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Overview Displacement Force Experimentation Summary Pre-Lab/LE References

Sensors and Instrumentation:Force and Displacement Measurement

Prof. R.G. Longoria

Department of Mechanical EngineeringThe University of Texas at Austin

July 10, 2014

ME 144L Dynamic Systems and Controls Lab (Longoria)


Overview

Force and displacement measurement methods are part of essentialengineering knowledge, especially for mechanical engineers. Thislaboratory is meant to provide familiarity with some of the most commonapproaches used to sense these quantities.

1 This lecture first reviews displacement sensing devices and systems

a strain gauges, classified as a resistive sensorb LVDTs, inductive sensorsc optical sensors

2 Sensing displacement is key to force sensing as well

3 The application of strain-gauges to force sensing in particular isdescribed in detail, especially using beam-type configurations

4 Laboratory study provides experience with both displacement andforce sensor systems and their usage



Sensing displacement, motion, or distance

Mechanical (gage blocks, rulers, etc.)

Strain gauges measure deflection

Contact sensors: LVDT, Inertialsensors (accelerometers, seismometers)

Non-contact sensors: Optical,magnetic, capacitive



Strain gauges are resistive type sensors

Strain gauges take advantage of change in resistance due to straining ofmaterial (see Appendix A).

Strain gauges exhibit piezoresistivebehavior, and are one of the mostcommon ways to measure strain, ε,the change in length per unitlength.

Strain gauges can be made in thefollowing forms:

unbonded wire - basically awire under strain (c. 1940s)

foil - type shown to left (c.1950s) are most common

semiconductor (c. 1960s)



Strain gauge sensitivity: gauge factor, G

A measure of the sensitivity of a strain gauge is given by the gauge factor,which is defined as,

G =fractional change in resistance

fractional change in strain

From the derivation detailed in Appendix A,

G =1

ε

dR

R= (1 + 2ν) +

1

ε

dρ

ρ

The first term on the right-hand side includes Poisson’s ratio, and thesecond term is a piezoresistive effect (temperature dependent).

Typical types/values: 1) 80% Ni, 20% Cr, G = 2, 2) 45% Ni, 55% Cu, G = 2, 3)Platinum, G = 4.8, 4) 95% Pt, 5% Ir, G = 5.1, 5) Semiconductor, G = 70 to 135



Strain gauge type selection

Strain gauges come in many specialized forms for detecting different typesof strain behaviors and should be matched to a material type (aluminum,steels, etc.)

All types have a calibrated gauge factor, G.

Semiconductor strain gauges have the highest values of G, from 70 to 135,and are typically very small. However, there are some disadvantages thatinclude: a) output not linear with strain, b) very temperature



Strain is measured by monitoring resistance change in astrain-gauge

Consider a situation where the strain is on the order of 1 microstrain.

For a metallic foil strain gauge with G = 2, R = 120 ohm,

∆R = G · ε ·R = 2 · 1 × 10−6 · 120 = 0.0024Ω

That represents a fractional change in resistance of 0.0024/120, or a 0.002%change in R.

Detecting this small a change in resistance is very difficult.

To use strain gauges practically, it is necessary to configure them within a circuit

and a structure so the change in resistance is detectable and has physical

meaning. This requires understanding concepts from signal conditioning and

structural design.



Signal conditioning circuits: monitor current or voltage

Strain gauges can be monitored by a simple voltage divider, as used tomonitor potentiometric sensors. Alternatively, small resistance changesmay require balancing potentiometric circuits, null bridges, or impedancebridges (next slide).



When all arms on a bridge are resistive, this is aWheatstone bridge

Typically, an input DC voltage is applied, say, across terminals A and B and theoutput voltage is measured across C and D.

The Wheatstone bridge is balancedwhen the ratio of resistances of anytwo adjacent arms is equal to the ratioof resistances of the remaining twoarms (taken in the same sense), i.e.,

R1

R2=R3

R4

R1

R3=R2

R4

In this case, the output voltage wouldbe zero.



Impedance bridges

Impedance bridges are used to detect changes in resistive, capacitive, or inductivesensors. There are many types of impedance bridges.

Most inductive and capacitive sensors rely on impedance bridges for signal

conditioning. AC type signals are used in these circuits.ME 144L Dynamic Systems and Controls Lab (Longoria)


Sensors based on changes in inductance (or reluctance)

Recall that electrical resistors can be used as sensors, taking advantage ofchanges in resistance due to changes in geometry and/or material properties. Anexample is how a potentiometer can be used as a motion/displacement sensor.

In the same way, some electrical inductors, which store magnetic energy, can alsobe used as sensors. These sensors take advantage of how changes in geometry ormaterial properties influence inductance (or reluctance).

Inductive or variable-reluctance sensorsrequire a magnetic flux to be set up sothat changes in electrical inductancecan be detected.

Examples are shown to the right thatshow how distances can be inferred bythis type of sensing mechanism.

Can you conceive a variable-reluctance pressure sensor?



LVDT sensors rely on detecting change in inductance

A linear variable differential transformer (LVDTs) senses displacement of acore that modulates the mutual inductance between two coils.

LVDTs are inductive (or variable-reluctance) sensors.



Typical LVDT wiring and calibration



Displacement sensors can be made using position sensitivedetectors

Position sensitive detectors (or PSDs) operate on the principle of photoeffect,relying on surface resistance of a silicon photodiode, with the output signalproportional to the continuous position of a beam incident on the sensitivesurface.

Digital PSDs have been made using CMOS and CCD sensors and these have adiscrete number of photodiodes in an array (not considered optimal but suitablefor some applications) [2]



Displacement measurement using (optical) triangularmethod [2]

From similar triangles: Lo/LB = f/x, where x = D/(P + 1) andP = IA/IB. Then, Lo = k(P + 1).



Commercially available sensors, systems

1 and 2D PSD sensor components(Hammamatsu photonics, Japan)

Industrial optical displacement sensor(CMOS-based. Keyence Corp.)



Simple force and weight measurement

Spring-based force sensors Classical weighing by balancing

Calibration weights are required.

Can you think of a force sensing devicethat does not infer force fromdisplacement?



Electromechanical force and torque sensors

Force and torque sensors provide an output in the form of an electrical signal(voltage, current).

Examples:Off-the-shelf strain-gauge sensors: Optical in-line torque sensor:



Force sensing in commercial products



Force is typically inferred from displacement sensing, whichrequires understanding the structural design

To measure force (or torque), it is usually necessary to design a compliantmechanical structure. This structure may itself be a sensing material.

Force will induce stress, leading to strain which can be detected in variousways, for example:

using strain gauges (piezoresistive effect)

using crystals or ceramics (piezoelectric effect)

displacement using contact or non-contact sensing



Example: keyboard force measurement

problem: measure low-level forcesrequired to trigger key-switch on acomputer keyboard

Here is a ‘home-made’ solution:

This design requires means for measuring the displacementsindicated.



Using beam configurations for strain-gauge-based forcesensors

The beam structure/geometry is used extensively in designing manytypes of force and torque sensors.

A beam offers certain advantages:I easy geometry for basic analysis and designI strain gauges can be mounted easily and configured in several different

ways to achieve different objectives

Example: a tip loaded cantilever is a common design element. The tip deflectioncan be related to applied tip force by a linear spring-like relation, where thestiffness is related to beam geometry and material.



Beams for bending, shearing, and axial and torsional loads



Beam configurations used for strain gauge sensors

The strain gauges are typically monitored as part of a Wheatstone bridge. Sometimes you may have a bridge with all activestrain-gauges (full-bridge), meaning they are all under strain. There may be cases where only 2 of 4 (half-bridge) or 1 of 4(quarter-bridge) are active. You typically have the same nominal resistance on all arms. Why?



The full-bridge beam sensor configuration

A very common force sensing design uses a cantilever beam with a full-bridge.

If all the gauges have the same resistance, it can be shown that:

dvoVs

=dR1 − dR2 − dR3 + dR4

R=G

4[ε1 − ε2 − ε3 + ε4]

This equation can be used to guide placement of gauges on a specimen.



A strain gauge measurement system

The strain gauge is part of a multi-stage process that generates a voltagesignal proportional to the strain.

The amplifier is required because the changes in the bridge circuit output voltageare usually in the millivolt range.



Laboratory set up and equipment

An aluminum cantilevered beam thathas been instrumented with straingauges in a full-bridge configurationwill be used in the laboratory.

1 Beam support and cantileverbeam

2 OMEGA DMD-465WB (straingauge amplifier)

3 LVDT support and LVDT sensor

4 Laser displacement sensor

5 Calibration weights/masses



Strain gauge amplifier

The amplifier used in lab provides the input voltage and then amplifies thebridge output. Adjustments are available to balance the bridge, to set theamplifier gain, and to set the input voltage level.



Laboratory objectives

1 Learn how to set up and use a strain-gauge beam, run calibrations and testfor force measurement

2 Learn about a laser displacement sensor and its proper and safe usage;assess calibration

3 Learn about and use a LVDT for displacement measurement; assesscalibration

4 Continue use of LabVIEW and myDAQ for data acquisition; create a VI forstatic calibration and for force and displacement sensing

5 Use the LabVIEW VI to conduct experiments and collect data to estimatethe beam stiffness and to estimate the Young’s modulus for the aluminumbeam



Summary

Force measurement takes advantage of the relationship between force,displacement and stiffness.

Strain gauges are a common basis for sensors that can measure forceor torque

We ‘wrap’ the sensor with signal conditioning to get a measurablesignal (voltage or current)

Strain gauges are core knowledge for mechanical engineers



Pre-Lab: Problem 1 – strain-gauge concepts

Review the derivation of the relationship between strain and strain gaugeelectrical resistance found in Appendix A, in order to understand the keyparameters that play a role in this sensing mechanism.

What is the term used to refer to ‘sensitivity’ of a strain-gauge?

Say you mounted strain gauges with R = 120 ohm and G = 2 as shown below onan axially loaded, aluminum beam. If thickness, b = 5 mm, and width h = 7 mm,what would you expect to see at the output of the bridge circuit if F = 5 N?



Pre-Lab: Problem 2 – beam mechanics

In a lecture example, the stiffness of a beam was given in terms of Young’smodulus, E, area moment of inertia, I, and beam length, L (or distance tolocation of force application). Conduct some research to justify this relation.

In completing this pre-lab, you should gain an understanding for how to relate thestrain at the strain gauge position on the beam (e.g., as shown on the lab setup)to the force applied at the end (tip) of the beam and to the deflection at the tipof the beam.



Pre-Lab: Problem 3 – full-bridge configuration

Sketch a three-dimensional drawing of a cantilevered beam and illustrate thelocation of the strain gauges for a full-bridge configuration.

Identify each gauge with its location within a circuit schematic of the Wheatstonebridge (use 1, 2, 3, and 4). Indicate on your sketch the input voltage terminalsand the output voltage terminals.

For the case where a force is applied at the tip of the beam in a downwarddirection, indicate the stress state of each strain gauge with T for tension and Cfor compression both on the sketch of the beam and gauges as well as on thecircuit schematic.



Pre-Lab: Problem 4 – estimating bridge outputs

Consider a beam with strain gauges in a full-bridge (as in Pre-Lab Problem 3)that has a balanced output when there is no applied force (unstrained). Under acertain load, the strain gauges on the top of the beam are strained by +5microstrain and those on the bottom side of the beam are strained by −5microstrain. Calculate the output voltage from the bridge assuming that eachgauge has a resistance of 350 ohms and the gauge factor is 2.0. Also assume thatthe input voltage is 8 volts.

Re-consider the case above, but now assume that after the bridge was balancedthe temperature changed and caused the resistance of each strain gauge todecrease by a factor of 10% (from the unstrained value). Show how this willaffect the output voltage (from zero volts in a balanced state).



Lab Evaluation

1 Develop a calibration relation for the beam sensor based on at least fivemeasured data points (force vs. deflection). The data should be recorded forthis LE in both tabular and graph form. Derive a measurement sensitivity(N/Volts) based on your force-voltage measurements.

2 Use your calibrated beam sensor to measure two unknown weights providedby the TA. You will be asked to compute the error in measuring each weight.

3 Using the measured force and displacement data, estimate the beamstiffness, kb, relating a vertically applied load to the beam tip displacement.Show how you can use this value to estimate the Young’s modulus for thebeam material (aluminum).



References

[1] Beckwith, Buck, and Marangoni, Mechanical Measurements,Addison-Wesley, 3rd ed, 1982.

[2] Fraden, Handbook of Modern Sensors: Physics, Design andApplications, 1996)


Appendix A: Piezoresistivity (1)

We know that for a conductor of uniform area, the resistance is given by,R = ρL/A, where ρ is the resistivity (cm ohm)., L is the length, and A isthe cross-sectional area. Under strain, the change in R is,

dR =∂R

∂ldL+

∂R

∂AdA+

∂R

∂ρdρ

which for uniform A is,

dR =ρ

AdL− ρL

A2dA+

L

Adρ

The fractional change of R is of more interest, so we find,

dR

R=dL

L− dA

A+dρ

ρ

where the fractional changes in length, area, and resistivity are given bydL/L, dA/A, and dρ/ρ, respectively.

For typical conductors, the resistivity values in units of ohm mm2/m are: Aluminum 0.0278, Pure Iron 0.1, Constantan 0.48,Copper 0.0172, Gold 0.0222, Tungsten 0.059, Manganese 0.423, Nickel 0.087.


Appendix A: Piezoresistivity (2)

For a linearly elastic body,

σxx = F/A = E · εx = E · dLL

where E is the Young’s modulus. Recall,

εx =dL

L, εy = −ν dL

L, εz = −ν dL

L

and for an area A = wt, the fractional change is,

dA

A=dw

w+dt

t= −2νεx

Recall that ν is Poisson’s ratio. Now the fractional change in R is,

dR

R= (1 + 2ν) · εx +

dρ

ρ


Appendix B: Belt testing machine

Marshek and Kim (UT-Austin, c. 1985), Flat belt test machine, US Patent4841783 A


Appendix B: Custom XY force sensors for flow-inducedforces on cylinders

Longoria (UT-Austin, c. 1989)


Appendix B: Four bar XY force sensor

This force sensor resolves a total forceapplied on the shaft into an X and a Ycomponent.

There are two full-bridge circuits, soeight total strain gauges.

There is some “cross-talk” (e.g., the Ycomponent is sensitive to forcesapplied in X) in the measurement, butthis can be almost completelyeliminated with a good calibration.


Download - Sensors and Instrumentation: Force and Displacement ...dsclab/leks/DSC_Force_Displacement_v1.pdf · Sensors and Instrumentation: Force and Displacement Measurement ... Strain gauge

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