Euler's Equations of Rigid Body Dynamics Derived | Qualitative Analysis | Build Rigid Body Intuition

Space Vehicle Dynamics 🤓 Lecture 21: Rigid body dynamics, the Newton-Euler approach, is given. Specifically, from the angular momentum rate equation, we derive the most common forms of Euler's equations of the rotational dynamics of a rigid body. A qualitative analysis of Euler's equation is also given to build intuition.

► Next: Free rigid body motion | precession of axisymmetric bodies | general motion

► Previous, Center of Mass & Moment of Inertia Matrix | Example Calculations

► More lectures posted regularly

► Dr. Shane Ross ➡️ aerospace engineering professor, Virginia Tech
Background: Caltech PhD | worked at NASA/JPL & Boeing
Research website for @ProfessorRoss

► Follow me on Twitter

► Space Vehicle Dynamics course videos (playlist)

► Lecture notes (PDF)

► References
Schaub & Junkins📘Analytical Mechanics of Space Systems, 4th edition, 2018

► Related Courses and Series Playlists by Dr. Ross

📚Space Vehicle Dynamics

📚3-Body Problem Orbital Dynamics Course

📚Space Manifolds

📚Lagrangian and 3D Rigid Body Dynamics

📚Nonlinear Dynamics and Chaos

📚Hamiltonian Dynamics

📚Center Manifolds, Normal Forms, and Bifurcations

► Chapters

0:00 Summary so far
0:37 Newton-Euler approach to rigid bodies
10:22 Qualitative analysis to build intuition about rigid bodies
11:06 Spinning top analysis
15:36 Spinning bicycle wheel on string
19:06 Fidget spinner analysis
22:01 Landing gear retraction analysis
24:53 Euler's equations of rigid body motion derived in body-fixed frame
29:09 Euler's equation written in components
30:56 Euler's equation in principal axis frame
35:33 Euler's equation for free rigid body
40:32 Simulations of free rigid body motion

- Typical reference frames in spacecraft dynamics
- Mission analysis basics: satellite geometry
- Kinematics of a single particle: rotating reference frames, transport theorem
- Dynamics of a single particle
- Multiparticle systems: kinematics and dynamics, definition of center of mass (c.o.m.)
- Multiparticle systems: motion decomposed into translational motion of c.o.m. and motion relative to the c.o.m.
- Multiparticle systems: imposing rigidity implies only motion relative to c.o.m. is rotation
- Rigid body: continuous mass systems and mass moments (total mass, c.o.m., moment of inertia tensor/matrix)
- Rigid body kinematics in 3D (rotation matrix and Euler angles)
- Rigid body dynamics; Newton's law for the translational motion and Euler’s rigid-body equations for the rotational motion
- Solving the Euler rotational differential equations of motion analytically in special cases
- Constants of motion: quantities conserved during motion, e.g., energy, momentum
- Visualization of a system’s motion
- Solving for motion computationally

#RigidBodyDynamics #EulersEquation #Gyroscope #SpinningTop #EulersEquations #RigidBody #FreeRigidBody #Axisymmetric #Prolate #Oblate #MomentOfInertia #CenterOfMass #InertiaMatrix #InertiaTensor #AttitudeDetermination #AttitudeEstimation #SpacecraftAttitudeControl #Spacecraft #SpacecraftOrientation #TRIAD #TRIADmethod #ADCS #MATLAB #MATLABdemo #MATLABtutorial #Numerics #NumericalDemonstration #EulerAngles #AngularVelocity #RigidBodyKinematics #AxisAngle #Quaternions #EulerParameters #RotationAxis #RodriguesFormula #RotationMatrix #RigidBody #YawPitchRoll #Rotation #BryanTait #aircraft #spacecraft #TaitBryan #Kinematics #DirectionCosineMatrix #Rigidity #EngineeringDynamics #SystemOfParticles #ConservationOfEnergy #WorkedProblem #WorkedExample #CenterOfMass #MatlabTutorial #Tutorial #Lecture #Matlab #NewtonsLaws #ConservationOfMomentum #TransportTheorem #RotatingFrame #InertialFrame #VectorDerivative #SatelliteGeometry #RotatingFrames #SpacecraftDynamics #SpaceVehicle #AttitudeDynamics #SpaceVehicleDynamics #RigidBodyDynamics #dynamics #LawsOfMotion #engineering #aerospace #elonmusk #SpaceX #Boeing #Satellite #Satellites #SpaceDomainAwareness #NewtonsLaw #EquationOfMotion #Newtons2ndLaw #NewtonianMechanics #AOE3144 #Caltech #NASA #VirginiaTech #engineering #dynamics #mechanics #physics #mathematics #science #aerospace #mechanicalengineering #spacecraft #openaccess #OnlineCourse #technology #robotics #space #spaceindustry #math #vehicledynamics #simulation #NewtonEuler #LinearMotion #NonlinearDynamics #DynamicalSystems #AppliedMath #DifferentialEquations #mathematics #Newton #math #FreeCourses #OnlineCourse #EquilibriumPoints #StablePoint #UnstablePoint #maths #mathematician

[n.aananthakrishnan]

Your space vehicle dynamics playlist is a grate playlist. Thank you professor!

[YoungjuPark]

I like the way that you keep mentioning which part this equation is positioning in the big picture of the concept. Thank you for sharing this mazing video.

[marcelo.griebeler]

Very in depth and still very didactically explained. Thanks for the amazig content, Dr. Ross!

[ivannoro]

congratulations for the videos. very clear and comprehensive. I would also put something on collisions between rigid bodies.

[AnasHawasli]

you make it sound very interesting thank you

[shravangulvadi]

Thanks for the great explanation, really helped clear some confusions I had on this. Please keep posting!

[pinkpickles]

Thank you very much --- the examples were so helpful

[lantonovbg]

you just derived gravitation using classical physics. The gyroscopic inertia of bicycle, gyroscope, and fidget spinner is the same as mass inertia, and gravitational inertia. The last one is the inertia of a body to leave its isolated (inertial) system.

[marcosrobertociaralofonsec]

Hi Ross! For the gyroscope example if the derivative of Hp (w.r.t. B) is not "0" the direction is not the same of torque. Am I make some conception mistake? Thanks for the amazing lectures! 👏

[andresjimenezmora]

If it is not torque-free motion and I want to design an attitude control using thrusters as a source of torque, how can I linearize the equations in order to obtain a state space representation?

[KK-fvbs]

Can you show how we can write the Lagrangian for this kind of motion