modeling and analysis of deformation on a flexure … · modeling and analysis of deformation on a...

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Int. J. Mech. Eng. & Rob. Res. 2014 Vennapusa V Sivareddy, 2014

MODELING AND ANALYSIS OF DEFORMATION ONA FLEXURE BEARING IN LINEAR COMPRESSOR

Vennapusa V Sivareddy1*

*Corresponding Author: Vennapusa V Sivareddy, [email protected]

Flexure bearing is a new concept and used for precision applications such as ProgrammableFocusing Mechanism (PFM), linear compressor, etc. These bearings are compact andinexpensive. A flexure bearing is designed for specific applications. These designed can usuallybe done with the advanced design tool like Fine Element Analysis. With the advent of computersFine Element Analysis has become the most suitable tool for the engineering analysis wherethe conventional approach is not suitable, geometric complexity are involved etc. This bearingcontains three slots having 120° apart and 12 peripheral holes are used to clamp the disc rigidlyonto a support structure. One central hole made for movement of shaft. The bearings arecommonly made by Aluminum material we calculate the deformations in theoretical and FEAmethod by using Ansys Software. In this paper we are modeling the flexure bearing by usingcopper material we are calculate the deformations in theoretical and FEA method by usingAnsys Software. We are analysis the deflection deformation on the both materials of bearing.We are analysis the deflection deformation on Flexure Bearing at different load (means 1 N to 5N). Using software’s like CATIA and PROE, modeling of flexure bearing done. Also make FineElement Method analysis on it by using Ansys software and lastly, this project considers thefatigue life criteria for flexure bearing and tries to optimize it. We are made design calculationand Fine Element Analysis for flexure bearing to make appropriate model and draw theperformance charts for linear compressor bearing by comparing the power consumption withdifferent temperatures, and capacity.

Keywords: Flexure bearing, Deformation, Linear compressor

INTRODUCTIONA bearing is any of various machineelements that constrain the relative motion

ISSN 2278 – 0149 www.ijmerr.comVol. 3, No. 1, January 2014

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Int. J. Mech. Eng. & Rob. Res. 2014

1 Tadipatri Engineering College, Kadapa Road, Veerapuram Village, Tadipatri Mandal, Anantapur Dist., Andhra Pradesh, India.

between two or more parts to only the desiredtype of motion. This is typically to allow andpromote free rotation around a fixed axis or

Research Paper

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free linear movement; it may also beto prevent any motion, such as by controllingthe vectors of normal forces. Bearings may beclassified broadly according to the motions theyallow and according to their principle ofoperation, as well as by the directions ofapplied loads they can handle.

The term “bearing” comes ultimately fromthe verb “to bear”,and a bearing is thus amachine element that allows one part to bearanother, usually allowing (and controlling)relative motion between them. The simplestbearings are nothing more than bearingsurfaces, which are surfaces cut or formed intoa part, with some degree of control over thequality of the surface’s form, size, surfaceroughness, and location (from a little control toa lot, depending on the application). Manyother bearings are separate devices that areinstalled into the part or machine. The mostsophisticated bearings, for the mostdemanding applications, are very expensive,highly precise devices, whose manufactureinvolves some of the highest technology knownto human kind.

Flexure BearingA flexure bearing is a bearing which allowsmotion by bending a load element. A typicalflexure bearing is just one part, joining two otherparts. For example, a hinge may be made byattaching a long strip of a flexible element to adoor and to the door frame. Another exampleis a rope swing, where the rope is tied to atree branch.

Flexure bearings have the advantage overmost other bearings that they are simple andthus inexpensive. They are also oftencompact, light weight, have very low friction,

and are easier to repair without specializedequipment. Flexure bearings have thedisadvantages that the range of motion islimited, and often very limited for bearings thatsupport high loads.

A flexure bearing relies on the bearingelement being made of a material which canbe repeatedly flexed without disintegrating.However, most materials fall apart if flexed alot. For example, most metals will fatigue withrepeated flexing, and will eventually snap. Thus,one part of flexure bearing design is avoidingfatigue. Note, however, that fatigue is importantin other bearings. For example, the rollers andraces in a rolling-element bearing fatigue asthey flatten against each other.

Flexure bearings can give very low frictionand also give very predictable friction. Manyother bearings rely on sliding or rolling motions,

Figure 1: A Living Hinge (a Type of FlexureBearing), on the Lid of a Tic Tac Box

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which are necessarily uneven because thebearing surfaces are never perfectly flat. Aflexure bearing operates by bending ofmaterials, which causes motion at microscopiclevel, so friction is very uniform. For this reason,flexure bearings are often used in sensitiveprecision measuring equipment.

Flexure bearings are not limited to lowloads, however. For example, the drive shaftsof some sports cars replace cardan universaljoints with an equivalent joint called a ragjoint which works by bending rubberized fabric.The resulting joint is lighter yet is capable ofcarrying hundreds of kilowatts, with adequatedurability for a sports car.

Many flexure bearings are combined withother elements. For example, many motorvehicles use leaf springs. The spring bothholds the position of the axle as the axle moves(flexure bearing) and provides force to supportthe vehicle (springing). In many cases it is notclear where flexure bearing leaves off andsomething else takes up. For example,turbines are often supported on flexible shaftsso an imperfectly balanced turbine can find itsown center and run with reduced vibration.Seen one way, the flexible shaft includes thefunction of a flexure bearing; seen another, theshaft is not a “bearing”.

Linear Motion BearingsA linear-motion bearing or linear slide isa bearing designed to provide free motion inone dimension. There are many different typesof linear motion bearings and this family ofproducts is generally broken down into two sub-categories: rolling-element and plane.

Rolling-Element BearingA rolling-element bearing is generallycomposed of a sleeve-like outer ring and

several rows of balls retained by cages. Thecages were originally machined from solidmetal and were quickly replaced by stampings.It features smooth motion, low friction, highrigidity and long life.

Ball Bearing SlidesAlso called “ball slides”, ball bearing slides arethe most common type of linear slide. Ballbearing slides offer smooth precision motionalong a single-axis linear design, aided by ballbearings housed in the linear base, with self-lubrication properties that increase reliability.Ball bearing slide applications include delicateinstrumentation, robotic assembly, cabinetry,high-end appliances and clean roomenvironments, which primarily serve themanufacturing industry but also the furniture,electronics and construction industries. Forexample, a widely used ball bearing slide in thefurniture industry is a ball bearing drawer slide.

Commonly constructed from materials suchas aluminum, hardened cold rolled steel andgalvanized steel, ball bearing slides consistof two linear rows of ball bearings containedby four rods and located on differing sides ofthe base, which support the carriage forsmooth linear movement along the ballbearings. This low-friction linear movementcan be powered by either a drive mechanism,inertia or by hand. Ball bearing slides tend tohave a lower load capacity for their sizecompared to other linear slides because theballs are less resistant to wear and abrasions.In addition, ball bearing slides are limited bythe need to fit into housing or drive systems.

Roller SlidesAlso known as crossed roller slides, rollerslides are non-motorized linear slides that

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provide low-friction linear movement forequipment powered by inertia or by hand.Roller slides are based on linear rollerbearings, which are frequently criss-crossedto provide heavier load capabilities and bettermovement control. Serving industries such asmanufacturing, photonics, medical andtelecommunications, roller slides are versatileand can be adjusted to meet numerousapplications which typically include cleanrooms, vacuum environments, materialhandling and automation machinery.

Consisting of a stationary linear base anda moving carriage, roller slides work similarlyto ball bearing slides, except that the bearingshoused within the carriage are cylinder-shapedinstead of ball shaped. The rollers crisscrosseach other at a 90° angle and move betweenthe four semi-flat and parallel rods that surroundthe rollers. The rollers are between “V” groovedbearing races, one being on the top carriageand the other on the base. The travel of thecarriage ends when it meets the end cap, alimiting component. Typically, carriages areconstructed from aluminum and the rods androllers are constructed from steel, while the endcaps are constructed from stainless steel.

Although roller slides are not self-cleaning,they are suitable for environments with lowlevels of airborne contaminants such as dirtand dust. As one of the more expensive typesof linear slides, roller slides are capable ofproviding linear motion on more than one axisthrough stackable slides and doublecarriages. Roller slides offers line contactversus point contact as with ball bearings,creating a broader contact surface due to theconsistency of contact between the carriageand the base and resulting in less erosion.

Plain BearingPlain bearings are very similar in designto rolling-element bearings, except they slidewithout the use of ball bearings.

Dovetail SlidesDovetail slides, or dovetail way slides aretypically constructed from cast iron, but canalso be constructed from hard-coataluminum, acetal or stainless steel. Like anybearing, a a dovetail slide is composed of astationary linear base and a moving carriage.a Dovetail carriage has a v-shaped, or dovetail-shaped protruding channel which locks into thelinear base’s correspondingly shaped groove.Once the dovetail carriage is fitted into itsbase’s channel, the carriage is locked into thechannel’s linear axis and allows free linearmovement. When a platform is attached to thecarriage of a dovetail slide, a dovetail table iscreated, offering extended load carryingcapabilities.

Since dovetail slides have such a largesurface contact area, a greater force isrequired to move the saddle than other linearslides, which results in slower accelerationrates. Additionally, dovetail slides havediff iculties with high-friction but areadvantageous when it comes to load capacity,affordability and durability. Capable of longtravel, dovetail slides are more resistant toshock than other bearings, and they are mostlyimmune to chemical, dust and dirtcontamination. Dovetail slides can bemotorized, mechanical or electromechanical.Electric dovetail slides are driven by a numberof different devices, such as ball screws, beltsand cables, which are powered by functionalmotors such as stepper motors, linear motors

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and hand wheels. Dovetail slides are directcontact systems, making them fitting for heavyload applications including CNC machines,shuttle devices, special machines and workholding devices. Mainly used in themanufacturing and laboratory scienceindustries, dovetail slides are not ideal for high-precision applications.

Why Flexural Bearings?A bearing is a rotating component which isplaced between to parts to allow them to moveeasily. A flexure bearing is just that it allows toparts to move with each other with no trouble.There are several different types of bearingsand they all use a different shape to move thetwo parts, for example ball bearings use littleballs, needle bearings use very thin needletubes, roller bearings use tubes as well but arelarger than needle size.

A flexure bearing allows the motion of twoparts by bending a load element. What doesthis mean? It means it takes two materials andfixes them together by a flexure bearing andthis bearing allows them to move. Think of adoor on a hinge. The door needs to beattached the door frame but it also needs tobe opened and closed and so a hinge is used.The door and door frame are the two materialsand the hinge is the flexure bearing allowing itto move freely.

Flexure bearings are great because theyare very simple to manufacture especiallycompared to some other types of bearingsand they are easy and cheap to replace andso maintenance isn’t a large issue. Flexurebearings have many advantages includingthey do not jitter or wobble as they are fixedinto place, this minimalists the risk of damage

to the bearing and the two rigid parts, theycan operate in a vacuum, they have virtuallyan unlimited life span if the atmosphere is notcorrosive, they can work in high and lowtemperatures, and they don’t make a noisewhen they are operating. Of course thesetypes of bearings have some limitations anddownfalls for example they can only be usedon materials that do not disintegrate afterbeing repeatedly flexed, their angularexcursion is limited, they are more expensivethan ball bearings, and they are harder toinstall.

All bearings have their up and down sidesto them and flexure bearings are the same,they are great for some applications where inothers another bearing is better suited.

DESIGN CONSIDERATIONToday still most of the linear coolers are drivenby a moving coil motor. Moving coils are wellknown form the loudspeaker technology andare a standard technology today. The motorcan easily be calculated according to theLorentz Force Law; also high motor efficienciescan be obtained as eddy current losses arelow. However, the moving coil concept isassociated with several drawbacks which canbe overcome when changing to movingmagnets.

The Benefits are

• Coil can be placed outside the heliumvessel, i.e., no out gassing into the workinggas

• No electrical feed through needed

• No flying current lead: eliminates potentialfailure mechanism and reducescomplexity.

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The goals for the compressor design were:

• Compliance with the One Watt Linearinterface (SADA2): diameter 60.45 mm,length < 123 mm

• Lowest complexity and less part for lowfabrication cost.

• Low amount of out gassing materials insidehelium vessel.

• Suitable for operation of all relevant Stirlingand Pulse Tube cold fingers.

• Implementation of efficient alignmentprocess for serial production.

• Motor impedance compatible to currentlinear coolers.

• Suitable as form, f it and functionreplacement for SADA2.

To allow an easy exchange with thecommon 1 W linear coolers the newcompressor SF100 has a cable exit for bothmotor halves on one end of the compressor,located on the same side as the fill port. This

concept also allows a compressor versionwith integrated cooler control electronics, seeFigures 2 and 3. The achieved reduction incomplexity can be seen in Table 1, some keyfigures are compared for the Flexure BearingMoving Magnet compressor SF100 with theconventional moving coil compressor SL100.Although achieving the same performancedata, the compressor length was reduced bymore than 4 mm, the weight was reduced alittle.

Number of Parts % 100 44

Thread Joints % 100 29

Adhesive Joint % 100 40

Welding Joint % 100 375

Weight Non-MetallicComponents inHelium Vessel % 100 3

Compressor Weight g 1720 1680

Compressor Length m m 122.2 117.9

Table 1: Comparison of DesignCharacteristics Between Linear

Compressor and Flexure BearingCompressor

Compressor Type UnitsSL100

(MovingCoil)

SF100: MovingMagnets,Flexure

Bearings

Figure 2: Flexure Bearing Spring and FEMSimulation for Max. Geometrical Stroke

Figure 3: Typical Flexure Unit of theSuspension System

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Flexure Bearing DesignA high end spring material was chosen for thesprings to achieve the highest radial stiffnessfor the given spring diameter and needed axialstiffness. The optimized spring design wasdeveloped by systematic FEM simulation atboth AIM and also external institutes. Figure 1shows the optimized spring design and a resultfrom FEM calculation. The highest strain withinthe spring at max geometrical piston stroke isat least 35% below the specified fatigue limitof the material. This ensures that the spring isoperated well within the endurance level of thematerial.

For highest compactness of the compressora welding joint for the spring was developed.All relevant parameters like material andtolerance for spring and adjacent compressorparts as well as welding details had to beconsidered to ensure robust welding seam inserial production. Also the heat distributionzone with impact on the materialcharacteristics inside the spring had to beevaluated.

To verify the simulation results the weldedspring was qualified and a Wohler curve was

generated. This diagram is shown in Figure4. For each stroke 20 springs were tested.During the tests the actual stroke wasmonitored with a laser stroke sensor. Theresults confirm more than 35% margin intension to the endurance level at maxgeometrical stroke. For nominal stroke at fullinput power (= 65% of max stroke) the safetyfactor is about 2.1 and thus sufficient.

Design Requirements ofCompressor Flexure BearingThe radial stiffness of the flexure bearingassembly (Figure 18) should be high enoughto support the clearance seal under the effectof the suspended mass which comprisesmainly of the piston-shaft sub assembly, coiland coil former. In order to evaluate the radialstiffness requirement for each of the flexurediscs that constitute the two stacks of the wholebearing assembly, a simplified modelproposed and piston deflection formulated.

Figure 4: Wohler Curve for FlexureBearing Spring (Mounting Condition

Identical to Cooler)

Figure 5: Assembly of the Flexure Bearing

Piston AlignmentTo ensure contact-less operation of the pistoninside the sleeve over full stroke and typicaloperating conditions an active alignmentprocess was developed. Heart of thealignment is a computerized alignment—and

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welding station, sees Figure 6. In the alignmentstation a precision of the adjustment of about1 m is achieved. After finishing the alignment,the components are welded directly in thealigned position. This process was optimizedto allow a fast alignment in serial production.

Figure 6: Computerized Alignment andWeldingStation

Figure 7: SF100A with 14 mm Pulse TubeCold Finger

Figure 8: Outline Dimension of CoolerSF100A

Figure 9: Flexure Bearing CompressorSF100A with 13 mm AIM Sleeve Coldfinger

Figure 10: Outline Dimension of CoolerSF100B

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PERFORMANCE DATAThe compressor is optimized for operationwith different types of Sterling and Pulse Tubecold fingers:

• ½” SADA cold finger

• 13 mm AIM sleeve cold finger

• 12 mm AIM standard cold finger

• 14 mm AIM Pulse Tube cold finger

Smaller cold fingers can be used as well,however, to keep resonance condition insidethe compressor and to achieve a high motorefficiency a reduction in piston diameter isneeded. The compressor design takesaccount for that. As part of the qualificationefforts a detailed performance mapping wascarried out. Performance data for the ½''SADA cold finger are shown in Figures 9 to12 show the cooling capacity vs. the AC inputpower for three dif ferent ambienttemperatures with curves for different cold tiptemperatures each.

All given data are measured with the samecharge pressure at constant operatingfrequency (55 Hz), i.e., without any adaptation

Figure 11: Flexure Bearing CompressorSF100B

Figure 12: Input Power for Constant HeatLoad of 400 mW vs. Ambient Temperature

for 77 K, 84 K and 95 K

Figure 13: Cooling Capacity for 71 °CAmbient Temperature, Constant Coldtip

Temperature


Temperature

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cold finger are slightly better compared to theSADA cold finger. This is achieved byimproved heat removal at the expander warmend and a reduced longitudinal conduction dueto reduced sleeve length. At 23 °C ambienttemperature more than 2 W cooling capacitiesat 80 K are achieved with 60 W of input power,at 71 °C more than 0.8 W@80 K are available.Compared to the SL100 cooler the cool downtime is improved.

The given ambient temperature is the airtemperature. Surface temperatures are upto 15 K higher. The given heat load is thepower of the electrical heater on a 1,440 Jcopper thermal mass. Data for thecompressor AC input power are shown inthe diagrams. The compressor is designedfor AC input power of up to 70 W. To allowthe cooler operation with typical powersupplies, the dynamic impedance ismatched to about 4 Ohm. Thus the coolercan be operated with the AIM controller AM7[2] and also the HTI board as used in theSADA program for instance. The AM7 canbe used in an external housing and also aspart of the SF100B compressor directlyattached to the compressor housing.

ANALYSIS OF FLEXUREBEARING IN ANSYSMaterial Data for Flexure Bearing

Copper AlloyYoung’s Modulus 1.1e+011 Pa = 1.1 e+5

N/mm2

Poisson’s Ratio 0.34

Density 8300. Kg/m³

Thermal Expansion 1.8e-005 1/°C

Figure 15: Cooling Capacity at 80 K,Curves for Constant Ambient

Temperature


Temperature

for ambient or cold tip temperature. Whenchanging these parameters, a furtherimprovement can be achieved. Theperformance data of the AIM 13 mm sleeve

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Tensile Yield Strength 2.8e+008 Pa

Compressive YieldStrength 2.8e+008 Pa

Tensile Ultimate Strength 4.3e+008 Pa

Compressive UltimateStrength 0. Pa

K1 0.238

Thermal Conductivity 401. W/m·°C

Specific Heat 385. J/kg·°C

Relative Permeability 1

Resistivity 1.724e-008 Ohm·m

Aluminum AlloyYoung’s Modulus 7.1e+010 Pa = 7.1

e+4 N/mm2

Poisson’s Ratio 0.33

Density 2770. kg/m³

Thermal Expansion 2.3e-005 1/°C

Tensile Yield Strength 2.8e+008 Pa

Compressive YieldStrength 2.8e+008 Pa

Tensile Ultimate Strength 3.1e+008 Pa

Compressive UltimateStrength 0. Pa

K1 0.238

Specific Heat 875. J/kg·°C

Relative Permeability 1.

Resistivity 5.7e-008 Ohm·m

Analysis of DeformationFlexural bearing under investigation in theproject is made up of stack of flexures each ofwhich is a disc having hole and the spiralshence, it is apt to analyze a sample disc using

the typical software to be used for furtheranalysis. It should be noted that Timoshenkohas given the empirical formulas for finding theaxial deflection of disc with hole if the axial loadis applied on the periphery of hole with theouter boundary of the disc is simply supported.Figure 17 shows a typical disc with hole forwhich parameter ‘a’ and ‘b’ have been defined.

Figure 17: Typical Disc

The Timoshenko have deduced the formulafor finding the deflection of disc which is asfollows:

Wmax = k*q*a4/E*h3

It was decided to analyze the disc with holeof the following specifications using analyticalas well as FEA approach.

Given Data for Copper

Outer Diameter a = 50 mm

Inner Diameter b = 10 mm

Thickness = t = 5 mm

Material Used = Copper

Loading Point = at center peripheral surface

Force Applied = 5 N

Co-efficient = K1 from data book is enclosedaccording to a/b

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Pressure = P = F/A for Force 5 N

= F/2 × × R × t

= 5/2 × × 5 × 5 = 0.0318 N/mm2

Wmax = k1*P*a4/E*h3

= 0.238*0.0318*254/1.1*105*53

= 0.00021501 mm


= F/2 × × R × t

= 4/2 × × 5 × 5 = 0.0254 N/mm2

Wmax = k1*P*a4/E*h3

= 0.238*0.0254*254/1.1*105*53

= 0.0001717 mm


= F/2 × × R × t

= 3/2 × × 5 × 5 = 0.0190 N/mm2

Wmax = k1*P*a4/E*h3

= 0.238*0.019*254/1.1*105*53

= 0.000128 mm


= F/2 × × R × t

= 2/2 × × 5 × 5 = 0.0127 N/mm2

Wmax = k1*P*a4/E*h3

= 0.238*0.0127*254/1.1*105*53

= 0.000085 mm


= F/2 × × R × t

= 1/2 × × 5 × 5 = 0.0063 N/mm2

Wmax = k1*P*a4/E*h3

= 0.238*0.0063*254/1.1*105*53

= 0.000043 mm

Given Data for Aluminum

Outer diameter a = 50 mm

Inner diameter b = 10 mm

Thickness = t =5 mm

Material Used = Aluminum

Loading Point = at center peripheral surface

Force Applied = 5 N

Co-efficient = K1 from data book is enclosed


= F/2 × × R × t

= 5/2 × × 5 × 5 = 0.0318 N/mm2

Wmax = k1*P*a4/E*h3

= 0.238*0.0318*254/7.1*104*53

= 0.0003331 mm


= F/2 × × R × t

= 4/2 × × 5 × 5 = 0.0254 N/mm2

Wmax = k1*P*a4/E*h3

= 0.238*0.0254*254/7.1*104*53

= 0.0002660 mm


= F/2 × × R × t

= 3/2 × × 5 × 5 = 0.0190 N/mm2

Wmax = k1*P*a4/E*h3

= 0.238*0.0190*254/7.1*104*53

= 0.0001990 mm


= F/2 × × R × t

= 2/2 × × 5 × 5 = 0.0127 N/mm2

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Wmax = k1*P*a4/E*h3

= 0.238*0.0127*254/7.1*104*53

= 0.000133 mm


= F/2 × × R × t

= 1/2 × × 5 × 5 = 0.0063 N/mm2

Wmax = k1*P*a4/E*h3

= 0.238*0.0063*254/7.1*104*53

= 0.0000659 mm

1. 5 N 0.0318 0.00021501 mm 0.0003331 mm

2. 4 N 0.0254 0.0001717 mm 0.0002660 mm

3. 3 N 0.0190 0.000128 mm 0.0001990 mm

4. 2 N 0.0127 0.000085 mm 0.000133 mm

5. 1 N 0.0063 0.000043 mm 0.0000659 mm

Table 2: Comparisition Tablefor Theoretical Calculations

S. N

o.

Forc

e

Pres

sure

N/m

m2 For Copper For

Aluminum

AxialDeformation

AxialDeformation

Figure 18: Graph Between Force vs. AxialDeformation

Graph between force Vs Axial Deformation,the force is taken in x-axis and AxialDeformation in y-axis.

1. 5 N 0.0318 0.00021501 mm 0.0003331 mm

2. 4 N 0.0254 0.0001717 mm 0.0002660 mm

3. 3 N 0.0190 0.000128 mm 0.0001990 mm

4. 2 N 0.0127 0.000085 mm 0.000133 mm

5. 1 N 0.0063 0.000043mm 0.0000659 mm

Table 3: Comparison Table for TheoreticalCalculations According to the Ansys

S. N

o.

Forc

e

Pres

sure

N/m

m2 For Copper

AxialDeformation

AxialDeformation

CONCLUSIONWe are design the bearing by using coppermaterial, the copper material is stronger thanthe aluminum according to the material data,presently the bearings are design by thealuminum, and in this project we design thebearing by using copper material. We analysisthe deformation on the bearing at different loads.

The deflection deformations are found outin theoretically and FEA Method by usingAnsys. Comparing the both results of theCopper and Aluminum, copper is having themore deflection deformations in the Flexurebearing at different loads. Finally the copperbearing is gives the more deflections.

REFERENCES1. Benschop A A J, Mullié J and Meijers M

(2000), “High-Reliability Coolers atThales Cryogenics”, Proc. SPIE 4130,pp. 385-648.

2. Groep vd W L, Mullié J C, Willems D W Jand Benschop T (2008a), “Developmentof a 15 W Coaxial Pulse-Tube Cooler”,Cryocoolers, Vol. 15, pp. 157-165.

3. Groep vd W, Mullié J, Benschop T,Wordragen v F and Willems D (2008b),

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“Design and Optimization of the CoaxialPulse-Tube Cooler”, Adv. in CryogenicEngineering, Vol. 53, pp. 1667-1674.

4. Linder M (2007), “ESA Cryogenics andFocal Plane Cooling”, ESAHarmonization Meeting.

5. Mullié J C, Bruins P C, Benschop T andMeijers M (2004), “Development of theLSF95xx 2nd Generation Flexure BearingCoolers”, Cryocoolers, Vol. 13, pp. 71-76.

modeling and analysis of deformation on a flexure … · modeling and analysis of deformation on a...

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