Rotational–vibrational coupling

The animation on the right shows ideal motion, with the force exerted by the spring and the distance from the center of rotation increasing together linearly with no friction.

This increasingly strains the spring, strengthening its pull and causing the circling masses to transfer their kinetic energy into the spring's strain energy, thereby decreasing the circling masses' angular velocity.

At some point, the pull of the spring overcomes the angular velocity of the circling masses, restarting the cycle.

The parametric equations (1) and (2) can be rewritten as: A transformation to a coordinate system that subtracts the overall circular motion leaves the eccentricity of the ellipse-shaped trajectory.

from the main center: That is in fact what is seen in the second animation, in which the motion is mapped to a coordinate system that is rotating at a constant angular velocity.

More precisely, the spring is oscillating between doing positive work (increasing the weight's kinetic energy) and doing negative work (decreasing the weight's kinetic energy) The centripetal force is a harmonic force.

This is a distinctive feature of motion under the influence of a harmonic force; all trajectories take the same amount of time to complete a revolution.

Both the centripetal force and the centrifugal term in the equation of motion are proportional to r. The angular velocity of the rotating coordinate system is adjusted to have the same period of revolution as the object following an ellipse-shaped trajectory.

It is only in very special circumstances that the vector of the centripetal force and the centrifugal term drop away against each other at every distance from the center of rotation.

Since the vector of the Coriolis term always points perpendicular to the velocity with respect to the rotating coordinate system, it follows that in the case of a restoring force that is a harmonic force, the eccentricity in the trajectory will show up as a small circular motion with respect to the rotating coordinate system.

However, this disregards the causal mechanism, which is the force of the extended spring, and the work done during its contraction and extension.

As the gas expands, its high pressure exerts a force on both the projectile and the interior of the barrel.

Two particles connected by a spring. In the absence of the spring, the particles would fly apart. However, the force exerted by the extended spring pulls the particles onto a periodic, oscillatory path.
The motion of the circling masses mapped in a coordinate system that is rotating at a constant angular velocity
Harmonic oscillation the restoring force is proportional to the distance from the center.
Motion due to a harmonic force described as circle + epi-circle motion