Problem setup: To understand the behavior of structures under the influence of disturbances such as strong

winds, earthquakes, etc. engineers often model them as mass$-$spring systems. The figure below shows such a repre$-$

sentation for a three$-$story building.

Considering only the horizontal motion of the structure, Newton's second law gives:

$m_1\frac{d^2{x}_1}{dt}=-k_1{x}_1+k_2(x_2-x_1)$

$m_2\frac{d^2{x}_2}{dt}=-k_2(x_1-x_2)+k_3(x_3-x_2),$

$m_3\frac{d^2{x}_3}{dt}=k_3(x_2-x_3)$

where $x_1,x_2,x_3$ are the displacements of each floor.

In this project, you will solve these equations using PYTHON to simulate the dynamics of the structure, going from

$t=0$ to $20$ seconds. The initial values of displacements and velocities are all zero, except the velocity of first floor is

$d\frac{x_1}{d}t=1\frac{m}{s}.$

Method: To solve the equations, you will use the two step midpoint method taught in class, which for a system

of equations: $d\frac{x}{d}t=F(x,t),$ is implemented as:

$x_{i+\frac{1}{2}}=x_i+\frac{h}{2}F(x_i,t_i)$

$x_{i+1}=x_i+hF(x_{i+\frac{1}{2}},t_{i+\frac{1}{2}})$

where $h$ is the step size.

Analysis: You will consider three different step sizes of $h=1,0\mathrm{.}5,0\mathrm{.}25$ to solve the equations. For each case,

analyze the behavior of all displacements and velocities as a function of time. In addition to that, also analyze if and

how the results change as you reduce the step size. Some portion of the score for the project will be awarded based

on the creativity of the analysis. For instance, you can calculate the relative error of the solution as the step size is

decreased to assess convergence.

How to answer: $1.$ The PYTHON code which solves the equations and writes out the solution. The entire source code, with all functions, variables, etc. A text$/$data file containing the solution for the case h $=0\mathrm{.}5.$ The file should contain $8$ columns, viz. the step number, time, the three displacements and the three velocities. Write the columns in a well formatted manner discuss the various cases and results, with appropriate figures. You are free to present the analysis in the manner you want, but the answer response should not be longer than $4$ pages $($standard formatting$),$ and you are allowed a maximum of $4$ figures. Your figures should study and analyze the behavior of various displacements and velocities as a function of time, also comparing the effect of different step sizes in a meaningful way with some kind of error analysis. Essentially, you have to present your findings in a clear and concise manner, but at the same time ensure all the important and relevant results are conveyed clearly. The answer response should end with a conclusions section $($of about half a page$),$ which should summarize your analysis and key findings.