Laboratory 4
Forced Vibration (Experiment)
Summary
This laboratory demonstrates the behavior of a sinusoidally
forced, single degree-of-freedom, spring-mass-damper system. Two di↵erent types of force can be imposed upon the
system: one arising from a base excitation, the other from a
rotating unbalance. Many of the physical parameters of the
system can be changed, and the influence of these changes
on the system’s response can then be studied. The system
is instrumented such that the absolute motion of the mass
and its motion relative to the base can be measured.
Particular attention is given to the phenomenon of resonance, the influence of damping, and the phase relationship
between input and output terms.
4.1
Base Excitation
Consider the system shown in Figure 4.1. The equation of motion governing
the absolute displacement of the mass is given by
mẍ + cẋ + kx = ky + cẏ,
(4.1)
where m is the mass of the system, c is the viscous damping coefficient, and
k is the linear spring sti↵ness. If we define a new coordinate z such that
x
m
k
c y
. .
k (x -y) c (x -y)
Figure 4.1 SDOF Mass-Spring-Dashpot Model with Base Excitation
z = x y, i.e. it is the displacement of the mass relative to the base, then
Equation (4.1) can be rewritten as
mz̈ + cż + kz =
mÿ.
(4.2)
If the base motion is given as y(t) = Y sin !t then Equation (4.2) becomes
z̈ + 2⇣!n ż + !n2 z = Y ! 2 sin !t
where
⇣=
is the damping ratio,
(4.3)
c
cc
p
cc = 2 km
is the critical damping coefficient, and
r
!n =
k
m
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Forced Vibration (Experiment)
is the natural frequency.
The complete solution of this equation comprises two parts: the transient
solution (obtained by solving the homogeneous equation) and the steady
state solution. The transient solution ztr can be shown to be
ztr (t) = X1 e
⇣!n t
sin(!d t +
1)
i↵ ⇣ < 1
where X1 and 1 are evaluated from the system’s initial conditions and !d ,
the damped natural frequency, is
p
!d = !n 1 ⇣ 2 .
It must be emphasized that the constants X1 and 1 are very di↵erent from
the amplitude Z and phase . The latter are fixed by the system’s parameters (see below), whereas the former are found from the initial conditions
of the problem.
The steady state solution zss can be shown to be
zss (t) = Z sin(!t
where
and
Y r2
Z=p
(1
tan
)
r2 )2 + (2⇣r)2
=
with
r=
2⇣r
(1 r2 )
(4.4)
(4.5)
(4.6)
!
.
!n
The complete solution is then
z(t) = ztr (t) + zss (t).
(4.7)
However, if the damping is nonzero, the transient solution will decay to zero
as time increases and we will simply be left with the steady state solution
z(t) = Z sin(!t
).
(4.8)
To obtain the absolute motion of the mass, x(t), we can either solve Equation (4.1) directly or use the relationship that x(t) = z(t) + y(t) to obtain
x(t) = X sin(!t
24
)
(4.9)
where
X=Y
s
tan
=
1 + (2⇣r)2
(1 r2 )2 + (2⇣r)2
(4.10)
2r3 ⇣
.
r2 ) + (2⇣r)2
(4.11)
and
(1
The variation of the steady-state amplitudes Z and X can be plotted as
a function of the forcing frequency. It is convenient to non-dimensionalize
such plots so that they become independent of a system’s parameters. This
is accomplished by plotting Equation (4.4) as Z/Y versus r in Figure 4.2(a)
3
3
ζ = 0 .1
ζ = 0 .1
2.5
2.5
Z
Y
ζ = 0 .2
ζ = 0 .2
X
Y
2
ζ = 0 .3
1.5
2
ζ = 0 .3
1.5
ζ = 0 .4
ζ = 0 .4
1
1
ζ = 0 .5
ζ = 0 .5
0.5
1
0.5
2
3
4
5
1
2
3
r
3
3
2.5
2.5
ζ = 0 .5
φ 2
[rad]
5
ζ = 0 .1
1.5
ζ = 0 .3
ζ = 0 .4
1
4
ζ = 0 .2
ψ 2
[rad]
ζ = 0 .4
ζ = 0 .3
1.5
r
ζ = 0 .5
1
ζ = 0 .2
0.5
0.5
ζ = 0 .1
1
2
(a)
3
r
4
5
1
2
3
r
4
5
(b)
Figure 4.2 Frequency Response Functions for Base Excitation. (a)
Z/Y and the associated phase, . (b) X/Y and the associated phase,
.
and Equation (4.10) as X/Y versus r in Figure 4.2(b), where r = !!n , with
!n constant. The functions are evaluated for various values of damping
ratios, ⇣. The associated phase angles
and
are also shown in these
figures.
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Forced Vibration (Experiment)
4.2
Rotating Unbalance
Unbalance in rotating machines is a very common source of vibration excitation. Consider the system shown in Figure 4.3. Mass m denotes the
x
m
ωt
e
M
k
Figure 4.3
c
SDOF Rotating Unbalance System
unbalance that is rotating at an angular velocity of ! and at a radius of e.
The total mass of the system is M . (Note that the m and M notation is
di↵erent than used in the previous section. However, it is consistent with
the class notes and the text book.) The equation of motion of this system
can be shown to be
M ẍ + cẋ + kx = me! 2 sin !t.
(4.12)
The right hand side of the equation originates from the angular acceleration
of the rotating unbalance in the x direction. As before, it is more convenient
to re-write Equation (4.12) as
ẍ + 2⇣!n ẋ + ! 2 x =
where
⇣=
is the damping ratio,
26
me! 2
sin !t
M
c
cc
p
cc = 2 kM
(4.13)
is the critical damping coefficient, and
r
!n =
k
M
is the natural frequency.
Comparing Equation (4.13) to Equation (4.2), we see that they are of the
same form and so the steady state component of the solution can immediately be written as
x(t) = X sin(!t
),
(4.14)
where
X=
p
M (1
mer2
r2 )2 + (2⇣r)2
(4.15)
and where the phase angle is unchanged and is given by Equation (4.6)
tan
=
2⇣r
.
(1 r2 )
Thus, the frequency response function for this system is the same as that
depicted in Figure 4.2(a), except that Z is replaced by X and Y is replaced
by me/M .
4.3
System Constants and General Hints for the Laboratory Experiment
The following list provides physical data associated with the laboratory experiment. Also, because the system you are going to use can have extremely
low damping (close to zero), transient solutions can last for a great deal of
time. Sometimes there is a need to wait for these transients to decay so that
a steady state amplitude can be observed. One way to speed-up this process
is to increase the damping, thus causing the transient to decay faster. The
damping can then be slowly reduced again. In general, it is good to change
parameter values (speed, damping) slowly and to carefully watch the resulting trends as you make the changes. This way, transient behavior does not
get introduced. Do these trends agree with, for example, the plots shown in
Figure 4.2?
One final point. The control labeled Base Motion Frequency is a multi turn
potentiometer that controls the frequency of the base motion by changing the
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