Chapter 10 Unique Behavior

7/30/2019 Chapter 10 Unique Behavior

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459

C H A P T E R 10

Unique Behavior 10

The common machinery malfunctions dis-cussed in cha pter 9 occur on a w ide var iety of ma chines. The ty pical frequencies

observed with those common malfunctions generally occur between one quarter

of rotative speed and twice running speed. Many process machines are subjectedto addit ional excita t ions tha t impose significan t dyna mic loads upon the ma chin-

ery at other frequencies. In chapter 10, the excitations produced within two ele-

ment and epicyclic gear boxes will be discussed. Common fluid excitations and

electrical phenomena will also be examined. Finally, the application of rotating

machinery technology to reciprocating compressors will be reviewed. As usual,

each of these topics will be highlighted with numeric examples, and actual

ma chinery case hist ories.

PARALLEL SHAFT - TWO ELEMENT GEAR BOXES

Speed increasing, or speed reducing gear boxes are devices that emit a dis-

tinctive set of excitations. Gear box elements move with definable static positionchanges, and t hey generate specific frequencies tha t m ay be used for mechanical

diagnosis. Due to the vast array of gear box configurations, the current discus-

sion will concentra te on the common t wo element, para llel shaft , single or double

helical gears used within the process industries. A review of the complex excita-

tions genera ted by epicyclic gears is in cluded in follow ing section of th is chapt er.

G ear boxes a re complicat ed ma chines tha t have evolved from slow speed wa ter

wheels to a vast array of industrial machines. In many respects, gear design,

configuration, fabrication, and application is a science unto itself. Due to the

complexity of this subject, the read er is encoura ged to examin e books by au th ors

such as Lester Alban 1, Darle Dudley2, and M.F. Spotts3 that go into specific

details regarding the mechanics of various types of gear boxes. There are also

numerous stan dar ds, handbooks, and design guides ava ilable from t he American

1 Lester E . Alban , Systemat ic Anal ysis of Gear Fai lu res, (Metals Park, Ohio: American Societyfor Metals, 1985).

2 Da rle W. Dud ley, Gear H andbook, (New York: McGr aw -Hill Book C ompan y, 1962).3 M.F. Spotts, Design of M achin e Elements, 6th E dit ion, (Englewood Cliffs , New J ersey: Pren-

tice-Ha ll, In c., 1985).


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460 Chapter-10

G ear Ma nufa cturers Associat ion (AG MA) in Arlington, Virginia . The gear OEMs

a lso produce some excellent t echnica l references on a ll a spects of gearing.

The discussion contained in this text is divided into a review of static ele-

ment shifts, the computa tion of the ma jor gear conta ct forces, an d t he dyna micvibratory characteristics. The position changes or static shifts of gear elements

are dependent upon rotation, the driver element, and the applied forces. It is

often difficult to maintain a clear perspective of the gear force directions under

normal conditions. This issue is clouded by the various potential variations in

gear set arrangements. Hence, it is reasonable to examine the expected types of

forces a nd th eir directions in both th e radia l and the a xial planes.

In order to describe the radial position characteristics of two element gear

sets Figs. 10-1 and 10-2 have been constructed. The sketches in Fig. 10-1

describe the expected radial load directions for a down mesh set. The left hand

diagra m depicts a speed decreasing box wh ere the pinion is the input element,

a nd t he bull gear is the reduced speed output . The right ha nd sket ch in Fig. 10-1

describes a speed increasing box where the bull gear is the input, and the pinion

is the high speed output. The heavy arrows on each sketch describe the general

load direction of the overall forces a cting a t ea ch respective bear ing.The t wo sket ches presented in F ig. 10-2 depict the expected ra dia l forces for

an up mesh gear box. The left ha nd dia gram describes a speed decreasing unit

where th e bull gear is the input element, and the pinion is t he increased speed

output. The right hand sketch in Fig. 10-2 shows a speed decreasing gear box

where the pinion is the input, and the bull gear provides the slow speed output.

These simple diagrams define the general load directions for each gear ele-

Fig. 101 Expected Radial Loading Of Down Mesh Gear Sets - View Towards Input

Fig. 102 Expected Radial Loading Of Up Mesh Gear Sets - View Towards Input

CCW

CW

InputOutput

CCW

CW

Input

Output

CCW

CW

InputOutput

CCW

CW

Input

Output


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Parallel Shaft - Two Element Gear Boxes 461

ment. This is important for the proper location of bearing thermocouples, pres-

sure dams, and the evaluation of radial shaft posit ion shifts as measured by

proximity probe DC gap voltages. Many OEMs now provide analytical calcula-

tions that predict the vertical and horizontal journal centerline posit ion at fullload. These radial positions are unique to each gear box design and should be

compared w ith a ctual sha ft centerline posit ion shifts. Note, this mea surement is

often diffi cult t o execute due to the t ooth enga gement between gears a t t he rest

position. Hence, an accurate zero speed starting point (particularly for the pin-

ion) may be diffi cult t o obta in.

Any gear box evaluation should always include a detailed examination of

the opera ting sha ft posit ions at each of the four ra dial bear ings. When a vaila ble,

the mea sured posit ion should be compared w ith t he ra dial locat ion calculated by

the OE M. An incorrectly posit ioned bearing w ill cause signifi cant distress w ithin

the gear box. Unless the journal locations are checked for proper running posi-

tion, the diagnostician ma y end up cha sing a va riety of abnorma l dynam ic char-

acterist ics when the real problem is easily identified by the radial journal

position da ta . This a lso reinforces th e argu ment for inst a lling X-Y rad ial proxim-

ity probes at all gear box bearings. Many facilit ies tend to install proximity

probes only a t t he input a nd output bearings, an d th ey often ignore the blind or

outboard end bea rings. This pra ctice can result in th e unava ilability of some crit-

ical journa l posit ion a nd vibra tion informa tion.

On a h elica l gear, the gea r tooth cont a ct force is typically r esolved into three

mut ua lly perpendicular forces. The tw o ra dia l forces consist of a ta ngent ial a nd a

separat ion force. The ta ngential force is based upon the t ran smitt ed torque an d

the pitch radius. Calculation of the torque is determined in equation (10-1), fol-

lowed by t he ta ngent ial force in equa tion (10-2):

(10-1)

(10-2)

where:

Torque

= Transmitted Torque Across Gear Teeth (Foot-Pounds)

HP

= Transmitted Power Across Gear Teeth (Horsepower)

RPM

= Gear Element Rotational Speed (Revolutions / Minute)

Force

Tan

= Tangential Force Across Gear Teeth (Pounds)

R

pitch

= Pitch Radius of Gear Element (Inches)

In t hese expressions th e speed and pitch ra dius must be for th e same gear

element. That is, if the bull gear speed is used to compute the torque, then the

bull gear pitch ra dius must be used to determine the ta ngential force. Similarly,

if the pinion speed is used to calculate th e tra nsmitt ed torque, then the pinion

pitch ra dius must be used to compute the correct ta ngential force. Note th at thetra nsmitt ed torque is different for th e pinion and the bull gear, but the t an gen-

tial force for both elements must be the same. As another check, the pitch line

velocity for both gear elements must also be identical.

To r q u e 33 000, H P

2 RPM

---------------------------------5 252, H P

RPM

------------------------------= =

Fo r ceT an12 To r q u e

Rp i t c h ---------------------------------

63 024, H PRp i t c h RPM---------------------------------------= =


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462 Chapter-10

The tangential force is the vertical force acting between the gears. Obvi-

ously, one gear element is subjected to a n upwa rd t an gential force, an d t he ma t-

ing gear element is subjected to a downward tangential force (necessary to be

equa l an d opposite). Ba sed upon th e gea r pressure an gle, a nd th e helix a ngle, thegear separation factor may be computed as in equation (10-3). Multiplying the

previously calculated t an gential force by this non-dimensiona l gear separat ion

fact or provides th e gear s epara tion force a s show n in equ a tion (10-4):

(10-3)

(10-4)

where:

SF

= Separation Factor (Non-Dimensional)

= Pressure Angle Measured Perpendicular to the Gear Tooth (Degrees)

= Helix Angle Measured from the Gear Axis (Degrees)

Force

sep

= Separation Force Between Gears (Pounds)

This separation force acts to the right on one gear, and to the left on the

ma ting gea r element. Again, a force balance must be achieved in the horizonta l

plane, an d the separa tion force must be less tha n th e ta ngential force. For sta n-

da rd gea rs, th e typical pressure angle is either 14.5 , 20 , or 25 . The most

common value encount ered for th e pressure a ngle is 20 . The helix a ngle typi-

cally varies between 15 an d 35 . Although these angles are similar , it is ma nda -

tory for the diagnostician to keep the numbers straight . Finally, the third

segment of the overall gear contact force is the axial component. The magnitude

of this thrust load is obtained from the following expression:

(10-5)

where:

Force

Thr

= Axial (Thrust) Force Between Gears (Pounds)

As a side note, if the helix angle is 0, the helical gear equations simplify

into spur gear equations. That is, the cosine of 0 is equal to 1, and the separa-

tion force is equa l to the t an gential force t imes the t an gent of the pressure angle.

Also, the tangent of 0 is equal to zero, and the thrust load is zero. Obviously,

spur gears ca nnot tra nsmit a n a xial force.

The axia l or thrus t loads on a double helica l (herringbone) gear a re theoret-

ically balanced by the two sides of the gear. If the gear is machined incorrectly,

an axia l force will occur on a double helical gear, an d t his ma y generat e signifi-

cant axial loads. However, on a single helical gear box, the thrust loads are

alw ays present. These axial forces must be accommodat ed by t hrust bearings for

each element of a single helical gea r set . I t is meaningful to understa nd t he nor-

mal versus the counter thrust directions for a single helical gear. This helps in

setting up the thrust monitors properly (i.e., normal versus counter), and it

allows a proper evalua tion of measured t hrust behavior.

SFt a ncos

--------------=

Fo rcesep Fo r ceT an SF Fo rceT ant a ncos

--------------= =

Fo rceT h r

Fo rceT an

( )t a n=


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The two diagrams presented in Fig. 10-3 describe the normal thrust direc-

tions for a single helical gear box equipped with down mesh gears. The drawings

in Fig. 10-4 depict the thrust directions for up mesh gears. In each case, the

gears a re identifi ed as either r ight-hand

or left-hand

. This is a common d esigna -

tion of how the teeth curve away from the mesh line. If the teeth lean or are

inclined to the right or the clockwise direction, the element is referred to as a

r ight-hand

gear. Conversely, if the teeth lean or are inclined to the left, or in a

counterclockwise direction, the element is identified as a left-hand

gear. In any

pair of mating helical gears, one element must be r ight-handed

and the other

gear e lement must a lways be left-handed

.

As mentioned earlier in this section, gear boxes emit many unique excita-

tions. The actual excitations vary from low to high frequencies. For example, a

typical parallel shaft, two element (bull gear and pinion) gear box, will normally

produce the follow ing gr oup of discrete frequencies.

Fig. 103 Normal Thrust Direction For Single Helical Down Mesh Gears

Fig. 104 Normal Thrust Direction For Single Helical Up Mesh Gears

CCWCW

InputOutput

Thrust

Thrust

LeftHand

RightHand

CCWCW

InputOutput

Thrust

Thrust

LeftHand

RightHand

CWCCW

InputOutput

Thrust

Thrust

LeftHand

RightHand

CWCCW

InputOutput

Thrust

Thrust

LeftHand

RightHand


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464 Chapter-10

r

Bull Gear Rotational Speed

F

bul l

r

P inion Rotat iona l Speed

F

pi n

r

G ear Mesh Frequency

F

gm

r

Assembly Pha se Passa ge Frequency

F

app

r

Tooth Repeat Frequ ency F

t r

r

G ear E lement Resonant F requencies

r

Ca sing Resona nt F requencies

The bull gear a nd pinion rotat ional speeds are th e actua l speeds of ea ch ele-

ment. These rota tive speeds ma inta in a fi xed rat io that is completely dependent

on the number of bull gear and pinion teeth. If at all possible, the diagnostician

should obta in an exa ct tooth count on both elements. This must be an exact nu m-

ber ( 0 allowa ble error). The gear m esh frequency is equa l to the speed times the

number of teeth a s show n in equ a tion (10-6):

(10-6)

where:

F

gm

= Gear Mesh Frequency (Cycles / Minute)

F

bull

= Rotational Speed of Bull Gear (Revolutions / Minute)

F

pin

= Rotational Speed of Pinion (Revolutions / Minute)

T

bull

= Number of Teeth on Bull Gear

T

pin

= Number of Teeth on Pinion

The gear mesh frequency must be the sa me for both th e bull gear an d th e

pinion. This commonality also provides a good means to verify the validity of a

presumed gear mesh frequency in an FFT plot. In other situations, if the bull

gear speed is known, the pinion speed may be determined from (10-7) when the

actual number of bull gear and pinion teeth are known. Obviously, the inverse

relationship is also applicable.

(10-7)

The gear m esh frequency provides genera l informat ion concerning t he gear

conta ct a ctivity an d forces. This ty pe of measur ement is usua lly obta ined with a

high frequency casing mounted accelerometer. Typically, the best data is

acquired at the gear box bearing housings, since this is the location where the

meshing forces a re tra nsmitt ed to ground. As a n example of this ty pe of informa -

tion, the data presented in case history 19 should be of interest.

The next two excitations of a

ssembl y ph ase passage fr equency

a nd toothr epeat fr equency

require an understanding of the concept of phase of assembly.

This is clearly expla ined by J ohn Wintert on

4

as follows:

M ath emat icall y, the num ber of un iqu e assembl y ph ases (N

a

) i n a gi ventooth combin ati on is equal to the product of th e pr im e factors common

to the num -

4 J ohn G . Winterton, Component identificat ion of gear-generated spectra , Orb i t

, Vol. 12, No. 2(J un e 1991), pp. 11-14.

Fgm Fb u l l Tb u l l Fp i n Tp i n= =

Fp i n Fb u l l Tb u l l Tp i n--------------=


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ber of teeth in th e gear and th e pin ion. The numbers 15 and 9 have the commonpr ime factor of 3. Therefor e, th r ee assembly pha ses exist. Th e number of assemblyphases determ i nes th e di str ibu ti on of wear betw een th e teeth of the gear a nd pi n-

ionWinterton goes on to define the assembly phase passage frequency as

shown in equation (10-8):

(10-8)

where: Fapp = Assembly Phase Passage Frequency (Cycles / Minute)

Na = Number of Assembly Phases (Non-Dimensional Prime Number)

This is followed by t he determ ina tion of the t ooth repeat frequency. This is

generally the lowest level excitation within the gear box. Typically it falls below

500 CPM, and sometimes it appears as an amplitude modulation. The tooth

repeat frequency ma y be computed in t he following ma nner:

(10-9)

where: Ftr = Tooth Repeat Frequency (Cycles / Minute)

Other investigators ha ve a tendency to consider only a true tooth hunting

combination where the number of assembly phases Nais equal to one. This is

proper a nd norma l for precision h igh s peed g ears. H owever, equa tion (10-9) rep-

resents the correct computation for all cases.

The natural frequencies of the bull gear and pinion are often located above

the normal operating speed range. Thus, examination of the transient Bode plots

will generally reveal a stiff shaft response of both gear elements. However, gearsdo exhibit natural frequencies that generally appear at frequencies above rota-

tional speed. Typically, a bull gear and a pinion will each display a stiff shaft

tra nslat iona l followed by a pivota l mode tha t a re both governed by bearing st iff-

ness. At h igher frequencies, the sha ft st iffness controls the resulta nt na tura l fre-

quencies. In a simplistic model, these are free-freemodes tha t a re dependent onmass and stiffness distribution across each respective gear element. These

higher order modes are independent of bearing stiffness, and they are often

excited durin g fa ilure conditions. Typically, discrete frequencies betw een 60,000

a nd 180,000 CP M (1,000 a nd 3,000 Hz) are d etecta ble. These gea r element reso-

nant frequencies are often frequency modulated by the running speed of the

problem element. For instance, if the bull gear is under distress, a higher level

bull gear na tura l frequency w ill be modulated by bull gear speed.

It should also be mentioned tha t m ost gear boxes display a variety of casingresonant frequencies. The distribution of these frequencies will depend on the

gear casing construction. A fabricated box will be lighter than an older cast box.

In general, the t hinner gear box casings will exhibit h igher nat ura l frequencies

FappFgmNa

------------=

Ft rFgm Na

Tb u l l Tp i n---------------------------------

Fb u l l Na

Tp i n-----------------------------

Fp i n Na

Tb u l l ---------------------------= = =


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466 Chapter-10

than heavier and thicker wall casings. Typically, an industrial helical gear box

may exhibit multiple casing na tura l frequencies, an d they may a ppear a nyw here

between 30,000 to 300,000 CPM (500 to 5,000 Hz). Various attachments to the

gear box may also appear as narrow band structural resonances. Items such asunsupported conduit, thermowells, small bore piping, and proximity probe hold-

ers may be detectable on the gear box. In one case, long unsupported stingers

were used on proximity probes in a large gear box installation. Unfortunately,

the na tura l resona nce of the probe stingers w as 3,580 CP M, which wa s excited

by th e synchronous ma chine speed of 3,600 RP M.

Parallel shaft gear boxes are also built with multiple gear elements. For

exam ple, an intermediat e idler gear ma y be installed between a bull gear a nd a

pinion to obta in a specifi c speed ra tio, or ma inta in a part icular direction of rota-

tion. Some gear boxes cont ain mult iple gear s, such a s th e seven element box dis-

cussed in case history 21 in chapter 9. These additional gear elements provide

addit iona l rotat iona l speed excita t ions. If th e unit contains direct m esh to mesh

conta ct a cross t he box, the gear mesh frequency w ill remain constant . However,

if the gear box contains any variety of stacked gear arrangements, the unit will

emit multiple gear mesh frequencies. These multiple rotational speeds and gear

mesh frequencies will often interact in a variety of signal summations, ampli-

tude modulat ions, and frequency m odulat ions.

Interactions of the multiple frequencies will depend on load, which influ-

ences the journal radial posit ions and the tooth contact between gears. In all

cases, the documentation of vibratory data with the box in good condition will be

benefi cial t owa rds a na lysis of a va riety of potentia l future malfunctions.

Case History 29: Herringbone Gear Box Tooth Failure

The speed increa sing gea r box shown is F ig. 10-5 is situa ted betw een a low

pressure a nd a high pressur e compressor. The norma l operat ing speed for th e LPcompressor and the bull gear is 4,950 RPM, and the pinion output to the HP

compressor runs at 11,585 RPM. A flexible diaphragm coupling is installed

between the LP compressor a nd t he bull gear, and an other diaphra gm coupling

is used between the pinion and the HP compressor. The axial stiffness for both

couplings are approximately equal, and the gear box has successfully operated in

this confi gurat ion for man y years.

The gear confi gura tion consist s of a d ouble helical, or herring bone, arra nge-

ment. This type of gear provides a generally ba lanced axial load between the tw o

sets of gear t eeth. Some axial load is inevitable, an d a thrust bearing is mounted

on the outboard end of the bull gear. Due to the dua l mesh int eract ion, the pinion

must follow the bull gear a xial posit ion, and a separat e pinion t hrust bearing is

not required. If th is unit wa s a single helical gea r, a separa te pinion th rust bea r-

ing would have been incorporated.The tooth failure problem on this unit was init iated during a routine top-

pin g offof the oil reservoir. For w ha tever rea son, excess oil wa s pumped int o thereservoir, and oil backed up the gear box drain line. This reverse flow filled the

gear box with lubricant, and oil began spewing from the gear box atmospheric


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vents. This external oil flow was ignited by a hot steam line, and a fire ensued.

The machinery train was tr ipped and the deluge system activated. These com-

bined actions extinguished the fir e with minima l externa l dama ge to the ma chin-

ery. Unfortuna tely, the tra in wa s resta rted to a fa st slow roll, and a llowed to run

at 1,500 RPM for approximately 90 minutes. Evidently the gear box was st ill

fi lled with oil during this abbreviated test run.

Following evaluation of the vibration, mechanical, and process data, the

train was shutdown for gear box disassembly and inspection. Upon removal of

the top half, it was visibly noted that the gears were in good condition. Following

removal of both gear elements it was clear that all four journal bearings were

da ma ged. The inside sections of the four journa l bearin gs th a t a re exposed to th e

interior of the gear box had melted babbitt , whereas the outside sections of allfour bearings r etained ba bbitt . This indicates tha t t he internal gea r box temper-

at ure wa s in excess of 500F to melt part of the bear ing babbitt .

Shop exam inat ion of the bull gear a nd pinion revealed tha t both elements

w ere coat ed wit h va rnish . This w as indicat ive of burnt or oxidized oil on th e sur-

face of the gears. After the varnish was removed, the gears appeared to be in

good physical condition with minimal surface wea r on t he teeth. A dye penetra nt

inspection did not reveal any cracks or discontinuities, and the shaft journals

w ere considered t o be in good condition.

Since the shop inspections revealed no evidence of physical damage to the

gear set , t he gear elements w ere reinsta lled in t he box. Although axia l spacing

between the bull gear shaft and the LP compressor shaft was maintained, the

bull gear coupling hub wa s mounted 0.25 further on t o the sha ft t ha n previous

installations. The effect of an axially mis-positioned hub on a diaphragm cou-pling would be the genera tion of an a xial preload on the bull gea r. This a xial load

could force one side of th e herringbone gear to carr y t he ma jority of the load.

The tra in wa s successfully restar ted, and m achinery behavior a ppear ed to

Fig. 105 General Arrangement Of Two Element Gear Box With Herringbone Gears

Bull Gear Input at 4,950 RPM

Pinion Output at 11,585 RPM

ThrustCollar

47 PinionTeeth

110 BullGear Teeth


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468 Chapter-10

be normal, and consistent with previous vibration data. After six days, a high

frequency vibration component around 75,600 CPM (1,260 Hz) was noticed on

the gear box. This frequency was approximately 15 times the bull gear speed,

an d it wa s intermitt ently tra nsmitt ed to both compressors. The unit operated inthis ma nner for a pproximat ely one month w hen a leaking thermowell forced the

train down to slow roll speeds for 45 minutes for thermowell replacement. Dur-

ing the subsequent resta rt , tw o compressor surges occurred, and this event m ay

have overloaded the gear teeth.

Two day s a fter the resta rt , sha ft vibrat ion a mplitudes experienced a series

of minor step cha nges in a gradua lly increasing tr end. The bull gear ra dial sha ft

vibration data revealed minor 1X vector changes. However, the largest change

occurred on the bull gear axial probes, where the synchronous 1X motion

increased from 0.48 to 1.29 Mils,p-p. This is certainly abnormal behavior for a

double helical gear with normally ba lanced axia l forces.

Simulta neously, the casing vibra tion am plitudes began to grow in t he vicin-

ity of 75,600 CP M (1,260 Hz) w ith peak levels rea ching 12.0 Gs,o-p

. I t should also

be noted that the previously dormant gear mesh frequency at 544,500 CPM

(9,075 Hz) had blossomed into existence, and it was modulated by bull gear rota-

tional speed. Furthermore, audible noise around the gear box had significantly

increased. FFT analysis of microphone data recorded on the compressor deck

revealed a dominant component at 1,238 Hz (15th harmonic), with sideband

modulat ion at bull gear r ota tiona l speed of 4,950 RP M (82.5 Hz).

Considering the available information, the bull gear distress was self-evi-

dent, and a controlled shutdown was the only reasonable course of action. Fol-

lowing an orderly shutdown, a visual inspection revealed 12 broken teeth on the

coupling side of the bull gear. Additional shop inspection revealed multiple

cracked teeth on the bull gear combined with an erratic and accelerated wear

patt ern on both gear elements.

In retrospect, the gears were probably solution annealed during the periodof high internal gear box temperatures. A micro-hardness survey revealed that

the broken gear teeth ha d a Rockw ell Csurface hard ness of 20 for th e fi rst 2 Mils(0.002 inches) of tooth s urfa ce. The ha rdness th en increas ed w ith depth t o levels

consistent with the original gear tooth heat treating. Normal surface hardness

for these gears should be 37 on the Rockw ell Cscale. This s oftening of th e gearteeth su rfa ces represents t he root cau se of this fa ilure. How ever, the tooth fa ilure

was logically due to a combination of the following events:

1. Init ia l fir e, an d the probable surface annealing of the gear teeth.

2. Potential axial load imposed by a mis-positioned bull gear coupling hub.

3. Potential impact loads suffered during compressor surges.

4. P robable high cycle fat igue of the heavily loaded an d soft gear teeth.

In all likelihood, the primary damage of softening the gear teeth occurred

during the fire. Based upon the metallurgical findings, it is clear that this gear

wa s destined for prema ture fa ilure. The a ctual infl uence of items 2 a nd 3 in t he

above list are difficult to quantify. In all probability, these contributors acceler-


11/76


ated the failure, but the life span of the bull gear teeth was greatly reduced by

the loss of tooth surface hardness.

The symptoms of this failure included minor changes in the shaft radial

vibration, significa nt chan ges in the bull gear axia l vibration, plus substant iallyincreased a ctivity a t t he gear m esh frequency. These condit ions are fully expla in-

able based upon the physical evidence of broken gear teeth. However, the casing

excitation at 75,600 CPM (1,260 Hz), and the dominant sound emitted by the

gear box at the sa me general frequency a re not immediat ely obvious.

In an effort to understa nd t he significa nce of this frequency component, a

simple impact t est wa s performed on the fa iled bull gear r esting in th e journal

bearings (with the pinion removed). The main component encountered during

th is test occurred a t a frequency of 76,800 CP M (1,280 Hz). Clea rly, this is q uite

close to th e frequency identifi ed during th e failure, a nd it could be a resonan t fre-

quency of the bull gear a ssembly.

Since addit ional testing on the bull gear w as not a viable option, a 28 sta-

tion und a mped critical speed model for t he gear element w a s developed. The cal-

culat ed first tw o modes include a st iff shaft t ra nslat iona l response at 8,500 RP M,

followed by a st iff shaft pivotal mode at 10,250 RPM. Both resonances have

greater than 93%of the strain energy in the bearings, with less than 7%of the

stra in energy in the sha ft . Hence, these first tw o modes would be infl uenced by

changes in bear ing st iffness a nd da mping chara cterist ics. The calculat ed higher

order modes a re bending m odes of th e bull gear rotor. These resonan ces a re com-

pletely dependent on shaft stiffness (i.e., bearing stiffness is inconsequential).

These higher order resonances were computed for a planar condition of zero

speed. This is equivalent to the bull gear sit t ing a t rest in t he gear box bearings

with out a ma ting pinion. This simplified a na lysis considers the case of a sta t ion-

ar y element w ithout external forces or rota tional inertia .

Of particular interest in this simplified analysis was the appearance of a

free-freemode a t a frequency of 73,200 CP M (1,220 H z). This frequency is close t oth e 74,280 CP M (1,238 Hz ) detected durin g opera tion, an d th e 76,800 CP M

(1,280 Hz) measured the stationary impact test on the failed gear. Additional

examinat ion of this relat ionship, and furt her refinement of the a na lytical model

would be academically interesting. However, this is not a cost-effective exercise,

an d it is necessary to dra w a logical conclusion ba sed upon th e ava ilable informa -

tion. In this case, it is reasonable to conclude that the frequency in the vicinity of

75,000 CP M (1,250 Hz) is a n a tur a l resonan ce of th e bull gear a ssembly. This fre-

quency appear s during t he sta t ic impact tests, and it is a lso excited during oper-

ation with failed gear teeth. In this condition, the rotating bull gear is

periodically subjected to multiple impacts due to the absence of various gear

tooth. It is postula ted th at these impacts during opera tion excite th is bull gear

resonance.


12/76

470 Chapter-10

EPICYCLIC GEAR BOXES

Other gear boxes are even more complex due to the internal configuration

of the gear elements. One of the most complicated industrial gear boxes is theepicyclic gear train. In these units, a moving axis allows one or more gears to

orbit about the central axis of the train. Simple epicyclic gear boxes contain a

central sun geartha t meshes with severa l plan et gear sthat are evenly spacedaround the sun gear. Both the suna nd plan et gearsare externa lly toothed spurgea rs. The plan et gearsalso mesh with a n interna lly toothed r in g gear. The inpu tan d output shaft s ar e coaxial. These shafts ma y rotat e in the same direction, an d

they may rotate in opposite directions. This is dependent on the actual mechani-

cal confi gurat ion of the individual gear box.

Epicyclic gear boxes derive their na me from t he fact th at the planet gea rs

produce epicycloidal curves during rotation. In actuality, there are three general

types of epicyclic gear boxes. Perhaps the most common type is the planetary

arr an gement tha t consists of a st at iona ry ring gear combined with a rota ting sun

gear, and moving planet carrier. The star configuration of a epicyclic gear boxconsists of a stat ionary planet carrier coupled with a rotating sun gear, and a

rotating outer ring gear. The third type, and probably the least common type of

arrangement, is the solar gear. This low ratio epicyclic box has a fixed sun gear

combined with a moving ring gea r, and planet carrier.

B efore ad dressing any specifi c details on these thr ee epicyclic gear arr an ge-

ments, it w ould be beneficial t o mention t he common chara cterist ics between the

three confi gurat ions. For insta nce, on a sun gear input, the ta ngential t ooth load

a t ea ch planet is derived by a n expans ion of equa tion (10-2) into th e following:

(10-10)

where: HP = Transmitted Power Across Sun Gear Teeth (Horsepower)Rpitch-sun = Pitch Radius of Sun Gear (Inches)

Fsun = Sun Gear Rotational Speed (Revolutions / Minute)

Np = Number of Planet Gears (Dimensionless)

Forcetan-plt = Tangential Force Across Planet Gear Teeth (Pounds)

The number of externa l teeth on the sun a nd planet gears, and t he number

of internal teeth on the stat ionary ring gear must maintain a particular tooth

ratio to allow assembly. Specifically, the following tooth assembly equations were

extracted from Dudleys Gear H and book5:

(10-11)

(10-12)

5 Da rle W. Dud ley, Gear H andbook, (New York: McG ra w -H ill B ook Compa ny, 1962), pp. 3-15.

Fo rce p l ttan63 024, H P

Rp i t ch su n Fsu n Np------------------------------------------------------------------=

Tr i n g Tsu n 2 Tp l t+=

Tr i n g Tsu n+Np

---------------------------------- In teger=


13/76

Epicyclic Gear Boxes 471

where: Tsun = Number of Sun Gear Teeth (Dimensionless)

Tplt = Number of Planet Gear Teeth (Dimensionless)

Tring = Number of Ring Gear Teeth (Dimensionless)

With three different gear configurations, epicyclic boxes may emit a variety

of excitat ions tha t va ry from low t o high frequencies. For exam ple, a generic epi-

cyclic box ha s t he potent ial t o produce the following a rra y of frequencies.

r Sun G ear Rota t ional S peed Fsunr P lanet G ear Rota t ional Speed Fpl tr P lanet Ca rr ier Rota t ional Speed Fcarr Ring Gea r Rotat iona l Speed Fr i ngr P lanet Pa ss Frequency Fplt-passr P lanet Absolute Frequency Fplt-absr One or More G ear M esh Frequ encies Fgmr G ear E lement Resonant Frequencies

r Ca sing Resona nt F requencies

For a planetarygear box the ring gea r speed Fr i ngis equa l to zero. On a star

confi gurat ion th e plan et carrier is fi xed, an d frequency Fcaris zero. Sim ilar ly, the

sun gea r speed Fsunis zero on a solargear. The individua l gear mesh fr equencies

are a bit more complicated, and they will be reviewed in conjunction with each

gear box discussion. The gear element natural resonant frequencies, and the cas-

ing resonant frequencies listed in the previous summary, originate from the

same sources discussed under two element gear boxes.

At t his point, it is mean ingful to exa mine the specifi c frequencies associat ed

wi th a planetary gear box. For instance, consider the typical planetary geartra in shown in F ig. 10-6. This is a basic ar ran gement tha t m ay be used as either

a speed increasing or a speed decrea sing device. For discussion purposes, a ssum e

that this gear box is used as a speed increaser. The input shaft is coupled to the

planet car rier, and it rota tes counterclockwise a t a frequency indicated by Fcar.

In t his example, three planet gea rs a re at ta ched to the carrier, and ea ch plan et

mat es with the sta t ionary r ing gear (Fr i ng= 0). As th e plan et carrier rotat es in a

counterclockwise direction, the planet gears must turn clockwise at a planet

rotational speed of Fpl t. The center output gear is the sun gear, an d it m at es with

the th ree planets. The sun gear ha s a rotat iona l speed of Fsunin a count erclock-

wise direction. In this case, the collinear input and output shafts rotate in the

sa me direction when view ed from one end of the gear box.

The planet spin or rotational frequency Fpl tis calculat ed based upon a gear

tooth rat io as follows:

(10-13)

The planet pa ssing frequency is determined by mult iplying the a ctual num -

P lane tSp i n Frequencyp l a n e t a r y F= p l t Fca rTr i n g Tp l t

--------------=


14/76

472 Chapter-10

ber of planet s Npby the plan et carrier speed Fcaras in t he next equa tion:

(10-14)

Since the planets are rotating or spinning on one axis, and that axis is

rotating in a circle, the planet absolute frequency is t he sum of the plan et carrier

a nd t he plan et spin speed as shown in equa tion (10-15). This fr equency is seldom

observed, but it is identified a s par t of the kinemat ics of the ma chine.

(10-15)

The most diffi cult calculat ion is associat ed wit h determina tion of the outputsun gear r otational speed Fsun. This is not a direct gear ratio due to the element

arr an gement wit hin t he gear box. The calculat ion ma y be performed by examin-

ing relative speeds and ratios of the various components. In direct terms, the

overa ll gear box rat io may be determined by equa tion (10-16):

(10-16)

The planet gear mesh frequency is another common excitation for this type

of machine. This frequency is easily envisioned as the planet rotational fre-

quency Fpl tt imes the num ber of planet teeth Tpl t. I t m ay also be computed based

upon the number of ring gear teeth Tr i ngan d the planet ca rrier input speed Fcara s sh own in eq ua tion (10-17):

(10-17)

Fig. 106 Typical Planetary Configuration Of Epicyclic Gear Box - Stationary Ring Gear

StationaryRing Gear

Fring=0

Sun

GearFsun

PlanetGearFplt

Planet

CarrierFcar

Fcar

Fplt Fplt

Fplt

PlanetGearFplt

P lane tPassFrequencyp l a n e t a r y Fp l t p a ss Np Fca r= =

P lane tAbso l u t e Frequencyp l a n e t a r y Fp l t a bs Fca r Fp l t+= =

Su nGearFrequencyp l a n e t a r y Fsu n Fca r 1Tr i n g Tsu n--------------+

= =

P lane tGearM eshp l a n e t a r y Fgm p l t Fp l t Tp l t Fca r Tr i n g = = =


15/76


Finally, the high speed sun gear mesh frequency is determined by the prod-

uct of the sun gear rotat iona l speed Fsunan d the number of sun gear teeth Tsuna s s hown in equ a tion (10-18).

(10-18)

It should be restated that equations (10-13) through (10-18) are directly

applicable only to a planetar ygear box configuration as shown in Fig. 10-6. Fordemonstra tion purposes, these equat ions w ill be applied on a planeta ry gear set

containing 214 ring gear teeth, 94 planet gear teeth, and 26 sun gear teeth.

Assume tha t t his is a speed increasing box with th ree planets, and th at the input

speed is 1,780 RP M. Before start ing th e gear frequency calculat ions, the va lidity

of this a ssembly m ay be checked w ith equa tions (10-11) and (10-12) as follows:

This calculation a grees w ith the a ctual r ing gear tooth count of 214 teeth.Next, the second a ssembly equa tion should be checked a s follows:

The final value of 80 is an integer number, and the equation is satisfied.

Hence, the basic epicyclic gear assembly equa tions are sa tisfi ed, and it is appro-

priate to commence the computation of the fundamental excitation frequencies.

The planet rotational frequency Fpl t, also known as the planet spin speed, is

determin ed from equa tion (10-13) as follows:

The planet passing frequency Fplt-passis comput ed from equa tion (10-14):

Next, the planet absolute frequency Fplt-absma y be determined from equa-

tion (10-15) in the following manner:

The sun gear rotational frequency Fsunma y be ca lculat ed w ith (10-16):

This fi na l ra tio of 9.2308:1 (= 1+ 214/26) ma y seem excessive, but for a pla ne-

tary gear box of this general arrangement it is quite common. In fact , overall

speed rat ios of 12:1 are a common a nd a cceptable pra ctice for pla neta ry boxes.

Su nGearM eshp l a n e t a r y Fgm su n Fsu n Tsu n( )= =

Tr i n g Tsu n 2 Tp l t+ 26 2 94+ 26 188+ 214= = = =

Tr i n g Tsu n+

Np----------------------------------

214 26+

3---------------------

240

3--------- 80 In teger= = = =

Fp l t Fca rTr i n g T

p l t

-------------- 1 780 RPM,214

94--------- 4 052 RPM,= = =

Fp l t p a ss Np Fca r 3 1 780 RPM, 5 340 RPM,= = =

Fp l t a bs Fca r Fp l t+ 1 780, 4 052,+ 5 832 RPM,= = =

Fsu n Fca r 1Tr i n g

Tsu n

--------------+

1 780, 1 214

26

---------+

16 431 RPM,= = =


16/76

474 Chapter-10

The planet gear mesh frequency Fgm-pltmay be determined from both por-

tions of equa tion (10-17) to yield the following ident ical result s:

Finally, the higher frequency sun gear mesh frequency Fgm-sunis easily cal-

culated from expression (10-18) as follows:

From this basic planetary gear train, a total of seven fundamental excita-

tions have been identified. Field vibration measurements on this gear box will

generally reveal various interactions and modulations between these frequen-

cies. Due to the potential narrow pulse width of some interactions, the resultant

vibration data should always be viewed in both the t ime and the frequencydomain to make sure tha t a ll of the significan t vibra tory motion is detected.

The second type of epicyclic gear box commonly encountered is the starconfiguration as illustrated in Fig. 10-7. For discussion purposes, assume thatthe input occurs at t he center sun gear tha t rota tes at a s peed of Fsunin a coun-

terclockwise direction. In this type of box, the planets are fixed in stationary

bearings, and t he planet carrier frequency Fcaris zero. Although the pla nets con-

tinue to rotate clockwise at a frequency of Fpl t, there is no translation of the

planet bearing centerlines. Also note that the sun gear is not restrained by a

bearing, an d it essentia lly floats w ithin the mesh of the planets.

The planet gears in Fig. 10-7 engage an outer ring gear that turns in a

clockwise direction at a rotational speed of Fr i ng. The ring gear then connects to

the output shaft directly, or it may mate with an outer coupling assembly

Fig. 107 Typical Star Configuration Of Epicyclic Gear Box - Stationary Planet Carrier

Fgm p l t Fp l t Tp l t( ) 4 052, 94( ) 380 900 CPM,= = =or

Fgm p l t Fca r Tr i n g ( ) 1 780, 214( ) 380 900 CPM,= = =

Fgm su n Fsu n Tsu n( ) 16 431, 26( ) 427 200 CPM,= = =

InternalRing Gear

Fring

SunGearFsun

StationaryPlanetCarrierFcar=0

PlanetGearFplt

Fplt

Fplt

Fsun Fring

Fplt


17/76


th rough a s pline arra ngement . Obviously, the speed of the outer coupling a ssem-

bly must be equal to the rotational speed of the internal r ing gear Fr i ng. Nor-

ma lly, the circumferentia l outer coupling a ssembly is connected to a common end

plat e, a nd t his plat e is secured to the output sh a ft, as illust ra ted in Fig. 10-10 (in

case history 30). As shown in this sketch, the input a nd output sh aft s rota te in

opposite directions for t his starconfiguration.The planet spin or rota tiona l frequency Fpl tfor this sta r arra ngement is cal-

culat ed from th e sun gear a s follows:

(10-19)

The planet passing frequency is equal to zero, since the planet carrier does

not rota te (i.e., Fcar= 0). By th e sam e logic, the pla net a bsolute frequency Fplt-absis identical to the planet rotational speed Fpl t. The output ring gear rotational

speed Fr i ngma y be determined by:

(10-20)

The planet gear mesh frequency is constant across all three gears, and it

ma y be computed w ith equa tion (10-21):

(10-21)

Equations (10-19) through (10-21) only apply to a star arrangement. A set

of example calculat ions for a sta r confi gura tion are presented in case history 30.

Prior to this machinery story, the third type of epicyclic gear box should be

reviewed. As previously stated, this is commonly known to as a solar configura-tion. This na me stems from the fa ct tha t t he sun gear remains fi xed (i.e. , Fsun= 0),

and all of the other gears are in motion, as shown in Fig. 10-8. For discussion

purposes, assum e tha t t he input to this gear tra in is the clockwise rotat ing ring

gear a t a frequency of Fr i ng. The enga ged planet s a re driven in a clockwise direc-

tion at a planet spin speed of Fpl t. The planet gears translate around the fixed

sun gear, and they drive the planet carrier in a clockwise direction at a speed of

Fcar. In this case, the input an d output shaft s rotat e in the same direction when

viewed from one end of the gear box.

The planet spin or rotational frequency Fpl tis calculated a s follows:

(10-22)

The planet pa ssing frequency is determined by mult iplying the a ctual num -

P lane tSp i n Frequencys ta r F= p l t Fsu nTsu nTp l t-------------=

R i n g GearFrequencys t a r Fr i n g Fsu nTsu nTr i n g --------------= =

GearM eshs ta r Fgm Fp l t Tp l t Fr i n g Tr i n g Fsu n Tsu n= = = =

P lane tSp i n Frequencyso la r

F=p l t

Fr i n g

Tr i n g

Tp l t--------------=


18/76

476 Chapter-10

ber of planet s Npby the plan et carrier speed Fcaras in t he next equa tion:

(10-23)

Since the planets are rotating or spinning on one axis, and that axis is

rota ting in a circle, the pla net a bsolute frequency is show n in equ a tion (10-24).

(10-24)

The determination of the output planet carrier rotational speed Fcaris pre-

sented in the following equation (10-25):

(10-25)

Note that the value result ing from the ratio of Tsun/Tr i ngwill be consider-

ably less th an one. When this value is summ ed with one, it is clear tha t t he over-

all gear ratio will be quite low. Hence, a solarconfiguration of an epicyclic gearbox is strictly a low speed ratio device. This type of machine exhibits only a sin-

gle gear mesh frequency th at is the planet rota tional speed Fpl tt imes the num-

ber of planet teeth Tpl t. It may also be computed based upon the number of ring

gear teeth Tr i ngan d the ring gear speed Fr i ngas shown in t he next equa tion:

(10-26)

Once ag a in, it should be n oted th a t equa tions (10-22) through (10-26) onlyapply t o a solararrangement of an epicyclic gear box. Furthermore, multiplebeats and signal modulations are possible on any epicyclic gear box due to the

Fig. 108 Typical Solar Configuration Of Epicyclic Gear Box - Stationary Sun Gear

InternalRing Gear

Fring

StationarySun GearFsun=0

PlanetCarrier

Fcar

PlanetGearFplt

Fplt

Fplt

Fring

Fplt

Fcar

P lane tPassFrequencyso la r Fp l t p a ss Np Fca r= =

P lane tAbso l u t e Frequencyso la r Fp l t a bs Fca r Fp l t+= =

Ca r r i e r Frequencyso la r Fca r Fr i n g 1Tsu nTr i n g --------------+

= =

GearM eshso la r Fgm Fp l t Tp l t Fr i n g Tr i n g = = =


19/76


interaction of a large variety of fundamental excitations. There is also the poten-

tial for considerable cross-coupling between the lateral and torsional characteris-

tics in these gear boxes. This becomes even more complicated when compound

epicyclic gear boxes a re exam ined tha t contain t wo planets on the sa me sha ft . Inother cases, two epicyclic gear boxes may be used in tandem (i.e., coupled

together) to achieve some very high speed ratios. In either case, the array of

mechanical excitations becomes quite large, and the potential for interaction

with one or more system resonances becomes significant. Specifically, consider

the situation described in the following case history.

Case History 30: Star Gear Box Subsynchronous Motion

The machinery train discussed in this case history consists of a 16,500 HP

gas turbine driving a synchronous generator through an epicyclic gear box. The

arrangement of the gear box and generator, plus the installed proximity probes

are shown in Fig. 10-9. The epicyclic gear box was configured in a star arrange-

ment similar to the previously discussed Fig. 10-7. This box contained 3 planet

gears with 47 teeth Tpl ton each gear. The power turbine input speed to the sun

gear Fsunwas 8,568 RPM, and the sun gear contained 25 teeth Tsun. The outer

ring gear must rotate at 1,800 RPM to drive the generator. This internally

toothed ring gea r cont a ined 119 teeth. As show n in F ig. 10-10, the ring gea r w a s

directly mated to an outer coupling assembly that drove the output gear box

sha ft . The gear box output sha ft wa s restra ined by an outer and a n inner journal

bearing as shown in Fig. 10-10. The sun gear floated on the planet mesh, and

each plan et wa s supported by a fi xed journa l bear ing.

The coupling between the epicyclic gear box and the generator was a close

coupled gear coupling. Although technical specifications were not available for

this a ssembly, it wa s clearly a hardcoupling w ith high t orsional an d lat eral st iff-

ness. Hence, any lateral or torsional excitations on the gear box could be easily

Fig. 109 Machinery And Vibration Transducer Arrangement For Star Gear and Generator

EpicyclicGear

4.760:1

Driven CCW

by 16,500 HP

Gas Turbine

at 8,568 RPM

SynchronousGenerator

9,500 KW

12,470 Volts, 550 Amps

1,800 Rpm, 3, 60 HzK

15

75

2V

2H

CW

45 45

6V 6H45 45

5V 5H

CW

45 45

4V 4H15

75

3V

3H

15

75

1V

1H


20/76

478 Chapter-10

tra nsmitt ed to the genera tor, and vice versa .

Before addressing the specific problems on this machinery, it would be

desirable to check the epicyclic gear box configuration, and identify the antici-

pated excitation frequencies. As before, the validity of this assembly may bechecked wit h eq ua tions (10-11) a nd (10-12), a s follows:

This calculat ion a grees w ith t he actua l r ing gear tooth count of 119 teeth.

Next, check the second as sembly equa tion a s follows:

The value of 48 is an integer number, and the equation is satisfied. Hence,

the ba sic epicyclic gear assembly equa tions are sa tisfi ed, and it is a ppropriate to

compute the fundamental excitation frequencies. The planet rotational or spin

frequency Fpl tis det ermined from equa tion (10-19) as follows:

The ring gear rotational frequency Fr i ngma y be verifi ed wit h (10-20):

The gear mesh frequency Fgmma y be determined from t he fi rst portion of

equation (10-21) to yield the following:

This starepicyclic gear configuration provides the proper output speed of1,800 RPM, and the above calculations also define the planet rotational speed

Fpl t, plus the gear mesh frequency Fgm. Under normal machinery behavior the

domina nt sh aft vibration frequency on both t he gear box output and t he genera -

tor should be 1,800 CP M. On the power turbin e sha ft (sun gear in put), th e ma jor

shaft vibration frequency should be 8,568 CPM. The planet rotational frequency

of 4,557 CPM probably would not appear unless one or more planets were in a

state of distress. Finally, the major high frequency casing vibration component

should logica lly occur a t t he gear mesh freq uency of 214,200 CP M. Although t his

is a complex mecha nical system, th e number of fundamenta l excita t ions ar e lim-

ited and definable.

Init ia l operat iona l tests on this unit revealed normal a nd a ccepta ble behav-ior at full load. Vibra tion levels w ere low, journa l positions were proper, bearin g

temperatures w ere norma l, and overall skid vibration wa s a ccepta ble. The tra n-

sient sta rtup a nd coas tdown B ode plots w ere likewise normal, and all sta t ic plus

dynamic measurements pointed towards a normal machinery train. However,

Tr i n g Tsu n 2 Tp l t+ 25 2 47+ 25 94+ 119= = = =

Tr i n g Tsu n+

Np----------------------------------

119 25+

3---------------------

144

3--------- 48 In teger= = = =

Fp l t Fsu nTsu nTp l t------------- 8 568 RPM,

25

47------ 4 557 RPM,= = =

Fr i n g Fsu nTsu nTr i n g -------------- 8 568 RPM,

25

119--------- 1 800 RPM,= = =

Fgm Fp l t Tp l t 8 568 RPM, 25 214 200 CPM,= = =


21/76


during a reduced load test , a n unusua l subsynchronous vibration component a t

1,240 CPM appeared on the generator. This frequency component at nominally

69% of rota tional speed w as forwa rd a nd elliptical. The highest a mplitudes

occurred at the inboard coupling end bearing. Further investigation revealed

that the same frequency appeared on the epicyclic gear box output bearing.

As the examination of the data progressed, it was clear that addit ional

mea surement points a long the low speed rotor elements w ould be benefi cial. The

ma chinery w as originally equipped w ith X-Y proximity probes mounted at 45

from vertical center. This included the gear box output (probes 4V & 4H), the

genera tor coupling end bea ring (probes 5V & 5H), a nd t he genera tor outboar d orexciter end bearing (probes 6V & 6H). All three sets of original transducers are

depicted on t he ma chinery a rra ngement dia gram Fig. 10-9.

In order t o obta in more informa tion about t his low frequency phenomena ,

three more sets of X-Y probes were installed through the top cover of the gear

box. These supplementa l probes were m ounted w ith th e vertica l or Y-a xis probe

at 15 to the left of the vertical centerline, and the horizontal or X-axis trans-

ducer wa s positioned a t 75 t o the right of vertica l. The physical location of these

a ddit ional probes a re show n in Figs. 10-9 a nd 10-10. The fi rst set of probes (1V &

1H) were positioned on the outer diameter of the ring gear. The next two sets of

pickups (probes 2V & 2H) were located on th e forwa rd pa rt of the outer coupling,

and probes 3V & 3H were located on the aft portion of the outer coupling. In

essence, probes 3V & 3H were in line with the inner bearing. The location and

axial distance between gear box transducer planes is described in Fig. 10-10.This diagram also depicts the outer coupling spline fit to the ring gear, and the

general a t ta chment of the outer coupling t o the gear box output s ha ft .

Armed w ith t his full arr ay of tra nsducers observing th e low speed rotors, a

series of detailed operational tests were conducted in an effort to quantify the

Fig. 1010 Low Speed Probes Installed In Star Configuration Epicyclic Gear Box

Output Shaft

OutputBearing

Ring GearProbes1V & 1H

OutputShaft

Probes4V & 4H

InnerBearing

Internal Ring Gear119 Teeth

CW Rotation at1,800 RPM

Outer CouplingWith Spline FitTo Ring Gear

CW Rotation at1,800 RPM

Outer CouplingForward Probes2V & 2H

Outer CouplingAft Probes3V & 3H

4.0" 12.2" 12.6"


22/76

480 Chapter-10

characteristics of this subsynchronous vibration component.

A graphical description of this subsynchronous vibration is presented in

Fig. 10-11. This stea dy sta te da ta wa s obtained w ith t he low speed sha ft operat -

ing at 1,800 RPM, and a 1,600 KW load on the generator. The shaft vibrationfrom each set of X-Y proximity probes was band-pass filtered at the maximum

subsynchronous frequency of 1,246 CP M, and the resulta nt orbits plotted at each

measurement sta t ion. Since the subsynchronous orbits a re a ll elliptical, a com-

parison of severity was achieved by identifying the vibration amplitude associ-

a ted w ith t he ma jor a xis of each ellipse. The subsyn chronous orbits present ed in

Fig. 10-11 also include a simulated timing signal at the tuned frequency of 1,246

CP M. This simulat ed t iming signa l is not related t o a st at iona ry reference point

like th e shaft once-per-rev Keypha sor signa l. How ever, the relat ive pha se rela-

tionship between a ll six measurement locat ions is a ccurat e and consistent.

From Fig. 10-11, it is noted that the subsynchronous motion on the ring

gear a nd t he forwa rd end of the outer coupling assembly a re in unison. Similar

orbit pat terns a re evident, and the respective am plitudes and phase ma rkers are

almost identical. At the aft end of the outer coupling assembly the subsynchro-

nous motion wa s a ttenua ted. The orbit a t t he output sha ft of the gear box reveals

a phase reversal plus a lower amplitude at the subsynchronous frequency. The

ma gnit ude of the subsyn chronous vibra tion contin ues to diminish a cross the cou-

pling to the generator, with the lowest amplitudes appearing on the outboard

end of the generator. In essence, the subsynchronous motion has a conical mode

shape with maximum amplitudes at the ring gear and outer coupling; with a

nodal point betw een t he outer coupling a ft , a nd t he gear box output.

Fig. 1011 Initial Mechanical Configuration Subsynchronous Shaft Orbits Filtered At1,246 CPM With Rotational Speed Of 1,800 RPM And 1,600 KW Load on Generator

GearBo

x

Cpl.

Generat

or

1.0 Mil / Div.

OuterCoupling FwdMajor Axis=3.4 Mils,p-p

OuterCoupling AftMajor Axis=0.8 Mils,

p-p

Output ShaftMajor Axis=1.2 Mils,p-p

Drive EndMajor Axis=1.0 Mils,p-p

Exciter EndMajor Axis=0.5 Mils,p-p

Ring GearMajor Axis=3.6 Mils,p-p


23/76


Closer examination of the variable speed startup data with cascade plots

revealed an independent lateral vibration component that migrates from 1,000

to 1,300 CPM. This behavior was dominant on the gear box, with much lower

amplitudes on the generator. In addit ion, the unfiltered shaft vibration datarevealed a n a mplitude modulat ion betw een runn ing speed a nd t he subsynchro-

nous component. This behavior was most visible on the ring gear, and the outer

coupling a ssembly. Fina lly, when t he subsyn chronous component w a s a ctive, cas-

ing a ccelerometers revea l a 1,240 CP M modulat ion of the 214,200 CP M gear

mesh frequency.

From this data array it is clear that the subsynchronous shaft vibration

encountered on the low speed end of this ma chinery t ra in originat es within the

epicyclic gear box. It was also determined that the subsynchronous excitation

occurs at a frequency th at only va ries betw een 1,220 an d 1,260 CP M. It wa s a lso

documented that the largest subsynchronous vibration amplitudes appear dur-

ing a limited torque range of 5,700 to 6,300 foot-pounds.

This information was substantiated during variable speed generator tests

at 1,500 and 1,800 RPM. Specifically, test data at 1,800 RPM and 1,600 KW

(6,260 foot pounds) revealed a subsyn chronous fr equency of 1,240 CP M. B y com-

parison, at a reduced generator speed of 1,500 RPM and a load of 1,212 KW

(5,690 foot pounds), the subsynchronous component appeared at essentially the

sa me frequency of 1,220 CP M. Any changes in genera tor load (up or down ) would

at tenuat e, or completely eliminat e th e subsynchronous vibra tion.

Load changes, or more specifically torque changes, are directly associated

with the repeatable appearance of this subsynchronous component. Since the

subsynchronous excitat ion appears at essentially a constant frequency, and the

maximum amplitude occurs within a limited torque range serious consider-

ation should be directed towards a resonant response in a twisting direction.

That is, the excitation of a lower order torsional resonance should be considered

as a realistic possibility. This concept was reinforced when the undamped tor-sional response calculations revealed a first mode at 1,320 RPM. Although some

of the ma ss elast ic data wa s questionable, the coincidence of the calculat ed tor-

sional critical speed and the measured behavior could not be ignored.

How ever, the correla tion of the mea sured subs ynchronous vibra tion compo-

nent w ith t he calculat ed torsiona l resona nce frequency wa s a n unpopular conclu-

sion. This w as complicat ed by t he fact tha t the subsynchronous component w as

sensitive to changes in the oil supply temperature, which could be related to

changes in damping. In addit ion, the machinery adversely responded to minor

alignment changes a cross th e low speed coupling between the gea r box and t he

generator. It appeared that raising the generator by 10 Mils unloaded the gear

output bearing, and allowed it to go unstable. Hence, the popular corporate the-

ory was that the observed behavior was nothing more than a bearing stability

problem. Since the m a jor a ctivity occurred w ithin th e epicyclic gea r box, the pa r-ties in charge of the machinery elected to suppress the subsynchronous instabil-

ity by unbalancing the outer coupling assembly. This change was implemented

by ad ding a 79 gram un bala nce weight t o the forwa rd a xial face of the outer cou-

pling assembly. The influence of this 79 gram unbalance weight on the subsyn-


24/76

482 Chapter-10

Fig. 1012 Unbalance Of 79 Grams On Outer Coupling - Subsynchronous Orbits FilteredAt 1,248 CPM With Rotational Speed Of 1,800 RPM And 1,600 KW Generator Load

Fig. 1013 Unbalance Of 79 Grams On Outer Coupling - Synchronous Shaft Orbits Fil-tered At Rotational Speed Of 1,800 RPM With 1,600 KW Generator Load

GearB

ox

Cpl.

Generat

or

0.5 Mils / Div.


Outer

Coupling AftMajor Axis=0.8 Mils,p-p





GearB

ox

Cpl.

Genera

tor

1.25 Mils / Div.


OuterCoupling AftMajor Axis=1.6 Mils,

p-p






25/76

Process Fluid Excitations 483

chronous motion is shown on Fig. 10-12. Note that the gear box vibration

amplitudes at 1,248 CPM on the ring gear and the outer coupling have been

attenuated from nominally 3.6 to 1.8 Mils,p-p. However, the vibration response

over on the generator at this frequency was virtually unaffected. Since the

amplitude of this subsynchronous vibration on the generator was the primary

concern of the OEM, the addition of the 79 gram unbalance to the gear box outer

coupling a ssembly wa s not an acceptable solution.

The inappropriat eness of this w eight ad dit ion w as further demonstra ted by

examining the synchronous 1X motion at each of the measurement planes as

presented in Fig. 10-13. The previous running speed vibration at 1,800 RPM

before installation of the 79 gram unbalance varied between 0.3 and 0.7 Mils, p-pat each in the measurement locat ion. However, after the 79 gram s wa s a tta ched,

th e 1X am plitudes increas ed to 4.3 Mils,p-p a t t he outer coupling, a nd 4.5 Mils,p-pon the ring gear. Clearly, these increased synchronous amplitudes would be det-

rimental to the long-term reliability of this gear box.

In the final assessment, the addit ion of unbalance weights to the gear boxdoes not represent a viable solution to the subsynchronous vibration problem. In

fact, it does impose additional dynamic forces upon the gear elements. The

proper engineering solution included a modification of the gear box output bear-

ing to cope with the occasiona l insta bility due to bearing unloading. In a ddit ion,

it w as necessary to recognize tha t the subsynchronous motion at nominally 1,240

CP M wa s logically a torsiona l na tura l frequency. This resona nce appeared a s a

lateral vibration due to cross-coupling between torsional and lateral motion

across the gear teeth. This torsional resonance was directly related to the st iff

gear coupling between the epicyclic gear box and the generator. In all cases,

units tha t conta ined the st iff gear coupling exhibited t his subsynchronous com-

ponent at reduced load, and identical units that had a torsionally softer flexible

disk coupling did notexperience th is subsyn chronous la tera l response.

PROCESS FLUID EXCITATIONS

Fluid handling machines invariably contain some arrangement of stat ion-

ar y a nd rota ting bla des or vanes. This a pplies to machines ha ndling incompress-

ible fl uids such as pumps, hydraulic turbines, an d extruders, plus machines tha t

ha ndle compressible fluids such as steam or gas turbines, centrifugal compres-

sors, expanders, and blowers. In many cases, the number of blades or vanes

times the shaft rotative speed provides a simple expression for computation of a

potentia l blade pa ssing frequency Fbas sh own in the next equa tion.

(10-27)

where: Fb = Blade Passing Frequency (Cycles / Minute)

Nb = Number of Blades or Vanes (Dimensionless Integer)

RPM = Rotative Speed (Revolutions / Minute)

B l a d e Pas g Frequencysi n F b

Nb

RPM= =


26/76

484 Chapter-10

An applica tion of this concept is presented in Fig. 10-14 th a t displa ys a pair

of spectrum plots obtained from a large single shaft gas turbine. The top FFT

plot in Fig. 10-14 wa s a cquired at th e inlet end bearing h ousing. It display s a fr e-

quency array that coincides with virtually all of the axial flow air compressorstages. The sixth stage was not evident and there is no guarantee that some of

the other components are attributable to only one stage. In addition, the compo-

nents that represent several stages (same number of blades on more than one

stage) might be due to an excitation from only one stage, or the interaction of

several s ta ges to produce one component a t a common freq uency.

The simpler plot a t t he bottom of Fig. 10-14 wa s obta ined from th e exhaust

end bearing housing. This diagram is dominated by the hot gas power turbine

fi rst a nd second st age blad e passing frequencies. Due to the clear dema rcation of

blade counts, this data carries more credibility than the complex spectra

extracted from the inlet end bearing. However, the diagnostician should still

review this data with caution. In most cases, the impedance path between the

rotor excitat ion and a bearing cap acceleration measurement is unknown. It is

therefore difficult to correlate these frequency components and amplitudes to

specific levels of severity within the machine. At best, the various components

may be identified in terms of harmonic order, and potentially associated with

specific mechanical elements (e.g., number of turbine blades on a particular

sta ge). From there on, the machinery diagnostician is fa ced w ith routine exami-

na tion of th e high frequency spectra , and a trending of the results. Nat urally, forthis type of progra m t o be effective, the sa me a ccelerometer m ust be mounted in

the same location, and the high frequency data acquired and processed in the

sam e man ner. Var iat ions of any of these steps would invalidat e the a ccumulat ed

da t abase .

Fig. 1014 Blade Passing Excitations On A Large Single Shaft Industrial Gas Turbine


27/76


It must also be recognized tha t high frequency blade passing excita t ions are

often infl uenced by st at iona ry objects. This intera ction betw een t he rotat ing a nd

the sta t ionary mechanical systems is a nuisan ce wh en dealing with compressible

fluids, and it forms a mandatory part of the analysis when examining machinestha t ha ndle incompressible liquids.

For insta nce, consider Ta ble 10-1 of ca lculat ed pump va ne pa ssing frequen-

cies. This data is for a centrifugal pump with a vaned diffuser. Typically, the

number of diffuser vanes exceed the number of impeller vanes. Due to the close

coupled configuration of this type of machinery, there is a definite interrelation-

ship and resultant excitat ion between stator and rotor parts. Thus, a six vane

impeller running inside of a nine vane diffuser will produce a blade passing fre-

quency at four t imes rotat ive speed. If the ma chinery dia gnostician is expecting

to see a 6X blade passing frequency on the pump, the appearance of a strong 4X

component ca n be m ost disconcert ing.

Table 101 Vane Pass Frequency For Various Combinations Of Impeller And Diffuser Vanes

PumpDiffuserVanes

Number of Pump Impeller Vanes

3 4 5 6 7 8 9

4 3 5 3 7 2 9

5 6 4 6 14 16 9

6 4 5 7 4 3

7 6 8 15 6 8 27

8 9 15 9 7 9

9 8 10 4 28 8

10 9 6 6 21 16 9

11 12 12 10 12 21 32 45

12 25 35 4 9

13 12 12 25 12 14 40 27

14 15 8 15 6 3 8 27

15 16 6 14 16 6

16 15 15 9 49 63

17 18 16 35 18 35 16 18

18 10 35 35 28

19 18 20 20 18 56 56 18

20 21 21 21 6 81


28/76


29/76


Blevins goes on to explain that vortex shedding from a smooth circular cyl-

inder in a subsonic flow is a function of th e Reynolds number. In t his cont ext, the

Reynolds number NReis defined in t he following m an ner:

(10-28)

where: NRe = Reynolds Number (Dimensionless)

V = Free Stream Velocity Approaching the Cylinder (Inches / Second)D = Cylinder Diameter (Inches) = Kinematic Viscosity (Inches2 / Second)

In cha pter 4 of this t ext, equa tion (4-5) identifi ed the va ria bles used to com-

pute the Reynolds through the minimum oil fi lm of a bearing. An init ia l compar-

ison between equations (4-5) and (10-28) reveals some differences. However, a

closer examination of (4-5) shows that the term xR is surface velocity of arotating shaft with units of inches per second. This is equivalent to the free

stream velocity approaching the cylinder Vshown in equation (10-28). The oilfi lm height in inches designated by Hin equa tion (4-5) is equiva lent t o the cylin-der diameter Dused in (10-28). Finally, the remaining terms are all associatedw ith the m oving fl uid viscosity. In equ a tion (10-28) the kinemat icviscosity w a sused, whereas the absoluteor dynamicviscosity was applied in (4-5). The twoviscosity forma ts a re directly relat ed, as show n in equa tion (10-29).

(10-29)

where: = Absolute or Dynamic Viscosity (Pounds-Seconds / Inch2),G = Acceleration of Gravity (386.1 Inches / Second2) = Fluid Density (Pounds / Inches3)

A dimensional analysis of equation (10-29) reveals that the units are cor-

rect. Fur th ermore, both Reynolds n umber equa tions (4-5) a nd (10-28) are equiva -

lent. Whereas (4-5) was used to define the ratio of inertia to viscous forces in a

fl uid fi lm bear ing expression (10-28) is a pplied to a fl uid str eam fl owing a cross

a smooth circular cylinder. In this application, the pattern generated by vortices

down st ream of the cylinder may be predicted ba sed upon t he value of the Rey-

nolds number. Specific flow regimes have been identified by various investiga-

tors, and t he reader is a gain referenced to the text by Robert B levins for detailed

informa tion. Some of the informa tion on this t opic is also ava ilable in the Shock

and Vibrat ion H andbook10 in the section a uthored by B levins. It should a lso bementioned that numerous studies have been conducted on vortex shedding, and

th e associat ed vortex induced vibra tion. Hence, th is is a w ell-document ed techni-

cal fi eld tha t incorporat es man y empirical st udies and an alyt ical solutions. Thereferences provided with in the B levins t ext reveal the tr ue breadth of this physi-

10 Cyril M. Harris , Shock and Vibr ation H andbook, Fourth edition, (New York: McGraw-Hill,1996), pp. 29.1 to 29.19.

NR eD V

---------------=

G

--------------=


30/76

488 Chapter-10

cal behavior tha t st retches across many technical fi elds.

Of particular importance within t he ma chinery business is the relat ionship

defined by the Strouhal number NSt r. This is a non-dimensional number that

allows computation of the fundamental or predominant vortex shedding fre-quency Fsas defined in the following expression:

(10-30)

where: NStr = Strouhal Number (Dimensionless)

Fs = Vortex Shedding Frequency (Cycles / Second)

The cylinder diameter D, and the constant velocity of the fluid stream Visthe same value used in equation (10-28). Thus, for a given cylinder diameter D,and flow velocity V, the vortex shedding frequency Fsmay be computed if the

Strouhal number NSt ris known. Fortunately, there are multiple empirical tests

tha t display a consistent relat ionship betw een the para meters specifi ed in equa -tion (10-30). For instance, on pages 48 through 51 of Bevins text, a variety of

charts describe the Strouhal number for a circular cylinder, an array of inline

and staggered cylinders, plus various other geometric cross sections. From this

dat aba se, it is clear tha t t he Strouhal num ber for the vast ma jority of cases w ill

vary between values of 0.1 and 0.8.

For the simple case of a circular cylinder with a Reynolds number between

500 and 1,000,000, this data suggests that the Strouhal number has a value of

nominally 0.2. Many technical references identify this value as 0.22. However,

across the range of Reynolds numbers specified, 0.2 represents a more realistic

avera ge for th e St rouha l number. If va lue this is s ubst itut ed into (10-30), the vor-

tex shedding frequency Fsma y be comput ed directly from:

(10-31)

As a pra ctical exam ple of the a pplica tion of these vort ex shedding concepts,

consider the situa tion of a ga s t urbine exha ust sta ck. If th e top cylindrical por-

tion of the stack has a diameter of 20 inches, and the environment consisted of

standard temperature (60F) and pressure (14.7 Psia), with wind gusts of 50

miles per hour it would be desirable to compute the anticipated vortex shed-

ding frequency. From various sources, the absolute viscosity of air under these

conditions is 0.018 centipoise. This is equal to 0.00018 poise. From the conver-

sion factors present ed in Appendix C of this t ext, 1 poise is equiva lent t o 1 dyne-

second /cent im eter 2. Thus, the absolute or dynamic viscosity of a ir a t standard

tempera tur e a nd pressure w ould be equa l t o 0.00018 dyne-second/cent imeter2.

Converting the metric viscosity units to English units may be accomplished in

the following ma nner:

NS t rFs D

V-----------------=

Fs0.2 V

D------------------=


31/76


From Table B -2 in the a ppendix of this t ext , the density of air a t st an da rd

temperature and pressure is equal to 0.07632 pounds per foot 3. This density

value ma y be converted to consistent units as follows:

Based on these physical properties, the kinematic viscosity of air may now

be computed from eq ua tion (10-29):

The peak wind velocity of 50 miles per hour may now be converted intocompa tible engineering units of inches per second in the followin g ma nner:

Suffi cient informat ion is now ava ilable to compute the R eynolds number of

th e air fl ow over the cylindrical st a ck with eq ua tion (10-28):

The Reynolds nu mber of 772,000 fa lls w ithin th e previously specified r a nge

of 500 a nd 1,000,000. This provides confi dence in using a St rouha l num ber of 0.2

for this case of a circular stack. More specifically, these conclusions allow the

direct a pplica tion of eq ua tion (10-31) as follows:

Hence, with a 50 mile per hour wind, the anticipated vortex shedding fre-

quency would be 528 cycles per minute (8.80 Hz). This is an appreciable fre-

quency tha t could infl uence the gas turbine, or a ny of the associat ed mecha nical

equipment. Furthermore, this frequency will change with wind speed. If any

combination of wind speed and associated vortex shedding frequency coincided

with a n at ura l frequency of the sta ck, the results could be devast at ing.

The traditional solution to this type of problem resides in the modification

of the stack outer diameter to disrupt the vortex shedding, and therefore elimi-

nate (or substantially minimize) the excitation source on the stationary cylinder.

Common modifications include helical strakes wrapped around the stack, or aseries of externa l slat s or shrouds tha t a re designed t o break up the vortices. In

some cas es, an an alyt ical m odel using Computat iona l Fluid Dyna mic (CFD ) soft-

wa re might be suffi cient to properly exam ine the system. In other situa tions, the

development a nd testing of a scale model in a w ind tunn el might be appropriate.

0.00018 Dyne-Sec/Cm2 2.2486

10 Pound/Dyne 2.54 Cm/Inch( )2

=

0.00018 2.2486

10 6.4516 2.6119

10 Pound-Sec/Inch 2= =

0.07632 Pound/Foot3 1 Foot 12 Inches( )3

4.4175

10 Pound/Inch3= =

G

--------------2.611

910 Pound-Sec/Inch 2 386.1 Inch/Sec2

4.4175

10 Pound/Inch3-------------------------------------------------------------------------------------------------------------- 0.0228 Inches

2/Sec.= = =

V 50 Miles/Hour= 5 280 Feet/Mile, 12 Inches/Foot 1 Hour 3 600 Sec, 880 Inch/Sec.=

NReD V

---------------

20 Inches 880 Inch/Sec.

0.0228 Inches2/Sec.

----------------------------------------------------------- 772 000,= = =

Fs0.2 V

D------------------

0.2 880 Inch/Sec20 Inches

------------------------------------------- 8.80 Cycles/Sec 60 Sec/Min 528 Cycles/Min.= = = =


32/76

490 Chapter-10

In a ll cases, the dia gnostician must be awa re of this vortex shedding phenomena

an d th e potentia l for induced vibration into the str ucture.

Another mechanism that periodically appears within fluid handling sys-

tems is the acoustic resonance problem. This is the classic organ pi pebehaviorthat appears in virtually every physics textbook. The traditional discussion of

sta nding w a ve theory relat es the velocity of sound (i.e., sonic velocity) in the fl uid

media Vswith the occurring acoustic frequency Faand the wa velength as pre-sented in the follow ing expression:

(10-32)

where: Vs = Sonic or Acoustic Velocity (Feet / Second)

Fa = Acoustic Frequency (Cycles / Second)

= Standing Wave Length (Feet)

The velocity of sound Vswill va ry a ccording t o the media. S olids w ill gener-

ally display the highest values, sonic velocity in liquids will generally be lower,a nd ga ses w ill display even low speeds. For example, Ta ble 10-2 summ a rizes

some common values for th e velocity of sound in a ssorted solids an d liquids.

The velocity of sound in solids will remain constant over a wide range of

conditions. This is due to the fa ct tha t t he ma terial density solan d modulus ofelasticity Erema in const a nt over a w ide ra nge of condit ions. The sonic velocity ina ny solid ma y be computed w ith t he following common expression:

Table 102 Typical Values For Sonic Velocity In Various Solids And Liquids

MaterialVelocity of Sound

(Feet/Second)

S olids Lea d 4,030

B ra ss 11,480

C opper 11,670

Iron & S t eel 16,410

Aluminum Alloys 16,740

G ra phit e 19,700

LiquidsAt 60F an d 14.7 Psia

Alcohol 3,810

Oil - S p. G r.= 0.9 4,240

Mercury 4,770

Fresh Wa t er 4,860

G lycerin 6,510

Vs Fa =


33/76


(10-33)

where: Vs-sol = Velocity of Sound in a Solid (Feet / Second)

G = Acceleration of Gravity (= 386.1 Inches / Second2)

E = Modulus of Elasticity (Pound / Inch2)sol = Solid Material Density (Pounds / Inch3)

To check the v a lidit y of (10-33), th e properties from Ta ble B -1 in a ppendix B

ma y be extra cted an d insert ed into this expression. For exam ple, pure copper has

a modulu s of elas ticit y equ a l to 15,800,000 pounds /inch2, and a density of 0.323

pound s/in ches3. Equa tion (10-33) ma y now be evalu a ted a s follows:

The result a nt va lue of 11,450 feet per minut e is compa ra ble to the velocity

of sound in copper list ed in Ta ble 10-2 of 11,670 feet per m inu te. The 2% va ria -

tion betw een velocit ies is due to the fact th at avera ge values to three significant

fi gures a re used for t he physica l properties in Ta ble B-1. By compa rison, experi-

ment a l result s provide t he sonic velocity listed in Ta ble 10-2.

The velocity of sound in liquids is determined with an expression equiva-

lent to equation (10-33). The difference between calculating the sonic velocity in

solids versus liquids is t ha t Youngs modulus of elas ticity Eis used for solids, a ndth e bulk modulus Bis used for liquids. Va ria tions in pressure and t emperat ure ofthe liquid will be compensated by using the bulk modulus and density at the

actual fluid operating conditions in the following expression.

(10-34)

where: Vs-liq = Velocity of Sound in a Liquid (Feet / Second)

B = Bulk Modulus (Pound / Inch2)liq = Liquid Material Density (Pounds / Inch3)

If a lubricat ing oil at at mospheric pressure an d 60 F ha s a bulk modulus of

219,000 pounds per inch 2, an d a density of 0.0327 pounds per inches3, the sonic

velocity is computed w ith equa tion (10-34) in t he followin g ma nner:

This s onic velocity in oil agrees d irectly w ith the va lue in Ta ble 10-2. Next,

the velocity of sound in gases must be addressed. It is w ell understood th at gases

are even more sensitive to variations in pressure and temperature. For perfect

ga ses, the sonic velocity m a y be computed w ith t he following expression:

Vsso lG E

144 so l-------------------------=

Vsso l386.1 Inches/Sec

215.8

610 Pounds/Inch2

144 Inches2/Foot

20.323 Pounds/Inch

3------------------------------------------------------------------------------------------------------ 11 450 Feet/Minute,= =

Vsl i qG B

144 l i q-------------------------=

Vsl i q386.1 Inches/Sec

2219 000 Pounds/Inch

2,

144 Inches2/Foot

20.0327 Pounds/Inch

3--------------------------------------------------------------------------------------------------- 4 240 Feet/Minute,= =


34/76

492 Chapter-10

(10-35)

where: Vs-gas = Velocity of Sound in a Gas (Feet / Second)

g = Acceleration of Gravity (= 32.17 Feet / Second2)k = Specific Heat Ratio of Cp/Cv (Dimensionless)

R = Universal Gas Constant (= 1,546 Foot-Pound force/ Pound mole -R)T = Absolute Gas Temperature (R = F + 460)z = Gas Compressibility Based on Temperature and Pressure (Dimensionless)

mw = Gas Molecular Weigh

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Chapter 10 Unique Behavior

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