# For an aircraft, what are the differences between nonlinear and linear equations of motion?

So, as far as I understand, linearized equations of motion are just nonlinear equations of motion converted to linear EoM's using approximations and assumptions to simplify/remove elements from the equations.

In an aircraft, what are these nonlinear EoM's? If, for example, I stuck my airplane in a wind tunnel to determine the entire set of aerodynamic coefficients, would that cover all of the nonlinear portions of the equations of motion?

Thanks!

• No, wind tunnel measures aerodynamics. EOMs capture inertial and dynamical effects on a mass. They stand apart.
– JZYL
Feb 13 '20 at 16:39
• Are you asking for a description of what "nonlinear equations of motion" means in the context of aircraft, or are you asking for the equations themselves? Feb 13 '20 at 22:20
• Improve which particular bit of the fidelity of the simulator? Are there any aircraft dynamic responses matched by simulator responses? Feb 15 '20 at 23:01
• @Koyovis, in particular, the post-stall behavior and the behavior in some high-speed, high-G maneuvers. The model generally has pretty good fidelity, we've done a relatively extensive study of flight data that matches the simulation pretty well. I'm generally okay with most aspects. Feb 18 '20 at 11:52

Equations of motion (EQM) define how a rigid-body accelerate and rotate given any arbitrary external forces and moments. They are no more and no less than Newton's Second Law. When expressed in body-frame, the EQMs are:

$$\bf{F}_B = m (\dot{\bf{V}}_b+\bf{\omega }_b \times \bf{V}_b)$$ $$\bf{M}_B=\bf{I}\dot{\bf{\omega }}_b$$

where $$\bf{F}$$ is body forces, $$\bf{M}$$ is body moments, $$m$$ is mass, $$\bf{V}$$ is inertial velocity, $$\bf{\omega}$$ is angular velocity, $$\bf{I}$$ is moments of inertia, and $$\times$$ is the cross-product operator.

The above are vector equations, which correspond to 6 scalar equations. At this point, there is no concept of aerodynamics. Notice how the equations are inherently nonlinear, so long as there are body forces/moments that depend on the state variables (e.g. velocity, orientation, etc). This is true for most forces, with the notable exception being gravity.

Aerodynamics is nonlinear in that aerodynamic forces and moments do not change linearly with flow incidence, rate of rotation and velocity, except if the changes are small. In general, quasi-steady aerodynamic force is a nonlinear function of flow incidence, rate of rotation, Mach number, Reynolds number, etc. To capture these, N-D lookup tables constructed from estimated/measured data are generally employed in simulation models.

Quasi-steady assumption itself may not be adequate in some flight conditions, such as during rapid maneuvering or stalled flight. In these circumstances, the description of aerodynamics may need to be time-variant.

• It would be a nice touch to not only list the formulas but also to explain what the letters mean. Not everyone is born with that knowledge. Feb 15 '20 at 20:04
• @PeterKämpf Amended with legend
– JZYL
Feb 15 '20 at 21:20
• @Koyovis where do you see force = mass x velocity?
– JZYL
Feb 15 '20 at 22:56
• Yep, dimensions are right. Feb 15 '20 at 23:03
• @synchh Normal force as an input? Is it to capture aeroelastics? This would not be a usual aerodynamic parameter. Unless your model has some formulation for time, hysteresis or other time-variant behavior, I would consider it quasi-steady.
– JZYL
Feb 18 '20 at 12:23