Reference: Nonlinear Dynamics and Chaos (Strogatz, Steven H.), Page 143 The moti

Question

Reference: Nonlinear Dynamics and Chaos (Strogatz, Steven H.), Page 143

The motion of a damped harmonic oscillator is described by nix + bx + kx - 0 , where b > 0 is the damping constant. Rewrite the equation as a two-dimensional linear system. Classify the fixed point at the origin and sketch the phase portrait Be sure to show all the different cases that can occur, depending on the relative sizes of the parameters. How do your results relate to the standard notions of over damped, critically damped, and under damped vibrations?

Explanation / Answer

x' = ax + b y

y'= cx + dy

We know that the behavior of this system is completely determined by the eigenvalues of the matrix A whose entries are a,b,c,d. These are the normal possibilities:

Under, Over and Critical Damping
1.   Response to Damping
As we saw, the unforced damped harmonic oscillator has equation
.. . mx + bx + kx = 0,   (1)
with m > 0, b ? 0 and k > 0. It has characteristic equation
ms
2 + bs + k = 0
with characteristic roots
?b

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