Problem $3\mathrm{.}3$ prob$3\_4.$mlx

set up

W$\}=35$;$\backslash$quad$\%\backslash$mathrm$\{$ weight $[$lb$]$

g $=32\mathrm{.}2*12$; $\%$ Gravitational acceleration $[$in$/$s$^2]$

m$\}=\backslash$textrm$\{$w$\}/\backslash$textrm$\{$g$\}$;$\backslash$quad$\%\backslash$mathrm$\{$ Mass $[$lb$*$s$^2/$in$]$

k $=6\mathrm{.}8$; $\%$ stiffness $[$lb$/$in$]$

x$0=-2$; $\%$ Initial position $[$in$]$

dxe $=6$; $\%$ Initial velocity $[$in$/$s$]$

Find the response analytically

wn $=$ sqrt $($k$/$m$)$;

A$\}=\backslash$operatorname$\{$sqrt$($x$0\}2)$

tA $=$ linspace$(0,3)$;

Find the response numerically

$[tN,zN]=$ ode$45($@trafficLightEOM, $[03],[x0,dx0])$;

Plot

figure

hold on

plot$($tA$,$xA$,$ 'DisplayName', 'Analytical'$)$

plot$($tN$,$zN$,\text{'}$:$\text{'},$ 'LineWidth',$3,$ 'DisplayName','Numerical'$)$

legend$(\text{'}$show$\text{'})$

xlabel$(\text{'}$Time $($s$)\text{'})$

ylabel$(\text{'}$x $($in$)\text{'})$ prob$3\_4.$mlx x trafficLightEOM.m

function dz $=$ trafficLightEOM$($t$,$z$)$

$\%$ First$-$order differential equation for an undamped free system to be used

$\%$ with ode$45.$

$\%$ Inputs:

$\%$ ~ t: the time, a single value

$\%$ ~ z: a vector of position and velocity. The first element is position and

$\%$ the second is velocity.

$\%$ Outputs:

$\%$ ~ dz: the derivative of z

$\%\%$ Set up

W$=35$; $\%$ Weight $[$lb$]$

g $=32\mathrm{.}2*12$; $\%$ Gravitational acceleration $[$in$/$s$^2]$

m$\}=\backslash$textrm$\{$W$\}/\backslash$textrm$\{$g$\}$;$\backslash$quad$\%\backslash$mathrm$\{$ Mass $[$lb$*$s$^2/$in$]$

k $=6\mathrm{.}8$; $\%$ Stiffness $[$lb$/$in$]$

$\%\%$ Equations of motion

z$1=$ z$(1)$; $\%$position

z$2=$z$(2)$; $\%$velocity

dz $=[$z$2$;$-$k$/$m$*$z$1]$

Just after a gust of wind, a $35$ lb traffic light has a position of $-2$ in and a velocity of $6i\frac{n}{s}$ relative to its

rest position. Suppose the equivalent stiffness is $6\mathrm{.}8l\frac{b}{i}n.$

a$)$ Find the response of the system analytically.

b$)$ Suppose a student were to calculate the phase shift $$ using their calculator. Explain why they

should or should not add $$ to the value from the calculator.

Partial answer: a$)x=(2\mathrm{.}12in)sin(8\mathrm{.}66t-1\mathrm{.}24)$

Problem $3\mathrm{.}4$

Download the files prob$3\_4.$mlx and trafficLightEOM.m from Canvas. These files will be used to find the

response of the system in Problem $3\mathrm{.}3$ numerically $($i$.$e$.,$ using ode$45).$

a$)$ There are several errors in the files. Find and fix all the errors. Also briefly explain why the error

is an error $($as if you were explaining it to your friend$).$ Circle, highlight, or otherwise clearly

indicate where you modified the code.

b$)$ Plot of the position of the traffic light with respect to time for both the analytical and numerical

responses. Does your plot seem correct? Why or why not?

Hint $1$: When you download the files, make sure you can see them in the Current Folder window in Matlab.

If you cannot, either move the files to your current folder or change the working folder. See the third video

in Module $3$E for additional details.

Hint $2$: For all problems that involve Matlab, submit the code you used$/$wrote as well as the output. You

can export $.$mlx files as a PDF$.$ You can print m files as a PDF$.$

Partial answer: b$)$

ONLY NEED $3\mathrm{.}4$ COMPLETED. $3\mathrm{.}3$ CONTAINS BACKGROUND INFORMATION AND NUMBERS RELEVANT TO $3\mathrm{.}4$