a procedure to predict influence of acceleration and damping of blades passing through critical...

7/29/2019 A Procedure to Predict Influence of Acceleration and Damping of Blades Passing Through Critical Speeds on Fatigue Life

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A Procedure to Predict Influence of Accelerationand Damping of Blades Passing Through Critical

Speeds on Fatigue Life

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]

S.SureshAltair Engineering

Mercury 2B Block, 5th Floor, Prestige Tech Park, Sarjapur Marathahalli

Outer Ring Road, Bangalore, Karnataka, 560103, Indiasuresh. rao@altaircom

Rejin RatnakarAltair Engineering

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

[email protected]

R. Narayan

Altair EngineeringMercury 2B Block, 5th Floor, Prestige Tech Park, Sarjapur Marathahalli

Outer Ring Road, Bangalore, Karnataka, 560103, [email protected]

www.altairproductdesign.com

copyright Altair Engineering, Inc. 2011


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Copyright Altair Engineering, Inc., 2011 2

Abstract

Turbine Blades suffer fatigue damage when they cross over a critical speed during start up

and shut down conditions. The stress response is usually determined from quasi-steadyanalysis through resonance with an assumed damping. This response above fatigue limit canbe divided into several steps to reach the peak value at critical speed and then fall afterpassing the critical. For a given acceleration of the rotor, one can then determine the numberof cycles at each of these stress levels and assess cumulative damage for one crossing. Inthis paper, the effect of acceleration and damping on the magnitude of peak stress and whereit occurs in the vicinity of critical speed is included in determining the damage suffered by ablade while passing through the critical speed.

1.0 Introduction

Fatigue is essentially due to alternating stresses; Wohler (I) conducted the earliest tests todefine S-N diagrams. The influence of the mean stress component is presented in Goodmandiagrams (2). Bagci (3) proposed a fourth order relation which is shown to give good resultsfor ductile materials. S-N diagrams and mean stress diagrams are combined to define thefatigue failure surface under elastic conditions, see (4). The fatigue failure surface definedthus is applicable when the stress field is entirely in the elastic region.

Ludwik (5) and Hollomon (6) defined the stress-strain relation in the plastic region, = Kn

where K is the strength coefficient and n is strength exponent For globally elastic and locally

plastic structures, Neuber (7) hypothesis is used for relating the nominal and local cyclic

stresses and strains, e.g., (8, 9). For fatigue loading, using fatigue stress concentration factorKf local stress and strain ranges are related to nominal stress range

Socie et al (10) suggest that Kf can be taken to be the theoretical stress concentration factor.Based on Basquin (11) for endurance and Manson (12) and Coffin (13) works, we have a liferelation for crack initiation given by

Palmgren (14) and Miner (15) defined a linear cumulative damage law. The nonlinear theoriesproposed include Marco and Starkey (16), Corten and Dolan (17), Haibach (18), Marin (19),Henry (19), Gatts (20) and Manson (21). Rao et al (22) made a comparison of lives fromdifferent nonlinear approaches and pointed out the relative merits of the different approaches.

Turbine blades suffer fatigue damage when they cross over a critical speed during start upand shut down conditions. The stress response is usually determined from quasi-steadyanalysis, i.e. determining response at different operating speeds through resonance with an


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assumed damping. This response above fatigue limit can be divided into several steps toreach the peak value at critical speed and then fall after passing the critical. For a givenacceleration of the rotor, one can then determine the number of cycles at each of these stresslevels and assess cumulative damage for one crossing. This process is adopted by Rao,Narayan and Ranjith (23) to determine the damage suffered by a blade during a cross-over ofthe critical speed.

Damping has been identified long ago as a key parameter in blade design. Rowett (24)conducted tests on elastic hysteresis in steel in 1914. Effects of friction and loose mountingwere studied by Hansen et al. (25). Sinha and Griffin (26) studied analytically the effects ofstatic friction on the forced response of frictionally damped turbine blades. Rao et al (27)developed a test rig to determine a nonlinear damping model that depends on strain amplitudeat a reference point in a given mode at a given speed. An analytical process to obtain such adamping model was developed by Rao et al (28).

Using this damping model, damage suffered by a blade during the crossover of a criticalspeed is determined This was a breakthrough in a true simulation of Iifing without dependingon assumed stimulus values and damping values. In the works above, the effect ofacceleration and damping on the peak stress and speed where it occurs in the vicinity ofcritical speed are ignored.

The influence of nonlinear damping on the life with angular acceleration is first demonstratedby Rao and vyas (29) using energy methods for defined blade geometries; the influence ofstress raisers in the blade root area was not included. This necessitates adoption of finiteelement methods from suitable commercial codes to achieve the required results.

1.1 Overview of the Process

A simplified outline of what is involved in the process is first given.

The blade or bladed-disk is first prepared as a CAD model and a FE model is prepared foranalysis. The steady loads usually can be centrifugal loads and mean gas loads. The steadygas loads can be negligible, particularly in low pressure stages and may be neglected.

The steady state stress field is then obtained using where possible cyclic symmetry conditionsand centrifugal loading. In gas turbines, thermo mechanical analysis may be conducted todetermine any thermal stresses and include them in the steady stress field where required.

We then need the alternating stress field to be able to determine fatigue life. Usually there willbe always some unsteady force on the blades due to load changes, speed changes etc atspeeds not necessarily in the vicinity of resonance. This alternating stress field is very smalland has very little influence on fatigue. The significant effect on life arises from alternatingstress field when the blade crosses a resonance.

Therefore, a modal analysis is made to determine the natural frequencies and plot a Campbelldiagram to identify the critical speeds where resonance occurs. At these speeds, thealternating stress field is determined and damage is calculated.


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The first step is to find the resonant stress at each of the critical speeds. We need twoadditional inputs for this, the magnitude of alternating force field and damping to perform alinear analysis. The alternating force field is determined from a CFD analysis.

Damping poses considerable problems in this analysis. Usually it is simplified as equivalentviscous damping without recognizing the influence of strain in the blade (which makes itnonlinear and this is one of the aspects of this paper).

The usual process is then to make a forced vibration analysis using the force field withexcitation at the critical speed under question and an assumed damping value. This analysisalso poses some difficulty since the exact natural frequency to a high accuracy of decimalplaces may be required at low values of damping and one may not pick up the correct value atresonance.

An alternate route to deteInline the resonant stress is to treat the alternating load as steadyload and obtain equivalent steady state stress field and then use quality factor 1/2 to get thecorrect peak stress. This can also pose problems since the pressure field may change phasefrom node to node in the vane. Therefore a forced vibration analysis may be resorted with analmost zero excitation frequency, say 0.0001 rad/s to obtain the equivalent steady stress field

and then use the quality factor.

The procedure adopted here to take into effect of acceleration and damping (for present onlymaterial damping is included) on the damage suffered by a blade during the crossover of acritical speed is as follows:

1. Treating the alternating loads on the blade as steady, the equivalent static response isfirst determined, see (30).

2. Lazan's law (31) is used to determine the material damping characteristic as a functionof strain amplitude at a reference point of the blade at the given operating speed andmode of vibration

3. A nonlinear approach, see (29) is adopted to determine the correct peak stress atresonance using an iterative approach to match the damping value and strain at thereference point.

4. While the blade accelerates through the critical speed, the peak occurs slightly to the

right of classical resonance and also the peak value of response may be lower thanthe classical resonant stress. While performing transient analysis with the influence ofacceleration included, it is initially assumed that the damping obtained in step 3 abovewithout including acceleration effect is valid

5. Then a transient analysis of the bladed-disk with the damping thus determined in step3 is obtained for a given acceleration through the critical speed.


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6. This response is filtered to capture the model component of interest and determine thedynamic stress as a function of instantaneous speed.

7. In step 6 with peak response having decreased with the given acceleration, thedamping is also changed An iterative process is adopted until the damping value andthe response have converged.

8. The resulting frequency response is then used to assess damage suffered by theblade while passing through the critical speed.

The Blade (now named Altair TurboManager) code reported in (23 and 28) is adopted andgeneralized to develop a simulation tool to include the acceleration effects as described aboveand determine life at design stage.

Nomenclature


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2.0 Lifing of Turbomachine Components

Turbomachinery blades pass through several resonances during start up and shut downoperations that cause cumulative damage leading to failures. In some cases, the bladesoperate very close to resonance at operating speed. It is important to estimate life of theseblades at design stage so as to reduce number of prototype tests. The lifing calculationsinvolve the following:

1. Mean stress field determination2. Natural frequencies, Mode shapes Campbell diagram3. Nonlinear modal damping4. Alternating pressure field from stage flow interference5. Peak stress and speed at which it occurs for a given acceleration of the rotor through

critical speed

6. Damage fraction through critical speed crossing and7. Cumulative damage during start-up and shut-down.

The Blade code is extended to consider the acceleration and damping effects as discussedbelow.

2.1 Blade ModelFor simplicity and saving computation time, only the blade of the bladed-disk model (32) isadopted for the analysis as shown in Fig. 1. There are 60 pre-twisted 290 mm long blades onthe disk. The blade root bottom radius is 248 mm. The operating speed is 8500 rpm. Thereare 70 nozzles.

Using mapped meshing options a solid element mesh with 8 nodes is generated, by capturingall the critical regions with finer mesh. Mesh around the singularities at root fillet regions wherepeak stresses occur are captured with 2 to 3 layers of elements with element size as low as0.235 rom. Total number of elements in the model are 31090 with 24649 nodes.


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Figure 1: Blade Model Showing the FE Mesh

2.2 Mean Stress FieldBlades are subjected to steady stress fields due to gas loads, thermal loads, and centrifugalloads under normal conditions of operation. The gas loads are determined from CFD analysis

of the gas path, which have a steady part and an unsteady part at nozzle passing frequencies.Because of the airflow in the compressor flow path or the hot gas path in turbine blades, theyare subjected to thermal loading during the transient period of start-up and shut-down, so theywill form a mean load at a given steady operation or an overall cyclic load for each start-upand shut-down operation. The blades are also subjected to mean loads due to centrifugalloading which could be substantial in low pressure compressor and turbine blades that willpush the structure into globally elastic and locally plastic conditions. Cyclic symmetry can beutilized in assessing these stress fields. The steady stress field determination is wellestablished and can be directly imported by the user at the start of life estimation in to Blade(23, 28).

Of particular interest is the result at the stress raiser location. Usually, the centrifugal loads

lead to local plastic conditions in the root; an elasto-plastic analysis can give a near truepicture in the stress raiser location. However, this is not necessary since true stress and strainconditions are assessed by Neuber hypothesis as given in equations (1) and (2) based on thesurrounding elastic field.

Alternatively, a user can perform the analysis directly using codes of his choice.


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2.3 Natural Frequencies and Mode ShapesThe natural frequencies and mode shapes are determined by using any well established code.Fig. 2 shows the Campbell diagram upto 1000 RPM speed. The intersection of II bendingmode (261.5 Hz) with I NPF excitation gives one critical speed 291 RPM. The transient effectstudy on life is illustrated for crossing this critical speed.

Figure 2: Campbell Diagram

2.4 Nonlinear Material DampingDamping is the key element in assessing life of the blade. Usually a simple viscous dampingmodel is used with equivalent damping determined from a test. Such a model is inadequatesince material damping is highly dependent of the state of stress in the blade. Nonlineardamping model was quantified through experiments by Rao et al (27); the equivalent viscousdamping is expressed as a function of strain amplitude at a reference point in a given mode ofvibration at a given speed of rotation. An analytical procedure using Lazan's hysteresis law

(31) to determine such a nonlinear model is briefly explained below.

The total damping energy D0 (Nm) is given by

where vthe volume. The Loss factornis


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where Wo is the total strain energy (Nm). Equivalent Viscous Damping is then given by

where C is the equivalent viscous damping (N-s/m), n is the natural frequency (rad/s) and Kis the modal stiffness (N/m).

For increased strain amplitudes, the orthonormal reference strain amplitudes, stress andstrain energy are multiplied by a factor F to obtain the equivalent viscous damping C e at

various strain amplitudes as given in (6).

For the critical speed under consideration, 291 rpm, the material damping obtained is shownin Fig. 3.


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Figure 3: 3Material Damping in Mode II at 291 RPM

It can be seen that even under low speed of operation at 291 rpm, the damping value morethan doubled from 0.00027 to 0.00057 with strain increased from 1 x 10-5 to 6 X 10-5 in thesame mode.

2.5 Excitation ForceAtransient CFD analysis for the flow path interference in the stage can be carried out in a

suitable code and the dynamic loads (or nodal pressures) can be calculated. The pressurefield used in this analysis is expressed by

The pressures are applied on both pressure and suction surfaces in four areas along the spanas shown in Fig. 4 and Table 1as given in equation (7).


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Table 1: Alternating Pressure Magnitudes Po

2.6 Resonant Stress under Quasi-Steady AnalysisTreating the alternating load as static, the equivalent steady stress distribution is firstobtained. Since the centrifugal loads should be included while determining this stress

distribution, it is convenient to perform a forced vibration analysis without damping at anexcitation frequency close to zero value, say 0.01 rad/s. The resonant stress is then obtained

by multiplying the steady stress field produced from treating the alternating loads as steady

field with quality factor = 1/2. An iterative approach is used to determine the correct valueof; and the resonant stress.

2.7 Effect of Acceleration


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Five different acceleration values are chosen, as given in Table 2. Peak stress values and thefrequency where it occurs are also given in this Table.

Figure 5: Quasi-Steady Response at Critical Speed

Table 2: Effect of Acceleration on Peak Stress and Critical Speed

It can be seen that the maximum stress decreases with high accelerations taking the bladethrough critical speed quickly without allowing it to build to conventional resonant conditionsand also that the peak stress occurs at a speed slightly beyond the conventional critical speedon the Campbell diagram.


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2.8 Effect of Acceleration and DampingNext the influence of damping for four values of = 0.0001, 0.005, 0.01 and 0.04 isconsidered and the transient response is obtained.

The peak stress value obtained as a function of angular acceleration for no damping and fourother dampings is given in Fig. 6. The peak stress value decreases rapidly for no dampingand very low damping of 0.0001. With damping the peak stress value decreases for a givenacceleration, however the acceleration has little influence when damping is high in the system.

Fig. 7 gives the critical speed (the speed at which peak response occurs) as a function ofacceleration for no damping and four other damping values. With acceleration the criticalspeed increases nearly 2\1% in the range of acceleration considered from 500 to 2000 rad/s2for all damping values. Also for a given acceleration, the critical speed is high with no or verylittle damping.

Figure 6: Stress vs. Acceleration


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Figure 7: Critical Speed vs. Acceleration

Table 3 gives the summarized results. The peak stress from no damping decreased gradually

from 168.65 MPa to 166.61, 99.67,67.71 and 20.76 MPa as damping is increased to 0.0001,0.005, 0.01 and 0.04 respectively for 500 rad/sec2 acceleration. The critical speed, speed atwhich peak stress occurred also shifted gradually from 26311 Hz to 268.16, 266.94, 26573and 262.09 Hz respectively. Similar trends are observed for higher acceleration values.


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Table 3: Influence of Damping and Acceleration

We note from Table 3 that for a given acceleration, say 500 rad/s 2 the maximum stress hasnot fallen by half from 166.61 to 83.305 MPa, when damping is doubled from 0.0001 to 0.5.Even though damping has doubled, the transient conditions with blade accelerating at 500rad/s2 did not allow a full reduction of stress by half as we would have obtained from quasi-steady analysis. This influence is predominant at higher angular accelerations. Therefore, thestress reduction is more influenced by acceleration rather than damping.

We also note that the peak response occurs at the farthest speed beyond the classical criticalspeed with low damping values and approaches critical speed with very high damping; alsothe speed at which peak response occurs increases with acceleration.

2.9 Estimation of Peak Transient Response and DampingThe determination of resonant response at critical speed under quasi-steady conditions ofoperation is presented in section 2.6. With acceleration of the blade, the correct value ofdamping and peak stress are determined as follows.

From quasi-steady analysis, the damping ratio at resonance is first determined; in this case itis 0.0011. With this damping, the strain amplitude under 1000 rad/s2 acceleration is obtainedas 8.1 x 10-5. The damping ratio correspondingly falls to 0.000625 as given in Fig. 3. For thisdamping, the peak value in transient response with the same acceleration 1000 rad/s 2 isdetermined. Since damping has decreased, the peak value goes up and for this new peak, thedamping value changes etc. Iteration is done until there is no change in the peak value anddamping value for this acceleration. Table 4 gives the iterations. It can be seen thatconvergency is obtained quickly.


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Table 4: Iterations for Correct Damping in Transient Conditions

Therefore the correct damping for 1000 rad/s2 is 0.000634. With this damping, the peak Stress

at the stress raiser point is obtained and given by 155.2 MPa.

Compared with quasi-steady response in Fig. 5 the effect of acceleration 1000 rad/s 2 is 10reduce damping from 0.0011 to 0.000634. Under quasi-steady conditions such a reduction ofdamping would have increased the stress; instead the 1000 rad/s2 acceleration did not allowthe stress build with this reduction in damping. Instead this acceleration has predominantlyinfluenced the peak stress and decreased it considerably from 275 MPa to 155.2 MPa.

The stress response a! stress raiser point for this condition is obtained as shown in Fig. 8.

Figure 8: Transient Response of Stress at Stress Raiser


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3.0 Cumulative Damage Through Resonance

The Blade code developed earlier is adopted here with modifications to account for theacceleration and considering the time domain response instead of frequency domain forquasi-steady response results.

The mean stress and alternating stresses are determined as described in section 2.

3.1 Material DefinitionThe fatigue material properties required for the analysis are defined as shown in Figs 9 and10.

3.2 Damage through ResonanceSince the response is in time domain as shown in Fig. 8, it is easier to count cycle by cycle

rather than frequency domain approach used earlier. Each cycle is counted with its stresslevel and fraction of damage for this cycle is determined. Counting all the cycles above thefatigue limit, the cumulative damage is obtained by simply adding these individual damagevalues.

The M.arco-S1arkey nonlinear damage rule (16) can be applied for decreasing stress levels

10 obtain nonlinear damage fraction.

Figure 9: Fatigue material definition


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Figure 10: Fatigue modification definition

Fig 11 gives the damage for four different values of damping ratio, 0.0, 0.0001, 0.005 and0.001 respectively. The damage decreases with increase in angular acceleration anddamping.

Both S-Nand E-Napproaches are applicable, here for illustration S-Nmethod is used.

Figure 11: Damage vs. Acceleration

3.3 Damage under Transient Stress ResponseOnce the damping under transient conditions through resonance is identified, e.g., =0.000634, with the response given in Fig. 8, cumulative damage in crossing this critical speedcan be estimated following section 3.2, which is 3.31x10 -7.


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4.0 Concluding RemarksA code is developed to determine the life of turbomachine blades using the HyperWorksplatform to help a designer speed up the design process. The special features included in thisversion of the code are:

1. Determining transient response under accelerating conditions of the blade.2. Determining correct damping ratio for a blade accelerating through a critical speed3. Accurate determination of resonant stress from the nonlinear damping model for a

given acceleration4. Determination of cumulative damage during stress raising and dropping through a

resonance by linear as well as nonlinear theories5. Adopt SN or EN method as appropriate in high or low cycle fatigue calculations6. Determine life in terms of start-up and shut-down operations or operation at a

continuous resonant conditions

A code like this can help the designer in complete simulation and reduction of number ofprototype tests and thus considerably decrease the design cycle time. In particular, it will helpthe designer in choosing possible acceleration of the rotor to achieve the required life targets.

AcknowledgementsThe authors acknowledge encouraging support from Altair India in this effort.

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a procedure to predict influence of acceleration and damping of blades passing through critical...

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