When a system oscillates over a fixed aperture, it can be considered to contain a mass$-$

spring$-$damper set$-$up$.$ This common model is often utilized in engineering applications

and mathematical simulations. Such model can be visualized in the following figure $($Fig

$1).$

Figure $1$: Example Mass Spring Damper.

In this project, you will create a program that takes the mass $($m$),$ damping ratio $($c$),$ spring

constant $($k$),$ and driving force $($F$)$ to find the reactionary motion of the mass over time.

The governing equation is a $2$nd order differential equation $($Eq $1).$

md$2$x

dt$2+$ cdx

dt $+$ kx $=$ F $($t$)(1)$

$1$

Specific Parameters

You are tasked to simulate the following trials and present the results:

Trail No$.$ mass $($kg$)$ k $($N$/$m$)$ c $($N$$s$/$m$)$

$132002$

$245045$

$3512550$

For this project, we are simulating the homogeneous response, where the forcing func$-$

tion F $($t$)=0.$ The initial position is always x$0=1$ m$.$

Required Function

Develop a function called:

function $[$xkp$1,$vkp$1]=$ VibrationPosition$($xk$,$vk$,$m$,$k$,$c$,$f$,$dt$,$type$)$

where you would approximate the position of a mass$-$spring$-$damper system from the

second order partial differential equation for an individual time$-$step. The input values for

the functions are:

xk$,$vk $-$ Iteration $($k$)[$Position$,$ Velocity$]$

m$,$k$,$c $-$ Model Parameters $($mass$,$ spring constant, damping constant$)$

f $-$ Forcing Function

dt $-$ Time$-$step

type $-$ Type of differentiation $(1-$ Forward Euler or $2-$ Runge Kutta$)$

$[$xkp$1,$ vkp$1]-$ Iteration $($k$+1)[$Position$,$ Velocity$]$

Incorporate your function into the following main topics:

$1.$ Forward Euler numeric scheme

Utilizing Forward Euler numeric schemes, e$.$g$.$ given dy

dt $=$ f $($t$,$ y$)$

df

dt $=$ f $($tk$+1,$ y$)$ f $($tk$,$ y$)$

$$t $(2)$

and using the given listed parameters:

$($a$)$ Solve for the homogeneous response of the mass$-$spring$-$damper ODE$$s and

plot the position of the mass vs$.$ time.

$2$

Hint: You should break down the second order differential equation into two first

order equations in order to apply the numeric schemes. I.e$.$ let v$($t$)=$ dx$($t$)$

dt $,$ then

d$2$x$($t$)$

dt$2=$ f $($t$)$ becomes:

dv$($t$)$

dt $=$ f $($t$)$

dx$($t$)$

dt $=$ v$($t$)$

$(3)$

$2.4$th Order Runge$-$Kutta

Utilizing $4$th Order Runge$-$Kutta numeric schemes, e$.$g$.$ given dx

dt $=$ f $($t$,$ x$)$

c$1=$tf $($tk$,$ xk$)$

c$2=$tf

$($

tk $+1$

$2$t$,$ xk $+1$

$2$c$1$

$)$

c$3=$tf

$($

tk $+1$

$2$t$,$ xk $+1$

$2$c$2$

$)$

c$4=$tf $($tk $+$t$,$ xk $+$ c$3)$

xk$+1=$ xk $+1$

$6$c$1+1$

$3$c$2+1$

$3$c$3+1$

$6$c$4$

$(4)$

and using the given listed parameters:

$($a$)$ Solve for the homogeneous response of the mass$-$spring$-$damper ODE$$s and

plot the position of the mass vs$.$ time.

$3.$ Animation of Time Evolution

Generate a video of the mass position as it evolves over time for a mass$-$spring$-$

damper system for each trial. Use the following timestep for your output frame.

Initial Time $($t$0)0$

Total Time $($tf$)10$ seconds

Timestep $($dt$)1/300$ seconds

Frames per Second $($fps$)30$ frames per second

Note: Make sure ALL your plots contain the necessary title$($s$),$ legend$($s$),$ and axis$-$

label$($s$).$ You need to submit three $10$ second videos along with your script on

Canvas. Use the naming convention $$LastName ID video #$$ where # is $1,2,$ or $3.$

$3$

$4.$ Extra Credit $(+20\%)$: Inhomogeneous Response

$$ For each of the three trials, also solve the inhomogeneous response, using

the forcing function F $($t$)=$ a$0$ sin $($t$/2\backslash$pi $),$ where a$0=25$ N$.$

$$ Solve for the inhomogeneous response of the mass$-$spring$-$damper ODE$$s and

plot the position of the mass vs$.$ time for each trial.

$$ Then change the three video animation to plot both responses. Each video

should contain two plots side$-$by$-$side, with the left subplot being the homoge$-$

neous free response, and the right subplot being the inhomogeneous forced

response. Use the same timestep as step $3.$

Hint: set f in the function input to an array: $[$f $($t$0),$f $($t$0+1$

$2$ dt$),$f $($t$0+$ dt$)]$

Note: If you completed the extra credit, you may replace the video submission in

part $3$ with the three videos in part $4$ instead.