Problem $3$

Consider the axial structural member shown, comprising of a rod $($ E$=$ const, A$=$const,rho$=$ const$),$ a spring,

and two point$-$masses, submerged in a fluid of viscosity mu$=$ const. The system $($initially at rest$)$ is subjected

to the loads shown below $($Figure $1)$: A statical axially distributed load and two time$-$dependent force$-$

controlled and velocity$-$controlled loads $($Figures $2-3),$ as well. To obtain the transient response of this

system, use the FEM to spatially discretize it and then employ the backward difference method $($with the

given time$-$step Deltat $),$ and obtain the following:

A plot of the displacement, velocity, and acceleration of the roller over time.

A plot of the reaction force at the fixed$-$end over time.

A plot of the axial stress diagram over the whole rod at the instant t$=0\mathrm{.}15$sec$.$

Figure $1$ Physical System

Notes:

Assign the system properties in the following table by yourself $($any convenient non$-$zero values,

along with their units$)$

to the loods shown below $($figuev $1)$: A statical awially distrituted load and two time$-$dependam farce$-$

contrefled and welocity$-$controlled loads $($Figures $2-33,$ as well. To obtain the transient response of this

opthem, une the FEM to spantialy discretive it and then employ the backword digerence method $($with the

given time$-$step all, and obtain the following:

A plot of the displocontent, velocity, and acculerotion of the roller over time.

A plot of the reaction force at the food$-$end ower time.

A plot of the eriol stress dogram over the whole rod at the instant t$=0\mathrm{.}15$sec$.$

Notes:

Assign the system properties in the following table by yourself $($ary cormaniam fan$-$bare vilues,

along with their units$)$

Take the excitation forcing function F$($t$)$ from the graph Below

Take the wibaif$-$contrallad loading fonction VIR$)$ from the graph balow

Show a clear schematic with al necessary lebefts $($whements$,$ nodes, DOF's, coord syoileme, etc.$).$

You cant outline your solution procensis in ipmbofic form first, and then parform all your fumwricar

colculotions is any computing sofhware $($fibes Marlab, Eneel, etc.$)$ of your choice.

$($Piots can be

left in Maelab, Exrel, etc.$]$ Problem $3[75$ Pts$]$

Consider the axial structural member shown, comprising of a rod $(\backslash (\backslash$mathrm$\{$E$\}=\backslash )$ const, $\backslash (\backslash$mathrm$\{$A$\}=\backslash$mathrm$\{$const$\},\backslash$rho$=\backslash )$ const$),$ a spring, and two point$-$masses, submerged in a fluid of viscosity $\backslash (\backslash$mu$=\backslash )$ const. The system $($initially at rest$)$ is subjected to the loads shown below $($Figure $1)$: A statical axially distributed load and two time$-$dependent forcecontrolled and velocity$-$controlled loads $($Figures $2-3),$ as well. To obtain the transient response of this system, use the FEM to spatially discretize it and then employ the backward difference method $($with the given time$-$step $\backslash (\backslash$Delta $\backslash$mathrm$\{$t$\}\backslash )),$ and obtain the following:

$-$ A plot of the displacement, velocity, and acceleration of the roller over time.

$-$ A plot of the reaction force at the fixed$-$end over time.

$-$ A plot of the axial stress diagram over the whole rod at the instant $\backslash (\backslash$mathrm$\{$t$\}=0\mathrm{.}15\backslash$mathrm$\{$sec$\}\backslash ).$

Figure $1$ Physical System

Notes:

$1.$ Assign the system properties in the following table by yourself $($any convenient non$-$zero values, along with their units$)2.$ Take the excitation forcing function $\backslash (\backslash$mathrm$\{$F$\}(\backslash$mathrm$\{$t$\})\backslash )$ from the graph below

$3.$ Take the velocity$-$controlled loading function $\backslash (\backslash$mathrm$\{$V$\}(\backslash$mathbf$\{$t$\})\backslash )$ from the graph below

Figure $2$ Velocity$-$Controlled Load Function

$4.$ Show a clear schematic with all necessary labels $($elements$,$ nodes, DOF's, coord systems, etc.$).$

$5.$ You can outline your solution process in symbolic form first, and then perform all your numerical calculations in any computing software $($like Matlab, Excel, etc.$)$ of your choice.

$6.$ Make sure to submit your spreadsheet or script file along with your exam solutions. $($Plots can be left in Matlab, Excel, etc.$)$