MET 487 Instrumentation and Automatic Controls. Lecture 13 Sensors

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1 MET 87 nstrumentation and utomatic Controls Lecture Sensors July 6-9, 00 Stress and Strain Measurement Safe Load Level monitoring Force (indirect measurement by measuring strain of a flexural element Pressure (by measuring strain in a flexible diaphragm Temperature (by measuring thermal expansion of a material Strain Gauge (SG Basic Device o Bonded metal foil a thin foil of metal (Constantan o grid pattern o Thin plastic backing material (polymide o Large metallic pads for soldering o Total resistance of the SG is given by where L ρ ρ foil metal resistivity (Ω-m L total length of the grid line (m the grid line cross-sectional area (m Figure. Typical Strain Gauge

2 Cantilever Beam Mounted with a SG SG mounted on a cantilever beam of uniform and symmetric cross section. push-down normal force (stress P is added at the unsupported end, as shown in Figure. P δ L x SG mounted on a cantilever beam of uniform and symmetric cross section Figure s the object is deformed, the SG foil is deformed, causing its electric resistance to change P L For the strength materials, the deflection at the unsupported end is δ, where P E normal force, L length of beam, moment of inertia, and E modulus of elasticity The force applied to a solid object resulting the deformation of the object. The effect of applied force is called Stress and the resulting deformation is called Strain. The Tensile Stress-Strain o tensile force applied to an object cause the object to elongate or pull apart o The stress (unit of pressure or tensile is defined as F Tensile Stress (N/m in S unit or lb/in in the English unit where F applied force in N cross-sectional area of the object in m o The resultant strain is defined as the fractional change in length: l Tensile Strain ε l

3 where l change in length in m ( or inch l original length in m (or inch The beam develops a tensile stress on the top surface, and total surface stress σ at location is where P x ( t / σ E ε t beam thickness, moment of inertia E modulus of elasticity or Young s modulus is given by Stress F / E Strain l / l For aluminum beam E 6.89 x 0 0 N/m Therefore we can find P in terms of other terms: P σ E ε x t x t P L Then substituting the applied stress P into deflection δ equation: δ E P L δ E E ε L x t E ε L x t Solve for strain ε ε x t δ L Gauge Factor (GF Electric Properties o ttached the SG to an object by a suitable adhesive o s the object is deformed, the SG foil is deformed, causing its electric resistance to change GF is given by / GF ε where change in resistance due to stress (Ω original resistance of strain gauge (Ω ε strain (in/in, m/m, cm/cm, etc

4 Example 9. f a 0 Ω strain gauge with a GF.0 is used to measure a strain of 00micro ε or 00με, how much does the resistance of the gauge change from the unloaded state to the loaded state? nswer: From the equation: GF / ε We can solve it for : GF ε ε GF (0*00*0-6 *.0 0*0.000* 0.0Ω Measuring with a Wheatstone Bridge Wheatstone bridge a circuit for measure small change in resistance four-resistor network with a DC voltage source For the static balanced mode, as shown in Figure. o and are precision resistors; precision potentiometer; the strain gauge o djust until voltage between and B is zero volt: B B 0 or B B o Since we will use a High mpedance olt Meter to measure the voltage B, so the meter will draw any current. This condition makes and. ex ex Substitute and into, we obtain the following expression:

5 So we know and accurately, and note the adjusted value for calculating unknown (strain resistance as : Strain gage ex B olt Meter High mpedance oltmeter Static Balanced Bridge Circuit Figure

6 Dynamic Unbalanced Bridge Circuit The circuit is shown in Figure Figure The put voltage can be expressed as The input voltage ex can be express as ( ( ex Solving for in terms of ex ( ex When the bridge is balanced: 0 and has a known value When change value, as the strain gauge is loaded, the is related to earrange the equation to give ( ( ex ex

7 eferences Strain Gages, ccessories and nstruments, Strength of Materials, Strain Gauge Measurement Tutorial, National nstruments pplication Note 078, Strain osette for Strain Measurement, efunda Engineering fundamentals, m Electrical esistance Strain Gage Circuits, Stress, Elasticity and Deformation, Stress and Strain Formulas, Engineering Edge, Beam Diagram and Calculator nput, efunda Engineering fundamentals, antilever_uniformload Structure Beam Bending Equation/ Calculation Cantilevered Beam with One Load pplied at End,

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