chapter 5 rotor modeling

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199

C H A P T E R 5

Analytical Rotor Modeling 5

M achinery shaft vibration characteristics

reflect the combined interaction of rotating assemblies with various fluids, and

stationary machine elements. Often these characteristics can be segregated and

quantified with appropriate measurement techniques. However, there are situa-tions where the required data cannot be obtained due to other restraints. For

instance, the critical speed of a rotor cannot be determined due to the economic

impact of shutting down the unit. In another case, the machine running speed

cannot be increased to investigate the effects of higher order modes. Similarly,

changing mechanical pa rts, such as bearings or couplings, ca nnot be realistically

evaluated based upon hardware trial and error substitutions. From a problem

solving standpoint, the machinery diagnostician may begin an investigation by

comparing the predicted machinery behavior with the measured vibratory char-

a cteristics. In these types of situa tions, the development a nd utiliza tion of ma th-

emat ica l models to simulate the mecha nical systems ma y be ma nda tory.

During t he latt er half of the tw entieth centur y, various computa tional tech-

niqu es ha ve been developed an d refi ned int o working tools. Techniq ues such a s

Transfer Matrix, Computational Fluid Dynamics (CFD), and Finite ElementAna lysis (FEA) provide signifi cant capability for modeling physical syst ems. In

order to provide any overview of some analytical techniques and the generic

types of ava ilable softw a re for th e evalua tion of rotat ing ma chinery behavior, the

follow ing descriptions a nd exa mples of ma chinery ca lculat ions a re presented.

MODELING OVERVIEW

The mathematics associated with simple mechanical systems were pre-

sented in chapter 2. For a single degree of freedom system consisting of a

undamped mass ha nging from a spring, it w as concluded tha t t he system natura l

frequency was a function of stiffness and effective mass. Paraphrasing equation

(2-44), it can be sta ted t hese three var iables a re relat ed in th e following ma nner:

(5-1)N a t u r a l Frequency St i f f ness

Mas s --------------------------≈



200 Chapter-5

From this expression it is clear that stiff elements have high natural fre-

quencies, and flexible parts have lower resonances. Similarly, heavy elements

will display low na tura l frequencies, and lighter components w ill exhibit higher

values. For exam ple, the na tura l frequency for a la rge steel girder ma y be 5 Hz,a nd a sma ll tuning fork ma y emit a tone equa l to a frequency of 500 Hz. In either

case, the geometry and mechanical configuration define a combination of stiff-

ness and m ass tha t yield a discrete nat ural f requency.

As system complexity increases, the intricacies of the descriptive equations

a lso expa nd. The simple expression of equa tion (5-1) is repla ced by a ma tr ix solu-

tion, and items such as inertia and rotational forces are included. It is apparent

that mass, and the distribution of that mass is critical to the solution. Further-

more, shaft stiffness must be determined, and combined with the mass proper-

ties. When these elements are defined, the calculation of undamped critical

speeds may be performed. These calculations do not include damping, they do

not allow asymmetric stiffness, and they do not consider specific forcing func-

tions. However, an undamped analysis provides an overview of the natural fre-

quencies associat ed with a ma ss distribut ion a t a selected support st iffness, plus

the sha ft mode sha pes for each resona nce and st iffness combina tion.

The next level of a na lytical m odeling progra ms incorporat es a symmetr ica l

stiffness coefficients, plus damping from the bearings, foundation, or process

fluid. The oil film coefficients are calculated for the bearing configuration, and

support coefficients are normally measured. It is important to include damping

into the calculations. This energy dissipater allows the examination of damped

critical speeds, the computation of rotor stability, plus damped mode shapes.

Although th is calculation refi nement does a credible job of fi nding t he Eigenva l-

ues (natural frequencies and damping), it does not accept actual forcing func-

tions such as unbalance, skewed wheels, or bowed rotors.

Adding forced vibration mechanisms requires another evolution of the pro-

gram structure. Within forced synchronous response programs, dimensionalforces are used to compute rotor response in displacement units. Hence, the

anticipated motion (vibration) at any speed, and at any position along the rotor

may be computed. The accuracy of these calculations is often determined by a

comparison with measured shaft vibration data at specific locations. This verifi-

cation of ca lculat ions by m easured vibra tion response chara cteristics is a n often

ignored step. In a ctua lity, the verifi cation of results is vita l to th e development of

confidence in the calculations. I t also helps to define areas where the custom

a na lytical progra ms require improvement or modificat ion.

Clear ly, the constr uction of a successful ana lytical model requires the int e-

gra tion of numerous calculat ions into a cohesive set of results. A single comput er

program does not contain the entire model. In fact, many calculations are per-

formed in separate environments from the rotor dynamics calculations. For

instance, cross-sectional inertias may be computed in a mathematical program,and rotor dimensional configuration may be initially established in a spread-

sheet progra m. In most ca ses, severa l different progra ms a re required to perform

the full ar ra y of ca lculat ions. In the rema inder of this cha pter, the primar y rotor

dyna mics programs w ill be discussed, and illustra ted w ith fi eld examples.



Undamped Critical Speed 201

UNDAMPED CRITICAL SPEED

In order to understand the fundamental behavior of a rotor system, it is

mandatory to determine the frequency of the system critical speeds, and thea ssociat ed mode shapes. One of the easiest tools to begin such a n investiga tion is

a n a na lysis of the unda mped critical speeds. To appreciat e the current a rra y of

computa tions it is meaningful t o briefl y r eview t he origin of these calcula tions.

Historically, one of the earliest procedures for critical speed calculations

was developed by A. Stodola circa 1925. This graphical construction technique

required t he estima tion of a rotor deflection curve, followed by the computa tion

of relat ive kinetic and potentia l energy. By equa ting t he kinetic to the potentia l

energy, a first critical speed was approximated. This technique assumed rigid

bear ings, gyroscopic effects w ere ignored, affects of coupling w eights could not be

determined, and higher order modes could not be successfully addressed.

In 1944 Myklestad1 published a pa per on calculating na tura l modes of a ir-

plane w ings a nd other beam s. Tha t w a s followed in 1945 by P rohl2

and his paperon critical speeds of flexible rotors. By 1954, Prohl and Myklestad init iated

development of calculation techniques for lateral critical speeds. This was fol-

lowed by the combined work of Holzer, Prohl and Myklestad for t orsiona l critical

speeds. In both cases, the t ra nsfer ma trix meth od wa s a pplied, and t his provided

a significant improvement in the ability to predict rotor resonant behavior.

Although the required matrix calculations are quite complex, the final results

reflect the sophistication of the model. Evolution of these techniques have pro-

gressed through various stages. During the 1980s, progress has been closely

associated with the development of smaller and faster computers that can ade-

qua tely process the ma trix calculations.

The undamped critical speed program (CRITSPD) used in this text was

developed by Edgar J. Gunter 3

.

In the formulation of this program, the transfer

matrix is divided into a point matrix containing the mass, inertia, and bearingpropert ies, plus a ma ssless field ma trix conta ining th e sha ft propert ies. The use

of a m a ssless field tra nsfer mat rix for sha ft properties has been used by Lund

4

in

rotor sta bility a nd unba lance computer codes, a nd it is a lso described by Thom-

son

5

. Although the ma ssless field ma tr ix is a considera ble improvement over the

hyperbolic continuum formulation, it suffers from numerical difficulties when

large numbers of stations are employed. Gunter

incorporated a unique auto-

ma tic scaling procedure to minimize the err ors, a nd t o allow t he successful mod-

1 N.O. Myklestad, “A New Method of Calculating Natural Modes of Uncoupled Bending Vibra-tion of Airplan e Wings a nd Ot her Types of Bea ms,” Jour nal of the Aeronau ti cal Sciences, Vol. 11, No.2 (April 1944), pp. 153-162.

2 M.A. Prohl, “A General Method for Calculating Critical Speeds of Flexible Rotors,” Journal of

Appl ied Mechan ics, Vol. 12, Tra nsa ctions of t he ASM E, Vol. 67 (Septem ber 1945), pp. A142-148.

3 E. J. G unter a nd C. Ga reth G aston, “CRITSP D-P C, Version 1.02,” Computer program in MS -DO S ® b y Rodyn Vibrat i on, Inc.

, Cha rlottesville, Virginia , August , 1987.

4 J.W. Lund, “Modal Response of a Flexible Rotor in Fluid Film B earin gs,” Transact ions Am eri -

can Society of M echani cal E ngin eers

, P a per No. 73-DE T-98 (1973).

5 Willia m T. Thom son , Theory of Vibrat ion wi th Appl icat ions

, 4th Edition, Prentice-Hall, Engle-wood C liffs, New J ersey, 1993.



202 Chapter-5

eling of 100 st a tion rotors on desktop computers.

The CRITSPD program includes the primary effects of rotor mass, shaft

and bearing flexibility, plus transverse and polar inertia. In addition, the

undamped analysis can incorporate synchronous or non-synchronous gyroscopiceffects, shear deformation, plus variable bearing and seal stiffness. Couplings,

impellers, thrust disks, and shaft spacers can be included by several different

modeling schemes. Hollow shafts plus variable material densities, and various

bounda ry conditions for each end of a rotor a re a ccommoda ted. It is possible to

comput e synchronous critica l speeds, pla na r m odes, plus order tr a cking. The pro-

gra m calculat es, a nd compar es tota l kinetic and potentia l energy for each mode,

plus the undamped critical speeds, and associated rotor mode shapes. In addi-

tion, the stra in energy distribution, defl ection, slope, moment, and shear a t ea ch

sta tion is computed a nd presented. Gra phica l outputs include the rotor cross sec-

tion, a summa ry mode shape diagra m for all criticals, and a mode sha pe for each

individual critical speed, as illustrated in Fig. 5-1.

Although undamped critical speed analysis can provide significant insight

into the beha vior of the m a chinery, it does ha ve inherent limita tions. For exa m-ple, it does not include the effects of forces such as mass unbalance or internal

synchronous mechanisms such as shaft bows. This analysis does not consider

damping, or cross-coupled influences from bearings, seals, or aerodynamics.

Undamped critical speed calculations are a simplification of the system mathe-

matical model, which in turn is a further simplification of the real mechanical

system. The parameters neglected in an undamped analysis can alter system

resonances. These parameters can control the amplitudes at the resonant fre-

quencies, and they may be responsible for instabilities. Hence, the results from

a n unda mped ana lysis must be considered in th e proper context.

One of the main utilizations for undamped critical speed calculations

resides in the ability to quickly compare the change in natural frequencies as a

function of support stiffness. As mentioned in chapter 3, the undamped critical

speeds may be computed for a large range of stiffness values, and the resultsplott ed a s a fam ily of curves. This ty pe of summa ry plot is normally referred t o

as an undamped critical speed map. For instance, Fig. 5-2 depicts this type of

plot for a 22,500 pound gas t urbine rotor. Ca lculat ions were performed a t 1, 2,

Fig. 5–1 UndampedSteam Turbine ModeShape Output FromCRITSPD Program




and 5 intervals to allow for even spacing of points on the logarithmic stiffness

scale. The stiffn ess calcula tion ra nge began a t 100,000 Pounds/Inch t ha t is cer-

ta inly less tha n a ny ma chine stiffness, a nd it extends t o 50,000,000 P ounds/Inch

tha t is greater tha n a ny potential ma chine element st i f fness.

The fi rst fi ve critica l speeds for each stiffness w ere plotted, and the points

connected for each natural frequency. In this manner, a log-log plot of stiffness

versus each critical speed is produced. The fi rst tw o modes reveal a var iat ion of

critical speeds with st iffness. I t is reasona ble to conclude tha t t hese are bearing

dependent m odes. This observa tion is confi rmed by the deta iled calculat ions t ha tshow the majority of the strain energy contained within the bearings. For the

higher order criticals, and t he stiffer portion of the 1st a nd 2nd modes, the na tu-

ral frequencies display minimal variation with support stiffness. These condi-

tions are indicative of resonant modes that are primarily controlled by shaft

stiffness. This is important information, since in the first scenario, bearing

changes could alter the rotor critical speed(s). This type of mechanical change is

reasonably inexpensive to perform. However, in the second situation, shaft modi-

fi cat ions w ould be required to change the critical speeds, a nd th is type modifi ca-

tion can be expensive a s w ell as technica lly complica ted.

The unda mped critical speed map is a lso used t o exam ine the relat ionship

between the calculated resonant frequencies, and the operating speed range as

shown in F ig. 5-3. In t his dia gra m, the rotor support stiffness is shown for thr ee

different condit ions. First, t he minim um or s oft condition of 500,000 Pounds perInch is shown. S econd, the horizontal bearing st iffness K

xx

is plotted, follow ed by

the t hird line of vertical bearing st iffness K

yy

. The support stiffness values help

to defin e the potential opera ting r a nge of th e ma chine. This informa tion a llow s

Fig. 5–2 Undamped Critical Speed Map For Single Shaft Gas Turbine

200

1,000

10,000

20,000

100,000 1,000,000 10,000,000 50,000,000

F r e q u e n c y ( C y c l e s / M i n u t e )

Stiffness (Pounds/Inch)

1st

2nd

3rd

4th

5th

Translational

Pivotal

Rotor Bending



204 Chapter-5

further quantification of the anticipated turbine characteristics.

The computation of natural resonant frequencies must also be compared

wit h the potent ial excita tion frequencies. I t is perfectly understa nda ble tha t na t-

ural frequencies will remain essentially dormant until some type of excitation

coincides wit h th ese crit ical speed(s). The applied excita tion ma y be a br oad ba nd

forcing function, such as a steam turbine subjected to a slug of water. The excita-

tion may also be a discrete frequency such as rotational speed mass unbalance

(1X), or an excitation due to specific geometry within the machine (e.g., pumpvan e pa ssing activity).

This type of evaluation is performed on an interference plot that is com-

monly referred to as a Ca mpbell diagra m. Fig. 5-4 depicts such a diagr a m ba sed

upon the previous critical speed map. In this diagram, the natural frequencies

are plotted on the vertical axis, and the excitation frequencies are shown along

the horizonta l a xis. For inst a nce, the fi ve critical speeds at normal opera ting con-

ditions were extracted from the critical speed map, and they are shown as hori-

zontal lines in the Campbell Diagram.

Excita tions at running speed 1X unbala nce, 2X misalignment, plus 5X, and

10X excita tions a re presented a s t he sla nted lines. The intersection betw een t he

horizonta l na tur al frequency lines, and t he slant ed excita tion lines represents a

potentia l case for the a ppea ra nce of th e specifi c resona nce. Nat ura lly, this t ype of

diagra m is often expa nded to include higher frequency excita tions, a nd other res-ona nt frequencies such a s str uctura l or torsiona l critica l speeds.

The number of resonant frequencies that exist on most large machinery

trains can be staggering. This is particularly true for bladed machines such as

Fig. 5–3 Undamped Critical Speed Map For A Single Shaft Gas Turbine With PrincipalBearing Stiffness Curves And Normal Operating Speed Range Identified

200

1,000

10,000

20,000

100,000 1,000,000 10,000,000 50,000,000

F r e q u e n c y ( C y c l e s / M i n u t e )

Stiffness (Pounds/Inch)

1st

2nd

3rd

4th

5th

OperatingSpeedRange

Kmin

Kxx

Kyy

Bearing Stiffness Curves

5th

4th3rd2nd

1st




steam or gas turbines where the individual blades and the segmented groups of

blades exhibit a var iety of ta ngentia l and a xial modes of vibrat ion. Other compli-

cat ions, such as separa te a nd distinct horizonta l and vert ica l rotor bala nce reso-

na nces, disk or impeller r esona nces, plus externa l resonances ca n subst a ntia lly

increa se the n umber of potentia l resonant frequencies.

Furthermore, when the sources of excitation are examined, the problem

becomes even more complicated. For instance, harmonics of the fundamental

excitations may be generated within the machine. In addition, specific frequen-

cies ma y int eract to form d istinct beat or m odulat ion frequencies. When a ll of th e

possible excitation frequencies are considered, the potential for exciting the

expa nded group of resonances increases dra ma tically.

In essence, an interference diagram such as the Campbell plot of Fig. 5-4

may become congested with all of the inherent excitations and resonances. I t

might even be concluded that the machinery cannot be operated at any reason-

able speed due to the coincidence of excitations and natural frequencies. Obvi-

ously, this is not an acceptable conclusion, and it is not representative of the

varied ar ray of operating ma chinery t ra ins.

In order to address this dilemma, it is suggested that machinery behavior

be exa mined in t wo different categories. The fi rst category w ould consist of the

major rotor balance resonances (lateral critical speeds), and the potential low

frequency excitations. This part of the analysis follows the scheme presented inthe C a mpbell plot of Fig. 5-4, but a dditional deta il is necessary to determine the

severity of the interference points.

For example, the rotor model should be expanded to include a forced

response analysis, as discussed later in this chapter. By varying the definable

Fig. 5–4 Campbell Diagram For Single Shaft Gas Turbine At Design Stiffness Values

0

2,000

4,000

6,000

8,000

10,000

12,000

10,000 20,000 30,000 40,000 50,000

N a t u r a l F r e q u e n c y ( C y c l e s / M i n u t e )

Excitation Frequency (Cycles/Minute)

1st

2nd

3rd

4th

5th

10X

5X

2X

1X

N a t u r a l F r e q u e n c i e s

E x c i t a t i o n s

Translational - 1st

Pivotal - 2nd

Rotor Bending - 3rd

4th

5th

1X

10X

5X2X



206 Chapter-5

excitations (e.g., rotor bow, unbalance at various rotor locations, disk skew, etc.),

the ma chinery dia gnostician should be able to eva luat e the vibrat ion severity for

a nticipa ted forcing functions. This a pproa ch will identify t he ma jor or signifi cant

resonances, and allow other interference points to be discounted.The second category of machinery behavior considers the higher frequency

chara cteristics a ssociat ed wit h t urbine blades or compressor w heels. In th is com-

plex mechanical domain, the traditional two-dimensional Campbell diagram

should be expanded into a three-dimensional SAFE

6

diagram (acronym for

Singh’s Advanced F requency Eva luat ion). This a na lytical tool combines the tw o-

dimensional Campbell plot with a third dimension of nodal diameters or mode

sha pes. The th ree-dimensiona l intersection of nat ura l resonan t fr equencies, exci-

ta tion frequencies, and n oda l diameters a re then used to identify potentia l reso-

nant conditions. The inclusion of the blade mode shape allows the diagnostician

to ignore the majority of the interfere points, and identify the frequencies and

modes of greatest potentia l vibrat ion.

Case History 10: Mode Shapes For Turbine Generator Set

Undamped critical speed calculations are relatively easy to setup and run.

As noted, they do not include synchronous forcing functions, and the support

stiffness characteristics represent a simple condition. However, these calcula-

tions can provide significa nt visibility int o the behavior of rotat ing syst ems.

For rotors supported between bearings, and for overhung assemblies, the

mode shapes discussed in cha pter 3 ma ke intuitive sense. Armed w ith the knowl-

edge of the general rotor configuration, and the relative bearing stiffness, the

anticipated mode shapes may be estimated. Even though the frequencies may

not be calculated, the mode shapes for simple systems can be deduced. However,

for more complicated systems, the shaft mode shapes may not be obvious.

For exam ple, consider th e tu rbine genera tor s et d epicted in Fig. 5-5. This isa thr ee bearing m a chine tha t r uns a t 3,600 RPM. The combined weight for both

rotors is 21,400 pounds, and a solid coupling is used bet w een the t w o sha fts. The

turbine is an extraction unit, with a surface condenser at the exhaust. The syn-

chronous generat or has collector rings mounted a t t he outboard end, and a sepa -

6 Mura ri P. Singh an d others, “SAFE D iagr am - A Design a nd Reliabilit y Tool for Turbine B lad-ing,” Proceedi ngs of th e Seventeenth Tur bomachin ery Symposium

, Turbomachinery Laboratory,Texa s A&M U niver sity, Colleg e St at ion, Texa s (November 1988), pp. 93-101.

Fig. 5–5 Rotor Arrangement For Three Bearing Turbine Generator Set




rate exciter. The entire machinery train is mounted on a mezzanine deck, and

the structure plus the bearing supports are compliant with stiffness values

a pproaching a minim um va lue of 600,000 Pound/Inch.

Over the operating history of this machinery train various problems haveoccurred. The majority of the difficulties have been traced to generator unbal-

ance problems, or high eccentricity at the solid coupling. Although successful

fi eld bala nce corrections ha ve been performed on both t he tur bine and the gener-

ator, the logic behind some of the weight corrections was not fully understood.

In a n effort to resolve some of the issues, the t ra in wa s retrofi tt ed with X-Y

proximity probes as shown in Fig. 5-5. During startup, these transducers

revealed critica l speeds t ha t were in direct contr a diction w ith h istorica l conclu-

sions. For example, the local personnel believed th a t t he T/G Set ha d a critica l

speed that began at 1,000 RPM, and lasted until well above 2,000 RPM. This

behavior wa s considered to be inconsistent w ith a ny expected response th rough a

single critical speed. In addition, the three planes of installed X-Y proximity

probes provided addit iona l contr a sting informa tion.

In order to addr ess these anomalies, the syst em wa s eventua lly subjected to

an undamped critical speed analysis. The computed mode shapes for the first

three critical speeds are presented in Fig. 5-6. The first mode was calculated to

Fig. 5–6 Undamped ShaftMode Shapes For The FirstThree Critical Speeds On AThree Bearing Turbine Gen-erator Set



208 Chapter-5

be 1,210 RP M, which a greed wit h th e measu red va lue of 1,250 RP M. This critica l

was always visible by the proximity probes installed at the #2 and #3 Bearings.

The probes at the #1 bearing are close to a nodal point, and this translational

resona nce is not part icular ly visible at the front t urbine bea ring.The measured second critical speed occurred at 1,650 RPM. This value is

virtua lly identica l wit h t he calcula ted second critical of 1,660 RP M. In a ddition,

the proximity probes mounted at bearings #2 and #3 alw a ys displayed a n out of

phase behavior. This wa s not fully understood until the a na lysis wa s performed,

a nd t he calculated pivota l mode sha pes produced. From t his second critical mode

sha pe, it is clear tha t the probes a t #2 a nd #3 bearings a re on opposite sides of a

shaft node, and a phase reversal must exist. Again, the analytical calculations

a re consistent wit h the fi eld sha ft vibrat ion response measurements.

Fina lly, eccentr icity problems a t the coupling a lwa ys caus ed high vibrat ion

amplitudes at slow speeds, and at speeds just below 3,600 RPM. The reason for

this behavior is evident from the calculated mode shape plots where a large

deflection in t he coupling a rea is visible for t he fi rst critica l a t 1,210 RP M, plus

the t hird critical speed at a computed va lue of 3,400 RPM. This resonance wa s

fully corroborated by the proximity probe transient data that exhibited a reso-

na nce a t 3,440 RP M.

Additional supportin g evidence concerning t he behavior of this t urbine gen-

erat or set w a s documented w hen bala nce weights w ere placed on each end of the

genera tor. The a nticipa ted modal r esponse with t he bala nce weights w a s consis-

tent with the shaft mode shapes described in Fig. 5-6. There are other correla-

t ions tha t may be extra cted from t his dat a set . However, the main point is tha t

undamped critical speed calculations provide an analytical tool that is directly

a pplica ble to existing ma chinery. In ma ny sit ua tions, it can provide the diagnos-

tician with significantly more insight into the dynamic behavior of the rotating

ma chinery, and it a lso provides valua ble moda l informa tion for fi eld bala ncing.

Case History 11: Torsional Analysis of Power Turbine and Pump

The same general techniques used for undamped lateral calculations may

a lso be applied t owar ds t he computa tion of undamped torsiona l frequencies and

mode sha pes. As sta ted in chapter 2, the ba sic equa tions for la tera l an d torsiona l

chara cteristics a re similar. However, the calculat ion scheme, and interpreta tions

of results are somewhat different. In a lateral system, stiffness is expressed as

Pounds per Inch, and ma ss carr ies the units of Pound-Seconds

2

per I nch. With in

a torsional analysis, the torsional stiffness is a torque per unit angle, with com-

mon units of Inch-Pounds per Radian. Polar inertia carries the units of Pound-

Inch-Seconds

2

per Ra dian , and t h is is analogous to mass in a latera l ana lysis .

Within a latera l ana lysis , the rotor sta t ions w ith ma ximum motion a re sig-

nificant (as in the previous case history). During a torsional analysis, the nodalpoints are meaningful since stress reversals occur across each torsional node.

Furthermore, in a lateral analysis, the mass and stiffness properties are typi-

cally confined to a single rotor, or rotor system with hard couplings. The influ-




ence of the flexible couplings between rotors are seldom included in a lateral

analysis. During a torsional analysis the inertia and torsional stiffness proper-

ties of the entire train are considered. Since inertia elements are essentially

fi xed, the mechanical element used for a ltera tion of torsiona l na tura l frequenciesoften revert s t o the coupling spool piece betw een ma chines.

For exa mple, consider t he ma ss elas tic da ta shown in Fig. 5-7. This dia gra m

describes the lumped inertia a t eight ma jor ma chinery sections, plus th e seven

interconnecting torsiona l springs. This da ta wa s ba sed upon OE M specified va l-

ues, plus independent calcula tions of cylindrical sections w ith the equa tions pre-

sented in chapter 3 of this text. The actual machinery in Fig. 5-7 represents a

processed wa ter injection pump tha t is gas t urbine driven via a str a ight th rough

gear box. The gas turbine is a two shaft unit, with no mechanical connection

between the gas generator and the power turbine. Hence, the drive end of this

tr a in begins with t he pow er turbine wheel.

Considering the relat ive size and ma ss of the ma chine elements, it is under-

sta ndable that a ma jori ty o f the system inertia is conta ined within t h is turbine

drive wheel. As noted, a gear box wa s a tt a ched to the turbine output sha ft. Since

turbine and pump speeds were compatible, the gear box consisted of a single

drive thr ough sh a ft element, w ith no speed change. The horizonta lly split pump

was originally a six stage unit that was de-staged to five stages to meet process

dema nds. The expected opera tin g speed ra nge for this pum p var ied from 5,180 to

6,800 RPM .

Va rious sta nda rds (e.g. , AP I 617) recommend a 10% separa tion betw eena ny torsiona l resonance an d t he operat ing speed ra nge. Applica tion of this crite-

ria expa nds t he above speed ra nge to include a m inimum t orsional frequency of

4,660 RPM, combined with a maximum of 7,480 RPM. Before performing any

extensive calcula tions, it w ould be reasonable to estimat e the fi rst torsiona l fre-

Fig. 5–7 Mass Elastic Data For Power Turbine, Drive Through Gear Box, And Pump



210 Chapter-5

quency based upon the a vaila ble ma ss elast ic da ta . For instan ce, the major iner-

tia occurs at the gas turbine wheel (40.13 Pound-Inch-Sec.

2

/Ra dia n), a nd t he

ma in coupling st iffness (6.67x10

6

Inch-Pound /Ra dia n) w ould norma lly be var ied

to contr ol the torsional resonance frequency. If t hese values a re placed in equa -

tion (2-103), a first torsional critical speed may be estimated as follows:

This est ima ted s peed of 3,890 CP M does fa ll below th e 4,660 to 7,480 RP M

exclusion range, but the differential is uncomfortably small. Obviously, a full set

of undamped torsional resonance calculations are required to obtain sufficient

precision in the torsional natural frequency calculations. The most significant

results of these computations are in Fig. 5-8. At the top of this diagram, the tor-sional mode shape at the calculated first critical speed of 5,030 RPM is shown.

This frequency is much higher than the simple model estimate of 3,890 RPM. In

addition, the computed first critical speed falls well within the exclusion speed

range of 4,660 to 7,480 RPM. Clearly, this deviation demonstrates that a very

simple model ma y not properly represent th e actua l mecha nical syst em.

The second undamped torsional critical speed appears at 16,990 RPM, as

indicated at the bottom of Fig. 5-8. This frequency is considerably higher than

Fig. 5–8 Undamped First

and Second Torsional ModeShapes For Power Turbine,Drive Through Gear Ele-ment, and Coupled FiveStage Centrifugal Pump

F c t or

1

2π------

K t or

J mass

----------------×1

2π------

6.676

×10 Inch-Pound/Radian

40.13 Pound-Inch-Second2 /Radian

-----------------------------------------------------------------------------×≈ ≈

F c t or

1

2π------ 166 210,× 64.89

Cycles

Second---------------- 60

Seconds

Minute------------------×=≈ 3 890 CPM,=




the opera ting speed ran ge, a nd it is beyond excita tion by tw ice rotat iona l speed

oscillation (2 x 7,480= 14,960 CP M). Hence, th is t orsional r esonan ce should not

cause any distress. Although higher order torsional frequencies are not shown,

they should also be computed, and compar ed aga inst potential excita tions with aCa mpbell or a SAFE diagra m.

From this da ta , it is evident tha t t he ma jor problem resides wit h th e coinci-

dence of th e fi rst torsiona l critica l a nd t he previously described exclusion ra nge

(opera tin g speed ra nge ± 10%). This predicted r elat ionship is una ccepta ble, a nd

physica l chan ges must be implemented t o correct t he defi ciency. U nfortuna tely,

the ma chinery under discussion included existing equipment tha t w a s in the pro-

cess of re-configuration to accommodate new operating conditions. Although

changes in these existing rotors were feasible, the economic considerations voted

heavily a ga inst a ny signifi cant changes to the tur bine or pump rotors. The drive

through gear box shaft was identical to other units at the same facility. Hence,

there w a s relucta nce to cha nge this drive sha ft t o a one-of-a -kind a ssembly.

The last candida te for modificat ion w a s t he loa d coupling t o the pump. The

ava ilable couplings a ll conta ined hollow spool pieces w ith outer diam eters t ha t

varied between 4.16, and 5.20 Inches. The torsional stiffness values for these

couplings ranged from 6.67x10

6

to 7.00x10

6

In ch-P ound/Ra dia n. This coupling

stiffness range was previously judged unacceptable due to the frequency inter-

ference. Hence, a new coupling must be provided that is both torsionally softer,

and still able to transmit the torque with acceptable stress values.

After the examination of potential stiffness changes, a nominal value of

2.00x10

6

Inch-Pound/Ra dia n w a s selected a s a ccepta ble. The complete a rr a y of

undamped torsional calculations were repeated, and the results summarized in

Ta ble 5-1. With t his reduct ion in torsional st iffness, the calculat ed unda mped

first critical was reduced from 5,030 to 4,160 RPM. This is approximately 20%

below the minimum operating speed, and outside of the exclusion speed range.

The higher order torsional resonances were influenced by this reduction in loadcoupling st iffness, but the va ria tions were insignifi cant . More importa ntly, there

was minimal interference between higher order excitations (e.g. , pump vane

passing) and t he unda mped torsiona l criticals.

The fi na l mecha nical design by t he coupling ma nufa cturer included a solid

coupling spool piece with an overall length of 16.80 Inches, and an outer diame-

ter of 2.27 Inches. The flanges at each end of this spacer were 0.53 Inches wide,

a nd 8.25 Inches in dia meter. I t is a lwa ys a good idea to check the torsiona l stiff-

ness provided by th e vendor. In t he va st ma jority of the cases, the stiffness pro-

Table 5–1 Comparison Of Undamped Torsional Critical Speeds For Two Different Couplings

Coupling TypeStiffness

(Inch-Lb/Rad

)

1st Mode

(RPM)

2nd Mode

(RPM)

3rd Mode

(RPM)

4th Mode

(RPM)

O r ig in a l H ol low Sp ool 6. 67 x 10

6

5,030 16,990 27,220 30,680

New Solid S pool 2.00 x 10

6

4,160 16,450 25,530 29,630



212 Chapter-5

vided by the coupling vendor is accurate. However, a quick check of this critical

parameter is always warranted. The torsional stiffness of this solid spool piece

can be closely approxima ted by computing t he stiffness of the ma in torque t ube,

and the mounting flanges. These values are summed in a reciprocal manner todetermine torsional stiffness of the spool piece. Since this is a solid assembly,

equa tion (3-66) may be used t o compute t he spool st iffness a s follow s:

Simila rly, the torsiona l stiffness of each end fla nge is ca lculat ed as:

These individual torsional stiffness values may now be combined in a recip-

roca l ma nner t o determine the overa ll or effective torsional st iffness of the entire

solid spool piece in t he follow ing m a nner:

From these calculations, it is clear that the spool piece torsional stiffness is

governed by t he center torque t ube. The end fl a nges a re very st iff, an d t he most

flexible member (center tube) controls the effective stiffness. This coupling tor-

sional stiffness value is consistent with the required 2.00x10

6

Inch-Pound/

Radian determined from the undamped analysis. Overall , this proved to be a

mecha nically a ccepta ble fi eld retrofit tha t performed w ith good reliability.

K t or spool

π G shear × D 4

×

32 L×-----------------------------------------

π 11.96

×10 Pounds/Inch2( )× 2.27 Inches( )

4×

32 16.80 2 0.53 Inches×–( )×-------------------------------------------------------------------------------------------------------------= =

K t or spool

992.666

×10

503.68----------------------------- 1.97

6×10 Inch-Pound/Radian= =

K t or f l a n g e

π G shear × D

4×

32 L×

-----------------------------------------π 11.9

6×10 Pounds/Inch

2( )× 8.25 Inches( )4

×

32 0.53 Inches( )×

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

K t or f l a n g e

173.196

×10

16.92----------------------------- 10.21

9×10 Inch-Pound/Radian= =

1

K t or ef f

-----------------1

K t or spool

----------------------1

K t or f l a n g e

------------------------1

K t or f l a n g e

------------------------+ +=

1

K t or ef f

-----------------1

1.976

×10-----------------------

1

10.219

×10--------------------------

1

10.219

×10--------------------------+ +=

or

K t or ef f

1.976

×10 Inch-Pound/Radian=



Stability and Damped Critical Speed Calculations 213

STABILITY AND DAMPED CRITICAL SPEED CALCULATIONS

The inclusion of dam ping into the an a lysis significa ntly expa nds t he useful-

ness of the analytical calculations. This manifests as improved correlationbetween the rotor dynamics computations, and the real machinery behavior. In

many ways, a damped analysis is s imilar to undamped calculat ions, with the

important inclusion of damping and cross-coupled stiffness from bearings, seals,

a nd a erodyna mic and/or fluid intera ctions. Most progra ms a llow the consider-

a tion of non sy mmetric bearing a nd support coefficients, plus a n output scheme

tha t displays representa tive vertical a nd horizonta l motion. This ty pe of a na lysis

also determines stability, and it provides a comparison of stability between

modes. Some programs, such as ROTSTB by Gunter 7 use a complex matrix

tra nsfer method, and other programs such a s DYROBE S by Chen, Gunter, and

Gunter8 are based upon finite element analysis (FEA) numerical methods. I t

should be noted that the matrix transfer method may skip roots that are closely

spaced, but this problem does not occur with FEA.Da mped ca lculat ions ma y be considered as somewha t of a pure ana lysis due

to the fact tha t t hese progra ms calculate a ll la tera l critical speeds of the mecha n-

ica l syst em, including a ll potentia l forw a rd, reverse, and m ixed modes. In a ddi-

tion, the damped analysis determines the stability characteristics of each mode.

This is quite significant since reverse modes and stability are not computed by

other methods. For a rotor mounted between bearings with small wheel diame-

ters, this feature may not be particularly important. However, for a rotor with

lar ge overhung w heels, the potentia l of various reverse modes is signifi cant , and

this behavior must be visible to the diagnostician. On the limitation side, this

type of program does not include external forces from mass unbalance, shaft

bows, or other external excitations.

Damped critical speed programs include multiple gyroscopic effects, shear

deformation, rotary inertia, plus a full set of eight bearing and support coeffi-cients a t defin ed speeds. Couplings, impellers, thr ust disks, shaft spacers, and

hollow shafts are accommodated. Hysteretic shaft damping may be included as

well as aerodynamic cross-coupling effects. These programs allow variable den-

sity of rotor materials, and various boundary conditions for the rotor. Forward,

backward, or mixed criticals are computed, plus the complex Eigenvalue for each

resona nce. The E igenva lues consist of real a nd imag inary portions, and the nor-

ma l output units a re Ra dians per S econd. The real portion of the Eigenvalue is

the modal damping or growth factor. The imag inary portion of the Eigenvalue

represents the da mped na tura l frequency in Ra dian s/Second. Multiplication by

30/π convert s th is va lue from Rad ia ns/Second to RP M a s shown in equa tion (5-2):

7

Edga r J . Gunt er, “ROTSTB, Sta bility P rogram by Complex Matrix Transfer Method - HP Ver-sion 3.3,” Computer Program in Hewlett Packard Basic by Rodyn Vibrat i on, Inc., C har lot tesville,Virginia, March, 1989, modified by Robert C. Eisenmann, Machinery Diagnostics, Inc., Minden,Nevad a, 1992.

8 W.J. Chen, E . J . G unter, and W. E. G unter, “DYROBE S, Dyna mics of Rotor B earing S ystems,Version 4.21,” Comput er Pr ogram in MS -DOS ® by Rodyn Vibrat i on, Inc., Charlottesville, Virginia,1995.



214 Chapter-5

(5-2)

This cla ss of ana lytical program a lso computes the L og Decr ement for eachresonance to allow an evaluation of rotor stability. The log decrement is deter-

mined by multiplying -2π times the ratio of real to imaginary portions of the

Eigenvalue a s follows:

(5-3)

A positive log decrement identifies a stable system, whereas a negative

value signifi es an unsta ble mode. Since the log decrement is a direct indica tion of

the damping and the decay rate through a resonance, it may also be used to

determine the a mplificat ion fa ctor of the resonance. Dividing π by t he log decre-

ment w ill result in the amplifica tion factor Q a s shown in the next expression:

(5-4)

On machines with split criticals, each individual resonance will be calcu-

lat ed, and t he dominant direction will be identifi a ble from the mode shape plots.

For insta nce, consider F ig. 5-9 tha t describes t he non-dimensiona l da mped verti-

cal and horizontal mode shapes (Eigenvectors) of a gas turbine rotor. Based on

the relative amplitudes of the vertical versus the horizontal mode shapes, it is

self-evident t ha t the described resonance is predomina ntly a vertical mode. On

some programs, two levels of normalization are provided for each non-dimen-

sional mode shape. Within these programs, the peak displacement for both

orthogonal directions is always 1.0, and there is no visibility of any dominant

motion in either th e vertical or t he horizonta l directions. The da ta presented in

Fig. 5-9 contains only one level of normalization, and the dominant direction ofthe computed motion is maintained. From this diagram, it is noted that the

Fig. 5–9 Damped GasTurbine Mode Shape

Damped C r i t i c a l Speed 30 I m a g ×

π----------------------------=

L og Decrement δ 2π– Real ×I m a g

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

Am pl i f i ca t i on Q πL og D ec ------------------------ π

δ---= = =




Eigenva lue for mode wa s calculated t o be:

Eigenvalue = -10.34 + 235.5 Ra dia ns/Second

The real portion of this E igenvalue is the fi rst term of -10.34, a nd t he ima g-

inar y par t is t he second t erm, or + 235.5. B a sed on th ese va lues, the last thr ee

equations may be applied to determine the natural frequency in RPM, the log

decrement for this mode, and the associated amplification factor. The damped

critica l speed is determ ined by convert ing unit s w ith (5-2) a s follows:

This value is identical to the damped frequency on the mode shape plot Fig.

5-9. Next, consider th e calcula tion of th e log decrement w ith equa tion (5-3):

This positive log decrement indica tes a st a ble mode, a nd t he va lue of the log

decrement may now be used to determine the amplification factor of the reso-

nance with equation (5-4) in the following manner:

The influence of bearing clearance upon the damped critical speeds is

always a question to be addressed. By calculating bearing oil film coefficients

under various clearance conditions, and combining this information with the

support coefficients, the anticipated machinery response may be computed. For

inst a nce, consider t he da ta presented in Ta ble 5-2.

These dam ped critical speed calculations w ere performed at avera ge, mini-

mum, and maximum allowable bearing clearances. These computations were

performed for a 40,000 HP ga s tur bine at 5,100 RP M. They reveal t ha t t he bear-ing dependent 1st a nd 2nd modes are moderat ely influenced, but th e frequency

of the 3rd critical is insensitive to bearing clearance variations. Generally,

changes in journal bearing clearances are not a major factor in the resonant

behavior of these machines. However, for long-term operation, it is always desir-

Table 5–2 Gas Turbine Damped Critical Speeds Versus Bearing Clearance

Journal BearingClearance

1st ModeTranslational

2nd ModePivotal

3rd ModeBending

Minimum 1,146 RP M 1,877 RP M 5,732 RP M

Avera ge 1,104 RP M 1,826 RP M 5,727 RP M

Ma ximum 1,073 RP M 1,760 RP M 5,719 RP M

Damped C r i t i c a l Speed 30 I m a g ×

π----------------------------

30 235.5×π

------------------------- 2 249 RPM,= = =

L og Decrement

2π– Real ×I m a g ------------------------------

2π– 10.34–( )×235.5------------------------------------- +0.276= = =

Amp l i f i c a t i o n π

L og D ec ------------------------

π0.276------------- 11.4= = =



216 Chapter-5

able to begin with the minimum bearing clearances to allow room for babbitt

a tt rition during t he run. Another perspective of the ma chine chara cteristics ma y

be obtained by examining the variations in log decrement of each mode at each

bear ing clea ra nce condition. For this ga s tur bine, Ta ble 5-3 summa rizes th esesta bility pa ra meters for the fi rst t hree critical speeds.

If t he log decrement is positive, vibra tion am plitudes w ill decay w ith t ime.Conversely, if the log decrement carries a negat ive sign, then t he mode is unst a -

ble, a nd a mplitud es will increa se wit h t ime. With in Ta ble 5-3, a ll values a re pos-

itive. Tha t indicat es sta ble modes wit hin t he opera ting speed doma in of the gas

turbine. The magnitude of the log decrement describes the rate of oscillation

decay. Specifically, a large positive log decrement delineates a well damped sys-

tem that will rapidly attenuate vibratory motion. A well damped resonance will

display a low amplification factor, and will persist over a broader frequency

ra nge. On th e other ha nd, a sm a ll log decrement identifi es a poorly da mped reso-

nance, with a h igher a mplificat ion factor, and a smaller bandwidth .

With respect to t he log decrement s present ed in Ta ble 5-3, it is clear th a t

the second pivotal critical speed exhibits the largest values. As such, this second

mode would be the most difficult to excite, and the resultant motion would be

quickly suppressed (i.e., damped out). The first translational mode has some-

what lower log decrement values. This critical would be slightly easier to excite,

a nd t he resultan t m otion w ould continue for a longer time. Fina lly, the rotor fir st

bending mode (3rd critical) has the lowest log decrement. This resonance is the

easiest t o excite, and the motion would decay a t a slower ra te. A low log decre-

ment is indicat ive of a high a mplificat ion fa ctor a t t he resona nce. This ma nifests

as rapidly increasing vibrat ion ampli tudes with minimal phase change as the

skirt of t he resonance is a pproached.

Table 5–3 Gas Turbine Log Decrement Versus Bearing Clearance

Journal BearingClearance


2nd ModePivotal

3rd ModeBending

Minimum 0.418 0.497 0.041

Avera ge 0.415 0.519 0.044

Ma ximum 0.338 0.496 0.046




Case History 12: Complex Rotor Damped Analysis

This par ticular r otor consists of a n overhung h ot ga s expander w heel, a pair

of midspan compressor wheels, and three stages of overhung steam turbinewheels9 a s described in F ig. 5-10. A series of axia l th rough bolts a re used t o con-

nect the expander stub shaft through the compressor wheels, and into the tur-

bine stub shaft. This type of assembly is similar to many gas turbine rotors.

However, in this machine, the rotor must be built concurrently with the inner

casing. Specifi cally, the horizonta lly split interna l bundle is assembled with the

tit a nium-a luminum compressor w heels, stub sha fts, plus bea rings a nd sea ls. The

end casings a re a tt a ched, the expa nder w heel is bolted into position, a nd t he tur-

bine sta ges are at ta ched wit h an other set of thr ough bolts.

The eight rotor segments a re joined wit h C urvic® couplings, identifi ed a s

#1 through #7 on Fig. 5-10. Even with properly ground and tight fitting Cur-

vics® , there is potent ia l for rela tive movement of rotor elements. Alth ough each

of the rotor segments a re component ba lanced, any minor shift betw een elements

will produce a synchronous force. Since this unit operates at 18,500 RPM, a few gra ms of unba lance, or a Mil or t w o of eccentr icity will result in excessive shaft

vibration, and strong potential for machine damage. Furthermore, the distribu-

tion of operating temperatures noted on Fig. 5-10 reveals the complexity of the

therma l effects tha t must be tolerat ed by th is unit. The 1,250°F expa nder inlet is

followed by compressor discharge temperatures in excess of 430°F. The steam

tur bine operat es with a 700°F inlet, and a 160°F exha ust.

By any definition, this must be considered as a complicated and difficult

rotor. On the positive side, this machine is a compact design that yields a high

therma l efficiency. Hence, when t he unit is properly a ssembled, an d ba lan ced, it

is very cost -effective t o opera te.

9 Robert C. Eisenman n, “Some realit ies of field bala ncing,” Orbi t , Vol.18, No. 2 (J un e 1997), pp 12-17.

Fig. 5–10 Combined Expander-Air Compressor-Steam Turbine Rotor Configuration

Balance Plane #120 Axial Holes

Balance Plane #220 Radial Holes

Balance Plane #320 Radial Holes

Balance Plane #430 Axial Holes

Expander

Wheel

Expander

Stub Shaft

2nd Stage

Compressor

1st Stage

Compressor

Turbine

Stub Shaft

1st Stage Turbine

C u r v i c # 1

C u r v i c # 2

C u r v i c # 3

C u r v i c # 4

Curvic #5

Curvic #6

Curvic #7

Journal 4.000" Ø5 Pads - LOP

6 Mil Diam. Clearance445# Static Load

ThrustFaces

Steam Inlet

700 °F

Exhaust

160 °F Ambient

Air Suction

1 , 2

5 0 ° F I n l e t

220 °F Suction Exhaust

700 °F

Rotation and AngularCoordinates Viewedfrom the Expander

Rotor Weight = 910 #Rotor Length = 77.00"

Bearing Centers = 45.68"

Journal 4.500" Ø5 Pads - LOP

7 Mil Diam. Clearance465# Static Load

2nd Stage Turbine

3rd Stage Turbine

4 3 0 ° F D i s c h a r g e

4 5 0 ° F D i s c h a r g e

60°30°

2X

2Y

60°30°

1Y

1X

CCW

Rotation



218 Chapter-5

A double overhung rotor wit h a n a pprecia ble midspan ma ss ha s th e poten-

tial for multiple resonances with either forward or reverse modes. In order to

better understa nd the behavior of this ma chine, va rious historica l dat a set s were

reviewed. It was noted that reverse orbits appeared around 7,000, and 17,000RP M. Field bala ncing activities on t his ma chine were generally successful when

a two step correction was used. The first step consisted of an intermediate bal-

ance based on transient data as the machine passed through 14,000 RPM. This

initial ba lance wa s a ccomplished using t he outboard planes #1 and #4. This wa s

follow ed by a trim a t 18,500 RP M on the inboard pla nes #2 and #3 located next t o

the compressor wheels. I t was evident that if the rotor was not adequately bal-

a nced a t 14,000 RP M, it probably would not run a t 18,500 RP M.

Further examination of h istorical data revealed that vibrat ion severi ty

changed in accordance with the machinery operational state. For instance, the

peak vibration amplitudes occur at a rotor critical that appears between 7,600

a nd 8,100 RP M. This resonance displa ys t he following va ria ble cha ra cteristics:

r Cold St a rt up to 14,500 RP M — Pea k Response of 2.0 to 5.0 Mils,p-p

r Wa rm C oast down from 14,500 RP M — Pea k Response of 4.0 to 5.0 Mils,p-p

r Hot C ra shdow n from 18,500 RP M — Pea k Response of 6.0 to 8.0 Mils,p-p

These amplitude varia tions ar e combined wit h changes in th e amplifica tion fac-

tor t hrough th e resona nce (potentia l change in d a mping). Clearly, this informa -

tion must be supplemented by an examination of the variable speed vibration

da ta — plus an underst an ding of the rotor critical speeds, and mode shapes.

Fig. 5–11 Bode Plot OfShaft Y-Axis ProximityProbes During A TypicalMachine Startup

450

400

350

300

250

200

4,000 6,000 8,000 10,000 12,000 14,000 16,000 18,000

P h a s e L a g ( D

e g r e e s )

40

90

Expander Bearing

Probe #1Y

Turbine Bearing

Probe #2Y

0.0

1.0

2.0

3.0

4.0

5.0

4,000 6,000 8,000 10,000 12,000 14,000 16,000 18,000

D i s p l a c e m e n t ( M i l s , p - p )

Rotational Speed Revolut ions/Minute)

Turbine Bearing

Probe #2Y

Expander Bearing

Probe #1Y

Mode @

7,800 RPM

Process Hold @

14,500 RPM

Full Speed @

18,500 RPM




A typical startup Bode plot of the Y-Axis response from each measurement

plan e is shown in Fig. 5-11. Both plots a re corrected for slow r oll run out a t 1,000

RP M, and the resulta nt da ta is representa t ive of the true dynam ic shaf t motion

a t ea ch latera l measurement pla ne. The major resona nce a ppea rs a t 7,800 RP M.A process hold point occurs at 14,500 RPM, and the unit displays various ampli-

tude a nd pha se excursions at this speed. Some of this behavior is logically due t o

the hea ting of the rotor and ca sing, plus varia tions in settle out of the operat ing

syst em (i.e. , pressures, tempera tur es, fl ow ra tes, and molecula r weights).

The Bode also exhibits additional vector changes between 14,500 and

18,500 RP M. Some of th ese cha nges a re due to t he infl uence of a ba ckwa rd m ode

a round 17,000 RPM. Oth er changes a ppea r a s t he ma chine a pproaches the nor-

mal operating speed of 18,500 RPM. This higher speed data is difficult to fully

comprehend in the Bode plot, but it becomes more definitive when replotted in

th e pola r forma t of Fig. 5-12.

The point of ma jor int erest on Fig. 5-12 is th a t a t full speed, the tur bine end

sha ft is moving towa rds t he 9 o’clock direction, a nd t he sha ft a t t he expander is

heading towards 4 o’clock. This behavior indicates a couple, and the presence of

some type of pivotal mode occurring at a frequency above the normal running

speed of 18,500 RP M. In ma ny ca ses, this ty pe of response w ould not be unus ua l.

However, for this unit, the machinery files had no indication of a resonance

around the operating speed. Due to the measured response of midspan balancew eights (pla nes #2 & #3) a t 18,500 RPM, it wa s clear t ha t t he vibra tion dat a wa s

correct. This also implies that the historical undamped mode shapes were not

fully representa tive of actua l ma chinery behavior.

As previously noted, there are only two lateral vibration measurement

Fig. 5–12 Polar Plot OfShaft Y-Axis ProximityProbes During A TypicalMachine Startup

0.0 1.0 2.0 3.0 4.0 5.0

0°

30°

60°

90°

120°

150°

180°

210°

240°

270°

300°

330°

Expander Bearing

Probe #1Y

Turbine Bearing

Probe #2Y

Mode @

7,800 RPM

Displacement (Mils,p-p)

Turbine

E x p a n d e r

Probes

#1Y & 2Y

I n c r ea s ing S

peed an d An g l e s

C C W

R o t a

t i o n




normalized deflections at both bearings are quite small. This indicates minimal

motion of the journals within their respective bearings. With small relative

motion, the velocity is low, and bearing damping is minimal. This behavior is

refl ected in th e low 0.0753 log decrement for t his mode.The validity of the analytical model is supported by correlation of the com-

puted r esonan t fr equency of 7,840 RP M (from Fig. 5-13), wit h t he mea sured r es-

onance of 7,800 RPM (Fig. 5-11). It is also clear from Fig. 5-13, that the rotor

balance response at this resonance can be effectively controlled by corrections at

the modally effective end planes #1 and #4.

With increasing speed, the da mped ana lytical model reveals a nother back-

ward mode at 17,710 RPM. The vertical and horizontal shapes for this reverse

mode are presented in Fig. 5-14. This pivotal mode is often visible as reversed

orbits on the transient vibration data. Immediately above the normal operating

speed of 18,500 RPM, a damped mode was calculated at a frequency of 21,940

RP M a s in F ig. 5-15. This forwa rd m ode ha s t he sa me deflection char a cteristics

a s th e previously ba ckwa rd mode at 17,710 RP M. In both ca ses, the inboard ba l-

a nce planes #2 a nd #3 a re the m ost modally effective correction plan es for t his

speed doma in. It is concluded th a t weight corrections a djacent to t hese compres-

sor w heels (plan es #2 and #3) should be out of pha se. This is due t o the fa ct t ha t

a nodal point exists at the middle of the rotor. The validity of this conclusion was

field tested on the machine. The installation of a pair of weights at the middle

planes a t the sa me a ngle resulted in excessive vibra tion. However, a couple shot

proved to be smooth, a nd support ive of the a na lytical mode sha pe at high speed.

Fina lly, the existence of a pivotal resona nce at slightly a bove running speed

was previously noted on the polar plot, Fig. 5-12. The damped mode shape pre-

sented in Fig. 5-14 support s t his observa tion. Once a gain, t he vibra tion mea sure-ments, and the analytical tools are combined to explain the behavior of a

complex machine. Although the variable behavior through the main critical

speed a t 7,800 RP M is st ill not tota lly clear, it is specula ted t ha t a loosening or

relaxa tion of the segmented rotor occurs w ith elevat ed tempera tur es.

Fig. 5–14 CalculatedDamped Mode Shapes OfBackward Resonance AtNominally 17,710 RPM



222 Chapter-5

This work provided an improved understanding of the shaft response and

damped mode shapes, plus a better appreciation of the process influence. Armed

wit h this informa tion, the rotor was ba lanced at t he intermedia te speed by using

transient data acquired during cold startups at 14,000 RPM. This intermediate

step was a two plane balance with weight corrections at the outboard planes #1

a nd #4. This a llowed t he rotor t o run at full speed of 18,500 RPM, a nd a fi na l tw o

plane trim balance was performed on the interior planes #2 and #3 after a full

heat soak. The synchronous shaft vibration amplitudes were significantly

reduced. The ma gnitude of the fi na l running speed vibrat ion vectors r a nged from

5 to 8%of the diametr ica l bearing clea ra nce. The suita bility of this ba lance sta te

is demonstr a ted by a n extended process run on this ma chine. Additional deta ils

on t he unba lance response of this ma chine are presented in case history 36.

FORCED RESPONSE CALCULATIONS

The programs used to calculate undamped critical speeds, stability, plus

damped critical speeds, all display the final results as non-dimensional ampli-

tudes. This is adequate for determining mode shape geometry, and identifying

the st a tions of ma ximum defl ection. However, these ba sic concepts m ust be sig-

nifi cant ly extended to duplicat e actua l rotor behavior.

This desirable simulation of rotor motion is addressed by a forced response

a na lysis of a da mped rotor syst em. Va rious forcing functions such a s ma ss unba l-

a nce, skew ed disks, or sha ft bows a re allowed in this t ype of a na lysis. In a ddition

to previous rotor modeling capabilities, the configuration for a forced response

analysis typically includes rigid plus flexible disks. This type of program accepts

constant coefficients for bearings and supports, or coefficients that vary as afunction of speed. Usua lly, the most a ccura te results a re obtained by employing

fifth or sixth degree polynomial functions that describe the stiffness and damp-

ing coefficients for the bearing oil film, and the housing as a function of rota-

tional speed. The program computes Bode and polar plots, elliptical orbits, two

Fig. 5–15 CalculatedDamped Mode ShapesOf Rotor ResonanceOccurring Above TheNormal Operating Speed



Forced Response Calculations 223

and three-dimensional mode shapes, plus bearing forces. Some programs, such

as UNBAL by Gunter10 use a Complex Ma tr ix Tra nsfer method, and other pro-

grams such as DYROBE S by Chen, Gunter, and G unter

11

a re based upon FiniteElement Ana lysis (FEA) numerical m ethods.

Since support coefficients may be calculated as a function of rotational

speed, sha ft d ispla cement response vectors ma y be computed w ith minima l dis-

continuity. One of the obvious applications for this information would be the

development of synchronous 1X vectors in B ode plots a s sh own in Fig. 5-16. The

bottom half of the Bode plot displays 1X vibration amplitudes in Mils, p-p a s a

function of rotative speed in RPM. The data in the top half of Fig. 5-16 depicts

the pha se lag in D egrees. The sam e cha ra cteristics a re shown in t he polar plot of

1X vectors in Fig. 5-17. In both ca ses, the da ta is observed from a t rue vert ical or

horizonta l perspective. The a ngular sta rt ing point for t he phase a ngles begins a tthe probe locat ion, and the convention follow s sta nda rd pha se lag logic with the

angles progressing against rotation.

These analytical data presentation are designed to be analogous to the

Bode and polar plots measured by proximity probe systems. The same 1X syn-

chronous vectorial data is presented on both types of plots. The Bode plot dis-

plays excellent visibility of a mplitude an d pha se changes w ith respect t o speed,

a nd th e pola r plot enha nces the va ria tions with r espect t o pha se. In a ddition, the

comput er solution is not limit ed to the physica l restr ictions imposed on th e phys-

ica l insta llat ion of the proximity probes. In fa ct, the ana lytical model allows the

generat ion of B ode and pola r plots a t a ny rotor sta tion over any speed doma in.

Since the entire rotor motion has been computed at numerous speeds, it is

possible to construct both two and three-dimensional mode shapes of the scaled

10 Edgar J. Gunter, “UNBAL, Unbalance Response of A Flexible Rotor - HP Version 4,” Com-puter P rogram in Hewlet t Pa ckard B as ic by Rodyn Vibrat i on, Inc., Cha rlottesville, Virginia, J uly,1988, modified by Robert C. Eisenmann, Machinery Diagnostics, Inc., Minden, Nevada,1992.

11 W.J. Ch en, E. J . Gunt er, and W. E. G unter, “DYROBES, D ynam ics of Rotor B earing S ystems,Version 4.21,” Comput er Pr ogram in MS -DOS ® by Rodyn Vibrat i on, Inc., Charlottesville, VA, 1995.

Fig. 5–16 Calculated GasTurbine Bode Plot



224 Chapter-5

rotor behavior. For example, Fig. 5-18 depicts a two-dimensional rotor mode

sha pe superimposed upon a n outline of th e ga s t urbine r otor. The solid lines rep-

resent th e predicted vertica l vibrat ion, an d t he dotted lines depict the horizonta l

sha ft m otion. In a ll cases, scaling is provided via the left ha nd a xis. This ty pe of

scaled mode shape allows the comparison of anticipated displacement ampli-

tudes w ith t he actua l ma chine cleara nces. On some machines, this t ype of infor-

mation may not be particularly useful. However, on industrial turbines with

close tip cleara nces on the a xial blad es, this t ype of displacement da ta a long t he

rotor ma y be extraordinari ly importa nt .The calculated shaft mode shape may also be viewed as a three-dimen-

siona l plot a s shown in Fig. 5-19. In ma ny ca ses this t ype of display is visua lly

more informative than the two-dimensional plot. This three-dimensional plot is

Fig. 5–17 Calculated GasTurbine Polar Plot

Fig. 5–18 Calculated Two-Dimensional Plot for Hori-zontal And Vertical Gas Tur-bine Rotor Mode Shapes




composed of shaft orbits at various locations, and it is scaled from a maximum

vector that is listed on each diagram. The bearing locations are identified, and

the speed is listed. This presentation provides an additional level of visibility to

th e ca lculat ed rotor mode sha pes. The entire display ma y be rota ted t o observe or

enhan ce the cha ra cteristics of a part icular plot. The determina tion of ca lculat ed

a mplitude at specifi c rotor sta tions is easier w ith t he tw o-dimensiona l plot of Fig.

5-18. How ever, th e th ree-dimensiona l dia gra m of Fig. 5-19 does provide a bett er

physica l rendition of th e sha ft m odal behavior.

The shaft orbits at any location and any rotative speed may also be

extra cted from the ca lculat ions a nd presented separa tely. Fig. 5-20 depicts a typ-ical pair of calculated orbits from opposite ends of the turbine. The orbits are

Fig. 5–19 Calculated

Three-Dimensional GasTurbine Mode Shape AtNormal Operating Speed

Fig. 5–20 Calculated GasTurbine Shaft Orbits AtProximity Probe Measure-ment Stations



226 Chapter-5

view ed from t he inlet end of the t urbine, a nd t he calcula ted orbits a re oriented to

be consistent with the measured shaft vibration data. The specific data used for

Fig. 5-20 is representa tive of the ant icipated sha ft vibra tion at the a ctua l probe

locations. By changing t he forcing function, such as var ious levels of unba lancea t different loca tions wit hin t he turbine, the a ffect upon t he overa ll mode sha pe,

and resultant loads plus shaf t vibrat ion at the journal bearings may be calcu-

lat ed. Similarly, the impact of skew ed wh eels or bent rotors ma y be examined on

paper before the machine is ever built.

Overall , it is evident t ha t t he computa tion of ant icipated vibra tory behavior

a long t he length of a rotor provides useful informa tion regar ding t he behavior of

the machinery. If it can be demonstrated that this computational information is

correct, and consistent with shaft vibration measurements, then a significant

tool is ava ilable for the ma chinery designer as w ell a s the diagn ostician .

Case History 13: Gas Turbine Response Correlation

At the conclusion of a set of a na lytical ca lculat ions, the issue of verifi cation

of the results must be addressed. This is not an easy topic since comparison of

a na lytical computa tions with fi eld vibrat ion measur ements is seldom performed.

As such, specific items of comparison are rarely defined, and tradition evalua-

tions are often filled with generalities. Within the context of this chapter, it

seems a ppropriat e to perform a compar ison on the ba sis of both q ua lita tive a nd

quantitative parameters. Specifically, synchronous characteristics of a single

shaft gas turbine will be reviewed, and definable items during an unbalance

response test will be correlated.

The 40,000 horsepow er ma chine under discussion cont a ins a 22,500 pound

rotor th a t norm a lly opera tes betw een 5,000 and 5,300 RP M. The unit is equipped

wit h elliptica l journal bearings, and a double acting Kingsbury t ype thrust bea r-

ing. This rotor contains seventeen stages in the axial flow air compressor, andtwo turbine stages as depicted in Fig. 5-21.

In nearly all situations, the measured shaft vibration is elliptical, with the

horizonta l motion exceeding th e vertical. In ma ny ca ses, the orbit is t ilted in thedirection of rotation. At normal speeds, the phase angles between inboard and

outboar d orbits a re a lmost identica l. The sam e genera l behavior is noted in th e

analytical computations. For example, the orbit plots presented in Fig. 5-20 were

extracted from a model of this machine. Hence, the general motion described by

Fig. 5–21 Gas TurbineRotor Configuration




the analytical model reflects the measured field vibratory characteristics.

Tra nsient speed chara cteristics a re often difficult t o underst a nd due to the

potential variation, and distribution of residual unbalance across the length of

the rotor. For example, a mid span weight will drive the first critical, but willhave minimal influence upon the pivotal mode. By the same token, a couple

unbalance may be sufficient to excite the pivotal second critical, but the first

tr a nslat iona l mode may experience minimal excita tion. Thus, the modal weight

distribution will influence the critical speeds stimula ted, and t he am ount of mea-

sured excita tion. Although it is diffi cult to duplica te a mplitude response t hrough

a series of resona nt frequencies, it is r easonable to compar e the mea sured criti-

cal speeds wit h t he calcula ted na tu ra l frequencies. For insta nce, Ta ble 5-5 com-

pares t he results for th e fi rst thr ee critical speeds of th is tur bine rotor. The top

row of calculated values displays the damped critical speeds with flexible sup-

ports. This analysis identifies a split horizontal (H) and vertical (V) critical for

the first and second modes. The second row of calculated critical speeds were

extra cted from a forced response an a lysis tha t includes an improved defi nition of

bearing and support characteristics. Finally, the bottom row summarizes the

range of measured criticals for several different machines during multiple star-

tup and coast down fi eld vibra t ion da ta sets .

Note that the first critical displays excellent agreement between the

da mped nat ura l frequencies, the forced response critica ls, a nd t he measur ed crit-

ical speeds. Similarly, the pivotal second mode also shows excellent agreement

between the calculated and measured resonant frequencies. The bending third

critical is visible in the calculations, but is somewhat elusive in the field mea-

surements. Since this mode is above the normal operating speed range, it can

only be r eached during over speed runs. These runs a re usua lly of rat her short

dura tion, a nd t he resona nce genera lly has minima l time to respond. Overall , this

agreement between the calculated and measured critical speeds providesincrea sed confi dence in th e validity of the computa tions.

Another check on the accuracy of the model may be performed by installing

a n easily defina ble excita tion on a r eal ma chine, an d adding t he same excita tion

to the model. A direct comparison of measured versus calculated vibration

Table 5–5 Comparison Of Calculated Versus Measured Critical Speeds

Origin ofCritical Speed


(RPM)

2nd ModePivotal(RPM)

3rd ModeBending

(RPM)

Ca lcula ted Da mped1,100 (H )1,400 (V)

1,830 (H )2,250 (V)

5,730

Ca lculated Forced1,100 (H )

1,300 (V)

1,800 (H )

2,360 (V)5,600

Field Measured1,000 to

1,4501,900 to

2,3005,600+



228 Chapter-5

response chara cteristics should provide a s uita ble test of the model. For t his test ,

consider the a ddition of unba lance calibrat ion w eights to each end of the turbine.

These weights would alter the 1X synchronous response, and the results

should be visible in the vibration measurements, plus the analytical computa-tions. For t he purposes of this response test, a n unba lan ce ca librat ion w eight of

77 Gram-Inches at 230° was added to the inlet coupling. The centrifugal force

from this w eight a t 5,100 RPM wa s 125 Pounds (0.6%of the rotor w eight). At the

exhaust coupling, an unba lance of 234 G ra m-Inches wa s a tt a ched a t 275°. This

weight produced a centrifugal force of 381 Pounds at 5,100 RPM (1.7%of rotor

weight). Since the rotor residual unbalance was low, the vertical shaft vibration

amplitudes were also small. Hence, the most meaningful data was extracted

from the horizontal proximity probes.

Sequential ly , an in i t ial data set was obtained at 5 ,100 RPM without any

extra unba lan ce. Next the machine wa s shutdown, the 77 G ra m-Inch weight wa s

installed at the inlet, and a second data set acquired. The turbine was again

shutdown, and the inlet weight was removed. Next, the 234 Gram-Inch exhaust

end weight wa s a dded, and a fi nal da ta set wa s acquired at 5 ,100 RP M. The 1X

vectors from the horizonta l probes were runout compensat ed, and t he results a re

summ a rized in Ta ble 5-6.

It is obvious that the small weight installed at the inlet end of the turbine

produced only minor chan ges, whereas the exhaust end weight r esulted in a sig-

nifi cant ly lar ger cha nge in shaft vibrat ion. Since the initia l synchronous 1X vec-

tors are quite small, a comparable analytical case was developed with minimal

shaft bow, and low residual unbalance. Specifically, a midspan shaft sag of 0.2

Mils (0.4 Mils TIR), was combined with a residual unbalance at the first com-

pressor sta ge of 100 G ra m-Inches. Another 100 Gra m-Inch residua l w a s loca ted

at the second stage turbine wheel. Calculations were performed at 5,100 RPM

Table 5–6 Measured X-Axis Vibration Response Vectors With Unbalance Weights

Weight Condition Inlet Bearing #1 Exhaust Bearing #2

No Weight Inst alled 0.85 Mils,p-p @ 32° 1.01 Mils,p-p @ 346°

Weight at Inlet End 1.04 Mils,p-p @ 15° 1.38 Mils,p-p @ 330°

Weigh t a t E xh au st En d 1.06 Mils,p-p @ 350° 1.65 Mils,p-p @ 337°

Table 5–7 Calculated Horizontal Vibration Response Vectors With Unbalance Weights

Weight Condition Inlet Bearing #1 Exhaust Bearing #2

No Weight Inst alled 0.93 Mils,p-p @ 281° 1.09 Mils,p-p @ 290°

Weight at Inlet End 1.10 Mils,p-p @ 260° 1.26 Mils,p-p @ 261°

Weigh t a t E xh au st En d 1.56 Mils,p-p @ 265° 1.67 Mils,p-p @ 274°




with the initial shaft bow, and the two residual unbalance locations. Two addi-

tional cases w ere run wit h th e previously identified coupling unba lance weights.

The comput ed 1X vibra tion vectors fr om th ese runs a re present ed in Ta ble 5-7.

The initia l rotor bow a nd r esidual unba lan ce vectors w ere selected t o mat chthe initial measured shaft vibration vector magnitudes. Since two sets of unbal-

ance weights were used, both the direct and the cross-coupled balance response

vectors m ay be calcula ted. The specific equa tions for these calculations a re listed

in chapter 11 of this text. For a two plane correction, equations (11-13), through

(11-16) may be used. For exam ple, the measu red sha ft vibra tion da ta a t the t ur-

bine exhaust bearing may be used to calculate the balance sensitivity vectors

from equation (11-16) as follows:

If the same calculations are performed for each set of primary and cross

coefficients, the measured versus calculated balance sensitivity vectors may be

generat ed as sh own in Ta ble 5-8. Note tha t t he calcula ted sensitivity a ngles were

a djusted by 45° t o correct for the t rue horizontal orienta tion of the a na lytical cal-

cula tions versus t he + 45° location of the proximity probe. Thus, the t a bulat ed

vectors in Ta ble 5-8 ar e directly compa ra ble in terms of a ngula r position.

The similarity between measured and calculated sensitivity vectors in

Ta ble 5-8 lends further credibility to th e va lidity of the a na lytical ca lculat ions.

Cert a inly the an a lytically derived ba lance sensitivity vectors are not of sufficient

accuracy to perform a refined field trim balance. However, they exhibit magni-

tudes that reflect the vibration response measurements, with reasonably consis-

tent vector angles. Again, it is concluded that the analytical model does an

excellent job of simulating the fi eld dyna mic beha vior of the ga s t urbine rotor.

Table 5–8 Comparison Of Measured Versus Calculated Balance Sensitivity Vectors

Vector Identification Measured Sensitivity Calculated Sensitivity

Inlet B earing - S 11 229 G ra m-Inches/Mil,p-p @ 263° 190 G ra m-Inches/Mil,p-p @ 340°

Inlet B earing - S 12 329 G ra m-Inches/Mil,p-p @ 338° 328 G ra m-Inches/Mil,p-p @ 346°

Exhaust B earing - S 21 156 G ra m-Inches/Mil,p-p @ 294° 126 G ra m-Inches/Mil,p-p @ 344°

Exhaust B earing - S 22 349 G ra m-Inches/Mil,p-p @ 312° 339 G ra m-Inches/Mil,p-p @ 342°

S 22

W 2

B 22 A 2–

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

234 Gram Inches 275°∠1.65 Milsp p– 337°∠ 1.01 Milsp p– 346°∠–----------------------------------------------------------------------------------------------------

= =

S 22234 Gram Inches 275°∠

0.67 Milsp p– 323°∠-------------------------------------------------------

349 Gram-Inches/Mil 312°∠= =



230 Chapter-5

Case History 14: Charge Gas Compressor with Internal Fouling

The centrifugal compressor depicted in Fig. 5-22 operates in cracked gas

service. A low stage double flow compressor is coupled to the discharge end ofthis machine, and a high stage compressor is coupled at the thrust end of the

subject compressor. This tr a in is stea m t urbine driven at a ma ximum operat ing

speed of 5,400 RPM. As noted, the rotor weighs 3,520 pounds, and it has a span

of 107 inches between bearing centers. This compressor contains six impellers,

a nd they a re equally divided between the 2nd and 3rd process sta ges.

The compressor ha d been opera tin g smoothly for a n extended period of tim e

wh en the horizonta l vibrat ion a t t he discha rge end began to increa se. The trend

plot in Fig. 5-23 documents th e vibra tion cha nge over a four m onth period. At th e

beginning of this data, the machine displayed low and acceptable vibration

amplitudes from all radial probes. A power outage in February resulted in an

increa se of vibra tion a mplitudes a t the discha rge end. Approximat ely one w eek

later, a problem with a seal pot float mechanism occurred, and vibration levels

increased again. The amplitudes remained fairly constant throughout March,

and t hen began a gradua l downw ard t rend towa rds the end of May.

I t should be mentioned that these data points w ere acquired manua lly w ith

a porta ble da ta collector on a w eekly route. Cha nges or va riat ions betw een these

periodic sam ples a re not visible. Hence, the t ra nsition between t he low vibrat ioncondition on or about May 28, and the 6.0 Mil, p-p value displayed on June 5 was

unknown. Furt hermore, the constit uent pa ra meters of rotat iona l speed vectors,

and radial position data was not available. The high vibration amplitude of 6.0

Fig. 5–22 Charge GasCentrifugal Compressor

Case Configuration

Compressor2nd & 3rd Stages

Rotor Weight = 3,520#Bearing Span=107"

6 Impellers

2nd StageSuct. Disch.

3rd StageSuct. Disch.

Axials

1 5 °

Horiz.

75°

Vert.

90°

2 0 °

Vert.

Horiz.

High Horizontal

Vibration

Low Horizontal &

Vertical Vibration




Mils,p-p on J une 5th continued t o increase unt il the horizonta l probe exhibited a n

unfi ltered a mplitude of 6.9 Mils,p-p. This behavior was documented in the orbit

and time base plots in Fig. 5-24. Simultaneously, the suction end displayed low

vibration amplitudes (1.3 Mils,p-p), and this data is shown in Fig. 5-25. It was

clear that the discharge journal was moving horizontally across the entire bear-

ing clearance. That is, the 7 Mil vibration, plus 1 or 2 Mils for the oil film thick-

ness is equivalent to the t otal dia metrical bear ing cleara nce of nominally 9 Mils.

The 1X vibration was reduced by unloading the compressor to allow opera-

tion at a lower speed. A furt her drop in vibra tion wa s a chieved by reducing the

oil supply temperat ure 7°F to increase th e da mping. This t empera ture r eduction

was accomplished by adding cold firewater to the water side of the oil cooler.

Since this is a dirty gas service, the issue of coke buildup should always be con-

sidered. In this case, it was understood that a drop in efficiency had occurred

during the past few months, but the specific decrease was not quantified. In ret-rospect, the plant personnel performed machinery efficiency calculations based

on a hea t a nd ma teria l bala nce. This wa s a poor method to determine compressor

efficiency, and it turned out to be extraordinarily inaccurate. The only realistic

a pproa ch to determine process ma chinery effi ciency is t o begin w ith a n a ccura te

measur ement of the input sh a ft t orque as discussed in chapter 6 of this t ext.

Wa sh oil ra tes w ere increased, with no measur a ble improvement. B a sed

upon th e ava ilable evidence, it w a s initially concluded tha t t he discha rge bea ring

was damaged. In addition, the suction end historical data was inconsistent. Spe-

cifically, the orbits in Figs. 5-24 and 5-25 describe a pivotal behavior across the

compressor. The suction end pha se ha d cha nged severa l times, a nd motion of this

compressor was considered to be abnormal. Finally, it was agreed to shutdown

the ma chinery, and prepa re for a rotor, bea ring, and sea l cha nge.

Follow ing a n orderly shutdown, t he subsequent disas sembly and inspectionof the compressor resulted in severa l surprises. First, th e discharge journa l bear-

ing wa s not dama ged. In fa ct, the disassembly clea ra nces were similar to the pre-

vious inst a llat ion clea ra nces. Second, the suction end bea ring displayed ba bbitt

Fig. 5–23 Charge GasCompressor VibrationAmplitude - Four MonthTrend Plot



232 Chapter-5

da ma ge on t he bott om pads. Third, the compressor ha d a ma jor a ccumula tion of

coke on th e sta tionar y, and t he rota ting elements. Inlet guide vanes, dia phra gms,

and return bends all exhibited various levels of coke deposits. In addition, the

last stage wheel revealed major coke clusters at random locations within the

impeller.

The compressor rotor contains three impellers for each process stage (2nd

and 3rd). In both sections of the compressor, the inlet wheel for the respective

stage was reasonably clean, and coke buildup increased progressively on the

next t wo w heels. This is t ypica l for a cracked ga s ma chine to display increa sing

coke deposits a s t he heat of compression increases a cross t he w heels t ha t form

the pa rt icular process st a ge. However, the a mount of buildup on t he last wh eel ineach process stage was substantial. Further examination of the casing revealed

that most o f the in terstage labyrinths, and the balance piston labyrinths were

completely filled with coke. At six locations, the mating surfaces on the rotor

were highly polished, an d t he evidence of close conta ct betw een t he rotor a nd t he

filled-in labyrinths wa s clear and unmista kable.

The condition of three of these surfaces is documented in Fig. 5-26. This

photogra ph of the t hird process St a ge shows the rela tively clean inlet w heel on

the left, and the heavily coked discharge end wheel on the right side. The pol-

ished shaft surfaces on the rotor are coincident with the coke filled interstage

labyr inths. On t he ba ck side of th e last impeller, the r otor ba lance piston resides.

Although photographic evidence of this element is not as clearly defined, the bal-

ance piston displayed most of the same characteristics as the coke filled inter-

sta ge labyr inths. The physical interpreta tion of this uniq ue mecha nical conditionwa s hypothesized as a machine tha t w as operat ing with a series of in ternal bear-

ings. Specifi cally, the tw o externa l oil fi lm t ilt pa d bear ings w ere supplemented

by six internal dry bearings. Five of these internal bear ings were associat ed with

Fig. 5–24 Compressor Discharge EndBearing Shaft Radial Vibration

Fig. 5–25 Compressor Suction End Bear-ing Shaft Radial Vibration




in terstage labyrinths, and t he sixth w as at the discharge end balance piston.

The hypothesis of th e development of six new int erna l bearings w a s exam-

ined in grea ter d eta il to determine if th is could be responsible for t he compressor

high vibr a tion problems. The only viable met hod to approa ch this problem w ould

be with an analytical simulation of the machinery. The arrangement of shaft,

impellers, spa cers, a nd couplings for a norma l rotor is depicted in F ig. 5-27. This

ma chinery sketch identifi es the proximity probe locations, a nd th e ra dial journa l

bearings at each end of the rotor. Stiffness and damping coefficients for the oil

film portion of these tilt pad bearings were computed. At a speed of 5,300 RPM,

the calculat ed horizonta l oil film stiffness K xx w a s 350,000 Poun ds/In ch. The ver-

tical st iffness K yy

w a s computed t o be 2,050,000 P ounds /In ch. The calcula t ed hor-

izonta l oil film da mping C xx wa s 1,100 Pounds-Seconds/Inch. F ina lly, th e vert ical

damping C yy wa s 3,000 P ounds-Seconds/Inch. Sin ce th ese ar e tilt pa d bear ings,

cross-coupling coefficients do not exist. The journal and thrust bearing housing

weight was approximately 200 Pounds, and the horizontal and vertical support

stiffness (K sxx & K syy ) for th is housing w ere estima ted a t 2,000,000 Pounds/Inch.

Housing damping was calculated at 10%of the critical damping to be 200

P ounds-Seconds/Inch for t he suction end bear ing h ousing (C sxx & C syy ) as per

equation (4-17). The discharge end housing contains only a journal bearing, and

the w eight of this housing w a s estima ted a t 100 pounds. The vertical st iffness of

this housing w a s set a t 2,000,000 Pounds/Inch, and the horizontal stiffness wa s

slight ly reduced t o 1,500,000 P ounds/Inch. The estim a ted vert ical a nd h orizont a l

da mping values w ere proportiona lly reduced in a ccordance with the chan ges in

stiffness and housing weight.

The norma l model included a residua l unba la nce of 30 G ra m-Inches a t 175°

on the suction end, and another 30 Gram-Inches at 145° on the discharge end.

This total residual unbalance was set to be somewhat less than the normal bal-

Fig. 5–26 CompressorThird Process Stage WithInternal Coke DepositsProducing Midspan

Pseudo Bearings



234 Chapter-5

a nce tolera nce (113.4W/N) of 74 Gr a m-Inches for t he ent ire r otor. This in it ia l

model allowed examination of the normal synchronous vibration response

between 500 and 5,500 RPM. The computed response at operating speed pro-

vided an acceptable duplication of normal machine behavior. In addition, the

transient calculations accurately predicted the first critical speed region cen-

tered at 2,500 RPM. Thus, the initial model (Fig. 5-27) successfully duplicated

the historical ma chinery behavior. I t w as now rea sonable to extend th is model to

th e abn orma l condit ion of a h eavily coked compressor as s hown in Fig. 5-26.

The rotor removed from the compressor was check balanced, and the resid-

ual unbalance determined at each end of the rotor. At the suction, the residual

was 488 Gram-Inches at 218°. A much higher unbalance was discovered at the

discha rge end of the rotor wit h a measur ed 2,074 Gr a m-Inches at 201°. This syn-

chronous excitation data was loaded into the model in conjunction with a 0.25

Mil midspan rotor sa g. The support condition for the a bnormal case required a

minor modifi cation of the existing bear ings, plus the add ition of the new interna l

bearings a t the fi lled la by loca tions. The previous bea ring housing chara cteristicswere held consta nt. Simila rly, the tilt pa d bearing oil film coefficients a t t he dis-

charge end w ere reta ined with out modificat ion. However, the suction end journa l

bearing coefficients were modified to reflect the demonstrated higher loads at

this location. Horizonta l stiffness K xx a t this locat ion w a s increased t o 1,500,000

Pounds /Inch, a nd t he vert ical st iffness w a s held a t 2,050,000 Pounds/Inch.

Fina lly, the oil damping at the suction bearing w a s held constan t.

Int erna l compressor bearings w ere placed at each of the loca tions where the

labyr inths were fi lled w ith coke, and there w a s obvious physical evidence of close

clearance contact between the shaft and the laby areas. These internal bearings

a re identified a s B rg. #2 thr ough B rg. #7. The normal r otor journa l bearings a re

shown as Brg. #1, and Brg. #8 on this model. The photograph in Fig. 5-26 shows

the thr ee interna l bea rings ass ocia ted wit h the 3rd process sta ge as B rg. #5, Brg.

#6, plus B rg. #7 at the ba lance piston. These interna l bearing locations a re iden-

tica l to th e locat ions on t he model diagr a m present ed in Fig. 5-28.

Computa tion of interna l bea ring coefficients w a s difficult due to the various

unknowns a ssociat ed with th e interna l beha vior of this unit. Using short bearing

Fig. 5–27 Normal Com-pressor Rotor Configura-tion With Normal BearingsAnd Typical ResidualUnbalance Levels




theory, ca lculat ions w ere performed a t va rious cleara nces, and h ydroca rbon gas

viscosities. The resulting minimum stiffness values varied between 85,000 and

325,000 Pounds/Inch. On th e high side of th e potentia l st iffness envelope, valuesof 5,000,000 to 18,000,000 P ounds /In ch w ere comput ed. These m a ximu m s upport

stiffness would be reduced by the actual structural rigidity of the casing itself .

Hence, the effective internal bearing stiffness would probably fall within the

ra nge of 400,000 to 1,500,000 P ounds /In ch.

In the final assessment, it was clear that a direct computation of support

st iffness w ould not be a tt a ina ble. A compromise va lue of 600,000 Pounds/In ch

wa s selected for t he vert ica l and h orizonta l support coefficients a t int ernal bear-

ings #2 through #6. The balance piston displayed less contact than the shaft

la bys, an d a st iffness of 400,000 Pounds /Inch w a s used for this locat ion. Cr oss-

coupling coefficients, and all damping coefficients for these internal bearings

were set to zero. Although this represents a simplistic model, the available

mechanical data allows no other realistic alternative.

The forced synchronous response ca lculat ions were repeat ed for t his a bnor-ma l case of multiple interna l bearings, plus high unba lan ce. The results of these

calculations at 5,300 RPM are presented in Fig. 5-29. The computed shaft vibra-

tion is compared with the measured 1X shaft orbits extracted from the previ-

ously discussed F igs. 5-24 and 5-25. In bot h set s of orbits, th e sa me sca ling of 2.0

Mils/Division wa s used, a nd both orbita l sets display a tr ue vertical/horizonta l

orientation.

Note that both discharge orbits in Fig. 5-29 are elliptical, and primarily

horizontal. Also note that the horizontal magnitudes are similar, and the

Keypha sor® d ots for th e discha rge orbits reside in the sa me qua dra nt. Vertical

magnitudes between the measured and computed discharge orbits show some

variation; and the suction end orbits are rotated approximately 90° between the

calcula ted a nd mea sured orbits. Nevertheless, the correlat ion betw een t he mea-

sured and computed vibration response is considered to be acceptable.

B a sed on this a bility t o ana lytically duplica te the fi eld machine behavior, it

is reasonable to conclude that the compressor experienced the physical changes

tha t w ere imposed upon t he a na lytical model. Specifica lly, the high vibra tion lev-

Fig. 5–28 Abnormal Com-pressor Rotor Configura-tion With Pseudo InternalBearings And MeasuredResidual Unbalance



236 Chapter-5

els a t t he discha rge plus the low suction end vibra tion amplitudes were a tt ribut-

able to the combined effect of an internal coke buildup on the stationary

internals (manifesting as internal bearings), plus large unbalance due to coke

accumulation on the rotor. This combination of abnormalities resulted in a

heavily loaded suction end bearing with low vibration (and pad damage), com-

bined with a generally unloaded (and undamaged) discharge end journal that

migrated across the available bearing clearance.

Once more, an analytical approach provides an acceptable simulation of a

mecha nical abnorma lity on a centrifuga l ma chine. In t his case, the physical evi-

dence was used to develop a model that explained the abnormal behavior

detected by the shaft sensing proximity probes. In all cases, it should be recog-

nized that measurement and calculation technologies are coexistent resources

tha t ca n provide significant ly improved understa nding of mechanical behavior.

Case History 15: Hybrid Approach To A Vertical Mixer

As demonstrated in the last two case histories, analytical solutions may be

effectively combined with fi eld vibra tion measurements to examine the ma chin-

ery beha vior from tw o different perspectives. This combina tion of techniques pro-

vides confidence in the individual technologies, plus the accuracy of the final

results. I t is clear t ha t a compar ison of ca lculat ed versus computed lat eral vibra -

tion behavior makes good engineering sense. However, some physical situations

cannot be properly examined by exclusively using only one technique. It these

situations, it is necessary to combine the computational techniques with the

physical measurements to arrive at a solution. This type of hybri d approach is

not a common pra ctice, but it does provide a wa y t o get the job done with a ccept-

a ble technical credibility.

As an exa mple of th is ty pe of problem, consider t he vert ical mixer rot or dis-played in Fig. 5-30. Cha rles J a ckson would probably classify t his a ssembly as the

proverbial mud bal l on a wi l l ow st ick . The long a nd slender sh a ft is supported by

two bearings at the top end, and two mixer wheels are located at the bottom of

th e rotor. A 30 inch eleva tion difference exists bet w een the upper a nd lower m ix-

Fig. 5–29 Comparison Of Calculated Versus Measured Compressor Shaft Orbits




ing blades. The distance between bearings is approximately 18 inches, and the

vertical length of unsupported shaft approaches 116 inches. This rotor is driven

by a variable speed motor via a belt and pulley configuration at the top of the

assembly. In operation, the mixer is used in a batch process where the rotor is

totally immersed in the process fluid, and pulley rotational speed is normallybetween 900 an d 1,200 RP M. The ra dial bea rings a re rolling element un its, and

a mecha nical seal is used t o contain the process fluids.

The dua l mixer bla des have a n outer diam eter of 20 inches, a nd a n a verage

thickness of 0.188 inches. Various perforations and raised lips are fabricated into

the blades to provide the necessary agitation action. This blade design was

empirically based, and proven successful over many years of operation. However,

due to process revisions, it would be necessary to install a thicker pair of mixer

blades for future mixtures. The maximum anticipated thickness for the new

blades w a s 0.488 inches. This bla de thickness increase could a dd a n a dditional

54 pounds to the rotor assembly. Since the initial rotor weight was 615 pounds,

the additional blade weight represented a nominal 9%increase in the assembly

w eight. In a ddition, this extra blade weight r epresented an a pprecia ble increa se

in the overhung ma ss.

During sta rtup of th is mixer with th in blades, i t w as observed that a crit ical

speed existed betw een 250 a nd 300 RPM. Since this frequency w a s considerably

below the operating speed range of 900 to 1,200 RPM, there was no interference

between the resonance an d norma l running speed excita tion. However, there wa s

concern that the heavier mixer blades might have a detrimental influence upon

th e rotor critica l speeds (especia lly t he higher order modes). There w a s no infor-

ma tion rega rding rotor na tur a l frequencies in the machinery files, a nd there wa s

limited opportunity for traditional vibration response testing. As displayed in

Fig. 5-30, the entire rotor is suspended from the two top bearings. During opera-

tion of the mixer, the only possible vibra tion meas urements m ust be ma de from

the exterior of the bearing housing. Obviously, this type of rotor will exhibit a

va riety of ca ntilevered modes, and vibra tory motion at the bearings w ill be mini-ma l under m ost conditions. Thus, direct casing vibra tion measur ements w ill not

be beneficial in solving this problem.

The undamped natural frequencies of the mixer rotor could be computed as

discussed earlier in this chapter. Unfortunately, internal shaft diameters were

Fig. 5–30 Physical Configuration Of Vertical Mixer Rotor



238 Chapter-5

not known , and th e shaft m a teria l properties were reasonably undefi ned. Hence,

a direct calcula tion of th e critical speeds could not be a tt empted due to a lack of

the funda menta l mecha nical informa tion on t he rotor.

The time honored bum p test technique of hi t th e stati onar y rotor w it h a 4x4 ti mber and measure th e vibr ati on response could be used, but this approach

leaves much to be desired. Although one or more natural frequencies would be

excited, there is minima l a bility to determine a ccura te mode sha pes for each res-

ona nce, an d virt ua lly no wa y t o sepa ra te out closely spa ced or coupled modes.

From ma ny a spects, a r ealistic engineering solution to th is problem m ight

seem to be unattainable. However, if the question is approached with multiple

tools instead of a single technique, a logical hybri d appr oach may be developed.

In this particular case, the initial step consisted of accurately measuring the

sta tic mode sha pe of the non-rota ting sha ft using a n H P -35670A Dyn a mic Signal

Ana lyzer plus an a ccelerometer, and a modally t uned impact ha mmer. The a ccel-

erometer was mounted close to the bottom mixer blade. The force hammer was

used to impact t he sha ft a t t welve different elevat ions a t 10 inch increments up

the length of the sha ft. Frequency response functions (FRF) were th en a cquired

between the accelerometer and each hammer location (acceleration /force). The

data was checked for proper phase shifts, plus acceptable coherence as discussed

in chapters 4 and 6. At this point, the FRF vectors at the various resonances

could th en be extra cted an d used t o constr uct representa tive mode shapes.

Performing the above tasks manually can be a time consuming process.

Handling a dozen FRFs is not impossible, but it is clear that a complex three-

dimensional model may prove to be quite challenging. Hence, it is appropriate to

consider methods of automating the field test, plus the associated calculations

and animation of the resultant mode shapes. Historically, this type of work has

been performed with large instrumentation systems operating under computer

control. These types of measurement and data processing systems are compli-

cated t o set up a nd operat e. In ma ny cases, the fi eld environment w ill not t oler-a te the t ime or expense associa ted w ith la rge sca le moda l tests.

A much more attractive approach resides in operating the DSA with soft-

ware that is dedicated to modal analysis. In this specific case, the DSA was con-

trolled with Hammer-3D 12 software that runs directly on the HP-35670A and

elimina tes t he need for externa l devices. Within this softw a re, the test element

geometry and transducer array are physical ly defined. FRFs were acquired

between t he a ccelerometer a nd ea ch ha mmer loca tion a s previously noted. Fol-

lowing a validity check of the a veraged FRF s, curve fi tt ing wa s a pplied to each of

the fi rst four r esona nt frequencies. The individual m odes w ere then a ssembled,

scaled, a nd presented a s a nima ted mode shapes on th e DSA. Since this is a sim-

ple and symmetrical rotor, the Hammer-3D software was used in a single plane

mode. The resultant mode shapes from these impact tests were committed to

hard copy, and the first two modes are presented in Fig. 5-31.As expected, the measur ed fi rst mode was a pure overhung cant ilever mode

12 Da vid Forrest , “Ham mer-3D Version 2.01,” Computer P rogram in H ewlett P ackard Instru-ment B asic by Sea tt le Sound an d Vibra tion, inc., Seatt le, Wash ington, 1997.




that appeared at a frequency of 234 RPM. As shown in Fig. 5-31, the second

mode display ed a zero axis crossing betw een th e mixer blades, and it h a d a mea-sured natural frequency of 1,812 RPM. This measured FRF data was obtained

only on th e exposed sha ft sections below t he sha ft sea l. There wa s no opport unit y

to acquire an y mean ingful FRF da ta in the vicinity of the bearings. Aga in, this is

sta t ic mode shape dat a with a non-rota t ing shaf t .

The next step consisted of generating an appropriate analytical model to

simulat e the measured behavior. This wa s a difficult ta sk since the specific sha ft

material was unknown, and the in ternal hollow shaf t d iameters were l ikewise

unknown. However, the tota l rotor weight w a s known t o be 615 pounds, and t he

external shaft dimensions were easily measured. It was also noted that the top

portion of the mixer shaft underneath the pulleys and bearings was solid. The

hollow port ion of the sha ft w a s in t hree steps with decreasing dia meters of 4.5,

4.0, and 3.5 inches. The weight of the pulleys and the mixer blades were mea-

sured on a shop scale, and the shaft material density was assumed to be 0.283pounds per cubic inch. This density of steel was used since the shaft was mag-

netic, and t herefore it wa s not a ny t ype of aluminum or sta inless steel.

A simple model of the sha ft w as then constructed on a Microsoft® Excel

spreadsheet. The externa l sha ft dimensions w ere combined w ith the known com-

Fig. 5–31 FRF Measured Static ModeShapes Of Vertical Mixer Rotor

Fig. 5–32 Calculated Planar ModeShapes Of Vertical Mixer Rotor

1 1

22

33

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

1212

Measured2nd Mode at1,812 RPM

Measured1st Mode at234 RPM

Exposed Mixer ShaftDirectly Below Seal

Bottom Mixer Blade

UpperMixer Blade



240 Chapter-5

ponent w eights, plus th e density of steel previously mentioned. It w a s a ssumed

that the wall thickness for each of the three sections of hollow shaft were con-

sta nt. This w a ll thickness for the hollow sections wa s t hen va ried until t he over-

all rotor weight matched the total physical weight of 615 pounds. This matchoccurred with a wall thickness of 0.5 inches, which seemed to be a reasonable

value for t his rotor a ssembly.

The dimensional rotor data from the spreadsheet was then loaded into the

unda mped critica l speed program CRI TSP D previously referenced in t his chap-

ter. In this software, a planar analysis was performed that consisted of setting

the polar inertia terms to zero. Basically this is used to simulate a stationary

non-rota ting sha ft. The bearing stiffness w ere then var ied between 400,000 and

1,000,000 pounds per inch. As expected, t his h a d lit tle infl uence upon t he calcu-

lat ed mode shapes or r esona nt frequencies. Certa inly, this is a reasona ble result

since better than 95%of the strain energy was contained in the shaft, and less

tha n 5% of the str a in energy w a s in the bearings. Hence, the sha ft properties

contr olled t he na tura l resona nt frequencies, plus t he a ssociat ed mode shapes.

The fi na l piece of unknown da ta for performing t he CRI TSP D ca lculat ions

was the modulus of elasticity E for the shaft material. Initially, the value for

steel of 30,000,000 P si wa s used. This produced pla na r modes th a t did n ot ma tch

the mea sured FRF results. A series of repetit ive runs were ma de, a nd t he value

of E was incrementally reduced for each run. At a level of 21,000,000 Psi for E,

the calculated plan a r results closely mat ched th e measured FRF modes. Specifi -

cally, th e fir st t w o computed modes a re show n in Fig. 5-32.

The similarities between the measured FRF modes in Fig. 5-31 and the cal-

culated CRITSPD modes in Fig. 5-32 are self-evident. The frequencies for both

fi rst a nd second modes are consistent, a nd th e compa ra ble mode sha pes a re vir-

tually identical. The analytical model covers the entire rotor up through the

drive pulleys, whereas the measured static model only addresses the exposed

sha ft. The lar gest deviat ion occurs in the frequency of the fi rst critica l. The mea -sured FRF data provided a value of 234 RPM, and the calculated planar mode

revealed a speed of 276 RPM for this first mode. Although the 42 RPM differen-

tia l is a n a pprecia ble percenta ge of th e resona nt frequency, it is st ill well below

the norma l opera ting speed ra nge.

Since the zero speed pla na r model mat ches the st a tic FRF r esults, it is con-

cluded that the analytical model is an acceptable representation of the mixer

rotor. The next step requires a ctivat ing th e polar moment terms in th e CRITSP D

program, and performing a normal synchronous analysis. This run indicated

tha t t he fir st mode of the rota ting sha ft w ould occur a t 277 RP M, a nd the second

critical would increase to 1,841 RPM. The predicted first critical of 277 RPM was

consistent wit h t he pla nt observat ions of a resonance between 250 and 300 RPM.

Furthermore, the calculated frequency of the second critical was considerably

above the normal running speed range of 900 to 1,200 RPM.At this point, the analytical model provided a good representation of the

real m a chine. This similar ity gave confi dence to pursue the fi na l step of increas-

ing t he t hickness of th e tw o mixing bla des from 0.188 to 0.488 inches. This pr o-

vided additional weight to the rotor, plus additional inertia due to the 20 inch




diameter of these blades. This change dropped the first mode to 244 RPM, and it

lowered the second critical to 1,705 RPM. Again, these frequencies are consider-

a bly removed from th e opera ting speed ra nge, a nd it is concluded tha t t he addi-

tional blade thickness will not adversely influence the natural frequency

chara cteristics of this vert ica l mixer.

For compar a tive purposes, the entire a rra y of mea sured an d calcula ted na t-

ura l frequencies of this vert ical rot or ar e summa rized in Ta ble 5-9. Addit ionally,the calculated mode shapes for the vertical mixer with the thicker mixer disks

a re present ed in Figs. 5-33 and 5-34 for th e fi rst a nd second modes respectively.

Table 5–9 Summary Of Measured And Calculated Natural Frequencies For Vertical Mixer

RotorResonance

Original Thin Mixer Blades Thick Blades

Static FRFMeasurement

PlanarCalculation

SynchronousCalculation

SynchronousCalculation

1st Mode 234 RP M 276 RP M 277 RP M 244 RP M

2nd Mode 1,812 RP M 1,808 RP M 1,841 RP M 1,705 RP M

Fig. 5–33 Calculated Synchronous FirstMode With Thicker Mixer Disks

Fig. 5–34 Calculated Synchronous Sec-ond Mode With Thicker Mixer Disks



242 Chapter-5

In many ways, this vertical mixer case history is a simplistic example of

interleaving measurements and calculations to achieve a realistic engineering

solution. At a ll times, the ma chinery dia gnostician must be cognizan t of the fa ct

that direct solutions are not always possible, and indirect or hybrid approachesare sometimes necessary to solve a problem. Furthermore, the accuracy of the

fi na l result does not ha ve to extend to the t hird decima l point. In ma ny cases, if

you ar e wit hin 5%, or perha ps 10% of the exa ct solution, tha t a nsw er is fully

a ccepta ble within t he fi eld environment. You do not wa nt to be inaccura te in your

work, but then again , you do not want to try and at tain some unreal ist ic mea-

sure of a ccura cy or precision.

BIBLIOGRAPHY

1. Chen, W.J. , E. J . Gunt er, and W. E. G unter, “DYROB ES, Dyn am ics of Rotor Bear ing

Sy stems, Version 4.21,” Computer P rogram in MS-DOS® by Rodyn Vibrati on, In c.,

Charlottesville, Virginia, 1995.2. Eisenmann, Robert C. , “Some realit ies of fi eld balancing,” Orbi t , Vol. 18, No. 2 (J un e

1997), pp. 12-17.

3. Forrest , David, “Hammer-3D Version 2.01,” Computer Pr ogram in H ewlet t Pa ckard

Inst rument B asic by S eat tle Sound a nd Vibra tion, inc. , Sea tt le, Wash ington, 1997.

4. G unter, Edgar J . , “ROTSTB, Sta bili ty P rogram by C omplex Matr ix Transfer Method

- HP Version 3.3,” Computer P rogram in Hew lett P ackar d B asic by Rodyn Vibrat ion,

Inc., Charlottesville, Virginia, March, 1989, modified by Robert C. Eisenmann,

Machinery Dia gnostics, Inc. , Minden, Nevada , 1992.

5. G unter, Edga r J . , “U NB AL, Unba lan ce Response of A Flexible Rotor - HP Version 4,”

Computer P rogram in Hewlet t Pa ckard B asic by Rodyn Vibrati on, In c., Charlottes-

ville, Virginia, J uly, 1988, modified by Robert C. Eisenma nn, Machinery Dia gnos-

tics, Inc., Minden, Neva da , 1992.

6. G unter, E. J . an d C. G areth Ga ston, “CRITSP D-PC, Version 1.02,” Computer pro-

gram in MS-DOS® by Rodyn Vibrati on, In c., Cha rlott esville, Virginia , August, 1987.7. Lund, J .W., “Modal Response of a F lexible Rotor in Fluid Film B earings,” Transac-

ti ons Am er ican Society of Mechan ical E ngi neers , Pa per No. 73-D E T-98 (1973).

8. Myklestad, N.O. , “A New Method of Ca lcula t ing Na tura l Modes of Uncoupled Bend-

ing Vibration of Airplane Wings an d Ot her Types of Bea ms,” Journ al of th e Aeronau-

ti cal Sciences, Vol. 11, No. 2 (April 1944), pp. 153-162.

9. P rohl, M.A., “A G eneral Method for Calculat ing Crit ical Speeds of Flexible Rotors,”

Jour nal of Applied M echani cs, Vol. 12, Tra nsa ctions of th e ASME , Vol. 67 (Sept em-

ber 1945), pp. A142-148.

10. Singh , Mura ri P. an d others, “SAFE D iagr am - A Design a nd Reliability Tool for

Turbine B lading,” Pr oceedi ngs of th e Sevent eenth Tur bomachin er y Symposium , Tur -

bomachinery Laboratory, Texas A&M University, College Station, Texas (November

1988), pp. 93-101.

11. Thomson, William T., Theory of Vibrat i on wi th A ppl icat ions , 4th Ed ition, Englewood

Cliffs, New J ersey: P rentice-Ha ll, 1993.

chapter 5 rotor modeling

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