analytical estimation of microslip damping in bladed disks

7/29/2019 Analytical Estimation of Microslip Damping in Bladed Disks

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Analytical Estimation of MicroslipDamping in Bladed-Disks

J.S. RaoChief Science Officer, Altair Engineering

Mercury 2B Block, 5th Floor, Prestige Tech Park, Sarjapur MarathahalliOuter Ring Road, Bangalore, Karnataka, 560103, India

[email protected]

Rejin RatnakarAltair Engineering


[email protected]

R. NarayanAltair Engineering


[email protected]

www.altairproductdesign.comcopyright Altair Engineering, Inc. 2011


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www.altairproductdesign.com

Copyright Altair Engineering, Inc., 2011 2

Abstract

Most usually damping in a blade is attributed to material and friction damping. Frictiondamping can be either due to Macroslip obeying Coulomb laws of friction or Microslip (alsocalled Fretting Fatigue) which occurs at high critical speeds with tight blade-disk junctions dueto heavy centrifugal loads. This paper is concerned with microslip damping which isconventionally determined through tests.

Oloffson and Hagman developed a theoretical model for micro-slip between flat surfacesbased on deformation of ellipsoidal elastic asperities and this model is verified experimentallyin Hitachi laboratory. In this paper Oloffson-Hagman microslip damping model is adopted todetermine analytically an effective friction coefficient which is a function of the tangential andnormal forces at the slipping surface junction, tangential contact stiffness and slip amplitude ata resonance. Here an iterative method is developed to determine the microslip dampingcoefficient.

Introduction

Turbine blades have very little damping and therefore when they go through a resonance atcritical speed, the stresses can easily get magnified by 100 times or even higher of steadystress. These very high resonant stresses are responsible for fatigue damage. Therefore,Damping has been identified long ago as a key parameter in blade design. Rowett (1914)conducted tests on elastic hysteresis in steel. Effects of friction and loose mounting werestudied by Hansen et al. (1953).

Lazan (1968), also see Nashif, Jones and Henderson (1986), measured hysteresis in tensiontests and defined the loss of energy per cycle by a simple relation.

A nonlinear damping model was quantified through experiments by Rao, Gupta and Vyas(1986); the equivalent viscous damping is expressed as a function of strain amplitude at areference point in a given mode of vibration at a given speed of rotation.

Centrifugal load is simulated by means of thermal expansion to avoid rotation and simplify thetest rigs. Rieger and Beck (1980) performed such tests for EPRI. Rao, Usmani andRamakrishnan (1990) presented a finite element method to study the friction dampingbetween blade root and disk by using contact elements. They have also designed and built atest rig simulating the centrifugal load by means of cryogenic liquid cooling on a blade pairmounted in the frame contracted by thermal cooling.

Rao and Saldanha (2003) developed an analytical procedure using Lazans hysteresis law(1968) to obtain a nonlinear relationship as observed in experiments of Rao, Gupta and Vyas(1986). Friction damping characteristic is obtained by determining the transient response dueto an impulse excitation at a suitable point on the blade to simulate the desired mode ofvibration and assessing the decay curve. Coulombs friction law is used assuming thatmacroslip occurs between the blade and disk interfaces.

Rao, Narayan and Ranjith (2008) developed process driven approach to incorporate materialdamping in determining an equivalent viscous damping ratio as a function of reference strain


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amplitude in the chosen mode of vibration to perform life estimation. Rao, Narayan, Ranjith,and Rejin (2008), determined the material damping and combined material friction damping ofa bladed disk in estimating blade life. A code TurboManager is developed to run onHyperWorks platform.

The response strain amplitudes decrease considerably as the speeds go up when the bladeand disk get locked in the root. Because of the tightness at high speeds, slip amplitudes getreduced and the friction is governed by contacts at asperity level rather than global Coulombslaws. This microslip is important from the point of fretting fatigue that is commonly observed inturbomachine blades, see Fig. 1.

Fig. 1 Fretting Fatigue due to Micro-Slip

Burdekin, Cowley and Back (1978) proposed a theoretical model for micro-slip based onreasonable physical properties, assuming that the contacting asperities are substituted byprismatic rods of equal stiffness. Hagman (1993) proposed a theoretical model with contactingasperities replaced by spherical bodies of constant radius. Olofsson (1995) and Olofsson and

Hagman (1997) expanded this model to include oscillating displacements and elliptical shapedasperities.

In this paper, Olofsson and Hagman model is used in developing an analytical procedure fordetermining the effective friction between blade and disk interfaces under microslip conditions.

Nomenclature

A apparent area of contacta, b semi-axes in x andy directions

C surface parameter

c major semi-axis

dslip slip displacementdstick stick displacement

dtotal total slip and stick displacement

E composite modulus of elasticity given by

2

22

1

21 11

'

1

EEE

F friction load


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Ft tangential (friction) load

Fn normal load

F*

initially applied loadG composite shear modulus given by

2

2

1

1 22

'

1

GGG

k ovality ratio a/b, a


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Fig. 3 Schematic of Contact

The following assumptions are made:

1. Shape of asperities is ellipsoidal2. Height distribution of asperities is uniform3. Surface contact is elastic and the behavior of an individual asperity follows Hertz

theory for elliptical contacts.4. All asperities have their semi-axes a and b in the same global x- and y- directions,

respectively5. Contacting asperities have the same constant ovality ratiok= a/b, a


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2

5

2

1

0

2

95

'4

Rk

ECAdzPCAP i

(4)

Deresiewicz (1957) expressed the force-displacement relationship for an individual asperity, i,as

(5)

(6)

Fiwill deflect uptoFi =Pi.Equation (5) gives the limit deflectionLi

'

'8

E

GzLi

(8)

The total frictional load becomes

Li

Li

z

i

z

i CAdzPCAdzFFFF0

slipspring

(9)

whereFspring is the frictional load from the active asperities which have not reached theirlimiting tangential deflection andFslip is the contribution from asperities which have reachedtheir limiting tangential deflection.

The total frictional load is obtained from using equations (1), (5) and (8) in equation (9)

2

5

'

3

1611

d

i

iiP

cGPF

ab

ee

ba

ee

,12

4

,1

2

4

2

2

2

22

2

2

2

2

2

2


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2

5

2

5

2

5

2

1

'

'811

'

'8

2

9

'

5

4

E

GP

E

G

Rk

ECAF

(10)

The above is valid until

(11)

Suppose that after reaching a valueF*, the frictional load Fis reduced; the force displacementrelationship under unloading for an individual asperity ican be expressed as, see Fig. 4.

2

3

32

'16112

i

diidi

P

cGPF

(12)

Fig. 4 Frictional Load vs. Displacement for an Individual Ellipsoidal Body

Corresponding limit deflection for unloading is twice that for loading. The maximum height ofthe asperitieszdlifor which they will slip is

'8

'max

G

E


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'

'4

E

GzdLi

(13)

The sense of slip must be reversed, see Mindlin and Deresiewicz (1953) but its absolutemagnitude is not altered during unloading. Then the slip part of the tangential load duringunloading is twice that for loading. The equation for the frictional load during unloading is

2

5

0

'

'4112

2

E

GP

CAdzPCAdzFF

d

z

i

z

did

dLi

dLi

(14)

where dis the reduction in the initially loaded displacement,*, andFdis the reduction in the

initially applied load,F*.

The equation for frictional load transformed to the original co-ordinate system is

2

5*

*

'

'4112

E

GPFFr

(15)

Suppose now that the frictional load is oscillating between F*andF*. The situation atF=F*is identical with that atF= F*, except for the reversal of sign. Hence the frictional load thenbecomes

2

5*

*

'

'4112

E

GPFFF rs

(16)

The area enclosed by the curvesFsandFrgives the energy dissipation during micro-slip percycle. Integration gives the energy dissipation,Was


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**

2

2

7

*

2

5

2

*

*

*

7'2

'

'

'8'

'2

'14

7

4

FG

EP

E

GE

G

EPP

dFFWrs

Now consider when= 1, a = b(asperities modeled as spheres, then= = equations(10), (14) and (17) become

(10a)

2

5

'

'2112

E

GPF dd

(14a)

**

2

2

7

*

2

5

2

5

2

3

*

7'

'

'4'

''

14

7

4

FG

E

GE

GE

PP

W

(17a)

These are same as Hagman (1993) and Olofsson (1995) for the case where the asperities arereplaced with spheres.

MARQUINA MODEL

For blade-disk junction case, Marquina et al (2008) adopted the above Olofsson andHagmans micro-slip model into their approach. First the Shear (tangential) stiffness isobtained from equation (10a) written as

n

NtE

GFF

'

'411

2

5

'

'411

E

GPF


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(18)

1

'

'41

'

'4

n

Ntt

E

G

E

nGF

d

dFK

(19)

Limitingto'4

'max

G

E the above equation (18) is simplified to give

2

5

11

N

tNt

nF

KFF

(20)

Multiharmonic balance-based prediction is used to obtain displacement results in thefrequency domain for non-linear analysis involving friction.

ASAI FORMULATION

Here the formulation is slightly different made in such a way to verify Olofssons formulation for blades.Their microslip damping model of two surfaces under contact with FtandFnas tangential and normalforces is given in Fig. 5, see Asai et al (2009); the tangential contact stiffness is

stick

ttc

d

FK

(21)

For the linear model without hysteresis, the material property is Imaginary Tangential ContactStiffnessKtc,imgiven by

total

timtc

d

FK ,

(22)

In Fig. 5, the total displacement is stick and slip as shown and given by

sliptc

t

slipsticktotal

d

K

F

ddd

(23)

Ifslipis the displacement due to the normal forceFn(slip per unit normal force) and tangentialstiffnessktc, we define a parameter

slip

slip

n

tcslip d

F

Kd

(24)


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Fig. 5 Micro-Slip Damping Model

Under constant normal load, as the displacements are increasing in Oloffsons oscillatingdisplacements model, the asperities are replaced by spheres with the same radius. It isassumed that the height distribution of the asperities is uniform and the behavior of anindividual asperity follows Hertz theory. The resulting contact model is

n

n

totaltc

n

t

nmF

dKm

F

F11

(25)

where n and m are constants. Using (23) the above becomes

n

n

tcslip

n

t

n

t

F

Kd

F

F

nmm

F

F 111

(26)

Asai et al (2009) verified the above experimentally for the parametern

tcslip

F

Kdas shown in Fig.

6 for three different test specimens. Microslip occurs for large values ofFn(Ft


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Fig. 6 Asai et al (2009) Experimental Result for Microslip

Asais experiments have shown that Hagman and Olofsson elasto-plastic theory of contactprovides a workable model for blades given by (10a). The problem however is not simple andhighly nonlinear.

MODEL IN THE PRESENT STUDY

A simple rectangular blade (Sandvik O&T steel) 5 cm by 1 cm with a T root shown in Fig. 7 isconsidered for the study. The blade length is 30 cm. The contact surfaces on either side are 5 cm by 1cm. The disk is taken to be 14 cm long and 25 cm wide with 1 cm thickness for the purpose of modelingand the contact surfaces are 4 cm from the top of the disk. The blade root is taken as 15 cm by 5 cm.The blade and its root are modeled using 606 SOLID 45 elements. The interfacial surfaces between theblade root and disk are modeled as, 20 CONTACT 173 and 20 TARGET 170 contact elements withnumbers as shown in Fig. 8.

The material density is 7800 kg/m3. The elastic modulus is taken as 210 GPa. The blade rotates at

20000 RPM.

m= 0.9, n= 2.5


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Fig. 7 Blade-Disk Model

Macro-friction study was made for this model using Coulombs friction coefficient = 0.4 by Raoand Saldanha (2003). This analysis was done by giving an impact load to determine the decaycurve at a given natural frequency for a given critical speed. From this decay curve thenonlinear model for equivalent viscous damping as a function of reference strain amplitude in

a given mode of vibration at a given speed of operation is obtained.

Blade

DiscContact elements


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(a) Contact Elements

(b) Left Side Contact Elements

(c) Right Side Contact ElementsFig. 8 Contact Elements with Numbers

ANALYSIS AND ITERATIVE SOLUTION

Here Olofssons relation is written as

25

'

'411

E

i

Gi

nF

tF

(27)

The friction coefficient is denoted instantaneous value with a subscript iwhich will vary duringiteration process proposed here. The first thing we notice that the following condition shouldbe satisfied

0'

'41

E

G

i

(28)

for microslip conditions to exist. If equation (28) is not satisfied, the friction follows globalconditions. First a check is made whether micro-slip according to Olofssons relation occurs by

checking 1'

'4

E

G

i

From the above we see that microslip occurs for lower values of

. The three quantities,

,andare dependent on each other.


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1. Larger value of is conducive for microslip and the penetration is higher with highspeeds (or high normal loads). This is why we have macroslip at low speeds when thetwo interfacial surfaces are well separated beyond asperity level. At high speeds,microslip can only occur with the two surfaces closer at asperity level.

2. The ratio

should be lower that can be conducive for microslip conditions to occur.

Nonlinear Steady Centrifugal Analysis: To get a feel of macroslip and microslip regimes, a

nonlinear steady state analysis for different rotational speeds (i.e., penetration ) and friction

coefficientsis performed. The material data is

xy = 0.3

E= 210 GPa

E= 115GPa

G = 80.7 GPa

G = 23.7 GPa

Using the resulting tangential displacement the parameter

'

'41

Ei

G in equation (27) is

plotted in Fig. 9 as a function of rotational speed and friction coefficient.

Fig. 9 Macro and Micro Slip Regions

Microslip regime is denoted by green and macroslip regime in red color.

For an oscillating case, the slip values depend on excitation on the blade.

Blade under Excitation: For the present problem, the blade is excited by a 1 sinusoidal forcewith a pressure distributed on the blade surface as shown in Fig. 10. The model wasconstrained at hub.

Material data is given individually to each element as micro-slip conditions are checked for allslipping surfaces.


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Because it is a nonlinear problem a transient analysis is performed to determine the slip

displacementand penetrationafter reaching steady state conditions.

Fig. 10 Pressure Loads in Pa

To determine penetration augmented Lagrangian algorithm approach developed byZavarise, Wriggers and Schrefler (1995) in Ansys is used. This method combines amicromechanical approach that takes the real stiffness of the asperities into account, with amean surface concept. The advantage in this method is in avoiding any numerical ill-conditioning as in Penalty Method.

To begin with a global friction value can be used; in this case 0.8 was used as initial value.

Macros were developed in Ansys to perform the iterations automatically. This iteration getsactivated only when Olofssons condition is satisfied at each step. The time step used intransient solution is 0.0002 sec.

The iterations are stopped once there is a convergency on the slip displacement.

RESULTS

All the elements are found to have microslip conditions. Two outer elements on the right sidecontact surface are considered here to observe the friction coefficient values. The initial run

with= 0.8 for the outermost element number 600 is given in Fig. 11. The converged value offriction coefficient under microslip is 0.046.

For element 611, the converged value for friction coefficient under microslip is 0.043 as givenin Fig. 12.

The converged friction coefficient is 0.0643 for element 699 as given in Fig. 13.


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Similarly Fig. 14 gives for element 688,= 0.0643. The right side contact surface has slightlyhigher friction coefficient compared to the left side.

Blade at 5000 RPM: In order to study the effect of speed on the occurrence of microslip in the

blade under consideration, the speed was reduced to 5000 RPM and the analysis repeated.Here the iterations are started with a coefficient of friction = 0.2.

In this case two outer elements 688 and 699 given above on the right contact surface arefound to be completely sticking with no change in the initial friction value 0.2. This can beobserved in Figs 15 and 16. We notice that there is no slip in these elements.

The remaining all elements are found to have microslip. The friction coefficient is found to bein the range 0.08 to 0.017.

CONCLUSIONS

A procedure for assessing micro-slip conditions in blades is presented. This along withmaterial friction and macro-slip with global Coulomb friction can define complete dampingmechanism in a turbomachinery bladed-disk.

ACKNOWLEDGEMENTS

The authors are thankful to Altair Engineering India in carrying out this work.

REFERENCES

1. Asai K, Sakurai S, Kudo T and Ozawa N, (2009) Evaluation of Friction Damping inDovetail Root Joints based on Dissipation Energy on Contact Surfaces, ASME TurboExpo, GT2009-59508

2. Burdekin N, Cowley A and Back N, (1978) An Elastic Mechanism for the Micro-SlidingCharacteristics between Contacting Machined Surfaces, J Mech Engng Sci., vol. 20, p.121

3. Deresiewicz H, (1957) Oblique Contact of non-spherical elastic bodies, J Appld Mech, vol.24, p. 623

4. Hagman L, (1993) Micro-slip and Surface Deformation, Licentiate thesis, Royal Institute ofTechnology, Stockholm, Sweden, TRITA-MAE, 1993:5

5. Hansen MP et al., (1953) A Method of Evaluating Loose-Blade Mounting as a means ofsuppressing Turbine and Compressor Blade Vibration, Proc. SESA, vol. 10, p.103

6. Lazan BJ, (1968) Damping of Materials and Members in Structural Mechanics, Per-gammon Press


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Copyright Altair Engineering Inc 2011 18

7. Marquina FJ, Coro A, Gutierrez A, Alonso R, Ewins DJ and Girini, G., (2008) Frictiondamping modeling in high stress contact areas using microslip friction model, Proceedingsof ASME Turbo Expo 2008, GT200850359.

8. Mindlin RD and Deresiewicz H, (1953) Elastic Spheres in Contact under varying Oblique

Forces, J Appld Mech, v. 20, p. 3279. Nashif AD, Jones DIG and Henderson JP, (1985) Vibration Damping, John Wiley & Sons10. Olofsson U, (1995) Cyclic Micro-Slip under Unlubricated Conditions, Tribology

International, vol. 28, p. 20711. Olofsson U and Hagman L, (1997) A model for microslip between flat surfaces based on

deformation of ellipsoidal elastic bodies, Tribology International, 30, 8, p. 599.12. Rao JS, Gupta, K and Vyas NS, (1986) Blade Damping Measurement in A Spin Rig with

Nozzle Passing Excitation Simulated by Electromagnets, Shock & Vib Bull, 56, Pt 2, p.10913. Rao JS, Narayan R and Ranjith MC, (2008) Lifing of Turbomachinery Blades A Process

Driven Approach, GT2008-50231, ASME Turbo Expo, Berlin, Germany14. Rao JS, Narayan R, Ranjith MC and Rejin R, (2008) Blade Lifing with Material and Friction

Damping, The Future of Gas Turbine Technology, 4th International Conference, Brussels,

Belgium.15. Rao JS and Saldanha A, (2003) Turbomachine Blade Damping, Journal of Sound and

Vibration, v. 262, Issue 3, p. 73116. Rao JS, Usmani MAW and Ramakrishnan CV, (1990) Interface Damping in Blade

Attachment Region, Proc. 3rd International Conference Rotor Dynamics, Lyon, p. 18517. Rieger NF and Beck CM, (1980) Damping Tests on Steam Turbine Blades, EPRI Project

RP-118518. Rowett FE, (1914) Elastic Hysteresis in Steel, Proc. Roy. Soc., vol. 8919. Zavarise G, Wriggers P and Schrefler BA, (1995) On augmented Lagrangian algorithms

for thermo-mechanical contact problems with friction", Int. Journal Numer. Methods Eng.,vol. 38, p. 2929

analytical estimation of microslip damping in bladed disks

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