An Examination of the Metallic Bonding of a Clad Material and 1.2

8/2/2019 An Examination of the Metallic Bonding of a Clad Material and 1.2

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An Examination of the M etallic Bon ding of a Clad Material and 1.2Tw o Gold Plating Systems under C onstant Force Fretting Conditions.Neil Aukland- Harry Hardee- Anna Wehr- Sean Brennan- Philip Lees*- New Mexico State UniversityAdvanced Interconnection LaboratoryPO Box 30001, MSC 3450Las Cruces,NM 88003

Abstract

* Technical Materials Inc.Five Wellington RoadLincoln, RI 02865

Metallic bonding (cold welding) comparisons were made between a clad and two different goldplated contact materials under constant force fretting conditions. Previous experimentsconducted at the Advanced Interconnections Laboratory showed that various material systemshad different tendencies for metallic bonding. Three different material systems were selected tostudy this phenomenon. They were a thick gold plating, a gold flash over 80% palladium 20%nickel and a clad material (WE#1 over R156).Four different normal forces: 20, 50, 100, and 200 grams were used in this research project.Contact resistance data were collected using a four-wire measurement system with the opencircuit voltage limited to 0.02 volts. Relative humidity was controlled to be 40%. Theprocedure for each experiment was as follows: tangential force was increased until the staticcoefficient of friction was exceeded, then a constant displacement was maintained until a weartrack was established, then a constant force was maintained throughout the remainder of theexperiment. Two sets of data were collected during each experiment: fret amplitude and contactresistance.The three material systems were statistically compared using analysis of variance techniques.They were compared on the number of fretting cycles needed to stop fretting motion, after awear track was established.Keywords: fretting, constant force, friction, clad, gold1.0 IntroductionWhenever two clean metallic surfaces arepressed together, free electrons can moveacross the interface and form metallic bonds[11. Highly conductive metals have manysuch free electrons [11. The strength ofthese interfacial bonds is of the same orderof magnitude as the bonds in a metalliccrystal [13. These bonds will increase instrength with time, due to the arrangementof atoms at the interface [l]. Soft metalstend to form metallic bonds so strong thatchunks of metal will actually be torn fromthe surface, if moved [11.0-7803-3968-197/$10.00 997 EE E

Before fretting can occur, the two metalsurfaces must experience a tangential forcethat exceeds the strength of the metallicbonds. There are two possible sources forthis tangential force: thermal stress andvibration. The force due to thermal stress is:F s = A E n AtWhere: A = area,E = Youngs modulus,n = coefficient of thermal expansion,

At =temperature change [ 2 ] .7


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The force due to vibration can bedetermined based on mass and acceleration.In either case, the tangential force mustexceed the static coefficient of frictionbefore fretting motion can occur.

I Load

Although the tendencies of connectormaterial systems to form metallic bondshave been observed in the laboratory, basicresearch into the phenomenonhas not beenpublished. The goal of this research projectis to begin to explore the tendency for amaterial system to form a strong metallicbond during fretting conditions.

Material' Heavy I Flash I W#

2. Material SystemsThree different material systems were usedin this research project. They were: a heavygold system, a flash gold system and a cladmaterial. All three systems used the samecopper base metal. The alloy was UNSC 17410 copper beryllium strip processed tothe '/zHT temper. The heavy gold systemconsisted of 30 microinches of cobalthardened gold plate, over 100 microinchesof nickel. The flash gold system consistedof 3-5microinches of cobalt hardened gold,over 40 micro-inches of 80% Pd - 20% Ni,over a 75 micro-inch layer of Ni. Thepercent of cobalt in the plated gold was 1%.Two different methods, atomic absorptionspectrophotometry and gravimetric analysis,were used to verify the actual compositionof the palladium-nickel layer. The resultwas 82% palladium and 18% nickel. Theclad material system consisted of 10microinches of 69% Au, 25% Ag, and 6% Pt( W # l ) over a 20 microinch layer of 60%Pd and 40% Ag (R156). Samples fiom theheavy gold system and clad system weresectioned. Individual layer thicknesseswere verified to be within 8% of the quotednominal values. The thickness of the flashgold layer was verified to be withintolerance using x-ray fluorescence.The gold plating bath composition was goldpotassium cyanide, citric acid and sodium

citrate with a complexing agent and additive(CO) . The pH level was held between 4 and4.2. The temperature of the bath was 35' C.Current density was 0.5A/dm2.The palladium-nickel alloy plating bathcomposition was palladium (as metal inPdClZ), nickel (as metal in NiS04,6HzO),(N&)2SO4) and hydroxycarboxylic acid.The pH level of this bath was 8.2. Thetemperature of the bath was 30" C. Currentdensity was 3.0A/dm2.The Vickers hardness of each materialsystem was measured at 20,50,100 and 200gram loads. The values are listed in Table 1.The hardness of all material systemsdecreased by about 10% over the load rangeevaluated.Table 1:Vickers hardness of the threematerial systems measured using variousloads. Units are ke/mm 2.

1 (grams) I Gold I Gold I over I

The surface finish of each material systemwas evaluated by measuring both & andR M ~ .he results are listed in Table 2.W # 1 over R156 exhibited the smoothestsurface. The gold flashed palladium-nickelexhibited a relatively smooth surface, buthad a large maximum amplitude. The heavygold samples exhibited a rougher surfaceand a larger amplitude.3.0 Experimental Procedure.Before motion can occur at the metallicinterface, the tangential force must exceedthe static coeEcient of friction. After thestatic coefficient of fiiction is exceeded, theactual &et amplitude is very high. We

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Material Attribute monitoring the fret amplitude and varying- RMAX the power delivered to the electrodynamicHard gold 0.3 1.2 drivers. Constant force fretting isFlash gold 0.2 1.1 accomplished by establishing a power level

WE#l over 0.1 0.5 for a specific fret amplitude andR156 then maintaining that power levelfound through preliminary experimentationthat first the tangential force must exceedthe static coefficient of friction, then thesamples must be fretted in a constantdisplacement mode to establish a wear track.If a wear track is not established before theconstant force mode begins, fretting motionwill immediately stop. During the periodthat a wear track was being created, all ofthe contact resistance and fret amplitudedata were collected. During the force drivenperiod, only enough data were saved tovisualize trends. Fret amplitude data werecollected to determine the cycles requiredbefore the samples stopped moving (stick).At the end of every experiment, the sampleswere separated and the voltage and unloadedfret amplitude was recorded. Thisinformation was used in the calculation ofthe dynamic coefficient of friction.All of the samples were cleaned using acommercially available contact solventbefore each experiment. This contactcleaner contains isopropanol, ethanol, 2-methyl-2-propanol and carbon dioxide.Exact proportions are not given on the label.The manufacture advertises that the contactcleaner does not leave a residue.3.1 Fretting m achineA force driven fretting machine was used forthis research project. This fretting machinehas been discussed in detail in previouspublications [3,4] and can be used in eithera constant force or a constant displacementmode. During each of the experiments inthis research project, the fretting machine

3.2Contact geometlyFor the hard gold and gold flashedpalladium-nickel samples, narrow strips ofcopper were pre-formed followed byelectroplating. The clad samples wereformed aRer the cladding was applied.The stationary component of a pseudo-crossed rod configuration was formed over a9 mm diameter rod to make the flat. Themoveable component was dimpled with a 3mm spherical diameter tool forming the rider. The difference in radii of curvaturewas intentional. First, the contact of the twosurfaces establishes a pseudo-crossed rodgeometry and assures contact. This systemwas discussed in an earlier paper [4].Second, during adhesive wear, for surfaceshaving no significant difference in hardnessbetween the two surfaces, material isnormally transferred from the flatter to thesharper surface. In this configuration, thesurfaces of the mating components wereidentical and optical examination of thewear track on the flat provided a quickscreening technique.3.3 Contact Resistance Measurem entsContact resistance measurements were takenusing a Hewlett Packard 4338Amilliohmeter. This milliohmeter limited theopen circuit ac voltage to 0.02 volt. Beforeeach fretting experiment, the milliohmeterwas zeroed through the pseudo-crossed rodconfiguration. This created a fixedreference point of zero resistance as the

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Experimental Normal Force Constant Fret AmplitudeSet (grams) Displacement (microns)1 20.0 40.0 20.02 50.0 50.0 25.03 100.0 150.0 37.54 200.0 1000.0 67.5

(cycles)

initial value, Therefore, the contactresistance should initially go slightlynegative as better metal to metal contact wasmade due to fretting motion.

Frequency(Hertz)4.04.04.04.0

3.4 Amplitude MeasurementsThe original fretting machine monitored fretamplitude using a single Keyence LC 2100laser displacement meter. The target for thisdisplacement meter was mounted on themoving rod in the crossed rod configuration.Another Keyence laser displacement meterwas used to monitor the displacement on

displacement fretting cycles, the sampleswere fretted in a constant force mode.During the constant force mode, the fretamplitude would vary, probably due tochanges in the levels of metallic bonding.All of the fret amplitude data sets had acommon characteristic during the constantforce regime. The fret amplitude would dip,increase, stabilize and suddenly decrease.Three characteristic fret amplitude data setsare shown in Figures 1 ,2 and 3 for heavygold, gold flashed palladium nickel andWE#l over R156, respectively. All of thedata in these figures were collected at 100the stationary rod. The difference of the twodisplacement readings was used to calculatethe relative motion at the metallic interface..

3.5 Experimental ConditionsOne set of experimental conditions could not

grams normal force.

be established for all three material systemsand for all four normal forces. Therefore, adifferent set of experimental conditions wasestablished at each of the normal forcelevels. So, all three material systems weretested under the same conditions at eachnormal force level. These conditions arelisted in Table 3 . Each set of experimentalconditions was replicated five times for eachmaterial system4.0 Results and discussionAll of the fret amplitude data patterns were

lo010 loo0 2ooo 3ooo

Cycles

similar. Fret amplitude was stabilized at aselected value based on normal force duringFigure 1: Fret am plitude data for heavygold at 100 grams normal force.

the constant displacement mode of eachexperiment. At the end of the constant10


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50 745-11

Figure 2: Fret Amplitude data for goldflashed palladium nickel at 100 gramsnormal force.

i0 loo0 2ooo

Count

Figure 3: Fret am plitude data for WE#1over R 156 at 100 grams normal force.All of the contact resistance data had asimilar pattern. Contact resistance wouldstart at an initial value of approximately zeroand steadily decrease throughout theremainder of the experiment. Figure 4shows a contact resistance data set for goldflashed palladium nickel at 100 gramsnormal force. The overall conditions appearto be similar to Stage 1 of Bryants model in

-1.250 5ooo loo00

Cycles

Figure 4: A contact resistance data set forgold flashed palladium nickel at 100grams normal force.which the contact resistance is steadilydecreasing from an initial value [5].One of the WE#1 over R156 experiments at100grams normal force was not stoppedimmediately after sticking. The data thatwas collected for this experiment iscontained in Figure 5. The constant forcefretting cycles were able to cause the sampleto unstick and motion resumed, but at asmaller amplitude and for fewer cycles.This phenomena was observed byWaterhouse [ 6 ] .Figure 6 contains the contact resistance dataset that correlates to the fret amplitude datain Figure 5 . Some interesting observationscan be made about contact resistance. First,contact resistance decreases throughout theentire experiment. Second if there is motionat the metallic interface, contact resistancevalues increase and have a larger variance.It is possible that the motion causes someaperities to lose contact, while new spots ofcontact are made at the asperity tips, whichwould explain the larger contact resistancevariance during motion.

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0 5ooo loo00Cycles

Figure 5: Another fret amplitude data setfor WE#1 over R156 at 100 grams normalforce.

-0.5-1

0 -1.5-2

-2.5

Ec.-z-0 5ooo loo00Cycles

Figure 6: Contact resistance values forthe data set in Figure 5.A heavy gold sample was fretted underconstant force conditions for an extendedperiod at 200 grams normal force.contact resistance data for this experiment isshown in Figure 7. Initially, contactresistance decreases rapidly as in Figure 4.Then the contact resistance values stabilizeand a minimum is reached at approximately

The

850,000 cycles. The region after thisminimum point is interesting because thecontact resistance values are increasing.This increase in resistance may mean thatsome debris is forming in the metallicinterface and causing the contamination ofsome of the touching asperity tips (Stage 2)[ 5 ] . Although no motion is detected duringthe first 1,600,000 constant force cycles, theasperity tips may be flexing. Moreover,some of the smallest asperity tips may evenbe making and breaking contact. The sharpincrease in resistance at about 1,600,000cycles is due to a short period of motion.These short periods of motion werediscussed and are shown in Figure 5.

I130 IOOOOOO MOOOOO

Cycles

Figure 7: Contact resistance data forheavy gold at 200 grams normal force.An overall statistical comparison betweenthe experimental sets was not possiblebecause the experimental conditionschanged between sets. Therefore, analysisof variance techniques could only beconducted on each experimental set. Astatistical comparison using linear contrastswas conducted for the material systems ateach normal force. The only significantstatistical comparison from this analysis wasat 50 grams normal force. The linearcontrast that compared heavy gold to theother two material systems was significant.All of the other statistical comparisons wereinconclusive.

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Because the statistical analysis wasinconclusive, a simple graphing techniquewas used to look for trends in the data. Anaverage was taken on the data for eachmaterial system at 20.0, 50.0, 100.0 and200.0 grims normal force. A plot of thisdata is given in Figure 8 . As expected, theheavy gold system maintained motion underconstant force conditions for a longer periodthan the other two systems. Although thestatistical comparison did not detect asignificant difference between the clad andflash gold system, a trend can be seen inFigure 8 . The clad material appears to beslightly less prone to weld when comparedto the gold flashed palladium-nickel.

20000,

16000

120000Uh-0 80W

4000

hI \~

n-A040 50 100 150 200

Grams

igure 8: The average numb er of cyclesrequired to cause the samples to stick forall three material systems.The dynamic coefficient of friction can beused as a figure of merit in the comparisonof different material systems. The tangentialforce developed by the fretting machine wasmodeled mathematically and simulated witha computer program. This development islengthy and is presented in Appendix 1. Theresults of these simulations were used tocalculate values for the dynamic coefficientof friction. The following data werecollected for each experiment: the amplitudeof motion, the amplitude of the sinusoidalvoltage sent to the motion driver, the normal

force on the fretting surface, and the contactresistance.Using the model developed in Appendix 1and the data collected at the end of eachexperiment, the dynamic coefficientsoffriction were calculated utilizing Equations(15 ) and (16) ofAppendix 1. Figures 9, 10and 11 contain friction force versus normalforce data for heavy gold, flash gold andWE#1 over R156, respectively.Linear regression was used to calculate thebest-fit straight line through these datapoints. The slope of this line is actually theexpected value of the dynamic coeficientof friction. Table 4 contains the expectedcoefficientof friction and the upper andlower values of the 95% confidence intervalfor each material system. These confidenceintervals can be used to make conclusionsabout the differences in dynamic coefficientof friction between material systems. The

025 ~I

O Z J

:8

0.05 1~0

040 0 5 1 1 5 2 2.5

N m a l ame

Figure 9: The friction force versus norm alforce data for heavy gold.

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0.25

0.2

0.15YfP$? 0.1

0.05

e 0.15 -Y

D5 0.1 -

ee44

0 0 5 -

0 0.5 1 1 5 2 2.5Nomalforce

** e88

Ggure 10:The friction force versus

MaterialSystemHeavy GoldFlash GoldWE#l overR156

0.25

Cldynamic Confidence IntervalLower Upper0.052 0.04 0.0640.086 0.072 0.0990.069 0.065 0.074

0.2Inormal force data for flash gold.

Figure 11: The friction force versus

0:

0 40 0.5 1 1.5 2 2.5

Normal orce

normal force data for WE#1 overR156.upper and lower values for gold lie outsidethe intervals for the other two systems.Therefore, it is likely that the dynamiccoefficient of friction for the heavy gold

system is different from the other twosystems. The upper and lower bounds for theflash gold and WE#l over R156 coincide.Therefore, no statistical difference isdetected.The values presented in Table 4 are similar topreviously published data [6].Duringfretting debris is produced and causes somerolling to take place [6]. Waterhouse notesthat rolling friction values are about 0.002[ 6 ] . His values were about 0.05 [ 6 ] . Sohetheorized that some sliding was taking placeP I .5.0 Surface Examination and D iscussionThe wear tracks were examined using a lightmetallograph. Since the same materid wason each surface, material transfer wasexpected to occur from the flat to the rider.In all cases, material appeared to betransferred from the flat component to therider.No discernable wear was detected on theflash gold (GF PdNi) samples after fretting at20 grams. The gold remained intact on bothcontact surfaces. After fretting at 50 and 100gram loads, the flash was worn off the flatsurface. Gold was worn off both surfaces at200 grams.For the clad WE#l over R156 samples, thecontact surface remained WE# 1 to WE# 1 atnormal loads of 20, 50 and 100 grams. R156was exposed after fretting at 200 grams.

Table 4: The dvnamic coefllcient of friction v alues a nd a 95% confidenn ce interval

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The metallographic observations wereconfirmed using nitric acid vapors. Nitricacid attacks both Pd-Ni and R156. Thecorrosion products have a distinct darkbrown to black color.The results for the clad WE#l over R156system and the gold flash (GF PdNi) systemare consistent with previous findings [7,8].At 20,50 and 100 gram loads, the thin goldflash and WE#l layers provided lubrication.This benefit is evident in both Fig. 10 andFig. 11. The tangential forces for bothsystems were nearly identical in this loadrange. At 200 grams, the gold flash wasrapidly worn away, typically within the firstfew hundred cycles whereas the WE#l layerhas proven to be more durable. Underidentical load and motion conditions, thecontact interface of a WE#lover R156system remains WE#l-WE#l long after thecontact interface of a GF PdNi system haseroded from Au-Au to Au-Pd to PdNi-PdNi.The result was a lower average tangentialload at 200 grams for the WE#l over R156system when compared to the gold flashsystem.The contact interface on the heavy goldsystem remained gold-gold after fretting atall contact loads. The microstructure ofcobalt hardened electroplated gold is notcomparable to either PdNi or WE#l overR156. Whereas the latter are true metals,electroplated hard gold is composite and canbest be described as fine gold grains with acobalt metal organic complex in the grainboundaries. The organic compound lowersthe interfacial shear strength of the asperityjunctions resulting in lower tangentialforces.No frictional polymers were detected duringthe material study. This is probably due tothe fact that the experiments were conductedin a very dry environment, free of polmerforming contaminants.

6.0 ConclusionsThe purpose of this research project was toexplore the tendency for a material systemto form a strong metallic bond (stick) duringfretting conditions. The dynamic coefficientof friction was chosen as a figure of merit todifferentiate between material systems. Inaddition, a mathematical model for theconstant force-fretting machine wasdeveloped. This model was used tocalculate the dynamic coefficient of frictionfor the material systems. Three differentmaterial systems were compared based ontheir tendency to form strong metallic bond(stick) and on their dynamic coefficient offriction.Four different normal forces were usedduring this research project. The fretamplitude of the pseudo-crossed rodconfiguration was monitored and data werecollected on fret amplitude using a laserdisplacement meter. The number of cyclesrequired to cause fret amplitude to decreaseto zero (stick) were recorded for all five ofthe replications at each normal force.The heavy gold system endured morefretting cycles before the pseudo-crossedrods stopped moving and had the lowestcalculated dynamic coefficient of friction.Under the fretting conditions presented inthis paper the heavy gold system is preferredover the over two material systems.Although WE#l over R156 endured morefretting cycles before sticking and had alower dynamic coefficient of friction thanthe flash gold system, the differences werenot statistically significant in thisexperiment7.0 References[11 Holm, R., Electric Contacts Handbook,Springer-Verlag, Berlin, 1958.

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[2] Seely, F., Smith, O., Resistance ofMetals, John Wiley and Sons, New York,1956.[3] Hardee, H. and Aukland, N. A NewConstant ForceDisplacement Fretting TestMachine, Proceedings of the 17*International Conference on ElectricalContacts, pp 17-24, 1994.[4 ] Aukland, N., Leslie, I., Hardee, H., Lees,P. A Statistical Comparison of a CladMaterial and Gold Flashed Palladium NickelUnder Various Fretting Conditions,Proceedings of the Fortv-First HolmConference on Electrical Contacts, pp 52-63 , 1995.[5] Bryant, M. Assessment of FrettingFailure Models of Electrical Connectors,Proceedings of the Fortieth HolmConference on Electrical Contacts, 1994, pp167-175.[6] Waterhouse, R.B.,Fretting Corrosion,Pergamon Press, 1972.[7] Lees, P.W., A Comparison ofElectroplated Gold Flashed Palladium-Nickel and Wrought Clad Inlay WE#1Capped Palladium-Silver, Proceedings ofthe 28* Annual Connector andInterconnection Symposium, pp 363-3791995.[8] Lees, P.W., Fretting Behavior of GoldFlashed Palladium, Palladium Nickel andPalladium Silver Contact Materials,Proceedings of the Thirty Seventh HolmConference on Electrical Contacts. pp 203-215, 1991.[9] Martin, J.B., Thornton, S.T. ,ClassicalDynamics of Particles and Systems,Harcourt Bruce College Publishers, 1995.

APPENDIX 1.The system is driven sinusoidally at anangular frequency, w. We assumed that thesystem motion occurs at the angularfrequency of the driving force and weignored any phase shifting between theapplied force and the position. Thus, theposition, velocity, and acceleration aregivenby:

x = A-sin(w.t) (1)v = A.w*cos(w.t) (2)a=-A-w2s in (w. t ) (3)

The forces in the system can be separatedinto three components: the applied forcefrom the electrodynamic driver, the resistiveforce due to the inherent friction in thedynamic system, and the friction force dueto the fretting contact between the twopseudo-crossed rods. In addition to theseforces, there is an additional inertial effect(mass times acceleration) because of non-zero accelerations given by Equation (3).Assuming that the friction force is a sinusoid(rather than constant), the sum of forces canbe written as:

Note that all the sinusoids cancel, so onlythe amplitudes are leR.Fapp~~ed-F*mping-Ff i ,*o ,=m [ -A*w21 ( 6 )Solving for the friction due to fretting:

The mass, amplitude of motion, andfrequency of vibration of the system areknown, but the remaining terms: appliedforce and damping response are not.16


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Determining these last two terms requiresadditional information about the system thatcan only be found through additionalmeasurements or modeling of the system.To determine the force applied to the systembased on our measured voltage values, wemust know how fret amplitude is affected bychanges in voltage applied to theelectrodynamic drivers. We believed thatthis relationship was linear. Therefore, weexamined the system without any frictionload, such that there wasn't any contact withthe fretting surface. At increasing voltagesto the driver, we measured the resultingamplitude of the unloaded system. Figure Acontains the resulting plot:

t1 ?=os99 i/zL /1

i0 Measured Data- egression it ~ 1I0 ~ ' ~ ~ ~ ~ ~ " " " " ~ ' ~ ~ ~ ~ ~ '0 50 100 150 200 250 300

System Amplitude, A (microns)

Figure A: The relationship between thevoltage applied to the system and thefretting amplitude.The relationship is of the form

where the slope, clfree= 0.0156 Volts/micron, A is the peak to peak amplitude ofthe sinusoidal motion, and V is the voltageapplied to the system. The subscript "free"is to denote that the system does not haveany load due to friction. We also note thatthe results were extremely linear, confirmingour early expectations.

The amplitude of the motion during frettingwill be less than the amplitude when there isno friction due to fretting. This decrease inamplitude will be due to the forces exertedby friction at the fretted surfaces. Based onEquation (8), we know that fret amplitudedepends on the voltage sent to the driver.To determine the forces in the system, wemust know how the applied force dependson the voltage presented to theelectrodynamic drivers. We decided againsttaking the fretting system apart to measurethese forces directly as the voltage isincreased. Instead we chose to model thesystem. From this model, we expected todevelop a relationship between driving forceand fret amplitude. The equation for thisrelationship would then be substituted intoEquation (8) to find the Voltage-Forceequation.To model this relationship, the mass of therod and the spring constant of the systemwas needed. The mass of the rod was 0.125kg. To find the spring constant of thesystem, a force was applied to the rod via aspring whose spring constant was alreadymeasured. FigureB contains the datacollected to determine the displacementforce relationship.

0 . 3 0 L , , , , , , , , ,i

i= k,*x + Bk,= -2190 N/m

0.05 B = 0.0157 N

measured Values-0 5 t I- egression fitt

-1 000e-4 0 oOoe*O 1 000e-4 2 000e-4 3 000e-4Displacement,AX (m)Figure B: The determination of the springconstan t for the system.

The spring constant was found to have twovalues: -323 N/m for positivedisplacements, and -2190 N/m for negative17


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displacements. The system was tested forhysterysis effects, and only minor hysterysiswas observed. An unexpected outcome ofthese measurements was that the springconstant depended on the direction the forcewas applied. Examining the system further,we found that each spring consisted of twomembranes. One membrane acted ondisplacements away from the driver, whileboth membranes affected displacements intothe driver. Thus, two separate springconstants were produced that weredependent on the direction of displacement.Because the mechanical system under studyquickly comes to rest after a displacement,the system must be loosing energy tointernal friction, which is called damping.Finding the amount of this damping,denoted p, is important to simulate thesystem correctly. Further, the damping andthe spring constants determine the resonancefrequency of the system. A damped systemwith no external forces can be modeled as:

(9)mx+2 .p . x+ - . x = 0~91.Note that the damping term effectivelyprovides a force proportional to velocity. Ifthe damping constant is small, the systemmay be underdamped and tend to oscillatelike a spring before coming to rest. If it isoverdamped, it will stop without anyoscillation. Examination of our systemreaction after a displacement indicated thatour system was overdamped. The solutionto Equation (9) for overdamping and aconstant k is well known:

(10)1+e-@ .[/J .e+ +A, e-Y'w = p--

~91.2c

We approximated our system assuming aconstant k value to use this solution. Todetermine this k value, the spring constantvalues in each direction were averaged, such

that k/m - 10000/sec2. Using experimentaldata collected from the system and thenfitting the data to Equation 10, we obtainedp values of 119.5, 117.4,and 121.9 sec-'.The average, 119.9 sec-', was used as thedamping factor for the system.After the spring constant and damping factorare known, the system is ready to bemodeled. With the sinusoidal driving forceadded, the system model becomes:x + 2 .p .x + -. x = -.o cos(w . ) (1 1)m m

[91.Euler integration techniques were used tomodel the system. A computer program waswritten to aid in the simulation of thesystem. The solution to the differentialEquation (1 1) has two parts: the particularsolution and general solution, whichcorrespond to the initial behavior and steadystate behavior of the system. The steadystate solution of the system is at thefrequency of the driving force. Thissupports our initial assumptions inEquations (1-3) that the displacement,velocity, and acceleration would all be atfrequencies corresponding to only thedriving frequencies.The system was simulated at four Hi with1,000 time steps per cycle. We confirmedconvergenceof solutions using 10,000 timesteps per cycle, which produced the sameresults. As the resulting displacementsversus time were graphed, we noted thenumber of cycles required to allow thesteady state solution to became dominant.We called the magnitude of the motion afterthis time the steady-state amplitude of thesystem. This generally occurred in 2 to 3cycles of motion. We then simulated thesystem with increasing driver force, andnoted the resulting steady-state amplitude ofthe system. The results of this simulationgive a relationship such that

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Data fittingFigure C in0 035-0 0030

0g 00020; tv OooZ5 I5 Oooi5 10 0 0 1 0 ~

Equation (12) was plotted inorder to measure ~2f iee .

0.00 0.25 0.50 0.75 I 00Force, F (NJ

FigureC: The graph of system amplitudeversus applied force for the free-motionsystem.was determined to be 2.943 x 10" m I

N. We now have a relationship between theapplied voltage to the driver and the forceexerted by the driver,

the fretting amplitude. The relationshipbetween Amplitude, A, and system force,Fhmphg was found to be:

Where c3 ~= -2.596xcan now determine the friction force fromEquation (7),Nlmicron. We

where M is the mass of the moving system,A is the amplitude, V is the voltage appliedto the system, and Clfree, ~ z f i e e , nd ~ 3 f i e e rethe constants previously discussed.Using this equation for the friction force, wecan determine the coefficient of friction foreach material system.

c- 'frictionNUam i c -where Pdynamic is the dynamic coefficient offriction, Ffridion s the friction force, and N isthe normal force on the plate.

At this point, the friction force as shown inEquation (7 ) is almost known. However, thesystem response, Fhmphg is unknown.Again, we can use the simulation to obtain arelationship between system response and

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Documents

An Examination of the Metallic Bonding of a Clad Material and 1.2