Part c$)$

Consider the following two ODEs for the parts c$-$g below, which approximate the $2$DoF Setups:

$++(+)=()$

$3$

$++(+)=$

The Laplace Transform $($assuming zero initial conditions$)$ would be$..\mathrm{.}6$

$1()12+1+(01+12)=()+122()$

$2()22+2+(12+23)=121()$

Setting two Polynomials, $12,$ to be$...$

$=++(+)$

$=++(+)$

The Laplace Transforms can be rewritten as$...$

$1()1=()+122()$

$2()=12$

$2$

$1()$

Substituting the second equation, $2(),$ into the first equation gives...

$1()1=()+12$

$12$

$2$

$1()=>1()112$

$12$

$2$

$=()$

$()$

$()=$

$$

Substituting back into the $2()$ equation gives...

$2()=12$

$2$

$()2$

$121212$

$()$

$()=$

$$

Summarizing the Transfer Function become...

$()=()$

$()=$

$$

$=$

$$

$()=()$

$()=$

$$

$=$

$$

$=++(+)$

$=++(+)$

Note: $21()$ refers to the Plant$$s Transfer Function for Disk $1$ in the $2$ DoF Setup.

The built$-$in function tf$()$ can be used to represent a Transfer Function, SYS$=10/(2$s$^2+10$s$+12),$

with the following code. Note: this is presented in further detail in the Background section.

s$=$tf$(\text{'}$s$\text{'})$

SYS$=10/(2*$s$^2+10*$s$+12)$

$4$

Write a brief script setting the values of m$1=$m$2=1$ C$1=$C$2=1$ K$01=0$ K$12=100$ and K$23=10$ then

create the Transfer Function P$21.$

Save this script as it will be used in the next parts and in future lab assignments, the values of

the m$$s c$$s and k$$s will be changed to match the experimental setups used in the lab.