ME 163 Vibrations
Lecture 1
Why study vibrations?
In engineering, most commonly we want to
suppress them.
Although, as the video shown in class indicates, we
can use our knowledge to produce some unusual effects with cars. You
wouldn’t want to buy a car that at 45 mph produces a howling sound
as mine currently does (and I am not selling cause I just need some
rubber inserts and the problem is fixed).
Car suspension
Figure 1: Car suspension diagram
In order to fix the problems more complicated than that, we need
to use some serious modeling, where we take just the essentials of the
vehicle description, and turn them into a set of equations, by utiliz
ing Newtonian mechanics.
In the car suspension figure 1 the basic
elements of the system are indicated, and we reduce analysis to just
four coordinates:
y
1
,
y
2
,
y
c
and
θ
, describing the position of the front
1
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wheel, rear wheel, center of mass, and the angular orientation of the car
with respect to the ground (
θ
is usually small, but not in that video...).
Then we solve these equations. Some of them we can solve analytically.
Having an analytical solution provides us with a lot of good intuition
about the problem at hand. Unfortunately, analytical solutions are not
always available. This is why we will make heavy use of software like
MATLAB in this class, to solve equations and evaluate the solutions
from vibrations engineering perspective.
Wave energy harvesting
In the video you saw in class, it becomes clear how complicated the
vibration problems can become when you are designing a device that
captures wave energy for electricity production. Sophisticated software
tools and vibration concepts are deployed. By the end of this class you
should be able to understand and perform much of the analysis that
goes into building a vibrationenergy capturing device. Of course, step
one is to be able to write the model of the ensuing motion. Recall from
dynamics: we do this using Newton’s laws (later in this class, we will
learn another approach , termed Lagrangian mechanics, that simplifies
this task somewhat at the expense of learning some advanced theory).
Newton’s law of motion
Newton told us that change in time of linear momentum of a body
equals the force applied to it. If mass is constant, this statement sim
plifies to ”mass times acceleration equals force”:
m
a
=
F
(
x
, t
)
(assuming mass
m
does not change in time
t
, and its position vector
x
whose acceleration is
a
= ¨
x
moves in 3D space under influence of force
F
). Being able to derive this equation in various coordinate systems
(Cartesian, polar in particular) using the freebody diagram approach
is the key background for this class. Remember that the force
F
needs
to be modeled.
The law of gravity tells you the expression for that
force in the case of two massive bodies at distance
r
from each other.
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 Winter '08
 Mezic,I
 Frequency, Natural Frequency, Complex number

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