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ROSE-HULMAN I NSTITUTE OF TECHNOLOGY
Department of Mechanical Engineering
ME 123 Computer Programming
Exercises for Day 35
Exercise 1. A simple robotic arm can be modeled as a pendulum, as in the picture to the right. Let ๐๐ be the angular displacement of the pendulum counterclockwise from the downward position, and denote the angular velocity by ๐๐. There is damping in the pivot, and a force ๐น๐น is applied horizontally to the lumped mass at the end of the pendulum.
The equations governing the behavior of a particular pendulum are as follows:
d๐๐ dt
= โ0.4 ๐๐ โ 20 sin(๐๐) + ๐น๐น cos(๐๐)
d๐๐ dt
where the units for ๐๐ and ๐๐ are rad and rad/s, respectively, and time ๐ก๐ก is in seconds. Suppose the applied force ๐น๐น is such that
๐น๐น = 0 when ๐๐ โฅ 0
๐น๐น = 2 when ๐๐ < 0.
The initial conditions for the pendulum are
๐๐(0) = 4 rad/s
๐๐(0) = 0 rad.
Complete the following:
a) Use MATLAB to plot the pendulumโs angular displacement ๐๐ as a function of time ๐ก๐ก, starting at ๐ก๐ก = 0 s and ending at ๐ก๐ก = 4 s. Use a time step ฮ๐ก๐ก = 0.01 s. Be sure to properly label your axes and to include a title.
b) Determine the values of ๐๐ and ๐๐ at ๐ก๐ก = 1.74 s. Print these values to a text file using the following format:
Angular displacement at t = 1.74 s is X.XXX rad. Angular velocity at t = 1.74 s is X.XXX rad/s.
(over)
ROSE-HULMAN I NSTITUTE OF TECHNOLOGY
Department of Mechanical Engineering
ME 123 Computer Programming
Rabbit
x
y
Dog
Exercise 2.
A dog, whose initial location is on the ๐ฆ๐ฆ-axis, spots a rabbit, initially located on the ๐ฅ๐ฅ-axis. The chase begins, with the rabbit running straight along the ๐ฅ๐ฅ-axis with a speed ๐ฃ๐ฃ๐
๐
. The dog, running at a constant speed of ๐ฃ๐ฃ๐ท๐ท, follows a โcurve of pursuit,โ meaning that he is running straight towards the rabbit from his present position. Your job is to generate the curve of pursuit from the beginning of the chase until the time the dog captures the bunny, or the bunny gets away.
The distance ๐๐ between the dog and rabbit at any given moment is given by
2 2
d = xR โ xD + y D
The dogโs ๐ฅ๐ฅ-position, ๐ฅ๐ฅ๐ท๐ท, varies according to
d xD vD^ ( xR^ xD )
d t d
The dogโs ๐ฆ๐ฆ-position, ๐ฆ๐ฆ๐ท๐ท , varies as follows:
d y D v D yD
d t d
The rabbitโs ๐ฅ๐ฅ-position, ๐ฅ๐ฅ๐
๐
, varies according to
R R.
d x
v
d t
Letโs say that the dog can run at 35 ft/sec for 9 seconds before tiring out. He starts at ๐ฆ๐ฆ = 100 ft. The rabbit starts at ๐ฅ๐ฅ = 100 ft and can run on and on at 20 ft/sec. If the dog gets within ๐๐ = 2 ft, he catches the rabbit. Using the Euler method and a time step of 0.05 seconds, solve this problem. Generate and plot the curve of pursuit. Have your program print out to a text file whether or not the dog captures the rabbit, and if he does, how long it took.
Hint: To test your program, just try giving the rabbit zero speed โ you can figure out what happens! Turn in the plot, your m-files, and the printout of whether the dog caught the rabbit.
