Eddy Current Braking - files.1337upload.netfiles.1337upload.net/MagneticDamping-e6ea39.pdf · The...

Eddy Current Braking

Geoff Baker

March 24, 2014

Eddy Current Braking - Geoff Baker 1

Abstract

Magnets can be used to induce eddy currents in conductive materials that create a force opposingthe material’s motion through the field, brakes that take advantage of this effect are known as eddycurrent brakes.

The aim of this experiment was to determine the relationships between the magnetic field strength,the linear damping coefficient, the material the non-magnetic component of the brake is made from,the time taken for the pendulum to come to rest and the velocity of the object for the purpose ofmodelling the damping effect, as well as investigating the effect changing any of the variables hason the efficiency of the brake.

The experiment consisted of moving a rigid conductive non-magnetic pendulum through a magneticfield of varying strength and with varying initial amplitude whilst recording the time the pendulumtook to come to rest for each configuration.

The experiment showed that

t ∝ ln

(1

Magnetic Field Strength

)and

α ∝ − 1

ln (Magnetic Field Strength)

where t is the time for the pendulum to come to rest and α is the linear damping coefficient. Theserelationships show that the damping factor is increased as the magnetic field strength increases,and the pendulum comes to rest faster as the magnetic field strength increases. The experimentalso showed that

Fd ∝ θ̇

where Fd is the braking force and θ̇ is the angular velocity of the pendulum, showing the brakesare more effective at higher velocities.

The results collected pertaining to the relationship between the damping effect and the material ofthe pendulum seem to contradict what would be expected. In all trials, the aluminium pendulumexperienced a greater damping effect than the copper pendulum, even though copper has a higherconductivity than aluminium[6] meaning that the magnets induce greater currents in the mate-rial and therefore greater magnetic fields opposing the motion, which should increase the dampingeffect. This discrepancy is most likely due to the physical dimensions of the pendulums beingdifferent, the aluminium being the larger, providing more surface area for the eddy currents topropagate on.

A model to predict the motion of the pendulum over time was created, but when used with the datacollected from the experiment it provided inaccurate results, most probably due to forces presentin the practical that were unaccounted for in the model.


Introduction

The purpose of this experiment is to determine the relationship between the strength of a magnet,the linear damping coefficient, the time taken for the pendulum to come to rest, the pendulum’svelocity and the damping effect it induces on a conductive non-ferrous object passing through thefield as well as produce a model to predict the behaviour of a damped system.This damping effect can be used as a brake in certain conditions, namely where the velocity of theobject moving through the magnetic field is high. The main advantages of an eddy current brakeover a friction based break is that no contact between the magnets and the object is required. Thismeans there is no wear, and the brake is silent. Example applications of this are on a rollercoasteror a train.

Brakes of ‘Goliath’ Rollercoaster at ‘Walibi World’ [1]

Theory

When a conductive material passes through a magnetic field, or vice versa, it experiences a forceopposing it’s motion due to the creation of eddy currents in the metal. These currents are namedafter eddies (swirling currents) produced on water when, for example, an oar is moved through it.The eddy currents induced in metal create a magnetic field which repels the magnet, creating aforce against it, and thereby damping it.

To calculate the damping coefficient, the equation for an undamped pendulum is required. Therestoring force on the pendulum is due to the parallel component of the pendulum’s weight, givenby −mg sin θ. Using Newton’s second law, this force will be equal to the mass of the pendulummultiplied by its tangential acceleration, mlθ̈, where θ is the angle of displacement in radians, θ̈ isthe second derivative of θ (radial acceleration).


−mg sin θ = mlθ̈

θ̈ = −gl

sin θ

θ̈ ≈ −glθ

The damping force acts opposite the motion and is proportional to the velocity of the pendulum.This is therefore given by the equation

Fd = −αθ̇

where α is the linear damping coefficient with units Nsm−1 and θ̇ is the derivative of θ (radialvelocity). Inserting this into the equation for an undamped pendulum gives the equation

θ̈ = −glθ − αθ̇

θ̈ + αθ̇ +g

lθ = 0

The solution of this differential equation will be of the form θ = ert[2]. Therefore, θ̇ = rert andθ̈ = r2ert.

r2ert + αrert +g

lert = 0

r2 + αr +g

l= 0

This equation can be solved to give all possible values of r[3]

r =−α±

√α2 − 4(g/l)

2

These solutions can then be used to create an equation for θ(t)

θ(t) = C1e−α+√α2−4(g/l)2

t + C2e−α−√α2−4(g/l)2

t


C1 and C2 can be calculated based on the starting conditions of the pendulum. Since the pendulumwill start at its maximum amplitude

θ(0) = a0

a0 = C1 + C2

where a0 is the initial amplitude of the pendulum. Assuming its motion is symmetrical, bothparameters of the equation, C1 and C2, are equal.

C1 = C2 =a0

2

The equation can be simplified further based on the above assumption

θ(t) =a0

2

(er1t + er2t

)Simplifying even further is possible through the use of Euler’s Formula[4] eix = cosx+ i sinx

ea+bi = eaebi = ea(cos b+ i sin b)

ea+bi + ea−bi = ea2 cos b

Substituting in the real and imaginary parts of r for a and b gives

θ(t) =a0

2e<(r)t2 cos=(r)t

θ(t) = a0e−α

2t cos

√α2 − 4(g/l)

2t

The amplitude of the pendulum at time t is controlled by a0e−α

2t. Since ex is never 0, this model

will never have a maximum amplitude of 0, therefore when the magnitude of a0e−α

2t is negligible

at time t the pendulum can be considered to be at rest.

1× 10−10 = a0e−α

2t

α = −2 ln

(1×10−10

a0

)t

The above equation can be used to calculate the linear damping coefficient given the time ittakes the pendulum to come to rest. Since the numerator of the equation remains constant, therelationship between α and t is

α ∝ 1

t

α is dependant on the strength of the magnetic field the pendulum is passing through and thematerial the pendulum is made of. Plotting t against the strength of the magnetic field will showthe effect increasing the strength of the magnetic field has on the time it takes the pendulum tocome to rest.


Initial Plan

Sheets of different metals will be moved through a magnetic field of varying strength on a rigidpendulum and the time taken for the pendulum to come to rest will be recorded. The experimentwill be repeated three times for each combination of magnets and metals, and each of these repeatedwith different starting displacements.

Experiment Setup

The pendulum is held in a clamp stand adjusted so that there is minimal resistance to the pen-dulum’s motion and is positioned so the metal will pass through the magnetic field at its point ofmaximum velocity (where θ = 0), to maximise the damping effect.The two large steel covered magnets are clamped to the desk with a distance of 30mm separatingthem. The size of the neodymium and ferrite magnets makes it difficult to hold them in place onthe desk, instead, the magnets are held in place by placing them on the two steel covered magnets.

Preliminary Results and Analysis

Table 1: Preliminary results with aluminium, amplitude π3 radians

Time (s)

Magnets 1 2 3 Average Ratio

None 17.81 17.68 18.15 17.88 1

Steel 9.14 9.65 10.04 9.61 0.54

Steel and Ferrite 6.01 5.54 5.98 5.84 0.33

Steel and Neodymium 1.26 1.21 1.43 1.30 0.07

These initial results show the expected outcome, as the strength of the magnetic field increases, thependulum comes to rest in less time: the damping factor increases with magnetic field strength.The ratio of the time taken to come to rest with no magnetic field and the time taken to come torest with the steel and neodymium magnets is incredibly small, more so than would be expected.This could be due to the distance between the magnets and the metal, since the magnetic field


strength diminishes at a rate of r−2[5]. The trial was repeated with a constant distance of 25mmbetween the magnets at their closest point to investigate whether the distance between the magnetsand the metal was affecting the results significantly.

Table 2: Trial with aluminium, constant magnet separation distance, amplitude 2π9 radians

Time (s)


None 3.46 3.36 3.18 3.33 1

Steel 2.26 2.15 1.98 2.13 0.639



These ratios are greater than the preliminary results, especially for the neodymium and steel magnetconfiguration. Repeating with another amplitude to confirm these ratios are more accurate thanthose provided by the preliminary trial

Table 3: Trial with aluminium, constant magnet separation distance, amplitude π6 radians

Time (s)


None 4.72 4.74 4.74 4.73 1

Steel 2.55 2.41 2.44 2.47 0.538



These results have a similar change in ratio between magnet configurations, about 0.1 from steelto steel and ferrite (Table 2: 0.639→ 0.533, Table 3: 0.538→ 0.417) and from steel and ferrite tosteel and neodymium a change of around 0.2 (Table 2: 0.533 → 0.332, Table 3: 0.417 → 0.238).Making this change removes distance as a factor in the damping effect caused by the magnets onthe pendulum. In all trials after these, the magnets were set to a constant 25mm apart.Table 1, 2 and 3 are the results of trials with initial amplitudes of π

3 , 2π9 and π

6 radian respectively.However, even the smallest amplitude, 2π

9 , is too large to work with the small angle approximationthis model relies on.

2π

9= 0.6981

sin 0.6981 = 0.6427 Difference: 7.94%

This is a large percentage difference , making any calculations done using these results inaccurate.To ensure the approximation is accurate to at least 1 decimal place, only amplitudes less than 0.3radians will be used in further trials.

sin 0.3 = 0.2955 Difference: 1.5%

Improvements and Final Method

The main problems with the initial plan were that it did not specify a region of acceptable initialamplitudes for the small angle approximation to be valid and that it did not ensure the magneticfield strength was dependant only on the magnets themselves. Changes made to correct these flawswere to limit a to less than 0.3 radians, where the small angle approximation is valid, and to makesure the distance between the two magnets is always 25mm.With these changes accounted for, the final method is as follows:


• Attach a flat piece of non-magnetic metal to one end of a rigid pendulum suspended by aretort stand.

• Secure two magnets to the work-surface, 25mm apart and either side of the pendulum at rest,using G-clamps.

• Using a protractor to measure the angle the pendulum makes with the vertical, set thependulum in motion with an initial amplitude less than 0.3 radians.

• Record the time it takes the pendulum to come to rest three times and take the average time.

• Repeat with different magnet configurations (ensuring the innermost magnets are 25mmapart) and different initial amplitudes.

The main risk associated with this experiment is to do with the use of strong magnets. If allowedto jump together, the magnets can very easily shatter and project shrapnel into eyes and flesh etc.The magnets can also pinch flesh and cause blood blisters if handled incorrectly.

Results


Time (s)


None 1.76 1.72 1.76 1.75 1

Steel 1.36 1.18 1.20 1.25 0.714



Table 2: Trial with copper, constant magnet separation distance, amplitude π18 radians

Time (s)


None 1.82 1.85 1.98 1.88 1

Steel 1.6 1.41 1.6 1.54 0.816




Graph 1

0 8 16 24 32 40 48 56 64 72 80

0.5

1

1.5

2

Plot of time(s) against relative magnetic strength. Amplitude π18 .

Copper = red, Aluminium = blue.


Time (s)


None 4.10 4.16 4.20 4.15 1

Steel 2.23 2.18 2.29 2.23 0.538



Table 4: Trial with copper, constant magnet separation distance, amplitude π12 radians

Time (s)


None 5.23 5.28 5.17 5.23 1

Steel 3.06 3.31 3.21 3.19 0.611




Graph 2

0 8 16 24 32 40 48 56 64 72 80

0.8

1.6

2.4

3.2

4

4.8

Plot of time(s) against relative magnetic strength. Amplitude π12 .

Copper = red, Aluminium = blue.

Curves of Best Fit

Aluminium π18 y = 1.7532e−0.0094x

Copper π18 y = 1.8999e−0.0096x

Aluminium π12 y = 4.1359e−0.0188x

Copper π12 y = 5.3071e−0.0166x

The curves of best fit can be used to predict the time to come to rest at any relative magneticstrength for any situation where the initial conditions are the same. However, for more generalpredictions, the data collected must be converted into a damping factor (α) specific to the mag-net/metal configuration, to be used in the equation

θ(t) = a0e−α

2t cos

√α2 − 4(g/l)

2t

Table 5: Damping Factor calculated for each configuration using results from π12 amplitude tests

Damping Factor

Magnets Aluminium Copper

Steel 19.45 13.6

Steel and Ferrite 25.07 16.62

Steel and Neodymium 43.81 35.26

Analysis

The results recorded in table 2 for trials with the steel magnet and steel and ferrite magnets appearto be anomalous on graph 1, due to an abnormally large change in time for a relatively small change


in magnetic strength. This change is not seen in the other three result sets, so is most likely dueto human error, either in recording the time or the initial amplitude.

Copper appears to be less affected by the damping as the ratios on trials involving copper aregreater than trials using aluminium. Since more conductive materials produce greater eddy cur-rents, copper would be expected to be damped more than aluminium. The fact that the opposite isobserved could be related the to physical dimensions of each material used in the experiment, thealuminium being slightly larger than the copper, meaning that the aluminium spends more timepassing through the magnetic field each time the pendulum oscillates.

The graphs and the curves of best fit show that

B ∝ e−t

t ∝ ln

(1

B

)where B is the magnetic field strength. Combining this relationship with the predicted relationship

between α and t(α ∝ −1

t

)gives

α ∝ 1

ln(

1B

)α ∝ − 1

ln (B)

Graph 3

-0.288 -0.28 -0.272 -0.264 -0.256 -0.248 -0.24 -0.232 -0.224 -0.216 -0.208

8

16

24

32

40

48

Calculated values of α for aluminium with initial amplitude π12 plotted against − 1

ln (B)Line of best fit y = 481.5871x+ 154.7262

Graph 3 confirms the relationship between α and B.

Comparing the gradients of the curves of best fit shows the impact the amplitude has on dampingeffect. For aluminium with initial amplitude π

18

dt

dB= −0.01648008e−0.0094B


and for aluminium with initial amplitude π12

dt

dB= −0.07775492e−0.0188B

Because the maximum velocity of the pendulum increases as the initial amplitude does, the gradi-ents of these two curves confirm that eddy current brakes are more effective as velocity increases,because a change in B produces a greater change in t when the amplitude is greater.

Plotting θ against t with α = 19.45 and a0 = π12 produces this graph

Graph 4: θ = π12e

19.452

t cos

√19.452−4∗(g/0.196)

2 t

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.08

0.16

0.24

0.32

0.4

The graph shows the pendulum start with an initial amplitude of θ = π12 , pass through the magnetic

field at θ = 0 and lose velocity before continuing to oscillate about θ = 0 with decreasing amplitude.This is an accurate representation of what the pendulum does in practice, however the model’samplitude seems to decrease and come to rest sooner than in practice. This is most likely dueto external forces acting on the pendulum in practice that the model does not account for whencalculating the damping factor, such as friction on the pivot and air resistance etc.

Conclusions

In conclusion, the experiment has shown that

t ∝ ln

(1

Magnetic Field Strength

)and therefore

α ∝ − 1

ln (Magnetic Field Strength)

This means that to increase the braking effect (decrease the time taken to come to rest) the magneticfield strength must be increased. The experiment also showed the damping force increases with thespeed of the object passing through the field.

Fd ∝ θ̇


Data from the experiment was able to be used to create a model for the pendulum’s motion, butit could not be used to accurately predict the system’s behaviour due to forces in the system thatwere not included in the calculation of the damping factor.

The experiment could be explored further through the use of a greater range of magnets with awider variation of magnetic field strengths. More results could be collected with a more preciseway of measuring the initial amplitude, so that velocity’s impact on the the damping could be morethoroughly investigated.

Bibliography

[1] Stefan Scheer, “Brakes of Rollercoaster Goliath at Walibi World”,http://en.wikipedia.org/wiki/File:Goliath Brakes.jpg

[2] MIT OpenCourseWare, “The Characteristic Polynomial”, http://ocw.mit.edu/courses/mathematics/18-03sc-differential-equations-fall-2011/unit-ii-second-order-constant-coefficient-linear-equations/modes-and-the-characteristic-equation/MIT18 03SCF11 s12 2btext.pdf

[3] Weisstein, Eric W. “Quadratic Formula.” From MathWorld–A Wolfram Web Resource.http://mathworld.wolfram.com/QuadraticFormula.html

[4] Arthur Mattuck, “Complex Numbers”, http://math.mit.edu/suppnotes/suppnotes03/c.pdf

[5] R Nave, “Magnetic Flux”, B = ΦA , http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/fluxmg.html

[6] Wikipedia, “Electrical Resistivity and Conductivity”,http://en.wikipedia.org/wiki/Electrical resistivity and conductivity

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