Underdamped System Response: Solving with Convolution Integral | Homework Help

[ganondorf29]

Homework Statement

$$<br /> x'' + 2\zeta \omega_{n} x' + \omega_{n}^2 x = u_{s}(t) <br />$$

zeta is underdamped and $$u_{s}(t)$$ is the unit step function and $$\omega_n$$ 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

$$<br /> u_{s}(t)=\left\{\begin{array}{cc}0,&\mbox{ if }<br /> t\leq 0\\1, & \mbox{ if } t>0\end{array}\right.<br />$$


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.

$$<br /> x(t) = exp(-\zeta \omega_{n}) * cos(\omega_{d} t)<br />$$



When t > 0

I'm not sure how to use the convolution integral to find the response

 

[vela]

Homework Statement

$$<br /> x'' + 2\zeta \omega_{n} x' + \omega_{n}^2 x = u_{s}(t) <br />$$

zeta is underdamped and $$u_{s}(t)$$ is the unit step function and $$\omega_n$$ 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

$$<br /> u_{s}(t)=\left\{\begin{array}{cc}0,&\mbox{ if }<br /> t\leq 0\\1, & \mbox{ if } t>0\end{array}\right.<br />$$


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.

$$<br /> x(t) = exp(-\zeta \omega_{n}) * cos(\omega_{d} t)<br />$$

You're not taking into account the initial conditions.

When t > 0

I'm not sure how to use the convolution integral to find the response

The idea is to find the impulse response h(t) of the system, which is the solution to the differential equation where the forcing function is the Dirac delta function and which satisfies the initial conditions. Then convolve h(t) with the given forcing function, u(t), to find the total system response.