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F A C T O R S A F F E C T I N G V I B R A T I O N O F A X I A L - F L O W C O M P R E S S O R B L A D E S

S. S. MANSON, A. J. MEYER, JR., H. F. CALVERT, and M. P. HANSON, National Advisory Committee for Aeronautics, Cleveland, Ohio.

SUMMARY

Measurements by means of wire-resistance strain gages of blade vibrations in an experimental 10-stage axial-flow compressor during engine operation a re used to present information relating to:

1, The effect of centrifugal force on natural frequency of blades.

2. The common modes of vibration present and orders of excitation producing them.

3, The effect of disturbances in the a i r flow originating in the inlet passage on vibration in each of the 10 stages.

4. The use of loosely mounted blades a s a means of vibration suppression (a supplemen- tary laboratory investigation on a rotating wheel was conducted in conjunction with the engine tests).

5. The importance of aerodynamic damping in limiting vibration.

Test methods and techniques for testing full- scale compressors under engine operation a re described in detail.

INTRODUCTION

The compressor is an important component of severaltypes of engines now under rapid de- velopment for aircraft propulsion. In the jet propulsion engine, for example, a schematic sketch of which is shown in Fig. 1, the function of the compressor is to supply large quantities of air a t relatively high density for combustion and ultimate expansion in the gas turbine and in the exhaust nozzle. While the axial-flow compressor is more efficient and requires less frontal area than its competitor, the centrifugal-type compressor, its adoption for aircraf t use was a t first retarded by numerous vibration problems encountered in early experimental units. Be-cause of the advantageous features of axial-flow compressors, the present trend, however, is towards their use while t l e vibration problems a rq be;ng ir.2ivirluz!!y ivvesti- gated and gradually minimized.

, COMPRESSOR / COMBUSTOR

TURBINE /

Fig. 1. Schematic sketch showing location of compressor in typical

jet propulsion engine.

Inorder to provide a better understanding of vibration of axial-flow compressor blades, the Lewis Flight Propulsion Laboratory of the National Advisory Committee for Aeronautics has been conducting a program of measurement of vibration s tresses and of the factors affecting them. This re-port presents some of the re - sults and is divided into four sections. In the f i rs t section, the test facilities and techniques a r e described; in the second and third sections the factors affecting the vibration excitation and suppression a re respectively discussed; and in the fourth section the important conclusions learned from the investigations a re sum- marized.

APPARATUS AND PROCEDURES

For the full-scale compressor investigations an experimental 10 stage axial flow unit, a cross-sectionof which is shown in Fig. 2, was drilled a s shown in the figure to permit lead wires to be taken from strain-gages mounted on the blades to a s e t of slip rings mounted on the nose of the engine. Threads were tapped in the radial passages to provide a footing for a polymerizing thermal setting plastic used a s a filler to hold the lead wires in place, and the machining operations were followed by liquid honing process which removed sharp edges that =igh"ltherwise h a w damaged the Eeaci- wire insulation,

1

2 E X P E R I M E N T A L S T R E S S A N A L Y S I S

Strain gages were then cemented to several biades of each stage, some gages bemg oriented to be particularly sensitive to torsion (450 to axis of blade), others to bending (gages on op-

+ on i s posite sides of blade). A typical instal1a.i shown in Fig. 3 for the f i r s t stage. The gages were made of Advance wire of 120 ohms r e - sistance, covered an a r e a of 1/8-inch square, and were cemented to the blades by Bakelite

COMPRESSOR ROTOR cement. A layer of very sheer pure silk was

cementedover the s t ra in gages and lead wires Fig. 2. Full-scale experimental principally to provide additional strength to

compressor rotor drilled for strain oppose the high centrifugal forces of rotation. gage lead wires. The 19 ring slip-ring assembly used in the

Fig. 3. Strain gages mounted on blades of f i rs t stage of experimental compressor.

FACTGRS AFFECTING VIBRATION O F AXIAL-FLOW COMPRXSSOR BLADES 3

Fig. 4 , Slip-ring assembly.

tests i s shown in Fig. 4. The rings were made of :none1 1-3/8 inch diameter and 3 f 6 inch wide, while the l /8 inch diameter brushes were made of 60 percent si lver and 40 percent graph- ite. The electrical connections between engine and sl ip rings were made by a multi-prong plug, thus permitting easy removal of ring a s - sembly when not in satisfactory operation, Ab- solute cleanliness of the rings was found es- sential for good results . The arrangement of the rings in the circuit is shown in Fig. 5. There were three groups of s ix rings, each associated with its own battery supply. Each grcup provided s t ra in indications from four bridges a t any one time. By simple re-wiring on the engine when stationary (e .g. changing all connections f roml to 2 o r 3 in Fig. 5) the same group of rings could be made to service eight other bridges. One grounded s l ip ring was common to a l l three groups of s ix active rings. Thus, the 19 rings serviced 12 bridges a t any one time, but a total of 36 bridges could be r e - cordedby successive tes ts . A 12-channel am- plifier and oscillograph was used to record the strain signals.

In the first s e r i e s of t es t s , the compressor was powered by a 2500 horsepower electric- drive motor, the principal value of these tes ts being the opportunity to increase the pressure ratio a t a given speed by throttling the exhaust. A schematic diagram of the t e s t setup is shown ifi Fig . 6. In the second s d r i ~ s G; ~ e s i s , the ComPressor was rebladed and mounted ina full- scale jet engine.

To obtain information on problems associated with root fastenings, a se r ies of tests were conducted on a flat disk machined at the r im to accept simulated blades of various designs. The setup i s shown in Fig, 7. The wheel was powered by a motor through a speed increaser. yiibrationof the blades was induced by a single stationary a i r nozzle, and measured by s t ra in gages mounted at the base of the blades. Slip rings s imilar to those of Fig. 4 were used to transmit s t ra in signals to stationary recording equipment.

FACTORS AFFECTING VIBRATION EXCITATIGN

Critical speed diagram. - In making any vibration analysis of compressor blades, it is very useful to plot a cri t ical speed diagram; a typical one is shown in Fig. 8. Along the horizontal axis is plotted the rotor speed in rpm, and along the vertical axis the vibrational f re - quency of the blade in cycles per second. Also drawnare a se r ies of radial lines called order lines. These l ines define points along which the vibrational frequency i s an integral multi- ple of the rotor speed; for example, along the order line 2 , the vibrational frequency i s every- where twice the rotor speed (although numeri- cally the two differ by a factor of 60 because frequency is customarily stated in cycles per second, while speed i s in revolutions per min- ute).

The question a r i s e s a s to which order lines


Fig. 5. Schematic diagram of bridge circuit installed on ten-stage experimental compressor. Three complete circuits

were used in tests.

DRIVE-MOTOR SETUP

DRIVE MOTOR

SPEED INCREASERS

Fig. 6. Experimental compressor test set-up.

n a r y p a r t s

FACTORS AFFECTING VIBRATION C F AXIAL-FLGW CGMPRESSOR ELADES 5

Fig. 7. T e s t se t -up for investigation of effect of mount looseness for single and double-ball root blades.

should be drawn, and which omitted. In some cases the reasonfor including a given line may be obvious. F o r example, if there a r e four front bearing supports, the blade encounters four in- terruptions of the a i r s t r e a m in each revolution; hence the fourth o r d e r and its harmonics, 3, 12, etc., should be included. I t has been found, experimentally, however, that o r d e r s f o r which there exists no rat ional explanation also induce vibrations. In the compressor tested, f o r example, every o rde r f rom 3 to 17 was found to induce vibrations in blades of one stage o r another, although the vibrational s t r e s s e s in most cases were smal l (more will be said in

another section about allowable s t r e sses ) . I t is probable that the source of the various o r d e r s of excitation i s the harmonic content of strong f i r s t order excitation, although the f i r s t o rde r line itself i s unimportant because i t does not intersect any of the natural frequency lines within the operating speed range of the engine. F o r the sake of completeness, therefore, it is desirable to include a l l order lines that may reasonably be expected to intersect a natural frequency line within the operating speed range.

The s e r i e s of nearly-horizontal l ines in the diagram a r e the natural frequency l ines, and they deiirle the variation with rotational speed


of thenaturalfrequency of each of the possible modes of vibration. Only for the fundamental bending mode is the frequency seriously affected by centrifugal force of rotational speed. The effect is given by the formula

ORDER OF ROTOR SPEED

where

f = natural frequency a t rotative speed N, cycles per second

f o = static natural f requency, cycles per second

p = constant depending on physical dimensions of blade

N = rotor speed, revolutions per second

ROTOR SPEED, R P M

FREQUENCY 860r o MEASURED

ROTOR SPEED, RPM

Fig. 9. Typical comparison between calculated and experimental effect of

centrifugal force on fundamental bending mode of frequency .

The results of the present investigation indicate that the constant p can be determined ana- lytically by the method outlined by Timoshen- ko.(l)* Fig. 9 shows a typical comparison between the calculated and the experimentally determined effect of centrifugal force. Num- erous checks of this type a t the National Ad- visory Committee for Aeronautic s on both laboratory rotating rigs and on fuii-scale compressor blades operating in engines have veri- fied the general validity of equation 1.

The static natural frequency, f,, can best be determined experimentally, i f a blade is available, by fixing the blade in a rigid mount and exciting it in the lowest resonant mode. Any looseness in the mount wil l tend to invalidate the results, however. If no blade is available, the method of ~ o r t ( l ) can be used to obtain a good fir s t approximation of the fundamental mode static natural frequency. Cnce fo , and p have been determined, the natural frequency line in Fig. 8 for f i r s t mode bending can be drawn. The natural frequency lines for each of the other modes is effectively horizontal.

While a blade may have numerous natural frequencies, there is a great difference in the ease with which the various modes can be excited. In the particular compressor tested in this investigation, only the first and second bending modes were at all excited, and only in

*Superiors in parentheses refer to bibliography

FACTORS AFFECTING VIBRATION OF AXIAL-FLOW COMPRESSOR BLADES 7

the f i r s t mode was the magnitude appreciable during normal operation of the engine. (Appre- ciable second mode vibration was artificially induced by partial blocking of the inlet passage.) For a practical cri t ical speed analysis in this case, it is, therefore, adequate to coniider only k e f i rs t bending mode of each of the stages of the compressor. Fig. 10 shows a critical speed diagram for the first bending modes of the ten stages of the compressor tested.

Each time an order line intersects a natural frequency line, there exists a potential critical speed. Whether o r not vibration is excited depends upon the amount of exciting force present of the particular order involved. In Fig. 10, the experimentally observed resonances a r e indi- cated by the various symbols, and i t will be noted that most resonant speeds were in fact, actual resonances. When it is considered that in a given stage a wide spread of natural f r e - quencies exists among the various blades, i t is evident that a t almost any speed some of the blades of the compressor a r e vibrating. As

ORDER OF ROTOR SPEED

STAGE

0 I + 2 0 3 x 4 0 5

6 v 7 0 8

9 0 10

ROTOR SPEED, RPM

Fig. 10. Crit ical speed diagram for fundamental bending mode of typical blades of each stage of experimental

compressor.

much a s 20 percent variation in static natural frequencies. has been observed among the individual blades of a stage. Undoubtedly, some of the variation i s due to different degrees of tightness of the blade in the mount, and the spread of natural frequency may narrow a t the high speeds of operation due t~ tightening by centrifugal force. But a t least some of the spread i s due to manufacturing tolerances, and persists throughout the speed range. Fig. 11 shows a plot of the variation of natural frequency wit!? rotative speed of two blades on the fourth stage of the compressor, the static natural frequencies of which were about 4 percentapart. If the difference in natural frequency were due to mount looseness, the frequencies would tend to converge under the high centrifu, oal forces of high rotative speed. Actually, in the plot of the square of the natural frequency against the square of rotative speed, both blades give essentially parallel lines, which indicates that the basic natural frequencies of the two blades

. a r e different. The possibility of vibration, although not necessarily cr i t ical vibration, a t a l l engine speeds, i s therefore substantiated. - -

Obstructions a s an exciting force: - One of the major sources of excitation in compressors is obstructions in the a i r passages leading to the compressor. Struts f o r supporting the front bearing, f o r example, break up the sniooth

BLADE I 2

Fig. 11. Variation of natural frequency with speed for two blades from fourth

stage.

passage of a i r , and subject the blades to fluctu- ating a i r forces. In the present compressor, there were four such s t ruts , thus providing a source of impulses a t 4 per revolution and its harmonics 8, 12, 16, etc., per revolution.

The inlet guide vanes which direct the flow of a i r a t the proper angle to the f i r s t stage of rotor blades constitute another effective source of a i r pulsation. In the present compressor, there were 56 inlet-guide vanes, and, in general thereare a large number of such vanes, hence the order of excitation due to this source i s


BLOCKS

0 NONE

loTH ORDER

d STAGE

Fig. 12. Effect of inlet disturbances on s t resses throughout compressor,

high. The corresponding order lines intersect the lower natural frequency modes a t speeds below the operating range of the engine. But they may intersect the higher modes near the rated speed where the likelihood of excitation is good and the amount of excitation energy available is high due to high a i r speeds. In the present compressor no higher m d e s were excited; but there is no reason to exclude the possibility of such excitation in other compressors.

While i t is easy to see how excitations originating in the inlet passage can readily excite vibrations in the f i r s t stage, there would be a question a s to how far into the compressor such excitations can pers is t . In order to obtain information on this problem, a s e r i e s of tests were conducted on the compressor in the jet engine in which sheet metal clips were fitted over adjacent pairs of inlet guide vanes, thereby blocking off the a i r flow through one o r more of the inlet passages. These t es t s were also intended to provide infor mation on the possible effects of severe icing conditions o r the effect of the lodging of any foreign matter in the in- l e t section of the compressor. When more than one passage was blocked, these passages were a s equally spaced a s posqible (exactly equal spacing for 3 blocked passages was not possible because of the indivisibility of 56 by 3).

Some of the results a r e shown in Fig. 12. The single amplitude s t r e s s e s in several stages for normal operation a r e compared with the cor- respondingstresses induced by blocking of the

ROTOR SPEED, RPM

Fig. 13, Effect of speed and order of excitation on vibratory s t r e s s of sixth

stage.

inlet, In each case, the order of excitation and the number of blocks reflecting the most pronounced effect of the blocking is shown, Fo r example, in the third stage little o r no s t r e s s wasobserved due to third order a t normal operation, Three blocks produced tne most pronounced effect a t third order by increasing the s t r e s s to + 37,000 psi. In the seventh stage, the most pronounced effect was produced by the four blocks a t fourth order excitation. Un- der normal operation, the s t r e s s was + 9,000 psi while four blocks increased the s t r e s s + 22,000 psi. -

It should be noted that the results of a test of this type depend, to a great extent, on the exact placement of the blocks relative to other disturbing influences such a s the bearing supports. Fig. 12 should, therefore, be construed to indicate that disturbances in the inlet section can affect the vibration in all the stages of the compressor ra ther than to indicate the quanti- tative effect of any particular number of blocks.

Dynamic pressure factor: - As the blade ro- tates, i t is subjected to forces which a r e proportional to the dynamic pressure factor p ~ 3 , where P is the a i r density and V the relative velocity between a i r and blade. Disturbing influences in the a i r s t ream al ter this dynamic pressure factor percentage-wise; hence, the amount of vibration excitation is roughly pro-

nd 0 -

or SS

'P - . - 'U - ;he $7

by Jn- 300 2s S

FACTORS AFFECTING VIBRATION O F AXIAL-FLOW COMPRESSOR BLADES 9

STAG E

PRESSURE RATIO, 1.17

PRESSURE RATIO, 1.54 Fig. 14. Strain gage signals from first four stages of 10-stage axial-flow compressor at low and increased

total pressure ratios.

portional to p,v2. Thus the same source of excitation will tend to produce greater vibration a t the higher speeds of operation than a t the lower speeds. Since the lower order lines intersect the natural frequency lines a t the higher speeds, resonances of the lower orders will, ingeneral, produce the highest stresses. Fig. 13 shows the variation of s t r e s s with speed and order for a blade in the sixth stage in first mode bending, which follows the general trend of increasing s t r e s s with decreasing order of excitation. The s t r e s s a t the fifth order which does not fall in the general trend curve is characteristic of exceptions that occurred in every stage. Excitations having particularly weak or strong origins can overcome the effect of the Speeds at which they occur , snci proJuce eitLier unusually low o r high s t resses . In this case,

a s well a s in most cases, no rational origin for the exceptional orders could be found.

Unusual operating conditions: - A s an example of the effect of unusual operating conditions on vibration of compressor blades, the following observation can be reported without drawing any particular conclusions: During one of the tests in which the compressor was powered by an electric motor, the speed was se t a t about one-half of top speed and in resonance with a blade from stage 2. With the valve in the exhaust line wide open, the pressure ratio of the whole compressor was only 1.17. As shown in the upper half of Fig. 14 only the blade from stage 2 was vibrating, the other three t races representing the noise level of the non-vibrating first, riiird, and iourLl stages. By throttling the exhaust valve, the pressure ratio was raised

10 E X P L R I M E N T A L S T R E S S A N A L Y S I S

to 1.54, at which time the blades f rom all four stages vibrated a s shown in the lower half of Fig. 14. The vibration was rather irregular. Fur ther throttling of the exhaust produced surging, a t which time al l vibrations ceased. When the surge condition was removed, the vibrations resumed.

FACTORS AFFECTING VIBRATIGN SUPPRESSIGN

Counteracting the exciting forces a r e damping forces that, under most conditions, prevent the vibration from building up to dangerous levels. These forces a r e the inherent damping of the material , the damping a t the root, and aerodynamic damping. In discussing damping, refer- ence isusually made to the logarithmic decrement. Basically, this t e rm refers to the ra te of decay of the blade amplitude in f ree vibration. It i s defined a s the logarithm of the ratio of the amplitude of one cycle to the amplitude of the next cycle of a blade in f ree vibration. Hence, the usual method of measuring logarithmic decrement i s from a die-away curve of the blade amplitude in f ree vibration. An al ter- nate method of measurement of this quantity makes use of the frequency response curve of the blade. If the blade i s excited by a force of constant amplitude, and the vibration amplitude is observed, the logarithmic decrement can be determined by the formula

where

b = logarithmic decrement of damping

~f = difference in frequencies a t which vibration amplitude response i s 50 percent of re - sponse a t resonance

fo = frequency a t resonance

Material damping. - The damping due to in- ternal friction in the material depends to a great extent on the material and the s t r e s s level. The materialused in the blades of the compressor investigated was 13 percent chrome iron common in steam turbine blading use. In a comprehensive repor t on damping capacity of

engineering m a t e r i a l s Hatfield: Stanfield, 2nd ~o the rha rn (2 ) point out the importance of chem- ical composition and heat treatment on damping capacity of stainless irons with chromium content in the neighborhood of 13 percent. Table I taken f rom Schabtach and ~ e h r ( 3 ) gives the damping capacity of 13 percent chrome iron a s obtained f rom tuning fork specimens a t different s t r e s s levels.

Incompressor blades which a r e subjected to a steady centrifugal s t r e s s and combined a l ter- nating s t r e s s the exact damping value cannot be readily determined. Values in the neighborhood of 2 percent o r 6 = .02 a r e not unreason- able, however, for appreciable amplitudes of vibration in proximity of failure. This value i s rather high and indicates the reason for the common use of this material for steam turbine blading and fo r its use in the experimental com - pressor of the present investigation. The value b = 0.02 agrees favorably with experimental values obtained f rom die-away curves and f requency response curves on typical blades from the compressor vibrating a t relatively high s t ress levels.

Blade root damping. - I t has been the practice in many designs to provide a very tight fit be-

m, tween the Made and rotor. 1.nis tight fit, to- gether with the tightening effect of centrifugal force, minimized the possibility of friction in the mount that might be beneficial in reducing the amplitude of vibration a t resonance. In order to determine the potential benefit that might be derived f rom loose insertion of the blades, two investigations were conducted. In the f i rs t , a loose mount was provided in one of the blades of the tenth stage of the jet engine compressor, and the vibration of this blade was noted during operation of the engine. It was found that the amplitudes of vibration of this blade were approximately the same a s those of a tight blade in the same stage. However, these amplitudes were, in general, small , a s there seemed to be no tendency to excite severe vibrations. I t was, therefore, not possible to determine the effect of mounting looseness in the case of interest- -the region of high amplitude approaching blade failure. The laboratory investigation on the disk described under "Ap- paratus and Procedures" was, therefore, un- dertaken. An additional objective of this investigation was the comparison of the single and double-ball roots f rom the vibration damping

4

I

1 1

T

n .t e n 1 f Le 1 S

3,s is of s e r e r i - to in li - ) rY {P - In - es - and ling


TABLE I

DAMPING CAPACITY G F 13 PERCENT CHROME IRON

Temp. Nominal bending s t r e s s , 1000 psi F 5 10 15 20 25 30 35 40 -

Damping: log decrement, percent

O R D E 9 OF ROTOR SPEED

LOOSE LOOSE

30

ROTOR SPEEG, RPM

Fig. 15. Crit ical speed diagram for single -ball root blades.

standpoint. The double-ball root is desirable from strength considerations of the disk rim.

Eachof the root designs investigated had two blades mounted loosely and one blade mounted tightly. The degree of looseness was repre- sented by the tip movement of the blade in i t s mount a t stat ic conditions. The two loose single-ball root blades had tip movements of .008 inch and ,023 inch and will hereafter be refer- red to a s loose and very loose, respectively.

Vibration was induced in the blades by means of a single stationary a i r jet which impinged on each blade once per disk revolution. Such a pulse has numerous harmonics of approximately equal amplitude, and is, therefore, cap- able of exciting blade vibration a t numerous speeds. Fig. 15 is a cr i t ical speed diagram of the loose and tight single-ball blades showing the speeds a t which vibrations were excited by the various harmonics. The rapid approach to parallelism of frequency-lines for the loose and tight blades is indicative of the rapidity with which the loosely mounted blades tighten a s a result of centrifugal force of rotation. The

4 5 ' 0 0 0 / ~ ~ ~ ~ ~ ~ ~ ~ ~ STRESS A T 11,300 RPM

RPM

11,300 11,250

a , 008 LOOSE

A ,023 LOOSE

I

0 1 2 3 4 5 6 7 8 EXCITING FORCE, L B

Fig. 16. Comparison of vibratory s t r e s s of tight and loose blades with single ball

roots under varying centrifugal force and exciting force.

diagram was obtained from data taken a t low exciting forces. The variation in the frequency is due to the slight differences in the blade tolerance, damping, and the apparent change in the effective length due to the degree of looseness . Actually, the very loose blade was sup- ported a t the ball r a ther than a t the neck. A check was made of the frequency with the blade tightly clamped a t the neck and a t the ball. The measured frequency difference was about 60 cycles per second which corresponds to the observed discrepancy. The figure also indicates the reason for limiting the speeds in the tes t to about 11,000 rpm, although i t would have been desirable to go to higher values. Intersection


of the frequency lines with the fourth-order a- citation line would have necessitated speeds greater than 15,000 rpm which was beyond the range of the dynamometer.

At any given speed a t which vibration was excited, the exciting force was varied by varying the velocity of the a i r issuing f rom the jet. Fig. 16 shows the s t r e s s amplitude built up a t sev- e r a l speeds a s a function of exciting force of the single-ball blade. The solid line applies to the tightly mounted blade and shows that the -

s t r e s s is directly proportional to the exciting force. The dotted lines apply to the blades of different degree of looseness. Fo r low excitS ing forces, the curves for the loose blades follow the tight blade line a t speeds above 8,000 rpm. Centrifugal force tightened the blade in the mount, and the low exciting forces were not sufficient to cause the blade to rub in the mount and provide added friction. As the exciting force was increased, however, the s t r e s s amplitudes fall off in the case of the loose blades, the lower the speed (centrifugal force) the more rapid the break-away for the individual blades. (The induced damping was also signi- fied by a decrease in the blade frequency with &creased excitb.g f ~ r c e . The decrease was sma l l a s compared to the frequency change due to centrifugal force and varied about 10 cycles per second for the complete range of exciting force). At 11,250 rpm, the loose blade was st i l l effectively tight in i t s mount a t an exciting force of about 7 pounds, the s t r e s s amplitude reach- ing +30,000 psi a s compared to the allowable vibratory s t r e s s (from a Goodman ~ i a ~ r a r n , ( 4 ) ) of + 43,000 psi. Limitations of the a i r nozzle, a n F the danger of blade failure, prevented the determination of the break-away point of the 11,250 rpm line for the loose blade. The very loose blade had more damping a t 10,150 rpm than the loose blade had a t 9,000 rpm; conse- quently, increased looseness had a definite advantage a t speeds below 10,000 rpm. It is apparent, however, that a t the higher speeds the curves for the loose blades approach that of the tight blade, while the allowable s t resses a r e reduced. Hence, any excitation that would cause the tightblade to fa i l a t high speed would likewise cause the loose blades to fail. The benefit to be derived f rom looseness depends, therefore, on the design and intended applica- tion. It should be noted, moreover, that the benefit of looseness is the resul t of rubbing of

l l OOr ORDER OF ROTOR SPEED

ROTOR SPEED, RPM

Fig. 17. Crit ical speed diagram for double-ball root blades.

45,OOOr

ALLOWABLE STRESS AT 12.200 RPM

A?' o TIGHT

RPM

12,200

9,180

11,850

6,525 9,360

U ,004 LOOSE A . 0 15 LOOSE

1

0 1 2 3 4 5 6 7 8

EXCITING FORCE, LB

Fig. 18. Comparison of vibratory s t r e s s of tight and loose blades with double ball roots under varying centrifugal force and

exciting force.

the blade and disk materials a t the base, and might produce a galling problem a t the high local pressures . In such an event, lubricants

, and high

cants


Fig. 13. Aerodynamic damping coefficients

may be useful to prevent galling. The characterist ics of the double-ball blades

were similar to those of the single ball blades. Here again, two loose blades were compared with a tight blade. Fig. 17 is a cri t ical speed diagram of the blades. The characteristic approach of the frequencies to parallelism a t increase speed is again present. The loose blade in this case had a ,004 inch tip movement and the very loose blade a tip movement of .015 inch. Fig. 18 shows the s t r e s s amplitude variation with exciting force. The frequencies of these blades were somewhat different from those of the single-ball type; hence, the speeds involved a r e different. I t will be seen in this case the loose blade derived considerably more benefit from looseness a t 11,850 rpm than the very loose blade a t 9,180 rpm.

Aerodynamic damping. - A blade vibrating in an air s t r e am imparts some of its energy to the a i r , the t ransfer of energy constituting aerodynamic damping. ~hannon(5) has pre- sented a simple method of determining analyti- cally the logarithmic decrement of aerodynamic damping resulting f rom the creation of vortices downstream of the blade. He derives the fo r - mula

where

P a = a i r density

Pm = blade metal density

V = a i r velocity relative to blade, fps

f = frequency of blade, cycles per second

h = amplitude of vibration a t point x along blade of length 1

A = a r ea of blade a t point x, square feet

D = a function of the frequency parameter

X = 2rrfc T T

i

c = chord, feet

If the blade is assumed to vibrate according to the shape

and the a r e a is assumed to taper linearly f r o m a base a r ea A. to a tip a r e a At, so that

then it can easily be shown that

Pa Vc s = k - - pm Aof

where 2 k = At

X D (.I85 + ,815 - ) A0


HASH L E V E L 0 I I I I I 900 9 2 0 940 960 9 8 0 1000

F R E Q U E N C Y , CPS L I 1 1

13,500 14,000 14,500 15,000

ROTOR SPEED, RPM

Fig. 20. Elade vibration response in region of resonance.

Using values of A D given by Shamon, the variation of K with frequency parameter x and a rea ratio A ~ / A , is given in Fig. 19.

As a typical case , consider a blade in the fifth stage for which A = 0.8, = 0.55. Hence from Fig. 19 K = 0.52. Substitution in equation (6) results in a value of 6 = 0.01. Di- r e c t verification of aerodynamic damping values in the engine is not possible, but by measuring the total damping and subtracting from it the sum of material and root damping, the aerodynamic damping may be approximated.

Determination of the total damping in the engine was approximated f rom a nominal f r e - quency response curve obtained by measuring the s t r e s s e s a t resonance and a t speeds slight- ly removed from resonance. The validity of the method depends upon the assumption that the exciting force is constant over the small speed range and that the aerodynamic forces involved can be treated a s those in the usual determination of frequency response. Fig. 20 shows the response curve for a blade in the 5th stage, operating in the engine and excited by blocking two passages of the inlet-guide vanes. The curve has two peaks characteris- t ic of many curves of this type obtained from various blades. The pr imary peak is assumed

STAGE

I 0 A

I

t

t- 0 NORMAL OPERATION (r

m 0 PARTIALLY BLOCKED INLET > 20,000

I 1 I I 1 0 4poO BpOO 12,000 16,000

ROTOR SPEED, WPM

Fig. 21. Comparison of measured and allowable peak s t resses .

to occur a t blade resonance. Applying equation (2) to the nominal frequency response curve composed of the pr imary wave, the total logarithmic damping is 0.08. If the damping decrement due to the mater ia i and blade root is taken to be .03, then the aerodynamic decrement must be .O5. Hence, assuming the validity of the f re - quency response method of measuring total damping, aerodynamic damping accounts for 62.5 %of the total damping. This value is in gen- e ra l accord with the value of 60 percent given by Shannon for a particular British compressor blade, although there is considerable discrepancy in the absolute magnitude of damping. ~hahnon gives a total damping decrement of approximately .O4 compared to .08 experimentally determined f rom the nominal response curve. Furthermore, the Brit ishblade was made of an aluminum alloy which would have higher aero - dynamic damping than the 13 percent chrome iron used in the blades of these tests. The comparison of the theoretical value of b = .O1 and ihe experimental value b = .05 for aerodynamic damping of the 5th stage blade would imply a further source of aerodynamic damping than the creationof vor t ices according to Shannon's equations, the assignment of inadequate damping to sources other than aerodynamic, or the invalidity of the frequency response curve in the engine a s a means of measuring total damp-

1

r - r 7 .

I - -

2.

tn 3 - ne n - nd I ic 6 a ;an n' s IP- the

in nP-


, Investigations a r e now in progress to clarify this question.

CONCLUDING REMARKS

From Fig. 10 it is seen that even considering only one blade f rom each stage, and only the fundamental bending mode of each blade, the speed range of the compressor i s well filled with resonance speeds, Considering the spread in natural frequencies among the blades of each stage (Fig. 11) and that vibration of a blade oc- curs not only a t exact resonance, but over a fairly wide speed range near resonance (Fig. 201, it follows that vibration of one o r more blades in the compressor is likely to occur a t all operating speeds. The vibrations need not be destructive, however. Fig. 21 presents a summary of the data obtained in the investigation. The curves in the upper part of the figure show the allowable vibratory s t resses for the f i r s t and last s tages a s obtained from a Good- mandiagram; the allowable vibratory s t r e s s e s for the remaining stages assume intermediate positions. The data points represent peak s t r e s s values measured anywhere in the engine at various speeds; the c i rc les cor res - ponding to the s t r e s s e s in the unaltered engine, the squares to s t r e s s e s artificially induced by blocking off passages between the inlet guide vanes. It can be seen from this figure that normal vibrations a r e well within allowable limits. Only in the case of the artifically induced vibration did the s t r e s s e s approach (point A, Fig. -21) the danger limit. This point is significant, however, in indicating possible detrimental effects of icing o r other foreign obstructions in the inlet section.

Assuming the validity of the frequency- response method of obtaining total damping the most important factor, limiting the vibration at resonance is aerodynamic damping. Since aerodynamic damping is inversely proportional to the density of the blade material, the use of light materials offers a distinct advantage f rom this standpoint. The choice of a blade material depends, however, on numerous factors other than aerodynamic damping. Further- more, the high damping tends to broaden the

curve, thereby increasing the speed range around the resonant speed at which high amplitudes may occur.

While aerodynamic damping may contribute

an important fraction of the total damping under normal operation, i t s effect may be appreciably diminished under certain conditions as , for example, in the vicinity of stall. To limit the amplitude under such conditions, i t is desirable to achieve either high material damping o r high blade root damping. Loose mounting of the blades offers an opportunity for increasing the root damping. However, careful design is necessary to insure the achievement of benefit from loose mounts. Because of the high centrifugal forces, the blades tend to tighten in the mount. Only when the exciting forces a r e high enough to induce some rubbing in the mount can added friction be realized. If the forces neces- sa ry to induce the rubbing a r e greater than those necessary to excite destructive s t r e s s e s , no benefit is derived from the loose mounts.

A gratifying result of the investigation was the excellent correlation between the theoretical predictions and the experimental resul ts on the effect of centrifugal force on natural f re - quency of blade vibration. The goal of resonant speed prediction in advance of engine operation is thereby brought closer to achievement. The studies of the effects of disturbance in the inlet section on the vibration of the blades in the various stages indicate in par t the reason for the numerous resonant speeds observed. if every sAage is to be affected by all the disturbances that precede it, many obscure sources of excitation can result.

BIBLIOGRAPHY

1. S. Timoshenko, Vibration Problems in Engineering, B. Von Nostrand Co., Inc., 2nd Ed,., 1937, pp. 382-388.

2. H. Hatfield, G. Stanfield, and L. Rother- ham, The Damping Capacity of Engineering Ma- terials, Trans . N. E. Coast Inst. of Engrs. & Shipbuilders, Vol. 58, 1941-42, pp. 290-292.

3. C. Schabtach and R. 0. Fehr , Measure- ment of the Damping of Engineering Materials During Flexural Vibration a t Elevated Temper - atures , Jour. App. Mech., Vol. 11, No. 2, June 1944, pp, A-88.

4. G. C. Noll, and C. Lipson, Allowable Working Stresses , Proc. S.E.S.A., Vol. ILI, No. 2 , 1946, pp. 89-101.

5. J. F. Shannon, Vibration Problems in Gas Turbines, Centrifugal and Axial Flow Com- pressors , R&M No. 2226, 1945, pp. 17-18.

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