Rigid$-$Body Dynamics Project

Instructions

There is a disk of radius $r$ and mass $m$ rotating inside half of a cylinder of radius $R$ and mass $M.$

$M=578$ grams

$m=162$ grams

$R=15\mathrm{.}625cm$

$r=3\mathrm{.}8cm$

Figure $1-$A disk of

radius $r$ rolls inside a

half$-$cylinder of radius

capital$-$R that rolls on

a horizontal surface.

$1.$Determine the center of mass and mass moment of inertia of the half of a cylinder

ANALYTICALLY. Show your work! This is a Statics problem.

$2.$Determine the EOM of just the half$-$cylinder if the disk does not exist. Plot the angular

position for $10$ seconds given an initial condition of $=\frac{}{9}$rad and ${}^{}=0ra\frac{d}{sec}.3.$Using the period of oscillations, what is the natural frequency of this system?

$4.$Determine the EOM of the disk if the cylinder is fixed. Plot the angular position for $10$

seconds given an initial condition of $=\frac{}{9}$rad and $=0ra\frac{d}{sec}.5.$ Using the period of

oscillations, what is the natural frequency of this system?

$6.$Determine the EOMs for the system, where both the cylinder and the disk move.

$7.$Integrate the full EOMs for the system using various initial conditions to show the different

styles of motion that the system exhibits.

a$.$ Initial conditions: $=0$rad,${}^{}=0ra\frac{d}{sec,}=\frac{}{18}$rad,${}^{}=0ra\frac{d}{sec}$

Plot the response for $6$ seconds.

b$.$ Initial conditions: $=\frac{}{18}$rad,${}^{}=0ra\frac{d}{sec,}=\frac{}{9}$rad,${}^{}=0ra\frac{d}{sec}$

Plot the response for $6$ seconds.

c$.$ Initial conditions: $=\frac{}{9}$rad,${}^{}=0ra\frac{d}{sec,}=-\frac{}{9}$rad,${}^{}=0ra\frac{d}{sec}$

Plot the response for $6$ seconds. FOR PLOTTING USE MATLAB PLEASE. THANK YOU!!!