Consider the production model $x=Cx+d$ for an economy with two sectors$,$ where

$C=\left[\begin{array}{cc}\mathrm{.}0& \mathrm{.}5\\ \mathrm{.}6& \mathrm{.}2\end{array}\right],d=\left[\begin{array}{c}50\\ 30\end{array}\right]$

Use an inverse matrix to determine the production level necessary to satisfy the final demand.

$6.$ Repeat Exercise $5$ with $C=\left[\begin{array}{cc}\mathrm{.}1& \mathrm{.}6\\ \mathrm{.}5& \mathrm{.}2\end{array}\right],$ and $d=\left[\begin{array}{c}18\\ 11\end{array}\right]$

AME $452-015,$ Spring $2023,$ HW$-8$

$8\mathrm{.}1$ Two rods with uniform mass distribution are connected to each other and to the ground by point$-$to$-$point spring and damper elements as shown. The corresponding data are given as: $m_1=0\mathrm{.}4kg,J_1=0\mathrm{.}3kg*m^2,m_2=0\mathrm{.}3kg,J_2=0\mathrm{.}25kg.m^2,k_1=3\mathrm{.}0\frac{N}{m},{?}^0{L}_1=0\mathrm{.}13m,d_{c-1}=1\mathrm{.}5N.\frac{s}{m},k_2=6\mathrm{.}5\frac{N}{m},{?}^0{L}_2=0\mathrm{.}2m,k_3=3\mathrm{.}5\frac{N}{m},{?}^0{L}_3=0\mathrm{.}14m,d_{c-3}=20N.\frac{s}{m}.$ Obtain any necessary geometric data from the figure. Assume that the only non$-$zero velocities are $y_1^{}=-0\mathrm{.}30$ and ${}_2^{}=-0\mathrm{.}15.$

In solving this problem you are asked to take a shortcut. Rather than following the standard procedures for determining the coordinates and velocities of points and vectors, you can determine some of these quantities directly from the figure.

$($a$)$ Determine the $x-y$ coordinates of points $O,Q,A,B,C,$ and $D$ from the figure.

$($b$)$ Determine the $x-y$ components of the following vectors from the figure: vec$(s{)}_1^A,$vec$(s{)}_1^B,$vec$(s{)}_2^C,$vec$(s{)}_2^D$

$($c$)$ Determine the velocity of points A and $D$ based on the given velocities

Now you need to develop a MATLAB program to perform the followings:

$($d$)$ Provide the constant data and all the computed coordinates and velocities from $($a$),($b$)$ and $($c$)$ to the program

$($e$)$ Determine the force of the spring$-$damper element s$-$d$-1$ and then construct the $31$ array of force that this element applies on body $(1)$

$($f$)$ Determine the force of the spring element s$-2$ and then construct two $31$ arrays of force that this element applies on body $(1)$ and on body $(2)$

$($g$)$ Determine the force of the spring$-$damper element s$-$d$-3$ and then construct the $31$ array of force that this element applies on body $(2)$

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

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

$($j$)$ Solve the equations for the accelerations

AME $452-015,$ Spring $2023,$ HW$-8$

$8\mathrm{.}1$ Two rods with uniform mass distribution are connected to each other and to the ground by point$-$to$-$point spring and damper elements as shown. The corresponding data are given as: $m_1=0\mathrm{.}4kg,J_1=0\mathrm{.}3kg*m^2,m_2=0\mathrm{.}3kg,J_2=0\mathrm{.}25kg.m^2,k_1=3\mathrm{.}0\frac{N}{m},{?}^0{L}_1=0\mathrm{.}13m,d_{c-1}=1\mathrm{.}5N.\frac{s}{m},k_2=6\mathrm{.}5\frac{N}{m},{?}^0{L}_2=0\mathrm{.}2m,k_3=3\mathrm{.}5\frac{N}{m},{?}^0{L}_3=0\mathrm{.}14m,d_{c-3}=20N.\frac{s}{m}.$ Obtain any necessary geometric data from the figure. Assume that the only non$-$zero velocities are $y_1^{}=-0\mathrm{.}30$ and ${}_2^{}=-0\mathrm{.}15.$

In solving this problem you are asked to take a shortcut. Rather than following the standard procedures for determining the coordinates and velocities of points and vectors, you can determine some of these quantities directly from the figure.

$($a$)$ Determine the $x-y$ coordinates of points $O,Q,A,B,C,$ and $D$ from the figure.

$($b$)$ Determine the $x-y$ components of the following vectors from the figure: vec$(s{)}_1^A,$vec$(s{)}_1^B,$vec$(s{)}_2^C,$vec$(s{)}_2^D$

$($c$)$ Determine the velocity of points A and $D$ based on the given velocities

Now you need to develop a MATLAB program to perform the followings:

$($d$)$ Provide the constant data and all the computed coordinates and velocities from $($a$),($b$)$ and $($c$)$ to the program

$($e$)$ Determine the force of the spring$-$damper element s$-$d$-1$ and then construct the $31$ array of force that this element applies on body $(1)$

$($f$)$ Determine the force of the spring element s$-2$ and then construct two $31$ arrays of force that this element applies on body $(1)$ and on body $(2)$

$($g$)$ Determine the force of the spring$-$damper element s$-$d$-3$ and then construct the $31$ array of force that this element applies on body $(2)$

$($h$)$ Construct the $61$ array of forces