21 B
Mathematics Math21b Fall 2010
Linear Algebra and Differential Equations
Exhibit: damped harmonic oscillator
Course Head: Oliver Knill
Office: SciCtr 434

The eigenvalues of the damped harmonic oscillator

The differential equation 
x'' = - x - a x'
is a damped harmonic oscillator. The left hand side is the acceleration, the right hand side a sum of the spring force and a damping term. This system can be written as a system of first order differential equations:
x' = y
y' = - x - a y
which is X' = A X for the matrix
A = | 0   1 |
    | -1 -a |
For a=0, the harmonic oscillator case, the eigenvalues of A are -i,i For positive a, the real part of the eigenvalues becomes negative and the eigenvalues move to the left in the complex plane. There is the moment, when the eigenvalues are both -1. This is the critically damped situation. For larger a, the eigenvalues are different. It is the overdamped situation.

Diplaying the acceleration
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Math21b (Exam Group 1)| Oliver Knill | Fall 2010 | Department of Mathematics | Faculty of Art and Sciences | Harvard University