SpinSpinSpin_Rotati​onVector

A Simple MATLAB Simulation for Unstable Free Rotation of a Rigid Body.

SpinSpinSpin

A Simple MATLAB Simulation for Unstable Free Rotation of a Rigid Body.

NOTE(s):

• Rotating object kinematics is "Rotation Vector" based.
• The projects "SpinSpinSpin_Quaternion_v1" and "SpinSpinSpin_Quaternion_v2" are similar but Quaternion based.
• Rotation Vector Definition:
  • A rotation vector is a one dimensional row or column vector of length three.
  • The norm of the rotation vector is the rotation angle.
  • The normalized rotation vector is a 3D unit vector which serves as the rotation axis.
• Ordinary Differential Equations (ODEs) are solved using the Matlab "ode45" function to apply the Dorman-Prince 4(5) explicit embedded adaptive time step Runge-Kutta method.
  • Error limits in Matlab "ode45" function set to 10^-13.
• State Vector Components:
  • State Vector components 1, 2 and 3 are the "rotation vector".
  • State Vector components 4, 5 and 6 are the angular velocity vector.
  • Initial Conditions:
    • Rotation Vector: [ 1.0; 0.0; 0.0 ].
      • 1 [radian] (57.3 [degrees]) rotation about the body X axis.
    • Angular Velocity: [ 0.10; 0.00001; 0.0 ] [radians/second]
      • Spin about body X axis with a small spin about the body Y axis.
    • Principal components of moments of inertia: [ 4.0; 1.0; 9.0 ]
      • Rectangular block dimensions determined by principal components of moments of inertia.
      • Spin will be about the rectangular block axis of intermediate length.
• Time rate of change of the rotation vector:
  • The time rate of change of the rotation vector is not the angular velocity vector.
  • To determine the time rate of change of the rotation vector, use Equation (42) on Page 265 of Reference [ 1 ].
• Note that the 2D plots of inertial angular momentum quantities reveal the constant nature of the inertial angular momentum over time as expected for free rotation.
  • Change is within 10^-13.
• Note that the 2D plot of rotational kinetic energy reveals the constant nature of this energy over time as expected for free rotation.
  • Change is within 10^-13.
• Note that the 3D animation video reveals the constant nature of the angular momentum vector over time as expected for free rotation.
  • Simulation generates a rotation vector solution every millisecond for 1000 seconds generating a total of one million rotation vector solutions.
  • Simulation creates a sequence of 3D animation frames, one 3D animation frame for every 200 th rotation vector solution. 3D simulation animation frames are at 5 Hertz rate.

CONTENTS:

• freeUnstableRigidBodyRotationDemonstration.m

  MATLAB source code file for top level simulation function.

• generateSolutionsForFreeRotationRigidBodyKinematics.m

  MATLAB source code file for non-real time solution of Ordinary Differential Equations (ODEs).

• generateInertialResults.m

  MATLAB source code file to convert state vector solutions at each time sample from body coordinates to inertial coordinates.

• generatePlots2D.m

  MATLAB source code file to generate 2D plots.

• generatePlots3D.m

  MATLAB source code file to generate 3D plots.

• performNonRealTimeAnimationPlayBack3D.m

  MATLAB source code file to generate 3D animation non-real time playback of solution applied to a rigid rectangular 3D block.

• generateFreeUnstableRigidBodyRotationDemoPurposeMessage.m

• generateFreeUnstableRigidBodyRotationDemoUsageMessage.m

• generateKinematicsSolutionsPurposeMessage.m

• generateKinematicsSolutionsUsageMessage.m

  Auxiliary MATLAB source code files.

• Utilities.

  Directory of utility MATLAB source code files.

EXAMPLES:

• nonRealTimeAnimationPartialPlayBackMovie.avi

  A non-real time 3D animation of a partial results play back AVI movie file.

• TwoDimensionalPlots

  Directory containing the two dimensional plots generated by this simulation.

REFERENCE(s):

- [ 1 ]  "The Kinematic Equation for the Rotation Vector",      
          Malcolm D. Shuster,      
          IEEE Transactions on Aerospace and Electronic Systems,      
          Vol. 29, No. 1, pp. 263 - 267,      
          January 1993      
          https://malcolmdshuster.com/Pub_1993c_J_RotVec_IEEE.pdf
          

- [ 2 ] "A Survey of Attitude Representations",
         Malcolm D. Shuster,      
         The Journal of the Astronautical Sciences,      
         Vol. 41, No. 4, pp. 430 - 517,      
         October-December, 1993      
         https://malcolmdshuster.com/Pub_1993h_J_Repsurv_scan.pdf
         

- [ 3 ] "The Bizarre Behavior of Rotating Bodies -    - 
        The Dzhanibekov Effect",      
        Derek Alexander Muller      
        YouTube channel Veritasium      
        Spinning objects have strange instabilities known as      
        "The Dzhanibekov Effect" or "Tennis Racket Theorem" -      
        this video offers an intuitive explanation.      
        https://www.youtube.com/watch?v=1VPfZ_XzisU
        
  
- [ 4 ] Elements of the following were used to help create this project:
  
    [ 4.1 ]  Gatech AE (2024). Euler Free Body Motion (https://github.com/alaricgregoire/EulerFreeBody/releases/tag/v1.0.0), GitHub. Retrieved July 15, 2024
    
    [ 4.2 ]   Ligong Han (2024). phymhan/matlab-axis-label-alignment (https://github.com/phymhan/matlab-axis-label-alignment), GitHub. Retrieved July 15, 2024. 
             
    [ 4.3 ]   Georg Stillfried (2024). mArrow3.m - easy-to-use 3D arrow (https://www.mathworks.com/matlabcentral/fileexchange/25372-marrow3-m-easy-to-use-3d-arrow), MATLAB Central File Exchange. Retrieved July 15, 2024.