We want to design a device that measures forces and torques. This device is a hollow cylinder with external and internal radii and length called r, and r, and L respectively. This cylinder is clamped at one end and we want to estimate the force F and the torque C that are applied at the other end. The cylinder is made of Aluminium: Young's Modulus E =70 000 MPa and Poisson Ratio p=0.3. We assume that the length L is much larger that the radius r. : we can then use the model of beam. 22 1 T -Te 0 x3 2 4 4 Te 0 3 31 4 10 FIGURE 1-Studied problem Sixteen strain gauges are glued on the external surface at some given locations. A strain gauge is a device that measures strain in a given direction. If is the unitary vector relative to this direction, the strain , measured by the gauge can be linked to the strain components : En=n.e.n=nen,. The following table lists the location of each gauge : Gange -Te 0 4 3L 4 TE 0 5 L 4 0 -Te X2 The direction for each gauge is given by : = for gauges from 1 to 8. 7=(+) for gauges 9 and 11, =(1-) for ganges 10 and 12, =(-es) for gauges 13 and 15. = ( + a) for gauges 14 and 16. 12 11 100 6 L 31 3L 4 4 0 0 7 8 9 10 40L 00 F Te -Te Te HINO 2/20 L 11 12 13 4 L LINO 2 0 0 -Te -Te Te 2220 2 9 LIN The goal is then to relate the forces and torques to the strains measured by the gauges. 21 L 14 15 16 L L F 2 2 2 -Te -Te 0 0 Te 0 Te 0 2.3 Stretching (Team 3) We consider in this case that F = F and C = 0. Questions a) Write the equations for this stretching problem: balance equations, constitutive equations, strain-displacement relationship and boundary conditions. b) Calculate () with respect to F and the other given data. c) Calculate 11 with respect to F and S. Show that the other components of the Cauchy stress tensor are zero. d) Calculate the area of the cross section S with respect to r, and r. e) Using the 3D constitutive relations, calculate the expression of the strain 11(1,2) with respect to F. f) We consider that the strains en are known for the gauges 1 to 8. What is the relation between the strains 11 and En, at these points? h) Conclusion: how can we estimate the force F by using the strains 5, that are measured at the gauges 1 to 8. Try to establish a formula that involves all these strain measures. 2.4 Twisting (Team 3) We consider in this case that F = 0 and C=Ce. Questions a) Write the equations for the twisting problem: balance equations, constitutive equations, strain-displacement relationship and boundary conditions. b) Calculate M (1) with respect to C and the other given data. c) Calculate 12 and 13 with respect to 1, 2, 3, C and J. Show that the other components of the Cauchy stress tensor are zero. d) Calculate the polar moment of area J with respect to re and ri e) Using the 3D constitutive relations, calculate the expression of the strain 12(1, 2) and 13 (1, 2) with respect to C and the other given data. Explain why the gauges 1 to 8 are not useful to estimate C. f) We consider that the strains E, are known for the gauges 9 to 16. Express the strain ,, for the gauges 9 and 10 with respect to the components at the point (,2,3)=(,0,.re). Show that the stain 12 can be related to the strains E measured at the gauges 9 and 10. Do the same things fot the couples of gauges 11-12, 13-14, and 15-16. g) Conclusion: how can we estimate the torque C by using the strains , that are measured at the gauges 9 to 16? Try to establish a formula that involves all these strain measures. 3 Summary Question By combining the results obtained for the four subproblems, describe a way to estimate all the components of the t force F and the torque C.