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Unformatted text preview: V IBRATION M EASUREMENT U SING AN A CCELEROMETER AND A L INEAR V ARIABLE D ISPLACEMENT T RANSDUCER (LVDT) Last Updated Dec. 11, 2007, M. Lucas Purposes of the Experiment 1) To learn to use LabVIEW to make measurements from a mechanical structural system 2) To learn the functioning of piezo-electric accelerometers and LVDT in conjunction with measurement of vibration in mechanical systems 3) To measure the natural frequency and damping ratio of a cantilever beam and to compare the experimental results with the theoretical analysis 4) To use statistical techniques to determine the range of values of the measured quantities 5) To know the issues involved in sensor selection. 6) To be aware of signal conditioning, and its importance in the acquisition of sensor data. Theory Theoretically Equivalent System: The system that will be examined in this lab is a simple cantilever beam with a mass attached to the beam tip. The fundamental mode of vibration of the cantilever beam can be modeled as a single degree of freedom vibrating system with lumped inertia and stiffness. The system is similar in many ways to a mass attached to a spring, shown in Figure 1. See Figure 8 for a schematic of the experimental setup used as well. L Ma equivalent system Meq Meq = 0.23Mb+Ma Figure 1: Equivalent System Model for Beam The assumption that will be made in this laboratory is that it would be possible to construct a mass- spring system such that the motion of the mass over time would be the same as the motion of the tip of our cantilever beam. This assumption will greatly simplify the modeling of our cantilever beam by allowing us to perform all of our mathematical derivations based on the simpler mass-spring system. We will call the value of the mass that would be required for this mass-spring system the equivalent mass of the cantilever beam, and we will call the required spring stiffness the equivalent stiffness. The equivalent stiffness could be found relatively easily by measuring the deflection of the beam tip due to a known load. However, in the absence of any displacement sensors we can estimate the equivalent stiffness of the beam using the following formula, which is based on our knowledge of mechanics of materials: 3 3 l EI K eq = ( 1 ) E = Youngs modulus, I = moment of area for the cross section of the beam, and l is the length of the beam. It can further be shown that the equivalent mass of the system is: a b eq m m * . m + = 23 ( 2 ) where m b is the mass of the cantilever beam itself, and m a is the added mass at end of beam, which includes the mass of the sensor itself. Based on these estimates for mass and stiffness, we can calculate the theoretical natural frequency of the system: 3 23 3 l ) m . m ( EI M K b a eq eq n + = = ( 3 ) The Impulse Response of the System : When the beam tip receives an impulse excitation (i.e., a quick blow or an initial deflection), the motion generated is a decaying sinusoid bounded by the envelope shown in Figure 2 and given by...
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- Fall '08