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University of California, Irvine
1
MAE 106 Laboratory Exercise #4

Vibration I: Lightly Damped Second Order Systems

University of California, Irvine
Department of Mechanical and Aerospace Engineering
Required Parts:
QTY PART


Oscilloscope

Trainer
Kit

EQUIPMENT
Vibrating beam experiment fixture

BNC to Alligator Clip Breakout
Accelerometer

BNC Cable
Accelerometer Amplifier

Strobe Light
24V DC Power Supply



1 Introduction


In this laboratory exercise you will look at the dynamic response of a cantilevered beam that
supports a motor with an unbalanced load attached at its end. If you use your imagination, this
system represents many typical problems in vibrations. For instance, a large rotating machine
attached to a building floor can be analyzed in the same manner as this experiment, with the floor
taking the place of the cantilevered beam. Alternatively, a building shaking during an earthquake
can also be represented by the same equations, with the building acting as the beam itself.

This lab is also the first lab that deals with a second order system. In other words the differential
equation that describes the system has second derivatives, and the transfer function has s
2
terms.
Many mechanical systems behave as second order systems because of Newtons second law (F =
ma) because acceleration is the second derivative of position. In fact, you can view a vast number
of mechanical and control systems as second order linear mass, spring, damper systems. Thus,
developing intuition about how second-order systems behave is very important. One key
difference between second and first order systems, as you will see in this lab, is that second order
systems can oscillate. First order systems cannot oscillate.


Instrum. Amp
(for unpainted
fixtures)

Scope
DC Supply
Variable
voltage
from trainer
kit
A
L


Figure 1 - Vibrating Beam Fixture:
Be gentle with the beam fixture. The accelerometer can
be easily damaged by impulsive forces. Also, place the beam fixture on the floor, being careful to
not to pull any short wires.
2
2 Time Domain Analysis (Transient Response)


In this part of the lab, you will measure how the beam responds to an impulsive input. This is
known as the transient response, and more specifically, as the impulse response of the beam,
and is a typical way to look at the time-domain response.
An accelerometer is mounted at approximately the center of mass of the vibrating load at
the beam end. For all the subsequent analysis, assume that the length of the beam is from the
clamped end to the center of the accelerometer. The following analysis also assumes that the
acceleration of the beam is a good measure of its position. The reason for this assumption is that
the equation of motion of the unforced system has the form
0
.
..
=
+
+
kx
x
c
x
m
,
and if c
0, then
x
m
k
x
)
/
(
.. =
.

Q1 Compute the theoretical natural frequency of the system. The motor weighs about 2.25 lb.,
the beam is made of carbon steel of dimensions 0.125 in. by 2.0 in. in cross-section. You
need to measure the length of your beam as the distance from the base to the center of the
accelerometer. (You can do the calculations at home but be sure to measure the length of
your beam, since they are all different!)

For the unpainted vibrating beam fixtures: Connect the accelerometer to the instrumentation
amplifier. Attach the amplifier output to the oscilloscope and set the amplifier gain to x5.

Q2 You need to calibrate the accelerometer to make sense of its output. Calibrating a sensor
refers to the process of measuring what voltage corresponds to what level of the measured
variable. For the accelerometer, you need to know how the output voltage and acceleration
correspond. You can use gravity as your known acceleration, and measure the voltage
output corresponding to gravity. Set the zero voltage adjustment on the instrumentation
amplifier to give zero volts on the oscilloscope. Then rotate the entire apparatus on its side
so that the accelerometer reads the acceleration of gravity (1 g). Report the accelerometer
output voltage corresponding to 1 g. You are now able to calculate actual acceleration by
measuring the accelerometer voltage. Give an example of how you would do this. What
assumptions are you making about the accelerometer and amplifier?

Q3 Twang the beam with your hand and set the oscilloscope so that a nice periodic waveform
appears on the screen. Report the frequency of vibration (both in rad/sec and Hz). Use the
stop button on the scope for this.

Q4 Estimate the damping using the logarithmic decrement method. Use the stop button on the
scope and a slow sweep rate to obtain a good scope trace. After twanging the structure,
measure the initial amplitude, the number of cycles, and the final amplitude. Repeat this
process a few times to be certain of your measurement. Hint: use Roll (horizontal mode)
and storage mode of the scope. Report you estimated value of
using both the exact and
approximate formulas in Equation 3 of the notes.

P1 Using the LabJack, record the impulse response of the beam. You will turn this plot in.

3
3 Frequency Domain Analysis (Forced Response)


In this part of the lab, you will determine how the beam responds when you apply sinusoidal forces
to it at different frequencies. A key phenomenon that you will observe is resonance. Resonance is
the increase in amplitude of oscillation of an electric or mechanical system exposed to a periodic
force whose frequency is equal or very close to the undamped natural frequency of the system.

You will use a motor with an off-balance load to apply the sinusoidal forces to the beam. The
motor is driven by a high gain, velocity control system. The voltage into the amplifier and controller
produces an angular velocity of the motor with a gain of about 300 rpm/volt (but you need to
measure this to find the exact value). So if you input zero volts (a short) to the amplifier, you should
get zero rpm out, while a 2 volt input would give 600 rpm out. Use the trainer kit to provide a
variable voltage input to the motor velocity control system.

Q5 It is important to know the relation of the motors velocity to input voltage accurately for the
system (i.e. to calibrate the motor). Using the strobe light and an input voltage of 2 volts,
determine the actual rpm of the motor. Be sure to hold the beam so that it does not vibrate
much. Repeat the measurement with an input voltage of 4 volts. Based on these
measurements, what is your estimate of the gain that relates voltage to velocity? State your
answer in rpm/volt and in (rad/sec)/volt. Now that youve calibrated the system, you can
estimate the motor velocity by measuring input voltage. Conversely, you can adjust motor
velocity by adjusting the input voltage.

Q6 Try to estimate the natural frequency, n
, for the system by getting it to resonate. The motor
angular velocity corresponding to the maximum amplitude response gives the resonant
frequency ( r
= n
). Try not to let the system shake too badly, i.e. do not let the system stay
in resonance too long. Record the accelerometer voltage amplitude at resonance. What is
the corresponding maximum acceleration? Also, estimate the amplitude of the tip beam
motion in inches at resonance with a ruler as shown in Figure 1.

Q7 Increase the voltage to the motor controller so that you are exciting the system well past its
resonant frequency, but not to the point where new, higher frequency resonance is occurring
(you can hear other things begin to shake). Record this input voltage. As above, record the
accelerometer amplitude voltage and estimate the amplitude of the beam tip motion, A
high
,
with a ruler by eye.

PRACTICAL EXAM: Demonstrate to the TA that you can drive your beam into resonance.

3
Problems to Consider at Home
You are now done with the experimental part of the lab. The rest is analytical.

Q8 It was shown in lecture that the forcing function on the mass has the form
)
sin(
)
(
2
t
a
t
f
=
.

Therefore, the output must be of the form
)
sin(
)
(
2
+
t
j
G
a
, (1)

where

2
2
2
2
)
(
n
n
n
s
s
w
s
G
+
+
=
.
4

When the beam is vibrating near resonance,
)
2
/(
1
)
(
=
n
j
G
, and the amplitude of the
output wave (that you measured with a ruler) is
2
2
n
res
a
A
=
.

In Q12, below you will prove that at higher frequencies, as ,
2
2
)
(
n
a
j
G
a
.
Therefore, the amplitude of the output wave (also measured with a ruler) is
2
n
high
a
A =
.
From these two facts, the damping ratio
can be estimated as
res
high
A
A
2
. (2)
Using this formula and the amplitudes measured by eye in Q6 and Q7, estimate the damping
ratio,
.

Q9 Prove that the resonant frequency r
which is the frequency of maximum output vibration
amplitude, is given by
707
.
0
,
2
1
2
=
n
r
.
Q10 Show that
2
1
2
1
)
(
=
r
j
G
.
Q11