Gyroscopic Whirl on a Compliant Mount: 2D Phase-Plane View

A flywheel's angular momentum doesn't just resist being yawed by an external drive — mounted on any real, slightly compliant support, it turns a simple two-axis tilt oscillator into a coupled gyroscopic system with its own forward and backward whirl frequencies. This simulator integrates that coupled equation of motion in real time (fourth-order Runge–Kutta, not a kinematic snapshot) and plots the platform's tilt state as a live phase-plane trace: kick a single flywheel's mount and the disturbance spirals between the two tilt axes as it decays, a direct, dynamical demonstration of gyroscopic coupling. Switch to a counter-rotating pair and the net angular momentum — and with it the coupling term in the equations — drops to exactly zero: the same kick now decays along a straight line with no cross-axis whirl, while the stored kinetic energy still doubles because both wheels are still spinning.