Fixed-Wing dynamics documentation

Nomenclature

c - Mean aerodynamic chord
b – Wingspan
S – wing area

u – component of velocity in the X­_b axis
v – component of velocity in the y­_b axis
W – component of velocity in the z­_b axis

V_x - component of velocity in the x­_E axis
V_y - component of velocity in the y_E axis
V_z - component of velocity in the z­_E axis

p - component of angular velocity around the x_b axis
q - component of angular velocity around the y_b axis
r - component of angular velocity around the z_b axis

Ψ - Yaw euler angle
φ - Roll euler angle
θ - Pitch euler angle

δa - Aileron deflection
δe - Elevator deflection
δr - Rudder  deflection

Reference Frames + Assumptions

Right handed reference frames are used for the aircraft simulation. The right hand rule is a useful tool when working with such frames and is shown below:

Earth reference frame:

A North-East-Down inertial reference frame is defined that will be used for expressing position, velocity, acceleration with respect to the ground. This assumes that the earth is flat.

It is important to consider that in this reference frame, positive z velocity points down which is counterintuitive.

Body-fixed reference frame:

The body reference frame is centred at the center of gravity of the aircraft with the axes defined in the following way:

• Xb pointing front (aligned with propeller rotation axis for prop planes)

• Yb perpendicular to Xb direction in line with the plane of the wings

• Zb pointing down normal to the plane defined by Xb and Yb.

This frame of reference allows intuitive interpretation of forces and moments and is convenient as moments of intertia and COG are easily defined in the body reference frame.

Aerodynamic reference frame (wind reference frame):

Orthogonal axis-system aligned with with the aerodynamic velocity V_a (velocity of plane with respect to undisturbed air). Denoted with a subscript.

Used for defining aerodynamic forces and the angle of attack (α) and sideslip angle (β).

Euler angles:

Euler angles are used to denote the aircraft attitude (orientation) with respect to an inertial reference frame (the relationship between body and earth axes). The following euler angle sequence is used: rotation around z, rotation around y, rotation around x.

This is denoted with the angles Ψ (yaw), θ (pitch), φ (roll)

Control Deflection convention:

The positive control surface deflection directions are defined with a feedback control scheme in mind so that positive control gains can be used. Specifically, the directions are chosen such that deflections produce moments that correct the setpoint error with the same sign. This is more clear with an example:

When the altitude is lower than the setpoint, a negative error is produced which yields a negative elevator deflection after multiplying with the positive controller gains. As such the negative elevator direction should correspond with the pitch up direction.

The positive control deflection directions are shown below:

Assumptions

• Vehicle is a rigid body.

• Vehicle mass is constant.

• Earth is flat.

• Earth is non-rotating.

• Body-fixed reference frame is chosen such that Ixy and Iyz are zero. ( XbZb-plane mass symmetry_\

• Effects of rotating masses are neglected.

• Thrust vector acts on COG and is colinear to Xb.

Inputs and State variables

The simulation takes the following as inputs for every time step:

The aircraft state is defined by the following variables:

Forces and Moments

In the earth reference frame newton's second law holds such that:

Where V_G is the velocity of the center of gravity of the aircraft. This however does not hold in the body axes as the body reference frame is a moving and rotating reference frame which results in the appearance of inertial forces like the Coriolis force. The above equation thus has to be transformed into the following:

Expressing the left hand side as the sum of the force due to gravity and aerodynamic forces the following equation of motion is obtained:

The equation of motion for rotation is found following a similar conversion from the earth to the body reference frame. The resulting equation is shown below:

Linearization

Model linearization 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 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:

Applying the first order Taylor expansion we get:

The same is done for the moments so that:

These equations allow the determination of external forces and moments due to perturbations at small deviations from the trim state. 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, coefficients of lift, and mach number for aircraft travelling fast enough)

Aerodynamic coefficients and derivatives

Aerodynamic coefficients are dimensionless parameters that relate the aerodynamic forces and moments acting on an aircraft to the dynamic pressure it is experiencing and account for the wing geometry (wing area S, wing span b, mean aerodynamic chord c). They are shown in the table below:

These coefficients are very useful in analysing the aerodynamic properties of an aircraft and for making comparisons between different aircraft/configurations.

Aerodynamic derivatives (also called stability derivatives) are values that describe the rate of change of aerodynamic coefficients with respect to changes in aircraft state variables, control inputs, or environmental conditions. They correspond to the dimensional force and moment components due to perturbations that are shown in the linearization section. They are described in the tables below:

Longitudinal Forces

Lateral Forces

Vertical Forces

Roll moment

Pitch moment

Yaw moment

Block diagram implementation

References:

[1] https://www.aircraftflightmechanics.com/

[2] https://www.aerostudents.com/courses/flight-dynamics/flightDynamicsFullVersion.pdf

[3]https://agodemar.github.io/FlightMechanics4Pilots/mypages/anatomy-conventional-aircraft/