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Rotational-vibrational coupling
Rotational-vibrational coupling occurs when there is a 1:2 ratio of rotation frequency of an object and a natural internal vibration frequency. The animation on the right shows the simplest example of this phenomenon. The motion depicted in the animation is for the idealized situation that the force exerted by the spring is proportional to the amount of extension. Note that in this demonstration the spring isn't alternating between pulling and pushing, the spring is exerting a contracting force all the time; given the chance the idealized spring would contract all the way down to zero length. Also, since the animation keeps on looping, the animation depicts what would occur if there would not be any friction.
In molecular physics it is recognized that there is a coupling of rotational and vibrational energy-levels. In molecular physics rotational-vibrational coupling is also called rovibronic coupling and Coriolis coupling. The physics of actual diatomic molecules is more complicated than the example in the animation, but because of its simplicity the animation is useful for illustrating the basic principles.
In molecular physics it is recognized that there is a coupling of rotational and vibrational energy-levels. In molecular physics rotational-vibrational coupling is also called rovibronic coupling and Coriolis coupling. The physics of actual diatomic molecules is more complicated than the example in the animation, but because of its simplicity the animation is useful for illustrating the basic principles.
During the phase in the cycle that the spring pulls the two weights closer to the center of rotation the angular velocity increases; the centripetal force is doing work, converting strain energy that was stored in the spring to kinetic energy of the weights. At some point contraction ends and the weights swing wide again. As the distance of the weights to the central axis of rotation increases kinetic energy is converted to strain energy of the spring. The angular velocity decreases during this phase, and from a certain point on there is a surplus of centripetal force. Eventually the surplus of centripetal force starts a new phase of contraction.
Analogy with harmonic oscillation Animation 2 provides a clearer view on the oscillation of the angular velocity. The motion as seen from a rotating point of view looks remarkably regular and symmetrical. It is in fact very regular; I will come to that further on in the article.
Harmonic oscillation is a cyclic process of energy conversion. 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 total energy of the system is conserved, but its form is oscillating back and forth between kinetic energy and potential energy.
In the motion pattern depicted in animation 2 there is, just as in the case of simple harmonic oscillation, a back and forth conversion between 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 its largest. The total energy of the system is conserved.
(With a real spring there is friction involved. With a real spring the vibration will be dampened and the final situation will be that the masses circle each other at a constant distance, with a constant tension of the spring. That is, in the final situation the ratio of kinetic energy to potential energy will be 1:1 .)
For a discussion of the role that momentum plays, see note on momentum
Mathematical derivation Cartesian coordinates Picture 4. Image The motion of the circling masses is planar. I'm applying the following simplifications: I'm taking the spring itself as being weightless, and I'm taking perfect spring; the centripetal force increases in a linear way as the spring is stretched out. That is, in this simplification the centripetal force is exactly proportional to the distance to the center of rotation. A centripetal force with this characteristic is called a harmonic force.
The motion of the weights in two dimensions of space can be decomposed in two harmonic oscillations, perpendicular to each other.
The following parametric equation of the position as a function of time describes the motion of the circling masses. The parametric equation provides a complete description: it describes the shape of the trajectory and the velocity at each point in time.
Animation 5 depicts this rearrangement. There is an overall circular motion, combined with motion along an epi-circle. Both the motion along the overal circle (counterclockwise) and the motion long the epi-circle (clockwise) are uniform circular motion. This is a remarkable symmetry of motion under the influence of a harmonic force; the eccentricity itself can be thought of as a uniform circular motion.
Transformation to a coordinate system that is rotating with angular velocity O. does the following: it subtracts the overall circular motion, and what is left is the eccentricity of the elliptical trajectory. The center of the eccentricity is located at a distance of (a + b) / 2 from the main axis of rotation.
The transformation to a rotating coordinate system is of course to a particular one: the coordinate system in which the eccentricty of the ellipse-shaped trajectory is a circle around a fixed point. Here and everywhere else in this article (and everywhere else on this web site): whenever I refer to "transformation to a rotating coordinate system" I'm referring to the particular rotating coordinate system that matches the period of rotation.
Rotational-vibrational coupling occurs when there is a 1:2 ratio of rotation frequency of an object and a natural internal vibration frequency. The animation on the right shows the simplest example of this phenomenon. The motion depicted in the animation is for the idealized situation that the force exerted by the spring is proportional to the amount of extension. Note that in this demonstration the spring isn't alternating between pulling and pushing, the spring is exerting a contracting force all the time; given the chance the idealized spring would contract all the way down to zero length. Also, since the animation keeps on looping, the animation depicts what would occur if there would not be any friction.
In molecular physics it is recognized that there is a coupling of rotational and vibrational energy-levels. In molecular physics rotational-vibrational coupling is also called rovibronic coupling and Coriolis coupling. The physics of actual diatomic molecules is more complicated than the example in the animation, but because of its simplicity the animation is useful for illustrating the basic principles.
In molecular physics it is recognized that there is a coupling of rotational and vibrational energy-levels. In molecular physics rotational-vibrational coupling is also called rovibronic coupling and Coriolis coupling. The physics of actual diatomic molecules is more complicated than the example in the animation, but because of its simplicity the animation is useful for illustrating the basic principles.
During the phase in the cycle that the spring pulls the two weights closer to the center of rotation the angular velocity increases; the centripetal force is doing work, converting strain energy that was stored in the spring to kinetic energy of the weights. At some point contraction ends and the weights swing wide again. As the distance of the weights to the central axis of rotation increases kinetic energy is converted to strain energy of the spring. The angular velocity decreases during this phase, and from a certain point on there is a surplus of centripetal force. Eventually the surplus of centripetal force starts a new phase of contraction.
Analogy with harmonic oscillation Animation 2 provides a clearer view on the oscillation of the angular velocity. The motion as seen from a rotating point of view looks remarkably regular and symmetrical. It is in fact very regular; I will come to that further on in the article.
Harmonic oscillation is a cyclic process of energy conversion. 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 total energy of the system is conserved, but its form is oscillating back and forth between kinetic energy and potential energy.
In the motion pattern depicted in animation 2 there is, just as in the case of simple harmonic oscillation, a back and forth conversion between 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 its largest. The total energy of the system is conserved.
(With a real spring there is friction involved. With a real spring the vibration will be dampened and the final situation will be that the masses circle each other at a constant distance, with a constant tension of the spring. That is, in the final situation the ratio of kinetic energy to potential energy will be 1:1 .)
For a discussion of the role that momentum plays, see note on momentum
Mathematical derivation Cartesian coordinates Picture 4. Image The motion of the circling masses is planar. I'm applying the following simplifications: I'm taking the spring itself as being weightless, and I'm taking perfect spring; the centripetal force increases in a linear way as the spring is stretched out. That is, in this simplification the centripetal force is exactly proportional to the distance to the center of rotation. A centripetal force with this characteristic is called a harmonic force.
The motion of the weights in two dimensions of space can be decomposed in two harmonic oscillations, perpendicular to each other.
The following parametric equation of the position as a function of time describes the motion of the circling masses. The parametric equation provides a complete description: it describes the shape of the trajectory and the velocity at each point in time.
Animation 5 depicts this rearrangement. There is an overall circular motion, combined with motion along an epi-circle. Both the motion along the overal circle (counterclockwise) and the motion long the epi-circle (clockwise) are uniform circular motion. This is a remarkable symmetry of motion under the influence of a harmonic force; the eccentricity itself can be thought of as a uniform circular motion.
Transformation to a coordinate system that is rotating with angular velocity O. does the following: it subtracts the overall circular motion, and what is left is the eccentricity of the elliptical trajectory. The center of the eccentricity is located at a distance of (a + b) / 2 from the main axis of rotation.
The transformation to a rotating coordinate system is of course to a particular one: the coordinate system in which the eccentricty of the ellipse-shaped trajectory is a circle around a fixed point. Here and everywhere else in this article (and everywhere else on this web site): whenever I refer to "transformation to a rotating coordinate system" I'm referring to the particular rotating coordinate system that matches the period of rotation.
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