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mae 3113_9

# mae 3113_9 - MAE 3113 3-1 Measurements and Instrumentations...

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School of Mechanical and Aerospace Engineering MAE 3113 3-1 Measurements and Instrumentations Stress and Strain Fall 2006

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School of Mechanical and Aerospace Engineering MAE 3113 3-2 If you recall the resistance of an electrical conductor can be expressed as A L R ρ = Where ρ is the resistivity, L is the length, and A is the cross sectional area. However since the wire typically has a circular cross section, the area can be expressed as; Strain Gage Principle: Variable Resistance 2 CD L R ρ = Where C is a constant.
School of Mechanical and Aerospace Engineering MAE 3113 3-3 Strain Gage Principle L dL d L dL D dD L dL R dR d D dD L dL R dR CD dD LD C dL Ld CD ρ ρ ρ ρ ρ ρ ρ + - = + - = - + = 2 1 2 ) ( 2 ) ( 2 2 dR The gage factor F is defined as L dL R dR F = L dL d R dR F ρ ρ ν ε + + = = 2 1

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School of Mechanical and Aerospace Engineering MAE 3113 3-4 Strain Gage Principle Finally the equation can be now solved for strain in terms of resistance and gage factor. L dL d R dR F ρ ρ ν ε + + = = 2 1 R dR F 1 = ε Unfortunately the changes in the resistance are extremely small.
School of Mechanical and Aerospace Engineering MAE 3113 3-5 Strain Gage Principle Example: A certain strain gage has a gage factor of 2.05 and a nominal resistance of 120 ohm. What resistance change should one expect for a strain of 1 µ-strain = = = = - μ ε ε 250 0.000246 ) 120 )( 05 . 2 )( 10 1 ( 1 6 dR x FR dR R dR F

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School of Mechanical and Aerospace Engineering MAE 3113 3-6 Strain Gage Principle Lets first take a look at using a voltage divider to measure the change in resistance. e i e o R R G G G i o R R R e e + = ( 29 ( 29 ε F R R RR e R dR R R RR e de G G i G G G G i o 2 2 + = + =
School of Mechanical and Aerospace Engineering MAE 3113 3-7 Strain Gage Principle e i e o R R G ( 29 ( 29 V 5.1 240 ) 120 )( 120 )( 10 1 )( 05 . 2 )( 10 ( 2 6 2 μ ε = + = - x de R R RR F e de o G G i o Example: Using the gage from the last example, with an excitation of 10 volts determine the change in output voltage 5 μ Volts is not that bad, however one has to recall that this is only the CHANGE in voltage from ‘no-load’. To determine the full picture we must also solve for the ‘no-load’ voltage. V e o 5 240 120 10 = =

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School of Mechanical and Aerospace Engineering MAE 3113 3-8 Bridge Circuit Bridge circuits of various types are employed widely for the measurement of resistance, capacitance, and inductance. While capacitance and inductance are important, the more widely used is the resistance bridge. (NOTE: More information on different types of Bridge circuits can be located in Frank E.,” Electrical Measurement Analysis”, McGraw-Hill New York, 1959) Lets Start with the basic Bridge circuit shown below.
School of Mechanical and Aerospace Engineering MAE 3113 3-9 Wheatstone Bridge Circuit Using KVL and KCL we can analyze the following circuit.

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