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Explain working of strain guage and draw the expression for guage factor.
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Answer: 

  • The strain Gauge may be defined as the passive resistive transducer based on the principle of conversion of mechanical displacement into the resistance change. The strain Gauge characteristics depensds on gauge sensitivity, accuracy, frequency and the enviromental condition.

Working of Strain Guage :- 

  • The basic principle of operation of an strain gauge is that the resistance of the wire changes as a function of strain, increasing with tension and reducing with compression. The change in resistance is measured with a Wheatstone bridge.
  • The strain gauge is attached to the specimen and hence the gauge is subjected to the same strain as that of the specimen under test.
  • The materials used for fabrication of electrical strain gauges must have some basic qualities to achieve high accuracy, excellent reproducibility, good sensitivity, long life and ability to operate under the required environmental conditions.
  • Some of these qualities are attained by selecting materials with high specific resistance, low temperature coefficient of resistance, constant gauge factor, and constant strain sensitivity over a wide range of strain values.
  • The bonding cement should have high insulation resistance and excellent transmissibility of strain, and must be immune to moisture effects.
  • The most common materials used for wire strain gauges are constantan alloys containing Nickel45% and55%Copper, as they exhibit high specific resistance, constant gauge factor over a wide strain range, and good stability over a reasonably large temperature range(from 0oC to 300oC). For dynamic strain measurements, Nichrome alloys, containing80% Nickel and20% Chromium are used. They can be compensated for temperature with platinum. Improper bonding of the gauge can cause many errors.

     

Derivation of Gauge Factor

  • The Gauge factor is defined as the unit change in resistance per unit change in length. Generally it is denoted byS. Thus the below figure shows:- ![](data:image/png;base64,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) **Fig Deformed resistance wire** * S=ΔR/RΔl/l Where S= Gauge factor R= Gauge wire resistance ΔR= Change in wire resistance l= length of the Gauge wire Δl= Change in the length of the Gauge wire * Let the resistance wire is in tensile stress and it is deformed byΔl. Thus ρ= Specific resistance in Ω.m l= Length of the wire in m A= Area of cross sections in m2 * When uniform stressσis applied to the wire along the length, the resistanceR changes toR+ΔR * Hence σ=Stress=Δll Δl/l=per unit change in length ΔA/A= per unit change in area Δρ/ρ = per unit change in resistivity * Now, we know that R=ρlA  dRdσ=d(ρlA)dσ =ρA.δlδσρlA2.δAδσ+lA.δρδσ * Thus δδσ(lA)=lA2.δρδσ * Multiply both sides by1R we obtain  1RdRdσ=ρRA.δlδσ1RρlA2.δAδσ+lRA.δρδσ * UsingR=ρlA, we obtain  1RdRdσ=1l.δlδσ1AδAδσ+lρ.δρδσ * Cancelingδσ from both sides from above equation , we obtain  dRR=dlldAA+δρρ i.e.=ΔRR=ΔlldAA+Δρρ........(1) * Thus for finite stress, total change in resistance is due to fractional change in length , are and resistivity * Thus for a circular wire, A=π4d2  δAδs=π4(2d)δdδs  1AδAδs=1Aπ4(2d)δdδs  1AδAδs=1d2(2d)δdδs * Cancelingδs from both sides from above equation , we obtain  δAA=2dδd * Hence  ΔAA=2Δdd * Now the Poisson's ratioμ=Δd/dΔl/l=Poisson's ratio  Δdd=μ(Δll)...........Equation3 * Using 2 and 3 in 1 we obtain i.e.=ΔRR=Δll2Δdd+Δρρ =ΔRR=Δll2[μΔll]+Δρρ =Δll[1+2μ]+Δρρ * Neglecting piezoelectric effect,Δρρ can be neglected =Δll[1+2μ] * Guage Factor S=ΔR/RΔl/l=1+2μ
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