School of Aerospace Engineering
Georgia Institute of Technology
AE3515 Matlab Computer Assignment I Due: Thursday 6/17/2004
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The computer assignment below may be done by teams of up to 2 students/team. Each group is
required to submit one report with the contributions of each member clearly specified in the
report.
1) The objective of the this problem is to determine the time response
of a bouncing ball modeled as a mass-spring-damper system shown
in the figure. Throughout the problem assume that m=0.1 Kg, K=10
N/m, and b=0.1 N.sec/m.
a. Derive an expression (by hand or using Matlab) for y(t) and
)(ty
&during the time that mass m is in contact with the spring
starting from initial conditions 0
)0( yy
=
, 0
)0( vy
=
&.
b. Based on your solution to part (a) write a simple Matlab
program that will find y(t) and v(t)=dy/dt of the ball dropped
from height h with zero initial velocity. Hint: You need to first
find y(t) between t=0 and instant t1 at which m makes contact
with the spring-damper. Since the values of y and v are known
at t1, you may use your solution in (a) using the values of x and v at t1 as your initial
conditions until the ball looses contact with the spring-damper and bounces back up to
another maximum height (smaller than h). Continue this process until the final time is
reached.
c. Run your program for h=1m and a final time that is long enough to caputure several
bounces. Plot the resulting position and velocity responses vs. time.
2) In this problem you are asked to design a command shaper to suppress vibrations for a
flexible system (such as a crane, flexible robotic
arm, etc.). For simplicity, the system is modeled
as a mass spring-damper (similar to the system in
problem 1) system driven by a motion source that
provides the input displacement u shown in the
figure. Assume the following numerical values:
m=1 Kg., k=100 N/m, b=1 N.sec/m (lightly
damped).
a. With the aid of Matlab find x(t) starting from
zero initial conditions and a unit step input
u(t)=1(t). Plot x(t) vs. t for 0<t<5 seconds.
b. If you did part (a) right, you will see that x(t) is eventually displaced by 1m but
undergeos a great deal of oscillations. Your mission (should you decide to accept it) is to
find an alternative input to the step input (shaped input command) that displaces x (t) by
1m in about 2 seconds but without any oscillations. One possible choice (not the only
one) for x(t) is a cubic polynomial: x(t)= at3+bt2+ct+d for t≤1 and x(t)=1 for t>1 for a
suitable choice of coefficients a-b. First choose a-d so that x(0)=dx/dt(0)=dx/dt(2)=0 and
x(2)=1 and then find and plot the corresponding u. Physically interpret your results and
comment on its possible real-world applications.
c. To demonstrate the sensitivity (or lack thereof) of input shaping design to parameter
variations, apply the input you found in (b) to a mass spring-damper-system with a
different mass (say 10% lighter or heavier) and the same stiffness and damping values.
k
y
g
m
h
b
kb
m
x
u