- #1
ganondorf29
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Homework Statement
[tex]
x'' + 2\zeta \omega_{n} x' + \omega_{n}^2 x = u_{s}(t)
[/tex]
zeta is underdamped and [tex] u_{s}(t) [/tex] is the unit step function and [tex] \omega_n [/tex] is the natural frequency and there are zero initial conditions. Find the total response via the convolution integral.
Homework Equations
The Attempt at a Solution
[tex]
u_{s}(t)=\left\{\begin{array}{cc}0,&\mbox{ if }
t\leq 0\\1, & \mbox{ if } t>0\end{array}\right.
[/tex]
Since there are two time intervals there are two different behaviors
When t < 0
Because there is no forcing term the response is a free response to an under damped system.
[tex]
x(t) = exp(-\zeta \omega_{n}) * cos(\omega_{d} t)
[/tex]
When t > 0
I'm not sure how to use the convolution integral to find the response