VTOL aircraft with a thrust-to-weight ratio smaller than 0.1

This invention relates to aircraft, and more particularly, to aircraft with VTOL technology.

Currently, in the traditional VTOL technologies, thrust vectoring technology, such as AV-8, Yak-36 and F-35B, and additional lift engines, such as Yak-38 and Yak-141, in a least efficient way that air doesn't flow over the upper surface of the aircraft but rather over the lower one during vertical take-off/landing, get directly lift from reaction of atmosphere to air jet of jet engine; the use of tilt rotor, such as Boeing's V-22, and rotary wing, such as Boeing's X-50, in a little more efficient way that air both flow over the supper and lower of the aircraft during vertical take-off/landing, get lift from velocity difference of air which both flow the upper and lower surface of aircraft. Because of these limitations, VTOL is impossible for aircraft, and more particularly for large aircraft with thrust-to-weight smaller than 0.1.

An aircraft to achieve vertical take-off/landing in a much more efficient way that air flow over the upper surface of the aircraft rather than the lower one, the aircraft comprising:

• An aileron, which one pair rotate in opposite way for controlling horizontal balance and another pair in one way for controlling vertical balance during take-off/lading, disposed on both sides of the aircraft axis;
• A turbofan engine, comprising a high-temp duct with an openable and closable propelling nozzle and a low-temp duct with an openable and closable propelling nozzle;
• An openable and closable inlet of high-temp bypass duct, which is connected to the high-temp duct and is opened as the propelling nozzle of the high-temp duct is closed for causing high-temp airflow into the high-temp bypass duct during vertical take-off/landing, on the contrary, is closed as one of the high-temp duct is opened for causing high-temp airflow into atmosphere during forward flight.
• An outer wall of high-temp bypass duct, which any cross-sectional area is large enough to keep velocity of high-temp airflow less than one of sound along its direction for avoiding congestion.
• A rectangle outlet of high-temp bypass duct, which height-to-width ratio is smaller than 0.1 enough to convert high-temp air into high-temp planar jet flowing above low-temp planar jet for keeping no air to flow over the lower surface of the aircraft during vertical take-off/landing;
• An openable and closable inlet of low-temp bypass duct, which is connected to the low-temp duct and is opened as the propelling nozzle of the low-temp duct is closed for causing low-temp airflow into the low-temp bypass duct during vertical take-off/landing, on the contrary, is closed as one of the low-temp duct is opened for causing low-temp airflow into atmosphere during forward flight.
• An outer wall and an inner wall of low-temp bypass duct, which any cross-sectional area is large enough to keep velocity of low-temp airflow less than one of sound along its direction for avoiding congestion.
• A rectangle outlet of low-temp bypass duct, which height-to-width ratio is smaller than 0.1 and area is small enough to provide the necessary velocity of the low-temp planar jet with features: concentrated energy, wide coverage and slow decay of shaft velocity, disposed below the rectangle outlet of the high-temp bypass duct and on the upper surface of the aircraft for spewing the low-temp planar jet to flow over the upper one comprising the ailerons, thereby generating the life and the power controlling vertical and horizontal balances.

According to the first characteristics of embodiments:

• The rectangle outlet of low-temp bypass duct, during vertical take-off/landing, is part which converts low-temp air provided by low-temp duct of the turbofan engine into low-temp planar jet having features: concentrated energy, wide coverage and slow decay of shaft velocity.
• The high-temp bypass duct, during vertical take-off/landing, causes the high-temp air above the low-temp planar jet for keeping no air to flow over lower surface of the aircraft during vertical take-off/landing.
• The aileron, during vertical take-off/landing, one pair rotate in opposite way to control the horizontal balance and another pair rotate in one way control the vertical balance. The attached drawings illustrate the invention:

  • Fig. 1 is a side view of an aircraft with this invention during vertical take-off and landing.
  • Fig. 2 is a top view of an aircraft with this invention during vertical take-off and landing.
  • Fig. 3 is a rear view of an aircraft with this invention during vertical take-off and landing.
  • Fig. 4 is J local view for Fig. 3.
  • Fig. 5 is A-A Section for Fig. 4
  • Fig. 6 is B-B Section for Fig. 4.
  • Fig. 7 is part Section for Fig. 4.
  • Fig. 8 is K local view for Fig. 3.
  • Fig. 9 is C-C Section for Fig. 8.
  • Fig. 10 is a side view of an aircraft with this invention during forward flight.
  • Fig. 11 is a top view of an aircraft with this invention during forward flight.
  • Fig. 12 is a rear view of an aircraft with this invention during forward flight.
  • Fig. 13 is M local view for Fig. 12.
  • Fig. 14 is D-D Section for Fig. 13.
  • Fig. 15 is E-E Section for Fig. 13.
  • Fig. 16 is part Section for Fig. 13.
  • Fig. 17 is N local view for Fig. 12.
  • Fig. 18 is F-F Section for Fig. 17.

Referring to these drawings, the aircraft comprising: an aileron (1, 2) which one pair(1) rotate in opposite way and another pair (2) rotate in one way; a turbofan engine (3) comprising a high-temp duct (4) with an openable and closable propelling nozzle (5), and a low-temp duct (6) with an openable and closable propelling nozzle (7); an openable and closable inlet (8) of high-temp bypass duct (9), connected to the high-temp duct (4); an outer wall (10) of high-temp bypass duct (9), which any cross-sectional area is large enough to keep velocity of high-temp airflow (11) less than one of sound along its direction for avoiding congestion; a rectangle outlet (12) of high-temp bypass duct (9), which height-to-width ratio is smaller than 0.1 enough to form high-temp planar jet (13); an openable and closable inlet (14) of low-temp bypass duct (15), connected to the low-temp duct (6); an outer wall (16) and an inner wall (17) of low-temp bypass duct (15), which any cross-sectional area is large enough to keep velocity of low-temp airflow (18) less than one of sound along its direction for avoiding congestion; a rectangle outlet (19) of low-temp bypass duct (15), which height-to-width ratio is smaller than 0.1 and area is small enough to provide the necessary velocity of the low-temp planar jet (20) with features: concentrated energy, wide coverage and slow decay of shaft velocity.

Referring to Fig.(1-9), the innovative combination being configured, during vertical take-off/lading, in such that closing the propelling nozzles (5, 7) of the high-temp duct (4) and low-temp duct (6) and opening the inlets (8, 14) of the high-temp bypass duct (9) and low-temp bypass duct (15), cause high-temp air (11) and low-temp air (18) in to flow over corresponding high-temp bypass duct (9) and low-temp bypass duct (15), then the low-temp planar jet (20) spews out of the outlet (19) of low-temp bypass duct (15), over upper surface of the aircraft including the aileron (1, 2), thereby generating lift for vertical take-off/landing and power for controlling the vertical and horizontal balances of the aircraft, and the high-temp planar jet (13) spews out of the outlet (12) of high-temp bypass duct (9), above the low-temp planar jet (20), into atmosphere for keeping no air to flow over lower surface of the aircraft.

Referring to Fig.(10-18), the innovative combination being configured, during forward flight, restore to normal states in order that high-temp air (11) and low-temp air (18) spew directly into atmosphere from the open propelling nozzles(5, 7) of the high-temp duct (4) and a low-temp duct(6).

All the formulas and calculations for this invention are listed in Annex 1.

All the formulas and calculations used to retrofit an Airbus A-380 based on this invention are listed in Annex 2.

This invention can be used to retrofit and existing aircraft to achieve VTOL or manufacture a VTOL aircraft with a thrust-to-weight ratio smaller than 0.1.

ANNEX 1

P

Atmospheric pressure

(Unit: Pa)

R

Ideal gas constant

(Unit:J. K^-1. mol^-1)

ρ₀

Low-temp jet density at the outlet

(Unit:kg/m³)

ρ₁

Atmospheric density

(Unit:kg/m³)

ρ

Jet density on the cross-section

(Unit: kg/m³)

ρ_m

Jet density of the shaft

(Unit: kg/m³)

T₀

Jet temp at the low-temp outlet

(Unit: K)

T₁

Atmospheric temp

(Unit: K)

T

Jet temp on the cross-section

(Unit: K)

V₀

Jet speed at the outlet of low-temp bypass duct

(Unit: m/s)

$V_0^{ʹ}$

Jet speed at the outlet of high-temp bypass duct

(Unit: m/s)

V

Jet speed on the cross-section

(Unit: m/s)

V_m

Jet speed of shaft

(Unit: m/s)

h₀

Jet height of the outlet

(Unit: m)

h_m

Jet height of the cross-section

(Unit: m)

h

Jet height on the cross-section

(Unit: m)

L

Width of the outlet

(Unit: m)

C

Gas specific volume

(Unit: m3/kg)

M

Molar mass

(Unit: kg/mol)

Q

Air inflow of turbofan engine

(Unit: kg/s)

B

Bypass ratio of turbofan engine

(Dimensionless unit)

G

Maximum take-off weight

(Unit: T)

X

Distance between any point and outlet of low-temp bypass duct in jet direction

(Unit: m)

X₁

Distance between wing root and low-temp outlet in jet direction

(Unit: m)

X₂

Distance between low-temp outlet and crossing of jet boundary and trailing edge of wing in jet direction

(Unit: m)

X₃

Distance between wingtip and low-temp outlet in jet direction

(Unit: m)

α

Angle between jet boundary and trailing edge of wing

(Unit: °)

β

Angle between chord line of wing and long side of outlet

(Unit: °)

γ

Dihedral angle of wing

(Unit: °)

δ

Sweep forward angle of jet

(Unit: °)

φ

Included angle between axis of low-temp jet And one of the high-temp jet

(Unit: °)

θ

Sweep back angle of jet

(Unit: °)

F

Total vertical lift

(Unit: T

F₁

Lift between wing root and low-temp outlet

(Unit: T)

F₂

Lift between low-temp outlet and crossing of jet boundary and trailing edge of wing

(Unit: T)

F₃

Lift between wingtip and low-temp outlet

(Unit: T)

F₄

Downward pressure generated by slant upward deflected jet

(Unit: T)

F₅

Thrust generated by jet of low-temp outlet

(Unit: T)

F₆

Thrust generated by jet of high-temp outlet

(Unit: T)

b

Constant

(Dimensionless unit)

d

Constant

(Dimensionless unit)

C

Constant

(Dimensionless unit)

n₁

S/N of turbofan engine

(Dimensionless unit)

n₂

S/N of turbofan engine used in vertical Take-off/landing

(Dimensionless unit)

n₃

S/N of outlet of low-temp bypass duct

(Dimensionless unit)

TWR

Thrust-to-weight ratio of aircraft

(Dimensionless unit)

According to thermodynamic, relative values of the enthalpy of the jet cross-section, in the case of equal pressure and the surrounding gas enthalpy acting as a starting point, is same.

And according to the feature of planar jet, it just spread on the flat which is perpendicular to the outlet section. $$P=\frac{{\rho }_0}{M}{\mathit{RT}}_0=\frac{{\rho }_1}{M}{\mathit{RT}}_1=\frac{\rho }{M}\mathit{RT}$$ $${\rho }_0{V}_0{h}_0\mathit{LC}\left(T_0-T_1\right)=\int \rho \mathit{VLC}\left(T-T_1\right)dh$$ $${\rho }_0{V}_0{h}_0\left(\frac{1}{{\rho }_0}-\frac{1}{{\rho }_1}\right)=\int \rho \left(\frac{1}{\rho }-\frac{1}{{\rho }_1}\right)dh$$

And according to the similarity of velocity and density distribution on the various jet sections, $$\begin{array}{l}\sqrt{\frac{V}{V_m}}=\frac{\rho -{\rho }_1}{{\rho }_m-{\rho }_1}=1-1-{\left(\frac{h}{h_m}\right)}^{1.5}\\ h_0{V}_0\left({\rho }_0-{\rho }_1\right)=\int \left(\rho -{\rho }_1\right)\mathit{Vdh}\\ \frac{h_0{V}_0\left({\rho }_0-{\rho }_1\right)}{\left({\rho }_m-{\rho }_1\right)V_m{h}_m}=\int \frac{\left(\rho -{\rho }_1\right)V}{\left({\rho }_m-{\rho }_1\right)V_m{h}_m}\mathit{dh}=\underset{0}{\overset{h_m}{\int }}{\left[1-{\left(\frac{h}{h_m}\right)}^{1.5}\right]}^3\cdot \frac{1}{h_m}\mathit{dh}\\ =\underset{0}{\overset{1}{\int }}{\left[1-{\left(\frac{h}{h_m}\right)}^{1.5}\right]}^{3}d\left(\frac{h}{h_m}\right)=0.3682\\ \left({\rho }_m-{\rho }_1\right)=\frac{h_0{V}_0\left({\rho }_0-{\rho }_1\right)}{0.3682\cdot V_m{h}_m}\end{array}$$

According to dynamic characteristic of jet, momentum of the various sections, in the case of equal pressure, is same. $$\begin{array}{l}{\rho }_0{h}_0{V}_0^2=\int \rho V^2\mathit{dh}\\ {\rho }_0{h}_0{V}_0^2=\int \rho V^2\mathit{dh}=\int \left(\rho -{\rho }_1\right)V^2\mathit{dh}+\int {\rho }_1{V}^2\mathit{dh}\\ \frac{{\rho }_0{V}_0{V}_0^2}{\left({\rho }_m-{\rho }_1\right)V_m^2{h}_m}\\ =\underset{0}{\overset{1}{\int }}\frac{\left(\rho -{\rho }_1\right)V^2}{\left({\rho }_m-{\rho }_1\right)V_m^2}\mathit{d}\left(\frac{h}{h_m}\right)+{\int }_0^1\frac{{\rho }_1{V}^2}{\left({\rho }_m-{\rho }_1\right)V_m^2}\mathit{d}\left(\frac{h}{h_m}\right)\\ =\underset{0}{\overset{1}{\int }}{\left[1-{\left(\frac{h}{h_m}\right)}^{1.5}\right]}^{5}d\left(\frac{h}{h_m}\right)+\frac{{\rho }_1}{\left({\rho }_m-{\rho }_1\right)}{\int }_0^1{\left[1-{\left(\frac{h}{h_m}\right)}^{1.5}\right]}^{4}d\left(\frac{h}{h_m}\right)\\ =0.2786+\frac{0.3156{\rho }_1}{\left({\rho }_m-{\rho }_1\right)}\\ \frac{0.3682V_0{\rho }_0}{V_m\left({\rho }_0-{\rho }_1\right)}=0.2786+\frac{0.1162h_m{V}_m{\rho }_1}{h_0{V}_0\left({\rho }_0-{\rho }_1\right)}\end{array}$$ Substitute (1) into (2): $$\begin{array}{l}\frac{0.1162h_m{\rho }_1}{h_0{\rho }_0}\cdot {\left(\frac{V_m}{V_0}\right)}^2+0.2786\frac{\left({\rho }_0-{\rho }_1\right)}{{\rho }_0}\cdot {\left(\frac{V_m}{V_0}\right)}^{}-0.3682=0\\ \frac{V_m}{V_0}=\frac{\frac{0.2786\left({\rho }_1-{\rho }_0\right)}{{\rho }_0}+\sqrt{{\left[\frac{0.2786\left({\rho }_1-{\rho }_0\right)}{{\rho }_0}\right]}^2+\frac{0.1711h_m{\rho }_1}{h_0{\rho }_0}}}{\frac{0.2324h_m{\rho }_1}{h_0{\rho }_0}}\\ V_m=\frac{1+\sqrt{1+\frac{0.1711h_m{\rho }_0{\rho }_1}{{0.2786}^2{h}_0{\left({\rho }_0-{\rho }_1\right)}^2}}}{\frac{0.2324h_m{\rho }_1}{0.2786h_0\left({\rho }_0-{\rho }_1\right)}}{V}_0\end{array}$$ Substitute (3) into (1) $$\begin{array}{l}\left({\rho }_m-{\rho }_1\right)=\frac{h_0{V}_0\left({\rho }_0-{\rho }_1\right)}{0.3682\cdot V_m{h}_m}\\ {\rho }_m={\rho }_1-\frac{\frac{0.2324}{0.3682\times 0.2786}{\rho }_1}{1+\sqrt{1+\frac{0.1711h_m{\rho }_0{\rho }_1}{{0.2786}^2{h}_0{\left({\rho }_0-{\rho }_1\right)}^2}}}\end{array}$$ When X ≤ X₁ or X ≤ X₂ $$F=P-\left(P-\frac{1}{2}\int {\rho }_m{V}_m^{2}Ldx\right)=\frac{1}{2}\int {\rho }_m{V}_m^{2}Ldx$$ Substitute (3), (4) into (2) $$\begin{array}{l}=\frac{1}{2}\underset{0}{\overset{X}{\int }}\left[{\rho }_1-\frac{\frac{0.2324}{0.3682\times 0.2786}{\rho }_1}{1+\sqrt{1+\frac{0.1711h_m{\rho }_0{\rho }_1}{{0.2786}^2{h}_0{\left({\rho }_0-{\rho }_1\right)}^2}}}\right]\cdot {\left[\frac{1+\sqrt{1+\frac{0.1711h_m{\rho }_0{\rho }_1}{{0.2786}^2{h}_0{\left({\rho }_0-{\rho }_1\right)}^2}}}{\frac{0.2324h_m{\rho }_1}{0.2786h_0\left({\rho }_0-{\rho }_1\right)}}\right]}^2\cdot V_0^2\mathit{Ldx}\\ =\frac{1}{2}\underset{0}{\overset{X}{\int }}\{{\rho }_1{\left[\frac{1+\sqrt{1+\frac{0.1711h_m{\rho }_0{\rho }_1}{{0.2786}^2{h}_0{\left({\rho }_0-{\rho }_1\right)}^2}}}{\frac{0.2324h_m{\rho }_1}{0.2786h_0\left({\rho }_0-{\rho }_1\right)}}\right]}^2\\ -\frac{0.2324}{0.3682\times 0.2786}{\rho }_1\cdot \left[\frac{1+\sqrt{1+\frac{0.1711h_m{\rho }_1{\rho }_0}{{0.2786}^2{h}_0{\left({\rho }_1-{\rho }_0\right)}^2}}}{\frac{0.2324h_m{\rho }_1}{0.2786h_0\left({\rho }_1-{\rho }_0\right)}}\right]\}\cdot V_0^2\mathit{Ldx}\\ =\frac{1}{2}\underset{0}{\overset{X}{\int }}{\rho }_1\{\frac{2+\frac{0.1711h_m{\rho }_1{\rho }_0}{{0.2786}^2{h}_0{\left({\rho }_1-{\rho }_0\right)}^2}+2\sqrt{1+\frac{0.1711h_m{\rho }_1{\rho }_0}{{0.2786}^2{h}_0{\left({\rho }_1-{\rho }_0\right)}^2}}}{{\left[\frac{0.2324h_m{\rho }_1}{0.2786h_0\left({\rho }_1-{\rho }_0\right)}\right]}^2}\\ -\frac{\frac{0.2324}{0.3682\times 0.2786}+\frac{0.2324}{0.3682\times 0.2786}\sqrt{1+\frac{0.1711h_m{\rho }_1{\rho }_0}{{0.2786}^2{h}_0{\left({\rho }_1-{\rho }_0\right)}^2}}}{{\left[\frac{0.2324h_m{\rho }_1}{0.2786h_0\left({\rho }_1-{\rho }_0\right)}\right]}^2}\}\cdot V_0^2\mathit{Ldx}\\ =\frac{1}{2}\underset{0}{\overset{X}{\int }}\left\{\frac{0.1711{\rho }_0}{{0.2324}^2\left(\frac{h_m}{h_0}\right)}-\frac{1.2655}{{\left[\frac{0.2324{\rho }_1}{0.2786\left({\rho }_1-{\rho }_0\right)}\right]}^2{\left(\frac{h_m}{h_0}\right)}^2}-\frac{0.2655\sqrt{1+\frac{0.1711{\rho }_1{\rho }_0}{{0.2786}^2{\left({\rho }_1-{\rho }_0\right)}^2}\cdot \left(\frac{h_m}{h_0}\right)}}{{\left[\frac{0.2324{\rho }_1}{0.2786\left({\rho }_1-{\rho }_0\right)}\cdot \left(\frac{h_m}{h_0}\right)\right]}^2}\right\}\\ \cdot V_0^2\mathit{Ldx}\end{array}$$ Because of $$\frac{h_m}{h_0}=2.44\left(\frac{0.24x}{h_0}+0.41\right)$$ When X ≤ X₂ or X ≤ X₃ $$\begin{array}{l}{F}_1=\frac{1}{2}\underset{1.0004}{\overset{2.44\left(\frac{0.24X_1}{h_0}+0.41\right)}{\int }}\{\frac{0.1711{\rho }_0}{{0.2324}^2\left(\frac{h_m}{h_0}\right)}-\frac{1.2655}{{\left[\frac{0.2324{\rho }_1}{0.2786\left({\rho }_1-{\rho }_0\right)}\right]}^2{\left(\frac{h_m}{h_0}\right)}^2}\\ -\frac{0.2655\sqrt{1+\frac{0.1711{\rho }_1{\rho }_0}{{0.2786}^2{\left({\rho }_1-{\rho }_0\right)}^2}\cdot \left(\frac{h_m}{h_0}\right)}}{{\left[\frac{0.2324{\rho }_1}{0.2786\left({\rho }_1-{\rho }_0\right)}\cdot \left(\frac{h_m}{h_0}\right)\right]}^2}\}\cdot \frac{h_0{V}_0^{2}L}{2.44\times 0.24}d\left(\frac{h_m}{h_0}\right)\\ =\frac{1}{2}\underset{1.0004}{\overset{2.44\left(\frac{0.24X_1}{h_0}+0.41\right)}{\int }}\{\frac{0.1711{\rho }_0}{{0.2324}^{2}x}-\frac{1.2655}{{\left[\frac{0.2324{\rho }_1}{0.2786\left({\rho }_1-{\rho }_0\right)}\right]}^2{x}^2}\\ -\frac{0.2655}{{\left[\frac{0.2324{\rho }_1}{0.2786\left({\rho }_1-{\rho }_0\right)}\right]}^2}\cdot \frac{\sqrt{+\frac{0.1711{\rho }_1{\rho }_0}{{0.2786}^2{\left({\rho }_1-{\rho }_0\right)}^2}\cdot x}}{\cdot x^2}\}\cdot \frac{h_0{V}_0^{2}L}{2.44\times 0.24}\mathit{dx}\\ \int \frac{\sqrt{1+bx}}{x^2}dx=-\left(\frac{\sqrt{1+bx}}{x}-\frac{b}{2}\mathit{ln}\frac{\sqrt{1+bx}-1}{\sqrt{1+bx}+1}+c\right)\\ F_1=\{\frac{0.1711{\rho }_1{\rho }_0}{{0.2324}^2}\ln x+\frac{1.2655}{{\left[\frac{0.2324{\rho }_1}{0.2786\left({\rho }_1-{\rho }_0\right)}\right]}^{2}x}+\frac{0.2655}{{\left[\frac{0.2324{\rho }_1}{0.2786\left({\rho }_1-{\rho }_0\right)}\right]}^2}\cdot [\frac{\sqrt{1+\frac{0.1711{\rho }_1{\rho }_0}{{0.2786}^2{\left({\rho }_1-{\rho }_0\right)}^2}\cdot x}}{x}\\ -{\frac{0.1711{\rho }_1{\rho }_0}{{2\times 0.2786}^2{\left({\rho }_1-{\rho }_0\right)}^2}\ln \frac{\sqrt{1+\frac{0.1711{\rho }_1{\rho }_0}{{0.2786}^2{\left({\rho }_1-{\rho }_0\right)}^2}\cdot x}-1}{\sqrt{1+\frac{0.1711{\rho }_1{\rho }_0}{{0.2786}^2{\left({\rho }_1-{\rho }_0\right)}^2}\cdot x}+1}]\}}_{2.44\times 0.41}^{2.44\left(\frac{0.24X_1}{h_0}+0.41\right)}\\ \times \frac{1}{2}\times \frac{h_0{V}_0^{2}L}{2.44\times 0.24}\end{array}$$ $$\begin{array}{l}{F}_2=\{\frac{0.1711{\rho }_0}{{0.2324}^2}\mathit{lnx}+\frac{1.2655}{{\left[\frac{0.2324{\rho }_1}{0.2786\left({\rho }_1-{\rho }_0\right)}\right]}^{2}x}+\frac{0.2655}{{\left[\frac{0.2324{\rho }_1}{0.2786\left({\rho }_1-{\rho }_0\right)}\right]}^2}\cdot [\frac{\sqrt{1+\frac{0.1711{\rho }_1{\rho }_0}{{0.2786}^2{\left({\rho }_1-{\rho }_0\right)}^2}\cdot x}}{x}\\ -{\frac{0.1711{\rho }_1{\rho }_0}{2\times {0.2786}^2{\left({\rho }_1-{\rho }_0\right)}^2}\mathit{ln}\frac{\sqrt{1+\frac{0.1711{\rho }_1{\rho }_0}{{0.2786}^2{\left({\rho }_1-{\rho }_0\right)}^2}\cdot x}-1}{\sqrt{1+\frac{0.1711{\rho }_1{\rho }_0}{{0.2786}^2{\left({\rho }_1-{\rho }_0\right)}^2}\cdot x}+1}]\}}_{2.44\times 0.41}^{2.44\left(\frac{0.24X_2}{h_0}+0.41\right)}\\ \times \frac{1}{2}\times \frac{h_0{V}_0^{2}L}{2.44\times 0.24}\end{array}\mathrm{.}$$ When X₂ ≤ X ≤ X₃ $$\begin{array}{l}{F}_3=\frac{1}{2}\underset{X_2}{\overset{X_3}{\int }}\{\frac{0.1711{\rho }_0}{{0.2324}^2\left(\frac{h_m}{h_0}\right)}-\frac{1.2655}{{\left[\frac{0.2324{\rho }_1}{0.2786\left({\rho }_1-{\rho }_0\right)}\right]}^2{\left(\frac{h_m}{h_0}\right)}^2}\\ -\frac{0.2655\sqrt{1+\frac{0.1711{\rho }_1{\rho }_0}{{0.2786}^2\left({\rho }_1-{\rho }_0\right)}\cdot \left(\frac{h_m}{h_0}\right)}}{{\left[\frac{0.2324{\rho }_1}{0.2786\left({\rho }_1-{\rho }_0\right)}\cdot \left(\frac{h_m}{h_0}\right)\right]}^2}\}\times V_0^2\left[L-\left(x-X_2\right)\times \mathit{tan}\alpha \right]\mathit{dx}\end{array}$$ When force is zero in body axis direction, $$F_4=F_5+F_6$$ $$\left(F_4+F_5+F_6\right)\sin \delta =\left(F_5+F_6\right)\sin \theta$$ $$\delta ={\sin }^{-1}\frac{1}{2}\sin \theta$$ $$F=n_2\left(F_1+F_2+F_3\right)\cos \beta \cos \gamma -n_2\left[F_4\cos \delta +F_6\sin \left(+\gamma \right)-F_6\sin \left(-\gamma \right)\right]$$

VTOL can be achieved once : F>G

ANNEX 2

• n₁=1∼4
• n₂ = 1∼2
• n₃ = 1∼4
• M = 29 kg/mol
• Q(Air inflow of Trent 900 turbo f an engine) = 1204 kg/s
• B(Bypass ratio of Trent 900turbofan engine) = 8.5:1 1
• G(Maximum take - of f weight of A380) = 560 T
• T₀ = 383K
• ρ₁ = 1.293kg/m²

Assuming:

• θ = 36.5° $$F_4=\frac{1}{n_2}Q{V}_0=1204\times 243=292572N=12.7T$$ $${\rho }_0=\frac{{\mathit{MP}}_0}{{\mathit{RT}}_0}=\frac{101325\times 29\times 0.001}{8.31\times 383}=0.9232\mathit{kg}/m^3$$
• h₀ = 0.4m
• L = 6m $$V_0=\frac{\frac{1}{n_2}\times Q\times \frac{8.5}{8.5+1}}{{\rho }_0\times L\times h_0}=\frac{\frac{1}{2}\times 1204\times \frac{8.5}{8.5+1}}{0.9232\times 6\times 0.4}=243m/s$$ $V_0^{ʹ}=V_0=243m/s$ $$F_6=\frac{1}{n_2}Q\times \frac{1}{8.5+1}\times V_0=15398.5363N=1.57T$$
• X₁=11 m
• X₂=18m
• X₃=32 m
• α = 14°
• β=5°
• γ=5°
• θ = 36.5°
• Ø = 18.75° $$\begin{array}{l}{F}_1=121002.0492[2.9246\mathit{ln}x+\frac{0.1488}{x}+0.0312\cdot [\frac{\sqrt{1+19.2419x}}{x}\\ -9.621\mathit{ln}\frac{\sqrt{1+19.2419x}-1}{\sqrt{1+19.2419x}+1}{]}_{2.44\times 0.41}^{2.44\left(\frac{0.24X_1}{h_0}+0.41\right)}\\ =121002.0492\{2.9246\mathit{ln}\frac{2.44\left(\frac{0.24\times 11}{0.4}+0.41\right)}{2.44\times 0.41}\\ +0.1488\left[\frac{1}{2.44\left(\frac{0.24\times 18}{0.4}+0.41\right)}-\frac{1}{2.44\times 0.41}\right]\\ +0.0312\cdot [\frac{\sqrt{1+19.2419\times 2.44\left(\frac{0.24\times 11}{0.4}+0.41\right)}}{2.44\left(\frac{0.24\times 11}{0.4}+0.41\right)}-\frac{\sqrt{1+19.2419\times 2.44\times 0.41}}{2.44\times 0.41}\\ -9.621\mathit{ln}\frac{\sqrt{1+19.2419\times 2.44\left(\frac{0.24\times 11}{0.4}+0.41\right)}-1}{\sqrt{1+19.2419\times 2.44\left(\frac{0.24\times 11}{0.4}+0.41\right)}+1}\\ +9.621\mathit{ln}\frac{\sqrt{1+19.2419\times 2.44\times 0.41}-1}{\sqrt{1+19.2419\times 2.44\times 0.41}+1}\}]\\ =121002.0492\left[8.3028-0.14-0.1072-0.1026\right]\\ =962329N\\ =98T\end{array}$$ $$\begin{array}{l}{F}_2=121002.0492\{2.9246\mathit{ln}\frac{2.44\left(\frac{0.24\times 18}{0.4}+0.41\right)}{2.44\times 0.41}\\ +0.1488\left[\frac{1}{2.44\left(\frac{0.24\times 18}{0.4}+0.41\right)}-\frac{1}{2.44\times 0.41}\right]\\ +0.0312\cdot [\frac{\sqrt{1+19.2419\times 2.44\left(\frac{0.24\times 18}{0.4}+0.41\right)}}{2.44\left(\frac{0.24\times 18}{0.4}+0.41\right)}\\ -\frac{\sqrt{1+19.2419\times 2.44\times 0.41}}{2.44\times 0.41}]-9.621\mathit{ln}\frac{\sqrt{1+19.2419\times 2.44\left(\frac{0.24\times 18}{0.4}+0.41\right)}-1}{\sqrt{1+19.2419\times 2.44\left(\frac{0.24\times 18}{0.4}+0.41\right)}+1}\\ +9.621\mathit{ln}\frac{\sqrt{1+19.2419\times 2.44\times 0.41}-1}{\sqrt{1+19.2419\times 2.44\times 0.41}+1})]\\ =121002.0492\left[9.6758-0.1358-0.1141-0.1047\right]\\ =1127884.301N\\ =115T\end{array}$$ $$\begin{array}{l}{F}_3=\frac{1}{2}\underset{2.44\left(\frac{0.24X_2}{h_0}+0.41\right)}{\overset{2.44\left(\frac{0.24X_3}{h_0}+0.41\right)}{\int }}\{\frac{0.1711{\rho }_0}{{0.2324}^2\left(\frac{h_m}{h_0}\right)}-\frac{1.2655}{{\left[\frac{0.2324{\rho }_1}{0.2786\left({\rho }_1-{\rho }_0\right)}\right]}^2{\left(\frac{h_m}{h_0}\right)}^2}\\ -\frac{0.2655\sqrt{1+\frac{0.1711{\rho }_1{\rho }_0}{{0.2786}^2{\left({\rho }_1-{\rho }_0\right)}^2}\cdot \left(\frac{h_m}{h_0}\right)}}{{\left[\frac{0.2324{\rho }_1}{0.2786\left({\rho }_1-{\rho }_0\right)}\cdot \left(\frac{h_m}{h_0}\right)\right]}^2}\}\\ \times \left[L-\left(\frac{h_0}{2.44\times 0.24}\times \left(\frac{h_m}{h_0}\right)-\frac{0.41h_0}{0.24}-X_2\right)\times \mathit{tan\alpha }\right]\frac{h_0{V}_0^2}{2.44\times 0.24}d\left(\frac{h_m}{h_0}\right)\\ =\frac{1}{2}\underset{2.44\left(\frac{0.24X_2}{h_0}+0.41\right)}{\overset{2.44\left(\frac{0.24X_3}{h_0}+0.41\right)}{\int }}\{\frac{0.1711{\rho }_0}{{0.2324}^{2}x}-\frac{1.2655}{{\left[\frac{0.2324{\rho }_1}{0.2786\left({\rho }_1-{\rho }_0\right)}\right]}^2{x}^2}\\ -\frac{0.2655\sqrt{1+\frac{0.1711{\rho }_1{\rho }_0}{{0.2786}^2{\left({\rho }_1-{\rho }_0\right)}^2}\cdot \left(\frac{h_m}{h_0}\right)}}{{\left[\frac{0.2324{\rho }_1}{0.2786\left({\rho }_1-{\rho }_0\right)}\cdot x\right]}^2}\}\\ \cdot \left[L-\left(\frac{h_0}{2.44\times 0.24}x-\frac{0.41h_0}{0.24}-X_2\right)\times \mathit{tan\alpha }\right]\frac{h_0{V}_0^2}{2.44\times 0.24}\mathit{dx}\\ =\frac{1}{2}\underset{2.44\left(\frac{0.24X_2}{h_0}+0.41\right)}{\overset{2.44\left(\frac{0.24X_3}{h_0}+0.41\right)}{\int }}\langle \frac{0.1711{\rho }_0}{{0.2324}^{2}x}-\frac{1.2655}{{\left[\frac{0.2324{\rho }_1}{0.2786\left({\rho }_1-{\rho }_0\right)}\right]}^2{x}^2}\\ -\frac{0.2655\sqrt{1+\frac{0.1711{\rho }_1{\rho }_0}{{0.2786}^2{\left({\rho }_1-{\rho }_0\right)}^2}\cdot x}}{{\left[\frac{0.2324{\rho }_1}{0.2786\left({\rho }_1-{\rho }_0\right)}\right]}^2{x}^2}\}\cdot \left[L+\left(\frac{0.41h_0}{0.24}+X_2\right)\cdot \mathit{tan}\alpha \right]\\ -\left\{\frac{0.1711{\rho }_0}{{0.2324}^2}-\frac{1.2655}{{\left[\frac{0.2324{\rho }_1}{0.2786\left({\rho }_1-{\rho }_0\right)}\right]}^{2}x}-\frac{0.2655\sqrt{1+\frac{0.1711{\rho }_1{\rho }_0}{{0.2786}^2{\left({\rho }_1-{\rho }_0\right)}^2}\cdot x}}{{\left[\frac{0.2324{\rho }_1}{0.2786\left({\rho }_1-{\rho }_0\right)}\cdot \right]}^{2}x}\right\}\\ \frac{h_0\mathit{tan\alpha }}{2.44\times 0.24})\frac{h_0{V}_0^2}{2.44\times 0.24}\mathit{dx}\\ \int \frac{\sqrt{1+dx}}{x^2}dx=2\sqrt{1+dx}-\mathit{ln}\frac{\sqrt{1+dx}-1}{\sqrt{1+dx}+1}+C\\ F_3=121002.0492\{10.6583\times [2.9246\mathit{lnx}+\frac{0.1488}{x}\\ +0.0312\cdot \left(\frac{\sqrt{1+19.2419x}}{x}-9.621\mathit{ln}\frac{\sqrt{1+19.2419x}-1}{\sqrt{1+19.2419x}+1}\right)]\\ -0.1703[2.9246x-0.1488\mathit{lnx}-0.0312(2\sqrt{1+19.2419x}\\ =121002.0492\{10.6625\times [2.9246\mathit{ln}\frac{47.8484}{27.3524}\end{array}$$ $$\begin{array}{l}+0.1488\left(\frac{1}{47.8484}-\frac{1}{27.3524}\right)+0.0312\left(\frac{\sqrt{1+19.2419\times 47.8484}}{47.8484}-\frac{\sqrt{1+19.2419\times 27.3524}}{27.3524}\right)\\ -9.621\mathit{ln}\frac{\sqrt{1+19.2419\times 47.8484}-1}{\sqrt{1+19.2419\times 47.8484}+1}+9.621\mathit{ln}\frac{\sqrt{1+19.2419\times 27.3524}-1}{\sqrt{1+19.2419\times 27.3524}+1})]\\ -0.1703[2.9246\left(47.8484-27.3524\right)-0.1488\mathit{ln}\frac{47.8484}{27.3524}-0.0312(2\sqrt{1+19.2419\times 47.8484}\\ -2\sqrt{1+19.2419\times 27.3524}-\mathit{ln}\frac{\sqrt{1+19.2419\times 47.8484}-1}{\sqrt{1+19.2419\times 47.8484}+1}\\ +\mathit{ln}\frac{\sqrt{1+19.2419\times 27.3524}-1}{\sqrt{1+19.2419\times 27.3524}+1})]\}\\ =121002.0492\{10.6625\times [1.6355-0.0023+0.0312(-0.205\\ -0.2044)]-0.1703[59.9426-0.0832-0.0312(14.7926\\ -0.0212)]\}\\ =866644.4884N\\ =88T\end{array}$$ $$\begin{array}{l}\delta ={\sin }^{-1}\left(\frac{1}{2}\mathit{sin}\theta \right)\\ ={\mathit{sin}}^{-1}\left(\frac{1}{2}\mathit{sin}36.5°\right)\\ =17.3°\end{array}$$

According to this sweep forward angle, low-temp planar jet can keep balances during vertical take-off/landing. $$\begin{array}{c}F=n_2\left(F_1+F_2+F_3\right)\mathit{cos\beta cos\gamma }-n_2\left[F_4\mathit{cos\delta }+F_6\mathit{sin}\left(+\gamma \right)-F_6\mathit{sin}\left(-\gamma \right)\right]\hfill \\ =2\left(98+115+88\right)\mathit{cos}5\mathit{°cos}5°-2[12.7\cos 17.3°+1.57\sin \left(18.75°+5°\right)\hfill \\ -1.57\mathit{sin}\left(18.75°-5°\right)]\hfill \\ =573T\hfill \end{array}$$ $$\begin{array}{l}F-G=573-560=13T>0\\ \mathit{TWR}=\frac{n_3\left(F_5+F_6\right)}{G}=\frac{n_3{F}_4}{G}=\frac{4\times 12.7}{560}=0.09<0.1\end{array}$$

It is clearly demonstrated above that VTOL is achievable on Airbus A380 once remodeled as shown, and more particularly, in case of using just two turbofan engines and thrust-to-weight ratio smaller than 0.1.