugoid | Short-Period | Dutch Roll | Roll | Spiral | |

$\lambda $ | $-{\zeta}_{p}{\omega}_{p}\pm i{\Omega}_{p}$ or ${\lambda}_{1}^{-}/{\lambda}_{1}^{+}$ | $-{\zeta}_{s}{\omega}_{s}\pm i{\Omega}_{s}$ or ${\lambda}_{2}^{-}/{\lambda}_{2}^{+}$ | $-{\zeta}_{d}{\omega}_{d}\pm i{\Omega}_{d}$ or ${\lambda}_{3}^{-}/{\lambda}_{3}^{+}$ | $-{\zeta}_{r}{\omega}_{r}={\lambda}_{4}^{-}$ | $-{\zeta}_{l}{\omega}_{l}={\lambda}_{4}^{+}$ |

1a | −0.102 ± i0.0374 | −0.624 ± i0.768 | −0.0759 ± i0.602 | −0.920 | −0.000397 |

1b | −0.000684 ± i0.0719 | −1.031 / 0.268 | −0.0605 ± i0.522 | −0.853 | −0.000382 |

1c | −0.00816 ± i0.0332 | −0.720 ± i0.877 | −0.0916 ± i0.657 | −1.065 | −0.000205 |

1d | −0.000146 ± i0.0641 | −1.495 / 0.308 | −0.0738 ± i0.576 | −0.985 | −0.000112 |

1e | −0.0727 ± i0.186 | −1.503 / 0.138 | −0.0458 ± i0.774 | −0.958 | −0.0000104 |

1f | −0.00150 ± i0.0695 | −2.172 / 0.804 | −0.0426 ± i0.748 | −0.956 | −0.0000110 |

1g | −0.114 ± i0.202 | −1.701 / 0.124 | −0.0595 ± i0.845 | −1.136 | −0.00000428 |

1h | 0.00227 ± i0.0771 | −2.429 / 0.798 | −0.0545 ± i0.819 | −1.136 | −0.0000258 |

1i | −0.00479 ± i0.0332 | −1.031 ± i1.346 | −0.141 ± i1.000 | −1.743 | −0.000181 |

1j | −0.0037 ± i0.0208 | −0.576 ± i1.181 | −0.0587 ± i0.848 | −1.087 | −0.000296 |

2a | −0.0130 ± i0.0386 | −0.652 ± i0.995 | −0.0305 ± i0.636 | −0.873 | −0.00226 |

2b | −0.0286 /−0.00306 | −0.555 ± i0.281 | −0.0267 ± i0.511 | −0.874 | 0.00160 |

2c | −0.00982 ± i0.0352 | −0.751 ± i1.131 | −0.0359 ± i0.632 | −1.010 | −0.00160 |

2d | −0.0241 /0.000679 | −0.642 ± i0.299 | −0.0325 ± i0.519 | −1.009 | 0.00112 |

2e | −0.00344 ± i0.00246 | −0.567 ± i1.051 | −0.00800 ± i0.644 | −1.193 | 0.000336 |

2f | −0.00419 ± i0.0155 | −0.706 ± i1.867 | −0.00271 ± i0.746 | −1.211 | 0.000471 |

2g | −0.00566 ± i0.00194 | −0.677 ± i1.142 | −0.0173 ± i0.700 | −1.407 | 0.000316 |

2h | −0.00194 /0.00459 | −0.841 ± i2.028 | −0.0144 ± i0.808 | −1.421 | 0.000442 |

**Table 8.**Eigenvalues of the stability matrix ($\omega $ —natural frequency;$\zeta $—damping ratio; $\xi $—amplification ratio).

Eigenvalue | Quantity | Symbol = Value | |
---|---|---|---|

Complex $\lambda =-\zeta \omega \pm i\Omega $ | Oscillation frequency | $\Omega \equiv \omega \sqrt{1-{\zeta}^{2}}$ | |

Real part $\mathrm{Re}(\lambda )=-\zeta \omega $ | Positive $\zeta >0$ | Damping ratio | $\zeta =-\xi =-\mathrm{Re}(\lambda )/\omega $ |

Negative $\zeta <0$ | Time constant | $\tau =1/\mathrm{Re}(\lambda )$ | |

Time to double amplitude | $T=0.693/\mathrm{Re}\left(\lambda \right)=0.693\hspace{0.33em}\tau $ |

**Table 9.**Parameters of flight modes: ${\lambda}^{\pm}$ –eigenvalues;$\omega $–natural frequency; $\zeta $–damping ratio; CAP = control anticipant parameter.

Stability | Longitudinal | Lateral | CAP | ||||||
---|---|---|---|---|---|---|---|---|---|

Mode | Phugoid | Short-Period | Dutch Roll | Roll | Spiral | ||||

Parameter | ${\mathit{\omega}}_{\mathit{p}}/{\mathit{\lambda}}_{1}^{+}$ | ${\mathit{\zeta}}_{\mathit{p}}/{\mathit{\lambda}}_{1}^{-}$ | ${\mathit{\omega}}_{\mathit{s}}/{\mathit{\lambda}}_{2}^{+}$ | ${\mathit{\zeta}}_{\mathit{s}}/{\mathit{\lambda}}_{2}^{-}$ | ${\mathit{\omega}}_{\mathit{d}}/{\mathit{\lambda}}_{3}^{+}$ | ${\mathit{\zeta}}_{\mathit{d}}/{\mathit{\lambda}}_{3}^{-}$ | ${\mathit{\zeta}}_{\mathit{r}}{\mathit{\omega}}_{\mathit{r}}=-{\mathit{\lambda}}_{4}^{+}$ | ${\mathit{\zeta}}_{\mathit{l}}{\mathit{\omega}}_{\mathit{l}}=-{\mathit{\lambda}}_{4}^{+}$ | |

Units | s^{−1} | −/s^{−1} | s^{−1} | −/s^{−1} | s^{−1} | -/s^{−1} | s^{−1} | s^{−1} | s^{−2} |

1a | 0.109 | 0.936 | 0.990 | 0.630 | 0.607 | 0.125 | 0.920 | 0.000397 | 0.0939 |

1b | 0.0719 | 0.00651 | /1.301 | /0.208 | 0.525 | 0.115 | 0.853 | 0.000382 | 0.0114 |

1c | 0.0342 | 0.239 | 1.135 | 0.647 | 0.663 | 0.138 | 1.065 | 0.000205 | 0.122 |

1d | 0.0641 | 0.00228 | /1.495 | /0.308 | 0.581 | 0.127 | 0.985 | 0.000112 | 0.0151 |

1e | 0.200 | 0.363 | /1.503 | /0.138 | 0.775 | 0.0591 | 0.958 | −0.0000104 | 0.00303 |

1f | 0.0695 | 0.0216 | /2.172 | /0.804 | 0.749 | 0.0569 | 0.956 | 0.0000110 | 0.103 |

1g | 0.240 | 0.475 | /1.701 | /0.124 | 0.847 | 0.0702 | 1.136 | 0.00000428 | 0.00245 |

1h | 0.0771 | −0.0295 | /2.429 | /0.798 | 0.820 | 0.0605 | 1.136 | 0.0000258 | 0.101 |

1i | 0.0335 | 0.143 | 1.695 | 0.608 | 1.010 | 0.140 | 1.743 | 0.000181 | 0.288 |

1j | 0.0211 | 0.175 | 1.314 | 0.438 | 0.850 | 0.0691 | 1.087 | 0.000296 | 0.222 |

2a | 0.411 | 0.0316 | 1.189 | 0.548 | 0.637 | 0.0479 | 0.873 | 0.000226 | 0.158 |

2b | /0.0286 | /0.00300 | 0.622 | 0.892 | 0.512 | 0.0521 | 0.874 | −0.00160 | 0.0126 |

2c | 0.0364 | 0.270 | 1.356 | 0.554 | 0.633 | 0.0567 | 1.01 | 0.00160 | 0.204 |

2d | /0.024 | /−0.000679 | 0.708 | 0.907 | 0.520 | 0.0625 | 1.009 | −0.00112 | 0.0142 |

2e | 0.0161 | 0.214 | 1.194 | 0.475 | 0.746 | 0.0125 | 1.211 | −0.000471 | 0.176 |

2f | 0.00422 | 0.993 | 1.996 | 0.354 | 0.644 | 0.00363 | 1.193 | −0.000336 | 0.555 |

2g | 0.0169 | 0.335 | 1.328 | 0.510 | 0.700 | 0.0247 | 1.421 | −0.000442 | 0.208 |

2h | /0.0194 | /0.00459 | 2.195 | 0.383 | 0.808 | 0.0178 | 1.407 | −0.000316 | 0.655 |

Mode | Level 1 | Level 2 | Level 3 | |
---|---|---|---|---|

Phugoid | ζ_{p} > 0.04 | ζ_{p} > 0 | T_{p} > 55 s | |

Short period | A + C | 0.35 < ζ_{s} < 1.30 | 0.25 < ζ_{s} < 2.30 | ζ_{s} > 0.15 |

B | 0.30 < ζ_{s} < 2.00 | 0.20 < ζ_{s} < 2.00 | ζ_{s} > 0.15 | |

Dutch Roll | A: ζ_{d} > 0.19B+C: ζ _{d} > 0.08 | ζ_{d} > 0.02 | ζ_{s} > 0.02 | |

A: Ω_{d}ζ_{d} > 0.35 rad/sB+C: Ω _{d}ζ_{d} > 0.15 rad/s | Ω_{d}ζ_{d} > 0.05 rad/s | - | ||

Ω_{d} > 0.40 rad/s | Ω_{d} > 0.40 rad/s | Ω_{d} > 0.40 rad/s | ||

Spiral Mode | T_{s} > 20 s | T_{s} > 12 s | T_{s} > 4 s | |

Roll Mode | τ_{r} < 1.4 s | τ_{r} < 3.0 s | τ_{r} < 10 s |

**Table 11.**Handling Qualities for all-natural modes (− means that not even level 3 criteria are met by at that mode or a sub-mode).

Mode | Phugoid | Short-Period | Dutch Roll | Roll | Spiral | CAP |
---|---|---|---|---|---|---|

1a | 1 | 1 | 2 | 1 | 1 | 2 |

1b | 2 | − | 2 | 1 | 1 | - |

1c | 1 | 1 | 2 | 1 | 1 | 2 |

1d | 2 | − | 2 | 1 | 1 | - |

1e | 1 | − | 3 | 1 | 1 | - |

1f | 2 | − | 3 | 1 | 1 | - |

1g | 1 | − | 2 | 1 | 1 | - |

1h | 3 | − | 2 | 1 | 1 | - |

1i | 1 | 1 | 2 | 1 | 1 | 1 |

1j | 1 | 1 | 2 | 1 | 1 | 1 |

2a | 2 | 1 | 3 | 1 | 1 | 1 |

2b | 3 | 1 | 3 | 1 | 1 | 3 |

2c | 1 | 1 | 3 | 1 | 1 | 1 |

2d | 3 | 1 | 3 | 1 | 1 | 3 |

2e | 1 | 1 | − | 1 | 1 | 1 |

2f | 1 | 1 | − | 1 | 1 | 1 |

2g | 1 | 1 | 3 | 1 | 1 | 1 |

2h | 3 | 1 | − | 1 | 1 | 1 |

Design | Case | Flight Condition | Manouver Point | ||||||
---|---|---|---|---|---|---|---|---|---|

Speed/ Mach | Altitude | Weight | Longitudinal | Lateral | |||||

kt | ×10^{3} ft | ×10^{3} kg | Range of Values | Estimated Value | Range of Values | Estimated Value | |||

BWB 1 | 1a/b | Minimum speed | 176 | 0 | 550 | 0.25 < x_{s} < 0.35 | x_{s} = 0.320 | x_{r} > 0.35 | x_{r} = 0.743 |

1c/d | approach | 200 | 0 | 550 | 0.25 < x_{s} < 0.35 | x_{s} = 0.320 | x_{r} > 0.35 | x_{r} = 0.779 | |

1e/f | Initial cruise | M = 0.85 | 39 | 670 | x_{s} < 0.35 | x_{s} = 0.345 | x_{r} > 0.39 | x_{r} = 0.992 | |

1g/h | Final cruise | M = 0.85 | 35 | 760 | x_{s} < 0.35 | x_{s} = 0.342 | x_{r} > 0.39 | x_{r} = 0.826 | |

BWB 2 | 2a/b | Minimum speed | 176 | 0 | 550 | x_{s} > 0.39 | x_{s} = 0.402 | x_{r} > 0.39 | x_{r} = 0.671 |

2c/d | approach | 200 | 0 | 550 | 0.35 < x_{s} < 0.39 | x_{s} = 0.390 | x_{r} > 0.39 | x_{r} = 0.772 | |

2e/f | Initial cruise | M = 0.85 | 39 | 670 | x_{s} > 0.39 | x_{s} = 0.553 | x_{r} < 0.35 | x_{r} = 0.330 | |

2g/h | Final cruise | M = 0.85 | 35 | 760 | x_{s} > 0.39 | x_{s} = 0.419 | x_{r} < 0.35 | x_{r} = 0.151 |

Manoeuver point | First kind | Second kind |
---|---|---|

Illustration | Figure 10 | Figure 11 |

Eigenvalue | $\lambda =-\zeta \omega \pm i\Omega $ | ${\lambda}_{1}\le {\lambda}_{2}<0$ |

At manouever point | $\lambda =\pm i\Omega $ | ${\lambda}_{1}\le {\lambda}_{2}=0$ |

Condition | $\mathrm{Re}\left(\lambda \right)=0$ | $\lambda =0$ |

Mode | oscillatory | non-oscillatory |

Example | BWB 1 | BWB 2 | |
---|---|---|---|

Fuselage | Length Width | Long Narrow | Short Wide |

Equal Fineless | Thickness Volume | Thick High | Thin Low |

Tail | Moment arm Elevator area | Long Small | Short Large |

Passenger motion | Pitch Roll | Large Small | Small Large |

Evacuation | Easy | Difficult | |

Conclusion | Conservative | Radical | |

Risk | Lower | Higher |

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## Share and Cite

**MDPI and ACS Style**

Campos, L.M.B.C.; Marques, J.M.G.
On the Handling Qualities of Two Flying Wing Aircraft Configurations. *Aerospace* **2021**, *8*, 77.
https://doi.org/10.3390/aerospace8030077

**AMA Style**

Campos LMBC, Marques JMG.
On the Handling Qualities of Two Flying Wing Aircraft Configurations. *Aerospace*. 2021; 8(3):77.
https://doi.org/10.3390/aerospace8030077

**Chicago/Turabian Style**

Campos, Luís M. B. C., and Joaquim M. G. Marques.
2021. "On the Handling Qualities of Two Flying Wing Aircraft Configurations" *Aerospace* 8, no. 3: 77.
https://doi.org/10.3390/aerospace8030077