Turbo Machinery Dynamics Part 8 pdf

A common theory due to Palmgren and Miner assertsthat damage fraction at any stress level is linearly proportional to the ratio of the number ofcycles of operation to the total number of

the disk had corrosion pits with nickel and cadmium plated on the pitted surface Onlyone other P & W JT8D engine experienced a failure of the seventh-stage disk during taxi

to take off in 1985, but the cause was not established because disk fragments were notrecovered

Problem 7.1 A gas generator turbine disk has a design speed of 10,970 rpm Thegeometry and material properties of the disk, similar to that shown in Fig 7.27, aredescribed in Table 7.4 Material density is 0.29 lb/in3 Stress at the rim due to the

TABLE 7.4 Disk Data

Element Inner radius, r i Outer radius, r o Axial length, h E n

no (in) (in) (in) (106lb/in2) –

attached blades and retainers is calculated to be 24,210 lb/in2Determine the radial andtangential stress distribution.

Solution The method and equations provided in Secs 7.7 and 7.9 will be used tocalculate the stresses The calculation sequence is carried out at design speed and at

no speed For the first iteration it will be assumed that the stress at the bore is 40,000lb/in2and the r2∆ value is 90,000 lb/in2Using Eqs (7.25) and (7.26) calculate the Σ

and r2∆ values at the inner and outer radii of the first, or inner, ring element Subtractthe inner radius value from the one at the outer radius to determine the difference val-

ues d Σ and dr2∆ Then at the outer radius Σ = 40,000 − 1559 = 38,441 and r2∆ =90,000 + 3234 = 93,234, from which ∆ = 93,234/2.252= 18,417 Thus, at the outerradius σt+σr= 38,441 and σt−σr= 18,417, from which σt= 28,429 and σr= 10,012

To account for the difference in axial hub length between two adjacent rings, thecondition of equality of radial growth at the interface will be applied Then

(σti−νσri)n= (σto−νσro)n−1or

(σti)n= (σto)n−1−ν{(σro)n−1− (σri)n}Thus, (σti)3= 26,252 lb/in2 Together with Eqs (7.27), (7.28), and (7.29) the proce-dure is repeated for the other elements to obtain stresses at the outer rim for the designspeed and at zero speed The results are shown in Tables 7.5 and 7.6

The value of σroobtained at the rim will not agree with the known rim radial stressbased on the rim load Denoting the true stress by σrimand the stress in the first and thesecond set of calculations by σ′rimand σ′′rim, a factor K can be obtained from the expres-

sion σrim=σ′rim+ Kσ′′rim The value of the factor is

K= {24210 − (−13959)}/19464 = 1.961Actual stresses in the disk are found by multiplying all values in the second set by

the factor K and adding them to the results of the first calculation set (Table 7.7) The

disk stress curves as a function of the radius will be discontinuous at the element faces Average values of the radial and tangential stresses should be obtained at the

inter-TABLE 7.5 Disk Sum and Difference Calculations at Design Speed (10,970 rpm)

interfaces to obtain a smooth curve through the data points The results are shown inTable 7.8, and the data are graphically shown in Fig 7.44.

The average tangential stress for the whole disk is determined by multiplying theaverage tangential stress over each ring element by the area of that element, adding the

values for all the M elements and dividing the summation by the sum of the cross

sec-tional areas of all the elements

(7.33)

The average tangential stress value thus obtained is of considerable interest It may

be used for comparison with experimental data to establish the speed at which the diskmay be expected to fail from burst when the tangential stress reaches the limit Alarge number of disks have been tested in a spin pit with the express purpose of deter-mining the speed at which failure occurs from burst, thus establishing the overspeed

σave= ∑=∑σ +σ − −

=

n M

n M

1 1

TABLE 7.7 Disk Stress Distribution after Modification by Factor K

limit of a rotating machine The failure may be either due to tangential forces acrossthe diameter or due to hoop stresses in the circumferential direction as a conse-quence of excessive radial forces As the disk overspeeds beyond the point at whichthe bore begins to yield, stress from the tangential forces increases at a rate less thanthat in the rest of the disk The nonyielded material away from the bore then picks

up more of the centrifugal loading, causing more material in the proximity of theinner radius to yield The process spreads toward the rim, and eventually the entiredisk yields The ultimate strength of the material is reached along the full cross sec-tion, and a further increase in speed will theoretically cause a complete fracture on

a diametral plane if the material is fully ductile and is not sensitive to notch effects

In practice, burst failures are encountered when the average tangential stress isbetween 75 and 100 percent of the material’s ultimate tensile strength In this exam-ple problem, the average tangential stress is calculated to be 64,857 lb/in.2

TABLE 7.8 Average Disk Stress Distribution at Element Interface

StressElement no Radial Height (in) Radial (lb/in2) Tangential (lb/in2)

Bursts from high radial stress generally cannot be predicted precisely since the tribution does not tend to follow a pattern This causes the radial direction stresses tovary nonlinearly at high speeds Growth in the radial direction causes the centrifugalforces in the disk and at the rim also to increase until failure takes place To avoid thissituation, a more desirable approach is to limit peak values of radial stress to the aver-age tangential stress in the disk.

dis-REFERENCES

Amedick, V., and Simon, H., “Numerical simulation of flow through the rotor of a radial inflow bine,” ASME Paper # 97-GT-90, New York, 1997

tur-Bladie, R., Jonker, J B., and Van den Braembussche, R A., “Finite element calculations and

experi-mental verification of the unsteady potential flow in a centrifugal pump,” International Journal of

Numerical Methods in Fluids 19(12), 1994.

Cairo, R R., and Sargent, K A., “Twin web disk—A step beyond convention,” ASME Paper # GT-505, New York, 1998

98-Childs, D., “Fluid-structure interaction forces at pump impeller-shroud surfaces for rotor dynamic

cal-culations,” Transactions, 111:216–233, ASME, New York, 1989.

Dambach, R., Hodson, H P., and Huntsman, I., “An experimental study of tip clearance flow in aradial inflow turbine,” ASME Paper # 98-GT-467, New York, 1998

Fatsis, A., Pierret, S., and Van den Braembussche, R A., “Three-dimensional unsteady flow and forces

in centrifugal compressors with circumferential distortion of the outlet static pressure,” ASME

Journal of Turbo-Machinery 119:94–102, 1997.

Hiett, G F., and Johnston, I H., “Experiments concerning the aerodynamic performance of inward radial

flow turbines,” Vol 178, Proceedings, Institute of Mechanical Engineers, England, Part 3I(ii), 1964.

Hillewaert, K., and Van den Braembussche, R A., “Numerical solution of impeller-volute interaction

in centrifugal compressors,” ASME Paper # 98-GT-244, New York, 1998

Japikse, D., “The technology of centrifugal compressors: A design approach and new goals forresearch,” VKI Lecture Series no 1987–1, 1987

Justen, F., Ziegler, K U., and Gallus, H E., “Experimental investigation of unsteady flow phenomena

in a centrifugal compressor vaned diffuser of variable geometry,” ASME Paper # 98-GT-368, NewYork, 1998

Kenny, D P., “A comparison of the predicted and measured performance of high pressure ratio trifugal compressor diffusers,” ASME Paper # 72-GT-54, New York, 1972

cen-Kerrebrock, J L., Aircraft Engines and Gas Turbines, MIT Press, Cambridge, MA, 1992.

Miner, S M., Flack, R D., and Allaire, P E., “Two-dimensional flow analysis of a laboratory

cen-trifugal pump,” ASME Journal of Turbo-Machinery 114:333–339, 1992.

Moore J J., and Palazzolo, A B., “Rotor dynamic force prediction of whirling centrifugal impellershroud passages using computational fluid dynamic techniques,” ASME Paper # 99-GT-334, NewYork, 1999

Mowill, J., and Strom, S., “An advanced radial component gas turbine,” ASME Journal of Engineering

for Power 105:947–952, 1983.

Nakazawa, N., Ogita, H., Takahashi, M., Yoshizawa, T., and Mori, Y., “Radial turbine ment for the 100 kW automotive ceramic gas turbine,” ASME Paper # 96-GT-366, New York,1996

develop-Orth, U., Ebbinh, H., Krain, H., Weber, A., and Hoffmann, B., “Improved compressor exit diffuser for

an industrial gas turbine,” ASME Paper # 2001-GT-323, New York, 2001

Rhone, K., and Baumann, K., “Untersuchungen der Stromung am Austritt eines offenenRadialverdichterlaufrades und Vergleich mit der klassichen Jet-Wake Theorie,” VDI Berichte

No 706, 1988

Saravanamuttoo, H I H., Rogers, G F C., and Cohen, H., Gas Turbine Theory, 5th ed., Prentice-Hall,

Harlow, England,1999

Sawyer, T., Gas Turbines, Vols I–III, International Gas Turbine Institute, Atlanta, ASME, 1982.

BIBLIOGRAPHY

Casey, M V., “The effects of Reynolds number on the efficiency of centrifugal compressor stages,”

ASME Journal for Engineering Power 107:541–548, 1985.

Childs, P R N., and Noronha, M B., “The impact of machining techniques on centrifugal sor impeller performance,” ASME Paper # 97-GT-456, New York, 1997

compres-Frischbier, J., Schulze, G., Zielinski, M., and Ziller, G., “Blade vibrations of a high speed compressorblisk-rotor,” ASME Paper # 96-GT-24, New York, 1996

Hall, R M., and Armstrong, E K., “The vibration characteristics of an assembly of interlock shroud

turbine blades,” in A V Srinivasan (ed.), Structural Dynamics Aspects of Bladed Disk Assemblies,

ASME, New York, 1976

MacBain, J C., Horner, J E., Stange, W A., and Ogg, J S., “Vibration analysis of a spinning disk

using image de-rotated holographic interferometry,” Experimental Mechanics, pp 17–22, SESA,

1978

Mikolajczak, A A., Snyder, L E., Arnoldi, R A., and Stargardter, H., “Advances in fan and

com-pressor blade flutter analysis and predictions,” AIAA Journal of Aircraft 12(4):325–332, 1975.

National Transportation Safety Board, “Uncontained engine failure/fire, valujet airlines flight 597,”NTSB Report # AAR-96/03, Aircraft Accident Report, Washington, D.C., July 30, 1996

National Transportation Safety Board, “Aircraft Accident Report—Uncontained Engine Failure,”NTSB Report # NTSB/AAR-98/01, Washington, D.C., 1998

Pfeiffer, R., “Blade vibrations of continuously coupled and packed steam turbine LP stages,” in R E

Kielband, N F Rieger (eds.), Vibrations of Blades and Bladed Disk Assemblies, ASME, New York,

1985

Simon, H., and Bulskamper, A., “On the evaluation of Reynolds number and relative surface ness effects on centrifugal compressor performance based in systematic experimental investigations,”

rough-ASME Journal for Engineering Power 106:489–498, 1984.

Srinivasan, A V., Lionberger, S R., and Brown, K W., “Dynamic analysis of an assembly of

shrouded blades using component modes,” ASME Journal of Mechanical Design 100:520–527,

Wadell, P., “Strain pattern experimentation,” Engineering and Materials Design, 17(3), 1973.

Wiesner, F J., “A new appraisal of Reynolds number effects on centrifugal compressor performance,”

ASME Journal for Engineering Power 101:384–396, 1979.

TURBINE BLADE AND VANE

The calculation of blade resonant frequencies and mode shapes of a bladed disk systemcalls for the definition of boundary conditions at shroud and dovetail interfaces The max-imum operating speed of the engine cannot be established unless vulnerable frequenciesand response amplitudes at resonance are known to a satisfactory degree of confidence.Research efforts have focused on the need to develop a fundamental understanding of top-ics such as Coulomb and viscous damping, coolant passage turbulators, and unsteady vis-cous flow in turbomachines Without adequate knowledge of damping, for example, there

is no alternative but to guess values for the coefficient in determining the response from aparticular excitation Note that damping may arise from material characteristics, friction,aerodynamic flow, and possibly from impact

Shrouds in the form of a protrusion are used in turbine blades to alleviate problems ing from dynamic motion in the blade Long and slender turbine blades in the last stages ofgas turbines can take advantage of the support provided by mating surfaces of shrouds onadjacent blades to reduce the flexural and twisting motion at the tip Even longer blades inlow-pressure steam turbines may be equipped with some form of dampening mechanism atmidspan and tip locations However, the shroud also imposes penalties in the form of amass at the tip of the blade, which requires body loads due to the centrifugal force field to

aris-be carried by the rest of the airfoil It also adds to the manufacturing cost of the blade In ashrouded blade system the protrusion constrains blade motion not only along the contactplane between the shrouds of adjacent blades but also along the normal direction of theplane In-plane tangential relative motion is mostly two-dimensional On the other hand,normal relative motion can cause variations in normal contact load and, in extreme cases,separation of the contact interface

Low-cycle fatigue (LCF) failures of turbine blades are of great significance in aircraft

engines Thermal gradients and mechanical stress from centrifugal loading rapidly increase

as the engine is started and goes to full speed during aircraft takeoff from the ground

CHAPTER 8

269

Copyright © 2005 by The McGraw-Hill Companies, Inc Click here for terms of use

Figure 8.1 shows a typical stress and temperature distribution in a freestanding and shroudedturbine blade The reverse pattern is repeated during landing and engine shutdown When bladerows in a turbine are operating close to resonance conditions, they are prone to failure fromhigh-cycle material fatigue In power generation turbines operating at a near constant speedblade resonance can occur during engine startup and shutdown conditions Excitation is

encountered due to two primary reasons: (i) flow path interference between stator and rotor blade rows at nozzle passing frequencies and (ii) manufacturing and assembly errors at per rev-

olution harmonics Interference between the stationary and moving blades is an aerodynamicphenomenon, and is based on potential interaction, wake interaction, and viscous effects.Mechanical excitation from manufacturing errors and mounting of stationary diaphragms can-not be readily simulated using mathematical formulations

Turbine blades and vanes constitute a considerable portion of the total cost of the ment, given the fact that thousands of airfoils are used in any turbomachine An accuratedetermination of the operating life of turbine and compressor blades plays a central role inthe design of aircraft power plants The rotating parts must be retired prior to failure, but

equip-Blade peakstressMetal temperatureAllowable stress

Percent of blade height

FIGURE 8.1 Stress and temperature distribution in freestanding

must still possess adequate life to be commercially acceptable to airline operators Life mation using stress-based theories is a multifaceted technology, and calls for the calcula-tion of mean steady stresses, dynamic stresses, failure surface, load history, and cumulativedamage The influence of mean stress may be described by a number of linear and nonlin-ear relations, for example, Goodman, Soderberg, Gerber, Marin, and Kececioglu Severalcumulative theories for alternating stresses of varying amplitudes are based on the damageaccumulated during the load cycles A common theory due to Palmgren and Miner assertsthat damage fraction at any stress level is linearly proportional to the ratio of the number ofcycles of operation to the total number of cycles that would produce failure at that stresslevel However, the order of application of different stress levels is not recognized, anddamage is assumed to accumulate at the same rate at a given stress level without the con-sideration of the history Experimental evidence indicates that fatigue damage accumulatesnonlinearly, depending on the alternating stress level Nonlinear theories proposed byMarco and Starkey, Corten and others rely on some exponent of the same ratio A problemsometimes encountered with the application of nonlinear theories is the lack of materialdata for the exponent at different stress levels

esti-The benefits of reduced fuel consumption and increased power arising from increasingthe turbine inlet temperature have been clearly brought out in Chaps 2 and 3 Despite lossesexperienced in cooling of blades and vanes, the gains are still considerable Methods to coolthe blades receive serious research attention Cooling of blades with liquids is difficultbecause of practical problems associated with the delivery and retrieval of the coolant inthe primary cooling system in the forced or free convection modes, or in a closed secondarysystem Difficulties are also encountered due to corrosion and deposits in open systems Inclosed systems it is difficult to obtain sufficient secondary surface cooling area at the base

of the blade Internal air cooling in the forced convection mode is more practical in engines.Turbine blade metal temperatures may achieve a reduction of 200 to 300°C by channeling1.5–2.0 percent of the airflow for cooling for each blade row The blades may be cast withinternal cooling passages in the core or forged and drilled with holes of any required shapeand size using the electrodischarge, electrochemical, or laser drilling process Outer surfacecooling is achieved by pushing cooling air out of holes in the blade walls In this processheat is extracted more uniformly from the surface and at the same time provides a layer ofcooler air isolating the metal from the hot gases of the main stream The concept of tran-spiration cooling requires porous blade walls for the cooler internal air to ooze out from theinternal blade cavity, but successful application will depend on the availability of appro-priate porosity in the skin material and manufacturing methods

Cooling of the rotating airfoils still represents unusual difficulties from engineering andmanufacturing considerations At elevated gas and metal temperatures oxidation and creepimpose limitations on the blade’s capabilities Unlike rotating airfoils, nozzle vanes do notexperience high stress levels In spite of it, cooling of annulus walls and stator vanes requiresspecial attention

A class of materials referred to as superalloys find application at a higher proportion oftheir actual melting point than any other group of commercial metallurgical materials Thealloys have made much of the very high temperature engineering technology possible, andare the leading edge materials of gas turbines in the air transport, power generation, andprocess industries In turn, gas turbines have been the prime driving force for the develop-ment and subsequent existence of superalloys The alloys respond to the need for materialswith creep and fatigue resistance at high temperatures Many of the alloys, perhaps 15 to 20percent, have been developed for utilization in corrosion-resistant applications Despite thehigher cost of superalloys, the economic implications of increased temperature at the tur-bine inlet are overwhelming through increased efficiency and power output by the applica-tion of this group of materials Figure 8.2 shows bladed disk assemblies for a steam turbinerotor, and Fig 8.3 provides examples of new steam turbine buckets

FIGURE 8.2 Steam turbine rotor (Courtesy: General

Electric/Toshiba).

FIGURE 8.3 Steam turbine buckets (Courtesy:

Thermal barrier coats represent perhaps the most promising and exciting development

in superalloy coatings Any mechanism by which the temperature limits can be raised byovercoming hot-section material restraints is of significant interest, and thermal barriercoatings offer this potential Coatings of ceramic or metallic or a combination of the twoare applied on the substrate of a superalloy to preclude or inhibit direct interactionbetween the substrate and a potentially damaging environment This damage can be eithermetal recession due to oxidation/corrosion or a reduction in the mechanical properties ofthe substrate due to the diffusion of harmful species into the alloy at elevated temperature.Coatings used on superalloys do not act as inert barriers Rather, they provide protection

by interaction with the oxygen in the environment to form dense, tightly adherent oxidescales that inhibit the diffusion of sulfur, nitrogen, and other damaging elements Hence,coatings tend to be rich in elements such as aluminum, chromium, and silicon that readilyparticipate in the formation of these protective scales This coat may be described as amultilayer coating system consisting of an insulating ceramic outer layer and a metallicinner layer between the ceramic and the substrate The ceramic layer insulates the metal-lic substrate from higher surface temperatures than it might otherwise be able to tolerate

Power density loading in turbines may be gauged from the production in excess of 70,000 hpfor some large aircraft engines, with the energy conversion process occurring in the limitedvolume of the turbine A single blade alone may generate 300 hp by extracting energy fromthe hot gases

The operating mechanism of a turbine does not differ substantially from that of a pressor While a compressor adds energy to the airflow by increasing the pressure, the turbineconversely absorbs energy from gases and converts it into mechanical power for the shaft.Compressor blades are designed to have an increasing flow cross section between adjacentblades in the downstream direction, so the flow is decelerated by the effects of diffusion toconvert its kinetic energy into pressure A reversed effect occurs in the turbine, where the flowcross section between adjacent blades narrows in the downstream direction, thereby creating

com-a nozzle effect to ccom-ause the flow to com-accelercom-ate com-and perform useful work In com-a constcom-ant sure, or impulse, turbine the expansion occurs in nozzle guide vanes, where gas potentialenergy is converted to kinetic energy (Fig 8.4) Exchange of momentum takes place as thegas impinges on the rotor blades at constant pressure, essentially as in a water wheel In a reac-tion turbine, on the other hand, the gas expands both in the nozzle vanes and through theblades of the rotating wheel Hence, the blades also feature a narrowing cross section ofthe flow passage between adjacent blades for further acceleration of the flow Analogous tothe lifting of an airplane’s wing, an aerodynamic force is imparted to the blades, causing thewheel to turn The reaction turbine offers better efficiency than the impulse turbine, but thelatter produces higher power output, which may permit reduction in the number of stages.Elevated gas temperatures in a turbine introduce material and cooling problems for the air-foils Temperatures beyond the capability of the material are possible only by using sophisti-cated cooling techniques and by applying ceramic coatings An external film of cooler airsurrounding the airfoil proves inadequate, so additional cooling is provided through internalchannels The first stages of nozzle vanes and blades are affected by impurities present in thefuel in addition to the extreme temperatures Hence they require protection from corrosionand thermal effects Nickel-, cobalt-, and iron-based superalloys are used at a higher propor-tion of their actual melting point than any other class of commercially available metallurgi-cal materials Superalloys stand at the material’s leading edge, and are responsible for makingthe unusually high-temperature engineering technology possible for gas turbines

pres-8.3 INDIVIDUAL BLADE VIBRATION

An in-depth study of the dynamic characteristics of a turbine blade is necessary for ing its capability to withstand the assigned loads What may appear at first sight a relativelysimple task of interpreting the modal characteristics of the blade is oftentimes hampered bythe geometric configuration—the root structure is thick, with a complex fir tree form forattachment to the disk; the airfoil trailing edge is thin; the cross-section profiles are varying

evaluat-at different radial heights; and the blade is twisted and leaned Initial blade geometry is trolled by aerodynamic and performance considerations Within the engine, however, it mustmeet fluid flow and structural criteria The chordwise motion of the airfoil is coupled with theflap, or perpendicular to the chord motion, primarily because of the twist Total bending ortorsional deformation is not encountered in the blade Varying degrees of the two movementsexist in each mode of vibration as a direct consequence of the coupling between the degrees

con-of freedom Aerodynamic and mechanical loads also sometimes tend to dramatically changethe blade’s profile The blade’s design may also go through modifications such as replacing

a shrouded configuration with a wide chord or large sweep blade without the shroud

FIGURE 8.4 Turbine stages: impulse (upper), reaction (lower).

A gas turbine’s rotating components are subjected to resonant vibration and flutter nomena Resonant vibrations result from a coincidence in the blade’s natural frequency andexciting force frequency Exciting forces arise from uneven pressure distributions aroundobstructions in the gas flow path For example, a prescribed number of support frame struts,nozzle vanes, combustor burners, and bleed ports will result in a corresponding number ofwakes in the gas stream past those obstructions Rotating stall in a compressor, rubbingcontact between rotor and stator, and meshing of gears are examples of nonintegral orders

phe-of obstructions in the gas stream Stationary components such as nozzle vanes and supportstruts, on the other hand, will experience excitation caused by the rotating blades or by thecompressor rotating stall Dynamic stresses, thus, must be adequately taken into consider-ation in the design of vanes and struts, but the absence of centrifugal force poses less strin-gent requirements than for rotating blades

When eight support struts are present upstream in a flow path, the turbine blades willencounter an equal number of wakes during one revolution of the rotor Consequently, 8E,

or eight times rotor frequency, and its second harmonic will be of significance The tion can be well summarized in the form of a Campbell diagram for identifying potentialresonant situations, as shown in Fig 8.5 The figure graphically depicts the extent of inter-ference between the blade’s natural frequency and the different excitation frequencies for

situa-an engine operating at situa-an essentially fixed speed Note that the blade’s natural frequencies,especially the bending mode frequencies, increase with the operating speed This observa-tion is consistent with the blade experiencing increased stiffness as a direct consequence ofthe higher centrifugal force, as the rotor speed increases The example shows the blade’s1F2T (combined first flexure/second torsion) and 2F modes to be above the operating speedfor 8E stimulus The first flexural mode (1F) also has the adequate margin for 16E excita-tion However, the 1F mode for the 8E and the 1F2T mode for the 16E stimulus are tooclose to the design speed line, which may be considered unacceptable for the blade design.The 1F mode of vibration is next to the first torsional (1T) mode, but a higher frequencylevel also implies increased excitation levels Blade tuning through modification of theblade’s taper characteristics may permit setting of the problem modes to above the rotorspeed The 24E stimulus intersects all the blade modes at a low rotor speed; hence it is not

of consequence

Note that this simplified illustration considers only the eighth multiple for excitationfrequency, and with the engine operating at a constant speed Other excitation sources andfrequencies will be generally present Aircraft engines operate in a wide operating speedrange as throttle movement goes from ground idle to flight idle, cruise and take off condi-tions Accuracy in the calculation sequence is also a problem for the blade’s higher modes

of vibration Fan blades are often subjected to distortion in the airflow from the inlet tion Low engine order excitations are generally the result, with force magnitude reducingfor increased orders To take care of the uncertainties one option is to use a ±10 to 15 percentmargin for the blade’s fundamental and second modes An increased engine operatingspeed range by a similar margin for the two modes may also be used to take care ofuncertainties

sec-Individual blades also need to be tested using modal analysis with impulse, white noiseand sinusoidal excitation applied at the leading edge and at the trailing edge Optical mea-surement systems based on Electronic Speckle Pattern Interferometry may also be employed

to determine natural frequencies, and even more important, mode shapes Very often each testprogram produces modes that may show small differences to totally new shapes, leading toambiguities that cannot be easily resolved Interpretation and classification of modes in thefamiliar flap, edge, torsional and stripe modes of vibration is not always feasible Finite ele-ment analysis may help to sort the problem Elements with one or two midside nodes and anappropriate master degrees of freedom can help in the identification, and may also reveal newmodes among those measured The blade may be assumed fully fixed at the root

As a rule, finite element procedures are reliable, and often used in the design process foroptimization studies In rare cases a combination of analytical and test may be necessitated.Not so negligible differences can sometimes be explained by the method of excitation,accelerometer mass, height location for analysis, or inappropriate boundary conditions.Modal analysis results may be deficient in content from motion perpendicular to a sensitivedirection, as may be the case in movement along an edge Location of an antinode may be

at fault if flap, or out of plane, motion takes place where the edge shows most of the ity Closely spaced modes can combine to appear as a natural mode, but in reality combineelements of the closely spaced modes Instead of a cursory look at measured or calculatedresults, it pays to be attentive to the details In this regard, note that when complete assem-blies of bladed disks are analyzed and tested in an engine or in a rig, the amount of data to

activ-be evaluated is considerably more

Suppression of flutter in an airfoil provides the primary inducement in introducing anelastic support in the form of a shroud Figures 8.2 and 8.3 show examples of midspan andtip shrouds on steam turbine blades manufactured by GE and Toshiba When operating atspeed the protrusions on the row of blades lock to form a continuous ring to couple theblades, resulting in a stiffer assembly Since the protrusions are in contact as the blades gothrough a certain amount of untwist under centrifugal action, the shrouds rub against eachother The extent of the rub and the consequent damping has been examined extensively todetermine suitable boundary conditions, with assumptions ranging from a fully locked to afreely slipping condition Precision in this matter comes into focus when analytically pre-dicted natural frequencies of vibration do not agree with measured data Slipping at shroudcontact points assumes that the degrees of freedom related to the motion occur without anaccompanying force at the interface A ball joint form of condition permits translationalmotion at the interface, but rotation can take place without the associated moments.Variations in the mode shapes obtained from the different sets of boundary conditions play

a substantial role in aeroelastic stability calculations, because the numbers are used directly

in work/cycle relations to predict flutter susceptibility

Z-shaped shrouds on many low-pressure gas turbine blades also perform the function ofsealing while maintaining a preload along the circumference during operation As the shroudcontact edges wear, a change in the dynamic characteristics of the rotating assemblymay be expected Coupled modes in a tight shroud condition degenerate into first flap and

first edgewise cantilever-type modes at lower frequencies when the shroud slackens (Halland Armstrong, 1976) Placing frequencies of integral order vibration in the low engineorders (2E, 3E, and occasionally 4E and 5E) outside the operating range or at low shaftspeeds finds favor among many gas turbine manufacturers Higher engine order resonancereaching in the 30 to 40E may also need scrutiny if excitation is of sufficient strength andblade damping is inadequate.

Wakes, potential pressure disturbances, circumferential flow distortions and shocks inpassage, and secondary flows produce pressure variations that result in time varying forces

on a rotating blade Flow of gas inside a turbine is inherently unsteady, and is far from form on both the upstream and downstream sides of a row Kielb and Chiang (1992) pro-vide a fine discussion on this subject of blade stimuli The complexity is further emphasized

uni-by the number of parameters affecting the aeroelastic stability of a turbine blade: number

of blades, blade chord, blade twist and geometry, aspect ratio, stagger, hub/tip ratio, shroudlocation and angle, incidence angle, blade load, tip speed, pressure distribution, shock posi-tion, inlet Mach number, natural frequency, mode shape, mechanical damping, and blademistuning Srinivasan (1997) explores the extent of influence of the parameters at specificaerodynamic conditions

high-throughout the industry Transient response of blades during startup and shutdown of machines

is another significant contributor Failure due to resonance at lower modes such as the firstbending and torsion may include fracture of an entire blade and its attachment to cause sub-stantial secondary damage, and even greater if the fragments are not contained in the engine

At the higher-second torsion or second- and third-bending modes, mostly the outer portion ofthe blade is separated At still higher modes only a portion of the tip maybe released

A number of cumulative damage theories have been proposed in making estimates ofthe life of a blade subjected to variable stress amplitude (Rao, Pathak, and Chawla, 1999)

A commonly used linear method in predicting fatigue life of a blade is based on work done

by Palmgren (1924) and Miner (1945) Dynamic stress around a critical speed may be splitinto a convenient number of steps on either side of peak stress Operation at a stress level

S i gives a life of N i cycles If the blade is subjected to n icycles, it suffers a damage fraction

of D i = n i /N i Failure is then predicted to occur when Σ(n i /N i) ≥ 1 The assertion is that thedamage fraction at any stress level is linearly proportional to the ratio of number of cycles

of operation to the total number of cycles that would produce failure at that stress level

Since the blade is subjected to a mean stress, the S − N plane is shifted to the location of the

applied mean stress level on the fatigue failure surface A drawback to this theory is that itdoes not recognize the order of application of various stress levels, and damage is assumed

to accumulate at the same rate at a given stress level without consideration of past history.Experimental evidence indicates that the nonlinear accumulation of the fatigue damagedepends on the alternating stress level

Marco and Starkey (1954) proposed that the damage for each level of reversed

sinu-soidal stress amplitude is expressed by D = (n/N) m , where the exponent m depends on the amplitude of the alternating stress A specimen is deemed to have failed when D reaches a value c irrespective of the sequence in which the stresses are applied Failure occurs when

Σ(n/N) is expressed by

(8.1)

where N i are cycles of completely reversed stresses S i to produce failure, with i denoting the order of application of stress levels, D is the damage ratio, r iis the ratio of exponents

m i /m1representing stress levels S i and Sl, and m iis the exponent in the damage equation

associated with the ith stress level Besides the difficulty in evaluating the integral, the mary difficulty is the necessity of generating a lot of experimental data for the exponent m i.Qualitatively, the procedure is illustrated by Fig 8.6 using available data for two stress lev-

pri-els s1and σ2, with σ1 >σ2 First, the specimen is loaded for n/N= 0.5 at stress level σ1(line

O − A), followed by the lower stress level σ2(line B – C), until damage occurs at D= 1 Then

= 0.5 + 0.05384

On the other hand, if the lower stress level σ2is applied first for n/N = 0.5 (OD in

Fig 8.6) and then with the higher stress level σ1 (line EC in Fig 8.6) until damage occurs at D= 1, then

= 0.5 + 0.94608

= 1.44608 (8.3)

n N

n N

n N

=   + 

∑

σ 2 σ 1

n N

n N

n N

N r r N N r r

N

r r i

i i i

1 2 2 2 1 3 3 3

1

2 1 3

LL

0 1

When the higher stress is applied first, it takes only a few more lower stress cyclesbefore damage occurs When the stress application order is reversed, the specimen requiresmany more higher stress level cycles before damage is encountered Experiments supportsuch a situation.

When stress levels are increased gradually, Palmgren-Miner’s rule provides satisfactoryresults When the stress levels are decreasing from a peak value, application of a factor sim-plifies the situation The average value of Σ(n i /N i) in this period may be multiplied by a fac-

tor f= 1.67 for the sum in the full period This implies that the damage is higher when stresslevels are decreasing

Corten and Dolan (1956) rely on the fracture mechanics theory for postulating initiation

of permanent damage Once initiated, fracture propagation is assumed to occur at stress els that are well below the minimum stress required to initiate a crack Damage per nucleus

lev-is given by D = mrN a where m is the total number of fatigue damage nucleii, r is the ficient of damage propagation rate and a function of stress level, and a is the damage prop-

coef-agation exponent, also a function of stress level From an experimental relation between the

stress-dependent ratio R = r2/r1, damage exponent a, and a material property d (given by

R 1/a = [S3/S1]d), failure can be expected when

(8.4)

where N1is the number of cycles to failure at the highest stress amplitude S1, and n iare the

number of cycles imposed at each stress S i For steels exponent d is found to be between 6.2 and 6.9 and mean value of 6.57 A modification of the S − N diagram calls for defining

a line N = N e × (S/S e)−k , where N eis the number of stress cycles endured at the fatigue limit

S e , and k is between 0.8 and 0.9 This yields a stress cycle with slope more than the cal S − N curve but continued into the range of fatigue strength

classi-Marin (1962) assumed that the equivalent number of cycles at a reference level S1

pro-duce the same damage as n i cycles of operation at S i level of stress can be expressed by nie=

n i × (S i /S1)y , where y is an exponent to be determined experimentally The damage ratio sponding to each operation is R i = n m /N1, so the condition of failure is R1+ R2+ … + R i= 1

corre-The exponent y is the same as Corten’s exponent d, so the S − N relation becomes S × N = k.

The failure criterion then becomes

(8.5)

where q = y − x Note that when q = 0, Marin’s theory takes the same form as Miner’s expression, and Corten’s expression for q = d.

Palmgren-Henry (1955) takes into consideration the reduction in the fatigue limit when a

spec-imen suffers damage, so the S − N curve shifts as a consequence This curve is an lateral hyperbola about the stress axis, and a line passing through S e parallel to the N axis as asymptotes Damage D and fatigue limit are related by D = S e − Sed/S e, where

equi-S e is the fatigue limit of the virgin material and Sedis the limit after the damage Byassuming no damage at cyclic stress levels below the fatigue limit, the curve may be

expressed by N r = N − n = k/S − Sed, where N ris the number of remaining cycles to

fail-ure at stress level S and k is a material constant The damage equation then takes the

n N

S S

n N

S S

n N

S S

i i i q

1 1 2 2 2 1 3 3 3

1

n N

n N

S S

n N

S S

n N

S S

d

1 1 2 1 2 1 3 1 3

1

The expression may be extended to a sequence of different alternating stress amplitudes

in the order of application

Gatts (1961) damage theory is based on the dependence of fatigue strength and fatiguelimit on the number of cycles of stress, and that this change is proportional to a damage

function D(S) = (−1/k1)(dS1/dN), with S i as the instantaneous value of strength, n as the number of applied stress cycles, and k1 as a constant of proportionality Damage is

expressed as a function of stress level D(S) = (S − S e) p , with p as a material constant k1and

p are related with strain energy associated with stresses exceeding the fatigue limit It is assumed that fatigue limit can be expressed by instantaneous strength S e = CS i where C is

a material constant By applying the boundary conditions, the S − N curve takes the form

(8.7)

where g = S/Seo; Seois the fatigue limit when n = 0, b = n/N, g e = S e /Seo, and K = kS e For

most steels C takes the value of 0.5 For C= 0 this expression is the same as that by Henry.Manson, Frecke, and Ensign (1967) recognize the role of crack initiation and propagation

during the damage sustained by a component The crack initiation period is denoted by N′,and crack propagation period is defined by the number of cycles for failure after the crack ini-

tiates Hence, N p = PN f p and N ′ = N f − PN f p where N f is the total number of cycles for failure

including crack initiation, P is the propagation coefficient, and p is the propagation exponent.

P = 14 and p = 0.6 from experiments Except for short life, Miner’s rule is adopted for crack

initiation and propagation phases separately Fatigue nucleii of critical size initiate when

(8.8)

Fatigue cracks then propagate to failure when

(8.9)

In both phases n is the number of cycles applied at ith or jth stress level Thus, Manson uses

a double linear damage rule

OF TURBINE BLADES

High-cycle fatigue of rotating turbine components is a serious problem since it has thepotential to cause substantial damage Highly loaded blades experience alternating stressesfrom aerodynamic excitation The blades are subjected to phenomena such as stator wake,blade flutter, rotating stall, and acoustic resonance, but the link between fluid dynamics andstructural mechanics must be established

Turbocharger turbine axial blades operating at variable speeds are exposed to unsteadydynamic forces σ The forces are set up by the engine stroke, charging system, gas entry atinlet, and nozzle vanes A research program initiated by ABB Turbo Systems ofSwitzerland aims at taking a combined computational fluid dynamics and finite elementanalysis approach to the problem (Filsinger, Szwedowicz, and Schafer, 2001) As a firststep the transient flow behavior in the turbine cascade is simulated using time-dependent

n N i Pj j q

m

′ =

=

∑1

temperature and pressure inlet boundary conditions Forced blade response u due to the sating pressure distribution p(t) is then obtained Coupling between the two numerical methods is achieved by a Fourier decomposition (amplitude P k and phase delay b k) of the

pul-time-resolved excitation forces F(t) acting on the rotating bladed disk

The fluid dynamics program is based on a two-dimensional time-accurate multiblockEuler/Navier-Stokes solver The integrated postprocessing offers a close link to mechani-cal integrity codes by determining blade forces, with calculations done in the absoluteframe of reference and using a moving grid for the rotor At the intersection between statorand rotor grids the cells overlap, and an interpolation technique is used Two-dimensionalEuler equations are valid for the flow simulation on a circumferential stream plane of aselected radius with constant radial thickness

Figure 8.7 shows details of the grid per block Unsteady inlet boundary conditions aredetermined by simulating the diesel engine’s behavior with respect to the exhaust pipe sys-tem and the turbocharger’s related performance In this procedure, items pertaining to thevarying ambient conditions, turbocharger specifications, and load acceptance must beaddressed Total pressure and temperature values are unsteady due to the pulsating nature of

the engine’s exhaust flow In the finite element routine disk assemblies containing N

sym-metric blades coupled tangentially through the rotor are analyzed The disk assembly is arotationally periodic structure with identical blades, and hence the cyclic wave theory may

be applied Static and dynamic deformations for the whole disk can then be represented by

a single blade with the application of complex circumferential boundary conditions

In the computation sequence for free vibrations, harmonic vibration of a single coupledblade (without damping) is represented by

(8.10)

where j= (−1)1/2and j = 2p/N is the circumferential periodicity of the disk sector The nodal diameter n assumes values 0, 1, 2, up to N/2 for even N and (N − 1)/2 for odd N [M] and [K] represent complex inertia and nonlinear stiffness matrices with respect to the rotational

speed Ω, and depend on the nodal diameter n Quantities{q} and {d2q/dt2}are complex placement and acceleration vectors of the blade Kinematic cyclic constraints of the form

[M e( jnϕ)]{d q dt2 / 2} [ (+ K ejnϕ,Ω)]{ } { }q = 0

FIGURE 8.7 Computational fluid dynamics grid (Filsinger, Szwedowicz, and

are applied between nodes on the right and left sector sides Rewriting the Euler function

in trigonometric notation, eigen frequencies of the cyclic finite element model can becomputed in the real domain Hence, the cyclic model has to be represented by two iden-tical finite element meshes, with nodal boundaries on the sector sides (Fig 8.8) con-

strained as given in Eq (8.11) For each mode i and nodal diameter n (n = 0 to N/2), two

identical eigen frequencies are computed, which refer to two possible mode shapes of the

bladed disk assembly Static calculations may be readily performed by substituting n= 0

in Eqs (8.10) and (8.11), so the static equation of the assembly rotating at Ω becomes

(8.12)

[ ( )]{ } { } { } { ( )}K Ω q = P o + T + F Ω

FIGURE 8.8 Boundary conditions at inlet: pressure (Upper), temperature (Lower) (Filsinger,