The double spring

A double spring consists of two springs, attached end$-$to$-$end, each with a point mass on the

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For this project, we will consider only the case where the two point masses are equal, $m,$ and

the springs are of equal rest length, $l$ and have equal spring constants, $k.$ In this case, the

motion of the double spring is governed by the following set of coupled ODEs:

$z_1^{}=-\frac{k}{m}(z_1-l)+\frac{k}{m}(z_2-z_1-l)$

$z_2^{}=-\frac{k}{m}(z_2-z_1-l)$

where $z_1$ and $z_2$ are the positions of the two masses. This can be rewritten as

$z_1^{}=-2{}^2{z}_1+{}^2{z}_2$

$z_2^{}={}^2{z}_1-{}^2{z}_2+{}^{2}l$

where $=\sqrt[\phantom{2}]{\frac{k}{m}}.$

Question $2$ Write this system of equations in matrix form: $Z^{\text{'}\text{'}}=-{}^{2}AZ+{}^{2}L,$ where $Z=$

$(z_1,z_2{)}^T.$

For Questions $3-8$ and Tasks $1-4,$ we will assume the rest length of each spring is zero. In

this case, $l=0,$ so that $L=0$ and the matrix equation is homogeneous, i$.$e$.Z^{\text{'}\text{'}}=-{}^{2}AZ.$

$($Although this is unrealistic, it will give us a good starting point.$)$

Since the point masses will oscillate back and forth, we should expect the solutions to this

system of equations to contain oscillatory functions.

Question $3$ Both $Z=xcos(\sqrt[\phantom{2}]{}t)$ and $Z=xsin(\sqrt[\phantom{2}]{}t),$ where $x=(x_1,x_2{)}^T,$ are solutions

to the system of equations given by $(4).$ By substituting each solution into the matrix equation

from Q$2($with $L=0),$ state what relationship $x$ and $$ have to $A.$

Let the point masses have mass $m=1kg,$ and let the springs have stiffness $k=2\frac{N}{m}.$Task $1$ In your MATLAB script file, write the code required to find the eigenvalues and

$\{$:${}_2)$ and the corresponding eigenvectors and $x_2)$ for the matrix $A.$ Use the corresponding

section of the 'skeleton' MATLAB script at the end of this assignment. Run your script file to

find ${}_1,{}_2,x_1$ and $x_2.$ Write your answers in your report.

The general solution of the system of equations $(4)$ will be a linear combination of all of the

solutions described above.

Question $4$ Write the general solution in terms of the eigenvalues and eigenvectors of the

matrix $A.$

Hint: The system of equations $(4)$ consists of two second order linear ODEs. This means

that we should expect the general solution to have four arbitrary constants, that is$,$ the general

solution should be a linear combination of four solutions. Use $c_1,c_2,c_3,c_4$ for the arbitrary

constants.

The values of $c_1,c_2,c_3,c_4$ will depend on the initial conditions of the system, that is$,$ the specific

initial value problem. We will consider the following specific initial value problem:

$z_1(0)=1m,z_2(0)=5m$

$z_1^{}(0)=0m{s}^{-1},z_2^{}(0)=0m{s}^{-1}$

Question $5$ Give a physical interpretation of the initial conditions listed above in $(5).$

Question $6$ To determine the unknown constants for any initial value problem, we can solve

a matrix equation of the form $Bc=d$ where $c=[c_1,c_2,c_3,c_4{]}^T.$ Write down $B$ and $d.$

Task $2$ Then, in your MATLAB script file, write the code required to find $c_1,c_2,c_3,c_4$ for the

specific initial value problem described above. Be sure, though, to set it up so that you can easily

change the initial value problem.

Our solution, $Z,$ describes the positions of the point masses at a particular time $t.$

Task $3$ Fill in the necessary MatLaB code required to calculate the positions of the masses at

times $0t40$ seconds $($use a stepsize of $0\mathrm{.}05$s$).$

To create the animation, we use the for loop within the skeleton MATLAB code provided. Copy

this to your MATLAB script file. Run the animation $-$ this will create MATLAB Figure $1.$