1
S.L. Phoenix 10/15/09
Mechanical Vibrations Summary
(An Alternate View of E&P Section 3.4)
Figure 1.
Simplified description of the mass-spring-damper system where motion is vertical.
Problem description.
We begin with a picture or cartoon (Figure 1.) of the physical system
consisting of a block of mass,
m
, connected to a spring with stiffness,
k
, and a damper or dashpot
with damping constant,
c
, and which moves up and down over time,
t
.
The bottom is fixed to the
ground.
This is called a
mass-spring-damper system
.
There is an equilibrium position about
which all displacements,
x
, are measured as shown in the figure, where the positive direction in
this picture is directed upward. The positive direction for velocity
vd
x
d
t
is also directed
upward.
Analogs of this system occur not only in mechanical engineering and civil engineering,
but also in electrical engineering and other fields where ‘systems’ are important.
By summing forces on the mass and setting them equal to mass times acceleration we obtain the
differential equation for the motion of the mass, which is
2
2
0
dx
d
x
mc
k
x
dt
dt
(
1
)
We also have initial conditions at time
0
t
,
which are
0
0
x
x
(initial position),
0
0
dx
v
dt
(initial velocity)
(2)
and we note that the initial values
0
x
and
0
v
can be positive or negative.
The problem is to
determine how the mass moves over time,
0
t
, when at time
0
t
it is initially lifted or pushed
down distance
0
x
and then released, or it is given an initial velocity
upward or downward
(i.e.,
it is already in motion) or it is given some combination of both.
Goal.
Our goal is to solve the differential equation (1) for all possible initial conditions (2).
However we also to describe the resulting motion,
x t
, in terms of two fundamental parameters

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for the system that give us physical insight into what governs the observed behavior in
engineering practice.
For instance we want to understand what combinations of mass,
m
, spring
stiffness,
k
, and damper resistance,
c
, give the same type of behavior.
A key parameter will be the ‘
undamped’ natural frequency
,
0
km
.
In the special case
where
0
c
, this is the sinusoidal frequency at which the system wants to oscillate up and down.
A second key parameter is the
damping factor
,
cr
cc
, where
2
cr
ck
m
and is called the
critical damping value
. It will turn out that the behavior of the system falls into three main
mathematical categories, depending on the magnitude of the damping factor,
, and whether it
is smaller or larger than one.
The
first category
is most important from an engineering point of view and it occurs when the
damping factor lies in the range
01
.
In this situation the system will exhibit something
called
underdamped vibration
and the mass will oscillate up and down.
When
0
, this
oscillation can theoretically go on forever (though in engineering practice there is always some
damping even if it’s just air resistance).
However, when
00
there will be a decay over
time in the amplitude of the oscillation.

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