# Mechanical System Transfer Function - Systems Modelling

First of all my english writing is not so good, but I can read it without problems. I've got this mechanical system and I need to get some transfer functions. We made all the force diagrams and got the respective equations, they're eight. Now we need to get the transfer function of $\dot{x}_1\over v_1$, $x_2\over v_3$, $\ddot{x}_3\over v_4$ and $x_4\over v_7$.

Now, assuming that the equations are correct, what we were trying to do was for the first T.F clear all $X_2(s)$ from the second equation and replace it on the first equation, then we clear all $sB_1$, $B_2X_1(s)$ from the new equation that we just got from replacing $X_2(s)$, then we're trying to make $sB_1$, $B_2X_1(s)\over V_1(s)$ but there's no way to clear $V_1(s)$.

It's very difficult to understand, and I don't know if I explained myself, but if someone can tell us what we're doing wrong I'll be really grateful.

## 1 Answer

First off, the equations you have are not correct. For example, if equations 1 and 5 are added it results in $V_1+V_5=0$. The l.h.s of the first four equations should be $V_1-V_5$, $V_2-V_6$, $V_3-V_7$, and $V_4-V_8$. Then you are have four equations in the four unknowns $X_1$, $X_2$, $X_3$, and $X_4$ after taking Laplace transforms with zero initial conditions.

The general method is to solve for these four unknowns and express them in terms of the $V_i$ and then you get the transfer functions for each input-output pair. The derivatives can then be obtained by multiplying with powers of the Laplace variable $s$.

Since you need just 4 transfer functions, I think it can be done a bit more easily.

For the first transfer function, you are interested only in $V_1$. Set all the other $V_i$ to zero and solve for $\frac{X_1}{V_1}$. Since you want the derivative just multiply the result by $s$.

For the second transfer function, set all the $V_i$ except $V_3$ to zero and solve for $\frac{X_2}{V_3}$. Multiply the result by $s^2$ to obtain the desired transfer function.

I hope you get the idea.