Department of Mechanical Engineering
ME 260W - Measurement Techniques
Spring 2005
Experiment #4: Beam Vibrations
1.
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OBJECTIVES
To determine the first three natural frequencies and mode shapes of a cantilever beam
To characterize a second-order system by it's damping frequency, and damping coefficient
To find the frequency response of a cantilever under sinusoidal excitation
2. EQUIPMENT
For this lab, we will use:
 Power amplifier, CE 2000
 Shaker, Model, VTS 100
 Strain gage conditioner, P-3500
 Digital storage oscilloscope, Tektronix TDS 210
 Accelerometer transducer PCB 482A22
 Signal conditioner, PCB 488A-01
 Computer station with data acquisition A/D board
3. THEORETICAL ANALYSIS
A shaker is driven, with displacement y, to displace the fixed end of a cantilever beam and
impart harmonic motion in the beam. Mounting two strain gages on the beam as shown in
Figure 1 allows the measurement of maximum strain in the beam as the end of the beam
oscillates in x direction. With virtual instrumentation, the computer operator can control the
motion of the shaker and simultaneously record data from the strain gages. This data can be
examined with two different approaches, through determination of the damping ratio and
determination of the deflection of the end of the beam.
Two metallic resistance type strain gages are used in conjunction with a half Wheatstone
bridge as shown in Figure 2. Briefly, the gages are mounted on a specimen and under pre-load
conditions; the balance potentiometer is adjusted such that eo is zero. Strain in the specimen
elongates the strain gage, altering the electrical resistance in the gage. This change in the gage
resistance unbalances the bridge, and results in a voltage at eo. This voltage at eo is proportional
to the strain. Using two gages in this configuration results in a doubled bridge output, as
compared to using a single gage, and also compensates for temperature effects, and torsional and
axial components. As the beam vibrates with harmonic motion, the output from the strain gages
is a sine wave with amplitude proportional to the strain and a period inversely proportional to the
frequency of the vibrations.
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Figure 1 Cantilever Beam Setup. Strain gages are mounted on the top and bottom of the beam.
The motion of the oscillator is y(t) and the deflection on the end of the beam is x(t).
The width of the beam is b, and the thickness is t.
Figure 2
Wheatstone Bridge. A half bridge with two strain gages. After the bridge is balanced
such that voltage across eo is zero, strain in gages results in voltage across eo.
By driving the shaker with a sine wave, the aluminum beam vibrates with simple harmonic
motion. Left to vibrate freely, without applied external forces, the beam will vibrate at its natural
frequency, n, and its amplitude of the response will decrease with time, as energy in the system
is lost. The rate at which this amplitude decreases is known as the logarithmic decrement. The
logarithmic decrement,, can be obtained by measuring two displacements separated by n
1 x 
number of complete cycles and applying:   ln  o  with x o and x n the amplitudes
n  xn 

separated by n cycles. From , the damping ratio, , can be found from:  
. For our system
2
with the displacement of the shaker, y, it can be shown that the deflection, x is approximated by:
2
x

y
1  2r 
2
1  r   2r 
2 2
2
with r 

. After impacting the beam, and allowing it to vibrate
n
freely, the measurement will determine the natural frequency of the beam, logarithmic decrement
and damping ratio of the beam. Along with the driving displacement, y, this data can be used to
determine the deflection, x, at the end of the beam for a given frequency.
The deflection in the beam can also be determined by measurements of strain, , in the beam.
The maximum deflection of a cantilever beam with a point load on the end of the beam
PL3
is: x 
with P, the load on the beam, L, the length of the beam, E, the modulus of elasticity
3EI
for the material, and I, the second moment of area of the beam. The maximum strain in a
Mc
t
cantilever beam is:  
, ( c  with t the thickness of the beam). From these two
EI
2
L2
equations, and with M  PL we can obtain the deflection, x, with: x 
. The computer
3c
station drives the oscillator at a series of determined frequencies and records the maximum strain
at each frequency. With the user input of data from the Beam Data VI and measurements of the
beam, along with the acquired driving displacement, the Frequency Data VI calculates the
deflection at the end of the beam and the magnitude ratios, (x/y), for each frequency, based on
the measured strain and based on the calculated damping ratio. The block
diagram for the Frequency Data VI is shown in Figure 4. We now can compare the deflection of
the end of the beam based on , n, and , with the deflection based on the measured strain in
the beam.
The resonant frequencies for a simple cantilever beam is given by
C
EI
n  n
2 L4
 n = natural frequency in cycles per second
E = modulus of elasticity of the material in psi
I = moment of inertia in inches
L = length in inches
 = mass per unit length
Cn = coefficient for the different resonant modes
(C1 = 3.52, C2 = 22.4, C3 = 61.7)
4. EXPERIMENTAL PROCEDURES
In this exercise, you are asked to determine the natural frequency, the damping coefficient,
phase angle, amplitude response and mode shapes for a cantilever beam.
In this experiment, virtual instruments created with LabVIEW software are used to
investigate a cantilever beam subject to forced harmonic vibration. The Freevib1 VI determines
the harmonic nature of the beam, and the Freqresponse VI drives the beam specified frequencies,
and at each frequency, determines the displacement at the end of the beam based on acquired
data and harmonic data.
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In this test, three VI’s are used. The first VI “FreeVib1 VI” is used to determine n, , and .
After setting the parameters on the front panel of the VI, the beam is impacted lightly and run the
VI. After sampling the signal from the strain gage conditioner for a pre-determined length of
time the sampled signal is displayed along with the values for n, , and .
The second VI “phaseAngle815 VI” is used to find the phase angle between the response and
the input excitations. After setting the parameters on the front panel of the VI. The VI will drive
the beam at a series of frequencies and the phase angles are calculated.
The third VI “Freqresponse VI” is used to determine the frequency response of the beam. On
the front panel of the VI, parameters are set for the input and output signals.
The values found by the Frequency Response VI for n, and  need to be entered along with
the length and half the thickness (c) of the beam. A strain conversion factor needs to be entered.
This factor depends on the strain gage(s) and bridge arrangement used. The VI will drive the
beam at a series of frequencies and measure the maximum strain and driving displacement at
each frequency. Values for the measured strain, driving displacement, calculated deflection
based on strain and based on , and magnitude ratios, are written to a specified file. This file can
later be opened with a spreadsheet program and examined.
Exercise 1: Natural frequency and damping coefficients Measurements
a. Mount a strain-gaged beam (beam A) and select a beam length. Zero the strain gage
conditioner as follows:
 Connect the red lead wire from the gage to the P+ terminal on the strain indicator, the
white lead wire to the S- terminal and the black lead wire to the D120 terminal. These
connections create a quarter bridge circuit by pairing the gage with an internal “dummy”
resistor. Make sure that the bridge switch indicates a quarter bridge arrangement.
 Turn the indicator power switch on and turn the sensitivity knob fully clockwise. Dial
the gage factor into the indicator and, with the indicator set for zero strain, adjust the
balance knob until the needle points to zero. This process zeroes the indicator at the
current state of the beam and no strain on the gage.
b. Get free vibration signal by impacting the beam by a light object such as a pen.
c. Observe the free oscillations on the oscilloscope and the computer. Measure the rate of
decay of free oscillations. The damping ratio can be found by measuring two amplitudes
separated by any number of complete cycles.
d. Use the logarithmic decrement  to determine the amount of damping present in the
 x 
beam. (   ln  n   2 ). Where xn and xn+1 denote the amplitudes corresponding to times tn
x 
 n 1 
and tn+1, respectively.
Exercise 2: Phase Angle measurements
In this section, the PhaseAngle815 VI is used to find the phase difference between the
forcing signal and the response signal of the beam. The driving signals and the signal from the
strain gage is sampled and saved in the specified file. A strain conversion factor needs to be
entered. This factor depends on the strain gage and bridge arrangement
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Taking Measurements
1. Open the Phase angle815VI.
3. Make sure the SCXI chassis is turned on.
4. Press the Save data button on the VI front panel (option). This stores the data in a file.
Set the following parameters on the VI
 Device:1
 Channel 0: ob0!sc1!md2!0 (strain)
 Channel 1: ob0!sc1!md2!1 (accelerometer)
 Scan rate: 1000
 Number of scans: 1000
 Filter Type: band Bass Low cutoff Freq: 30
 High cutoff freq: 60
 Open Functiongenerator.vi
 Select sine wave signal
 Dual cycle 50%
 Offset: 0
 Amplitude 1.0 v
 Channel: 1 or 0 (check the channel number on the scxi 1302)
 Run the function generator.vi
5. Start the phaseangle815.VI by pressing the Run button in the top left corner of the VI.
Run the VI and measure input voltage and out voltage (can be Vp-p or Vp or Vavg as long as
Vin and Vout are consistent. Measure the time difference t between the two waveforms Phase
angle = 360 (t/T)
a. No filter
F (HZ)
Vin
Vout
Vout/vin
Vout/vin
(db)
t (time between
the two waves)
T
( period)
Phase
angle=360(t/T)
T
( period)
Phase
angle=360(t/T)
Plot Vout/Vin vs f and phase (degrees) vs Frequency (HZ)
Repeat for low- and high-pass filters with the theoretical behavior
b. Low-pass filter
F (HZ)
Vin
Vout
Vout/vin
Vout/vin
(db)
5
t (time between
the two waves)
c. High-pass filter
F (HZ)
Vin
Vout
Vout/vin
Vout/vin
(db)
t( time between
the two waves)
T
( period)
Phase
angle=360(t/T)
How well do the measured amplitude responses you have obtained for low- and high-pass filters with the theoretical
behavior.
Exercise 3: Frequency Response Measurements
The freqresponse.Vi is used to determine the frequency response of the beam. The driving
signals and the signal from the strain gage is sampled and saved in the specified file. Strain and
accelerometer conversion factors need to be entered. These factors depend on the strain gage
and bridge arrangement and the accelerometer sensitivity.
The VI will drive the beam at a series of frequencies and measure the maximum strain level
and driving displacement at each frequency
Taking Measurements
1. Open the Freqresponse VI. File path: c:\me260w\Lab6_beamvib lab\freqresponse.vi
2. Make sure the SCXI chassis is turned on.
3. Press the Save data button on the VI front panel (option). This stores the data in an array.
Set the following parameters on the VI
Shaker settings







Device:1
Channel: 0
Level (volts): 0.6
Start frequency (HZ): 4
Stop frequency (HZ): 10
Freq incr: 0.2 ( or .4)
Actual frequency: 5 Hz
Beam Settings
 Damping: .007
 Natural Frequency: 5
 Length: 25
 Thickness: 0.125
Acquisition settings
 Device: 1
 Channels 0: ob0!sc1md2!0 (default)
 Channels 1: ob0!sc1md2!1 (default)
 scan rate 1000
 number of scans 1000
6

strain or acc 1: (select strain)
Filter settings
Filter type ch 1: band pass
Filter type ch 2: band pass
strain or acc2 ( select accel)
low cut off ch1: 3, high cut off ch 1: 100
low cut off ch2: 3, high cut off ch 2: 100
Run the VI. All data will be written to the specified file (Y/X vs frequency).
Y: strain gage output
X: accelerometer output
Compare the results with the theoretical values
Exercise 4: Find the first three mode shapes
-
-
-
-
Apply a sinusoidal voltage to the shaker, proceeding from low to high frequencies. It may
take a short time for the beam to respond to the excitation, particularly at the low frequencies.
It takes time to transfer energy to the beam, even at a resonant frequency. Adjust resonance
level near resonance, so that both the excitation and the response signals are reasonably
sinusoidal.
To find a natural frequency, slowly increase the frequency of excitation until a large beam
response is observed. At each natural frequency the mode shape is found by locating the
position of the nodes. Run a light object as a pencil point along the surface of the beam.
When the pencil point ceases to vibrate, it is resting on a node.
Repeat the above procedure for three different beam lengths and record the corresponding
natural frequencies.
Make a log-log plot of the measured natural frequency vs. Beam length showing three lines
representing the first, second, and third natural frequencies. Add three lines to the plot,
which show the relationship between theoretical natural frequency and beam length.
Plot the node location from the fixed end of the cantilever vs. Beam length for the second and
third modes.
Compare the resonant frequencies of the beam with predicted values. Explain the
discrepancy, if any.
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Figure 1: FreeVib VI – Front Panel
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Figure 2 – PhaseAngle VI – Front Panel
Figure 3: Frequency Response VI -Front Panel
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