EXPERIMENT A3
Vibrations
Summary:
In this lab you will be using a model test bed to perform experiments illustrating the behavior of some
simple vibrating systems. This experiment will provide you with an understanding of some of the basic
principles and equations necessary to understand dynamic systems and data collection. You will also explore
the effects of sampling rate on dynamic data acquisition.
(Note: Portions of the following are taken directly from the manuals provided with the ECP experimental
systems you will be using in this laboratory.)
Before coming to the lab:
Before coming to the lab be sure to read this handout thoroughly and complete the questions
on the pre-lab (preliminary questions) sheet. Also, make sure that at least one calculator is brought to the lab
period to perform calculations during experimentation.
Vibration has been covered in ME 121 and ME 163. Textbooks for these courses are useful references
in preparing for, performing, and writing-up this lab. In addition, in this handout a number of background
sections have been assembled from the equipment manuals to provide a brief review of the physics and
mathematics of vibrations as well as to aid you in understanding the experimental system.
Before beginning work:
Review the basic safety information for the ECP systems, which is given on pages A3-A6, copied
from the manual (pages 31-33 of the rectilinear system manual).
In addition you need to be careful during lab not to change or manipulate the systems in any way
unless specifically instructed to do so.
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Background:
Overview of the rectilinear test system (taken from pages 1, 2, 25, 26 of the rectilinear system
manual)
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NOT
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Theory:
One degree of freedom systems (58-60 of the rectilinear system manual)
Linear Systems
Free Vibration
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 A1  A2 t
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Response to an Applied Step Force
Consider the response of a spring, mass, damper system when the force F(t) in Equation (5.1-1) is a
step function as shown in Figure 5.1-2. The particular solution xp has a form based on the forcing function. So,
for a constant force, expect xp=A, where A is a constant. Find the constant A by substituting this form back in to
Equation 5.1-1. In your prelab, you will derive the total solution x(t)=x c(t)+xp(t) for a step load and zero initial
conditions for underdamped, critically-damped, and overdamped systems.
F
Fo
t
Response to an Applied Harmonic Force (88-89 of the rectilinear system manual)
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Torsional Systems
The most general form of the torsional 1-degree-of-freedom system is shown in Figure 5.1-3. Using the
free-body diagram of the figure and summing torques acting on the mass, we have via the rotational for of
Newton’s second law:
Ý
Ý c 
Ý
J
t  kt  T(t) .
Note that this equation for  has the same form as Equation 5.1-1 for x. Therefore, the system response for a
torsional system and be obtained from the equations derived for the linear system, just by substituting J, c t, kt,
and T for m, c, k, and F, respectively. For example, the steady-state response to a harmonic applied torque is
governed by the equation
Ý
Ý c 
Ý
J
t  kt  To sin( t) .
The steady-state response will be at the same frequency as the excitation, but have a phase lag as given by
 t   sin( t  ) .
From equation 6.4-3 and 6.4-4, the amplitude and phase are given as

To
kt  J   ct 
2 2
2
,
and
 c  
t

  arctan

2 .
kt  J 
Ý
ct
kt
kt
ct
Figure 5.1-3. 1-degree-of-freedom torsional system
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Experimental procedure:
(rectilinear system)
The following procedures are taken from pages 79, 95, and 96 of the rectilinear system manual.
spring constant (soft spring)
= 175 N/m
spring constant (medium spring)
= 400 N/m
spring constant (hard spring)
= 800 N/m
mass (brass weights)
= 500 g each
carriage mass
= 500 g
A)
Step response of damped oscillator
In this experiment, you will obtain the response of rectilinear 1-degree-of-freedom systems to a step (square
wave) pulse applied force. Three systems will be studied: 1) underdamped (=0.1), critically damped ( =1.0),
and overdamped ( =2.5). From the response of the underdamped system, you will determine the natural
frequency and damping ratio and compare them to theoretical values.
1. Verify that the system has 4 brass weights on the first carriage and that the medium spring is attached from
the support structure to the first carriage.
2. Set the damping constant c to the value calculated in your prelab for the underdamped system using
Force+Spring+Damper in the Setup Driving Function dialog box. Set k=0 within the driving function.
Since you have a real spring attached to the system, you do not need the system to simulate a spring.
3. Set up a unidirectional Step input shape of 9 N and 2?? second duration. Set the data acquisition function
to collect Encoder 1 and Drive Input data every 4 servo cycles. Note that a servo period T s=0.0042 s.
4. Execute the step input and plot and save your Encoder 1 Position data. Be sure to choose cm as the output
units for the displacement.
5. Check that the results are in close agreement with your expectations and prelab plots. If they are not, verify
that all settings are correct.
6. For the underdamped system only, change the data acquisition function to collect data every 40?? servo
cycles. Execute the step input and plot your Encoder 1 Position data.
7. Repeat for the damping coefficients corresponding to the critically damped and overdamped systems.
Due to nonlinearities in the drive motor and amplifier, a more ideal response is usually obtained in the return
portion of the forward/return step maneuver because the drive is only supplying the damping and not the
greater step force during this period. Therefore, use the return system response in analyzing the test results,
but be prepared to show how this is dynamically equivalent to the forward (driven) motion in an ideal system.
B) Harmonic response of damped oscillator
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Verify that the system has 4 brass weights on the first carriage and that the medium spring is
attached from the support structure to the first carriage.
(use value of c calculate in the
prelab)
freq
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.
f=2
Not Req’d
Not Req’d
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C)
Step response: effect of mass, stiffness, & sampling rate
1. Carefully remove the 4 brass weights on the first carriage and replace the spring that is attached from the
support structure to the first carriage with the spring with the highest spring constant.
2. Calculate the natural frequency and damping coefficient for =0.05. Determine a step pulse duration that
will last ~ 5-10?? cycles of vibration. Do not apply a force for longer than 5 seconds. Determine a
sampling rate that will be adequate to represent the response.
3. Set the damping constant c to the value calculated using Force+Spring+Damper in the Setup Driving
Function dialog box. Set k=0 within the driving function.
4. Set up a unidirectional Step input shape of 1?? N. Set the duration of the pulse and the sampling rate to
values calculated.
5. Execute the step input. If results are not as you expected, check your calculations for the sampling rate
and time duration of the step pulse. Plot and save your final Encoder 1 Position data.
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(torsional system)
D) Harmonic response of damped oscillator
The following procedures are taken from pages 89 and 90 of the torsional system manual. Note that
you will be performing only part of the experiment outlined in the manual (i.e. starting with step 5) and that the
system will have been put into "Configuration 8" for you. The disk is attached to a torsional spring at its center
point. The other end of the torsional spring is fixed.
centerline distance
brass mass
(2)
brass mass radius
disk moment of inertia
torsional stiffness
d = 7.8 cm
m = 500 g apiece
r = 2.5 cm
Jo = 0.00236 kg m2
kt = 1.37N-m
disk
mass
d
r
1.
Verify that the two brass masses are attached as specified above.
2.
Apply damping to the system via Setup Force + Spring + Damper by setting c equal to the value
calculated for =0.1. Set k=0. Setup a logarithmic sine sweep from 0.5 to 10 Hz,
150 mN-m
amplitude, and 120 second sweep time. Set the data acquisition function to collect Encoder 1 and Drive
Input data every 4?? servo cycles.
3.
Execute the sine sweep maneuver and plot the data using linear frequency horizontal axis scaling and
linear vertical axis scaling with Remove DC bias selected. (This removes any offset from zero of the
low amplitude, high frequency data.) Save your plot. If the centerline of the response plot wanders
excessively during the maneuver, you may wish to run the maneuver a second time.
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Report:
In your report be sure to include copies of all of your results (graphs, etc.) and describe the conclusions
you reached about the system performance. In particular, you should be sure to discuss/comment upon the
following points:
A) Step response of damped oscillator in the rectilinear system:
1) What are the prominent response characteristics of the three damping ratios tested?
Compare with theoretical plots from your prelab (include these plots in your report). What is the
oscillation frequency in the underdamped case? What is the steady state position (the position that the
system settles to in response to the 9 N step force). Compare both of these measurements to
theoretical predictions.
2) For the underdamped case, calculate the damping ratio from the logarithmic decrement and
compare to the value set in the experiment. (Use the response for the return portion of the step, as per
the footnote on page 3-11.)
3) Explain any significant differences between the observed and theoretically
predicted responses.
B) Harmonic response of damped oscillator:
1) Describe the differences between the three damping cases at low, medium (near n), and
high frequencies. Compare the experimentally measured plots to theoretically predicted plots of the
magnitude expression. In particular, discuss the limit of these responses as frequency approaches
zero.
2) For the underdamped case, compare the damping ratio calculated from the quality factor to
that calculated in part A2) above. Also, calculate f n from measuring the frequency at the peak and
2
setting it equal to fn 1  2 .
3) Plot and compare the theoretical and experimental values of phase (use units of degrees)
versus frequency. What is the ideal phase at  = 0+,  = n, and  = -?. Compare your measured
results (Table 6.4-2) with the ideal values at 0.5, 2.0, and 5.0 Hz.
4) Explain any significant differences between the observed and theoretically
predicted responses.
C) Effect of mass, stiffness, and sampling rate:
1) Discuss how you chose the sampling rate and pulse duration used in this experiment. How
would you choose the sampling rate in other experiments? In particular, consider measuring the
response of a machine where the first three natural frequencies f 1=200 Hz, f2=280 Hz, and f3=500Hz
dominate the response.
D) Torsional system:
1) Based on your experimental results, is this system underdamped, critically damped, or
overdamped? Explain. If it is underdamped, calculate the damping coefficient from the quality factor.
Compare theoretical and experimental values for the resonant frequency and damping coefficient. Note
2
that the peak frequency occurs at fn 1  2 .
2) Explain any significant differences between the observed and theoretically
predicted responses.
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EXPERIMENT A3
VIBRATIONS
PRELIMINARY QUESTIONS
Group number (names):_____________________________
Date:______________
1) What are potential safety concerns for this experiment?
2) For an underdamped spring-mass system with viscous damping, write down the equations for

the natural frequency, n, (in terms of the spring constant, k, and mass, m)

the damped natural frequency, d, (in terms of the natural frequency and
damping ratio)

the damping ratio, , (in terms of the spring constant, the mass, and the
coefficient of viscous damping, c)
Also, calculate the damping coefficient c corresponding to =0.1, =1.0, and  =2.5 needed for experiments of
part A (with 4 brass masses on the carriage and the medium spring). Calculate the natural frequency n and
the frequency of oscillation d for the underdamped system
3) Derive the solutions for an underdamped, critically damped, and overdamped system to a step force of
amplitude Fo, assuming the system is initially at rest. Plot (using a program such a Mathematica) the expected
behavior for the underdamped, critically damped, and overdamped systems of experiment A (with 4 brass
masses on the carriage and the medium spring) to a step force of 9 N. (Use a separate sheet of paper for this
question and plot all three responses on one graph.)
4) The dynamic behavior of rotational systems can be analyzed in a manner paralleling to that of rectilinear
systems. Consider the governing equation for a torsional system as
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Ý
Ý c 
Ý
J
t  kt  T(t) .
(Note that I have added a subscript t to emphasize that the torsional damping and stiffness coefficients do not
have the same units as their linear counterparts.) What parameter in a rotational system plays a role analogous
to that of mass in a rectilinear system? Calculate the mass moment of inertia of the system of experiment D
(page A3-16). Use this value to calculate the natural frequency and the damping constant cT for =0.1.
5) If the torsional system described in question 4) is subject to an applied harmonic torque T(t)=T osin(ft), then,
after the transients decay, the steady state solution will oscillate at the forcing frequency but with a phase lag
(t)= osin(ft -). Plot, using a computer program, the amplitude o and phase versus forcing frequency
ff=f/2, for 0<ff<10 Hz and To=150 mN-m. From the amplitude versus frequency plot, use the definition of the
quality factor to calculate the damping coefficient and compare this value to the value used to create the plot.
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