$8\mathrm{.}2$ Two rods are pinned to the ground as shown. The lengths of the rods are $2\mathrm{.}0$ m and $1\mathrm{.}5$ m respectively. A spring$-$damper$-$actuator element is connected between the free ends of the rods having the following characteristics: $k=120\frac{N}{m},{?}^{0}L=1\mathrm{.}8m,d_c=20N.\frac{s}{m},{?}^{(a)}f=25N.$ In the shown configuration, the angular velocities of the two bodies are, respectively, ${}_1^{}=1\mathrm{.}0ra\frac{d}{s}CW$ and ${}_2^{}=2\mathrm{.}5ra\frac{d}{s}CCW.$ Gravity acts on the system. The mass and moment of inertia of each body are given as $m_1=0\mathrm{.}8,m_2=1\mathrm{.}2kg,J_1=0\mathrm{.}15,J_2=0\mathrm{.}40kg.m^2.$

In solving this problem you are asked to follow the standard procedures for determining the coordinates and velocities of points and vectors without taking shortcuts as you did in the previous problem. You can only determine the body coordinates $(r_1,{}_1,r_2,{}_2)$ and velocities $(r_1^{},r_2^{})$ in the shown configuration from the figure $({}_1^{}$ and ${}_2^{}$ are given$).$

Develop a MATLAB program for the followings:

$($a$)$ Extract from the figure the $-$ components for the following vectors and enter them in the program: vec$(s{)}_1^A,$vec$(s{)}_1^O,$vec$(s{)}_2^B,$vec$(s{)}_2^Q[$you need to define the $x-y$ frame and attach a $-$ frame to each body$]$

$($b$)$ Enter in the program all other constant data

$($c$)$ Enter the extracted values for $r_1,{}_1,r_2,{}_2$ and $r_1^{},{}_1^{},r_2^{},{}_2^{}$

$($d$)$ Determine the $x-y$ components for the following vectors: vec$(s{)}_1^A,$vec$(s{)}_1^O,$vec$(s{)}_2^B,$vec$(s{)}_2^Q[$check the computed values against the figure$]$

$($e$)$ Determine the coordinates of points A and $B[$check the computed values against the figure$]$

$($f$)$ Determine the velocity of points A and $B$

$($g$)$ Determine the force of the spring$-$damper$-$actuator element

$($h$)$ Construct two $31$ arrays of force that this element applies on body $(1)$ and on body $(2)$

$($i$)$ Construct the $61$ array of forces for the entire system $($do not include gravitational force$)$

$($j$)$ Construct the mass matrix for the system

$($k$)$ Construct a $46$ Jacobian matrix for this system Not with MATLAB:

$(1)$ Write down the EQM

$($m$)$ Can we solve these EQM for the accelerations? Why?