Storyboard

Yaw control is the mechanism that allows the aircraft to rotate around its vertical axis, turning the aircraft to the right or left. This control is achieved by deflecting the rudder, located at the tail of the aircraft. By moving the rudder, a lateral force is generated, creating a torque that causes the aircraft to rotate around an imaginary axis perpendicular to the fuselage, known as the yaw axis.

>Model

ID:(2115, 0)

Concept

>Top

To perform turns in an aircraft, the rudder is used. It generates a force on the rudder ($F_r$), which, combined with a center of mass and rudder distance ($d_r$), induces a force on the rudder ($F_r$). The rudder is located at the tail of the aircraft to maximize the center of mass and rudder distance ($d_r$) and achieve a greater a force on the rudder ($F_r$).

The pilot controls this movement using the pedals. The direction of the turn is determined by the pedal's direction.

ID:(15162, 0)

Description

>Top

The wing mass ($m_w$) can be approximated as the volume of a rectangular parallelepiped multiplied by the density of the aircraft:

The volume can thus be calculated from the surface that generates lift ($S_w$) and the wing height ($d$).

Therefore, the wing mass ($m_w$) is determined using the aircraft body density ($\rho_a$), the surface that generates lift ($S_w$), and the wing height ($d$), as follows:

ID:(15989, 0)

Concept

>Top

ID:(11077, 0)

Description

>Top

The aircraft body mass ($m_p$) can be approximated as the volume of a cylinder multiplied by the density of the aircraft:

The volume can thus be calculated using the total object profile ($S_p$) (the radius or diameter) and the distance along the wing ($l$) (the height of the cylinder).

Therefore, the aircraft body mass ($m_p$) is determined from the aircraft body density ($\rho_a$), the total object profile ($S_p$), and the distance along the wing ($l$), as follows:

ID:(15990, 0)

Description

>Top

The vertical axis moment of inertia ($I_r$) can be approximated as the sum of the moment of inertia of a cylinder representing the aircraft fuselage, rotating around an axis perpendicular to its longitudinal axis, and the moment of inertia of a rectangular parallelepiped representing the wings, rotating around an axis perpendicular to them:

If, for the estimation of the vertical axis moment of inertia ($I_r$), it is assumed that the radius of the fuselage cylinder is much smaller than the distance along the wing ($l$) and the wing width ($w$) is much smaller than the wing span ($L$), the moment of inertia of the cylinder primarily depends on the aircraft body mass ($m_p$) and the distance along the wing ($l$), while the moment of inertia of the parallelepiped depends on the wing mass ($m_w$) and the wing span ($L$).

Therefore, the vertical axis moment of inertia ($I_r$) is calculated using the aircraft body mass ($m_p$), the wing mass ($m_w$), the wing span ($L$), and the distance along the wing ($l$) as follows:

$I_r = \displaystyle\frac{1}{12}( m_p l ^2 + m_w L ^2)$

Conditions

Variables

Relation

ID:(15993, 0)

Equation

>Top, >Model

$T_r = d_r F_L$

$T_r = d_r F_r$

Conditions

Variables

$d_r$

Center of mass and rudder distance

$m$

10213

$F_r$

$F_L$

Lift force

$N$

6120

$T_r$

Rudder generated torque

$N m$

10216

Relation

ID:(15165, 0)

Equation

>Top, >Model

$T_r = I_r \alpha_r$

Conditions

Variables

$T_r$

Rudder generated torque

$N m$

10216

$\alpha_r$

Vertical axis angular acceleration

$rad/s^2$

10224

$I_r$

Vertical axis moment of inertia

$kg m^2$

10221

Relation

ID:(15168, 0)

Equation

>Top, >Model

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 ($$). The pressure over the wing, the lift force ($F_L$), can be estimated using the density ($\rho$), the surface that generates lift ($S_w$), the coefficient of lift ($C_L$), and the speed with respect to the medium ($v$) through the following formula:

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

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

Conditions

Variables

$\rho$

Density

$kg/m^3$

5342

$F_L$

Lift force

$N$

6120

$C_L$

Simple Model for Sustainability Coefficient

$-$

6164

$v$

Speed with respect to the medium

$m/s$

6110

$S_w$

$S_r$

Rudder surface

$m^2$

6118

Relation

Development

The lift force ($F_L$), along with the wing span ($L$), the density ($\rho$), the wing top speed factor ($c_t$), the wing bottom speed factor ($c_b$), the upper wing length ($l_t$), the bottom wing length ($l_b$), and the speed with respect to the medium ($v$), is found in

$F_L = \rho L ( c_b l_b - c_t l_t ) v ^2$

If we consider the surface that generates lift ($S_w$), given by the wing span ($L$), the upper wing length ($l_t$), and the bottom wing length ($l_b$),

$S_w = \displaystyle\frac{1}{2} L ( l_t + l_b )$

and for the coefficient of lift ($C_L$), defined as

$C_L = 4\displaystyle\frac{ c_t l_t - c_b l_b }{ l_t + l_b }$

we obtain

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

ID:(4417, 0)

Equation

>Top, >Model

From measurements, it is concluded that the lift coefficient $C_L$ is proportional to the angle of attack $\alpha$:

$C_L = c \alpha$

Conditions

Variables

$\alpha_s$

Angle required for lift

$rad$

6167

$c$

Proportionality constant coefficient sustainability

$1/rad$

6165

$C_L$

Simple Model for Sustainability Coefficient

$-$

6164

Relation

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, 0)

Equation

>Top, >Model

The vertical axis moment of inertia ($I_r$) is calculated from the wing mass ($m_w$), the wing span ($L$) and the distance along the wing ($l$), as follows:

$I_r = \displaystyle\frac{1}{12}( m_p l ^2 + m_w L ^2)$

Conditions

Variables

$m_p$

Aircraft body mass

$kg$

6340

$l$

Aircraft length

$m$

10469

$I_r$

Vertical axis moment of inertia

$kg m^2$

10221

$m_w$

Wing mass

$kg$

6339

$L$

Wing span

$m$

6337

Relation

ID:(15988, 0)

Equation

>Top, >Model

The wing mass ($m_w$) is calculated from the aircraft body density ($\rho_a$), the surface that generates lift ($S_w$), and the wing height ($d$), as follows:

$m_w = \rho_a S_w d$

Conditions

Variables

$\rho_a$

Aircraft body density

$kg/m^3$

6220

$S_w$

Surface that generates lift

$m^2$

6117

$d$

Wing height

$m$

6338

$m_w$

Wing mass

$kg$

6339

Relation

ID:(15984, 0)

Equation

>Top, >Model

The aircraft body mass ($m_p$) is calculated from the aircraft body density ($\rho_a$), the total object profile ($S_p$), and the distance along the wing ($l$), as follows:

$m_p = \rho_a S_p l$

Conditions

Variables

$\rho_a$

Aircraft body density

$kg/m^3$

6220

$m_p$

Aircraft body mass

$kg$

6340

$l$

Aircraft length

$m$

10469

$S_p$

Total object profile

$m^2$

6123

Relation

ID:(15985, 0)

Equation

>Top, >Model

The center of mass and rudder distance ($d_r$) is defined as half of the distance along the wing ($l$), expressed as follows:

$d_r = \displaystyle\frac{ l }{2}$

Conditions

Variables

$l$

Aircraft length

$m$

10469

$d_r$

Center of mass and rudder distance

$m$

10213

Relation

ID:(15996, 0)

Equation

>Top, >Model

The thickness to span ratio ($\gamma_d$) is defined as the ratio of the wing height ($d$) to the wing span ($L$), represented as follows:

$\gamma_d =\displaystyle\frac{ d }{ L }$

Conditions

Variables

$\gamma_d$

Thickness to span ratio

$-$

6344

$d$

Wing height

$m$

6338

$L$

Wing span

$m$

6337

Relation

ID:(15976, 0)