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Compute aerodynamic forces and moments using aerodynamic coefficients, dynamic pressure, center of gravity, center of pressure, and velocity

Aerodynamics

The Aerodynamic Forces and Moments block computes the aerodynamic forces and moments about the center of gravity. By default, the inputs and outputs are represented in the body axes.

Let α be the angle of attack and β the sideslip. The rotation from body to stability axes:

$${C}_{s\leftarrow b}=\left[\begin{array}{ccc}\mathrm{cos}(\alpha )& 0& \mathrm{sin}(\alpha )\\ 0& 1& 0\\ -\mathrm{sin}(\alpha )& 0& \mathrm{cos}(\alpha )\end{array}\right]$$

can be combined with the rotation from stability to wind axes:

$${C}_{w\leftarrow s}=\left[\begin{array}{ccc}\mathrm{cos}(\beta )& \mathrm{sin}(\beta )& 0\\ -\mathrm{sin}(\beta )& \mathrm{cos}(\beta )& 0\\ 0& 0& 1\end{array}\right]$$

to yield the net rotation from body to wind axes:

$${C}_{w\leftarrow b}=\left[\begin{array}{ccc}\mathrm{cos}(\alpha )\mathrm{cos}(\beta )& \mathrm{sin}(\beta )& \mathrm{sin}(\alpha )\mathrm{cos}(\beta )\\ -\mathrm{cos}(\alpha )\mathrm{sin}(\beta )& \mathrm{cos}(\beta )& -\mathrm{sin}(\alpha )\mathrm{sin}(\beta )\\ -\mathrm{sin}(\alpha )& 0& \mathrm{cos}(\alpha )\end{array}\right]$$

Moment coefficients have the same notation in all systems. Force coefficients are given below. Note there are no specific symbols for stability-axes force components. However, the stability axes have two components that are unchanged from the other axes.

$${F}_{A}^{w}\equiv \left[\begin{array}{c}-D\\ -C\\ -L\end{array}\right]={C}_{w\leftarrow b}\cdot \left[\begin{array}{c}{X}_{A}\\ {Y}_{A}\\ {Z}_{A}\end{array}\right]\equiv {C}_{w\leftarrow b}\cdot {F}_{A}^{b}$$

Components/Axes | x | y | z |
---|---|---|---|

Wind | C_{D} | C_{C} | C_{L} |

Stability | — | C_{Y} | C_{L} |

Body | C_{X} | C_{Y} | C (–_{Z}C)_{N} |

Given these definitions, to account for the standard definitions
of *D*, *C*, *Y* (where *Y* =
-*C*), and *L*, force coefficients
in the wind axes are multiplied by the negative identity *diag*(-1,
-1, -1). Forces coefficients in the stability axes are multiplied
by *diag*(-1, 1, -1). *C*_{N} and *C*_{X} are,
respectively, the normal and axial force coefficients (*C*_{N} =
-*C*_{Z}).

**Input Axes**Specifies coordinate system for input coefficients:

`Body`

(default),`Stability`

, or`Wind`

.**Force Axes**Specifies coordinate system for aerodynamic force:

`Body`

(default),`Stability`

, or`Wind`

.**Moment Axes**Specifies coordinate system for aerodynamic moment:

`Body`

(default),`Stability`

, or`Wind`

.**Reference area**Specifies the reference area for calculating aerodynamic forces and moments.

**Reference span**Specifies the reference span for calculating aerodynamic moments in

*x*-axes and*z*-axes.**Reference length**Specifies the reference length for calculating aerodynamic moment in the

*y*-axes.

The first input consists of aerodynamic coefficients (in the chosen input axes) for forces and moments. These coefficients are ordered into a vector depending on the choice of axes:

Input Axes | Input Vector |
---|---|

Body | (axial force C_{x},
side force C_{y}, normal
force C_{z}, rolling moment C_{l},
pitching moment C_{m}, yawing
moment C_{n}) |

Stability | (drag force C_{D(β=0)},
side force C_{y}, lift force C_{L},
rolling moment C_{l}, pitching
moment C_{m}, yawing moment
C_{n}) |

Wind | (drag force C_{D},
cross-wind force C_{c}, lift
force C_{L}, rolling moment C_{l},
pitching moment C_{m}, yawing
moment C_{n}) |

Input | Dimension Type | Description |
---|---|---|

Second | Contains the dynamic pressure. | |

Third | Contains the center of gravity. | |

Fourth | Contains the center of pressure. This can also be taken as any general moment reference point as long as the rest of the model reflects the use of the moment reference point. | |

Fifth (For inputs or outputs in stability or wind axes) | Three-element vector | Contains the velocity in the body axes. |

Output | Dimension Type | Description |
---|---|---|

First | Contains the aerodynamic forces (in the chosen output axes)
at the center of gravity in x-, y-,
and z-axes. | |

Second | Contains the aerodynamic moments (in the chosen output axes)
at the center of gravity in x-, y-,
and z-axes. |

The default state of the block hides the *V*_{b} input
port and assumes that the transformation is body-body.

The center of gravity and the center of pressure are assumed to be in body axes.

While this block has the ability to output forces and/or moments in the stability axes, the blocks in the Equations of Motion library are currently designed to accept forces and moments in either the body or wind axes only.

See Airframe in `aeroblk_HL20`

for
an example of this block.

Stevens, B. L., and F. L. Lewis, *Aircraft Control
and Simulation,* John Wiley & Sons, New York, 1992