Question

give the second order differential equation for a harmonic oscillator with mass 1, damping constant 4, and spring constant 3. Then convert this equation to a system.

Answer 1

the DE for a harmonic oscillator:
\[\ddot{x}+{k\over m}x=0\]

Answer 2

when damped, you add a first derivative component..
\[\ddot{x}+c\dot{x}+{k\over m}x=0\]

Answer 3

k=spring const
m=mass
c=damping const.

Answer 4

x = displacement as a function of time
and \(\dot{x}\) is the first derivative with time
and \(\ddot{x}\) is the second

Answer 5

okay. whats the difference with damped undamped overdamped? do I memorize certain equations?

Answer 6

this becomes a simple homogenous differential equation that you can solve

Answer 7

the sign of "c"

Answer 8

if c>0, over dampled -> amplitude increases with time
if c=0, critically damped -> amplitude is const. -> no damping
c<0 underdamped, amplitude slowly decreases to 0 (at infinity)

Answer 9

this you can easily derive from the force equation
\[a=\ddot{x}\\
-kx=m\ddot{x}\]
rearrange it and you get it