Steps for

Problem $1$: State$-$Space Representation and Markov Parameters for a $4$ DOF System

Consider a $4$ Degrees of Freedom $($DOF$)$ mechanical system with the following mass, stiffness, and damping matrices:

M=[200020001.50

0001]

$0$

$0$

$0$

$0$

$0$

$K=\left[\begin{array}{ccccccccccccc}6& -2& 0& 0& 5& -1& 0& -1& 4& -1& 0& -1& 3\end{array}\right]$

$0$

$0$

$-2$

$C=\left[\begin{array}{ccccccccccccc}0\mathrm{.}5& -0\mathrm{.}1& 0& 0& 0\mathrm{.}4& -0\mathrm{.}1& 0& -0\mathrm{.}1& 0\mathrm{.}3& -0\mathrm{.}1& 0& -0\mathrm{.}1& 0\mathrm{.}2\end{array}\right]$

$0$

$0$

$-0\mathrm{.}1$

Input: A harmonic force $u(t)=F_{0}sin(t)$ is applied to the first degree of freedom$($DOF $1).$

Let $F_0=1N($Newton$).$

Let $=22ra\frac{d}{s}$

Output: Consider $2$ different system's response is measured as:

Displacements at each DOF, $y(t)=x(t)$

Accelerations at each DOF, $y(t)=x^{}(t)$

Using this information, complete the following tasks:

a$)$ Derive the State$-$Space Representation$-$find the state$-$space matrices A$,$B$,$C and D that describe the system once using displacement measurements as outputs and once using accelerations

b$)$ Calculate the Markov Parameters using the state$-$space matrices derived in part a$.$ Calculate the first five Markov parameters for both cases $($displacement measurements and acceleration measurements$).$

c$)$ Natural Frequencies and Mode Shapes. Calculate the natural frequencies and mode shapes of the system $($both undamped and damped$).$

d$)$ Using the state$-$space matrix A derived, compute the eigenvalues of $A$ to analyze the stability of the system. Determine whether the system is stable, marginally stable, or unstable based on the location of the eigenvalues.

$1$