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Prof. Phoenix, 10/21/09
Forced Mechanical Vibrations: A Summary
(An Alternate View of E&P Section 3.6)
Figure 1.
A periodically forced mass-spring-damper system where motion is vertical (left).
What’s left of the brand new Tacoma Narrows Bridge that collapsed in heavy driving winds in
1940 caused by periodic forces stemming from the shedding of vortices (right).
Many engineering systems, when subjected to “shaking forces”, have oscillation behavior that
can be adequately modeled by a mass-spring-damper system subject to a harmonic forcing
function.
Such a system is shown in Figure 1, where the applied force,
0
cos
F
t
, causes the
mass to move up and down relative to its equilibrium position in the absence of a force.
We will
study this system and two variations (applications) that are almost the same mathematically.
The
system itself can be relevant to periodic wind forces as come from “vortex shedding” as caused
the spectacular collapse in 1940 of the then new Tacoma Narrows Bridge.
A variation we will study (see Figure 2 given later) is rotating imbalance in a machine (such as a
refrigerator) or in an automobile tire and the annoying shaking that can result.
The second
variation is the effect of ground disturbances on a building during an earthquake or the effect of a
bumpy “washboard” road on an automobile (see Figure 3 given later).
All of these situations can
result in very serious engineering problems with catastrophic consequences if not understood and
dealt with in design.
So we shall try to explain what goes on from a mathematical point of view.
One important aspect is that the most important behavior from an engineering perspective results
mainly from the particular solution,
p
x
t
, whereas the vibrations considered in E&P Section
3.4 were free vibrations under an initial displacement
0
x
and velocity
0
v
at time
0
t
, and drew
on the complementary solution
c
x
t
.
In most cases of interest in this section, these initial
values are zero (though paradoxically this does not mean there is no transient component when
the unknown constants of the complementary solution are evaluated).
The general solution
combining the two does play a role in the response here too (especially in homework problems)
but only early in time since the complementary solution, now called the transient solution, dies
out fairly rapidly.

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Problem description.
We begin with 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
subject to an applied external force,
0
cos
F
t
, acting on the mass as shown in Figure 1.
The
bottom of the system is fixed to the non-moving ground (though in the last application we
change this assumption).
Over time the mass moves up and down distance
x t
relative to its
equilibrium position
0
x
and this motion is what we are interested in. This system is called a
periodically forced
mass-spring-damper system
.

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