ME 322: InstrumentationLecture 7

February 1, 2016

Strain Gage Introduction, Demonstration, Wheatstone bridge, Temperature

compensation

Announcements/Reminders• HW 2 due now (before lecture starts)• This week in Lab: – Lab 3 Pressure Transmitter Calibration– Acquire your own data and use it to calibrate the

pressure transmitter you’re issued– There are only 4, two-port pressure standards

which can be used by two groups at a time. There are 16 groups in lab, so please be patient (and prepared).

– Please bring an electronic copy of the Excel Workbook you created for homework

Pressure Gage

• Device –Pressure difference causes diaphragm

deformation–Deformation can be measured using a Strain Gage

Strain Gauge Construction

• Thin substrate (0.0002”)• Firmly bounded to surface• Metal or Semiconductor foil on or within substrate• Surface deformation stretches or compresses the foil and

changes its resistance.– Measure DR = R - RI

– However, are small• Demo: Gage resistance on an aluminum beam in bending

∆ 𝐿

Stain Gage Applications

• Can measure Strain of a deformed part (or material elastic modulus)• Can be incorporated in devices to sense

– Applied force or weight– Acceleration– Pressure

What is Strain?

• Unit strain – Microstrain

• Strain is caused by stress

• For Linear elements – E ≡ Elastic or Young's Modulus

• Material property = fn(Temperature, material)• Describes material stiffness• In Lab 5 we measure E for aluminum and steel beams

Lδ

F

F

To use a Strain Gage, firmly bond it to a Surface

• Do this in Lab 4 (next week)• The gage will experience nearly the same stain

and the object’s (specimen’s) surface – eGAGE = eSURFACE

–Does not measure internal stains• The gage deformation will affect the gage

resistance, R–The initial (un-deformed) resistance is RI

• How to predict DR = R – RI?

gage

specimen

Wire Resistance

• Depends on length, area and material property– r ≡ Electric Resistivity• Material property = fn(Temperature, Strain, …)

• Stretching wire changes L, D & ρ by small amounts, and results in small changes in R

A D

L, R

Effect of Small Changes in L, D & ρ on R

• DR = R – RI • For small changes DR = dR, use the chain rule: – dR= – • Divide by R to get fractional change

– • Relates fractional change in to fractional changes in L, D,

• How do the terms relate to strain e ?

Evaluate Terms• axial strain (assuming = eSpeciman)• This is what we are tying to measure!

• Transverse strain = – ≡ Gage material Poison’s ratio• ~ 0 – 0.5• ~ 0.3 for metals

• – Cstrain ≡ Strain Coefficient of Resistivity• Material property = fn(Temperature, …)• Can be large, and > or < 0 for semiconductors

Combined Effects

• • =

– Dimensionless– Metal Foils: S = 1.6 to 4 (typical 2.07)– Semiconductors: S = -140 to 175

Example• Apply 1000 lbf to diameter D = 0.25” steel rod.

– E = 207 GPa = 30x106 psi

– For a strain gage with: RI = 120 Ω, S = 2.07

• Find final resistance: R = RI + ∆R• Solution

• R = 120 + 120(0.001406) = 120.17 W– Very small fractional change!

L

D

Undesired Temperature Sensitivity• Gauge resistivity is affected by temperature• Thermal expansion of specimen and gage may be different– This can stain the gage

• Resistance is determined by measuring the voltage across the gage while passing a current though it, which can heat the gage!

• Temperature factor ST:–

• Can a circuit “automatically” compensate?Gage Temperature Change = undesirable sensitivityDesired measurand

Wheatstone Bridge Circuit• Two voltage dividers

• Use strain gages for some or all of these resisters• If R1R3 ~ R2R4, then the output voltage will be close

to

R3

R3

Initial State

• Choose initial resistances so that initial VO,I ~ 0– R1R3~ R2R4 (For example, all could be ~equal)

• Small changes in each Ri will cause a relatively large fractional change in VO (compared to VO,I ~ 0)

• To increase – Increase or and/or decrease or

+

+ -

-

Effect of small resistance changes on VO

• Chain Rule:– ++– Find all four partial derivatives and plug in – use R1R3= R2R4 and = 0 = , – Simply ...

• If all four resistances start off roughly the same (satisfies )

– and

• For precision resistors, , but not for stain gages

R3

Incorporate gages into some bridge legs

• To increase – Increase and and/or decrease and

• Install stain gages in some or all legs– – Use gages with the same characteristics• Ri=R, Si =S, and STi = ST

• If identical gages are installed in all four legs, thenR3

• In general–

• For quarter bridge with a gage only at 3 – – undesired sensitivity

Quarter Bridge

R3 +

+ -

-

• For this half bridge

–

– If Gage 3 is placed on a deformed specimen and Gage 2 is placed on an identical but un-deformed specimen, then

– , – (Automatic Temperature Compensation)

Half Bridge

R3 +

+ -

-

• Place gage 2 on the side opposite of gage 3, so ε2 = -ε3

•

– Twice the output amplitude of a quarter bridge, and with temperature compensation

Beam in Bending: Half Bridgeε3

ε2 = -ε3

ε2 = -ε3

Full Bridge

• All four legs are identical stain gages

• In bridge, opposite legs (1, 3) and (2, 4) reinforce – Adjacent legs (1, 2) and (3, 4) oppose

R3 +

+ -

-

Beam in Bending: Full Bridge

• V0 is 4 times larger than quarter bridge– And has temperature compensation.

3 1

2 4R3 +

+ -

-

= DT3 = e3 = -e3 = -e3 = DT3 = DT3

Tension

ε4=-υ ε3

ε2=-υ ε3

ε1=ε3

2 3

4 1

R3

General Guidelines for HWs• Use your Course ID numbers (which you can find in MyNevada), not your

name or student ID number, when submitting your HWs• Units and significant digits always! • No hand-drawn plots! Starting from HW3, whenever you are asked to plot

your data, plot them using computer software, i.e. Excel, Matlab, Mathcad etc.– Include labels for both axes with the units. If necessary, include legends too. Make it

look good!• Show you work! Do not skip the steps and just write your final answer.

Whenever applicable, list your assumptions, write out your formulas and work through to your final answer. If you use a Table (or graph) on your solutions, give reference to that table in your book, i.e from Table 6.3, z=-1.28.

• Be clear with your solutions and work neatly! If the grader needs to spend more than 3 minutes to figure out what you write, you may not get even partial credit.