Damped Harmonic Oscillators

by David Walker

Submitted : Fall 2012

After obtaining the needed equations (universal harmonic oscillator and sinusoidal driving force taken from driven harmonic oscillators), and being given the solutions to the equations, through second order homogeneous differential equations, the work needed to be shown that the solutions were indeed solutions to each respective equation. The first and second derivatives were taken from the transient solutions, plugged into the universal harmonic oscillator equation and set equal to zero. The same process occurred in the steady state solution with its respective equation (sinusoidal driving force) but, it was set equal to a constant multiplied by a sinusoid.

Upon completion of the solutions and proving each solution worked with the respective equation, each solution was graphed based on certain constants with a varying degree of dampening in the transient solutions and varying degree of dampening and driving frequency in the steady state solutions, and in each solution there was a variable t for time. The results vary depending on which system is being used and for what purpose.

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