1. Particle pendulum with air-resistance   (2D dynamics) The figure to the right shows a particle Q of mass m = 100 kg
at the distal end of a straight cable B of length L = 50 m.
Earth's local gravitational acceleration is g = 9.8 m/s2.
Air-resistance of force   -2*v   resists the pendulum motion.
This problem shows how to use MotionGenesisô to determine
the pendulum angle q and cable tension as a function of time.
 Dynamics: F = m a,   M = I 𝛂,   Lagrange, Kane Program responses Dynamics: With quaternion Program responses
To simulate/plot with MATLABÆ, edit   MGParticlePendulumDynamics.txt,   and change
ODE() ParticlePendulumDynamics     to     ODE() ParticlePendulumFma.m
This causes MotionGenesisô to auto-generate the file ParticlePendulumFma.m
Alternately, use this MotionGenesisô   file   to auto-generate a
MATLABÆ   .m file   with plant dynamics   (e.g., for Simulink®). Enlarge image  2. Rigid-body pendulum   (2D dynamics)
.
The figure to the right shows a non-uniform rigid rod A attached
by a frictionless pin joint to Earth N (a Newtonian reference frame).
The only external forces on rod A are contact forces at No
(the point of N in contact with A) and Earth's gravitational force.
 Rigid-body pendulum problem statement Student (.pdf) Instructor (.pdf)
 Dynamics: M = I 𝛂, Lagrange, Kane Program responses Enlarge image

 3. Metronome   (2D Dynamics -- Euler, Lagrange, Kane)
 Metronome problem statement Student (.pdf) Instructor (.pdf) Form equations of motion with the methods of Euler (angular momentum principle), Lagrange, ...
 Dynamics: M = I 𝛂, Lagrange, Kane Program responses
Video: Metronomes synchronize each other   (B)  4. Helicopter retrieval   (2D variable-length pendulum)   See detailed description.
 Dynamics: F = m a, Kane Program responses  .

 5. Swinging spring   (2D pendulum, particle on spring)
 Dynamics: F = m a, Lagrange, Kane Program responses  6. Foucault pendulum   (3D motion with spinning Earth)
 Dynamics at North Pole: F = m a, Lagrange, Kane Program responses Dynamics at latitude: F = m a, Lagrange, Kane Program responses   7. Spherical pendulum   (3D tether/wrecking-ball forces & motion)
 Dynamics: F = m a Program responses  8. Tether ball   (2D/3D particle wrapping around a cylinder)
 Rigid-body pendulum problem statement Student (.pdf) Instructor (.pdf)
 2D: F = ma Program responses 3D: F = m a Program responses  9. Swinging beam on two cables   (construction hoist)
 Swinging beam problem statement (.pdf)   10. Trifilar pendulum   (3D aircraft)
 Statics: F = 0    M = 0 Program responses Statics: Kane augmented Program responses Statics: Kane embedded Program responses Dynamics: F = m a    M = dH/dt Program responses Dynamics: Kane augmented Program responses Dynamics: Kane embedded Program responses Constraints: Yaw/pitch/roll Program responses  