ME 365 EXPERIMENT 9
A FORCE MEASUREMENT TRANSDUCER
Objectives:
This two week experiment examines the performance characteristics of a device used
to measure force. The force transducer consists of a cantilever beam fitted with three
types of transducers: a strain gage, a linear variable differential transformer (LVDT), and
a proximity sensor. The static and dynamic performance characteristics of each
beam/sensor combination must be evaluated before a specific configuration may be
chosen for a particular application.
The procedure in this laboratory is deliberately vague. You may use any of the VIs
and techniques from previous labs to accomplish your goals. It will be useful to review
previous labs and the VIs that were used in each. Study the material on LVDT and
Proximeter sensors in Appendix B. An additional exercise may be assigned.
After completing this lab you should be able to:
Experimentally determine static and dynamic performance characteristics for the
three different configurations of a simple force transducer.
Determine and measure the first four natural frequencies of the force transducer
using various system identification techniques.
Determine the usable frequency range for a transducer and explain why this is
important.
Procedure:
The Cantilever Beam Force Transducer (CBFT):
The beam and transducers will be referred to as a cantilever beam force transducer
(CBFT). A schematic of the CBFT assembly used in this experiment is shown in Figure
1. Take a moment to identify the different components of the assembly.
Figure 1: The Cantilever Beam Assembly
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Background information on strain gages may be found in your course notes. Information
pertaining to proximeter sensors and LVDTs is contained in Appendix B of this
laboratory. The shaker consists of a coil, a permanent magnet, and a spring mounted in
an assembly. Applying an alternating current to the coil caused the magnet, attached to a
linkage, to move up and down. We will be using the shaker to excite the beam at various
frequencies during the dynamic response portion of this lab.
All three sensors require additional signal conditioning before an output voltage
suitable for data acquisition may be obtained. Figure 2 depicts the necessary connections
between these components.
Figure 2: Instrumentation Connection Diagram
Make the appropriate connections, and examine each signal on the oscilloscope while
lightly tapping on the beam to make sure everything is working properly. You may wish
to use a potentiometer to reduce particularly large signal magnitudes. The potentiometer
on the bridge circuit should be used to zero the strain gage output with zero load. During
the static calibration portion of the lab, you will be applying known weights to the end of
your beam. You may wish to apply them now in order to properly adjust the distance
between the proximeter and the beam. It is important that a small air gap remain between
the proximeter and the beam under a full* load.
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* Note: The shaker linkage may only be displaced a certain distance before it “bottoms
out”. Taking static or dynamic performance measurements when this occurs will yield
erroneous results. Be sure to adjust any applied weight or input amplitude accordingly.
Static Calibration of the CBFT (Week 1):
Begin the static calibration portion of this experiment by completely filling out
worksheet number one. This information will be of use in subsequent calculations.
When you are satisfied with the output from the three sensors, connect them to the data
acquisition interface panel as follows:
Table 1: ADC Channel Assignments for the CBFT
Signal
Strain Gage
Shunt
LVDT
Proximeter
Channel #
3
4
5
6
ADC Input Range
±20 mv
±10 V
±10 V
Launch LabVIEW, and load BEAM LOGER.VI from the Desktop/ME365 directory.
This VI collects static calibration data from the three sensors each time you click the
ENTER button the front panel. Take care to check that the input ranges for both channel
groups are set properly. Type in the washer mass (in grams) available at your workbench
in the input box labeled “mass”. The slider underneath this box should indicate the
number of washers that have been placed on the end of the beam whenever the ENTER
button is clicked. Use the VI to collect at least three complete sets of static calibration
data for all of your sensors. ENTER each data point several times so that the effects of
random measurement errors may be quantified. Note: Make sure the VI is set to run
continuously.
A thorough static performance analysis will address: sensitivity, bias, input resolution,
repeatability, hysteresis, and nonlinearity for each sensor. Present these results in a table.
In addition, you should complete worksheet number two. Graphically compare these
theoretical strain gage results with what you observed in the laboratory. Be prepared to
discuss your static calibration results at the beginning of the second lab period.
Dynamic Response of the CBFT (Week 2):
Complete laboratory worksheet number three prior to performing the dynamic
response experiments during the second week in lab. The information in Appendix A
will be helpful in computing the theoretical natural frequencies of your beam. Based on
your static calibration results and class discussion, select one of the sensors for dynamic
response testing. Your TA may offer additional credit for performing dynamic response
tests using one or both of the remaining sensors.
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Make the appropriate connections between the beam, signal conditioning hardware,
oscilloscope and data acquisition interface. You will also need to connect the output
from the oscillator to the shaker input during parts of this experiment. Perform the
following dynamic response tests and analyses:
1) Record the snap back response of the beam and determine the damping ratio and
natural frequency of the first mode of the beam by using SYSTEM ID.VI.
2) Lightly tap the end of the beam with a pen or pencil. Study the frequency content
of this impulse response using the SIMPLE SPECTRUM ANALYZER.VI.
3) Apply a sinusoidal input to the shaker, observing the input and output signals on
the oscilloscope. Use a sine wave with an amplitude of 10 Volts peak to peak.
Sweep through a range of input frequencies and identify as many natural
frequencies as possible using the Lissajous figure. You may need to adjust the
input signal amplitude to keep the shaker from bottoming out.
A thorough dynamic performance analysis will compare the theoretical natural
frequencies for the CBFT with those found using the different dynamic tests. In addition
to presentation of your damping ratio and natural frequency results, you should discuss
how these parameters relate to the CBFT’s ability to make accurate measurements of
dynamic forces. Extend this discussion to present suggestions for improving the CBFT’s
accuracy.
REMEMBER: The cantilever beam force transducer (CBFT) is a mechanism for
measuring applied forces. The CBFT is composed of both the sensors and the beam.
Static and dynamic performance analysis describes how well the entire system conveys
information about an input of interest.
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Cantilever Beam Force Transducer Worksheet 1
Measure the resistances of the strain gages and bridge resistors while they are
disconnected from the circuit.
Gage 1 ____________ Gage 2 ____________ R1 ____________ R 2 _____________
Record the strain gage factor off of your beam (shown as G.F.).
G = __________
Having hooked up the bridge circuit, measure the voltage across the battery terminals.
Battery Voltage __________
Measure the beam dimensions.
Length __________ Width __________ Thickness __________
Measure the location of the sensors from the fixed end of the beam.
Strain gages __________ LVDT __________ Proximeter __________
We are using a 16-bit ADC. Depending on the sensor and channel number, the input
range is either ±0.02 or ±10 Volts. Calculate the ADC resolution corresponding to each
sensor.
Strain gages _____________ LVDT _____________ Proximeter _____________
Complete the following checklist to verify that all sensor outputs lie within the voltage
range for each ADC channel.
Input Range
Strain gages
LVDT
Proximeter
No Load Voltage
Adjust to 0 V
5
Full Load Voltage
Notes
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Cantilever Beam Force Transducer Worksheet 2
F
d
I = bh3/12
Strain Gages
h
L
b
Figure 3: Cantilever Beam with Strain Gages
Referring to Figure 3, the theoretical strain in a cantilever beam at the location shown is
given according to:
1
E
For your beam:
6(L d)
Ebh 2
My
EI
F(L d)h
3
2E
bh
12
6F(L d)
Ebh 2
_____________
The output voltage from a 2 active arm bridge in terms of strain is:
Vout
For your setup:
G
Vbattery
2
G
Vbattery
2
____________
Write down the equation relating the applied force to the output voltage for your setup.
Complete the following table of theoretical strain and voltage values corresponding to the
masses you used in the static calibration experiment.
Weight (g)
Force (N)
Theoretical Output Voltage
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Cantilever Beam Force Transducer Worksheet 3
Using the results presented in Appendix A, calculate the first four theoretical natural
frequencies of your beam.
3.52
1
EI
l4
_______________ Rad/sec
f1
_______________ Hz
Higher natural frequencies of the beam:
2
f2
3
f3
4
f4
Appendix A also contains an expression for the first natural frequency, assuming a
point mass is located on the end of the beam. This is, perhaps, a more realistic
description of the CBFT set up in the lab. Use this expression to find alternate values for
first natural frequency of the beam.
1
__________________ rad/sec
f1
7
____________________ Hz
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Appendix A: Dynamic Properties of Beams
Some of the dynamic properties of beams are shown in these figures taken from the
Shock and Vibration Handbook by Harris and Crede. For the beam used in the lab, the
values shown in Figure 5 are not exactly correct because of the added mass of the shaker
linkage at the end of the beam. Figure 4 provides an expression for the first natural
frequency of a beam with a mass (m) at its end.
Massive Springs (Beams) with Concentrated Mass Loads
m
ms/mb
k
l
I
E
n
= Mass of load, kg
= Mass of spring (beam, kg)
= Stiffness of spring, N/m
= Length of beam, m
= Area moment of inertia of beam cross-section, m 4
= Young’s modulus, Pa
= Natural frequency, rad/sec
Figure 4: Natural frequency expressions for different setups
Beams of Uniform Section and Uniformly Distributed Load
n
A
EI
l4
rad/sec
where:
E
I
l
A
= Young’s modulus, Pa
= Area moment of inertia of beam cross-section, m 4
= Length of beam, m
= Mass per unit length of the beam, kg/m
= Coefficient from figures below
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Figure 5: Coefficients for computation of the natural frequency for different setups
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Appendix B: Notes on the Motion Sensors
The Linear Variable Differential Transducer (LVDT):
The linear variable differential transformer, or LVDT, is one of the most popular
methods of measuring mechanical position in use today. It has found use as a primary
transducer in the measurement of a variety of variables such as: force, pressure, torque,
and acceleration. In each of these transducers, the position of an elastic member
undergoing deflection due to an applied force is measured by the LVDT. The
measurement of interest is then inferred from the position signal. The word “linear”
which appears in the name of the LVDT, describes the measurement of straight line
motion – not a linear calibration curve.
The construction and operation of an LVDT is deceptively simple. Three coils of
electrically conducting wire are wound on an insulating form as shown in Figures 6 and
7. An AC voltage applied to the terminals of the primary coil induces a voltage of the
same frequency in each of the two secondary coils. If the movable ferromagnetic core is
centered, the secondary voltages will have the same amplitude. A positive displacement
of the core results in a greater amplitude in the 1st secondary coil than in the 2nd. Thus,
the two secondary voltages are amplitude modulated according to the core displacement.
Demodulation of the two secondary signals yields a DC signal related to the core
position.
Figure 6: A Linear Variable Differential Transformer
One of the most advantageous attributes of the LVDT is the fact that the core does not
touch the internal bore of the transformer. Therefore, the problems of mechanical wear,
friction loading of the displacement being measured, and electrical noise as a result of
rubbing action are all eliminated. In this sense, an LVDT is a “non-contact” sensor,
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however, the core must be secured to the part of the system being measured. This may be
inconvenient, or even impossible, as in the case of measuring the transverse vibration of a
spinning shaft. If the core cannot be secured to the moving element which is to be
sensed, but may be placed in contact with it, a gage head LVDT may be selected. This
type of LVDT, illustrated in Figure 8, is similar to the conventional LVDT, but has two
additional features: a means of guiding the core along the axis of the transformer and a
mechanism for maintaining contact between the tip and the moving system.
Figure 7: Cutaway view of an LVDT (Courtesy Shaevitz Engineering, Camden, NJ)
The outer diameter of a typical LVDT transformer is approximately 1.9 cm and the
length varies from 4 cm for a ±0.13 cm stroke unit to 50 cm for a ±13 cm stroke unit.
Because of the frictionless operation of the conventional LVDT and its basis on the
induction principle, the resolution of the LVDT is said to be infinite. However, external
demodulation circuitry may impose limitations on the resolution. Also, recall from the
course notes chapter on carrier systems that the frequency response range (bandwidth) of
the LVDT is limited by the frequency of the carrier signal applied to the primary coil.
Figure 8: A Typical LVDT Gage Head (Courtesy of Schaevitz Engineering, Camden NJ)
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Inductive Proximity Probe Displacement Measurement Systems:
An inductive proximity probe is an electro-mechanical transducer which has the
capability, without contact, of electronically sensing small distances to a conductive
surface. The transducer is primarily used for displacements no greater than 2.5 mm,
although some units are capable of measuring up to 125 mm of displacement.
Proximeters are used in many applications as shown in Figure 9.
Figure 9: Applications of the Bently D-152 Distance Detector System
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The proximity probe, model D-152, manufactured by Bently Nevada Corporation is a
cylindrical jacket with a pancake-wound coil of wire implanted very close to the surface
of the probe face. Upon application of power from a suitable source, any conductive
element near the coil will load it. The loading is very nearly proportional to the distance
from the coil to the loading surface and the conductivity of that surface. Loading changes
are converted to DC voltages from which the distance may be determined.
Sensitivity data, relating the distance to a conductive surface and the detector output
voltage, is not provided by the manufacturer. This is because the sensitivity will vary
depending on several factors, including: the material of the sensed surface, the voltage to
the coil, the ambient temperature, and the type of sensor head. Typical calibration curves
obtained from proximeter measures with different surface materials are shown in Figure
10. Specifications for a Bently D-152 distance detector system are given in Table 2.
Figure 10: Typical Proximeter Calibrations with Steel, Brass, and Aluminum
Table 2: Proximity Probe Specifications (Courtesy of Bently Nevada Corp)
Input
Scale Factor
Output Voltage Range
Dynamic Frequency Response
Output Impedance
Normal Operating Temperature
-15 Volts DC at 1 to 5 mA
Variable [10 V/in to 1000 V/in]
0 to -14 Volts DC
0 to 10 KHz
10 KW
10 to 40 C
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