Gyroscopic effects

In this example, we demonstrate how a rotating body creates a reaction torque when its axis of rotation is changed.

The system consists of a pendulum suspended in a spherical joint, a joint without any rotational constraints. The tip of the pendulum is a cylinder that is rotating around a revolute joint in its center. When the pendulum swings, the rotation axis of the rotating tip is changed, this causes the entire pendulum to rotate around the axis through the pendulum rod.

using Multibody
using ModelingToolkit
using Plots
using JuliaSimCompiler
using OrdinaryDiffEq

t = Multibody.t
D = Differential(t)
world = Multibody.world

systems = @named begin
    spherical = Spherical(state=true, radius=0.02, color=[1,1,0,1], quat=true)
    body1 = BodyCylinder(r = [0.25, 0, 0], diameter = 0.05)
    rot = FixedRotation(; n = [0,1,0], angle=deg2rad(45))
    revolute = Revolute(n = [1,0,0], radius=0.06, color=[1,0,0,1])
    trans = FixedTranslation(r = [-0.1, 0, 0])
    body2 = BodyCylinder(r = [0.2, 0, 0], diameter = 0.1, color=[0,0,0.5,1])
end

connections = [
    connect(world.frame_b, spherical.frame_a)
    connect(spherical.frame_b, body1.frame_a)
    connect(body1.frame_b, rot.frame_a)
    connect(rot.frame_b, revolute.frame_a)
    connect(revolute.frame_b, trans.frame_a)
    connect(trans.frame_b, body2.frame_a)
]

@named model = ODESystem(connections, t, systems = [world; systems])
model = complete(model)
ssys = structural_simplify(multibody(model))

prob = ODEProblem(ssys, [model.world.g => 9.80665, model.revolute.w => 10], (0, 5))

sol = solve(prob, FBDF(), abstol=1e-8, reltol=1e-8);
@assert SciMLBase.successful_retcode(sol)

import GLMakie
Multibody.render(model, sol; x=1, z=1, filename = "gyro.gif") # Use "gyro.mp4" for a video file

Try setting model.revolute.w => 0 and plot this variable using plot(sol, idxs=model.revolute.w) and you will notice that the swinging of the pendulum induces a rotation around this joint, even if it has no rotational velocity from the start.