MAE 2055: Mechetronics I Mechanical and Aerospace Engineering Fall 2013
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MAE 2055: Mechetronics I Mechanical and Aerospace Engineering Fall 2013 Homework 3 1) Determine I1, I2, I3, and V1 in the following circuit by performing either a nodal or mesh analysis. Choose the analysis method that will require that you solve the fewest number of equations. 200 Ω 300 Ω V1 I3 I1 100 Ω 8V 4V I2 2) Perform a mesh analysis to determine i1, i2, i3, i4, i5, V1, and V2 in the following network. What is the power dissipated by the 4β¦ resistor? 2Ω 5V 2Ω V1 + V2 i1 8V + i3 3V + i2 6Ω i4 i5 4Ω 3) One type of electrical sensor that is of particular interest to mechanical engineers is a strain gauge. A strain gauge is essentially a resistor, whose resistance value varies as a function of mechanical deformation. A common type of strain gauge is a foil strain gauge. Foil strain gauges consist of a serpentine resistor patterned onto a thin foil or film. The resistive pattern typically looks similar to that shown in Figure 1. A picture of an actual strain gauge attached to a metal test specimen is shown in Figure 2. The thin foil strain gauge is attached securely to a solid object whose strain is to be measured. If that object elongates under tension, the strain gauge will deform similarly. As it is stretched the crosssectional area of the resistive traces decreases and the overall gauge resistance increases. If the underlying material and the strain gauge mounted on it undergo compression, the cross-sectional area of the resistive traces on the gauge increases and the overall resistance decreases. contacts Sensitive to strain in this direction Figure 1. Strain gauge resistive pattern. Electrical connection is made to the strain gauge at the contacts on the right. As the serpentine resistive pattern stretches, its resistance increases. As it is compressed, resistance decreases. Figure 2. A foil strain gauge attached to a metal test specimen. The strain gauge has some nominal (under zero-stress conditions) resistance of π πΊ , which is typically on the order of tens to hundreds of ohms. When the gauge experiences strain, its resistance changes by some amount, Δπ , which is proportional to the strain as follows. Δπ = ππΊπΉ π πΊ where π is the strain, (1) π= and πΊπΉ is the gauge factor, given by πΊπΉ = ΔL L (2) Δπ /π πΊ π (3) Gauge factors for foil strain gauges are typically in the range of 2-5. So, for a strain gauge with π πΊ = 100β¦ and πΊπΉ = 2, experiencing a strain of π = 0.01 (1%), the change in resistance would be Δπ = 2β¦. Small changes in resistance are not easy to accurately measure with a direct resistance measurement. A circuit commonly used to facilitate strain gauge measurements is the Wheatstone bridge circuit. The Wheatstone bridge comprises four resistors connected as shown in Figure 3. A constant DC voltage, ππ , is supplied to the bridge. If all resistors are equal, or more precisely, if the ratio of resistors in each of the two legs is equal: π 1 π 3 = π 2 π 4 (4) then the bridge is said to be balanced and ππ = 0. However if one of the resistors deviates from its nominal value, then the bridge becomes unbalanced, and ππ ≠ 0. + R1 R3 + Vo - Vs R2 R4 - Figure 3. A Wheatstone bridge circuit. A Wheatstone bridge is useful for measuring the small changes in resistance typically associated with strain gauges. One of the resistors in the bridge would be replaced with the strain gauge, a voltage ππ is applied to the bridge, and strain is measured as a function of the output voltage, ππ . A typical strain gauge measurement circuit is shown in Figure 4. Here, π 3 has been replaced with the strain gauge, whose nominal resistance is π πΊ . All other resistors have a resistance π , which is equal to the nominal gauge resistance, π = π πΊ . + RG + ΔR R + Vo - Vs R R - Figure 4. Bridge circuit configured for a strain gauge measurement. If, for example, the strain gauge were being used to measure the strain of a bending beam, it may be mounted as shown in Figure 5. Strain gauge RG + ΔR Figure 5. Strain gauge measuring the strain of a bending beam. a) Show that the output voltage of the circuit in Figure 4 can be approximated by the following expression ππ ≈ ππ π πΊπΉ 4 (5) Note that equation (5) is an approximation. You will need to make a simplifying assumption to arrive at this approximation. (Hint: think about relative values of terms in your expressions.) b) Calculate the output voltage, ππ , for the following conditions. π = π πΊ = 250β¦ πΊπΉ = 3 π = 1.5% ππ = 12π The measurement configuration illustrated by Figure 4 and Figure 5, in which one of the four bridge resistors is replaced with a strain gauge is referred to as a quarter-bridge implementation. An alternative to the quarter bridge is the full-bridge configuration, in which four strain gauges are used, each replacing one of the resistors in the bridge circuit. A full-bridge strain gauge circuit is shown in Figure 6. + RG + ΔR RG - ΔR + Vo - Vs RG + ΔR RG - ΔR - Figure 6. Full-bridge strain gauge circuit. The four strain gauges used for a full-bridge measurement may be mounted to the objectunder-test as illustrated in Figure 7. Notice that the two gauges mounted on the top of the bending beam experience an equal compression and a corresponding equal decrease in their resistances. Similarly, the two gauges on the bottom of the beam experience an equal tensile stress and a corresponding equal resistance increase. RG - ΔR RG - ΔR RG + ΔR RG + ΔR Figure 7. Four strain gauges mounted in a full-bridge configuration to measure the strain of a bending beam. c) Show that the sensitivity of the full-bridge circuit has improved by approximately a factor of four compared to the quarter-bridge circuit. That is, show that the output of the full-bridge strain gauge measurement is given by ππ = ππ ππΊπΉ (6) d) Now, recalculate the output voltage, ππ , of the full-bridge circuit under the following conditions. π = π πΊ = 250β¦ πΊπΉ = 3 π = 1.5% ππ = 12π What advantage is provided by the full-bridge measurement?
