Laplace transform of a mass-spring-damper system


Apply the Laplace transform to both equations, term by term (since the Laplace transform is a linear operator). Then you will have two equations in terms of $X (s) $ and $U (s) $. Combine the equations to eliminate $U (s) $. Then perform the algebra to isolate $X (s)$ on one side of the equation.


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Updated on December 02, 2020


  • James
    James almost 3 years

    We consider integral control of a mass-spring-damper system, that is a coupled system

    $$\ddot x(t) + 5\dot x(t) + 4x(t) = u(t),$$ $$\dot u(t) = k(r - x(t))$$ where k is a positive parameter and r is a desired set point.

    Verify that if the initial conditions are zero $($i.e. $x(0) = 0$, $\dot x(0) = 0 $ and $u(0) = 0$$)$, then, $$X(s) = \frac{k}{s(s^2+5s+4)+k}\cdot \frac{r}{s}$$ How do I go about reaching this solution?