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
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