Rotational–vibrational coupling
In physics, rotational–vibrational coupling occurs when the rotation frequency of a system is close to or identical to a natural internal vibration frequency. 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. In rotational-vibrational coupling, angular velocity oscillates. By pulling the circling masses closer together, the spring transfers its stored strain energy into the kinetic energy of the circling masses, increasing their angular velocity. The spring cannot bring the circling masses together, since the spring's pull weakens as the circling masses approach. At some point, the increasing angular velocity of the circling masses overcomes the pull of the spring, causing the circling masses to increasingly distance themselves. 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. In helicopter design, helicopters must incorporate damping devices, because at specific angular velocities, the rotorblade vibrations can reinforced themselves by rotational-vibrational coupling, and build up catastrophically. Without damping, these vibrations would cause the rotorblades to break loose. Energy conversions The animation on the right provides a clearer view on the oscillation of the angular velocity. There is a close analogy with harmonic oscillation. When a harmonic oscillation is at its midpoint then all the energy of the system is kinetic energy. When the harmonic oscillation is at the points furthest away from the midpoint all the energy of the system is potential energy. The energy of the system is oscillating back and forth between kinetic energy and potential energy. In the animation with the two circling masses there is a back and forth oscillation of kinetic energy and potential energy. When the spring is at its maximal extension then the potential energy is largest, when the angular velocity is at its maximum the kinetic energy is at largest. With a real spring there is friction involved. With a real spring the vibration will be damped and the final situation will be that the masses circle each other at a constant distance, with a constant tension of the spring. Mathematical derivation This discussion applies the following simplifications: the spring itself is taken as being weightless, and the spring is taken as being a perfect spring; the restoring force increases in a linear way as the spring is stretched out. That is, the restoring force is exactly proportional to the distance to the center of rotation. A restoring force with this characteristic is called a harmonic force. The following parametric equation of the position as a function of time describes the motion of the circling masses: where is the major radius is the minor radius is 360° divided by the duration of one revolution The motion as a function of time can also be seen as