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Investigation of Magnet Oscillations in Vertical Configuration

This report explores the oscillations of a moving magnet arranged vertically with a fixed magnet, analyzing forces, force fields, experimental setups, and results. The study includes analysis, experimental procedures, and finite damping effects.

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Investigation of Magnet Oscillations in Vertical Configuration

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  1. Problem 14Magnetic Spring Reporter: Hsieh, Tsung-Lin

  2. Question • Two magnets are arranged on top of each other such that one of them is fixed and the other one can move vertically. • Investigate oscillations of the magnet.

  3. Outline • Horizontal Dimension (Force field)‏ • Experimental Setup • Experimental Result • Vertical Dimension • Analysis • Summary

  4. Horizontal Dimension (Force field)‏ • Experimental Setup • Experimental Result • Vertical Dimension • Analysis • Summary

  5. Forces • Magnetic force • Gravitational force • Dissipative force

  6. Force Field • Cylindrical magnet can be interpreted by a magnetic dipole. • When the upper magnet is at the unstable equilibrium position, the separation is said to be r0. Fig. Potential diagram for the upper magnet

  7. Horizontal Dimension Experimental Setup • Experimental Result • Vertical Dimension • Analysis • Summary

  8. Tube Tube Confinement • Large friction • Start with large amplitude Top view Side view

  9. String Confinement String • Large friction • Start with large amplitude Top view Side view

  10. Beam Confinement • Almost frictionless • Start with small amplitude

  11. Experimental Procedures • Perturb the upper magnet • Record by camera • Change initial amplitude • Change length (l)‏ • Change mass (m)‏

  12. Horizontal Dimension • Experimental Setup Experimental Result • Vertical Dimension • Analysis • Summary

  13. Tube Confinement • C=6.4*10-4 J-m • m=5.8 g • l=1.00 cm • y0=12.2 cm • v0=0 cm/s

  14. String Confinement • C=5.4*10-5J-m • m=5.7 g • l=1.00 cm • y0=23 cm • v0=0 cm/s

  15. Experimental Results • with Period • The curve at the bottom turning point is sharper • Amplitude decays • Period reduces

  16. Beam Confinement • C=6.4*10-4J-m • l=1.00 cm • mmagnet=5.8 g • mbeam=10.0 g • Beam length=31.9 cm • y0=0.88 cm • v0=0 cm/s

  17. Experimental Results • T=0.17 ±0.00 s • Almost frictionless • Periodic motion

  18. Horizontal Dimension • Experimental Setup • Experimental Result Vertical Dimension • Analysis • Summary

  19. Magnetic Force vs. Separation

  20. Verifying the Equation l r l

  21. Horizontal Dimension • Experimental Setup • Experimental Result • Vertical Dimension Analysis • Analytical • Numerical • Summary

  22. : Moment of Inertia Equation of Motion

  23. Small Amplitude Approximation The force can be linearized. Small oscillation period Ts =

  24. Finite Amplitude , Thus, there are only three parameters , , .

  25. Numerical Solution Finite oscillation period T=f (Ts, ,)‏

  26. Comprehensive Solution of • y0↑，T↑ • y0→0， T→Ts • l →large，TXl 1.0 1.0 1.4 2.2

  27. Period (T)‏ Usage of the Solution Diagram • C=6.39*10-4 J-m • l=1.00 cm • mmagnet=5.8 g • mbeam=10.0 g • Beam length=31.9 cm • y0=0.88 cm • v0=0 cm/s

  28. Finite Damping

  29. Horizontal Dimension • Experimental Setup • Experimental Result • Vertical Dimension • Analytical Modelling • Numerical Modelling Summary

  30. Summary • Confinements • Tube • String • Beam • Analytical Modelling • Numerical Modelling 1.0 1.4

  31. Thanks for listening!

  32. , where Thus, Small Amplitude Approximation • S.H.O., • Damping force proportional to velocity: Fig. Analytical result Fig. Tube confinement result

  33. Finite Amplitude Damping force proportional to velocity Constant friction Both term

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