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 Experimental Result Vertical Dimension Analysis Summary

Horizontal Dimension (Force field)Experimental SetupExperimental ResultVertical DimensionAnalysisSummary

Forces

Magnetic force Gravitational force Dissipative force

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.

Force Field

Fig. Potential diagram for the upper magnet

Horizontal Dimension

Experimental SetupExperimental ResultVertical DimensionAnalysisSummary

Tube Confinement

Large friction Start with large

amplitudeSide view

Top view

Tube

String Confinement

Large friction Start with large

amplitudeSide view

Top view

String

Beam Confinement

Almost frictionless Start with small amplitude

Experimental Procedures

Perturb the upper magnet Record by camera Change initial amplitude Change length (l) Change mass (m)

Horizontal DimensionExperimental Setup

Experimental ResultVertical DimensionAnalysisSummary

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

String Confinement

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

Experimental Results

withPeriod

The curve at the bottom turning point is sharperAmplitude decays Period reduces

Beam Confinement

C=6.4*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

Experimental Results

Almost frictionlessPeriodic motion

T=0.17 ±0.00 s

Horizontal DimensionExperimental SetupExperimental Result

Vertical DimensionAnalysisSummary

Magnetic Force vs. Separation

Verifying the Equation

l

l

r

Horizontal DimensionExperimental SetupExperimental ResultVertical Dimension

Analysis AnalyticalNumerical

Summary

Equation of Motion

: Moment of Inertia

Small Amplitude Approximation

Small oscillation

period Ts =

The force can be linearized.

Finite Amplitude

,

Thus, there are only three parameters , , .

Numerical Solution

Finite oscillation period

T=f (Ts, , )

Comprehensive Solution of

y0 ↑ ， T↑

y0 →0 ， T →Ts

l →large ， T X l

1.0

1.4

1.0

2.2

Usage of the Solution Diagram

Period (T)

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

Finite Damping

Horizontal DimensionExperimental SetupExperimental ResultVertical DimensionAnalytical ModellingNumerical Modelling

Summary

ConfinementsTubeStringBeam

Analytical ModellingNumerical Modelling

Summary

1.0

1.4

Thanks for listening!

S.H.O.,

Damping force proportional to velocity:

Small Amplitude Approximation

teyty d

tb

cos)( 20

22 bod , where

Finite Amplitude

Constant friction Damping force proportional to velocity

Both term