Introduction
to Experiment 3: Vibrations
1.0 Overview:
From jackhammers to buildings,
automotive suspension systems to aircraft gas turbine engines, it is obvious to
even the most casual observer that vibrations are extremely important. In this experiment
you will examine the vibrations of the spring, mass, damper system. In the
experimental investigation you will look at the effect of sinusoidal excitation
on a cantilever system with small damping. The thrust will be to gain an
understanding of the natural frequency of a system. You will vary the
dimensions (length) of the beam, hence affecting its frequency response
characteristics. You will also use a delta impulse function for each case to
determine the system’s damping coefficient. In the simulation portion of this
investigation you will verify the theory behind the experimental investigation
by comparing the system response using a simulated single degree of freedom
system. You will expand your database by repeating the simulation using various
materials, end masses and damping coefficients to quantify the system’s phase
and amplitude response characteristics. This part of the lab provides insight
into how the engineer and scientist can use damping to minimize the negative
effects of vibrations. Before you begin this lab you should review second order
differential equations with constant coefficients. Keep in mind what quantities
in the governing second order differential equation for the spring mass damper
system that you are changing throughout the experiment. This should be
discussed in some detail in the lab write-up.
1.1
Elementary Theory:
1.1.1 A Simple Spring-Mass System
Many
oscillating systems can be modeled as a spring-mass system using the
differential equation of motion. The displacement, y(t), of such systems can be
found using
(1)
where
m is the mass of the object, c is the coefficient of viscous damping, k is the
spring constant, and F(t) is some forcing function. Each term in this expression
is actually a time-dependent force: my,, is the
inertial force, cy, is the frictional or damping force, and ky
is the spring force. Modeling the damping of a system in this way assumes that
the damping force is proportional to the velocity of the mass; this is called
viscous damping.
It
is convenient to express damping non-dimensionally by calculating the viscous
damping factor, , using
(2)
Here,
is the
natural frequency of the system as defined below. For , can be
estimated by
(3)
where
Y1 and Y2 are the values of any two consecutive maximum
displacements from the time response that are one cycle apart.
The
natural frequency of a system is the frequency at which an undamped system will
freely vibrate, and can be calculated by using
(3)
(4)
Elementary Beam Theory and The Solution to
the Governing Differential Equation
1.2 Required Equipment:
- Shaker (single degree of freedom)
- Power supply for shaker
- B & K function generator
- 2 accelerometers
- 3 BNC to SMB coaxial cables
- Carbon steel bar
- Charge Amplifier (power supply for the
accelerometers)
- National Instruments Data Aquisition system with LabVIEW
- NI cDAQ-9172 series controller.
- NI 9234 A/D card with 4 channel, +/- 5
volts, 24 bit IEPE and AC/DC Analog Input Module.
- 24 bit resolution
- Analog low pass filters
- 50 kHz max sampling rate.
- Measuring tape or ruler
- Instrumented Hammer w/ built in
accelerometer
1.3 Experiment Apparatus:
The
apparatus consist of a steel cantilever beam mounted on a single degree of
freedom shaker. Two accelerometers are mounted to the beam, one at the base
(input) and one at the free end (response) of the beam. The charge amplifiers
supply excitation to the transducers. In addition, the base of the beam is
fixed using a removable clamp that can be used to adjust the length of the beam
(which will change the natural frequency of the system).
The
LabVIEW software will be used to acquire the experimental data. The format for
some of the files is found following the "Topics for Discussion".
2.0 Experiment
Procedure:
Measure
the dimensions of the beam for use in calculation of the theoretical natural
frequency. We will be performing this test for three different bar lengths. Be
sure to include the uncertainty associated with the measuring device. Let’s
concentrate on the first bar length for now. Also, note the material of the
beam (most likely carbon steel). The mass of the accelerometer is small and has
little effect on the system’s response. Be sure to measure only the portion of
the beam that will be vibrating. Do not include the portion of the beam located
in the clamp.
2.1 The Shaker Test
In
this experiment, the bars forced frequency is
investigated. When the forcing frequency matches the beam's natural frequency,
resonance is observed.
We
begin by sweeping through several input functions to see where the different
modes of the system’s natural frequency exist. .
- Make sure that the bar is clamped
securely for a given bar length.
- Turn on the accelerometer’s charge
amplifier and wait approximately 5 minutes for the accelerometers to
stabilize.
- Turn on the power supply to the shaker
and power on the input function generator (B & K unit).
- Set the output function on the function
generator to sweep mode. This will increase the input frequency of the
shaker table from 2Hz to 30Hz and back to 2Hz at a constant rate of 1
Hz/second with a given amplitude.
- Record the resonance frequency (s)
observed.
2.2 The Accelerometer Test
In this test,
the natural frequency and damping of the bar’s free response is investigated.
- Ensure that the first accelerometer (at
the beams free end) is connected to channel 1 of the charge amplifier’s
input. Use the coaxial cable to connect the output from the power supply
(channel 1) to channel 0 of the PXI system. Do the same for the second
accelerometer located at the base of the cantilever beam. That is channel
2 of the charge amplifier is connected to channel 1 of the PXI system.
- Open the VI (located on the desktop)
"VIBEXP2013.vi" and make an observation of the various tabs and
controls on the screen. The top plot is the time series response of the
two accelerometers, whereas the bottom two plots are the frequency
spectra of the accelerometers 1 and 2 respectively.
- We will be sampling at 3 KHz, low pass
filtering at approximately 1 KHz, and acquiring 8192 samples per scan.
The "Save Spectra & Time series Data?" and "Save Peak
Data" can be activated and/or deactivated at any time during the experiment.
2.2.1 Recording the Natural
Frequency of the System
We
will be running this experiment for three different beams lengths – 16”,
20”, and 24”. Therefore, be sure to properly name the output file accordingly,
as not to rewrite over previous data.
- From your observations made during the
shaker test, determine the frequency’s incremental spacing that would
allow you to best capture the systems natural frequency. (For example: if
you observed large excitations at 15Hz from the shaker table test, then
perhaps you should choose 6 total increments; that is 2 decrements of 1Hz
each below the 15Hz observation, and 2 increments of 1Hz each above 15
Hz. Therefore the total range over which you are acquiring data is from 12Hz
to 17Hz, with 1Hz increments.)
- Change the Input Function Generator’s
output to Sine Wave and dial the input frequency to your first desired
setting. Be aware that the beam is now vibrating and should not be
touched!
- Activate the "Continue Acquisition" switch to "YES" and run the
"VIBEXPnew.vi" by activating the arrow in the top left corner
of the screen.
- Type in the excitation frequency that
you selected on the B&K function generator device and press the
"Press to Acquire Data"
button.
- Wait for the PXI system to fully
acquire the data before moving to the next incremented frequency.
- Be sure to activate the "Save Peak Data" button.
- When you have moved through all of the
input frequencies, depress the "Continue
Acquisition" button so that is says "NO" and acquire 2
more data points. These last two data points may be disregarded.
2.3 The Impulse Test
Remove
the beam from the clamp and place it in the similar style clamp mounted to the
solid bar located above the shaker. Try at best to preserve the same length
used in the previous study, as you will be trying to compare natural
frequencies between these two investigations. As an impulse, you will strike
the end of the beam with an instrumented hammer. In this part it is best that you practice the
timing between the impulse and the data acquisition system before saving any
data to file as you have only about 2
second window to get things right!
- Turn off the "Save
Peak Data" and the ‘Continue
Acquisition" buttons on the vi interface and activate the "Save Spectra & Time Series Data"
button.
- Acquire the input signal by pressing
the "Press to Acquire Data" button, while immediately impacting
the beam with the impact hammer.
- You will notice that the time series
will appear at a frequency equal to that of the natural frequency of the
system, with additional peaks (seen in the bottom spectra plot) located
at higher frequencies. These higher frequency peaks are the other modes
of the system’s natural frequency. For this lab we will only concern
ourselves with the first mode of vibration.
- Repeat the entire procedure starting
from the section titled "The Shaker Table Test" but with two
different lengths of the beam.
2.4 Exercises
- Calculate the theoretical natural
frequency of the beam for the different lengths used.
- Calculate the damping coefficients for
each of the cantilever systems from the hammer test (delta function).
- Calculate the natural frequency of the
different length beams using the data from the accelerometer test and
compare.
- On the same graph, plot the theoretical
natural frequency vs. mass curve for the cantilever beam system and
all of the experimental data points.
- Be sure to include a full uncertainty
analysis with your results!
2.5 Topics for Discussion
- Describe the system’s response (i.e.
the amplitude and frequency) to different forcing frequencies.
- Compare the various measures of natural
frequency.
- How does the geometry of the system
effect the natural frequency of the system?
- Discuss the type of damping (if any)
that is present in the cantilever.
3.0
Simulation: Experiment validation, Effects of Changing Damping and End Mass
Note: Every time you
save a run two .txt files will be
created. The first will have the system’s response in voltage vs. time and the
second in voltage vs. frequency.
3.1 Experimental Validation
In
this simulation, the results obtained from the experimental part of the lab
will be duplicated and compared. If one is performing this part of the lab
first, it is advisable to first measure the dimensions and take note of the
material types of the beam used in the experimental part of this lab.
- Launch the vi labeled
"VIBSIMv9" from the desktop.
- Run the vi by pressing the arrow
located in the top left corner of the screen.
- Choose the Simulation portion of the lab and press “Continue”.
- Turn
off the "Save Data to File" button so
that it reads "NO"
- Input the required fields as they apply
to the experimental characteristics just performed.
- Length: begin with 16 in (406.4 mm)
and upon completing the simulation do 20 in (508 mm) and 24 in (660.4
mm).
- Width: 1 in (25.4 mm)
- Thickness: 0.25 in (6.35 mm)
- End-weight = 0
- Damping coefficient (c): assume 0.01
- Material type: Carbon Steel
- If the required input fields are
correct, press the "Acquire"
button.
- Toggle between the "Magnitude Ratio"
and "Phase Angle" button located above the bottom graph and
take note of the system’s natural frequency and phase lag, respectively.
This will have more relevance in the next section.
- To continue press the "Continue" and "Continue Acquisition" buttons.
You will have to update the
simulation by running through this routine twice before obtaining
accurate results.
- Activate the "Save Data to File" button and re- "Acquire"
the signal.
- Perform
this for the other two lengths used in the experimental portion of this
lab.
3.2 The Effects of Damping
We
will now quantify the sensitivity of the systems frequency response to varying
damping coefficients.
- Set
the length to 20 in (508 mm).and
the width and thickness to match the geometry of section 3.1. End-weight
should be zero.
- Choose
a damping coefficient of 1.00.
- Choose
Material Type: Carbon Steel
- If the required input fields are
correct, press the "Acquire"
button.
- Toggle between the "Magnitude
Ratio" and "Phase Angle" button located above the bottom
graph and take note of the system’s natural frequency and phase lag,
respectively.
- If the response looks correct, change
the "Save Data to File"
to "YES".
- Repeat
the above procedure for damping coefficient values of 3, 5, 7, and 9.
3.3 The Effects of End Mass and Material Type
We
will now quantify the sensitivity of the systems frequency response to varying
weights applied to the end of the cantilever beam and to the beam’s material.
3.3.1 Effect of End-Mass
- Set the length to 20 in (508 mm) and the width and thickness to match the
geometry of section 3.1. Damping
Coefficient should be .01 (approximately zero).
- Choose
an "End Weight" of 0.25 kg.
- Choose
Material Type: Carbon Steel
- If the required input fields are
correct, press the "Acquire"
button. (You will have to update
the simulation by running through this routine twice before obtaining
accurate results as was previously done in the "Experimental
Validation" section of this lab.)
- Toggle between the "Magnitude
Ratio" and "Phase Angle" button located above the bottom
graph and take note of the system’s natural frequency and phase lag,
respectively.
- If the response looks correct, change
the "Save Data to File"
to "YES".
- Repeat
the above procedure for end masses of 0.40kg and 0.65kg.
3.3.2 Effect of Material Type
- Set
the length to 20 in (508 mm) and
the width and thickness to match the geometry of section 3.1. Use a damping coefficient of .01 and
an end-weight of zero.
- Choose the “Material” to be Carbon
Steel.
- If the required input fields are
correct, press the "Acquire"
button. (You will have to update
the simulation by running through this routine twice before obtaining
accurate results as was previously done in the "Experimental
Validation" section of this lab.)
- Toggle between the "Magnitude
Ratio" and "Phase Angle" button located above the bottom
graph and take note of the system’s natural frequency and phase lag,
respectively.
- If the response looks correct, change
the "Save Data to File"
to "YES".
- Repeat the above procedure for
Stainless steel and Aluminum.
3.4 Exercises
- Calculate the theoretical natural frequency
of the beam for the different materials and end masses used, and compare
to the simulation data.
- On the same graph, plot the system’s
frequency response to different end masses and discuss.
- On the same graph, plot the system’s
frequency response to different damping coefficients and discuss.
- On the same graph, plot the system’s
phase lag / lead to different damping coefficients and discuss. What
effect does the damping coefficient have on the amplitude of the system’s
natural frequency? How may this play a role (critical) in designing
engineering systems? Discuss.