Mechanical
Engineering Projects
The purpose of these projects
is to relate linear algebra concepts to mechanical vibrations, especially
resonance. While they use differential
equations, they do not assume any knowledge of
differential equations on your part, only calculus 1 and 2.
Project #1 Forced vibrations
due Tuesday Nov 5
Your
group should turn in a paper which covers mass-spring systems without damping
but with an optional external force. After
Newton’s Second Law is applied, this
amounts to a differential equation of the form
mx’’ +
kx = f(t)
where
x
stands for displacement from rest and t is time. The external force is f(t), the
mass m
and the spring constant k.
The ‘
indicates a derivative with respect to time.
(the word “show” is where work by
you is required)
Case #1: homogeneous. Assume no external force f(t)
Show
that solutions to this are cos(w0t) and
sin(w0t)
where w0 =
sqrt(k/m) is called the “natural
frequency”
Show
that this problem is linear by showing that for
any constant c that
1) Ccos(w0t) and
Csin(w0t) are also
solutions
2) C1 cos(w0t) + C2
sin(w0t) is also a
solution
Question: in physics and mechanics, what is the Principle
of Superposition?
Case #2: nonhomogeneous. External force f(t) present.
Note: this part requires only careful algebra
and differential calculus.
Assume an external, oscillating force f(t) = cos(wt) and that w ¹ w0, above.
Show that
a particular solution is
x(t) = cos(wt)/(m(w02
– w2))
Having
done this, show that the general solution to the problem is
x(t) = C1 cos(w0t) + C2 sin(w0t) + cos(wt)/(m(w02
– w2)) (c1 and c2 arbitrary)
Next,
the solution must also satisfy initial
conditions:
x(0)
= 0 (initially at the origin)
x’(0)
= 0 (initially at rest – no
velocity)
Pick C1 and C2 so that
these are satisfied
Turn
in a paper with a cover page listing all group members, the group number, and
project number and title. The body may be a mix
of typing of
text and hand printing of equations. List any resources (books, web, people).
x(t) = (cos(wt)
- cos(w0t) )/(m(w02
– w2))
This is the basis for the second project.
First, consider the trig identities
cos(A+B) = cos(A)cos(B) –
sin(A)sin(B)
and cos(A-B) = cos(A)cos(B)
+ sin(A)sin(B)
Use them to write the solution from part 3 into one
single term, instead of 2. (hint: use the substitution A = (w0 + w)/2
and B = (w0 – w)/2
and do some algebra) I will be happy to tell you when you have it right!
5. To make things
interesting, lets try out some specific values for the frequencies. Take w0 = 2
and w = 2.1. What is significant here is
that they are relatively close to one another (remember that w0 is
the natural frequency and w the external or forcing frequency)
Graph the solution (easy in Maple with the plot function) and describe what you see in
general terms. (suggestion t= 0 to 20 Pi or so
to begin to get a sense of what’s going on).
The name for this phenomena is a beat.
6.
Resonance: this happens when the natural and forcing
frequencies match. We will find out what happens by considering w0 fixed and taking the limit as w -> w0. Note that this results in an Indeterminate Form (remember those
from Calc III??). Use LHopital’s Rule and
show that the solution becomes
t sin(w0t)/(2mw0)
(the t in front
is significant!)
7.
Graph the resonant
solution and describe what you see
Project #3: Examples involving Resonance due Tuesday Dec 3
The basis for this project is two video tapes (available
in the library on reserve ) as well as a talk on resonance
by Professor Dimentberg of the Mechanical Engineering
Department. Your paper should
a)
summarize his talk
b)
explain how resonance takes place in both of the physical
situations videotaped