3-analytical techniques for evaluation of compressor-manifold response - jcw&frs

8/12/2019 3-Analytical Techniques for Evaluation of Compressor-manifold Response - Jcw&Frs

1/10

Divisions or Sections, or p rin ted in its publications

$1.50 PER75C TO SME

COPY

MEMBERS

Discussion is ~ r i n t e d nly if the paper is publiin an SME journa l or Proceedings.

Released fo r general pu blication upon presentation

Analytical valuation of

Compressor-Manif oldJ. C. WACHELF. R. SZEN SI

Engineering Dynamics, Inc.16117 University OakSan Antonio, X 78249

This paper discusses the types of vibration and stress problem encountered in reciprocating compressor -manifold piping systems and mathematical analysis methods and digital

computer programs developed in the research sponsored by the Pipeline and CompressorResearch Council of the Southern Gas Association. These techniques can be used tocalculate natural frequencies, mode shapes, vibrational amplitudes, and dynamic stressescaused by acoustical and mechanical excitation forces in the system, and they have beenused in the design of domestic and foreign compressor installations representing well over10 mil lion installed horsepower. Comparison of these predictions wi th experimental fieldresults has shown that accurate estimates of dynamic stress and impending failures can beobtained.

Contributed by the Petroleum Division of The American Society of Mechanical Engineers for presentation at the SME Petroleum Mechanical Engineering Conference, Tulsa, Okla. September 211969. Manuscript received at SME Headquarters June 27, 1969.Copies will be available until July 1, 1970.

THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS, UNITED ENGINEERING CENTER, 345 EAST 47th STREET, NEW YORK, N.Y.


2/10

J. C. WACHEL

E x ce s si v e v i b r a t i o n s a nd s t r e s s e s i n t h ep i p i n g o f r e c i p r o c a t i n g c om pr es so r u n i t s a r e aDroblem i n t h e n a t u r a l g a s a nd p e t r o ch e m i ca l i n -

dus t ry. Compresso r cy l ind e r s and th e i r man i fo ld -

in g bo t t l e s form a complex mechanical system sub -

j e c t ed t o dynamic p u l s a t i o n f o r c e s i n t he b o t t l e sand c y l i n d e r s . S ou th we st R e se a rc h I n s t i t u t e h a sd ev el o pe d a n a l y t i c a l t e c h n i q u e s f o r s o l v i n g t h e s ev i b r a t i o n and s t r e s s problei i is . This r e sea rch wassponsored by the Pipel ine and Compressor ResearchCouncil (PCRC) of th e Sou ther n Gas As soc iat ion .

The complete compressor-

mani fold sys tem i ss i m u l at e d m a t h e ma t i c a ll y b y d e f i n i n g t h e s t i f f h e s sa nd ma ss c o n t r i b u t i o n o f a l l s t r u c t u r a l members .N a t u r a l f r e q u e n c i e s a nd mode s h a p e s , b o t h p a r a l l e la nd p e r p e n di c u l a r t o t h e e n g i ne c r a n k s h a f t , a r ec a l c u l a t e d . Vi b r a t i o n s a nd r e s u l t a n t s t r e s s e sin t roduced i n t o the nozz l es can a l so be de te rmined .R ec en t work on t h e f l e x i b i l i t y a t n o z z l e - b o t t l eJ u n c t i o n s h a s s i g n i f i c a n t l y i mp ro ve d t h e a c c u ra c yo f t h e a n a l y t i c a l c a l c u l a t i o n s . C om pr es so r i n -

s t a l l a t i o n s r e p r e s e n t i n g w e l l ov e r 10 m i l l i o ni n s t a l l e d h or s ep ow er h av e b ee n d es i g n ed u t i l i z i n gthese t echn ique s . Th i s paper w i l l d e s c r i b e t h e

b a s i s o f t h e a n a l y s i s an d how i t i s u sed i n i n -dus t r y t o des ign compresso r sys t ems.

COMPRESSOR - MANIFOLD SYSTEM

A t yp ica l compresso r - manifold system shown

F i g . 1 Typical compressor - manifold system

i n F i g . 1 i s a g a s e n g in e w hi ch i s d r i v i n g f o u rc om pr es so r c y l i n d e r s i n t h e h o r i z o n t a l p l a n e .The gas en te r s t he r ec ip ro ca t i ng compresso r cy l -

i n d e r s t h ro u gh t h e s u c t i o n ma n if o ld b o t t l e ; i t i sthen compressed and discharged i nt o a disch argemani fo ld bo t t l e . These suc t ion and d i scha rgem a n i fo l d b o t t l e s a r e u s u a l l y de s i g ne d t o r e du cet h e p u l s a t i o n s i n t r o d u c e d i n t o t h e g a s e s b y t h erec ip roc a t ing compress ion p rocess . The nozz lesw h i c h a t t a c h t h e m a n i f o l d b o t t l e s t o t h e c o m -

p r e s s o r c y l i n d e r s a r e u s u a l l y d e s i gn e d t o mi ni mi ze

the load ing on the cy l inde r s and p rov ide p rope ra c o u s t i c a l f i l t e r i n g , w i t h c o n si d e r at i o n s a l s og i ve n t o t h e i r f l e x i b i l i t y a nd p r e ss u r e d ro p c h ar -

a c t e r i s t i c s .The pr imary s t re s s problem in compressor

m a ni f ol d i n s t a l l a t i o n s i s t he no z zl e s i n c e i t

Fig.2 Comparison of measured vi br at io n andp u l s a t i o n s

PULSATION

UNmALANCcD DYNAMIC rAlLURC" .RATION

MCCYANICAL r o ~ c r s S T R C ~ S P R O B A ~ I L I T YCNCROY

I I II I I II I I ! I

NATURAL

&rLkINGs CRLOUCNCbLS

F i g . 3 Flow c h a r c f o r a n a l y s i s o f s ys te m r e l i a -

b i l i t y


3/10

connects the major masses of the system i . e . t h es u c t i o n b o t t l e , d i s c ha r g e b o t t l e , an d co mp re ss orcy l i nd er s . When a vibr at io n resonance of one oft h e s e m as se s o c cu r s , r e l a t i v e d e f l e c t i o n s be -

tween the masses cause dynamic s t re ss es i n t hen o zz l es . The r e s u l t a n t s t r e s s i n t h e n o zz l e s i sa combina t ion of the f l ex ur a l bending and to r -s i o na l s t r e s s e s .

Experience based upon approximately 400f i e l d e v a l u a t i o n s t u d i e s a t r e c i p r o c a t i n g com -pres so r sys t ems has shown th a t , whenever f a i lu re so c c u r, t h e c a u s e c a n u s u a l l y be t r a c e d t o t h eex c i t a t i on o f a mechan ical r e sonance . Rec ipro -c a t i n g p i s t o n m o ti o n g e n e r a t e s p u l s a t i o n e n erg y

a t e v e ry e n g in e harmonic ; however, acous t i ca lresonances due t o the combinat ion of manifoldvolumes nozz ies , chokes , i n t e rna i passage geomet r y, a nd t h e s t a t i o n p i p i ng c an ca us e c e r t a i n f r equency component s t o be ampl i f i ed . Qu i t e o f t ent h e a c o u s t i c a l r e s o n a n c e w i l l c o i n c i d e w i t h amechan ical na tu ra l fr equency o f the sys t em. Th i sr e s o n a n c e c o n d i t i o n u s u a l l y c a u s e s e x c e s s i v e v i -

bra t i ons . However, h igh ampl i tude pu l s a t i onsaway from a mechanical natural frequency may notc a us e e x c e s s i v e v i b r a t i o n s . T he se f a c t s a r e b o rn eo u t by t h e d a t a g i v e n i n F i g . 2 .

When a mechanical na tu ra l f requency i s ex -

c i t e d b y ac o u s t i c a l p u l s a t i o n s r e s u l t i n g i n ex-

NOMENCLATURE

A~~~= c r o s s - s e c t i o n a l a r e a o f c r o s sh e a d g ui d e

ADP = c r o s s - s e c t i o n a l a r e a o f d i s t a n c e p i e ce

Ai = c o n s t a n t s

BJ = b o l t e d j o i n t i n e r t i a o f c r o s sh e ad g ui de1

BJ2 = b o l t e d j o i n t i n e r t i a o f d i s t a n c e p i ec e

D = d i a m e t e r o f b o t t l e

d = d iamete r o f nozz le

E = m o d u l u s o f e l a s t i c i t y

FCHG=

s h e a r form f a c t o r o f c r o ss h e a d g u i d e

FDP = s h e a r f or m f a c t o r o f d i s t a n c e p i e c e

G = m o d u l u s o f r i g i d i t y

I C H ~ = moment of i n e r t i a of crosshead guide

IDP = moment o f ine r t i a o f d i s t ance p iece

K. = g e n e r a l i z e d s t i f f n e s s e s1

K i j = s t i f f n e s s m a tr i x

K ~= f l e x i b i l i t y m a tr i xi

L = l e n g t h o f n o z z l e

- l e ng t h of c rosshead gu ideL~~~LDP = l e n g t h o f d i s t a n c e p i e c e

M = gen era l i zed moments

M. = genera l i zed masses

P, F = g e n e r a l i z e d f o r c e s

x = d e f le c t io n p a r a l l e l t o x a x i s

xi = g e n e r a l i z e d d i s p la c e me n t s

y = d e f le c t io n p a r a l l e l t o y a x i s

z = d e f le c t i on p a r a l l e l t o z a x i s

B = j o i n t f l e x i b i l i t y f u n c t i o n

= Kronecker de l t a

= damping ra t io

= e igenva lues o f s t i f f ' nes s - mass ma t r ix

= f u n c t i o n

SUBSCRIPTS

C = c y l i n d e r

CHG = c rosshead gu ide

DB = d i s c h a r g e b o t t l e

DN = d i scha rge nozz le

DP = d i s t a n c e p i e c e

SB = s u c t i o n b o t t l e

SN = s u c t i o n n o z z l e

7 [ = dummy i n d i c e s


4/10

F i g . 4 SGA c om pr es s or i n s t a l l a t i o n d es i g n f a c i l i t y

c e s s i ve v i b r a t i o n s and s t r e s s e s , a l t e r a t i o n o f t h ed e si g n i s n e ce s sa r y t o e l i m i n a t e the problem. Thisso lu t ion can be accompl i shed in e i t he r o f two ways.One i s t o r e d uc e t h e m a gn i tu de o f t h e p u l s a t i o n sa nd u n ba l a nc e d f o r c e s s u f f i c i e n t l y t o l o we r t h ev i b r a t i o n a m p l i t ud e s t o a s a f e l e v e l . The se co ndmet hod i s t o c ha ng e t h e m e c ha n ic a l n a t u r a l f r e -quency so t ha t co inc idence wi th the ma jo r acou s t i cf r e q u e n c i e s d o e s n o t o cc u r. T h i s w i l l r e s u l t i na r e d u c t i o n o f t h e v i b r a t i o n a m p l i t ud e s p r op o r -t i o n a l t o t h e m ec ha ni c al a m p l i f i c a t i o n f a c t o r ( Q ) .C ha ng in g t h e n oz z l e s i z e , l e n g t h , w a l l t h i c k n e s s ,o r t h e m a s s e s o f t h e m a n i f o l d b o t t l e s t o c h a n g ethe mechan ical na t u r a l f r equency o f the sys t emmay a l so a f fe c t t he acou s t i c r e sponse . Wheneverc ha ng es a r e made i n t h e s y s t e m, t h e e f f e c t s o f t h eacou s t i c r e sponse must be r e - e v a l u a t e d t o b e s u r eth a t no new co inc idence o f the a cous t i c and me -

c h a n i c a l f r e q u e n c i e s o c cu r. As d i s c u s s e d i n ap r e v i o u s p a p e r L), i t i s n e c e s s a r y t o make a

l u n d e r l i n e d n um be rs i n p a r e n t h e s e s d e s i g n a t eR e f er e nc e s a t t h e e nd o f t h e p a p e r.

c om pl e te s y st em a n a l y s i s i n o r d e r t o i n s u r e t h ea de qu ac y o f t h e s ys te m r e l i a b i l i t y. F i g .3 g i v e st h e f lo w c h a r t f o r a n a l y s i s o f t h e s ys te m r e l i -a b i l i t y. The u l t i m a t e c o nc e rn o f t h e e n g i n ee ri s w h et h er o r n o t t h e s ys te m w i l l f a i l . In o r d e rt o d et e rm in e t h i s f a i l u r e p r o b a b i l i t y o r t h es a f e t y f a c t o r of t h e i n s t a l l a t i o n , i t i s n e ce s sa r yt o d e f i n e t h e p u l s a t i o n a nd me c ha n ic a l e ne rgy i nt h e s ys t e m, t h e n a t u r a l f r e q u e n c i e s , t h e un b al -

a nc ed f o r c e s , t h e v i b r a t i o n a m p l i t u d e s , an d t h er e s u l t a n t d yn am ic s t r e s s e s . T he se a r e a s of co n -

cern have been emphasized by the Southern GasAss oci at i on ' s PCRC re se arc h over the pas t 1 5 y e a r

The Southern Gas Ass oci at i on Compressor In st a l -l a t i o n D es i gn F a c i l i t y h a s become t h e s t a n d a r dd e s i g n t e c h n i q u e f o r d e t e r m i n i n g t h e a c o u s t i c a le n e rg y i n p i p i n g s y st e m s. T h i s f a c i l i t y, showni n F i g . 4 h a s b e e n d e s c r i b e d i n s e v e r a l p a p e r s

2-2). T h i s s y s t e m u s e s a n e l e c t r i c a l a n a l o gmode l of t h e a c o u s t i c p u l s a t i o n s i n t h e pi p i n gs y s te m , t h e r e b y e n a b l i n g o ne t o a c c u r a t e l y mea s -u r e p u l s a t i o n s a nd un b al a nc e d f o r c e s i n t h e l a b -

o r a t o r y. T h i s i n f o r m a t i o n can the n be used wi th


5/10

K - A X I A L STIFFNESSK - F L E X U R A L STIFFNESSK T - TORSIONAL STIFFNESS

N G I ~ ~

F i g . 5 Spring and mass n o d e l o f compressor-mani -fo ld sys tem

+MOTION

rnii + P - F l - F z 0

mi. Fs+Fq+FeO

B + M - P O + T , + T ~ - S , ~ , - F ~ ~ - O

F i g . 6 Free - body diagram of c yl i nde r

the mechanica l na tura l f requencies and mode shapedat a of th e compressor - manifold sys tem to pre - COMPRESSOR CYLINDERSd i e t v i b r a t i o n a nd s t r e s s a m p li t u d es .

Fo r t he cy l i nde r equa t i ons o f mo t ion , s t i f f

CALCULATIONS OF NATURAL FREQmNCIES

A mathemat ica l model of the e l a s t i c and mass

p ro pe r i t e s o f a compres so r - manifold system hasbeen developed w h i c h s i m u l a t e s the response of thephys i ca l sy s t em. The sy s tem i s concep tua l l y r e -duced t o masses and spr i ngs a s shown i n F i g . ? .The nozz l e s and c ro s shead gu ide a r e r ep r e sen t eda s f l e x u r a l , a x i a l , a nd t o r s i o n a l s p r i n g s wh il et h e c y l i n d e r s a nd b o t t l e s a r e m as s es . The d e -g r e e s o f f re ed om o f t h e b o t t l e s a nd c y l i n d e r s a r ei l l u s t r a t e d i n F ig s .6 , 7 , and 8 . T h e n a t u r a l f r e -que nci es and mode shape s of th e system can be ob -t a i n e d b y s o l v i n g the d i f f e r e n t i a l e q ua t io n s ofmotion f o r t h e masse s o f t he sy s t em.

The equat ions of mot ion which were wr i t tenf o r t h e s y s t e m a r e o f t h e form:

M. z + C i i i t K i x i 0I 1)

where :

nes s o f t he c ro s shead gu ide and d i s t ance p i ece

mus t be de f i ned . The s t i f f ' n e s s e s of t he c r o s s -h ea d g u i d e , d i s t a n c e p i e c e , a nd b o l t e d j o i n t s a r ec a l c u l a t e d a s s p r i n g s i n s e r i e s t o o bt a i n t h ee f f e c t i v e s p r i n g c o n s t a n t s f o r t he f l e x u r a l , r o -t a t i o n a l a n d e x t e n s i o n a l s p r i n g s F i g . 5 ) . Theexpress ion for the combined s t i f f h e s s o f t he sebeams was de termined by cons i der i ng the r es ul t i ngd e f l e c t i o n c au se d b y a u n i t f o r c e a c t i n g a t t h ec e n t e r o f g r a v i t y o f t h e c y l i n d e r ( F i g . 6 ) . Thec a l c u l a t i o n o f d e f l e c t i o n c o n s i d e r s t h e s h e a rf o r c e s an d t he s l o p e s a t e a c h o f t h e j u n c t i o n sbe tween t he d i f f e r en t component s . The r e s u l t i ngexp re s s ion s f o r de f l e c t i on and s l ope can be com -b i n e d i n t o t h e f o l l o w i n g e q u a t i o n s :

i = i t v i b r a t i n g m as s where

K - l K - l K - lCi = damping on i

t hmass w 15 61 ~

F or s i m p l i c i t y i n s o l v i n g f o r t h e n a t u r a l The f l e x i b i l i t y m a t r ix , ~ 5 1 w h i c h r e l a t e sf req uenc ies the damping terms ar e a l lowed to van- t h e d e f l e c t i o n s x an d t o t h e f o r c e s P and i s

i sh . The p e ~ , c e n t c r i t i c a l dampi ng i n c o mp re ss or - symmetric. In m a t r i x form t he equ ati on becomes:manifo ld sys tems h as been determined by exper i -m e nt a l f i e l d t e s t s t o be a pp r o xi m a t el y 0 . 0 5 A = K I Fr l 5 5 4( Q - 10 ) . The d i f fe re nc e be tween the damped andundamped n a t u r a l f r e q u e n c i es i s t r i v i a l . where A,, r e p r e s e n t s t h e d i s p l a c e m e n t v e c t o r ; ~ i i


6/10

M,e s s

U, Xsm- Xc+b,8+ FM

T+= J

J+-M,-M2-F,R+F2R+Kr+-

DISCHARGE BOTTLE EOUATIONS

Fx,, - M D , ~ , ,FY~, - Msa YD

X F ~ O rn

XM-

r n 2 , - F q = O

J i T I F e a t 0

~ i g . 8 Free - body diagram of bottles

DB

Fig.7 Free - body diagram of system determined as a function of the stiffnesses. The

resulting equations for zSB and asB can be solved

as in equations (2) and 3 ) . For these vectors

the flexibility matrix; and F;. , the force andmoment vector. K ~ ; 5 , 5 K t ; = (LsN. I S N . ASN, D S N . PSB. E , G 7)

The calculation of the flexibility matrix

inverse which is the stiffhess matrix K allowsl All of the equations o f motion are now in

the solution of the force vector. the form:

The forces and moments resulting from de -

flections of the crosshead guide and distance

piece spring are now determined for use in the

equation of notion.

The differential equations of motion wereconverted into matrix form for simplicity of

solution. The general form of the equation is

MANIFOLD BOTTLES

For the manifold bottles the major springs

Are the suction and discharge nozzles. The noz -

zles have forces and moments which cause torsional

and bending deflections (Fig.7) . An expressionfor the deflections xsB and C can be written interms of the forces and moments imposed upon thenozzle from the cylinder similar to equations (2)

and (3). For these displacement vectors

K KC: K t 5 = + ( L s N . I S N , F S N DSN, P S N , E , G

(6)The equations for the discharge bottle can bewritten similarly.

The other degrees of freedom of the bottles

are illustrated in Pig.8. The motions are acombination of linear and angular movements from

the equilibrium position. The calculation of theforces and torques on the bottles require thedetermination of the torsional and flexural spring

stiffnesses. The forces acting on the bottles as

a result o f the linear and angular motion can be

where M is the diagonalized mass matrix and K isthe stiffhess matrix. The form of these equationslends itself quite readily to the eigenvalue-

eigenvector solution method.2The relationship between x and y i i = w x)

is used to simplify the equation of motion:

where the w 2 represents the diagonalized eigen-

value matrix which is called lambda ( A ) .

The diagonalized mass matrix inverse is ob -

tained by forming a matrix whose diagonal i s theindividual inverse of the individual masses.

Multiplying both sides of the matrix equation byM inverse ( iT1) gives


7/10

C Y L l CYL 2

F i g . 9 E x c i t a t i o n f o r c e s i n co mp re ss o r - manifold F i g . 1 0 Comparison of ca lc ul at ed and measureds ys te m s u c t i o n b o t t l e v i b r a t i o n s

Rear rang ing g ives :

T h i s f or m o f a m a t r i x e q u a t i o n i s t h e f a m i l -i a r e i g e n va l u e e q u a t i o n . The v a l u e s o f A f o rw hi ch t h e e q u a t i o n i s s o l u b l e a r e known a s t h ec h a r a c t e r i s t i c v a l u e s o r e i ge n v a l ue s of t h e m a t r i x .The p ro bl em o f f i n d i n g t h e v e c t o r s w h ic h s a t i s f yt h e e qu a t i o n i s t h e r e f o r e c a l l e d t h e e i g e nv a lu eprob lem f o r the g iven ma t r ix . Cor respond ing ly,t h e v e c t o r s o l u t i o n s a r e t h e e i g e n v ec t o r s o f t h em a t r i x ( T K ) , which i s r e f e r re d t o a s t h e s t i f f -ness - m a ss m a t r i x .

P h y s i c a l l y, t h e e i g e n v e c t o r s r e p r e s e n t t h emode s h ap e s o f t h e p a r t i c u l a r v i b r a t i o n c o r r e s -ponding t o the e igenva lue which r ep re sen t s the v i -b r a t i o n a l n a t u r a l f r e q u en c y. The e q u a t i o n ob-

Table 1 C a l c u l a t e d v e r s u s m e a s u r e d n a t u r a l f r e -quencie s

Mode f c a l c . m e a s

Unit A r i g ~ dbody x ) 25 4

suction bott le c ) 42 38

cylinder (x) 58 58

Unit B r o t a r y x ) 6 7 73

cylinder x) 39 39

suction bott le 2) 28 9

d ischarge bo tt l e 2 ) 1 9 2 0

Unit r i g ~ dbody x ) 18 15

r o t a r y x ) 76 72

discharee bo tt l e 2 ) 33 3 1

t a i n e d when t h e de t e r m i na n t o f t h e c o e f f i c i e n tm a t r i x v a n is h e s i s known a s t h e c h a r a c t e r i s t i ce q u a t i o n o f t h e m a t r i x an d t h e v a l u e s o f A f o rwhi ch t h e e qu a t i o n i s s a t i s f i e d a r e t h e d e s i r e de i g e n va l u e s . I n g e n e r a l , t h e c h a r a c t e r i s t i c eq uat i o n w i l l have n r o o t s w i t h n u e i g e n v e c t o r s .

The so lu t ion fo r the e igenva lues o f a sys t emw i t h n u degrees of freedom would be an 'nth'' o r dequ at i on whose so lu t i on would give n u r o o t s o ft h e c h a r a c t e r i s t i c e q u a t i o n . The n roots wouldt h e n r e p r e s e n t t h e v i b r a t i o n a l f r e q u e n c i e s s q u a r e d .The ch a r ac te r i s t i c equa t i on o f the example wouldb e a s f o l l o w s :

where Ai, = 0 n r e p r e s e n t s f u n c t i o n s o f m a s sa n d s t i f f n e s s .

When t h i s c h a r a c t e r i s t i c e q u a t io n i s s o l ve df o r t h e r o o t s o r e i g e n v a l u e s , t h en t h e e i g e n v e ct o r scan be ob ta ine d . The e ig enve c to r s a r e ob ta inedu s i n g e q u a t i o n ( 1 2 ) . U si ng t h e s t i f f n e s s - m a s sm a t r i x M - ~ K )a n d m u l t i p l y i n g b y a n e i g e n v e c t o rX a nd f o r c i n g t h i s t o e q u a l a n e i g e n va l u e t i m e st h e unlmown e i g e n v e c t o r w i l l r e s u l t i n a s e t o fe q u a ti o n s . T h is s e t of e qu a t i o ns i s s o l u bl e f o ra un ique e igenvec to r d i r ec t i on ; however. t he mag -n i tude r emains u n d e t e r m i n e d . These e igenvec to r srep res en t the mode shape o f v ib ra t ion co r res -p on di ng t o t h e e i g e n v a l ue o r n a t u r a l f r e q u en c y o ft h a t v i b r a t i o n .

MODES OF VIBRATION

A res onan t mode shape normal l y occurs f o reach degree of freedom. The lowest vi br at io nmode i n a t y p i c a l s ys te m i s g e n e r a l l y t h e z r e -


8/10

CYLlNDER.M E S U R E D

C A L C U L AT E D

F i g . 11 Comparison of calculated and measured Fig .12 Comparison of calculated and measuredc y l i n d e r v i b r a t i o n s d i sc h ar g e b o t t l e v i b r a t i o n s

s on an ce o f t h e m a ni f o ld b o t t l e s . I f t h e b o t t l ei s n o t c o m p l e t el y s ym me tr ic w i t h r e s p e c t t o t h ecy l inde r and nozz le lo ca t i on s , a combined a - - z

r es on an ce o f t h e b o t t l e w i l l o c c u r.I n t he x d i r e c t i o n w hich i s p a r a l l e l t o t h e

bo t t l e ax es , s eve ra l r e sonan t modes occur whicha r e o f i n t e r e s t i n p r a c t i c a l p ro bl em s. A lowf r e q u e n c y r i g i d b o d y m o t i o n o f t h e m a n i f o l d b o t t l e sa nd a l l c y l i n d e r s i s u s u a l l y t h e p r e do mi n an t modeshape. Another x r e sonan t mode i s t he ro ta ry modei n which the two b o t t l e s move i n oppos i t e x d i -r e c t i o n s and t h e c y l i n d e r s r em ai n a t r e s t . I na d d i t i o n a c y l i n d e r r e s o na n ce mode o c c u r s i nw hi ch t h e m a n if o l d b o t t l e s r em ai n a t r e s t w hi l et h e c y l i n d e r s a r e a t r e s o n an c e.

Othe r r e sonances which occur a t h i ghe r f r equenc ies a r e the c y l ind e r z r e sonan t mode whicho c c u r s c o l i n e a r w i t h t h c c y l i n d e r l o n g i t u d i n a lax i s . The cy l in de r r e sonance i s a mode shapei n w hi ch t h e c y l i n d e r r o t a t e s a b ou t i t s l o n g i -t u d i na l a x i s .

CALCULATION OF VIBRkiTIClN ND STRESS

A computer program which uses the e i g e nv e c t o r m eth od f o r s o l v i n g t h e m e ch a ni c al n a t u r a lf r equenc ies and mode shapes o f the compressor

m a n i f ol d s ys t em e n a b l e s on e t o c a l c u l a t e t h ef o r c e d v i b r a t i o n r e s p on s e o f t h e s y st e m u s i n gv a r i o u s f o r c i n g f u n c ti o n s a t d i f f e r e n t m ass l o -c a t i o n s . A Fourier expansion of any complexf o r c i n g f u n c t i o n , i n c l u d i n g t h e p h a s i n g, c a n b eapp l i ed a t each mass lo ca t i on . Complex waves oft y p i c a l u nb al a nc ed f o r c e s i n t h e s u c t i o n an d d i s -c h arg e b o t t l e s a n d c y l i n d e r s a r e shown i n F i g .9 .

s t a l l a t i o n d es ig n f a c i l i t y .The harmonics of the complex for ci ng func -

t i o n s s i m u l a t i n g t h e m a g n i t u d e a n d d i r e c t i o n o ft h e a c t u a l f o r c e s e nc o un t er e d i n t h e s y st em a r eapp l i ed t o the mathematical model . The dynamicd i sp lacemen t s r e su l t in g from the app l i ed ha rmonicf o r c e s o f a l l t h e ma ss es ca n b e c al c u l a t e d . Vi -b r a t i o n s a t v a r i o u s h ar mo ni c f r e q u e n c i e s c a n b erecombined as a complex vib ra t i on waveform. S t r

These fo rc ing func t ions can be ob ta ined f rom e x F i g . 1 3 E xp er i me nt al f a c i l i t y f o r b r a n ch c o n ne c ti o np e r i m e n t a l f i e l d dat a o r t he SGA compressor i n r e s e a r c h


9/10

v a l u es ba s ed u pon t h e r e l a t i v e d e f l e c t i o n s b e -

tween connec t ing members a re ca l c u la t e d f o r eachharmonic f r equency. F lexur a l bend ing and to r s ion a ls t r e s s e s i n t h e n o z zl e s c a l c u l a t e d f o r e ac h ty peof mot ion a re then combined in to the r e su l t an tmaximum she a r i ng and p r in c i pa l s t r e s s es by thecombined s t r e s s equa t ions .

CORRELATION WITHFIELD

STUDIES

Mechanical natural f requencies and modes h ap e s m ea su re d d u r i n g f i e l d e v a l u a t i o n s t u d i e sof the compressor - mani fo ld sys t em have co r re l a t edw el l w it h c a l c u l a t e d r e s u l t s . A comparison ofm ea su re d a nd c a l c u l a t e d n a t u r a l f r e q u e n c i e s o fthe most commonly occ urr ing modes i s g iven i nTable 1.

I n a d d i t i o n , v i b r a t i o n d e f l e c t i o n s a nd dy -

n ami c s t r e s s e s a r e r o u t i n e l y m ea su re d on f i e l de v a l u a t i o n s t u d i e s a nd a r e c omp ar ed w i t h t h e c o r -

r es po nd in g v i b r a t i o n s an d s t r e s s e s c a l c u l a t e d b y

t h e p r e v i o u s l y d e s c ri b e d a n a l y t i c a l t e c h ni q u e s .A compar i son o f the ca lcu la t ed and measured va lueso f v i b r a t i o n s f o r a t y p i c a l f i e l d e v a l ua t i o ns t ud y a r e p r e s en t e d i n F i g s . 1 0 , 11 , and 12. Whilec o r r e l a t i o n o f i n d i vi d u a l d a t a p oi n t s v a r i e s , t h eove ra l l t r e nds compare f avorab ly.

When f i e l d da ta on the v ib ra t i on a l ampl i -

tudes of the compressor - mani fo ld sys t em a r e ava i l -

a b l e , t h i s a n a l y t i c a l t e ch n iq u e c an b e us ed t op r e d i c t t h e s t r e s s e s o c cu r r i n g i n t h e sy st em . Them ea su re d v i b r a t i o n a l a m p l i t ud e a t a p o i n t c a n b eu se d t o d e t e r m i ne t h e v i b r a t i o n a l a m p l it u d e s an ds t r e s s e s o f t h e s ys te m a t a p a r t i c u l a r f r e q u e r c y.

FLEXIBILITI I N NOZZm-BOTTLE JUNCTIONS

To ac cur a t e l y c aj ~ u l ?e n a t u r a l f r e q u e n c i e sa n d s t r e s s e s i t i s n e ce s s a r y t o c o n s i d e r t h en o z z l e - b o t t l e j o i n t f l e x i b i l i t y f a c t o r . The s h o r t ,s t i f f n o z zl e s t y p i c a l l y used i n compressor-man-i f o l d systems c,ause t he b o t t l e w a ll t o d e f l e c ta nd t h e j o i n t t o r o t a t e . A n al y si s o f t h e s e d e -

f l e c t i o n s i s c om pl i ca te d b y t h e u se o f r e i n f o r c e db r a n ch c o nn e c t i on s s u ch a s s a d d l e s , p a d s , sweepo-l e t s , w e l d o l e t s , d r a w n o u t l e t s , e n c i r c l e m e n ts a d d l e s , a nd t e e s . E x pe r i me n ta l d a t a f o r e a c hr e i n f o r c e d b r a n c h c o n n e c t i o n a r e n e c e s s a r y t od et er mi ne t h e j o i n t f l e x i b i l i t y f un c t i on , s i n c et h e c ompl ex g e o me t ry i s n o t r e a d i l y s o l v a b l e b ys i mp le s h e l l t h e o r y. D at a o n s e v e r a l t y p e s o fb ranch connec t ion des igns has been ob ta ined byo t h e r s 6 ) .

The au t hor s have conduc ted s t a t i c and dy -

nam ic t e s t s on s e v e r a l n o z z l e - b o t t l e r e i n f o r c e db r a n ch c o n ne c t i o n s i n c l u d i n g s a d d l e s , s w e e p ol e t s ,

f l e x i b i l i t i e s a n d s t r e s s i n t e n s i f l c a t l o n f a c t o r sf o r b e nd i ng , s h e a r , and t o r s i o n a l l o a d s. I na d d i t i o n t h e e f f e c t of t h e v a r i o u s r e i n f o r c e m e n t supon the mechan ica l na t u r a l f r eque nc ies o f thesys tem were ob ta in ed . The exper imen ta l f a c i l i t yused t o make th ese measurements i s shown i n F i g . 1 3 .

Equa t ions which can be used t o ca l cu l a t et h e j o i n t f l e x i b i l i t y f u n c t i o n s ha ve b ee n o b t a i ne d

f ro m a n a l y s i s o f t h e m ea su re d d a t a .

CONCLUSIONS

This pape r has des c r ibed methods which a r ep a r t o f a c o n t i n u i n g e f f o r t b y t h e S o u t he r n GasA s s o c i a t i o n a n a S o u t h w e s t R e s e a r c h I n s t i t u t e t op red ic t t he r e sponse o f compresso r - manifold sys -

tems. Some of t he major conc lus ions which can bemade are:

1 Comparison between calculated and rneas-ured mechan ical n a t u r a l f r equen c ies has shown

agreemen t wi t h in accuracy r equ i remen t s needed f o rd e s i g n o f r e c i p r o c a t i n g c om pr es so r i n s t a l l a t i o n s .

2 v i b r a t i o n s o f t h e c o mp re ss o r - manifolds ys t em c an b e c a l c u l a t e d b y u s i n g t h e a c o u s t i c a lfo rc es i n the mathematical model . Measured andpred ic t ed v ib r a t i on ampl i tudes compare f avorab l y.

The j o i n t r o t a t i o n a t t h e n o z z l e - b o t t l ec o nn e ct i on g r e a t l y a f f e c t s t h e c a l c u l a t i o n o fn a t u r a l f r e q u e n c i e s , mode s h a p e s , v i b r a t i o n , a nds t r e s s e s . R e se a r ch c on du ct ed d u r i n g t h e p a s ty e a r h a s r e s u l t e d i n im pr ov ed e q u a t i o n s f o r p r e -

d i c t i o n o f t h i s f l e x i b i l i t y , and met hods f o r t h es e l e c t i o n o f t h e optimum f i t t i n g f o r p a r t i c u l a r

load ing cond i t ions have been evo lved .The a n a l y t i c a l t e c h n i q u e s d e s c r i b e d ca n

b e us e d i n t h e d e s i g n s t a g e i n c o n j u n c t i o n w i t hthe Sou the rn Gas Assoc ia t ion compresso r in s t a l -

l a t i o n d e s i g n f a c i l i t y t o s t u d y t h e combinede f f e c t s o f a c o u s t i c a l a n d m e c h a n i c a l c h a n g e s .The dy na mi c s t r e s s e s a n d f a t i g u e l i f e o f t h e com -

p r e s s o r n o z z l e s c a n b e p r e d i c t e d .5 In e x i s t i n g i n s t a l l a t i o n s i t i s p o s s i b l e

t o e s t i m a t e t h e s t r e s s e s i n t h e c om pr es so r n oz z l e scaused by v i b ra t i ons measured on the ac tu a l i n -

s t a l l a t i o n b y f o r c i n g t h e m at he ma ti c al s ys te m t ohave the same v ib ra t i on ampl i tudes . By us ingt h i s t e ch ni qu e a n i n c i p i e n t f a i l u r e c an be c o r -

r e c t e d b e f o r e i t o c c u r s .

1 Wachel, J . C . , " Consideration of Mechan -

i c a l System Dynamics i n P lan t Design, ASME PaperNo. 6 7 - ~ ~ ~ - 5 .

ads , and we ldo le t s . These t e s t s de te rmined the 2 N i m i t z , Walter, and Damewood. Glenn, Com-


10/10

p r e s so r I n s t a l l a t i o n D es ig n U t i l i z i n g a n E l e c t r o -Acoust ical System Analog, " ASME Paper No. 61 - WA -

290.3 Henderson, E.N. , "Gas P u l s a t i o n s T h e

Problem, Southern Gas A s s o c i a t i o n l s Approach,R e s u l t s , " O i l G a s J o u r n a l , Vo l . 56, No. 1 9

1958.4 Hughes, J.V. and Sharp, J . M . , " The

Des ign and Opera t ion o f an ~ l e c t r i c a l Analog of

a R e c i p r o c a t i n g Conpressor I n s t a l l a t i o n , ASMEPaper No. 56 - A- 200, 1956.

5 Damewood, Glenn, " S GA and Southwest Rese ar ch Team Up t o Take th e P ul se of Compressorp i p i n g , " P i p e l i n e I n d u s t r y, J u l y, 1 9 5 6 , p p 30 -3

6 R od ab au gh , E .C ., e t a l . , " Evaluat ion ofExpe r imen ta l and The ore c t i ca l Da ta on Rad ia lN oz z le s i n P r e s s u r e Ve s s e l s , " S e c t i o n V I , USAEC,TID-24342.

3-analytical techniques for evaluation of compressor-manifold response - jcw&frs

Documents