# Electrical and mechanical oscillations, an introduction

PHYS 5.5: The mathematics of oscillations

Since there is no resistance in the circuit, no energy is lost through Joule heating; thus, the maximum energy stored in the capacitor is equal to the maximum energy stored at a later time in the inductor:. Since the inductor resists a change in current, current continues to flow, even though the capacitor is discharged. This continued current causes the capacitor to charge with opposite polarity.

From the law of energy conservation, the maximum charge that the capacitor re-acquires is. Then, in the last part of this cyclic process, energy flows back to the capacitor, and the initial state of the circuit is restored. We have followed the circuit through one complete cycle. Its electromagnetic oscillations are analogous to the mechanical oscillations of a mass at the end of a spring. In this latter case, energy is transferred back and forth between the mass, which has kinetic energy , and the spring, which has potential energy.

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Analysis and mitigation of ferroresonant oscillations based on harmonic balance method and bifurcation theory are presented in [ 5 ]. You can do it, too! Resnick, D. More About Us. Next: Write a comment.

With the absence of friction in the mass-spring system, the oscillations would continue indefinitely. The equivalence of the two systems is clear. To find the maximum current, the maximum energy in the capacitor is set equal to the maximum energy in the inductor.

### 1.1 Module introduction

Electrical and Mechanical Oscillations, an Introduction [D S Jones] on Amazon. com. *FREE* shipping on qualifying offers. Electrical and mechanical oscillations: an introduction. Front Cover. Douglas Samuel Jones. Routledge, Jan 1, - Science - pages. 0 Reviews.

At sufficiently large displacements, the stiffness of the wing dominates to provide the restoring force that enables an oscillation. The harmonic oscillator and the systems it models have a single degree of freedom.

## 11.5 Oscillations in an LC Circuit

More complicated systems have more degrees of freedom, for example two masses and three springs each mass being attached to fixed points and to each other. In such cases, the behavior of each variable influences that of the others. This leads to a coupling of the oscillations of the individual degrees of freedom.

For example, two pendulum clocks of identical frequency mounted on a common wall will tend to synchronise.

Physics - Mechanics: Simple Harmonic Motion (1 of 5) Introduction 1

This phenomenon was first observed by Christiaan Huygens in More special cases are the coupled oscillators where energy alternates between two forms of oscillation. Well-known is the Wilberforce pendulum , where the oscillation alternates between an elongation of a vertical spring and the rotation of an object at the end of that spring.

Coupled oscillators is a common description of two related, but different phenomena. One case is where both oscillations affect each other mutually, which usually leads to the occurrence of a single, entrained oscillation state, where both oscillate with a compromise frequency. Another case is where one external oscillation affects an internal oscillation, but is not affected by this.

In this case the regions of synchronization, known as Arnold tongues. The latter case can lead to highly complex phenomena as for instance chaotic dynamics. As the number of degrees of freedom becomes arbitrarily large, a system approaches continuity ; examples include a string or the surface of a body of water. Such systems have in the classical limit an infinite number of normal modes and their oscillations occur in the form of waves that can characteristically propagate. The mathematics of oscillation deals with the quantification of the amount that a sequence or function tends to move between extremes.