magnetorheological fluid analysis final report sa vrednošću relativne permeabilnosti za aisi 1018

7/30/2019 Magnetorheological Fluid Analysis Final Report sa vrednošću relativne permeabilnosti za AISI 1018

http://slidepdf.com/reader/full/magnetorheological-fluid-analysis-final-report-sa-vrednoscu-relativne-permeabilnosti 1/26

of 26

Magnetorheological

Fluid Analysis:

Final ReportJulia Cossé

Molly Glenn

Brendan Tracey

Abstract: We attempted to characterize Magnetorheological fluids (MRF) while

exposed to a fluctuating magnetic field. We built an AC power source out of two

DC power supplies. We acquired Hall Effect chips with which we measured the

magnetic field during testing. We calibrated the magnetorheometer as well as the

Hall Effect chips. We then attempted to characterize the magnetic field as a

function of frequency. Unfortunately we were never able to run tests on the MRF

while under the fluctuating magnetic field due to problems with the AC power

supply and torque measurements. Given more time, and resolving these problems

it would be possible to find relationships between frequency and viscosity, and to

create a mathematical model for MRFs under an AC driven magnetic field.



of 26

1. IntroductionThe goal for this semester was to examine a magnetorheological fluid (MRF) under the

influence of a magnetic field generated by an electromagnet powered under alternating current.

MRFs are fluids that are made up of a mixture of 5-10μm iron particles suspended in a carrier fluid such as silicon oil, mineral oil or glycol. The iron particles in the carrier fluid give the

MRF properties that are uncharacteristic of general fluid properties while the MRF is exposed toa magnetic field. These properties include the rapid increase in the magnitude of viscosity of

the fluid as well as displaying properties characteristic of solids, such as rigidity and ability tohold a shape.

MRFs have been used in a number of areas in technology. Applications include

polishing lens, due to their ability to control shape, and as shock absorbers because themovement of the fluid can be finely tuned. Generally speaking, more is known about MRFs

when they are used in constant magnetic fields. These fields can be created by electromagnets

that are driven by direct currents. However, little is known about the characteristics of MRFsunder a fluctuating magnetic field.

In the spring of 2006 and following fall, Dean Kleissas, Matt Roe, and Phillip

Scheupbach (from now on referred to as the KRS group) did a series of experiments using amagnetorheometer in order to aid them in the characterization of MRFs. By running the

magnetorheometer on either a DC or an AC power source they were able to look at the changes

in shearing, and viscosity properties of the fluid. However, their results in characterizing the

field under a fluctuating magnetic field were incomplete.

2. Experimental Procedure

2.1 The Magnetorheometer A few years ago, Dr. Aric Shorey of QED technologies lent a magnetorheometer to the

University of Rochester. In the spring of 2006 the KRS group reassembled the machine (See

Figure 1). The magnetorheometer is made up of an electromagnet, a motor in order to turn theshaft, and a device which measures the torque applied by the motor to the shaft. Theelectromagnet is made up of a large coil of 1018 steel, and is powered by plugging the coil into

a power source. Surrounding the coil is an enclosure of steel, which helps contain the magnetic

field to the testing region. Attached to the electromagnet is a variable mount on which the IKAEurostar is attached. The Viscoklick torque meter is attached to the Eurostar by means of a

coaxial cable, and sits behind the magnetorheometer on the table.

The Eurostar rotates the shaft on which the gear head is attached, and also measures the

torque. It is connected to the computer via a RS232 cable and read by a program calledLabWorldSoft. The Eurostar rotates at a speed specified by an arbitrary scale that goes from 0

to 2000. In order to calibrate the Eurostar scale to rotations per minute (RPM), we used a

tachometer and plotted RPM versus the Eurostar Scale (See Figure 2).



Figure 1: The Magnetrheometer

RPM vs. Eurostar Scale

y = 0.0666x - 0.0445

-20

0

20

40

60

80

100

120

140

0 500 1000 1500 2000 2500

Eurostar Scale

R P M

Figure 2: Rotational Speed Calibration

of 26



It was previously determined through a pulley test by the KRS group that the torque

measured by the Eurostar was incorrect. This was addressed by using the Viscoklick tomeasure the torque. The KRS group also ran pulley tests on the Viscoklick and determined that

they were measuring a reasonable value for torque. We therefore ignored any output of torque

by the Eurostar and originally assumed that the Viscoklick was performing properly.

2.2 Calibration TestsAfter familiarizing ourselves with the machine, we tested two calibration fluids with

certified viscosities that were purchased from Brookfield Industries. To perform these tests we put the calibration fluids into the load plate and let them sit there for an extended period of time

so that they would level out. We then lowered the gear head onto the load plate, taking care to

measure the height of the gear head above the plate, and ran a 3 minute test usingLabWorldSoft. The program within LabWorldSoft was designed so that the rotational speed

would slowly ramp up from rest to 40 RPM over a 45 second period of time. After 2 minutes

and 15 seconds at full speed, the ramp would bring the rotational speed back to rest. Whenlooking at our data we took the value of the average torque during the steady state portion of the

test. Using this value for torque we calculated the viscosity by using equation (1), which isderived in Appendix A.

( )4 4

2

o i

h T

r r μ

π ω

⋅ ⋅=

⋅ ⋅ −(1)

Where, μ is the dynamic viscosity, h is the gap height in meters, T is the torque in Newton

meters, ω is the rotational speed in radians per second, r o is the outside radius of the gear headand r i is the inside radius of the gear head. In comparing our calculated value of viscosity with

the certified value we found our values did not match the certified values. For the 200,000 cP

fluid, we found a viscosity of 69,407 cP, and for the 400,000 cP fluid, we found a viscosity of 103,077.6 cP. The viscosities are certified at 25°C, whereas our measurements were taken at

approximately 21°C. We do not think this difference in temperature is the reason for the

inaccuracies in our viscosities, especially since in general viscosity increases with lower temperature. Our values were also consistently inconsistent, as we ran several tests on the

calibration fluid and each time acquired the same results. Our next step was to reexamine the performance of the Viscoklick.

2.3 Pulley Tests When our calibration fluids didn’t have the expected viscosity values, we wanted to see

if the Visocklick was giving us accurate data. To do this we ran the same pulley test that the

KRS group did last spring to check the functionality of the Eurostar torque meter. A string was

attached to the gear head of the magnetorheometer and was then run over a pulley. On the other side of the pulley, a weight was attached to the string and hung in the air. By using the radius of

the gear head and weight of the pulley, we compared the actual torque value to the torque value

given by the Viscoklick. From these tests we determined that the Viscoklick was not readingaccurately. However, the torque reading on the Viscoklick was linearly related to the actual

torque applied (See Figure 3). Thus when we look at the data collected by the Viscoklick we

will use the calibration curve produced by the pulley test in order to convert the Viscoklick reading to actual torque.

of 26



Pulley Test

y = 0.7636x - 0.5123

R2 = 0.9995

-5

0

5

10

15

20

25

30

35

40

0 10 20 30 40 50 6

Viscoklick Torque (N cm)

T h e o r e t i c a l T o r q u e ( N c m )

0

Figure 3: Calibration of Viscoklick Torque Reader

2.4 The Magnetorheological Fluid We measured out a mixture of 75% by mass iron particles and 25% by mass 1000

Centistoke Silicon oil using an OHAUS precision scale. The iron particles and the silicon oil

were among the equipment that came with the magnetorheometer from QED technologies. Theiron particles were 5-10μm in diameter.

2.5 Hall Effect Chips In order to analyze the behavior of the MRF as a function of the magnetic field, we

wanted to know the strength of the magnetic field at the time of each measurement as we weregoing to be using a fluctuating magnetic field. In order to do this Professor Gans suggested that

we use Hall Effect chips so that we could create a VI in LabVIEW in order to get real time

measurements of the magnetic field. We ordered samples of Hall Effect chips from AllegroMicroSystems Inc. The Hall Effect chips were chosen because of their very fast response time.

Therefore, as the magnetic field fluctuates, the Hall Effect chips can continue to read the field

output accurately. Professor Gans then helped us build a VI through LabVIEW to read thevoltage output from the Hall Effect chips. Each Hall Effect chip was wired into a separate DAQ

assistant by running the power into AO0, the ground into GND and the Vout into AI0. Due to

grounding issues and to have a common node, there were also connecting wires between AI4 to

GND, AI1 to AO0, and from AI5 to GND.

2.6 Skin Depth The unequal charge distribution within the wire during alternating current flow is

referred to as the skin effect. The skin effect affects the output of the magnetic field. In DC

there is no skin effect as the skin depth goes to infinity. However, once we start testing our

of 26



circuit under an AC power source, this becomes an important variable. We calculated the skin

depth of the magnet at various frequencies in order to see if there was any relationship betweenthe skin depth and the drop off in magnetic field (See Appendix B). We found that at a

frequency of 1 kHz, the skin depth was 0.2758 mm and at a frequency of 10 Hz the skin depth

was 2.758 mm. This is a very significant result because we want the skin depth to be as large as

possible, ideally large compared with the thickness of the coil. However, the skin depth quickly becomes very small, which is possibly one of the reasons for a drop off in magnetic field at

higher frequencies.

2.7 Magnetic Field Characterization – MagnetorheometerLast year the KRS group characterized the magnetic field versus the current applied to

the magnet and also of the magnetic flux as a function of height within the load plate region.However, they assumed the magnetic field was constant across the load plate. This assumption

is contrary to our own observation of the magnetic field, and thus we wished to do our own

characterization of the magnetic field above the load plate. To do this we powered the magnetwith the DC power source at about 10 amps and put a transparency sheet on the load plate of the

magnet. On the transparency sheet we drew 8 lines from the center of the magnet outward. Thelines were spaced at approximately 45° angle to each other. On each of the lines we drew 9 dots

spaced 0.15 inches apart. The 9th

dot was in the center of the magnet and the 1st

was on theoutside of the load plate. We then tested the magnetic field at each of the points using the Hall

Effect chips (See Figure 4).

Magnetic Field Characterization

-1000

-500

0

500

1000

1500

2000

2500

0 2 4 6 8

Radial Location

M a g n e t i c F i e l d ( G )

10

Figure 4: Characterization of Magnetic Field across the Load Plate

of 26

As seen above, the magnetic field across the load plate is not constant. The location of

the MRF on the load plate is between points 2 and 4, the most inconsistent area of the field. It

was then brought to our attention by Professor Gans that there were multiple configurations of the magnetorheometer, and that it must currently be in the configuration where the magnetic



field was parallel to the load plate, rather than perpendicular to the load plate as we had

previously assumed. We used MATLAB in order to create a 3D rendering of the magnetic field(See Figure 5 and Appendix C).

Figure 5: 3D Rendering of Magnetic Field Characterization

2.8 AC Circuit Modifications Last semester the KRS group built an AC circuit to drive the magnetorheometer out of

two DC power supplies, two transistors and a frequency generator (See Figure 6). One of the

DC power supplies was the power source that the KRS group bought in the spring of 2006 andthe other power supply was donated by Professor Derefinko. The power supply that was

purchased could was an 80V 20A power source, and the supply that was donated by Professor

Derefinko was made up of two 10V power supplies connected together in series, and then wiredso that it would output negative 20V instead of the usual 20V. These two power supplies were

then hooked up to the collectors of the transistors, the negative source to the PNP transistor and

the positive source to the NPN transistor. The bases of both transistors were connected to the

positive end of the frequency generator and the emitter of each of the transistors was connectedto the positive lead of the electromagnet within the magnetorheometer. The 80V 20A power

supply was set so that it operated at an equivalent 20V as the second power supply. By turning

on the frequency generator, the transistors would then control whether the magnet was receivinga positive or a negative voltage, thus forcing the electromagnet to output an oscillating magnetic

field. The transistors were found to overheat as the high currents needed to drive the magnet

were forced through them, so they were placed on heat sinks in order to aid in the dispersion of heat.

of 26



Figure 6: Original AC Circuit

The KRS group determined that there was a problem with this circuit set up because

when they tried to run tests at frequencies above 2Hz the current dropped off much morequickly than they were expecting. They attributed this problem to the response time within the

power supplies. As they tried to obtain higher frequencies, the power supplies were not able to

output the required current to operate the magnet quickly enough. To remedy this problem, theKRS group thought up a modification to the circuit that they thought would fix the problem

(See Figure 7). The idea behind the circuit was to use a dummy load that matched the magnet

in order to keep the power supplies on, which should fix the problems with response time. Atany given time, one of the power supplies would be powering the magnet, and the other would

be powering the dummy load. As the frequency switched, each power supply wouldrespectively switch to powering the other load. This way, at any given time there would be a

load to draw the current from the power supply. If the power supply was always outputting acurrent, there wouldn’t be a response time for the power supply to slowly ramp up deliver the

needed magnitude of current.

of 26



Figure 7: AC Circuit Modification by KRS Group

We tested this hypothesis by testing the response time of the power supplies. To do this

we built a circuit designed by Professor Derefinko. The circuit consisted of a 1Ω resistor inseries with a switch and the power supply. Using an oscilloscope as a visual to look at what

was going on across the resistor; we could flip the switch open and closed in order to look at theresponse time of each of the power supplies. After running this test, we thought that we did seea large enough response time to account for the current drop that was seen by the KRS group.

However, it is also important to note that the electromagnet is a large coil of wire, so that it also

has a large inductance. As higher frequencies are run across it, it would have more difficulty inforcing the same amount of current to flow across it. However, we believe that this would start

happening at frequencies much higher than 2Hz.

Since we attributed the drop in current to the response time of the power supplies, we

went ahead and started to build the AC circuit designed by the KRS group. This involved

of 26



buying two more transistors of the same variety (one NPN and one PNP) and creating a dummy

load that matched the properties of our electromagnet. In order to create a dummy load, weneeded to measure the resistance and inductance of the magnet. To do so we brought the

magnetorheometer to an LCR meter in the Hopeman ECE lab and found that the inductance was

46.7mH and that the resistance was 1.4Ω.

Scott Russell found a few RF doughnuts for us to use as the inductors of our dummyload. Using the LCR meter we used 4 of them in series to create an inductance of 51.4mH. For

the resistive load we used the same 1Ω resister we had used to measure the response of the

power supplies. Though not perfect, the dummy load was close to the same as themeasurements of the magnetorheometer.

We ordered the transistors and heat sinks from Goldcrest Electronics and after the parts

came in we put together the circuit. After we had it up and running we did a few tests using theHall Effect chips to look at the output from the magnetorheometer and it appeared that we were

in fact outputting a sign wave. Unfortunately, due to an operator error, we short circuited the

circuit within 5 minutes of having it up and working. It took us a while to pinpoint what hadgone wrong, but it was eventually determined that the transistors had blown. We then ordered

new transistors and rebuilt the circuit. The circuit, however, was still not working.While going through and troubleshooting the circuit other issues were identified. First

of all, it appeared that instead of the dummy load and the magnet being out of phase with eachother, they would be in phase. The circuit design that was drawn up by the KRS group lacked

one important component, an inverter that needed to be used in order to drive the dummy load

at the opposite point of the magnet. A second problem was that the frequency generator didn’tappear to be outputing enough current in order to drive the transistors. After discussion with

Professor Derefinko, he designed an amplifier for us to use that included an inverter (See

Figures 8 & 9). In order to test the circuit we only hooked up one load and one set of transistors. Using this amplifier we eventually got the magnet to function under the AC circuit

without the dummy load. We immediately began to look at the magnetic field that themagnetorheometer was producing.

Figure 8: Circuit Diagram Designed by Professor Derefinko

of 26



Figure 9: Picture of Built Circuit

2.9 Characterization of AC Driven Magnetic FieldDuring the last week of testing, the AC circuit was finally working properly so we took

some measurements using the Hall Effect chips in order to start characterizing the magnetic

field while it was being driven by the AC power source. One Hall Effect chip was set up in the

center of the magnet on the load plate and another Hall Effect chip was secured on the edge of

the load plate. The second Hall Effect chip was to remain in place during MRF characterizationin order for us to match up the magnetic fields. The first Hall Effect chip was placed in the

center of the magnet because of the extreme difference in magnetic field within the loading

area.After the Hall Effect chips were secured we ran a long series of tests. The tests were run

at intervals of 0.5Hz between 0Hz (DC field) and 5Hz. Each test was repeated with thefrequency generator set to generate a sine wave, a square wave, and a triangular wave. After thetests were completed, the data from LabVIEW were transferred into an Excel file. We

constructed graphs displaying the magnetic field in gauss versus the time in seconds. We also

looked at the average magnetic field and the maximum magnetic field as a function of

frequency.Unfortunately, before we were able to collect data on the MRF response in the AC

driven magnetic field, our AC circuit, once again, stopped working. From the display on the

oscilloscope it looked as though the op amps that we had been using in order to get enough

of 26



voltage out of the frequency generator had saturated, and thus we were no longer getting a

consistent output from our power source.Due to time constraints, we never used the dummy load during testing. Instead we

simply used the AC circuit without the dummy load hookup. The circuit designed by professor

Derefinko, however, has a place to hook up the dummy load. Thus further testing can easily be

done by using this setup.

3 Results

3.1 Hall Effect Chip Calibration In order to calibrate the Hall Effect chips, we used a digital gauss meter that Scott

Russell borrowed from the ECE department for us. We then put the Hall Effect chips at thesame location as the gauss meter, and ran a program in LABVIEW. The program took the 1000

voltage reading from the Hall Effect chip over 1 second, and recorded the average value. At the

same time we recorded the reading from the gauss meter. Using this data, we were able to finda linear relation between the Hall Effect chip voltage and the magnetic field (See Figure 10).

By making a trendline, we were able to use the calibration curves to determine the magneticfield during our experiments.

Hall Effect Chip Calibration

y = 811.45x - 1273.9

y = 747.84x - 1190.2

y = 682.82x - 1073.3

0

500

1000

1500

2000

2500

0 1 2 3 4 5

Hall Effect Chip Outp ut (Volts)

M a g n e t i c F i e l d ( G u a s s )

A1373-1

A1373-2

A1373-3

Linear

(A1373-1)

Linear

(A1373-3)

Linear

(A1373-2)

Figure 10: Hall Effect Chip Calibration Curves

of 26



3.2 Magnetorheological Fluid Testing Under DC Driven Magnetic FieldBefore moving on to testing the MRF under an AC driven magnetic field, we wanted to

determine the response of the MRF under the DC drive magnetic field. This would allow us to

compare any AC data with DC data. It would also allow us to compare our data to that of the

KRS group. The KRS group reported their data as shear stress versus the magnetic field so we

took out torque measurements, calibrated the torque measurement using the calibration curvegenerated by the pulley test (See Table 1 and Figure 11). We then calculated the shear stress by

using the following equation

r SA

T StressShear

⋅= (2)

Where T is the torque, r is the average radius of the gear head, and SA is the surface area, where

SA = π (r o2-r i

2).

Table 1 – DC Driven Magnetic Field Test Results

Magnetic Field (G) Average Torque(N-cm)

Calibrated Torque(N/cm) Shear Stress (kPa)

106.63 17.20 12.62 33.37390.95 29.96 22.36 59.11

581.19 27.80 20.71 54.75

751.62 43.88 33.00 87.22

865.84 52.64 39.69 104.90

951.98 57.11 43.10 113.92

1009.74 59.89 45.22 119.53

1054.80 62.01 46.84 123.80

1089.89 64.08 48.42 127.99

1120.14 65.24 49.31 130.33

1146.43 66.11 49.97 132.08

Shear Stress vs. Applied Magnetic Field at 40 RPM

0

20

40

60

80

100

120

140

0 200 400 600 800 1000 1200 1400

Magnetic Field (Gauss)

S h e a

r S t r e s s

( k P a )

Figure 11: Shear Stress vs. Applied Magnetic Field for 40 RPM

of 26



In comparing our DC data with that of the KRS group, it becomes clear that there are

many inconsistencies. They didn’t run tests at 40 RPM; they did, however, run tests at 35 RPMand 45 RPM. One would expect that our test results would lie in-between those two sets of

data. Unfortunately they didn’t. In fact our lowest calculated value for shear stress was higher

than their highest value. One cause of this could be that our values for magnetic field are very

different from that of the KRS group. We were measuring the magnetic field at the time of testing whereas they were measuring the current, and thus assumed a value for magnetic field.

We also made a plot of the magnetic field as a function of current going into the magnet.

We then added a trendline to the data. When we looked at the data under a cubic trendline therewas systematic error. With a quintic trendline, the error was much more random so we assumed

we had a quintic fit; represented in the equation displayed below (See Figure 12).

232731041451 1052.71010.21075.21073.51016.41052.2 −−−−−− ⋅−⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅= x x x x xY (3)

Current Vs Magnetic Field

y = 2.5173E-16x5

+ 4.1640E-14x4

+ 5.7281E-10x3

- 2.7475E-07x2

+ 2.9998E-03x - 7.5200E-02

-6

-4

-2

0

2

4

6

8

10

-1500 -1000 -500 0 500 1000 1500 2000

Magnetic Field (Gauss)

A

m p e r a g e ( A m p s )

Figure 12: Current versus Magnetic Field

3.3 Characterization of AC Driven Magnetic FieldIn order to characterize the magnetic field, we used two Hall Effect chips positioned on

the magnet in the same locations as those used in the DC testing. After making graphs of each

of the tests, we also took the average and the maximum value of magnetic field from each HallEffect chip and plotted those values as a function of frequency for each of the three waves.

While looking at each of these graphs, we saw an interesting trend. The graphs

generally had sinusoidal properties but when looked at more closely there appeared to be an

additional function within the data. This systematic phenomenon was most apparent in datataken during the square wave tests (See Figure 13).

of 26



3 Hz Square Wave

200

220

240

260

280

300

320

340360

380

400

0 2 4 6 8

Time (s)

M a g n e t i c F i e l d ( G a u

s s )

10

Figure 13: 3 Hz Square Wave – Magnetic Field Characterization

Looking back over the work done by the KRS group, this trend seems to fit with the data

they collected of the torque versus time. In the data they collected, they also experienced a verystrange phenomenon in that the data they collected also seemed to have a periodic variation

within the expected periodic function. We don’t know why this fluctuation is occurring, but it

is important to note that it exists before continuing into the analysis of the magnetic field.Other interesting trends were found while looking at the magnetic field as a function of

frequency. At higher frequencies, the magnetic field started to drop hyperbolically. As we

were using the same type of AC circuit setup as the KRS group had the year before, we still had

three options of what caused this drop off. At higher frequencies it is expected that themagnetic field will drop off, as a result of the drop in current due to the overall inductance of

the electromagnet. Another reason for the drop off at higher frequencies could be the skin depth

of the electromagnet. Finally, the drop off could have something to do with the response timeof the power sources (See Figure 14).

of 26



Magnetic Field vs. Frequency

200

250

300

350

400

450

500

0 1 2 3 4 5 6

Frequency (Hz)

M a g n e t i c F i e l d ( G a u

s s ) Sine Wave

Square WaveSpiked Wave

Figure 14: Magnetic Field as a function of Frequency

We decided that it was unlikely that the drop off was due to the inductance of themagnet. To prove this, we used the previously measured inductance and resistance to calculate

the theoretical maximum current that would be in the electromagnet (assuming the

electromagnet is a resistor in series with an ideal inductor). Next, we converted the maximum

values of the magnetic field in our tests into the DC amperage necessary to create this fieldusing Equation 3. We then compared these values with the theoretical current going to the

electromagnet (See Figure 15, 16 & Appendix D).

Theoretical Maximum Current Through Magnet

0

2

4

6

8

10

12

14

16

0 1 2 3 4 5

Frequency (Hz)

C u

r r e n t ( A m p s )

6

of 26

Figure 15: Theoretical Maximum Current for a Sine Wave due to Inductance of theMagnet



Necessary DC Current to Produce Max Magnetic Field

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5 6

Frequency (Hz)

C u r r e n t ( A m p s )

Sine Wave

Square Wave

Spiked Wave

Figure 16: Calculated Current Entering Magnet

In order to compare the theoretical and actual current that should be able to be enteringthe magnet, we superimposed the sine wave values with the theoretical values. The two values

differed significantly; the theoretical current hovered around 14A whereas the calculated current

was under 1A. To examine the differences in slopes between the graphs, we scaled thecalculated value of current by a factor of 14.(See Figure 17).

of 26



Comparison of Actual and Theoretical Currents

8

9

10

11

12

13

14

15

0 1 2 3 4 5

Frequency (Hz)

C u r r e n t ( A m p s )

6

Theoretical Current

Scaled DC Calculated Current

x14

Figure 17: Overlaid Theoretical Current versus Calculated Current

We see that the slope of the actual current is significantly different than that of the

theoretical current; the calculated current is not only much less than the theoretical, but it also

drops off much faster than predicted. Therefore, there must be a factor other than theinductance of the magnet that is reducing the magnetic field.

We then tried to find a mathematical formula that would relate the magnetic field tofrequency. We first looked to see if an exponential curve would fit the graph of the magnetic

field versus frequency, but the fit was very poor. We then compared the natural log of the

magnetic field with the natural log of frequency in order to see if this relation had the same proportionality as the skin depth to frequency; the skin depth is inversely proportional to the

square root of frequency. (See Figure 18).

of 26



LN of Frequency vs LN of Magnetic Field

y = -0.1606x + 6.1545

R2

= 0.971

y = -0.2101x + 5.8688

R2

= 0.9788y = -0.267x + 5.8057

R2

= 0.932

5.3

5.4

5.5

5.6

5.7

5.8

5.9

6

6.1

6.2

6.3

0 0.5 1 1.5 2

LN of Frequency

L N

o f M a g n e t i c F i e l d

Sine Wave

Square Wave

Spiked Wave

Linear (Square Wave)

Linear (Sine Wave)

Linear (Spiked Wave)

Figure 18: Natural Logarithmic Comparison

4 Discussions and Conclusion By using the Hall Effect chips, we found a very effective and straightforward way to

measure the magnetic field during our experiments. In comparing our DC data to that of the

KRS group there were many inconsistencies. Not only did we find that the magnetic field is a

quintic rather than cubic function of the current input, but we also found that our calculatedshear stresses were much higher than those found by the KRS group. Even though the

magnitudes of the shear stresses measured by the KRS group are significantly lower than those

that we measure, a linear trend is seen in both sets of data.In looking at the magnetic field as a function of frequency, it is clear that the magnetic

field is dropping at higher frequencies due to something other than the inductive resistance.

This is clear because of the significant difference between the theoretical maximum current thatcould enter than magnet due to the inductance, and the current that is actually making its way

into the electromagnet. We hypothesize two additional causes of the drop off; skin effect and

the response time of the power supplies. In looking at the skin depth, magnetic field andfrequency we were unable to determine conclusively any mathematical relation to explain the

current drop. However, we formulated equations that describe the magnetic field as a functionof frequency, as shown below.

8688.52101.0 e f WaveSinea for Field Magnetic ⋅= − (4)1545.61606.0

e f WaveSquarea for Field Magnetic ⋅= − (5)8057.52670.0 e f WaveSpiked a for Field Magnetic ⋅= − (6)

These equations seem to fit well for the sine and square waves, though they don’t fit as well for

the spiked wave. Whereas the skin depth is inversely proportional to the square root of the

of 26



of 26

frequency, we find that the magnetic field is roughly, though not exactly inversely proportional

to the fifth root of the frequency for a sine wave input, the sixth root for a square wave, and thefourth root for a spiked wave. Thus, if the cause of the drop in magnetic field is due to skin

effect, there is not a linear relation between skin depth and field strength. These relations are not

complete descriptions, because at a frequency of 0 Hz they predict an infinite magnetic field,

instead of the magnetic field produced by 20V DC.Despite all the data we collected having to do with the magnetic field as a function of

frequency, we are unable to determine a complete relation. We ruled out that the inductive

resistance was the whole reason for the drop off in magnetic field, though we theorize that thedrop off could result partially from the skin effect. Sadly, we were not able to examine the

magnetic field produced by the circuit including the dummy load. If we had done so, we would

have been able to find out if the response time of the DC power supplies was also a factor in thedrop in magnetic field.

5 References

1. ME 241 and ME 242 final reports submitted by Dean Kleissas, Matt Roe, and PhillipSchuepbach

2. http://en.wikipedia.org/wiki/Hall_Effect_Sensor

3. http://www.allegromicro.com/sf/1373/

4. http://en.wikipedia.org/wiki/Skin_depth

5. http://www.mecheng.ohio-state.edu/~dapino/Malla_Thesis.pdf

6. http://www.ndt-ed.org/GeneralResources/MaterialProperties/ET/ET_matlprop_Iron-Based.htm

7. http://en.wikipedia.org/wiki/Magnetorheological_fluid

http://en.wikipedia.org/wiki/Hall_Effect_Sensor

http://www.allegromicro.com/sf/1373/

http://en.wikipedia.org/wiki/Skin_depth

http://www.mecheng.ohio-state.edu/~dapino/Malla_Thesis.pdf

http://www.ndt-ed.org/GeneralResources/MaterialProperties/ET/ET_matlprop_Iron-Based.htm


http://en.wikipedia.org/wiki/Magnetorheological_fluid

http://en.wikipedia.org/wiki/Magnetorheological_fluid



http://www.mecheng.ohio-state.edu/~dapino/Malla_Thesis.pdf

http://en.wikipedia.org/wiki/Skin_depth

http://www.allegromicro.com/sf/1373/

http://en.wikipedia.org/wiki/Hall_Effect_Sensor



6 Appendices

Appendix ADerivation of Viscosity Equation

A torque T, can be expressed as a function of shear stress τ as follows

of 26

A A

T rd τ = ∫ (0.1)

where r is the radius. The shear stress can be related to the effective viscosity, μ, by the equation

( )dU z

dz τ μ = (0.2)

where z is the axis perpendicular to the load plate with z = 0 defined at the bottom of the test

plate, and U(z) is the velocity profile in the z direction. If we assume a linear profile for U(z),and a no slip condition at z = 0

( )dU z v

dz h= (0.3)

where v is the velocity at the surface (z = h). We can express the linear velocity in terms of

angular velocity ω in rad/s by

r ω ν = (1.4)

Plugging in this value for v in equation (1.3), and plugging into equation (1.2)

h

r

μ τ = (1.5)

Plugging this into equation (1.1)

dAh

r T

A

∫=2

ω μ (1.6)

We have a circular area varying from r 1 to r 2, and assuming symmetry about the z-axis

2dA rdr π = (1.7)

dr

h

r T

r

r

∫=

2

1

32ωπ μ (1.8)

Integrating

)(2

4

1

4

2 r r h

T −=π

μ (1.9)

And thus

)(

24

1

4

2 r r

hT

−=

πω μ (1.10)



Appendix B

Skin Depth Calculation

The skin depth can be written4

as:

f r ⋅⋅

⋅=

μ

ρ

μ π δ

0

1 (1.1)

where

μ =0 4π ×10-7

H/m

μ =r the relative permeability of the mediumρ = the resistivity of the medium in Ωm

f = the frequency of the wave in Hz

For 1018 steel, the resistivity is 1.590E-07 ohm-m, and the relative permeability is about 529.Using the above formula, it can be calculated that at 10 Hz δ = 2.758 mm and at 1 kHz, δ =0.2758 mm.

Table I: Excel Calculations of Skin Depth

mu0 1.25664E-06

rho 1.59E-07

mur 529

f delta =rho^(1/2)/(pi*mu0*mur*f) in mm

1 0.008725511 8.725511

1.5 0.00712435 7.12435

2 0.006169868 6.169868

2.5 0.005518498 5.518498

3 0.005037676 5.037676

3.5 0.004663982 4.663982

4 0.004362755 4.362755

4.5 0.004113245 4.113245

5 0.003902167 3.902167

6 0.003562175 3.562175

7 0.003297933 3.297933

8 0.003084934 3.084934

9 0.002908504 2.908504

10 0.002759249 2.759249

of 26



of 26

Appendix C3D Rendering of Characterization of Magnetic Field – Matlab File

clc clear all A = 0.15; %Distance between points (inches) B = [1.0001 -583.421252 %Matrix of points and magnetic field at the point 2.0001 33.28131069 %The number in the ones place is the radial dot 3.0001 1745.447135 %number (the measure of how close or far it is from 4.0001 2021.26258 %the center, where 9 is the center, and 1 is on the

5.0001 1927.112207 %outer edge. The number in the thousands place is

6.0001 1737.047728 %the marker of which row of dots it is a part. 7.0001 1692.980377 8.0001 1721.346212 9.0001 1743.524988 1.0002 -350.6147816 2.0002 -149.594407 3.0002 411.7694444 4.0002 1404.619423

5.0002 1802.469688 6.0002 1715.527877 7.0002 1661.192233 8.0002 1659.146495 9.0002 1633.227215 1.0003 -227.5154422 2.0003 -285.0064117 3.0003 -57.2794619 54.0003 799.1656876 5.0003 2021.09324 6.0003 2021.020178 7.0003 1993.188994 8.0003 1804.594633 9.0003 1716.762421 1.0004 -191.0130876 2.0004 -248.8782441 3.0004 -153.331497 54.0004 654.0132122 5.0004 2020.961455 6.0004 2021.11987 7.0004 1980.36626 8.0004 1855.690964 9.0004 1772.172829 5.0005 2020.997645 6.0005 2020.987403 7.0005 1907.603952 8.0005 1800.512717

9.0005 1758.200221 1.0006 -526.8268291 2.0006 -698.0727046 3.0006 -224.359434 4.0006 1855.102371 5.0006 2021.09324 6.0006 1859.368649 7.0006 1720.163563 8.0006 1724.021513 9.0006 1747.981091 1.0007 -407.1846228 2.0007 109.4468078



of 26

3.0007 1134.700837 4.0007 1931.206414 5.0007 1876.026801 6.0007 1753.245657 7.0007 1736.152547 8.0007 1742.413352 9.0007 1773.087129 1.0008 -711.9278671 2.0008 -68.34392665 3.0008 2020.903416 4.0008 2020.877468 5.0008 1962.099377 6.0008 1792.480669 7.0008 1748.75609 58.0008 1749.40068 9.0008 1795.859278];

%B2 isolates the row of each point

B2 = (B(:,1)-round(B(:,1)))*10000 %B3 isolates just the location in each row

B3 = B(:,1)-B2/10000; %C is the combined matrix in the form (row, location, field strength) C = [B2,B3, B(:,2)]; % Converts the row and location into rectangular coordinates,

%and stores this information into matrix D D =[A*(9-C(:,2)).*cos((pi/4 .* (C(:,1)-1))),A*(9-C(:,2)).*sin((pi/4 .*

(C(:,1)-1))),C(:,3)] %The next lines break up D into an individual matrix for each line of %points. This is done for graphical reasons for i = 1:9

D1(i,:)=D(i,:); end for i = 10:18

D2((i-9),:) = D(i,:); end

for i = 19:27 D3((i-18),:) = D(i,:);

end for i = 28:36

D4((i-27),:) = D(i,:); end for i = 37:41

D5((i-36),:) = D(i,:); end for i = 42:50

D6((i-41),:) = D(i,:); end

for i = 51:59 D7((i-50),:) = D(i,:);

end for i = 60:68

D8((i-59),:) = D(i,:); end

%Plots the data points plot3(D1(:,1),D1(:,2),D1(:,3)) xlabel('x position (in)') ylabel('y position (in)')



of 26

zlabel('magnetic field (G)') hold on plot3(D2(:,1),D2(:,2),D2(:,3)) plot3(D3(:,1),D3(:,2),D3(:,3)) plot3(D4(:,1),D4(:,2),D4(:,3)) plot3(D5(:,1),D5(:,2),D5(:,3)) plot3(D6(:,1),D6(:,2),D6(:,3)) plot3(D7(:,1),D7(:,2),D7(:,3)) plot3(D8(:,1),D8(:,2),D8(:,3)) hold off



Appendix DCalculation of the Maximum Theoretical Current

The impedance of the electromagnet: L = 46.7 mH = .0467 H

The resistance of the electromagnet: R = 1.4 Ω

The voltage produced by our AC circuit: V (t) = 20cos (ωt) = °∠020 The angular frequency (dependant on , our input frequency): f f π ω 2=

The equivalent resistance:

)0467(. j1.4Zeq ω += = )4.1

0467.*(tan)0467.(4.1 122 ω

ω −∠∗+

Current as defined by Ohm’s law:eq Z

t V )(I(t) =

)

4.1

0467.*(tan)0467.(4.1

020)(

122 ω ω −∠∗+

°∠=t I

is the only unknown variable. We plugged in the same values for into this equation for

current to see if our experimental data matched our theoretical maximum current as seen below:

Frequency ω (2*pi*f)Magnitude(amps)

Angle(degrees)

0.1412 0.887185765 14.27946265 -1.695115538

0.5 3.141592654 14.20791238 -5.982449928

1 6.283185307 13.98191865 -11.83722935

1.5 9.424777961 13.6280998 -17.45229592

2 12.56637061 13.17503519 -22.74236973

2.5 15.70796327 12.65388702 -27.65331461

3 18.84955592 12.09374028 -32.160339943.5 21.99114858 11.51881579 -36.26235236

4 25.13274123 10.94749724 -39.97500286

4.5 28.27433388 10.39260995 -43.32422737

5 31.41592654 9.862330825 -46.34113717

magnetorheological fluid analysis final report sa vrednošću relativne permeabilnosti za aisi 1018

Documents