# MCE380: Measurements and Instrumentation Lab. Chapter 9: Force, Torque and Strain Measurements

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1 MCE380: Measurements and Instrumentation Lab Chapter 9: Force, Torque and Strain Measurements Topics: Elastic Elements for Force Measurement Dynamometers and Brakes Resistance Strain Gages Holman, Ch. 10 Cleveland State University Mechanical Engineering Hanz Richter, PhD MCE380 p.1/11 Elastic Elements for Force Measurement Material elasticity allows the conversion between force and displacement according to Hooke s law: F = ky The elastic constant k can take various forms according to the geometry of the elastic body, the direction of force application and the direction of the displacement readout. In a straight bar (Fig. 10.3), the constant has the form k = AE L where E is the Young s modulus, A is the area of the cross section and L is the length. In a cantilever beam with the force at the tip and the displacement at the tip (Fig. 10.4), the constant is k = 3EI L 3 where I is the moment of inertia of the cross section. Eq in Holman has a typo. MCE380 p.2/11

2 Elastic Torsion in Torque Measurement Torsion bars work in an entirely analogous way as linear elastic elements. The basic relationship is M = kφ where M is torque (N-m) and φ is angular deflection (rad). The constant for a hollow cylinder of outer and inner radii r o and r i is k = πg(r4 o ri 4 ) φ 2L where G is the shear modulus (Pa). The shear modulus is related to the Young s modulus and Poisson ratio µ by G = E 2(1 + µ) If the angular deflection is small, and can be read using strain gages at 45 degrees from the axis ( see Fig. 10.7). The strain gages will indicate strains of Mr o ǫ 45 = ± πg(ro 4 ri 4) MCE380 p.3/11 Rotary Torque Sensor This device allows the measurement of torque on rotating shafts. We will be using this device in the lab. MCE380 p.4/11

3 Reaction Dynamometer The rotor of a DC machine is connected to the shaft whose torque is to be determined. The housing of the DC machine is mounted on bearings. An arm attached to the housing rests against a force-measuring device (scale). The torque on the housing equals that of the rotor by action-reaction principle. MCE380 p.5/11 Uniaxial Stress and Strain Consider a uniform elastic bar of area A and length L under the action of an axial force (Fig ). The stress is given by σ = F A while the strain obeys linear elasticity: ǫ = σ E ǫ is the engineering strain, that is, it doesn t account for the reduction in area that arises as a consequence of elongation. The engineering strain is also equal to when is small. ǫ = L L MCE380 p.6/11

4 Bending Stress and Strain We will need to use the bending stress and strain formulas for some lab experiments. For pure beam bending, the axial stress due to bending and the moment are related by σ = My I The strain at the surface of the beam can be obtained as ǫ = Mȳ EI where ȳ is half the thickness of the beam. The bending moment M can be obtained by examining the mounting conditions (cantilever, simply supported, etc.) and the loading conditions. MCE380 p.7/11 Resistance Strain Gages A strain gage is made of a thin wire or foil embedded in an insulating matrix which is very flexible. The wire deforms the same as the surface to which it s bonded, changing its resistance. Once more, a Wheatstone bridge can be used to find the resistance and, therefore, the strain. A strain gage reader is a signal-conditioning unit containing the bridge and other necessary electronic elements allowing a direct readout of strain. MCE380 p.8/11

5 The Gage Factor The resistance is related to length as follows: R = ρ L A where L, A and ρ are the length, cross-sectional area and resistivity of the wire. Taking differentials, the following can be derived: dr R = ǫ(1 + 2µ) + dρ ρ The gage factor F is defined by We can now express the strain as F = ǫ = 1 F dr R ǫ R R That is, the gage factor simply expresses proportionality between percent elongation (strain) and percent change in resistance. MCE380 p.9/11 Strain Gage Bridges Suppose the strain gage of resistance R 1 is one of the branches of a Wheatstone bridge. The voltmeter voltage is E D, while the excitation voltage is E (see Fig. 4.25). When strain is applied and the resistance of the gage changes to R 1 + R 1, the voltage detected at the voltmeter is: E D E = R 1 + R 1 R 1 + R 1 + R 4 R 2 R 2 + R 3 The resistance change can be obtained from this equation as R 1 R 1 = R 4 R 1 ( E D E + R 2 R 2 +R 3 ) 1 E D E R 1 2 R 2 +R 3 MCE380 p.10/11

6 Example A strain gage is bonded 10mm from the fixed end of a cantilever beam of L = 100mm. The beam is made of steel (E=210 GPa) and has a cross section of 1 by 10 mm. The gage factor is 2.05 and the resistance is 120 Ω. A Wheatstone bridge with R 2 = R 3 = R 4 = 120Ω is used with an excitation voltage of E = 10V. An unknown deflection is applied at the tip of the beam. The voltage at the terminals of the Wheatstone bridge is E D = 3.5mV. Determine the deflection and the strain. MCE380 p.11/11

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