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Bead in a Rotating Hoop, Part 1- Deriving Equations of Motion

Equations of motion are derived for a bead in a rotating hoop -- that is, an idealized system where a particle slides along a circular wire frame which rotates via motor about the vertical direction. Newton's laws are used. To simplify analysis, we nondimensionalize the governing equation next    • Nondimensionalization of Equations, P...  , then analyze the overdamped limit    • Bead in a Rotating Hoop, Part 2: High...   ► Next: Nondimensionalizing equations of motion    • Nondimensionalization of Equations, P...   ► From 'Nonlinear Dynamics and Chaos' online course playlist https://is.gd/NonlinearDynamics ► Subscribe https://is.gd/RossLabSubscribe​ ► Dr. Shane Ross, Virginia Tech professor (Caltech PhD) http://chaotician.com​ ► Course lecture notes (PDF) https://is.gd/NonlinearDynamicsNotes ► Chapters 0:00 Introduction of bead in rotating hoop 0:30 Deriving the bead's equations of motion using Newton's laws 1:33 Motion only along the hoop 1:55 Centripetal force 2:25 Modeling friction of bead in hoop 3:30 Tangential direction equation (for theta 𝜃) ► Courses and Playlists by Dr. Ross 📚Attitude Dynamics and Control https://is.gd/SpaceVehicleDynamics 📚Nonlinear Dynamics and Chaos https://is.gd/NonlinearDynamics 📚Hamiltonian Dynamics https://is.gd/AdvancedDynamics 📚Three-Body Problem Orbital Mechanics https://is.gd/SpaceManifolds 📚Lagrangian and 3D Rigid Body Dynamics https://is.gd/AnalyticalDynamics 📚Center Manifolds, Normal Forms, and Bifurcations https://is.gd/CenterManifolds ► References: Steven Strogatz, "Nonlinear Dynamics and Chaos", Chapter 3: Bifurcations Video at the beginning is from a Wolfram Demonstration by Ryan K. Smith (Wolfram Research)    • Bead on a Rotating Wire   Wolfram demonstration: Bead on a rotating wire https://demonstrations.wolfram.com/Be... marble in hula hoop bead in circular hoop rotating planar pendulum problem example system bead in a rotating hoop bifurcation subcritical bifurcations buckling beam model change of stability nonlinear dynamics dynamical systems differential equations dimensions phase space Poincare Strogatz graphical method Fixed Point Equilibrium Equilibria Stability Stable Point Unstable Point Linear Stability Analysis Vector Field One-Dimensional 1-dimensional Functions centrifugal centripetal force viscous damping over-damped under-damped overdamped bead on a rotating hoop marble in hula hoop #NonlinearDynamics #DynamicalSystems #EquationsOfMotion #NewtonianMechanics #RotatingFrame #CriticalPoint #SaddleNode #buckling #snapthrough #beadinhoop #beaddynamics #marbleinhulahoop #beadinrotatinghoop #pitchforkbifurcation #robust #StructuralStability #DifferentialEquations #dynamics #dimensions #PhaseSpace #Poincare #Strogatz #graphicalmethod #FixedPoints #EquilibriumPoints #Stability #Wolfram #StablePoint #UnstablePoint #Stability #LinearStability #LinearStabilityAnalysis #StabilityAnalysis #VectorField #OneDimensional #Functions