Druyd:Knowledge

Take off

Storyboard

The key to taking off is to modify the wing in a way that achieves sufficient lift at lower speeds, allowing for a successful takeoff on a given runway length.

>Model

ID:(1464, 0)

Mechanisms

Description

<druyd>mechanisms</druyd>

ID:(15173, 0)

Lift coefficient

Description

The lift coefficient is a function of the angle of attack and typically follows the trend indicated in the following figure: <druyd>image</druyd> In the illustrated case, the slope is approximately 1.5 per 15 degrees, which corresponds to 0.1 1/degree or 5.73 1/radian.

ID:(7148, 0)

Variation of the lift coefficient

Description

Both airplanes and birds can modify the shape of their wings. Airplanes achieve this through the use of flaps, while birds adjust the position of their primary and secondary feathers. This allows them to achieve high lift at low speeds during takeoff and landing, and a reduced lift coefficient at high speeds. <druyd>image</druyd> Additionally, airplanes are equipped with spoilers that assist in braking during landing.

ID:(11072, 0)

Model

Description

<druyd>model</druyd>

ID:(15186, 0)

Take off

Description

The key to taking off is to modify the wing in a way that achieves sufficient lift at lower speeds, allowing for a successful takeoff on a given runway length.

Variables

Symbol

Text

Variable

Value

Units

Calculate

MKS Value

MKS Units

$\alpha_s$

alpha_s

Angle required for lift

rad

$m$

Body mass

$C_W$

C_W

Coefficient of resistance

$V2$

Critical speed $V2$

m/s

$\rho$

rho

Density

kg/m^3

$v_0$

v_0

Initial speed

m/s

$F_L$

F_L

Lift force

$a_p$

a_p

Maximum acceleration

m/s^2

$v_p$

v_p

Maximum speed

m/s

$n$

Number of engines

$s$

Path taken on the strip

$c$

Proportionality constant coefficient sustainability

1/rad

$F_p$

F_p

Propulsion force

$Vr$

Rotation speed $Vr$

m/s

$C_L$

C_L

Simple Model for Sustainability Coefficient

$v$

Speed with respect to the medium

m/s

$S_w$

S_w

Surface that generates lift

m^2

$t$

Takeoff time

$S_p$

S_p

Total object profile

m^2

Calculations

Equation

Number

First, select the equation:

, then, select the variable:

Symbol

Equation

Solved

Translated

Calculations

Symbol

Equation

Solved

Translated

Variable

Given

Calculate

Target :

Equation

To be used

Equations

$ F_L =\displaystyle\frac{1}{2} \rho S_w C_L v ^2$

F_L = rho * S_w * C_L * v ^2/2

<var>6120</var>, along with <var>6337</var>, <var>5342</var>, <var>10201</var>, <var>10202</var>, <var>10199</var>, <var>10200</var>, and <var>6110</var>, is found in <druyd>equation=15156</druyd> If we consider <var>6117</var>, given by <var>6337</var>, <var>10199</var>, and <var>10200</var>, <druyd>equation=15154</druyd> and for <var>6119</var>, defined as <druyd>equation=15155</druyd> we obtain <druyd>equation</druyd>

(ID 4417)

$ C_L = c \alpha $

C_L = c * alpha

(ID 4441)

$ C_L =\displaystyle\frac{2 m g }{ \rho S_w }\displaystyle\frac{1}{ v ^2}$

C_L =2* m * g /( rho * S_w * v ^2)

<var>6120</var> along with <var>5342</var>, <var>6117</var>, <var>6119</var>, and <var>6110</var> is represented as <druyd>equation=4417</druyd> which, along with <var>6150</var> and <var>5310</var>, must be equal to: <druyd>equation=14515</druyd> that is: <meq>\displaystyle\frac{1}{2}\rho S_wC_Lv^2=mg</meq> resulting in: <druyd>equation</druyd>

(ID 4442)

$ \alpha =\displaystyle\frac{2 m g }{ c \rho S_w }\displaystyle\frac{1}{ v ^2}$

alpha =2* m * g /( c * rho * S_w * v ^2)

<var>6119</var> is calculated with <var>6150</var>, <var>5310</var>, <var>6117</var>, <var>5342</var>, and <var>6110</var> as follows: <druyd>equation=4442</druyd> Therefore, with <var>6165</var> and <var>6121</var>, <druyd>equation=4441</druyd> we have <druyd>equation</druyd>

(ID 4443)

$ Vr = \sqrt{\displaystyle\frac{2 m g }{ \rho C_L S_w }}$

Vr = sqrt( 2* m * g /( rho * C_L * S_w ))

(ID 14474)

$ V2 = \sqrt{\displaystyle\frac{ n -1}{n}} v_p $

V2 = sqrt((n -1)/ n )* v_p

(ID 14477)

$ a_p = \displaystyle\frac{ F_p }{ m }$

a_p = F_p / m

(ID 14506)

$ v_p = \sqrt{\displaystyle\frac{2 F_p }{ \rho S_p C_W } }$

v_p = sqrt( 2* F_p /( rho * S_p * C_W ))

If we equate <var>10078</var> with <var>6124</var> with <var>6123</var>, <var>6122</var>, <var>5342</var>, and <var>6110</var> in <druyd>equation=4418</druyd> we obtain, for a <var>10075</var>, <meq>F_p = \displaystyle\frac{1}{2} \rho S_w C_L v_p ^2</meq> which, when solved for the maximum velocity, results in <druyd>equation</druyd>

(ID 14507)

$ v = v_0 + a_p t $

v = v_0 + a_p * t

<var>6110</var> for an airplane taking off satisfies the equation with <var>10076</var>, <var>10075</var>, and <var>10209</var>: <druyd>equation=15158</druyd> Upon integration, we obtain the following expression: <meq>\log(v_p + v) - \log(v_p - v) - \log(v_p + v_0) + \log(v_p - v_0)= \displaystyle\frac{2 a_p}{v_p} t</meq> If <var>6110</var> is much smaller than <var>10075</var>, the logarithms can be expanded into a Taylor series, resulting in a first-order approximation: <druyd>equation</druyd>

(ID 14508)

$ l = v_0 t + \displaystyle\frac{1}{2} a_p t ^2$

l = v_0 * t + a_p * t ^2/2

Since the velocity as a function of time is given by <druyd>equation=14508</druyd> we can express velocity as the rate of change of distance with respect to time: <meq>\displaystyle\frac{ds}{dt} = \sqrt{2 a_p v_p t }</meq> This equation can be integrated, yielding the relationship between the distance traveled and time: <druyd>equation</druyd>

(ID 14509)

$ F_g = m g $

F_g = m * g

(ID 14515)

Examples

Mechanisms

<druyd>mechanisms</druyd>

(ID 15173)

Lift coefficient

(ID 7148)

Variation of the lift coefficient

(ID 11072)

Model

<druyd>model</druyd>

(ID 15186)

Flight condition

In order for a spacecraft or a bird to remain in flight, <var>10204</var> must counteract the force of gravity, which is defined by <var>6150</var> and <var>5310</var>. In other words, it must be: <druyd>kyon</druyd> <warning>This is a simplified situation that does not take into account that the force of resistance can also generate a lift force.</warning>

(ID 14515)

Lift force

To generate higher pressure below than above the wing and generate lift, Bernoulli's principle is employed, correcting for the lack of energy density conservation using <var>6119.1</var>. The pressure over the wing, <var>6120</var>, can be estimated using <var>5342</var>, <var>6117</var>, <var>6119</var>, and <var>6110</var> through the following formula: <druyd>kyon</druyd>

(ID 4417)

Equilibrium lifting coefficient

The condition for achieving flight is met when <var>6120</var> equals the weight of the aircraft or bird, which is calculated from <var>6150</var> and <var>5310</var>. This is achieved with sufficient values of <var>6110,0</var>, <var>6117</var>, and <var>6119</var>, where the latter coefficient is the adjustable factor. In the case of aircraft, pilots can modify the value of <var>6119</var> using flaps, whose value must satisfy: <druyd>kyon</druyd> Flaps are adjusted by changing the angle that the wing makes with the direction of flight, known as the angle of attack.

(ID 4442)

Lift constant

From measurements, it is concluded that the lift coefficient $C_L$ is proportional to the angle of attack $\alpha$: <druyd>kyon</druyd> After a certain angle, the curve decreases until it reaches zero. This is because beyond that critical angle, the vortices fully cover the upper surface of the wing, leading to a loss of lift. This phenomenon is known as \\"stall\\".

(ID 4441)

Attack angle

Since the lift coefficient $C_L$ is proportional to the angle of attack $\alpha$, we can calculate the necessary angle to achieve sufficient lift for a given velocity $v$: <druyd>kyon</druyd> where $m$ is the mass, $g$ is the gravitational acceleration, $\rho$ is the density of the medium, $S_w$ is the wing area, and $c$ is the proportionality constant between the lift coefficient and the angle of attack.

(ID 4443)

Speed $V2$

If during takeoff an airplane with <var>10465</var> loses an engine, <var>10078</var> decreases, which could mean the aircraft might not be able to take off and would be forced to abort the procedure. However, if <var>6110</var> exceeds the new value of <var>10075</var>, it is possible to continue the takeoff safely. For this reason, the pilot monitors the parameters and informs the flying pilot once the aircraft has surpassed the so-called <var>10206</var>, which is the critical speed calculated by: <druyd>kyon</druyd>

(ID 14477)

Rotation speed $Vr$

<var>10207</var> is reached when the airplane can take off by rotating to the necessary climb angle. In other words, it corresponds to the case where, with the values of <var>6150</var>, <var>5310</var>, <var>6165</var>, <var>5310</var>, <var>6123</var>, and <var>6121</var>: <druyd>kyon</druyd>

(ID 14474)

Initial acceleration

At the beginning of takeoff, aerodynamic resistance, which depends on velocity, is minimal. Therefore, <var>10076</var> is determined solely by <var>10078</var> and <var>6150</var>: <druyd>kyon</druyd> As aerodynamic resistance starts to reduce the propulsion force, this initial acceleration will be the maximum possible.

(ID 14506)

Maximum speed

<var>10078</var> counteracts <var>6124</var> by generating velocity, which in turn increases the same resistance force, as described in <var>6123</var>, <var>6122</var>, <var>5342</var>, and <var>6110</var> in <druyd>equation=4418</druyd> This process continues to increase the velocity until the point where the propulsion force equals the resistance force, representing the maximum achievable speed. By equating the propulsion force with the resistance force and solving for velocity, we obtain <var>10075</var>: <druyd>kyon</druyd> As aerodynamic resistance starts to reduce the propulsion force, this initial acceleration will be the maximum possible.

(ID 14507)

Takeoff speed

<var>6110</var> for an airplane taking off satisfies the equation with <var>10076</var>, <var>10075</var>, and <var>10209</var>: <druyd>equation=15158</druyd> When integrated in the limit <var>6110,0</var> much smaller than <var>10075,0</var>, we obtain: <druyd>kyon</druyd> Typically, the takeoff speed of an aircraft is significantly lower than the maximum speed <var>10075</var>. Therefore, the equation can be solved analytically, as explained in the development.

(ID 14508)

Path taken when taking off

Since the takeoff velocity $v$ as a function of time $t$ is described by <druyd>equation=14508</druyd> we can calculate the distance traveled along the runway by integrating this equation with respect to time: <druyd>kyon</druyd> On the other hand, by considering the required takeoff velocity, we can determine the time needed to achieve it, and using the distance traveled, we can calculate the runway length required for takeoff.

(ID 14509)

ID:(1464, 0)