Piezoelectric Lab. Module 3: Dr. Yirong Lin. MECH 4395/5390 and IE 4395 Conducted: 7/29/14. Group 4. Jose Luis Coronel Jr. Italia VallesOchoa


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1 Piezoelectric Lab Module 3: Dr. Yirong Lin MECH 4395/5390 and IE 4395 Conducted: 7/29/14 Group 4 Jose Luis Coronel Jr. Italia VallesOchoa Abstract
2 The purpose of this lab report is to specify the process of testing the frequency given by a beam s vibration in comparison to the theoretical frequency obtained from using equations. The report will begin with a brief introduction to piezoelectric material, then proceed to detail the experimental setup, gathering of data (including calculations), and conclude with a comparison of the first natural frequency obtained theoretically with the actual experimental frequency. Introduction: Pierre and Jacquez Cuie discovered the piezoelectric effect in the late 19 th century. They discovered that materials such as tourmaline and quartz could convert mechanical vibration energy to alternating electric energy. Such materials become electrically polarized when a strain is applied. Piezo is Greek for pressure, which induces a voltage. The piezo used in this experiment detects vibration when a small force is applied to the free end of the cantilever beam. An oscilloscope is used to determine the vibration frequency of the beam acting with the piezo and creating a voltage. This form of energy harvesting is an effective form of power generation because it requires little acceleration to produce a high power output. Experimental Setup: h
3 Data Analysis: The dimensions of the beam we selected, beam 4, are taken from the average of three measurements. This would give us a more accurate value for our calculations. The values were all converted to SI units to facilitate our analysis and are presented in the table below. Length (L) Base (b) Height (h).3145 m mm 3.04 mm From these values, we proceeded to calculate the theoretical frequency by utilizing the equation for angular frequency ω 1 and the equation that relates angular frequency with first natural frequency f 1. Equation 1: ω 1 = α 1 2 EI ml 4 Equation 2: ω 1 = 2πf 1 The beam is made from aluminum, whose properties were utilized in the calculations. The Young s modulus for aluminum is E = 69GPa, and the density is ρ = 3000 kg m 3. The value of alpha is a constant and was given as α 1 = The mass of the beam was determined by multiplying the density of the beam, by the cross sectional area. That calculation is shown below: m = ρa = ρ(bh) = (3000 kg m 3) ( m)( m) = kg m Another parameter that must be calculated is the moment of inertia of the beam. Utilizing the dimensions of the beam the moment of inertia was determined by: I = 1 12 bh3 = 1 12 ( m)( m)3 = 5.94x10 11 m 4 After having obtained all the values necessary, the angular frequency can be calculated. ω 1 = (69x10 9 Pa)(5.94x10 11 m 4 ) (0.231 kg = rad m ) s (0.3145m)4 Then finally, the theoretical first natural frequency is found by manipulation of Equation 2. f 1 = ω 1 2π = rad s = Hz 2π
4 For the calculation of the experimental frequency, the experimental setup mentioned before was used. The piezoelectric was attached to the aluminum beam, and also to the oscilloscope. This allows the mechanical deformation of the beam to be converted to a voltage that can be read by the oscilloscope. The beam was clamped down to ensure to movement other than the bending, and the oscilloscope was turned on. The experiment proceeded by flexing the beam and allowing oscillation in the vertical direction. The voltage produced by the piezoelectric was graphed versus time by the oscilloscope. That graph is shown below s.0518 s.0944 s.1350 s In order to calculate the frequency using the graph presented above, there is a simple equation that utilizes time at the peak values of two adjacent waves on the graph, subtracting them and finding the inverse. This equation for frequency is: 1 f = x 2 x 1 To ensure a proper calculation, there were three sets of data points taken, and the average of those values was utilized to calculate the final results. The equation, as it was applied to the values obtained from the waves is shown below. 1) f = 1 = ) f 1 = = ) f = 1 = The average of these three values was taken to ensure a more accurate comparison to the theoretical value. This is shown below.
5 f avg = = Hz 3 The results of both the theoretical and experimental values of frequency will be compared in the next section. Comparison and Discussion: From the values gathered from data analysis, there is enough information to make a comparison of theoretical, to applied results. Utilizing the properties of aluminum, and the dimensions of the cantilever beam, the first natural frequency was found to be Experimentally, through the use of a piezoelectric to measure the mechanical deflection through voltage output, the frequency was found to be on average The percent error as a comparison between the two methods is: % Error = % =.25% As can be determined by the percent error calculation, there is not much difference between the theoretical and experimental value. This shows that the piezoelectric is effective at converting mechanical stress to electricity. The accuracy of the piezoelectric is one of the reasons it is becoming so widely used. It can sense small changes and as demonstrated in this experiment, can perform very efficiently. References: Lin, Yirong, Mechanical Engineering Professor, UTEP
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