Pendulum with Vibrating Base Energy Manifold Team Members Project Description Thomas Bello Emily Huang Fabian Lopez Kellin Rumsey Tao Tao Theoretical Models Methodology 1. The Pendulum is treated as a point mass at its center. 2. The equations of motion for the vertical, horizontal, and arbitrary cases are determined by finding the Lagrangian for each case. 3. Determine stability conditions. 4. Using averaging, determine the Effective Potential. 5. Use the Effective Potential to determine ranges of stability. Results β’ Vertical Equations Experimental Model 3. 4. Measure the dimensions of the pendulum and find the center of mass. Using a high speed camera, determine the minimum frequency of stability for the vertical, horizontal, and arbitrary case. Compare experimental results with theoretical expectations. Determine error. Measurement Length of Pendulum (m) Diameter of Pendulum (m) Amplitude (m) Minimum Maximum Frequency (rad/s) Angle of Base Momentum of Inertia .187 0.009525 0.020 0.010 0.030 275.62 51° πΏπππππππππ πΏ = πΎππππ‘ππ πΈπππππ¦ β πππ‘πππ‘πππ πΈπππππ¦ π ππΏ ππΏ πΈπ’πππ β πΏπππππππ πΈππ’ππ‘πππ ππ πππ‘πππ: β =0 ππ‘ ππ ππ 1 πΏ = π π 2 + π sin π π0 π2 cos ππ‘ + ππππ (π) 2 π0 π2 π πβ cos π cos ππ‘ + sin(π) = 0 π π 2 π 1 ππ π2 2 ππππ = β cos π β sin (π) π 4 π2 ππ‘ππππππ‘π¦ πΆπππππ‘πππ: π > 2ππ 2ππ β1 β 2 2 2 , πΆπππ‘ππππ π΄ππππ: ππ = cos π0 π0 π Potential Applications β’ Arbitrary Equations 1 2 πΏ = π π + π sin π β π π0 π2 cos ππ‘ + ππππ (π) 2 π0 π2 π πβ cos π β π cos ππ‘ + sin(π) = 0 π π 2 2 π 1 ππ π ππππ = β cos π β sin2 (π β π) 2 π 4 π Results and Comparison Raw Data Table Theoretical Definitions β’ Horizontal Equations Methodology 1. 2. 1 πΏ = π π 2 + π sin π π0 π2 sin ππ‘ β ππππ (π) 2 π0 π2 π π+ cos ππ‘ β sin π = 0 π π π 1 ππ2 π2 2 ππππ = β cos π + sin (π) π 4 π2 Error Analysis The theoretical model we used is for a simple pendulum. The pendulum used in the experiment was a physical pendulum. To develop a more accurate model, we account for the moment of inertia of the rod, which is summarized in the table below under the βCorrected Theoreticalβ column Measurement Theoretical Experimental Corrected Theoretical Vertical Stability Angle Critical Angle Horizontal Stability Angle Arbitrary Stability Angle 180° 97° 83° 136° 180° 113° 76° 109° 180° 98° 82° 135° Error Analysis Measured Value Absolute Error Percent Error Length of Pendulum (π) π = 0.187 meters Ξ΄π = 0.001 meters .54% Amplitude of Base (dΛ³) dΛ³ = 0.020 meters Ξ΄dΛ³ = 0.001 meters 5.0% Period for 60 Oscillations (T*) T* = 1.35 seconds Ξ΄T* = 0.05 seconds 3.7% *Note that angular frequency could not be directly measured. T* is the length of time required for the base of the pendulum to make 60 oscillations. T* 60 can be related to Angular Frequency in the following way: π = 2π β πβ πΏπ = ππ πΏπ ππ 2 ππ β + πΏπ ππ β 2 ππ + πΏπ0 ππ0 ΞΈc = 97° ± 0.86 ° = 97° ± 0.88% ΞΈs = 83° ± 0.86 ° = 83° ± 1.04% 2 A Segway is an example of an Inverted Pendulum with an oscillatory base, while Magnetic Levitation is a system that uses the interaction of two oscillatory forces with different speeds. Reference 1) Landau L. D. & Lifshitz E. M., Mechanics, Second Edition: Volume 1, (Course of Theoretical Physics), (Oxford ; New York : Pergamon Press, 1976), pp 665-70. Oscillations of systems with more than one degree of freedom. Acknowledgement This project was mentored by Prof. Ildar Gabitov, whose help is acknowledged with great appreciation. We would also like to acknowledge support from the Bryant Bannister Tree-Ring Lab for access to their equipment. Support from a University of Arizona TRIF (Technology Research Initiative Fund) grant to J. Lega is also gratefully acknowledged.
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