Introduction

New version of System Modeler, new version of the Aircraft Library!
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  • Wolfram System Modeler 14.0: A drag-and-drop modeling and simulation environment for cyber-physical systems based on the Modelica language.
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  • New in the Aircraft Library 2.0: RotaryWing (drones) and redefined structure for fixed-wing airplanes.

    • RotaryWing Package

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  • Model an existing quadrotor, your own design or tune an autopilot.
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  • Possibilities: Model a 76-rotor drone.

    • Physical Fixed-Wing Package

    Contrary to the conventional state-space linearized model, airplanes in the Aircraft Library 2.0 use the Modelica MultiBody elements.
    Aircraft Library 2.0 allows:
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  • Complex task done in a simple way.
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  • Get insights into your airplane from the very beginning!

    • Modeling of Today: The Italian Frecce Tricolori

    The aerobatic demonstration team of the Italian Air Force.
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  • Today’s model: Aermacchi MB339.
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  • The aircraft is a two-seat advanced trainer military jet aircraft.
    • Image credits: Aeronautica Militare. Retrieved from: https://www.aeronautica.difesa.it/home/noi-siamo-l-am/personale-e-mezzi/pattuglia-acrobatica-nazionale

    Modeling Aerobatics

    Aerobatics is the practice of flying maneuvers involving aircraft attitudes that are not used in conventional flights.
    Video credits: Aeronautica Militare - Il volo folle del solista delle Frecce Tricolori. Publicly available at: https://www.youtube.com/watch?v=DUkbDvWcfwI

    Forces in an Airplane in Equilibrium

    On straight and level unaccelerated flight, forces are in equilibrium.
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  • Lift equals weight, and thrust equals drag.

    • G-Forces

    Because of the interplay among lift, weight and centrifugal forces during maneuvers, the pilot and the airframe are subjected to acceleration changes.
    This is measured in g’s, where at 1 g the airplane and pilot “weigh” the same as static, at 2 g they weigh double, etc.
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  • Pilots in a high-G maneuver feel as if “an elephant is sitting on top of them.”
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  • Pilots in zero G experience the same as astronauts.
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  • Pilots in negative G feel they are being pulled down (my personal favorite), similar to a rollercoaster drop.
    • The load factor measures the relationship between lift (changing dependant on maneuver) over weight (always constant, pointing to the ground).
    Doing some algebra, load factor becomes only a function of the bank angle.
    But why does this happen?

    Forces during a Turn

    Turning is done when banking: rolling the airplane around the axis that runs from nose to tail using ailerons.
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  • When banking, the lift force (always perpendicular to the wings) decomposes into a vertical and a horizontal.
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  • By Newton’s third law, the force that opposes the horizontal component of lift is the centrifugal force.
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  • While weight remains the same, turning the aircraft effectively increases the load applied to it.
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  • The higher the bank, the higher the load!
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  • Valid for equilibrium, at constant velocity and neglecting control surface coupling interactions.

    • Pilot’s Point of View during a Turn

    Linköping SAAB’s JAS-39 Gripen performing tight turns. The G-meter shows the acceleration the pilot and airplane are subjected to.
    Video credits: Gripen and the G Force. SAAB. Publicly available at: https://www.youtube.com/watch?v=eXaPfUs6sQw

    Forces during a Pull-Up: Positive G

    A sudden increase in lift will make the airplane go upward.
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  • The tighter the turn radius, the higher the positive loads.

    • Forces during an English Blunt: Negative G

    A sudden pull-down will increase lift in the weight direction.
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  • The tighter the turn radius, the higher the negative load.

    • Pilot’s Point of View during Pull-Up and Pull-Down

    SAAB’s JAS-39 Gripen performing loops.
    Video credits: Gripen and the G Force. SAAB. Publicly available at: https://www.youtube.com/watch?v=eXaPfUs6sQw

    The V-n Diagram

    A velocity vs. load factor diagram allows pilots to know where it is structurally safe to fly.
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  • The curved lines show aerodynamic limitations, while the straight lines show structural limitations.
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  • It is not symmetric: both the pilot and the airplane withstand positive loads better.

    • Conclusions

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  • The new capabilities of the Aircraft Library 2.0 were presented.
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  • These include the RotaryWing (drone) package and the redesigned structure of the library.
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  • The 6 degrees of freedom rigid body modeling allows testing for structural limit integrity when performing maneuvers.
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  • This can be coupled with all the resources from Wolfram System Modeler’s existing libraries!

    • References

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  • Anderson, J. D. (2012). Introduction to Flight (7th ed.). McGraw-Hill Education.
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  • Nelson, R. C. (1998). Flight Stability and Automatic Control. 2nd ed. McGraw-Hill.
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  • Cook, M. A. (2012). Flight Dynamics Principles. (3rd ed.) Butterworth Heinemann.
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  • Erä-Esko, N. (2022). Development and Use of System Modeler 6DOF Flight Mechanics Model in Aircraft Conceptual Design, master’s thesis.
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  • SAAB. Gripen and the G Force. Publicly available at https://www.youtube.com/watch?v=eXaPfUs6sQw.

    • CITE THIS NOTEBOOK

    Simulating high-G jet aerobatics with Wolfram System Modeler's Aircraft Library 2.0​
    by Erick Espinosa-Juárez and Vedat Şenol​
    Wolfram Community, STAFF PICKS, June 24, 2024
    ​https://community.wolfram.com/groups/-/m/t/3198862