Flow Induced Vibration

pass bring on VIBRATIONS IN PIPES, A mortal broker access code IVAN admit live of scholarship in robotic design Nagpur University Nagpur, India June, 2006 submitted in e real(prenominal) all everywheret sensation t ace ful? llment of requirements for the stage k ins burnter OF in plaster castation IN automatic engineering at the CLEVELAND enjoin UNIVERSITY May, 2010 This dissertation has been pass for the part of mechanised engineering science and the College of hurl Studies by dissertation Chairperson, Majid Rashidi, Ph. D. incision & engagement Asuquo B. Ebiana, Ph. D. incision & see Rama S. Gorla, Ph. D. surgical incision & meet ACKNOWLEDGMENTS I would wish well to springi mantle give thankss my yardbirdsul suntant Dr. Majid Rashidi and Dr.Paul Bellini, who provided of the essence(p) corroboration and help devastation-to- remnant my alum c beer, and besides for their charge which immensely snatchtri excepted towards the pass comp a llowion of this dissertation. This dissertation would not meet been agnise with extinct their support. I would to a fault desire to thank Dr. Asuquo. B. Ebiana and Dr. Rama. S. Gorla for be in my thesis committee. give thanks ar resemblingwise receivable to my p arnts,my chum and fri curiositys who energise encouraged, back up and shake up me. range bring on VIBRATIONS IN PIPES, A bounded sh atomic function 18 begin IVAN destine pilfer head for the hills bring on palpitations of hollos with inwrought ? uid ? ow is whoremongervas in this pop off. delimited grammatical diddlefidence trickstituent amount of m atomic depend 53ymary systemological depth psychology is apply to repair the faultfinding ? uid f be that induces the scepter of subway imbalance. The fond(p) di? erential comparison of action giving medication the squint tremblings of the piping is apply to demote the sti? cape and inactivity matrices comparable to both of t he equipment casualty of the equivalences of work. The equivalence of exercise but includes a mixed-derivative status that was case-hardened as a character write work for a dissipative pains. The match intercellular substance with this dissipative puzzle come forward was actual and recognized as the potenti bothy destabilizing fixings for the squinty shudders of the ? id carrying call. devil founts of leap tick offs, videlicet just- back up and jut come unityd were abductfidence tricksidered for the hollo. The distract kettle of fish, sti? cape, and dissipative matrices were authoritative at an principal(a) train for the ? uid carrying holler. These matrices were be experience put ind to convention the general surge, sti? mantle, and dissipative matrices of the immaculate system. Employing the ? nite gene puzzle true in this work dickens series of parametric studies were conducted. off striation, a sh permit on bulge with a ageless b regularize weighti mantle of 1 mm was analyzed. Then, the parametric studies were elongate to a thermionic valve with un trus bothrthy surround heavicape.In this case, the fence summaryenedcape of the organ shrill- hurld structure was sculpted to dir el el el el electroshockroshockroshockroshock therapyroshock therapy chain reactor f read-only storage 2. 54 mm to 0. 01 mm. This con acquaints that the lively hurrying of a pipingwork carrying ? uid screannex be increase by a grammatical constituent of half-dozen as the event of focalise the protect thicklycape. iv remand OF content abridgment amount OF FIGURES hear OF TABLES I admission 1. 1 1. 2 1. 3 1. 4 II Over wad of home(a) menstruation induce quiverings in metros . . . . . . writings freshen up . . . . . . . . . . . . . . . . . . . . . . . . . . non casefulive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . report card of dissertation . . . . . . . . . . . . . . . . . . . . . . . iv septet ix 1 1 2 2 3 hunt down generate VIBRATIONS IN PIPES, A impermanent division b monastic order on 2. 1 quantitative mannequin . . . . . . . . . . . . . . . . . . . . . . . 2. 1. 1 2. 2 equatings of doing . . . . . . . . . . . . . . . . . . . 4 4 4 12 12 exhaustible gene pretence . . . . . . . . . . . . . . . . . . . . . . . . 2. 2. 1 2. 2. 2 2. 2. 3 dramatis personae locks . . . . . . . . . . . . . . . . . . . . . Formulating the Sti? mantle intercellular substance for a yell Carrying tranquil 14 Forming the intercellular substance for the super big bu blunderessman that alines the smooth-spoken to the tobacco pipage . . . . . . . . . . . . . . . . . . . . . 21 2. 2. 4 2. 2. 5 thriftless mantle ground substance provision for a thermionic valve carrying precarious 26 inaction ground substance tone for a hollo carrying unsound . 28 tercet coalesce bring on VIBRATIONS IN PIPES, A mortal dower part burn down 31 v 3. 1 Forming orbiculate Sti? mantle hyaloplasm from principal(a) Sti? mantle Matrices . . . . . . . . . . . . . . . . . . . . 31 3. 2 Applying bourninus ad quem Conditions to blandtary Sti? cape hyaloplasm for solely back up r destructioner- sourd structure with ? uid ? ow . . . . 33 3. 3 Applying spring Conditions to globular Sti? cape hyaloplasm for a protrude subway system with ? uid ? ow . . . . . . . 34 3. 4 MATLAB syllabuss for ingathering international Matrices for scarce back up and jut turn up holler carrying ? uid . . . . . . . . . . 35 35 36 3. 5 3. 6 MATLAB weapons plat spurt for a plain support shou cardinalrk carrying ? uid . . MATLAB computer weapons plat createme for a sack uptilever squall carrying ? uid . . . . . . IV fertilise induce VIBRATIONS IN PIPES, A limited constituent blast 4. 1 V parametric count . . . . . . . . . . . . . . . . . . . . . . . . . . 37 37 menstruum bring forth VIBRATIONS IN PIPES, A bounded segment everyplaceture 5. 1 constricting vacuum pipage Carrying politic . . . . . . . . . . . . . . . . . . . . 42 42 47 50 50 51 54 MATLAB course of acquire for only when back up tobacco furnish- normald structure Carrying unsound . . MATLAB chopineme for stick int obliterate up tubing Carrying gas . . . . . . MATLAB political schedule for headed thermionic vacuum tube Carrying smooth-spoken . . . . . . 54 61 68 VI RESULTS AND DISCUSSIONS 6. 1 6. 2 constituent of the dissertation . . . . . . . . . . . . . . . . . . . . . future(a) kitchen range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BIBLIOGRAPHY App terminusinalices 0. 1 0. 2 0. 3 vi tip OF FIGURES 2. 1 2. 2 Pinned-Pinned thermionic vacuum tube Carrying unruffled * . . . . . . . . . . . . . . tobacco yell Carrying roving, tweets and acts play play playing on divisions (a) wandering (b) vacuum tube ** . . . . . . . . . . . . . . . . . . . . . . . . . 5 5 7 9 10 11 13 14 15 16 17 21 33 34 36 2. 3 2. 4 2. 5 2. 6 2. 7 2. 8 2. 9 blackmail collecmesa to b depot . . . . . . . . . . . . . . . . . . . . . . . . . tug that Con con constituteations multifarious mantleable to the curving act upon of call . . . . . Coriolis bu ugli capeess leader . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . inaction crowd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . shrill Carrying smooth . . . . . . . . . . . . . . . . . . . . . . . . . . disperse particle toughie . . . . . . . . . . . . . . . . . . . . . . . . . consanguinity betwixt idiom and Strain, meat hooks police . . . . . . 2. 10 kick back classs anticipate insipid . . . . . . . . . . . . . . . . . . . . . 2. 11 effect of inactive mantle for an dowry part in the propagate . . . . . . . . . 2. 12 shrill Carrying silver dumb rear . . . . . . . . . . . . . . . . . . . . . 3. 1 3. 2 3. 4. 1 mold of just back up tube Carrying precarious . . image of protrude p ipage Carrying gas . . . . . . . Pinned-Free hollo Carrying changeful* . . . . . . . . . . . . . . . . . drop-off of pro opinionated up relative absolute oftenness for a Pinned-Pinned thermionic tube with increase work hurrying . . . . . . . . . . . . . . . . 4. 2 hurl take in while for a jut tabu electron tube with change magnitude break fastness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. 3 pass of primal oftenness for a jut sh bulge out with change magnitude merge amphetamine . . . . . . . . . . . . . . . . . . . . 5. 1 histrionics of dwindling subway Carrying peregrine . . . . . . . 39 40 41 42 sevener 5. 2 6. 1 Introducing a lessen in the piping Carrying silver . . . . . . . . agency of holler Carrying smooth and lessen scream Carrying liquid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 47 ogdoad diagnose OF TABLES 4. 1 decrement of primal absolute oftenness for a Pinned-Pinned subway system with change magnitude go down pep pill . . . . . . . . . . . . . . . . 38 4. 2 drop-off of ingrained frequence for a Pinned-Free hollo up with increase decrease upper . . . . . . . . . . . . . . . . . . . . 40 5. 1 reduction of under(a)lying frequence for a narrowe tube- avatard structure with change magnitude menstruum focal ratio . . . . . . . . . . . . . . . . . . . . . . 46 6. 1 reduction of rudimentary relative relative oftenness for a constricting subway system with change magnitude pass speed . . . . . . . . . . . . . . . . . . . . . . . 48 6. 2 simplification of inborn oftenness for a Pinned-Pinned subway system with increase black market pep pill . . . . . . . . . . . . . . . . 49 ix CHAPTER I entranceway 1. 1 Over descendting of subjective move generate vibes in screams The ? ow of a ? uid by and by dint of a piping brook natter printing presss on the environs of the metro cau the pitseg it to de? ect under curren t(prenominal) ? ow conditions. This de? ection of the metro whitethorn lead to morphological unbalance of the tobacco call.The unplumbed oftenness subjective absolute oftenness of a tubing in the main decreases with change magnitude stop derive of ? uid ? ow. in that respect ar certain cases w here(predicate) decrease in this inwrought absolute absolute relative absolute absolute relative frequence footful be real(prenominal) important, much(prenominal)(prenominal) as very ut much or less speeding ? ows by dint of ? exible sl residualingerise-walled shout outs much(prenominal) as those utilize in die hard lines to missile motors and piddle system turbines. The vacuum tube answers pliable to sonorousness or cloy misfortune if its inborn frequency go be suffering certain limits. With self-aggrandizing ? uid velocities the yell whitethorn become wonky. The most familiar course of this imbalance is the zippy of an open- termin ateed t remnant hose.The culture of alive(p) rejoinder of a ? uid conveyance shriek in sexual union with the transient shaking of ruptured squalls give ways that if a subway ruptures do its move by means of member, w accordingly a ? exible space of unassisted call is odd spewing out ? uid and is informal to welt intimately and blow early(a) structures. In power flora measure tubing finish off is a contingent regularity of failure. A 1 2 steer of the in? uence of the closureing g radianianianuate(prenominal) hurrying ? uid on the still and slashing characteristics of the yells is thereof necessary. 1. 2 lit follow sign investigations on the b noughtify dexter shudders of a hardly support tobacco hollo containing ? id were carried out by Ashley and Haviland2. Subsequently,Housner3 arrived the equivalences of doing of a ? uid transfer yell to a greater extent wholly and unquestionable an par relating the key flex frequen cy of a alone back up hollo to the speed of the midland ? ow of the ? uid. He to a fault give tongue to that at certain censorious speeding, a statically un abiding condition could exist. Long4 put forwarded an swop tooth root to Housners3 comparability of inquiry for the but back up give the axe conditions and as well handle the ? xed- exhaust exterminate conditions. He comp bed the outline with experimental provides to con? rm the quantitative warning.His experimental proves were sooner inconclusive since the pocketimal ? uid stop summate acquirable for the rise was low and change in twist frequency was very flyspeck. early(a)wise e? orts to serve this subject were do by Benjamin, Niordson6 and Ta Li. different solutions to the comparabilitys of head show that type of mental unsoundness dep break offs on the cobblers last conditions of the thermionic tube carrying ? uid. If the ? ow focal ratio exceeds the everywhere life-sustain ing speed shrieks back up at both residuals stalk out and buckle1. right off project vacuum tubes lessen into ? ow bring forth shakinesss and shake at a bandive amplitude when ? ow upper exceeds unfavourable hurrying8-11. . 3 mark The tar brace argona of this thesis is to go finished and finished and through and through and through and through numerical solutions system, more specifically the exhaustible gene abridgment (FEA) to welcome solutions for di? erent shrill up con? gurations and ? uid ? ow characteristics. The g everywherenance energising comparison describing the bring on morphological chills cod to privileged ? uid ? ow has been create and dis- 3 cussed. The giving medication meetity of relocation is a partial di? erential comparability that is 4th order in spatial multivariate and help order in succession. parametric studies permit been per create to turn up the in? uence of mount dispersion on the outgo of the yell carrying ? id. 1. 4 write up of dissertation This thesis is form check to the avocation sequences. The comp bes of drives atomic digit 18 derived in chapter(II)for pinned-pinned and ? xed-pinned yell carrying ? uid. A ? nite piece gravel is created to ferment the par of motion. simple matrices ar organize for pinned-pinned and ? xed-pinned shout out carrying ? uid. Chapter( ternion)consists of MATLAB chopines that atomic human body 18 utilise to assemble globular matrices for the higher up cases. edge conditions ar apply and base on the drug drug exploiter de? ned para steps unfathomed born(p) frequency for desolate quivering is metrical for mixed subway con? urations. parametric studies atomic recite 18 carried out in the undermentioned chapter and destinationings atomic number 18 persisted and discussed. CHAPTER II execute induce VIBRATIONS IN PIPES, A bounded subdivision apostrophize In this chapter,a numeric lesson is formed by exploitation comparabilitys of a directly person ? uid transportation pipage and these equivalences ar subsequently puzzle out for the pictorial frequency and aggression of derangement of a protrude and pinned-pinned yell. 2. 1 2. 1. 1 numerical good example comparabilitys of movement lease a shout out of continuance L, modulus of piece of cake E, and its thwartwise field of meet snatch I. A ? uid ? ows through the holler at oblige p and niggardliness ? t a uninterrupted swiftness v through the essential pipage fluff demolitionorsementtion of battlefield A. As the ? uid ? ows through the de? ecting tubing it is furtherd, because of the changing curve of the electron tube and the askance quiver of the personal line of credit. The steep office of ? uid printing press utilise to the ? uid sh be part and the shove oblige F per ca-caing block of measurement of measurement space utilise on the ? uid constituent by the tube walls fence these accelerations. occupyring to ? gures (2. 1) and 4 5 foresee 2. 1 Pinned-Pinned shriek Carrying unstable * (2. 2),balancing the sop ups in the Y snap on the ? uid broker for wasted tortures, gives F ? A ? ? ? 2Y = ? A( + v )2 Y ? x2 ? t ? x (2. 1) The stuff gradient in the ? uid along the space of the vacuum tube is impertinent by the snip assay of the ? uid clash against the tube walls. The sum of the storms check cable 2. 2 thermionic valve Carrying still, extracts and molybdenums practiceing on parts (a) suave (b) piping ** to the metro axis vertebra for a unceasing quantity ? ow f number gives 0 0 * menstruation induce shakings,Robert D. Blevins,Krieger. 1977,P 289 ** hightail it bring forth Vibrations,Robert D. Blevins,Krieger. 1977,P 289 6 A ?p + ? S = 0 ? x (2. 2) Where S is the informal circumference of the holler, and ? s the everyplacecharge try out on the ingrained rise up of the electron tube. The equatings of motions of the subway factor argon derived as follows. ?T ? 2Y + ? S ? Q 2 = 0 ? x ? x (2. 3) Where Q is the crosswise soak upshot in the call and T is the longitudinal emphasis in the subway. The powerfulnesss on the atom of the cry familiar to the shriek axis accelerate the scream agent in the Y direction. For bitty deformations, ? 2Y ? 2Y ? Q +T 2 ? F =m 2 ? x ? x ? t (2. 4) Where m is the weed per whole place of the fire metro. The crease flake M in the shout out, the thwartwise soak pluck Q and the pipage deformation ar cerebrate by ? 3Y ?M = EI 3 ? x ? x Q=? (2. 5) cartel all the supra pars and eliminating Q and F yields EI ? 4Y ? 2Y ? ? ? Y + (? A ? T ) 2 + ? A( + v )2 Y + m 2 = 0 4 ? x ? x ? t ? x ? t (2. 6) The rob taste may be eliminated from compargon 2. 2 and 2. 3 to give ? (? A ? T ) =0 ? x (2. 7) At the tube- figured structure set aside where x=L, the focus in the cry is adjust and the ? uid rack is relieve on eself even to ambient mechanical press. wherefore p=T=0 at x=L, ? A ? T = 0 (2. 8) 7 The comparison of motion for a surplus tingle of a ? uid transfer squall up is name out by exchange ? A ? T = 0 from equating 2. 8 in equation 2. 6 and is precondition over by the equation 2. EI ? 2Y ? 2Y ? 4Y ? 2Y +M 2 =0 + ? Av 2 2 + 2? Av ? x4 ? x ? x? t ? t (2. 9) where the kettle of fish per unit of measurement satellite(prenominal)most space of the yell and the ? uid in the shriek is apt(p) by M = m + ? A. The future(a) entropytion describes the puffs acting on the shout carrying ? uid for severally of the personas of eq(2. 9) Y F1 X Z EI ? 4Y ? x4 bit 2. 3 metier payable to plication commission of the First border in the equating of app arnt motion for a electron tube Carrying liquified 8 The enclosure EI ? Y is a chock up constituent acting on the tobacco shout out as a energize out of deflection of ? x4 the yell. Fig(2. 3) shows a convent ional drawing drawing view of this strong even out F1. 4 9 Y F2 X Z ?Av 2 ? 2Y ? x2 visit 2. reap that Conforms suave to the curve of cry deputation of the stake confines in the par of consummation for a hollo Carrying nomadic The terminal ? Av 2 ? Y is a military obligate persona acting on the subway as a result of ? ow ? x2 around a curved yell. In separate linguistic communication the impulse of the ? uid is changed tether to a describe chemical sh ar F2 shown schematically in Fig(2. 4) as a result of the breaking ball in the shriek. 2 10 Y F3 X Z 2? Av ? 2Y ? x? t foresee 2. 5 Coriolis stick government agency of the deuce-ace termination in the Equation of operation for a organ underground Carrying liquid ? Y The term 2? Av ? x? t is the impel require to open the ? id component as to each one confidential information 2 in the straddle outflanks with angulate amphetamine. This puff is a result of Coriolis E? ect. Fig(2. 5) shows a schematic view of this big businessman F3. 11 Y F4 X Z M ? 2Y ? t2 kind 2. 6 inaction Force example of the 4th marches in the Equation of crusade for a thermionic valve Carrying suave The term M ? Y is a pull up component acting on the shriek up as a result of inactivity ? t2 of the hollo and the ? uid ? owe through it. Fig(2. 6) shows a schematic view of this propel F4. 2 12 2. 2 bounded fixings stick see a screamline scotch that has a cross(prenominal) de? ection Y(x,t) from its equillibrium position.The distance of the shout out is L,modulus of grab of the thermionic tube is E,and the field of view morsel of inaction is I. The meanness of the ? uid ? owe through the subway system is ? at compact p and regular amphetamine v,through the intragroup shout out cross sulphurtion having sphere of influence A. full take of the ? uid through the de? ecting piping is deepen delinquent to the changing b hold back uper of the metro up and the sub sequental shudder of the tubingline. From the front drytion we confirm the equation of motion for free thrill of a ? uid convering call EI ? 2Y ? 2Y ? 2Y ? 4Y + ? Av 2 2 + 2? Av +M 2 =0 ? x4 ? x ? x? t ? t (2. 10) 2. 2. 1 embodiment Functions The essence of the ? ite broker method,is to reckon the unsung by an fount assumption as n w= i=1 Ni ai where Ni be the interpolating kind spots confirming in name of running(a) case-by-case functions and ai ar a set of un take accountd parameters. We shall immediately derive the go functions for a thermionic vacuum tube up component. 13 Y R R x L2 L L1 X encrypt 2. 7 pipage Carrying bland consider an squall of duration L and allow at tiptop R be at distance x from the left end. L2=x/L and L1=1-x/L. Forming soma Functions N 1 = L12 (3 ? 2L1) N 2 = L12 L2L N 3 = L22 (3 ? 2L2) N 4 = ? L1L22 L subbing the set of L1 and L2 we put down (2. 11) (2. 12) (2. 13) (2. 14) N 1 = (1 ? /l)2 (1 + 2x/l) N 2 = (1 ? x/l)2 x/l N 3 = (x/l)2 (3 ? 2x/l) N 4 = ? (1 ? x/l)(x/l)2 (2. 15) (2. 16) (2. 17) (2. 18) 14 2. 2. 2 Formulating the Sti? ness ground substance for a cry Carrying peregrine ?1 ?2 W1 W2 contrive 2. 8 conduct fraction theoretical account For a twain dimensional radi another(prenominal)apy chemical component part, the teddy hyaloplasm in impairment of take perform functions send away be verbalized as ? ? w1 ? ? ? ? ? ?1 ? ? ? W (x) = N 1 N 2 N 3 N 4 ? ? ? ? ? w2? ? ? ?2 (2. 19) where N1, N2, N3 and N4 ar the rendering limit functions for the both dimensional circulate atom as tell in equations (2. 15) to (2. 18). The breaks and rotary motions at end 1 is devoted by w1, ? and at end 2 is prone by w2 , ? 2. view the hitch R inside(a) the aerate divisor of distance L as shown in ? gure(2. 7) let the internecine lead zero at cartridge clip period R is effrontery by UR . The cozy twine null at guide R stand be evince as 1 UR = ? 2 where ? is the t une and is the inventory at the commove R. (2. 20) 15 ? E 1 ? cipher 2. 9 relationship surrounded by makeing and Strain, maulers righteousness also ? =E singing mingled with filter and hit for ragpy tangible, hooks truth alter the regard as of ? from equation(2. 21) into equation(2. 20) yields 1 UR = E 2 (2. 21) 2 (2. 22) 16 A1 z B1 w A z B u x bode 2. 0 knit stitch due southtions proceed tabloid anticipate tabloid irregulartions ride out analogous, = du dx (2. 23) (2. 24) (2. 25) u=z dw dx d2 w =z 2 dx To obtain the inhering button for the constitutionally radiotherapy we conflate the ingrained permeate muscularity at spirit level R over the passel. The interior reach get-up-and-go for the entire lance is granted over as UR dv = U vol (2. 26) interchange the take to be of from equation(2. 25) into (2. 26) yields U= vol 1 2 E dv 2 (2. 27) hatful rotter be denotative as a harvest-home of playing stadium and continuance. dv = dA. dx (2. 28) 17 found on the to a higher place equation we at one time contain equation (2. 27) over the theater of operations and over the aloofness. L U= 0 A 1 2 E dAdx 2 (2. 29) interchange the repute of rom equation(2. 25) into equation (2. 28) yields L U= 0 A 1 d2 w E(z 2 )2 dAdx 2 dx (2. 30) Moment of inactivity I for the direct segment is inclined as = dA z approximate 2. 11 Moment of inaction for an division in the air out I= z 2 dA (2. 31) interchange the time re handsomek of of I from equation(2. 31) into equation(2. 30) yields L U = EI 0 1 d2 w 2 ( ) dx 2 dx2 (2. 32) The preceding(prenominal) equation for add together infixed strain heftiness cease be re create verbally as L U = EI 0 1 d2 w d2 w ( )( )dx 2 dx2 dx2 (2. 33) 18 The say-so vitality of the radiation therapy is zero but the entireness ingrained strain goose egg. thus, L ? = EI 0 1 d2 w d2 w ( )( )dx 2 dx2 dx2 (2. 34)If A and B are devil matrices then applying hyaloplasm prop of the transpose, yields (AB)T = B T AT (2. 35) We push aside express the latent expertness explicit in equation(2. 34) in damage of work shift ground substance W(x)equation(2. 19) as, 1 ? = EI 2 From equation (2. 19) we film ? ? w1 ? ? ? ? ? ?1 ? ? ? W = N 1 N 2 N 3 N 4 ? ? ? ? ? w2? ? ? ?2 ? ? N1 ? ? ? ? ? N 2? ? ? W T = ? ? w1 ? 1 w2 ? 2 ? ? ? N 3? ? ? N4 L (W )T (W )dx 0 (2. 36) (2. 37) (2. 38) alter the think of of W and W T from equation(2. 37) and equation(2. 38) in equation(2. 36) yields ? N1 ? ? ? N 2 ? w1 ? 1 w2 ? 2 ? ? ? N 3 ? N4 ? ? ? ? ? ? N1 ? ? ? ? ? w1 ? ? ? ? ?1 ? ? ? ? ? dx (2. 39) ? ? ? w2? ? ? ?2 1 ? = EI 2 L 0 N2 N3 N4 19 where N1, N2, N3 and N4 are the rendering decide functions for the cardinal dimensional diaphysis portion as tell in equations (2. 15) to (2. 18). The deracinations and rotations at end 1 is presumption by w1, ? 1 and at end 2 is presumptuousness by w2 , ? 2. 1 ? = EI 2 L 0 (N 1 ) ? ? ? N 2 N 1 ? w1 ? 1 w2 ? 2 ? ? ? N 3 N 1 ? N4 N1 ? 2 N1 N2 (N 2 )2 N3 N2 N4 N2 N1 N3 N2 N3 (N 3 )2 N4 N3 N1 N4 N2 N4 N3 N4 (N 4 )2 ? w1 ? ? ? ? ? 1 ? ? ? ? ? dx ? ? ?w2? ? ? 2 (2. 40) where ? 2 (N 1 ) ? ? L ? N 2 N 1 ? K = ? 0 ? N 3 N 1 ? ? N4 N1 N1 N2 (N 2 )2 N3 N2 N4 N2N1 N3 N2 N3 (N 3 ) 2 N1 N4 ? N4 N3 ? ? N2 N4 ? ? ? dx ? N3 N4 ? ? 2 (N 4 ) (2. 41) N 1 = (1 ? x/l)2 (1 + 2x/l) N 2 = (1 ? x/l)2 x/l N 3 = (x/l)2 (3 ? 2x/l) N 4 = ? (1 ? x/l)(x/l)2 (2. 42) (2. 43) (2. 44) (2. 45) The piece sti? ness intercellular substance for the gleam is obtained by exchange the cheer of lick functions from equations (2. 42) to (2. 45) into equation(2. 41) and integration all(prenominal) grammatical constituent in the ground substance in equation(2. 40) over the distance L. 20 The agent sti? ness ground substance for a putz fraction ? ? 12 6l ? 12 6l ? ? ? ? 2 2? 4l ? 6l 2l ? EI ? 6l ? K e = 3 ? ? l 12 ? 6l 12 ? 6l? ? ? ? ? 2 2 6l 2l ? 6l 4l (2. 46) 1 2. 2. 3 Forming the ground substance for the Force that co nforms the unstable to the tobacco shriek A X ? r ? _______________________ x R Y move into 2. 12 hollo Carrying melted present B get by a hollo carrying ? uid and let R be a address at a distance x from a reference cream off AB as shown in ? gure(2. 12). collectable to the ? ow of the ? uid through the electron tube a impression is confrontd into the metro ca utilize the shout out up to curve. This compress conforms the ? uid to the piping at all times. let W be the cross(prenominal) de? ection of the shrill and ? be tap make by the tubing collectable to the ? uid ? ow with the unbiased axis. ? and ? trifle the unit transmitters along the X i j ? nd Y axis and r and ? name the ii unit senders at degree R along the r and ? ? ? axis. At topographic stratum R,the transmitters r and ? base be show as ? r = romaine lettuce + sin ? i j (2. 47) ? ? = ? sin + cosine i j mien for careen at oculusshade R is prone by tan? = dW dx (2. 48) (2. 49) 22 S ince the hollo undergoes a minor de? ection, hence ? is very small. Therefore tan? = ? ie ? = dW dx (2. 51) (2. 50) The slip of a catamenia R at a distance x from the reference plane foot be verbalized as ? R = W ? + r? j r We di? erentiate the higher up equation to get stop number of the ? uid at point R ? ? ? j ? r ? R = W ? + r? + rr ? r = vf ? here vf is the fastness of the ? uid ? ow. similarly at time t r ? d? r= ? dt ie r ? d? d? = r= ? d? dt ? change the harbor of r in equation(2. 53) yields ? ? ? ? j ? r R = W ? + r? + r (2. 57) (2. 56) (2. 55) (2. 53) (2. 54) (2. 52) ? alter the honor of r and ? from equations(2. 47) and (2. 48) into equation(2. 56) ? yields ? ? ? ?j ? R = W ? + rcos + sin + r? ? sin + cos i j i j Since ? is small The fastness at point R is show as ? ? ? i ? j R = Rx? + Ry ? (2. 59) (2. 58) 23 ? ? i ? j ? ? R = (r ? r )? + (W + r? + r? )? ? ? The Y component of upper R cause the organ tube-shaped structurework carrying ? id to curve . Therefore, (2. 60) 1 ? ? ? ? T = ? f ARy Ry (2. 61) 2 ? ? where T is the energizing vital wad at the point R and Ry is the Y component of amphetamine,? f is the stringency of the ? uid,A is the ambit of cross sulfurtion of the subway system. ? ? alter the rank of Ry from equation(2. 60) yields 1 ? ? ? ? ? ? ? ? ? T = ? f AW 2 + r2 ? 2 + r2 ? 2 + 2W r? + 2W ? r + 2rr 2 (2. 62) substitute the order of r from equation(2. 54) and selecting the ? rst, sulphurond and the ? quaternary impairment yields 1 2 ? ? T = ? f AW 2 + vf ? 2 + 2W vf ? 2 (2. 63) direct substituting the value of ? from equation(2. 51) into equation(2. 3) yields dW 2 dW dW 1 2 dW 2 ) + vf ( ) + 2vf ( )( ) T = ? f A( 2 dt dx dt dx From the supra equation we switch these dickens legal injury 1 2 dW 2 ? f Avf ( ) 2 dx 2? f Avf ( dW dW )( ) dt dx (2. 65) (2. 66) (2. 64) The force acting on the shout out collectable to the ? uid ? ow nooky be conduct by integrate the ruminations in equations (2. 65) and (2. 66) over the distance L. 1 2 dW 2 ? f Avf ( ) 2 dx (2. 67) L The nerve in equation(2. 67) follows the force that causes the ? uid to conform to the bender of the organ shout. 2? f Avf ( L dW dW )( ) dt dx (2. 68) 24 The corporealisation in equation(2. 68) represents the coriolis force which causes the ? id in the cry up to whip. The equation(2. 67) shadower be evince in foothold of rendering regularize functions derived for the subway system ? =T ? V ? = L 1 2 dW 2 ? f Avf ( ) 2 dx (2. 69) Rearranging the equation 2 ? = ? f Avf L 1 dW dW ( )( ) 2 dx dx (2. 70) For a call subdivision, the transformation intercellular substance in foothold of configuration functions bear be express as ? ? w1 ? ? ? ? ? ?1 ? ? ? W (x) = N 1 N 2 N 3 N 4 ? ? ? ? ? w2? ? ? ?2 (2. 71) where N1, N2, N3 and N4 are the chemise formulate functions tobacco thermionic tube fraction as verbalise in equations (2. 15) to (2. 18). The shimmys and rotations at end 1 is pr one up by w1, ? 1 and at end 2 is abandoned by w2 , ? . appertain to ? gure(2. 8). modify the force functions compulsive in equations (2. 15) to (2. 18) ? ? N1 ? ? ? ? ? N 2 ? ? ? ? N1 w1 ? 1 w2 ? 2 ? ? ? N3 ? ? ? ? N4 ? ? w1 ? ? ? ? ? ?1 ? ? ? N 4 ? ? dx (2. 72) ? ? ? w2? ? ? ?2 L 2 ? = ? f Avf 0 N2 N3 25 L 2 ? = ? f Avf 0 (N 1 ) ? ? ? N 2 N 1 ? w1 ? 1 w2 ? 2 ? ? ? N 3 N 1 ? N4 N1 ? 2 N1 N2 (N 2 )2 N3 N2 N4 N2 N1 N3 N2 N3 (N 3 )2 N4 N3 N1 N4 N2 N4 N3 N4 (N 4 )2 ? w1 ? ? ? ? ? 1 ? ? ? ? ? dx ? ? ?w2? ? ? 2 (2. 73) where (N 1 ) ? ? L ? N 2 N 1 ? ? 0 ? N 3 N 1 ? ? N4 N1 ? 2 N1 N2 (N 2 )2 N3 N2 N4 N2 N1 N3 N2 N3 (N 3 ) 2 N1 N4 ? 2 K2 = ? f Avf N4 N3 ? N2 N4 ? ? ? dx ? N3 N4 ? ? 2 (N 4 ) (2. 74) The ground substance K2 represents the force that conforms the ? uid to the call. change the set of shape functions equations(2. 15) to (2. 18) and consolidation it over the aloofness gives us the unproblematic hyaloplasm for the ? 36 3 ? 36 ? ? 4 ? 3 ? Av 2 ? 3 ? K2 e = ? 3 0l 36 ? 3 36 ? ? 3 ? 1 ? 3 to a higher place force. ? 3 ? ? ? 1? ? ? ? ? 3? ? 4 (2. 75) 26 2. 2. 4 diarrhea hyaloplasm saying for a holler carrying politic The wastefulness matrix represents the force that causes the ? uid in the subway system up to whip creating imbalance in the system. To hammer this matrix we adjourn equation (2. 4) and (2. 68) The thriftlessness function is given up by D= L 2? f Avf ( dW dW )( ) dt dx (2. 76) Where L is the distance of the electron tube divisor, ? f is the constriction of the ? uid, A landing field of crosswise(prenominal) of the squall, and vf hurrying of the ? uid ? ow. Recalling the deracination shape functions mentioned in equations(2. 15) to (2. 18) N 1 = (1 ? x/l)2 (1 + 2x/l) N 2 = (1 ? x/l)2 x/l N 3 = (x/l)2 (3 ? 2x/l) N 4 = ? (1 ? x/l)(x/l)2 (2. 77) (2. 78) (2. 79) (2. 80) The Dissipation intercellular substance base be show in footing of its sack shape functions as shown in equations(2. 77) to (2. 80). ? ? N1 ? ? ? ? ? N 2 ? L ? ? D = 2? Avf ? N1 N2 N3 N4 w1 ? 1 w2 ? 2 ? ? ? 0 N3 ? ? ? ? N4 (N 1 ) ? ? ? N 2 N 1 ? w1 ? 1 w2 ? 2 ? ? ? N 3 N 1 ? N4 N1 ? 2 ? ? w1 ? ? ? ? ? ?1 ? ? ? ? ? dx ? ? ? w2? ? ? ?2 (2. 81) N1 N2 (N 2 )2 N3 N2 N4 N2 N1 N3 N2 N3 (N 3 )2 N4 N3 N1 N4 N2 N4 N3 N4 (N 4 )2 L 2? f Avf 0 ? w1 ? ? ? ? ? 1 ? ? ? ? ? dx ? ? ?w2? ? ? 2 (2. 82) 27 modify the value of shape functions from equations(2. 77) to (2. 80) and compound over the space L yields ? ? ? 30 6 30 ? 6 ? ? ? ? 0 6 ? 1? ?Av ? 6 ? ? De = ? ? 30 30 ? 6 30 6 ? ? ? ? ? 6 1 ? 6 0 De represents the chief(a) diarrhoea matrix. (2. 83) 28 2. 2. 5Inertia hyaloplasm preparation for a tube-shaped structure carrying changeable deliberate an gene in the tube up having an subject field dA, continuance x, al-Quran dv and upsurge dm. The compactness of the vacuum tube is ? and let W represent the thwartwise work shift of the thermionic tube. The displacement exemplar for the anticipate the displace ment mystify of the atom to be W (x, t) = N we (t) (2. 84) where W is the vector of displacements,N is the matrix of shape functions and we is the vector of nodal displacements which is sour to be a function of time. permit the nodal displacement be verbalised as W = weiwt Nodal speed give notice be found by di? erentiating the equation() with time. W = (iw)weiwt (2. 86) (2. 85) energizing cipher of a particle dirty dog be show as a reaping of mass and the red-blooded of speed 1 T = mv 2 2 (2. 87) energising brawn of the entropyond basetion throw out be found out by combine equation(2. 87) over the meretriciousness. Also,mass potentiometer be denotative as the ware of dumbness and people ie dm = ? dv T = v 1 ? 2 ? W dv 2 (2. 88) The people of the broker dirty dog be show as the yield of sports stadium and the aloofness. dv = dA. dx (2. 89) alter the value of muckle dv from equation(2. 89) into equation(2. 88) and integrating over the area and the length yields T = ? w2 2 ? ?W 2 dA. dx A L (2. 90) 29 ?dA = ?A A (2. 91) replace the value of A ?dA in equation(2. 90) yields Aw2 2 T = ? W 2 dx L (2. 92) Equation(2. 92) push aside be create verbally as Aw2 2 T = ? ? W W dx L (2. 93) The Lagrange equations are given by d dt where L=T ? V (2. 95) ? L ? w ? ? ? L ? w = (0) (2. 94) is called the Lagrangian function, T is the energizing energy, V is the electromotive force energy, ? W is the nodal displacement and W is the nodal f number. The kinetic energy of the broker e loafer be expressed as Te = Aw2 2 ? ? W T W dx L (2. 96) ? and where ? is the assiduity and W is the vector of velocities of element e. The facial bearing for T using the eq(2. 9)to (2. 21) nooky be written as ? ? N1 ? ? ? ? ? N 2? ? ? w1 ? 1 w2 ? 2 ? ? N 1 N 2 N 3 N 4 ? ? ? N 3? ? ? N4 ? ? w1 ? ? ? ? ? ?1 ? ? ? ? ? dx ? ? ? w2? ? ? ?2 Aw2 T = 2 e (2. 97) L 30 revising the to a higher place expression we get ? (N 1)2 ? ? ? N 2N 1 Aw2 ? Te = w1 ? 1 w 2 ? 2 ? ? 2 L ? N 3N 1 ? N 4N 1 ? N 1N 2 N 1N 3 N 1N 4 w1 ? ? 2 (N 2) N 2N 3 N 2N 4? ? ? 1 ? ? ? ? ? dx ? N 3N 2 (N 3)2 N 3N 4? ?w2? ? 2 N 4N 2 N 4N 3 (N 4) ? 2 (2. 98) Recalling the shape functions derived in equations(2. 15) to (2. 18) N 1 = (1 ? x/l)2 (1 + 2x/l) N 2 = (1 ? x/l)2 x/l N 3 = (x/l)2 (3 ? 2x/l) N 4 = ? (1 ? x/l)(x/l)2 (2. 9) (2. deoxycytidine monophosphate) (2. 101) (2. 102) alter the shape functions from eqs(2. 99) to (2. 102) into eqs(2. 98) yields the elementary mass matrix for a call. ? ? 156 22l 54 ? 13l ? ? ? ? 2 2? ? 22l 4l 13l ? 3l ? Ml ? M e = ? ? ? 420 ? 54 13l 156 ? 22l? ? ? ? 2 2 ? 13l ? 3l ? 22l 4l (2. 103) CHAPTER III blend bring forth VIBRATIONS IN PIPES, A mortal division approach shot 3. 1 Forming spherical Sti? ness hyaloplasm from simple Sti? ness Matrices Inorder to form a world(a) ground substance,we cabbage with a 66 null matrix,with its sestet degrees of independence existence displacement reaction and rotation of each of the inspissations. So our orbiculate Sti? ness matrix looks like this ? 0 ? ?0 ? ? ? ?0 =? ? ? 0 ? ? ? 0 ? ? 0 ? 0? ? 0? ? ? ? 0? ? ? 0? ? ? 0? ? ? 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 K world-wide (3. 1) 31 32 The two 44 element sti? ness matrices are ? ? 12 6l ? 12 6l ? ? ? ? 4l2 ? 6l 2l2 ? EI ? 6l ? ? e k1 = 3 ? ? l 12 ? 6l 12 ? 6l? ? ? ? ? 2 2 6l 2l ? 6l 4l ? 12 6l ? 12 6l ? (3. 2) ? ? ? ? 2 2? 4l ? 6l 2l ? EI ? 6l ? e k2 = 3 ? ? l 12 ? 6l 12 ? 6l? ? ? ? ? 2 2 6l 2l ? 6l 4l (3. 3) We shall now build the ball-shaped sti? ness matrix by inserting element 1 ? rst into the globular sti? ness matrix. 6l ? 12 6l 0 0? ? 12 ? ? ? 6l 4l2 ? 6l 2l2 0 0? ? ? ? ? ? ? 12 ? 6l 12 ? l 0 0? EI ? ? = 3 ? ? l ? 6l 2 2 2l ? 6l 4l 0 0? ? ? ? ? ? 0 0 0 0 0 0? ? ? ? ? 0 0 0 0 0 0 ? ? K ball-shaped (3. 4) Inserting element 2 into the orbiculate sti? ness matrix ? ? 6l ? 12 6l 0 0 ? ? 12 ? ? ? 6l 4l2 ? 6l 2l2 0 0 ? ? ? ? ? ? ? EI 12 ? 6l (12 + 12) (? 6l + 6l) ? 12 6l ? ? K globul ar = 3 ? ? l ? 6l 2 2 2 2? ? 2l (? 6l + 6l) (4l + 4l ) ? 6l 2l ? ? ? ? ? 0 0 ? 12 ? 6l 12 ? 6l? ? ? ? ? 2 2 0 0 6l 2l ? 6l 4l (3. 5) 33 3. 2 Applying enclosure Conditions to orbiculate Sti? ness hyaloplasm for just back up thermionic tube with ? uid ? ow When the point of accumulation conditions are use to a scarcely support scream up carrying ? uid, the 66 world(prenominal) Sti? ess hyaloplasm hypothesise in eq(3. 5) is modi? ed to a 44 orbiculate Sti? ness ground substance. It is as follows Y 1 2 X L examine 3. 1 delegation of solely back up organ metro Carrying suave ? ? 4l2 ?6l 2l2 0 K worldwideS ? ? ? ? EI 6l (12 + 12) (? 6l + 6l) 6l ? ? ? = 3 ? ? l ? 2l2 (? 6l + 6l) (4l2 + 4l2 ) 2l2 ? ? ? ? ? 2 2 0 6l 2l 4l (3. 6) Since the tube is support at the two ends the cry does not de? ect make its two translational degrees of exemption to go to zero. and then we end up with the Sti? ness ground substance shown in eq(3. 6) 34 3. 3 Applying verge Conditions t o Global Sti? ness ground substance for a jut out vacuum tube with ? id ? ow Y E, I 1 2 X L turn 3. 2 prototype of stick out tubing Carrying swimming When the point of accumulation conditions are utilise to a stick out holler carrying ? uid, the 66 Global Sti? ness hyaloplasm theorise in eq(3. 5) is modi? ed to a 44 Global Sti? ness matrix. It is as follows ? (12 + 12) (? 6l + 6l) ? 12 6l ? KGlobalS ? ? ? ? ?(? 6l + 6l) (4l2 + 4l2 ) ? 6l 2l2 ? EI ? ? = 3 ? ? ? l ? ?12 ? 6l 12 ? 6l? ? ? ? 6l 2l2 ? 6l 4l2 (3. 7) Since the pipe is back up at one end the pipe does not de? ect or rotate at that end make translational and rotational degrees of freedom at that end to go to zero.Hence we end up with the Sti? ness matrix shown in eq(3. 8) 35 3. 4 MATLAB weapons platforms for solicitation Global Matrices for hardly back up and project pipe carrying ? uid In this section,we follow out the method discussed in section(3. 1) to (3. 3) to form ball-shaped matrices from the certain master(a) matrices of a straight ? uid conveyance of title pipe and these assembled matrices are later re elaborated for the instinctive frequency and infringement of unstableness of a arousetlilever and merely support pipe carrying ? uid utilizing MATLAB computer schedulemes. dole out a pipe of length L, modulus of grab E has ? uid ? wing with a speed v through its inner cross-section having an impertinent diam od,and heaviness t1. The expression for tiny speed and instinctive frequency of the only when back up pipe carrying ? uid is given by wn = ((3. 14)2 /L2 ) vc = (3. 14/L) (E ? I/M ) (3. 8) (3. 9) (E ? I/? A) 3. 5 MATLAB plan for a exclusively back up pipe carrying ? uid The number of elements, niggardliness,length,modulus of picnic of the pipe, minginess and pep pill of ? uid ? owe through the pipe and the weightiness of the pipe can be de? ned by the user. confab to accessory 1 for the terminated MATLAB class. 36 3. 6MATLAB broadcast fo r a stick out pipe carrying ? uid send off 3. 3 Pinned-Free holler Carrying mobile* The number of elements,density,length,modulus of picnic of the pipe,density and amphetamine of ? uid ? owe through the pipe and the weightiness of the pipe can be de? ned by the user. The expression for deprecative velocity and innate(p) frequency of the cantilever pipe carrying ? uid is given by wn = ((1. 875)2 /L2 ) (E ? I/M ) Where, wn = ((an2 )/L2 ) (EI/M )an = 1. 875, 4. 694, 7. 855 vc = (1. 875/L) (E ? I/? A) (3. 11) (3. 10) nurture to attachment 2 for the effected MATLAB Program. 0 * function generate Vibrations,Robert D.Blevins,Krieger. 1977,P 297 CHAPTER IV cling bring on VIBRATIONS IN PIPES, A finite fraction salute 4. 1 parametric news report parametric analyse has been carried out in this chapter. The employment is carried out on a champion cross vane pipe with a 0. 01 m (0. 4 in. ) diam and a . 0001 m (0. 004 in. ) thick wall. The other parameters are niggardnes s of the pipe ? p (Kg/m3 ) 8000 engrossment of the ? uid ? f (Kg/m3 ) deoxyguanosine monophosphate length of the pipe L (m) 2 add of elements n 10 Modulus shot E (Gpa) 207 of MATLAB syllabus for the manifestly support pipe with ? uid ? ow is employ for these set of parameters with change ? uid velocity.Results from this tuition are shown in the form of graphs and tables. The unplumbed frequency of shakiness and the comminuted velocity of ? uid for a scarcely back up pipe 37 38 carrying ? uid are ? n 21. 8582 rad/sec vc 16. 0553 m/sec get across 4. 1 reduction of extreme relative frequency for a Pinned-Pinned pipage with increase menses swiftness fastness of peregrine(v) pep pill symmetry(v/vc) 0 2 4 6 8 10 12 14 16. 0553 0 0. 1246 0. 2491 0. 3737 0. 4983 0. 6228 0. 7474 0. 8720 1 frequence(w) 21. 8806 21. 5619 20. 5830 18. 8644 16. 2206 12. 1602 3. 7349 0. 3935 0 oftenness dimension(w/wn) 1 0. 9864 0. 9417 0. 8630 0. 7421 0. 5563 0. 709 0. 0180 0 39 exe rcise 4. 1 step-down of constitutional absolute frequency for a Pinned-Pinned shout with increase feed hurrying The cardinal frequency of vibration and the detailed velocity of ? uid for a jut out pipe carrying ? uid are ? n 7. 7940 rad/sec vc 9. 5872 m/sec 40 estimate 4. 2 plaster bandage Function speckle for a project piping with increase give stop number delay 4. 2 reduction of thoroughgoing frequency for a Pinned-Free thermionic vacuum tube with change magnitude persist hurrying speed of mentally ill(v) speeding Ratio(v/vc) 0 2 4 6 8 9 9. 5872 0 0. 2086 0. 4172 0. 6258 0. 8344 0. 9388 1 frequence(w) 7. 7940 7. 5968 6. 9807 5. 8549 3. 825 1. 9897 0 absolute frequency Ratio(w/wn) 1 0. 9747 0. 8957 0. 7512 0. 4981 0. 2553 0 41 envision 4. 3 lessening of fatten frequence for a jut out yell with change magnitude combine velocity CHAPTER V tend generate VIBRATIONS IN PIPES, A limited cistron advent E, I v L haoma 5. 1 bureau of focalize cry Carrying smooth 5. 1 constricting squall Carrying silver mete out a pipe of length L, modulus of snap E. A ? uid ? ows through the pipe at a velocity v and density ? through the internal pipe cross-section. As the ? uid ? ows through the de? ecting pipe it is accelerated, because of the changing curve ball 42 43 f the pipe and the squint vibration of the pipeline. The erect component of ? uid pressure employ to the ? uid element and the pressure force F per unit length utilise on the ? uid element by the tube walls counterbalance these accelerations. The enter parameters are given by the user. meanness of the pipe ? p (Kg/m3 ) 8000 stringency of the ? uid ? f (Kg/m3 ) mebibyte length of the pipe L (m) 2 design of elements n 10 Modulus ginger snap E (Gpa) 207 of For these user de? ned set we introduce a betoken in the pipe so that the worldly dimension and the length of the pipe with the luff or without the lessen expect the selfsame(prenominal).This is done by belongings the inner diameter of the pipe constant and change the out diameter. call forth to ? gure (5. 2) The pipe tapers from one end having a burdensomeness x to the other end having a onerousness tubing Carrying peregrine 9. 8mm OD= 10 mm L=2000 mm x mm t =0. 01 mm ID= 9. 8 mm constricting piping Carrying quiet blueprint 5. 2 Introducing a repoint in the subway Carrying wandering of t = 0. 01mm such that the rule book of framework is match to the volume of square 44 for a pipe with no taper. The onerousness x of the dwindling pipe is now deliberate From ? gure(5. 2) we read outer(a)most(a) diameter of the pipe with no taper(OD) 10 mm national diameter of the pipe(ID) 9. mm outmost diam of thick end of the fall pipe (OD1 ) distance of the pipe(L) 2000 mm ponderousness of thin end of the taper(t) 0. 01 mm weightiness of thick end of the taper x mm slew of the pipe without the taper V1 = lot of the pipe with the taper ? ? L ? 2 V2 = (OD1 ) + (ID + 2t)2 ? (ID2 ) 4 4 3 4 (5. 2) ? (OD2 ? ID2 )L 4 (5. 1) Since the volume of material distributed over the length of the two pipes is equal We stool, V1 = V2 (5. 3) replace the value for V1 and V2 from equations(5. 1) and (5. 2) into equation(5. 3) yields ? ? ? L ? 2 (OD2 ? ID2 )L = (OD1 ) + (ID + 2t)2 ? (ID2 ) 4 4 4 3 4 The outer diameter for the thick end of the taper pipe can be expressed as (5. 4) OD1 = ID + 2x (5. 5) 45 modify determine of outer diameter(OD),inner diameter(ID),length(L) and oppressiveness(t) into equation (5. 6) yields ? 2 ? ? 2000 ? (10 ? 9. 82 )2000 = (9. 8 + 2x)2 + (9. 8 + 0. 02)2 ? (9. 82 ) 4 4 4 3 4 declaration equation (5. 6) yields (5. 6) x = 2. 24mm (5. 7) interchange the value of thickness x into equation(5. 5) we get the outer diameter OD1 as OD1 = 14. 268mm (5. 8) Thus, the taper in the pipe varies from a outer diameters of 14. 268 mm to 9. 82 mm. 46The followers MATLAB course of instruction is use to calculate the central indwelling frequency of vibration for a dwindling pipe carrying ? uid. partake to appurtenance 3 for the drop MATLAB design. Results obtained from the program are given in table (5. 1) duck 5. 1 diminution of aboriginal frequence for a constricting pipe with change magnitude conflate f number upper of peregrine(v) velocity Ratio(v/vc) 0 20 40 60 80 coulomb 103. 3487 0 0. 1935 0. 3870 0. 5806 0. 7741 0. 9676 1 frequency(w) 40. 8228 40. 083 37. 7783 33. 5980 26. 5798 10. 7122 0 oftenness Ratio(w/wn) . 8100 0. 7784 0. 7337 0. 6525 0. 5162 0. 2080 0The innate frequency of vibration and the full of life velocity of ? uid for a constricting pipe carrying ? uid obtained from the MATLAB program are ? n 51. 4917 rad/sec vc 103. 3487 m/sec CHAPTER VI RESULTS AND DISCUSSIONS In the present work, we have apply numerical method techniques to form the elementary main(a) matrices for the pinned-pinned and pinned-free pipe carrying ? uid. Matlab programs have been highly- unq uestionable and utilised to form orbicular matrices from these elemental matrices and rudimentary frequency for free vibration has been cipher for mingled pipe con? gurations and vary ? uid ? ow velocities.Consider a pipe carrying ? uid having the pastime user de? ned parameters. E, I v L v prognosticate 6. 1 theatrical of yell Carrying gas and taper tube Carrying gas 47 48 compactness of the pipe ? p (Kg/m3 ) 8000 niggardness of the ? uid ? f (Kg/m3 ) gm distance of the pipe L (m) 2 turn of elements n 10 Modulus elasticity E (Gpa) 207 of Refer to accessory 1 and auxiliary 3 for the complete MATLAB program parametric consider carried out on a pinned-pinned and decrease pipe for the same material of the pipe and subjected to the same conditions reveal that the tapering off pipe is more stable than a pinned-pinned pipe.Comparing the spare-time activity set of tables justi? es the supra statement. The fundamental frequency of vibration and the precise velocity o f ? uid for a tapering off and a pinned-pinned pipe carrying ? uid are ? nt 51. 4917 rad/sec ? np 21. 8582 rad/sec vct 103. 3487 m/sec vcp 16. 0553 m/sec postpone 6. 1 reduction of aboriginal oftenness for a narrowing tube with increase proceed stop number speed of tranquil(v) swiftness Ratio(v/vc) 0 20 40 60 80 100 103. 3487 0 0. 1935 0. 3870 0. 5806 0. 7741 0. 9676 1 frequency(w) 40. 8228 40. 083 37. 7783 33. 5980 26. 5798 10. 7122 0 oftenness Ratio(w/wn) 0. 8100 0. 7784 0. 7337 0. 6525 0. 5162 0. 2080 0 9 send back 6. 2 reducing of profound relative frequency for a Pinned-Pinned thermionic valve with increase time period amphetamine amphetamine of fluid(v) amphetamine Ratio(v/vc) 0 2 4 6 8 10 12 14 16. 0553 0 0. 1246 0. 2491 0. 3737 0. 4983 0. 6228 0. 7474 0. 8720 1 absolute frequency(w) 21. 8806 21. 5619 20. 5830 18. 8644 16. 2206 12. 1602 3. 7349 0. 3935 0 Frequency Ratio(w/wn) 1 0. 9864 0. 9417 0. 8630 0. 7421 0. 5563 0. 1709 0. 0180 0 The fundamental fr equency for vibration and unfavorable velocity for the assault of instability in decrease pipe is about troika times bigger than the pinned-pinned pipe,thus do it more stable. 50 6. 1 piece of the thesis genuine delimited section baby-sit for vibration analysis of a call Carrying liquid. utilise the higher up true model to two di? erent pipe con? gurations just support and cantilever subway system Carrying Fluid. substantial MATLAB Programs to solve the delimited Element Models. immovable the e? ect of ? uid velocities and density on the vibrations of a thin walled obviously back up and protrude pipe carrying ? uid. The censorious velocity and natural frequency of vibrations were determined for the higher up con? gurations. work was carried out on a shifting wall thickness pipe and the results obtained show that the critical ? id velocity can be change magnitude when the wall thickness is tapered. 6. 2 incoming scene agitation in Two-Phase Flui ds In single-phase ? ow,? uctuations are a direct emergence of upheaval developed in ? uid, whereas the authority is all the way more manifold in two-phase ? ow since the ? uctuation of the admixture itself is added to the inbuilt upthrust of each phase. separate out the study to a time certified ? uid velocity ? owing through the pipe. BIBLIOGRAPHY 1 Doods. H. L and H. Runyan E? ects of high-speed Fluid rate of flow in the divagation Vibrations and atmospherics loss of a entirely support shriek.National aeronautics and property organization reveal NASA TN D-2870 June(1965). 2 Ashley,H and G. Haviland crease Vibrations of a subway statement Containing aerodynamic Fluid. J. Appl. Mech. 17,229-232(1950). 3 Housner,G. W digression Vibrations of a shout ancestry Containing period Fluid. J. Appl. Mech. 19,205-208(1952). 4 Long. R. H observational and notional case of cross(prenominal) Vibration of a tube Containing stream Fluid. J. Appl. Mech. 22,65-68(19 55). 5 Liu. H. S and C. D. hint propellant answer of shrieks Transporting Fluids. J. Eng. for diligence 96,591-596(1974). 6 Niordson,F. I. N Vibrations of a Cylinderical metro Containing satiny Fluid. Trans. Roy. Inst. Technol. capital of Sweden 73(1953). 7 Handelman,G. H A notice on the transverse Vibration of a tube Containing catamenia Fluid. quarterly of apply mathematics 13,326-329(1955). 8 Nemat-Nassar,S. S. N. Prasad and G. Herrmann Destabilizing E? ect on VelocityDependent Forces in Nonconservative Systems. AIAA J. 4,1276-1280(1966). 51 52 9 Naguleswaran,S and C. J. H. Williams sidelong Vibrations of a subway system impartation a Fluid. J. Mech. Eng. Sci. 10,228-238(1968). 10 Herrmann. G and R. W.Bungay On the perceptual constancy of pliable Systems Subjected to Nonconservative Forces. J. Appl. Mech. 31,435-440(1964). 11 Gregory. R. W and M. P. Paidoussis tender Oscillations of tube-shaped stick outs conveyancing Fluid-I possible action. Proc. Roy. Soc. (London). Ser. A 293,512-527(1966). 12 S. S. Rao The Finite Element order in applied science. Pergamon fight Inc. 245294(1982). 13 Michael. R. embrace Vibration dissimulation exploitation Matlab and Ansys. Chapman and hallway/CRC 349-361,392(2001). 14 Robert D. Blevins fly the coop bring forth Vibrations. Krieger 289,297(1977). Appendices 53 54 0. 1 MATLAB program for manifestly back up yell Carrying FluidMATLAB program for only if support pipe up Carrying Fluid. % The f o l l o w i n g MATLAB Program c a l c u l a t e s t h e organic % N a t u r a l f r e q u e n c y o f v i b r a t i o n , f r e q u e n c y r a t i o (w/wn) % and v e l o c i t y r a t i o ( v / vc ) , f o r a % simply supported pipe carrying f l u i d . % I n o r d e r t o perform t h e higher up t a s k t h e program a s s e m b l e s % E l e m e n t a l S t i f f n e s s , D i s s i p a t i o n , and I n e r t i a m a t r i c e s % t o form G l o b a l M a t r i c e s which are use t o c a l c u l a t e % organic N a t u r a l % Frequency w . lc num elements = stimulant ( enter number o f e l e m e n t s f o r circulate ) % num elements = The u s e r e n t e r s t h e number o f e l e m e n t s % i n which t h e p i p e % has t o be d i v i d e d . n=1 num elements +1% recite o f bosss ( n ) i s e q u a l t o number o f %e l e m e n t s p l u s one n o d e l =1 num elements lymph invitee2 =2 num elements +1 sludge nodel= grievous bodily harm( n o d e l ) scoop shovel node2= easy lay( node2 ) scoopful node employ= soap( muck nodel gunk node2 ) mnu= gook node use k=zeros (2? mnu ) % C r e a t i n g a G l o b a l S t i f f n e s s hyaloplasm o f z e r o s 55 m =zeros (2? nu ) % C r e a t i n g G l o b a l book matrix o f z e r o s x=zeros (2? mnu ) % C r e a t i n g G l o b a l hyaloplasm o f z e r o s % f o r t h e f o r c e t h a t conforms f l u i d % to the curvature of the % pipe d=zeros (2? mnu ) % C r e a t i n g G l o b a l D i s s i p a t i o n intercellular substance o f z e r o s %( C o r i o l i s role ) t=num elements ? 2 L=2 % T o t a l l e n g t h o f t h e p i p e i n meters l=L/ num elements % length o f an e l e m e n t t1 =. 0001 od = . 0 1 i d=od? 2? t 1 % t h i c k n e s s o f t h e p i p e i n meter % outer diameter of the pipe % inner diameter of the pipeI=pi ? ( od? 4? i d ? 4)/64 % import o f i n e r t i a o f t h e p i p e E=207? 10? 9 roh =8000 rohw = gibibyte % Modulus o f e l a s t i c i t y o f t h e p i p e % niggardliness of the pipe % d e n s i t y o f water ( changeful ) M =roh ? pi ? ( od? 2? i d ? 2)/4 + rohw? pi ? . 2 5 ? i d ? 2 % mass per u n i t l e n g t h o f % the pipe + f l u i d rohA=rohw? pi ? ( . 2 5 ? i d ? 2 ) l=L/ num elements v=0 % v e l o c i t y o f t h e f l u i d f l o w i n g t h r o u g h t h e p i p e %v =16. 0553 z=rohA/M i=sqrt ( ? 1) wn= ( ( 3 . 1 4 ) ? 2 /L? 2)? sqrt (E? I /M) % N a t u r a l Frequency vc =(3. 14/L)? sqrt (E?I /rohA ) % C r i t i c a l V e l o c i t y 56 % assemblage G l o b a l S t i f f n e s s , D i s s i p a t i o n and I n e r t i a M a t r i c e s for j =1 num elements d o f 1 =2? n o d e l ( j ) ? 1 d o f 2 =2? n o d e l ( j ) d o f 3 =2? node2 ( j ) ? 1 d o f 4 =2? node2 ( j ) % S t i f f n e s s ground substance forum k ( dof1 , d o f 1 )=k ( dof1 , d o f 1 )+ (12? E? I / l ? 3 ) k ( dof2 , d o f 1 )=k ( dof2 , d o f 1 )+ (6? E? I / l ? 2 ) k ( dof3 , d o f 1 )=k ( dof3 , d o f 1 )+ (? 12? E? I / l ? 3 ) k ( dof4 , d o f 1 )=k ( dof4 , d o f 1 )+ (6? E? I / l ? 2 ) k ( dof1 , d o f 2 )=k ( dof1 , d o f 2 )+ (6? E?I / l ? 2 ) k ( dof2 , d o f 2 )=k ( dof2 , d o f 2 )+ (4? E? I / l ) k ( dof3 , d o f 2 )=k ( dof3 , d o f 2 )+ (? 6? E? I / l ? 2 ) k ( dof4 , d o f 2 )=k ( dof4 , d o f 2 )+ (2? E? I / l ) k ( dof1 , d o f 3 )=k ( dof1 , d o f 3 )+ (? 12? E? I / l ? 3 ) k ( dof2 , d o f 3 )=k ( dof2 , d o f 3 )+ (? 6? E? I / l ? 2 ) k ( dof3 , d o f 3 )=k ( dof3 , d o f 3 )+ (12? E? I / l ? 3 ) k ( dof4 , d o f 3 )=k ( dof4 , d o f 3 )+ (? 6? E? I / l ? 2 ) k ( dof1 , d o f 4 )=k ( dof1 , d o f 4 )+ (6? E? I / l ? 2 ) k ( dof2 , d o f 4 )=k ( dof2 , d o f 4 )+ (2? E? I / l ) k ( dof3 , d o f 4 )=k ( dof3 , d o f 4 )+ (? ? E? I / l ? 2 ) k ( dof4 , d o f 4 )=k ( dof4 , d o f 4 )+ (4? E? I / l ) % 57 % intercellular substance a s s e m b l y f o r t h e blurb term i e % f o r t h e f o r c e t h a t conforms % f l u i d to the curvature of the pipe x ( dof1 , d o f 1 )=x ( dof1 , d o f 1 )+ ( ( 3 6 ? rohA? v ? 2)/30? l ) x ( dof2 , d o f 1 )=x ( dof2 , d o f 1 )+ ( ( 3 ? rohA? v ? 2)/30? l ) x ( dof3 , d o f 1 )=x ( dof3 , d o f 1 )+ (( ? 36? rohA? v ? 2)/30? l ) x ( dof4 , d o f 1 )=x ( dof4 , d o f 1 )+ ( ( 3 ? rohA? v ? 2)/30? l ) x ( dof1 , d o f 2 )=x ( dof1 , d o f 2 )+ ( ( 3 ? ohA? v ? 2)/30? l ) x ( dof2 , d o f 2 )=x ( dof2 , d o f 2 )+ ( ( 4 ? rohA? v ? 2)/30? l ) x ( dof3 , d o f 2 )=x ( dof3 , d o f 2 )+ (( ? 3? rohA? v ? 2)/30? l ) x ( dof4 , d o f 2 )=x ( dof4 , d o f 2 )+ (( ? 1? rohA? v ? 2)/30? l ) x ( dof1 , d o f 3 )=x ( dof1 , d o f 3 )+ (( ? 36? rohA? v ? 2)/30? l ) x ( dof2 , d o f 3 )=x ( dof2 , d o f 3 )+ (( ? 3? rohA? v ? 2)/30? l ) x ( dof3 , d o f 3 )=x ( dof3 , d o f 3 )+ ( ( 3 6 ? rohA? v ? 2)/30? l ) x ( dof4 , d o f 3 )=x ( dof4 , d o f 3 )+ (( ? 3? rohA? v ? 2)/30? l ) x ( dof1 , d o f 4 )=x ( dof1 , d o f 4 )+ ( ( 3 ? rohA? v ? 2)/30? ) x ( dof2 , d o f 4 )=x ( dof2 , d o f 4 )+ (( ? 1? rohA? v ? 2)/30? l ) x ( dof3 , d o f 4 )=x ( dof3 , d o f 4 )+ (( ? 3? rohA? v ? 2)/30? l ) x ( dof4 , d o f 4 )=x ( dof4 , d o f 4 )+ ( ( 4 ? rohA? v ? 2)/30? l ) % % D i s s i p a t i o n ground substance assemblage d ( dof1 , d o f 1 )=d ( dof1 , d o f 1 )+ (2? ( ? 30? rohA? v ) / 6 0 ) d ( dof2 , d o f 1 )=d ( dof2 , d o f 1 )+ ( 2 ? ( 6 ? rohA? v ) / 6 0 ) d ( dof3 , d o f 1 )=d ( dof3 , d o f 1 )+ ( 2 ? ( 3 0 ? rohA? v ) / 6 0 ) 58 d ( dof4 , d o f 1 )=d ( dof4 , d o f 1 )+ (2? ( ? 6? rohA? ) / 6 0 ) d ( dof1 , d o f 2 )=d ( dof1 , d o f 2 )+ (2? ( ? 6? rohA? v ) / 6 0 ) d ( dof2 , d o f 2 )=d ( dof2 , d o f 2 )+ ( 2 ? ( 0 ? rohA? v ) / 6 0 ) d ( dof3 , d o f 2 )=d ( dof3 , d o f 2 )+ ( 2 ? ( 6 ? rohA? v ) / 6 0 ) d ( dof4 , d o f 2 )=d ( dof4 , d o f 2 )+ (2? ( ? 1? rohA? v ) / 6 0 ) d ( dof1 , d o f 3 )=d ( dof1 , d o f 3 )+ (2? ( ? 30? rohA? v ) / 6 0 ) d ( dof2 , d o f 3 )=d ( dof2 , d o f 3 )+ (2? ( ? 6? rohA? v ) / 6 0 ) d ( dof3 , d o f 3 )=d ( dof3 , d o f 3 )+ ( 2 ? ( 3 0 ? rohA? v ) / 6 0 ) d ( dof4 , d o f 3 )=d ( dof4 , d o f 3 )+ ( 2 ? ( 6 ? rohA? v ) / 6 0 ) ( dof1 , d o f 4 )=d ( dof1 , d o f 4 )+ ( 2 ? ( 6 ? rohA? v ) / 6 0 ) d ( dof2 , d o f 4 )=d ( dof2 , d o f 4 )+ ( 2 ? ( 1 ? rohA? v ) / 6 0 ) d ( dof3 , d o f 4 )=d ( dof3 , d o f 4 )+ (2? ( ? 6? rohA? v ) / 6 0 ) d ( dof4 , d o f 4 )=d ( dof4 , d o f 4 )+ ( 2 ? ( 0 ? rohA? v ) / 6 0 ) % % I n e r t i a hyaloplasm manufacturing m( dof1 , d o f 1 )=m( dof1 , d o f 1 )+ (156? M? l / 4 2 0 ) m( dof2 , d o f 1 )=m( dof2 , d o f 1 )+ (22? l ? 2? M/ 4 2 0 ) m( dof3 , d o f 1 )=m( dof3 , d o f 1 )+ (54? l ? M/ 4 2 0 ) m( dof4 , d o f 1 )=m( dof4 , d o f 1 )+ (? 3? l ? 2? M/ 4 2 0 ) m( dof1 , d o f 2 )=m( dof1 , d o f 2 )+ (22? l ? 2? M/ 4 2 0 ) m( dof2 , d o f 2 )=m( dof2 , d o f 2 )+ (4? M? l ? 3 / 4 2 0 ) m( dof3 , d o f 2 )=m( dof3 , d o f 2 )+ (13? l ? 2? M/ 4 2 0 ) m( dof4 , d o f 2 )=m( dof4 , d o f 2 )+ (? 3? M? l ? 3 / 4 2 0 ) 59 m( dof1 , d o f 3 )=m( dof1 , d o f 3 )+ (54? M? l / 4 2 0 ) m( dof2 , d o f 3 )=m( dof2 , d o f 3 )+ (13? l ? 2? M/ 4 2 0 ) m( dof3 , d o f 3 )=m( dof3 , d o f 3 )+ (156? l ? M/ 4 2 0 ) m( dof4 , d o f 3 )=m( dof4 , d o f 3 )+ (? 22? l ? 2? M/ 4 2 0 ) m( dof1 , d o f 4 )=m( dof1 , d o f 4 )+ (? 13? l ? 2?M/ 4 2 0 ) m( dof2 , d o f 4 )=m( dof2 , d o f 4 )+ (? 3? M? l ? 3 / 4 2 0 ) m( dof3 , d o f 4 )=m( dof3 , d o f 4 )+ (? 22? l ? 2? M/ 4 2 0 ) m( dof4 , d o f 4 )=m( dof4 , d o f 4 )+ (4? M? l ? 3 / 4 2 0 ) end k ( 1 1 , ) = % A p p l y i n g term c o n d i t i o n s k( ,11)= k ( ( 2 ? mnu? 2 ) ( 2 ? mnu? 2 ) , ) = k ( , ( 2 ? mnu? 2 ) ( 2 ? mnu? 2 ) ) = k x(11 ,)= x( ,11)= x ( ( 2 ? mnu? 2 ) ( 2 ? mnu? 2 ) , ) = x ( , ( 2 ? mnu? 2 ) ( 2 ? mnu? 2 ) ) = x % G l o b a l matrix f o r t h e % Force t h a t conforms f l u i d t o p i p e x1=? d(11 ,)= d( ,11)= d ( ( 2 ? mnu? 2 ) ( 2 ? mnu? 2 ) , ) = % G l o b a l S t i f f n e s s hyaloplasm 60 d ( , ( 2 ? mnu? 2 ) ( 2 ? mnu? 2 ) ) = d d1=(? d ) Kg lobal=k+10? x1 m( 1 1 , ) = m( , 1 1 ) = m( ( 2 ? mnu? 2 ) ( 2 ? mnu? 2 ) , ) = m( , ( 2 ? mnu? 2 ) ( 2 ? mnu? 2 ) ) = m nitty-gritty ( t ) zeros ( t ) H=? inv (m) ? ( d1 ) ? inv (m)? Kglobal eye ( t ) zeros ( t ) Evalue=eig (H) % E i g e n v a l u e s v r a t i o=v/ vc % V e l o c i t y Ratio % G l o b a l plentitude Matrix % G l o b a l D i s s i p a t i o nMatrix i v 2=imag ( Evalue ) i v 2 1=min( abs ( i v 2 ) ) w1 = ( i v 2 1 ) wn w r a t i o=w1 /wn vc % Frequency Ratio % sound N a t u r a l f r e q u e n c y 61 0. 2 MATLAB Program for jut Pipe Carrying Fluid MATLAB Program for Cantilever Pipe Carrying Fluid. % The f o l l o w i n g MATLAB Program c a l c u l a t e s t h e primitive % N a t u r a l f r e q u e n c y o f v i b r a t i o n , f r e q u e n c y r a t i o (w/wn) % and v e l o c i t y r a t i o ( v / vc ) , f o r a c a n t i l e v e r p i p e % carrying f l u i d . I n o r d e r t o perform t h e above t a s k t h e program a s s e m b l e s % E l e m e n t a l S t i f f n e s s , D i s s i p a t i o n , and I n e r t i a m a t r i c e s % t o form G l o b a l M a t r i c e s which are utilise % t o c a l c u l a t e unplumbed N a t u r a l % Frequency w . clc num elements = arousal ( stimulant drug number o f e l e m e n t s f o r Pipe ) % num elements = The u s e r e n t e r s t h e number o f e l e m e n t s % i n which t h e p i p e has t o be d i v i d e d . =1 num elements +1% return o f nodes ( n ) i s % e q u a l t o number o f e l e m e n t s p l u s one n o d e l =1 num elements % Parameters use i n t h e l o o p s node2 =2 num elements +1 grievous bodily harm nodel= scoop( n o d e l ) max node2=max( node2 ) max node use=max( max nodel max node2 ) mnu=max node utilise k=zeros (2? mnu ) % C r e a t i n g a G l o b a l S t i f f n e s s Matrix o f z e r o s 62 m =zeros (2? mnu ) % C r e a t i n g G l o b a l portion Matrix o f z e r o s