Part I $(50$ pt$)$ We analyze a static mechanical spring$-$mass system with four masses $($similar to the

textbook example$),$ as illustrated in Figure $1.$

$($a$)$

$($b$)$

Figure $1.$ Static mechanical spring$-$mass system

Performing a static force balance on the four masses yields the following system of four linear

algebraic equations:

$(K_1+K_2+K_3+K_4)x_1-K_2{x}_2-K_3{x}_3-K_4{x}_4=W_1$

$-K_2{x}_1+2(K_2+K_3)x_2-K_2{x}_3-K_3{x}_4=W_2$

$-K_3{x}_1-K_2{x}_2+(2K_2+K_3)x_3-K_2{x}_4=W_3$

$-K_4{x}_1-K_3{x}_2-K_2{x}_3+(K_1+K_2+K_3+K_4{)}_4=W_4$

Here, the spring constants and the weights are as below:

$K_1=80,K_2=40,K_3=20,$ and $K_4=10$;$W_1=100,W_2=80,W_3=60,$ and $W_4=100$

You will find the displacements of four weights, $x_1,x_2,x_3,$ and $x_4.$ Present your answers using at

least eight decimal places.

$($a$)$ Convert the system of equations for this problem into the matrix form, i$.$e$.,Ax=b.$ Find

the coefficient matrix A and the nonhomogeneous vector $b.$

$($b$)$ Solve the matrix equation by Cramer's rule using MATLAB. You can find the required

determinants using 'det' command.

$($c$)$ Find the solution $(x_1,x_2,x_3,$ and $x_4)$ using $($i$)$ Gauss Elimination and $($ii$)$ Gauss$-$Jordan

elimination, using MATLAB.

$($d$)$ Factorize the coefficient matrix A into a low triangular matrix L and an upper triangular

$U.$ Show $L$ and $U,$ and find the solution $(x_1,x_2,x_3,$ and $x_4)$ using the matrices $L$ and $U.$

Write a MATLAB code based on the LU algorithm, not using the command $\text{'}$lu$\text{'}.$