Harmonic Oscillator Paper
- 1. Hunter Rose, R10572598 10/20/2014 Differential Equations Harmonic Oscillator A harmonic oscillator is a special case where, when a systemis displaced from its equilibrium conditions, it will experience a resultant force that is proportional to its displacement; x. Harmonic oscillators can be used to find displacement as a function of time as well. This will relate to us in Differential Equations in multiple ways, two of which are in relation to Hooke’s law and Newton’s second law, as well as almost anytime we encounter a spring or something similar to one. Harmonic oscillators appear in numerous places in the world including pendulums, masses connected to springs, and acoustical systems. One of the simplest harmonic equations we will see is y’’+ky=0, which is a simple harmonic function, otherwise known as free un- damped motion. An un-damped harmonic function is one in which your force or you (ky) is the only thing acting on your system. If this is not the case then you encounter what is called a damped harmonic function in which there are more forces acting on your system, such as friction. Dampening can be broken up into different sub categories such as over-damped in which the function will decay to the equilibrium positon without oscillating, underdamped in which the frequency and size of the oscillations decreases with time, or possibly critically damped. We are looking at the case of a simple harmonic oscillator in which there is no dampening occurring. In this scenario a Laplace transformation could be used to solve an initial value problem involving a harmonic oscillator equation. For example, in our book on page 233 there is an example of solving an initial value problem that describes the forced, un-damped and resonant motion of a mass on a spring, which is a simple harmonic oscillating equation x’’+16x=cos(4t). After using a Laplace transformation on this problem you will find that x(t)=(1/4)sin(4t)+(1/8)t*sin(4t). Another example of using a Laplace Transformation on an Initial value harmonic oscillator problem is question number 14 in section 4.6 of our book. This question asks you to derive a systemof differential equations describing the motion of some springs then use a Laplace transformation to solve them, which will be a pretty hefty amount of work but can be done.
- 2. References Advanced Engineering Mathematics, Fifth Edition, by Dennis G. Sill and Warren S. Wright. NCSU department of mathematics slides on harmonic oscillator equations http://www.ncsu.edu/crsc/events/ugw05/slides/root_harmonic.pdf http://scipp.ucsc.edu/~haber/ph5B/sho09.pdf http://en.wikipedia.org/wiki/Harmonic_oscillator#Simple_pendulum