mae 3272 - lecture 4 notes supplement - strain gages

WSachse; 2/2012;

Load Measurement System:

Force

Force

M&AE 3272 - Supplementary Lecture Materials: Strain Gages

Elastic Member

1

M&AE 3272: Mechanical Propertyand Performance Laboratory

Excitation

Signal Conditioning and Processing

Display and Analysis via LabVIEW

Strain Gage

Load Cell

WSachse; 2/2012;

Resistance Strain Gage – Brief History:

M&AE 3272 - Supplementary Lecture Materials: Strain Gages 2

The electrical resistance of a conducting wire increaseswith elongation and decreases with compression.

Lord Kelvin’s Experiments:Strain ε and electrical resistance of wires

1856

WSachse; 2/2012;

Resistance Strain Gage – Brief History 2:

1936-1938 Ruge (MIT) and Simmons (CalTech) plus 2 Students!

SR-4 (Simmons+Ruge+4 Others) joined with DeForest

Ruge-DeForest Partnership; SR-4 Gages distributed

SR-4 Strain Gages distributed by Baldwin


Simmons Patent, 1944

“…He’s a familiar figure around the CalTech campus, which he considers a ‘suitable local amusement park.’”Simmons: Near Genius; Brilliant EE; “Lab Rat”;64 μV/με ! Failed to realize the significance of his invention.

Ruge: Bonded wire gage; Stymied by low-level signals;

Realized at once the significance of their

invention.

WSachse; 2/2012;

Resistance Strain Gage – Brief History 3:

1952 Development of foil gage by Saunders-Roe, UK

1960-70’s Improved Control and understanding of gage materials,design, photolithography, chemical etching; vacuumdeposition; manufacturing

Today Used in most applications; Many, many configurations


Single-element foil gages

WSachse; 2/2012;

Foil Strain Gages - Various:

(a)-(c) Single-element gages

(d)-(e) Two-element rosette

(f) Two-element, stacked rosette

(g)-(h) Three-element rosette

(i) Three-element, stacked rosette

(j) Torque gage

(k) Diaphragm gage

(l) Stress gage

(m) Gages for use on concrete


WSachse; 2/2012;

Semi-conductor Strain Gage – Brief History:

1954 Piezoresistive properties of Si and Ge discovered

1957 Mason and Thurston (Bell Labs); Transducer development (theory and experiment)

1960 Commercial piezoresistive strain gages available

1990’s -Current

Development of MEMS strain gages with electronics (analog/digital); telemetry


• Usually have a larger Gage Factor (-50 to -200) than foil gages (typically +2.0 to 2.5)

• Highly non-linear resistance/strain behavior (Calibration?)• More expensive• More sensitive to temperature changes• More fragile than foil gauges.

Characteristics:

WSachse; 2/2012;

Strain Gage Specifications:


WSachse; 2/2012;

Strain Gage Operation:


http://www.rdpe.com/ex/hiw-sglc.htm

Essential Assumption:

The deformation of the gage accurately mimics the deformation of the material to which it is attached.

• Minimal loading effect of the gage on the test specimen

• Strain sign tensile/compressive)

• Strain magnitude

• Secondary effects negligible or accounted for

WSachse; 2/2012;

Strain Gage Applications:• Material property sensor

• Monitor and control loads/deformations in mechanical systems; e.g. Scales, Tools, Thermal sensor, Flow, Motion, etc.; Multi-B$ industry.


“ … Real-time Computer Graphics for Character Animation. … we use strain gages as the input device. By using this, we can get the relative moving data between two human surfaces with no pains.”テレビジョン学会技術報告 17(55) pp.31-36 19930930

Alinghi; America’s Cup (IEEE Trans Neural Sys Rehab Eng, 2009)New Minneapolis I-35W BridgeGreen: Strain Gage Monitoring System

WSachse; 2/2012;

Strain Gages – Desired Characteristics:


• Low mass: Minimal loading effect in dynamic

measurements

• Low stiffness: Minimal loading effect on deformation

• Gage calibration stable wrt temperature and time

• Wide operating temperature range

• High Gage resolution: ±1 μm/m

Large Dynamic range: ±5% strain (±50, 000 μm/m)

(High ε-sensitivity)

• Gage length small ⇒ point-like measurement

• Linear response: Simplified data process

• Good fatigue life - in dynamic measurements

WSachse; 2/2012;

Strain Gage Sensitivity:


Factional Change of Gage Resistance with Strain:

ΔR

R= (1 + 2ν)εaxial︸︷︷︸

Dimensional

+Δρ

ρ︸︷︷︸piezo−resistive

The fractional change of gage resistance per unit strain –Strain gage Sensitivity :

ΔR

R· 1

εaxial⇒ Gage Factor ≡ Sgage = (1 + 2ν) +

Δρ

ρ· 1

εaxial

When ν ≈ 0.3 : the Gage Factor is given by

Sgage � 1 + 0.6 + (0.4 to 2.0) Metallic conductors

Sgage � 1 + 0.6 + (−125 to 175) Semiconductors

P − type (e. g. Boron) Sgage > 0

N − type (e. g. Arsenic) Sgage < 0

Sgage > 0 → Rg ∝ +ε > 0 [T] Sgage < 0 → Rg ∝ −ε < 0 [C]

• Numerical Example : Metal foil gage, 120 Ω ; Sgage ≈ 2.0 ,then for εaxial = 1 με (i. e. 1 × 10−6 in/in) :

ΔRg = Sg Rg εaxial � 2·120·10−6 � 2.4×10−4 [Ω] = 240 [μΩ]

WSachse; 2/2012;

Strain Gage: Performance Factors - 1


Installation - It is assumed that a properly selected gage hasbeen correctly bonded to the material under test.

Transverse Sensitivity - Sensitivity of a gage to transversestrains (non uniaxial)

ΔR

R= Sgage(εaxial + Kt εtrans) where: Kt ≡ Strans

Saxial

Sgage = Saxial (1 − ν Kt)

True : εaxial =ΔR/R

Sgage

1 − ν Kt

1 + Kt(εaxial/εaxial)App : ε′axial =

ΔR/R

Sgage

Error in neglecting εtrans : Error =εaxial − ε′axial

εaxial100 %

Percent error of actual axial strain as a function of εtrans/εaxial

WSachse; 2/2012;



Cyclic Straining -May result innonlinearity, hysteresisand zero-shift.

Possible resultsof strain cycling

Temperature Sensitivity - Important if measurements aremade over a large ΔT . Possible effects:1. Gage Factor Saxial changes2. Gage dimensions change: ΔL/L = α ΔT3. Specimen dimensions change: ΔL/L = β ΔT4. Gage resistance changes: ΔR/R = γ ΔT#1 is relatively small; Mismatch between #2 and #3 leads tothermal straining of gage (unable to separate from specimen).

Apparent

strain for

two gage

alloys

Corrections - For measurements over a broad range oftemperature measured strains must be corrected.

WSachse; 2/2012;



Power Dissipation - Depends on gage size; design; material properties;adhesive/thickness; specimen material/thermal properties; coating; cooling.

WSachse; 2/2012;



Loading Effects - The gage/backing has an effective modulus,Egage = 7 ∼ 20 GPa (1 to 3 × 106 psi).

Effect of mechanical behavior of specimen under test isaffected locally and globally .

• Example of local effect:

Effective gage modulus: 1.15×106 psi (8.0 GPa);

Thickness of gage installation: 0.0023 in (0.06 mm)

• Global effects of the gage also arise affecting the entirecross-section of the specimen.

• Solution - Use the lowest modulus gage; smallest in size –or – use optical, capacitive techniques.

WSachse; 2/2012;

Wheatstone Bridge Circuit – Static Measurements:


• Constant voltage (or current) excitation; Resistors R1, R2, R3 and

R4 and load resistance RM → ∞ .

Constant voltage circuit

Output Voltage:

E0 =R1 R3 − R2 R4

(R1 + R2))R3 + R4)Ei

At balance :

E0 = 0 when R1 R3 = R2 R4

⇒ Static Measurements

WSachse; 2/2012;

Wheatstone Bridge Circuit – Dynamic Measurements:


• Dynamic Measurements: R1 → R1 + ΔR1;R2 → R2 + ΔR2; R3 → R3 + ΔR3 and R4 → R4 + ΔR4

then . . .

ΔE0 =R1 R2

(R1 + R2)2

(ΔR1

R1− ΔR2

R2+

ΔR3

R3− ΔR4

R4

)Ei + h. o. t.

The omitted higher-order-terms lead to an error given by

Error : =∑4

i=1 ΔRi/Ri∑4i=1 ΔRi/Ri + 2

• When ΔR1 = −ΔR4 and ΔR2 = ΔR3 = 0

– or : ΔR2 = −ΔR3 and ΔR1 = ΔR4 = 0 ⇒ Error equals zero.

WSachse; 2/2012;

Common Strain Gage Wheatstone Bridge Circuits:


4-Arm Active (4X Output)

1-Arm Active (Quarter Bridge)

2-Arm Active (Temp Comp)

2-Arm Active (Temp Comp)

WSachse; 2/2012;

Common Strain Gage Wheatstone Bridge Circuits:


Dummy Gage: Temperature

Compensation

WSachse; 2/2012;

Amplification of Bridge Signals:


Pressure Sensor Application:

Circuit compensates for sensor-to-sensor offset and gain variations

Functional Block Diagram:

WSachse; 2/2012;

Measurement/Analysis of Dynamic Effects with Strain Gages:


• Strain gage

detecting a

stress pulse :

Transmission of

dynamic strains

from specimen

into gage

• Dynamic response

of the strain gage

• Time-spatial signal

convolution:

ΔR(t) = Sgage(�0/C) ∗ ε(t) R0

• Examples - Dynamic, Axial Impact Loadings

Shock tube generated stress waves in rods; (a) Measurementsystem; (b) Longitudinal strain record at 1.51 m from impact

(Fox and Curtis, 1958)

Pneumatic rifle pellet excitation of stress wave in a rod; (a)Measurement system; (b) Longitudinal strain record showing

compressive and tensile pulses (Pao and Kowal, 1965)

WSachse; 2/2012;

Strain Gage as a Dynamic Pressure Sensor:


Experiment : Two strain gages were bonded to the sides of a full can (Aluminum) ofsoda. The can was opened and the voltage signals from each gage were recorded using adigital waveform recorder.

Recorded unloading strains when opening a can of soda.

We only used one gage measuring the hoop strain, εhoop , of the can during unloadingto evaluate the internal pressure p prior to opening. The relationship is

Released Pressure, p =E t

r (1 − ν/2)εhoop

where

εhoop Measured hoop strain difference � 800 μεE Material’s Young’s modulus � 10.5 × 106 [psi]ν Material’s Poisson’s ratio ν � 0.33t Can wall thickness = 0.0040 [in]r Can inside radius = 1.3125 [in]

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

=⇒ 28.8 [psi]

Principle : A fluid (gas or liquid) under pressure inside of a can resultsin stresses and strains (deformations) in the material making up the can.

Stresses in a cylindrical pressure vessel. (from Gere, Mechanics of Materials (2004))

σhoop → ← σaxial

Biaxial State of Stress:

Hoop stresses : σhoop

2 · σhoop (t · Δx) � p (2r · Δx)

σhoop =p r

t

Axial stresses : σaxial

σaxial (2πr · t) � p (πr2)

σaxial =p r

2 t

Procedure : A strain gage is used to measure the hoop strain, εhoop , of the can.

εhoop =σhoop

E− ν

σaxial

EFor an aluminum can :

ν � 0.33E � 10.5 × 106 [psi]

Gives . . .

Pressure, p =E t

r (1 − ν/2)εhoop

WSachse; 2/2012;

Measuring Large (Plastic) Strains with Elastic Strain Gages:


• Permits measurement of large, inelasticdeformations.

• Useful for measurements in hostile environments.

• Re-useable gage.

Semi-circular,thin beam

extensometer

Applied Load, P , deflection δ :

Bending Stress: σb =6 P R

b h2

Bending Strains: εb =6 P R

b h2 ECastigliano’s 2nd Theorem to findbending strain energy:

U =∫ π0

(P R sin θ)2

2 E IR dθ =

3 π P 2 R3

b h3 E

Axial deflection: δ =∂U

∂P=

π 6 P R3

b h3 E

Deflection Sensitivity:

δ

εb=

π R2

2 h

When connected to aquarter-bridge Wheatstonebridge for which the excitationis Ei gives the Output Signal :

ΔE0

Ei=

⎛⎜⎝Sg h δ

2π R2

⎞⎟⎠

Extensometer Sensitivity:

Sδ =ΔE0

δ

=Sg h

2π R2

WSachse; 2/2012;

Hot, New Ideas with Strain Gages:


Silver Ink: Before/After Sintering 350^C, 60-min

A Digital MEMS-Based Strain Gage for Structural Health Monitoring

B. J. MacLean, M. G. Mladejovsky, M. R. Whitaker, M. Oliver, S. C. Jacobsen

Mat Res Soc Symposium Procedings, 503, 309-320 (1998)

Arthroscopically Implantable Force Probe: μ-Forces

Fiber-Optic Strain Gage

WSachse; 2/2012;

Back to us and M&AE 3272:


We’re going to learn how to mount strain gages onto an elastic member in order to

fabricate a Load cell, or Force transducer.

mae 3272 - lecture 4 notes supplement - strain gages

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