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In the earth reference frame newtons newton's second law holds such that:
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Linearization
Model linearization consists is the process of simplifying the equations of motion using a first order taylor expansion at some trim state. In this simulation, the nonlinearity of the state equations is not an issue as they can be reliably solved using numerical integration. The nonlinearity of the forces, however, poses a significant modelling challenge however as “They are not part of the state of the aircraft. Instead they also depend on the state of the aircraft. And they don’t only depend on the current state, but on the entire history of states! (For example, a change in angle of attack could create disturbances at the wing. These disturbances will later result in forces acting on the tail of the aircraft.)” [2]
In practice many of these nonlinear force relationships can be neglected, however, several significant non-linear relations remain. Mathematically this is shown by:
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The same is done for the moments so that:
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These equations allow the determination of external In this form, the net forces and moments are described by their components due to perturbations at the different perturbations in the state which makes them much easier to find. Due to the linearization, however, these equations are only valid for small deviations from the trim state. To ; to accurately simulate the aircraft in the full flight envelope these linearized force and moment values have to be determined at multiple trim conditions (usually at different angles of attack for symmetric derivatives (Cx, Cm, Cm), and at different coefficients of lift , and mach number for aircraft travelling fast enough)for asymmetric derivatives (Cy, Cl, Cm)).
Aerodynamic coefficients and derivatives
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Coefficient | Description |
|---|---|
Cx0Positive | X component of the drag and lift force |
Cx_alpha | Cx derivative with respect to angle of attack (Usually included in Cd calculation) |
Cx_de | Cx derivative with respect to elevator deflection (usually negligible) |
Cx_q | Cx derivative with respect to pitch rate (negative, but usually negligible) |
Lateral Forces
Coefficient | Description |
|---|---|
Cy0 | Y component of the drag and lift force |
Cy_beta | Cy derivative with respect to sideslip angle (should be negative) |
Cy_p | Cy derivative with respect to roll rate (usually negligible) |
Cy_r | Cy derivative with respect to yaw rate (usually negligible) |
Cy_dr | Cy derivative with respect to rudder deflection (small positive value) |
Cy_da | Cy derivative with respect to aileron deflection (usually negligible) |
Vertical Forces
Coefficient | Description |
|---|---|
Cz0 | Z component of the drag and lift force |
Cz_alpha | Cz derivative with respect to angle of attack (usually included in CL calculation) |
Cz_alphadot | Cz derivative with respect to rate of change of angle of attack (usually negligible) |
Cz_q | Cz derivative with respect to pitch rate (should be negative) |
Cz_de | Cz derivative with respect to elevator deflection (small negative number) |
Roll moment
Coefficient | Description |
|---|---|
Cl0 | Component of moment about Xb due to aerodynamic moment in Xs |
Cl_p | Cl component due to roll rate (should be negative for stability) |
Cl_r | Cl component due to yaw rate (small positive or negative number depending on slip vs skid tendency) |
Cl_beta | Cl component due to side slip angle (usually small negative number) |
Cl_da | Cl component due to aileron deflection (negative number) |
Cl_dr | Cl component due to rudder deflection (small positive number) |
Pitch moment
Coefficient | Description |
|---|---|
Cm0 | Component of moment about Xb due to aerodynamic moment in Xs |
Cm_alpha | Cm component due to angle of attack (should be negative) |
Cm_de | Cm component due to elevator deflection (should be negative) |
Cm_q | Cm component due to pitch rate (should be negative for stability) |
Cm_alphadot | Cm component due to rate of change of angle of attack (should be negative but hard to find in practice) |
Yaw moment
Coefficient | Description |
|---|---|
Cn0 | N component due to aerodynamic moment in stability frame |
Cn_p | Cn component due to roll rate (small positive or negative number) |
Cn_r | Cn component due to yaw rate (should be negative for stability) |
Cn_beta | Cn component due to sideslip (should be small positive number for stability) |
Cn_da | Cn component due to aileron deflection (non-zero if differential ailerons used) |
Cn_dr | Cn component due to rudder deflection (Should be negative) |
Block diagram implementation
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