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Problem 14 Magnetic Spring

Problem 14 Magnetic Spring. Reporter: Hsieh, Tsung -Lin. 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. Outline. Horizontal Dimension (Force field) ‏ Experimental Setup

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Problem 14 Magnetic Spring

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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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