Project Task Requirements In the upcoming Beam I and Beam II projects, students will instrument a cantilever beam with a strain gauge and use this gauge, with an accompaning circuit, to characterize the beam's dynamic behavior. First, it will be necessary to calculate the first resonant frequency of a cantilever beam 1. Freely Vibrating Cantilevered Beam ww Time Figure 1: Freely Vibrating Cantilevered Beam An excited cantilevered beam vibrates at a set of frequencies corresponding to its natural modes. These frequencies depend on the physical and material properties of the beam. The first five natural frequencies can be computed from the equation fi = 3.526 22.03 f2 - ₤3 61.70 ΕΙ 2π £4 ML 120.91 fs=199.85 Eqn. 1 where is the mass per unit length of the beam, E is the Young's modulus of the beam material and I is the moment of inertia of the beam (I = bh³/12). See Fig. 2. Calculate the theoretical first resonant frequency (fi) of this beam using Mathcad. Use the data in the table below. The beam is made of 7075 aluminum, so you will need to look up the density and Young's modulus of this material. [Remember that mass is weight divided by gravity]. Table 1 Beam Dimensions Beam Length (L) Beam Width (b) Distance to Strain Gauge (x) Beam Thickness (h) Distance to Applied Force (x) 11.5 inches 1.0 inch 0.125 inch 1.0 inch 11.0 inches. 2. Theoretical Strain Gauge Circuit Output Assume that that simple beam bending theories are applicable. Based on these theories, the strain, (x), of a cantilever beam subject to a point force, F, applied at the free end in a direction perpendicular to the longitudinal axis can be calculated. Assume that there is a strain gauge attached to the beam at point X Xs X XF F Force applied by a micrometer. L dx Figure 2: Cantilevered Beam The deflection of the beam, dx, at any point, x, (0 <x<XE) is given by Fx² dx 6EI (3XF-x) Eqn. 2 where E is the Young's modulus of the beam material and I is the moment of inertia which is given by bh³ Eqn. 3 At the point of application of the force, FXF³ dr - 3EI Eqn. 4 and at points between the force and beam end Fx² dx 6EI (3L-XF) Eqn. 5 The strain at X is given by & (3dph(XF-X)) 2xF³ Eqn. 6 You do not know the force applied to the beam when it is depressed by the micrometer. However, you do know that the maximum depression at the point where the force is applied will be no more than 25 mm.Using the equations given above, calculate the expected strain on your strain gauge for a micrometer depression values (dr) of 0, 5, 10, 15, 20 and 25 mm. Plot the strain as a function of dr. Then, using the gauge factor given in Appendix C, calculate the resistance change expected in your strain gauge as a function of dr and plot. Appendix B: Strain and the Strain Gauge Strain is defined as the amount of deformation of a body due to an applied force. More specifically, strain (s) is defined as the fractional change in length. ε = SL/L When a downward force is applied to a beam, the surface length will increase, therefore stretching the attached strain gauge, Figure B-1. Tension increases resistance Insensitive to lateral force Figure B-1: Strain Gauge Measured resistance As the strain gauge is stretched, its resistance increases in proportion to the applied strain. That is Where: • · SR/Ro GF S1/lo = GF & SR is the strain induced change in the resistance of the strain gauge Ro is the original resistance of the strain gauge Sl is the strain induced change in the length of the strain gauge • lo is the original length of the strain gauge • GF is the gauge factor (sensitivity) of the strain gauge • Є is the actual strain induced in the strain gauge. The strain gauge used in the Vibrating Beam I and II projects has the nominal values Ro 120 2 and GF = 2.1.