Given the system is modeled by passive I.V.P: $$y''+6y'+13y=f(t), y(0)=y'(0)=0$$
Write the transfer function and the weight function for the IVP classify the mass spring system as over damped, damped, undamped, critically damped
What are the transfer and weight functions? I missed class. What do they do and how do you find them? Is there a formula to determine if a mass spring system is under damped, over damped....etc,etc. What is a unit step response?
Let $F(s)=\mathcal{L}\{f\}(s)$ be the Laplace transform of the input function and $Y(s)=\mathcal{L}\{y\}(s)$ the Laplace transform of the solution (output) function $y$. The transfer function $H(s)$ is defined as: $$ H(s) = \frac{Y(s)}{X(s)}. $$ In order to find $H$, take the Laplace transform of the differential equation.
If your calculations are correct, you will see that the denominator of $H$ is equal to $p(s)$, where $p$ is the characteristic polynomial of the differential equation. From the roots of this denominator, you can deduce the behaviour of the system.
Instead of giving a full revision of all possible cases, I will just give two examples.
If the roots are $-1+2i$ and $-1+2i$ the solution of the IVP is a combination of $e^{-t}\cos 2t$ and $ e^{-t}\sin 2t$. In this case we have a damped solution.
If the roots are $-3$ and $-3$, the solution is a combination of $e^{-3t}$ and $te^{-3t}$, and so the system is critically damped.