Force and Displacement Measurement

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Erik Hudson
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1 Force and Displacement Measurement Prof. R.G. Longoria Updated Fall 20

2 Simple ways to measure a force

3 Example: Key Force/Deflection measure low-level forces required to trigger key-switch on a computer keyboard x key x in F = k x x key

4 Force and torque sensors A force or torque sensor provides an output in the form of an electrical signal (voltage, current). Examples: Force Force Sensing Device Voltage Sensitivity = voltage/force Strain-gauge force sensors Optical in-line torque sensor

5 Force sensing in products Postal scale, ~$30 Economical bending beam scale, ~$65 Strain gauge bending beam sensor Omega Engineering

6 Sensing Mechanism To measure force (or torque), it is usually necessary to design a compliant mechanical structure. This structure may itself be a sensing material. Force will induce stress, leading to strain which can be detected in various ways, for example: using strain gauges (piezoresistive effect) using crystals or ceramics (piezoelectric effect) optically Sometimes force can be measured using other types of displacement sensing devices.

7 Lab study: beam sensor Experimental setup Beam configuration with a full-bridge strain gauge configuration Wheatstone bridge 3 2 4

8 Strain gauge concepts Matrix length Terminals Gage width Matrix width Strain gauges exhibit piezoresistive behavior, and are one of the most common ways to measure strain. Connecting wires Gage length Insulating layer and bonding cement specimen under strain Types unbonded wire - basically a wire under strain (c. 940s) foil - type shown to left (c. 950s) are most common semiconductor (c. 960s)

9 Strain gauge Sensitivity The 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 Using the derivation in Appendix B, Typical values: 80% Ni, 20% Cr, G = 2 45% Ni, 55% Cu, G = 2 Platinum, G = % Pt, 5% Ir, G = 5. Semiconductor, G =70 to 35 G dr d ρ = = ( + 2 ν ) + ε R ε ρ Piezoresistive effect

10 More on gauge Types Strain gauges come in many specialized forms and typically include a calibrated gauge factor, G. Semiconductor strain gauges have the highest values of G. These strain gauges can have G values of 70 to 35, and they are typically very small. However, there are some disadvantages which include: output is not linear with strain, very temperature dependent, usually have a much lower strain limit than metallic type, more expensive than metallic type.

11 Strain Detection Order of Magnitude Calculation Consider a situation where the strain is on the order of microstrain. For a metallic foil strain gauge with G = 2, R = 20 ohm, 6 R G ε = R = = Ω You need to measure a 0.002% change in R! How would you detect such a change?

12 Beam Sensors The beam structure/geometry is used extensively in designing many types of force and torque sensors. A beam offers certain advantages: easy geometry for basic analysis and design strain gauges can be mounted easily and configured in several different ways to achieve different objectives See beam configurations in Appendix C.

13 Wheatstone Bridge Configuration Output DC voltage v o R R R R = ( R + R )( R + R ) V Null condition is satisfied when: s R R = R R V s If all the gauges have the same resistance, you can show: dvo dr dr dr + dr G V R 4 s [ ε ε ε ε ] 2 3 = = This equation can be used to guide placement of gauges on a specimen.

14 Balanced Bridge The bridge is balanced when the ratio of resistances of any two adjacent arms is equal to the ratio of resistances of the remaining two arms (taken in the same sense). R = R 2 R R C or, R = R 3 R R D A B

15 Signal Conditioning: Impedance Bridges Impedance bridges are used to convert the output of resistive, capacitive, or inductive sensors into a voltage signal IMPEDANCE VOLTAGE Many types of impedance bridges exist; see examples in Appendix D.

16 Strain Gauge Measurement System The strain gauge is part of a multi-stage process that generates a voltage signal proportional to the strain. ε G R R v o v amplified Sensing Mechanism Bridge Amplifier The amplifier used in our lab experiments is described in Appendix E.

Inertial sensors (accelerometers, seismometers) Non-contact

17 Displacement measurements Mechanical (gage blocks, rulers, etc.) Strain gauges measure deflection Contact sensors LVDT Inertial sensors (accelerometers, seismometers) Non-contact sensors Optical There are MANY ways to measure displacement

18 LVDT displacement sensor Linear variable differential transformer (LVDT) monitors displacement of a core which modulates the mutual inductance between two coils. - Motion Coil (secondary) Primary coil Coil 2 (secondary) Core Primary core Secondary v Difference voltage v o = v v 2 Constant AC voltage Insulating form or bobbin v 2 Secondary Difference voltage v o = v v 2 (a) This sensor type is described briefly in Appendix G. (b)

20 Summary Force measurement takes advantage of the relationship between force, displacement and stiffness. Strain gauges are a common basis for sensors that can measure force or torque We wrap the sensor with signal conditioning to get a measurable signal (voltage or current) Strain gauges are core knowledge for mechanical engineers

21 Appendix A: Example force and torque sensors Appendix B: Piezoresistivity and gauge factor Appendix C: Beams as basis for strain-gauge sensors Appendix D: Summary of impedance bridges Appendix E: DMD-465W strain-gauge amplifier Appendix F: Examples of home-made force sensors Appendix G: Inductive-type sensors

22 Appendix A: Example force sensors Strain-gauge force sensors Omega Piezoelectric Force and acceleration Bruel and Kjaer Omega Sensotec XYZ force sensor from PCB Piezotronics General purpose PCB Piezotronics

23 Appendix A: Example torque sensors Reaction torque sensor In-line torque sensor

24 Appendix B: Piezoresistivity () 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 is the cross-sectional area. Under strain, the change in R is, R R R dr = dl + da + d ρ l A ρ which for uniform A is, ρ ρl l dr = dl da + d ρ 2 A A A For typical conductors, the resistivity values in units of ohm mm 2 /m are: Aluminum , Pure Iron 0., Constantan 0.48, Copper 0.072, Gold , Tungsten 0.059, Manganese 0.423, Nickel

25 Appendix B: Piezoresistivity (2) The fractional change of R is of more interest, so we find, dl = fractional change in length l da = fractional change in area A d ρ = fractional change in resistivity ρ dr dl da d = + R l A ρ ρ

26 Appendix B: Piezoresistivity (3) For a linearly elastic body, σ dl = F / A = E ε = E l xx o x where E is the Young s modulus. Recall ε x = dl, ε dl y = ν, ε dl z = ν l l l And for an area A = w t, the fractional change is, da dw dt = + = A w t 2νε x Recall that n is Poisson s ratio. Now the fractional change in R is, Input a strain dr = ( + 2 ν ) ε x + R Geometric d ρ ρ Material Output a resistance change

27 Appendix C: Strain gauge orientations for axially-loaded and cantilevered Beams Axially-Loaded Beam SIDE K Bending Temperature SIDE Cantilevered Beam K Axial/ Torsion Temperature A TOP Sensitive Compensated with dummy gage in arm 2 or arm 3 E TOP Sensitive to both axial and torsion loads Compensated with dummy gage in arm 2 or arm 3 B SIDE 4 2 Compensated Compensated (with four-arm bridge and dummy gages in arms 2 and 3) SIDE F 2 2 Compensated for both axial and torsion loads Compensated TOP TOP SIDE C TOP SIDE D TOP 2A 2B 2A A B A 2 independent bridges one four arm bridge +ν 2(+ν) Compensated Compensated Compensated with two-arm bridges. Compensated through fourarm bridge. NOTES: All axially-loaded beams sensitive to torsion. Requirement for null: R/R2 = R3/R4 K = Bridge constant = (output of bridge)/(output of primary gage) SIDE G TOP SIDE H TOP 4 2 a 2 2 b 2 independent bridges 3 4 one four arm bridge +bν/a 4 Compensated for both axial and torsion loads Compensated for both axial and torsion loads Compensated Compensated

28 Appendix D: Impedance Bridges From Beckwith, Buck, and Marangoni, Mechanical Measurements, Addison-Wesley, 3rd ed, 982.

29 Appendix E: Omega Engineering DMD 465WB

30 Appendix F: Home-made force sensors Belt force measurement (Kim and Marshek, c. 98) This apparatus used two different types of home-made force sensors to measure the forces induced by a grinding belt on the disk.

31 Appendix F: Measuring tangential belt forces This beam has a full-bridge of strain gauges.

32 Appendix F: Measuring normal belt forces

Appendix F: Using displacement to measure low-range forces Measuring the difference in displacement between the key and the push tube allows force measurement (assuming stiffness is known). g 0?

33 Appendix F: Using displacement to measure low-range forces Measuring the difference in displacement between the key and the push tube allows force measurement (assuming stiffness is known). g 0? x in x key m t gm k m = 2.6 mm Rubber dome Push tube x key data := (mm) (gm) m = 6.5 gm t x in F (gm) depress x key x key (mm) release

34 Appendix G: Inductors store magnetic energy Inductors store magnetic energy An inductive (or variable-reluctance) sensor design takes advantage of how magnetic flux passes through the circuit, and this can be detected as a change in inductance.

35 Appendix G: Variable-reluctance sensors Reluctance is inversely related to inductance In all of these devices, the inductance, L, is changing based on geometry. You can also have material change to affect reluctance.

Strain Gages. Approximate Elastic Constants (from University Physics, Sears Zemansky, and Young, Reading, MA, 1979

Strain Gages. Approximate Elastic Constants (from University Physics, Sears Zemansky, and Young, Reading, MA, 1979 Material Strain Gages Approximate Elastic Constants (from University Physics, Sears Zemansky, and Young, Reading, MA, 1979 Young's Modulus, Y Shear Modulus, S Bulk Modulus, B Poisson's Ratio 10 11 N/m