Raˇcunalniˇsko simuliranje dinamike rotorjev Computer ...lab.fs.uni-lj.si/ladisk/data/pdf/clanek-dinamika_rotorjev.pdf · Dinamika rotorjev se je v zadnjih letih razvila vˇsiroko

STROJNISKI VESTNIK - JOURNAL OF MECHANICAL ENGINEERING, LJUBLJANA (42) 1996/9 − 10 1

Racunalnisko simuliranje dinamike rotorjev

Computer Simulation of the Dynamics of Rotors

Robert Cokan, Miha Boltezar, Anton Kuhelj

Univerza v Ljubljani, Fakulteta za strojnistvo, Laboratorij za dinamiko strojev in konstrukcij.University in Ljubljana, Faculty of Mechanical Engineering, Laboratory for the dynamics of

machines and structures.

Povzetek

Dinamika rotorjev se je v zadnjih letih razvila v siroko paleto specialnosti. Tudi strokovnjakitezko prepoznajo in povezejo vse vplive in pojave, ki se pojavljajo v praksi. V ta namenje bil razvit programski paket, ki omogoca obravnavo upogibnih nihanj linearnih sistemovin vrednotenje vpliva razlicnih parametrov na dinamiko rotorjev. Posebna pozornost je bilaposvecena ziroskopskemu efektu in modeliranju drsnih lezajev ter njunemu vplivu na lastnefrekvence in kriticne hitrosti. Numericni del razvitega programskega paketa je zasnovan nametodi koncnih elementov. Program je namenjen konstrukterjem za dolocitev dinamicnegaobnasanja rotorja v zacetni fazi razvoja.

Abstract

The dynamics of rotors has in recent decades developed into a wide range of specialties. Evenfor the experts it is difficult to recognize and draw a correlation among all phenomena that occurin practice. For that purpose a computer program has been developed that enables evaluationof bending vibrations of linear systems and assessment of influence of various parameters onrotordynamics. Special attention has been given to the gyroscopic effect and modeling of oil-film bearings and its influence on eigenfrequencies and critical speeds. Numerical part of thedeveloped software program is based on the finite element method. Program is intended to beused as an efficient and easy to use tool in early stage of design.

0. UVOD 0. INTRODUCTION

Pojem dinamike rotorjev pokriva zelo The term rotordynamics covers a widesiroko podrocje problematike; od balansira- range of topics; from balancing of rigid ro-nja togih rotorjev do obravnave nestabilnosti tors to evaluation of instability due to non-zaradi nelinearnosti v sistemu. Zaradi kom- linearity of the system. Rotordynamics haspleksnosti posameznih sklopov se je veda v because of complexity of individual topics de-zadnjih desetletjih razvila v disciplino z ve- veloped in a wide range of specialties. Evenliko specialnostmi. Tudi strokovnjaki tezko for the experts is difficult to evaluate wholeprepoznajo in ovrednotijo celotno paleto pro- range of problems and draw the correlationblemov in povezejo vplive bistvenih parame- among significant parameters. The followingtrov. V nadaljevanju smo obravnavo ome- discussion has been restricted to the determi-jili na dolocanje lastnih frekvenc in kriticnih nation of eigenfrequencies and critical speedshitrosti rotorjev. Glede na dejstvo, da ana- of rotors. Due to the fact that analytical so-liticne resitve pokrivajo zelo ozko podrocje lutions are available only for a limited areaproblematike, je bil v preteklosti razvit niz of problems a range of discrete methods hasdiskretnih metod, ki omogocajo obravnavo been developed that make the evaluation of


problematike dinamike rotorjev. Danes se rotordynamics possible. Nowadays two me-v glavnem uporabljata dve metodi in sicer thods has been broadly used. These are firstmetoda koncnih elementov (v nadaljevanju the finite element method (FEM in the follo-MKE) in metoda prenosnih matrik. Kljub wing) and second the transfer matrix method.vse bolj splosni uporabnosti in razsirjenosti Despite the fact that FEM is more and moremetode koncnih elementov se metoda preno- used the transfer matrix method is still usedsnih matrik se vedno uporablja za obravnavo for evaluation of some specific cases.nekaterih specificnih primerov. Basic principles of the transfer matrix me-

Sodobno zasnovo metode prenosnih ma- thod, based on the use of computer, can betrik, ki temelji na uporabi racunalnika, lahko found in [1]. Somewhat modified formula-najdemo v [1]. Nekoliko spremenjeno formu- tion of the method that enables evaluationlacijo metode, ki omogoca popis ziroskopskega of gyroscopic effect and general description ofefekta in splosen opis tirnice gibanja poljubne the path of the arbitrary point in the systemtocke v sistemu, lahko najdemo v [2] in [3]. can be found in [2] and [3]. The method wasMetoda omogoca dolocitev vpliva deformabil- extended to include the influence of disk’s fle-nosti diska na lastne frekvence in kriticne hi- xibility on eigenfrequencies and critical speedstrosti [4], kot tudi obravnavo veclinijskih sis- [4] as well as evaluation of multi-line systemstemov (rotorja s podporno konstrukcijo) [5]. (systems with support construction) [5].

Glede na izkusnje z metodo prenosnih ma- Considering our own experiences with thetrik [6] se je izkazalo, da je omenjena metoda transfer matrix method [6] it has become evi-manj primerna za obravnavo veclinijskih kon- dent that the use of the method is less suita-strukcij oz. rotorjev, kjer je potrebno po- ble, where besides the rotor itself the supportleg osnovne linije modelirati tudi podporno construction has to be modeled as well. Forkonstrukcijo. V ta namen razvijamo v La- that purpose a computer program, based onboratoriju za dinamiko strojev in konstrukcij the FEM, has been developed in the Labora-(LADISK) na Fakulteti za strojnistvo v Lju- tory for dynamics of machines and structuresbljani programski paket, ki temelji na MKE, (LADISK) at the Faculty of mechanical en-in bo v koncni obliki omogocal obravnavo upo- gineering in Ljubljana that provides the me-gibnih nihanj rotorjev skupaj s podporno kon- ans for evaluation of various parameters andstrukcijo, vrednotenje razlicnih parametrov in their influence on rotordynamics. The detailsnjihov vpliv na dinamiko rotorjev. Podrob- of current development are shown in [7] andneje je trenutno stanje lastnega razvoja pri- [8].kazano v [7] in [8]. A wide range of use of the FEM in rotor-

V literaturi lahko zasledimo siroko upo- dynamics can be found in the literature. Arabnost metode koncnih elementov pri obrav- scope of issues is covered and discussed usingnavi dinamike rotorjev. V [9] je podana the FEM in [9]. The eigenfrequencies [10] andobsirna problematika na osnovi omenjene me- critical speeds [11] of the pre-twisted rotor,tode. Metoda omogoca tudi doloccitev la- described by Timoshenko’s equation, can bestnih frekvenc [10] in kriticnih hitrosti [11] determined using the FEM. For more accuraterotirajocega zvitega nosilca na osnovi Timo- description of the rotor the axisymmetric so-shenkove enacbe. Za bolj tocen popis rotorja lid finite elements can be used [12]. Conside-lahko uporabimo osno simetricne koncne ele- ring coupled systems such as driving machine-mente [12]. Pri pogonskih agregatih pride rotor the additional influences on rotordyna-zaradi zamika pogonske gredi motorja in ro- mics can be detected due to misalignment oftorja do dodatnih vplivov na dinamiko rotorja the driving shaft and rotor [13]. Linear for-[13]. Linearna formulacija problema je na- mulation of the problem is usually sufficientvadno zadostna, seveda pa metoda koncnih but the FEM can also be used for evaluationelementov omogoca tudi popis nelinearnih ni- of non-linear vibrations [14]. Application ofhanj [14]. Pri delu z metodo koncnih elemen- the FEM, in contrast to the transfer matrix


tov imamo, za razliko od metode prenosnih method where the size of transfer matrices ismatrik, kjer je velikost prenosnih matrik kon- constant, implies the use of relatively largestantna, opravka z relativno velikimi matri- matrices. Matrix reduction is therefore re-kami, zato je pogosto priporocljiva redukcija commended [12]. Considering the gyroscopicmatrik [12]. Z upostevanjem ziroskopskega effect the dynamic matrix becomes asymme-efekta postane dinamicna matrika sistema ne- tric. Solving the eigenvalue problem impliessimetricna. Za resevanja problema lastnih the use of time consuming general algorithmsvrednosti moramo zato uporabljati splosne al- [15] and [16]. The problem can be avoidedgoritme [15] in [16], ki pa so casovno zahtevni. with some specific cases using the transfor-Pri obravnavi nekaterih specificnih primerov mation of the dynamic matrix in symmetricse problemu lahko izognemo s transformacijo form [17].dinamicne matrike v simetricno [17].

1. TEORETICNO OZADJE 1. THEORETICAL BACKGROUND

Za popis linearnih upogibnih nihanj, oz. The FEM has been used to describe fle-za dolocitev matricne gibalne enacbe (1), je xural vibrations and to determine the matrixbila uporabljena metoda koncnih elementov. equation of motion (1). Basic principles of theOsnove uporabljene metode, kot tudi nacini method used as well as the mechanism to de-dolocitve matrik posameznih elementov, so termine the matrices of distinct elements aresplosno znane. Med drugim jih lahko najdemo well known. Among others they can be foundv [9] in [18]. in [9] and [18].

1.1 Problem lastnih vrednosti 1.1 The eigenvalue problem

Probleme lastnih nihanj lahko splosno The problem of natural vibrations can bepopisemo s homogeno gibalno enacbo v obliki: generally described by the homogeneous equa-

tion of motion in the form:

M x + B x + Kx = 0 (1)

kjer so M masna, B vsota dusilne in where M, B and K are mass, the sum ofziroskopske matrike ter K togostna matrika. damping and gyroscopic matrices and stiff-Pomike in njihove casovne odvode oznacimo ness matrices respectively. Displacements andz x, x in x. Resitev gibalne enacbe (1) vodi their time derivatives are denoted by x, xdo problema lastnih vrednosti. Kot rezul- and x. Solution of the equation of motiontat dobimo lastne frekvence in pripadajoce la- (1) leads to the eigenvalue problem. As a re-stne oblike obravnavanega sistema. Ob za- sult one can obtain eigenfrequencies and cor-nemarljivem ziroskopskem efektu in dusenju, responding mode shapes of the system underpri B = 0, lahko zapisemo klasicen problem consideration. With negligible gyroscopic ef-lastnih vrednosti v obliki: fect and damping, with B = 0, we can write

the classic eigenvalue problem in the form:

A− λI = 0 (2)

kjer predstavlja A dinamicno matriko sistema where A represents the dynamic matrix of thein I enotsko matriko. V primeru, ko sta ma- system and I is an identity matrix. In thesna in togostna matriki simetricni, lahko z case of symmetric mass in stiffness matricesdekompozicijo ene od matrik na spodnjo in the symmetricity of dynamic matrix A can bezgornjo trikotno matriko zagotovimo tudi si- guarantied through the decomposition of onemetricnost dinamicne matrike A. Taksen pro- of the matrices in lower and upper triangular


blem je relativno enostavno resljiv [15, 16, 19]. matrix. That kind of problems are relativelyTudi v splosnem primeru, kjer je B 6= 0, in easy to solve [15, 16, 19].

so matrike lahko tudi nesimetricne, moramo In the general case, with B 6= 0, and gene-enacbo (1) zapisati v obliki (2). Mnozenje rally asymmetric matrices, the equation (1)enacbe (1) z M−1 pripelje, ob enakosti x = x, has to be written in the form of (2). Multipli-do enacb: cation of the equation (1) by M−1 and using

identity x = x leads to the equations:

x = I xx = −M−1Kx−M−1B x

Z uvedbo Denoting

z =

xx

in

andC =

[0 I

−M−1K −M−1B

](3)

lahko tako transformirano gibalno enacbo one can write the equation of motion in thezapisemo v obliki: form:

z −Cz = 0 (4)

Lastne vrednosti so v splosnem kompleksno Eigenvalues are generally complex conju-konjugirani pari in jih lahko zapisemo v obliki: gate pairs and can thus be written in the form:

λk = αk ± iωk (5)

kjer ωk predstavlja k-to lastno frekvenco sis- where ωk represents the k-th eigenfrequencytema in αk koeficient rasti nihanj. Za ne- of the system and αk the coefficient governinggativne vrednosti αk se amplituda nihanj s the growth of vibrations. For negative valuescasom zmanjsuje, medtem ko se za pozitivne of αk vibrations decrease with time and forvrednosti povecuje. To pomeni, da je sistem positive values vibrations increase with time.pri pozitivnih vrednostih koeficienta rasti ni- That means that for positive values the sy-hanj nestabilen. stem becomes unstable.

Podobno kot lastne vrednosti so tudi la- Similar as the eigenvalues the mode shapesstni vektorji v splosnem kompleksni. Lastna are generally complex. The natural vibrationsnihanja celotnega modela lahko popisemo z of the whole model are described by the mo-gibanjem njegovih vozlisc. V uporabljenem tions at certain nodes. In the mathematicalmatematicnem modelu je gibanje sestavljeno model used these motions consist of displa-iz pomikov, pravokotnih na os rotorja in od- cements perpendicular to the rotor axis andgovarjajocih zasukov. Za numericno analizo the corresponding rotations. For the nume-zadosca zgolj obravnava pomikov, saj lahko rical analysis it is sufficient to evaluate onlyzasuke dolocimo iz upogibnice rotorja. displacements, because rotations can be cal-

Pomike v poljubnem vozliscu lahko culated from the bent form of the rotor.zapisemo [9] z enacbo: Displacement of an arbitrary node can be

written [9] by the equation:

xik = Ckeαkt [rik sin(ωkt + γk) + sik cos(ωkt + γk)] (6)

kjer so: αk, ωk - realna in imaginarna kompo- where: αk, ωk - are the real and imaginary


nenta lastne vrednosti, rik, sik - ustrezna re- parts of an eigenvalue, rik, sik - are the realalna in imaginarna komponenta lastnega vek- and imaginary parts an eigenvector and Ck, γk

torja in Ck, γk - konstanti, ki ju dolocimo iz - are constants found from initial conditions.zacetnih pogojev. When considering only natural vibrations

Pri obravnavi lastnih nihanj lahko za- one can neglect the change in amplitude andnemarimo spremembo amplitude nihanj in put αk = 0. Further we can with no seriouszapisemo αk = 0 . Nadalje lahko zanema- restrictions put Ck = 1 and γk = 0. In thisrimo konstanti, odvisni od zacetnih pogojev way simplified equation has the form:in zapisemo Ck = 1 in γk = 0. Tako poeno-stavljeno enacbo lahko zapisemo v kompleksniobliki kot:

z(t) = R′ej(ωt+α′) + R′′e−j(ωt−α′′) (7)

Opazimo lahko, da je elipticna tirnica na It can be seen that the elliptical path insliki 1 sestavljena kot vsota dveh rotirajocih Figure 1 consists of the sum of two rotatingvektorjev, ki imata oba kotno hitrost ω. Vek- vectors, both having angular velocity ω. Onetor dolzine R′ rotira sinhrono s smerjo rotacije vector of length R′ rotates in the positive di-rotorja, medtem ko drugi vektor, dolzine R′′ , rection while the other of length R′′ rotates inrotira asinhrono. the negative direction.

a

Re(z), x1

Im(z), x2

W

R'

a' R''a''

Slika 1: Elipticna tirnicaFigure 1: Elliptical path

Za dolocitev smeri opletanja vozlisca vpe- To find the direction of rotation we intro-ljemo rotacijsko stevilo duce the rotation number

u =R′ −R′′

R′ + R′′ , (8)

ki doloca smer opletanja obravnavanega vo- which indicates the direction of rotation of


zlisca. Glede na rotacijsko stevilo u je opleta- the node considered. Considering the rota-nje lahko: tion number u the rotatioin can be:

−1 ≤ u < 0 asinhrono opletanjeu = 0 gibanje vzdolz premice ali

0 < u ≤ 1 sinhrono opletanje

−1 ≤ u < 0 asinhrono opletanjeu = 0 gibanje vzdolz premice ali

0 < u ≤ 1 sinhrono opletanje

1.2 Resevanje problema lastnih vrednosti 1.2 Solving the Eigenvalue Problem

Za resevanje problema lastnih vrednosti To solve the eigenvalue problem a range ofsmo testirali in primerjali niz numericnih me- numerical routines has been tested and com-tod. Za simetricne matrike smo uporabili pared. For symmetric matrices the Househol-Hauseholderjevo metodo. Da bi obdrzali si- der method has been used. To retain symme-metricnost dinamicne matrike A, definirane v tric properties of the dynamic matrix A defi-enacbi (2), smo za masno matriko M upora- ned in equation (2), Cholesky decompositionbili dekompozicijo Choleskega. Za resevanje of mass matrix M has been used. Conside-splosnega problema lastnih vrednosti, defini- ring the general eigenvalue problem defined byranega z enacbo (4), program uporablja za equation (4) the HQR (Hessenberg) algorithmdolocitev lastnih vrednosti HQR (Hessenber- has been used to calculate eigenvalues whilegovo) metodo in inverzno iteracijo vektorjev inverse vector iteration has been applied toza dolocitev lastnih oblik. Podrobneje so ome- obtain eigenvectors. Details of all aforementi-njene metode predstavljene v [15], [16] in [19]. oned methods can be found in references [15],

[16] and [19].

1.3 Teorija kratkega drsnega lezaja 1.3 Oil-film Bearing

Osnove MKE, kot tudi nacini dolocevanja The FEM as well as the mechanism to de-matrik posameznih elementov, so splosno termine the matrices of distinct elements areznani [9] in [18]. V nadaljevanju je zaradi po- well known [9] and [18]. Because of the si-membnega vpliva na dinamicno obnasanje ce- gnificant influence on the dynamic behaviorlotnega rotorja zato na kratko predstavljen le of entire rotor the model of the oil-film bea-uporabljeni model drsnega lezaja [9]. Primer ring is shortly discussed in the following. Antaksnega lezaja je prikazan na sliki 2 example is shown in Figure 2.

zx

yy

g

j

eJ

W

L

h0 h

R

rJ

CB

CJCJ

CB

Slika 2: Drsni lezajFigure 2: Oil-film bearing

Tlacne razmere v drsnem lezaju navadno The pressure conditions in the oil-film be-popisemo Reynoldsovo enacbo [9]. Za pri- aring are usually described by the Reynolds’


mer drsnega lezaja s krozno lezajno blazi- equation [9]. For the case of circular bearingnico ima Reynoldsova enacba, ob zanemaritvi and neglecting the change in pressure in cir-spremembe tlaka v obodni smeri, obliko: cumferential direction we can write the Re-

ynolds’ equation:

∂2p

∂z2=

6η

h3(ϕ, t)[ej(Ω− 2γ) sin(ϕ− γ)− 2ej cos(ϕ− γ)] (9)

kjer so: η - dinamicna viskoznost maziva, Ω - where: η - is the dynamic viscosity of the lu-kotna hitrost gredi oz. cepa in p - tlak maziva. bricant, Ω - is the angular velocity of the shaft

Govorimo o teoriji kratkega drsnega lezaja and p - is the pressure of the lubricant.oz. o linearnem modelu drsnega lezaja. The theory is known as short oil-film be-Za staticno obremenjen horizontalno podprt aring theory or linear oil-film bearing model.rotor lahko togostne in dusilne koeficiente For a horizontally supported rotor with staticdolocimo z odvajanjem komponent rezulti- load one can obtain stiffness and damping co-rajoce sile glede na pomike: efficients by differentiating component forces

with respect to displacements:

kik =∂Fi

∂xk

= γikF0

δ, dik =

∂Fi

∂xk

= βikF0

δΩ(10)

kjer sta γik in βik brezdimenzijska togostna in where γik and βik are dimensionless stiffnessdusilna koeficienta. and damping coefficients respectively.

0.0

0.1

1.0

10.0

100.0

0.0 0.2 0.4 0.6 0.8 1.0

gik

e

g11

g22

g12 -g12

-g21

0.1

1.0

10.0

100.0

0.0 0.2 0.4 0.6 0.8 1.0

bik

e

b11

b22

-b12, -b21

Slika 3: Brezdimenzijski togostni koeficientFigure 3: Dimensionless stiffness and damping coefficients

Na opisani nacin dolocena brezdimenzijska Coefficients obtained in the described waykoeficienta sta prikazana na sliki 3 v odvi- are shown in Figure 3 as functions of thesnosti od razmernika ekscentricnosti drsnega eccentricity ratio. Latest is defined by thelezaja. Le-ta je definiran z enacbo: equation:

ε =e3

R− r3

(11)

Iz diagramov je razvidno, da sta togo- From the diagrams one can observe thatstna in dusilna matrika drsnega lezaja nesi- the stiffness and damping matrices of the oil-metricni. Kot bo prikazano v nadaljevanju, film bearing are asymmetric. In the case oflahko neugodna kombinacija parametrov dr- unfavorable combination of parameters con-


snega lezaja in celotnega rotorja privede tudi cerning oil-film bearing and the whole rotordo nestabilnosti delovanja. the aforementioned property can lead to the

unstable operation as it will be shown in thefollowing.

2. PROGRAMSKI PAKET 2. PROGRAM PACKAGE

Za ovrednotenje vpliva razlicnih parame- To evaluate the influence of various para-trov na dinamiko rotorjev smo v Laboratoriju meters on rotordynamics the computer pro-za dinamiko strojev in konstrukcij (LADISK) gram, based on the finite element method, hasrazvili racunalniski program, ki temelji na me- been developed in the Laboratory for dyna-todi koncnih elementov. Namen programa mics of machines and structures (LADISK).je zapolniti vrzeli komercialnih programov, ki The program is intended to fill the gapsvecinoma (z redkimi izjemami) ne zadoscajo of general FEM programs, which (with rareposebnim zahtevam obravnave dinamike ro- exemptions) do not meet the special require-torjev (ziroskopski efekt ter nesimetricne to- ments of rotordynamics (gyroscopic effect andgostne in dusilne matrike). asymmetric stiffness and damping matrices).

Izdelani program vkljucuje numericno vre- The developed program incorporates nu-dnotenje matematicnega modela kot tudi merical processing of the mathematical modelgraficno pripravo podatkov in graficni pri- as well as graphical pre- and post-processingkaz dobljenih rezultatov. Groba programska of given numerical information. A rough pro-shema je prikazana na sliki 4. gram chart is shown in Figure 4.

Elementi

Elements

Grafièni modeler

Graphic modeller

Izraèuni

Calculations

Lastne vrednosti

Eigenvalues

Dinamièni odziv

Dynamic response

Kritiène hitrosti

Critical speeds

Delo z datotekami

File management

Popravljanje

Editing

Pregled rezultatov

Results owerview

Nastavitve

Options

Matematièni model

Mathematical model

Pomoè

Help

Disk

Disk

Zunanja obdelava

External processing

Tiskalnik

Printer

Slika 4: Programska shemaFigure 4: Program chart

Zaradi poenostavitve dela s programom je To simplify handling of the program, abil razvit graficni uporabniski vmesnik, ki graphics interface has been created that ena-omogoca enostavno modeliranje in obdelavo bles elegant modeling and processing of data.podatkov. Vnos podatkov je omogocen s sis- Input of data is supported through the sy-temom dialogov in menujev. Primer vnosa stem of dialogs and menus. An example ofpodatkov za gredni element je prikazan na data input is shown in Figure 5 for the beamsliki 5. Koncni uporabnik tako ne potrebuje element. The user thus does not need anynobenega znanja o ukazih za modeliranje ma- knowledge about the syntax of a modelingtematicnega modela. Interpretacijo vnesenih language. Interpretation and translation ofpodatkov in pretvorbo v racunalniku razu- input data to computer-understandable lan-


mljiv jezik opravi sam program v ozadju, ne- guage is done in the background by the pro-odvisno od uporabnika. Kjer je bilo to mogoce gram itself. Where possible the checking ofje vgrajeno preverjanje vnesenih podatkov, input information has been implemented ins cemer se izognemo nepotrebnim logicnim order to avoid any logical mistakes. As shownnapakam pri vnosu podatkov. Kot je raz- in Figure 5 the program enables work with se-vidno s slike 5, program omogoca hkratno veral models simultaneously. Transfer of datadelo z vec modeli. Prenos podatkov med between loaded models is made possible thro-modeli je omogocen preko internega ”odla- ugh the program’s internal clipboard.galisca”podatkov.

Slika 5: Uporabniski vmesnik - vnos podatkovFigure 5: User interface - data input

Uporabnik lahko prilagodi program svo- The user can adjust the program to his re-jim potrebam preko niza nastavitev. Mozne quirements with a set of options. Appearanceso nastavitve izgleda uporabniskega vmesnika of the user interface as well as some numericalkot tudi nekaterih numericnih parametrov. parameters can be adapted to meet the user’s

Poleg interaktivnega dela s programom je requests.mozna tudi zunanja paketna obdelava podat- Besides interactive work with the program,kov. Slednja je mozna s programiranjem v external processing of data is also possible.poljubnem programskem jeziku, seveda pod The latest is possible by programming in ar-pogojem, da uporabnik pozna obliko zapisa bitrary computer language, assuming the filedatotek in so mu na voljo potrebne numericne format used by the program and numericalrutine. Zunanja paketna obdelava podatkov routines are available to the user. That kindima velikokrat prednost pred interaktivnim of work is sometimes preferred to interactivedelom s programom vendar od uporabnika but it requires detailed knowledge about thezahteva dosti vec znanja o samem programu program structure and general numerical ana-kot tudi o splosni numericni analizi. lysis.

Za namen prikaza numericnih rezultatov je To review the calculations a postprocessorbil narejen graficni ”post-procesor”. Na sliki has been developed which enables graphic re-


6 je predstavljen primer prikaza izracunanih presentation of results. An example is shownlastnih frekvenc in pripadajocih lastnih oblik in Figure 6 where calculated eigenfrequenciesmodela. and corresponding mode shapes are shown.

Kot smo ze omenili program omogoca As mentioned before the program supportstudi zunanjo obdelavo podatkov. Le-ta je further processing of numerical results. Latestomogocena tudi preko izvoza podatkov in re- has been made possible through the export ofzultatov v ASCII obliki. Tako zapisane po- data in the ASCII file format. Results, writ-datke lahko uporabljamo z vecino komercial- ten in that form, can be accessed and furthernih programskih paketov. Omogocen je iz- processed by almost any commercial compu-voz koncnih rezultatov (lastne vrednosti in ter program. Export of results (eigenvalueskriticne hitrosti) kot tudi posameznih matrik and critical speeds) as well as individual ma-v enacbi (1). trices in equation (1) is possible.

Slika 6: Uporabniski vmesnik - prikaz rezultatovFigure 6: User interface - data output

Program je napisan za PC kompatibilne The program is written for PC compati-racunalnike z operacijskim sistemom Win- ble computers with Windows 3.1 operatingdows 3.1. Za smotrno delo s programom je systems. For better performance at least 486zazelen vsaj 486 DX procesor. DX processor is preferred.

Trenutno program omogoca dolocitev: At the present stage of development the- lastnih vrednosti in pripadajocih lastnih vek- program enables calculation of:torjev, - eigenvalues and eigenvectors,- kriticnih hitrosti, - critical speeds,- odziva modela v ustaljenem stanju zaradi - response due to dynamic force andzunanje harmonske dinamicne obremenitve in - response to given kinematics of one or more- odziva modela zaradi predpisane kinematike arbitrary points of the system.ene ali vec tock v sistemu.

3. DINAMICNA ANALIZA MODELAVENTILATORJA

3. DYNAMIC ANALYSIS OF FANMODEL


Za analizo posameznih vplivov na dina- For the analysis of distinct influences on ro-miko rotorjev, predvsem ziroskopskega efekta tordynamics, mainly the gyroscopic effect andin vpliva drsnih lezajev, oz. za predstavitve the influence of the oil-film bearing, and toprogramskih zmogljivosti, smo oblikovali mo- demonstrate the program capabilities a venti-del ventilatorja. Osnovni model je prikazan lator model has been built. The basic modelna sliki 7. Ventilator ima celotno maso 436 is shown in Figure 7. The ventilator has totalkg in dolzino 1280 mm. Opazovano obmocje mass of 436 kg and length of 1280 mm. Theobratovanja ventilatorja je od 0 do 1000 rad/s. range of operation in interest is from 500 to

1000 rad/s.

Slika 7: Model ventilatorjaFigure 7: Ventilator model

Za vrednotenje posameznih vplivov na la- To identify distinct influences on eigenfre-stne frekvence in kriticne hitrosti smo primer- quencies and critical speeds, several variationsjali nekaj variacij osnovnega modela. Osnovni of the basic model were considered. Geome-geometrijski in materialni podatki so enaki za tric and material properties are the same forvse obravnavane modele. Izvedba podprtja in all the tested models. Description of the su-opis upostevanih dodatnih parametrov je za pport conditions and additional model para-vse modele zbran v tabeli 1. meters are given in Table 1.

Tabela 1: Opis modelnih parametrovTable 1: Description of model parameters

model 1 2 3 4 5 6 7 modeltogi lezaji × × × rigid bearingselasticni lezaji × × elastic bearingsdrsni lezaj × × oil-film bearingstrizne deformacije × × × × × × share deflectionziroskopski efekt × × × gyroscopic effekt

Matematicni model sestavlja 13 elementov. The numerical model consists of 13 ele-Modeli z elasticnimi podporami in drsnimi ments. Models with elastic supports andlezaji (modeli 4-7) imajo tako 14*4=56 pro- oil-film bearings (models 4-7) have thereforestostnih stopenj. Modeli s togo izvedbo pod- 14*4=56 degrees of freedom. Rigidly suppor-prtja (modeli 1-3) imajo na mestu podprtja ted models (models 1-3) have at the supportonemogocene pomike v vseh smereh in imajo point disabled motions in all directions andtako 52 prostostnih stopenj. have therefore 52 degrees of freedom.

3.1 Rotor s togim podprtjem 3.1 Rotor with rigid bearings

Za modela 1 in 2 smo predpostavili togo For models 1 and 2 rigid bearings were as-


podprtje rotorja. Model 1 predstavlja osnovni sumed. Model 1 represents the basic modelmodel rotorja brez dodatnih vplivov, medtem with no additional influences, while the eigen-ko smo lastne frekvence modela 2 dolocili z values for model 2 were calculated consideringupostevanjem striznih deformacij. Za oba mo- shear deflection. First 4 eigenfrequencies aredela so prve stiri lastne frekvence podane v given in Table 2 for both models.tabeli 2.

Tabela 2: Lastne frekvence modelov 1 in 2Table 2: Eigenfrequencies of models 1 and 2

Model 1 Model 2ω01 137.02 136.43ω02 299.43 297.18ω03 2566.64 2467.2ω04 3792.25 3619.60

Zaradi striznih deformacij lahko opazimo One can observe small changes in naturalmajhne spremembe lastnih frekvenc. Razlika frequencies due to shear deflection. The diffe-lezi v obmocju od -0.4% za prvo do -4.5% za rence lies in the range from -0.4% for the 1stcetrto lastno frekvenco. Vpliv striznih defor- to -4.5% for the 4th eigenfrequency. Due tomacij je za obravnavani model majhen, saj the fact that the model is relatively slenderimamo opravka z relativno vitkim rotorjem. the influence of shear deflection is small. ForZa taksne rotorje je omenjeni vpliv majhen in such rotors the aforementioned influence is ge-ga lahko zanemarimo, ne da bi s tem naredili nerally small and can be neglected with no se-preveliko napako. Vpliv striznih deformacij je rious restrictions. The influence of shear de-bolj opazen pri kratkih rotorjih, katerih raz- flection is more noticeable with rotors wheremerje dolzine modela proti premeru gredi je the ratio of rotor length to the shaft diametermanjse. gets smaller.

Za ovrednotenje ziroskopskega efekta smo Model 3 has been used to estimate the in-uporabili model 3. Na sliki 8 je prikazana od- fluence of gyroscopic action. In Figure 8 thevisnost lastnih frekvenc od krozne hitrosti ro- dependence of eigenfrequencies on rotor speedtorja. Opazimo lahko, da se stevilo lastnih is shown. One can observe that the number offrekvenc podvoji, ce je krozna hitrost rotorja eigenfrequencies doubles with the shaft speedΩ 6= 0. Oznake lastnih frekvenc na sliki 8 Ω 6= 0 so the notation in Figure 8 has beensmo zato ustrezno popravili. Opazimo lahko adjusted accordingly. Forward and backwardsinhrono in asinhrono opletanje rotorja. whirl of rotor can also be noticed.

Kot je razvidno s slike 8 je vpliv As it can be seen from the graph, the in-ziroskopskega efekta velik in ga nikakor ne gre fluence of gyroscopic action can not be ne-zanemariti. Zakljucek velja splosno za rotorje glected. This is generally the case with rotorsz rotacijskimi telesi, katerih masni vztrajno- having rotating bodies with mass moments ofstni momenti so veliki v primerjavi z masnimi inertia relatively large compared to the one ofvztrajnostnimi momenti same gredi modela. the shaft itself. Calculations have also beenOpravili smo tudi primerjavo lastnih frekvenc done for the shaft of the model 3 alone. Inza samo gred modela 3. V tem primeru lahko that case, the influence of gyroscopic actionvpliv ziroskopskega efekta zanemarimo, saj je can be neglected since the changes of eigen-razlika lastnih frekvenc z in brez upostevanja frequencies in the whole range of operationziroskopskega efekta manjsa od 1%. are less than 1%.

Pri Ω = ΩS lahko opazimo zanimiv pojav. At Ω = ΩS an interesting phenomenon canZa vrednosti krozne frekvence Ω > ΩS je na- be observed. For Ω > ΩS the 3rd eigenfre-


0

50

100

150

200

250

300

350

400

450

0 200 400 600 800 1000

W [rad/s]

w0 [rad/s]

Wcr,1 Wcr,2 Wcr,3 Wcr,4 WS

w01

w03

w02

w04

Slika 8: Lastne frekvence modela 3Figure 8: Eigenfrequencies of model 3

Tabela 3: Kriticne hitrosti modela 3Table 3: Critical speeds of model 3

Ωcr,1 = 94.42 rad/sΩcr,2 = 168.02 rad/sΩcr,3 = 249.60 rad/sΩcr,4 = 321.57 rad/s

mrec tretja lastna frekvenca modela manjsa quency is smaller than the 2nd. Both re-od druge lastne frekvence. Obe frekvenci tain their direction of rotation; that is forwardohranita svojo smer opletanja; t.j. sinhrono whirl for the 2nd and backward whirl for theopletanje za 2. in asinhrono opletanje za 3. 3rd eigenfrequency. The phenomenon can belastno frekvenco. Omenjeni pojav je opazen observed with rotors having rotating bodiespri modelih z velikimi rotacijskimi telesi, ka- with mass moments of inertia relatively largeterih masni vztrajnostni momenti so veliki v compared to the one of the shaft itself. Fourprimerjavi z masnimi vztrajnostnimi momenti critical speeds, marked in Figure 8 and listedsame gredi rotorja. V opazovanem obmocju in Table 3, can be found in the observed rangelahko opazimo stiri kriticne hitrosti, ki so of shaft speed.oznacene na sliki 8 in prikazane v tabeli 3.

3.2 Rotor z elasticnim podprtjem 3.2 Rotor with flexible bearings

Za dolocitev vpliva elasticnosti podpor na To study the influence of bearing flexibilitylastne frekvence in kriticne hitrosti rotorja on eigenfrequencies and critical speeds modelssmo uporabili modela 4 in 5. Uporabili smo 4 and 5 have been considered. Bearings werepoenostavljen model izotropnih podpor. Obe assumed isotropic. Both bearings have thepodpori sta modelirani z enakimi vzmetnimi same stiffness coefficients. Model 4 has beenkonstantami. Model 4 smo uporabili za oceno used to assess the influence of bearing stiffnessvpliva togosti lezajev na prve stiri lastne fre- to the first four eigenfrequencies. The ratios ofkvence. Razmerje lastnih frekvenc modela 4 eigenfrequencies over reference frequencies arein referencnih lastnih frekvenc je prikazano na shown in Figure 9. As the reference frequen-sliki 9. Kot referencno vrednost smo uporabili cies, the values of model 2 have been used.lastne frekvence modela 2. It is to be emphasized that the program

Pri tem gre poudariti, da program si- enables the use of elastic supports but the va-cer omogoca modeliranje elasticnih podpor, lues for the stiffness coefficients of the bea-vendar pa so vrednosti za togost lezajev rings are difficult do determine. Usually theyzelo tezko dolocljive. Obicajno jih moramo have to be determined experimentally or ba-dolociti s poskusi ali pa jih ocenimo na pod- sed on experiences.lagi izkusenj.

Povecanje togosti lezajev vpliva na The increase of bearing stiffness results inpovecanje lastnih frekvenc, ki limitirajo k vre- increased eigenfrequencies which in the limitdnostim za model s togimi podporami (model approach the values of the model with rigid2). Zmanjsanje togosti lezajev pa se odrazi z bearings (model 2). On the other hand thenizjimi vrednostmi lastnih frekvenc, ki limi- decrease of bearing stiffness results in decrea-


0.0

0.2

0.4

0.6

0.8

1.0

0E+0 1E-3 1E-2 1E-1 1E+0 1E+1 1E+2 1E+4 1E+6

k [N/m]

w

w

0i

iR

w01w02

w03

w04

Slika 9: Lastne frekvence modela 4Figure 9: Eigenfrequencies of model4

tirajo k vrednostim nepodprtega modela. Za sed eigenfrequencies which in the limit appro-zelo majhne togosti lezajev se prvi lastni fre- ach the values of the free rotor with respectkvenci priblizata vrednosti nic. Pri teh vre- to support conditions. For very small valuesdnostih celoten rotor niha kot togo telo. of bearing stiffness coefficients of bearings the

first two eigenfrequencies approach zero. Atthat value the entire rotor vibrates as a rigidbody.

Na sliki 9 so lahko razvidna 3 locena po- Three distinct areas can be seen in the Fi-drocja. V obmocju na zacetku diagrama, kjer gure 9. In the initial part of the diagramje togost lezajev v obmocju 0 ≤ k ≤ 10−2N/m where the bearing stiffness lies in the range, so spremembe lastnih frekvenc, v odvisnosti 0 ≤ k ≤ 10−2N/m the changes of eigenfre-od spremembe togosti podprtja, zelo majhne. quencies are small with regard to the changeZa 3. in 4. lastno frekvenco je to obmocje of the bearing stiffness. The area is even widerse vecje 0 ≤ k ≤ 1N/m. Podobno lahko for the 3rd and 4th eigenfrequency and amo-zakljucimo za obmocje na desnem robu dia- unts to 0 ≤ k ≤ 1N/m. Similar can be saidgrama, kjer se vrednosti lastnih frekvenc uje- for the area at the right side of the diagrammajo z vrednostmi togo podprtega modela. where the values correspond to the eigenfre-

quencies of rigidly supported rotor.V srednjem delu diagrama, kjer je togost In the central part of the diagram where

lezajev v obmocju 0 ≤ k ≤ 102N/m, pa so la- the bearing stiffness lies in the range the ei-stne frekvence zelo obcutljive na spremembo genfrequencies are very sensitive to the change

togosti lezajev. Ze majhna sprememba to- of bearing stiffness. Even the small change ofgosti lahko pomeni veliko spremembo lastnih the stiffness results as a large change of ei-frekvenc. Problem pri modeliranju elasticnih genfrequencies. The problem of modeling thepodpor je v dejstvu, da so togosti lezajev, ki elastic supports lies in the fact that the valuesse obicajno pojavljajo v praksi, ravno v osre- that occur in the practice lie in the centraldnjem obmocju diagrama. Izracunane lastne part of diagram. Calculated eigenfrequenciesfrekvence so tako zelo odvisne od izbranih vre- are therefore highly dependent on the choicednosti za togost lezajev. of the values for the bearing stiffness.

Vpliv ziroskopskega efekta na dinamiko ro- The influence of gyroscopic action to ro-torjev pri modelu z elasticnimi podporami tordynamics for elastic supported rotor has


0

50

100

150

200

250

0 200 400 600 800 1000

W [rad/s]

w0 [rad/s]

WS

w01

w02

w03

w04


Tabela 4: Kriticne hitrosti modela 5Table 4: Critical speeds of model 5

Ωcr,1 = 72.25 rad/sΩcr,2 = 92.13 rad/sΩcr,3 = 168.34 rad/sΩcr,4 = 188.35 rad/s

smo opazovali pri togosti obeh lezajev k = been estimated using the stiffness coefficients107N/m. Odvisnost prvih stirih lastnih fre- k = 107N/m for both bearings. Dependencykvenc od krozne hitrosti rotorja je prikazana of the first four eigenfrequencies on the shaftna sliki 10. Opazimo lahko, da so vrednosti la- speed is shown in Figure 10. It can be noticedstnih frekvenc malo nizje kot so bile za primer that the first 4 eigenfrequencies are a bit lowertogega podprtja. Vpliv ziroskopskega efekta than those of rigid support. The influence ofje pricakovano manjsi, saj je togost celotnega the gyroscopic action is as expected smallerrotorja manjsa zaradi elasticnih podpor. since the stiffness of entire rotor is decreased

Podobno kot za model s togimi podporami, due to elastic supports.lahko opazimo stiri kriticne hitrosti, ki so Similar to the model with rigid bearingsoznacene na sliki 10 in zbrane v tabeli 4. Za- four critical speeds, marked in Figure 10 andradi spremembe lastnih frekvenc pa so ustre- listed in Table 4, can be found. Since the ei-zno nizje tudi kriticne hitrosti. genfrequencies have changed the critical spe-

eds are correspondingly lower.

3.3 Rotor z drsnimi lezaji 3.3 Rotor with oil-film bearings

Za ovrednotenje vpliva drsnih lezajev na Models 6 and 7 have been used to studydinamiko rotorjev smo obravnavali modela 6 the influence of the oil-film bearings on rotor-in 7. Uporabljena je bila teorija kratkih dr- dynamics. Short oil-film bearing theory hassnih lezajev, podrobneje predstavljena v po- been used as described in the chapter aboutglavju o rotorjih z drsnimi lezaji. Predposta- rotors with oil-film bearings. Dynamic visco-vljena dinamicna viskoznost maziva je η = sity of the lubricant was η = 0.01Ns/m2 for0.01Ns/m2 za oba lezaja. Glede na staticno both bearings. Considering the static load theobremenitev znasa tlak v lezajih 140 kPa za bearing pressures are 140 kPa and 1.25MPalevi in 1.25 MPa za desni lezaj. for left and right bearing respectively.

Pri obravnavi drsnih lezajev, katerih raz- Considering the oil-film bearings with themerje dolzine in premera je manjse od 1, se ratio of the bearing length to the bearing di-vrednosti togostnih in dusilnih koeficientov, ameter smaller than 1 the values of stiffnessdolocenih s teorijo kratkega drsnega lezaja, le and damping coefficients determined with themalo razlikujejo od realnih vrednosti. Napaka short bearing theory differ only slightly fromse povecuje z narascanjem razmernika ekscen- the real ones. The difference increases withtricnosti. Rezultati dobljeni pri zelo velikih the eccentricity ratio. Results obtained atvrednostih razmernika ekscentricnosti, ε ≈ 1, very high values of eccentricity ratio, ε ≈ 1,so zato dvomljivi. can therefore be considered as doubtful.

Ker se razmernik ekscentricnosti hitro Since the eccentricity ratio decreases rapi-


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 500 1000 1500 2000W [rad/s]

e L

e R

e

Slika 11: Razmernik ekscentricnosti lezajevFigure 11: Eccentricity ratio of bearings

zmanjsuje s povecevanjem krozne hitrosti ro- dly with shaft speed W the range of the shafttorja Ω, je v nadaljevanju pri obravnavi speed rad/s is thus not evaluated in all the fol-izpusceno obmocje krozne hitrosti 0 ≤ Ω ≤ lowing. As can be seen in Figure 11 the values200rad/s. Kot je prikazano na sliki 11 so of the eccentricity ratio are especially high forvrednosti razmernika ekscentricnosti v tem the right bearing. The omitted range can notobmocju velike predvsem za desni lezaj. be taken generally since the eccentricity ratioIzpuscenega obmocja seveda ne gre jemati for the specific case depends on geometric pa-splosno, saj je razmernik ekscentricnosti za rameters of the bearings as well as the rotorkonkreten primer odvisen od geometrijskih itself.parametrov lezaja kot tudi od rotorja samega. As expected the eccentricity ratio of both

Po pricakovanjih se razmer- bearings decreases with rotor speed. In Figurenik ekscentricnosti obeh lezajev zmanjsuje s 11 εL and εR denotes the eccentricity ratiopovecevanjem kotne hitrosti rotorja. Na sliki of left and right bearing respectively. Depen-11 je z εL oznacen razmernik ekscentricnosti dency of stiffness and damping coefficients onza levi, z εR pa za desni lezaj. Odvisnost togo- rotor speed has been calculated after equationstnih in dusilnih koeficientov od kotne hitrosti (10) and are shown in Figure 12 for both be-rotorja smo dolocili z enacbama (10). Vre- arings. It can be noticed that all stiffness co-dnost koeficientov je za oba lezaja prikazana efficients for the right bearing converge towardna sliki 12. Opazimo lahko, da vsi togostni definite value. For the left bearing the incre-koeficienti desnega lezaja konvergirajo k neki ase of off-diagonal stiffness coefficients withkoncni vrednosti. Pri levem lezaju pa lahko rotor speed can be noticed. The rotor repre-opazimo povecevanje izvendiagonalnih togo- sents therefore highly cross-coupled system.stnih koeficientov s kotno hitrostjo rotorja. The vibration of cross-coupled systems canRotor je zaradi tega mocno vezan sistem. Kot became unstable as will be shown in the fol-bo razvidno v nadaljevanju, lahko nihanje ve- lowing. Such behavior can be explained withzanih sistemov postane tudi nestabilno. Dej- higher static load for the right bearing whichstvo lahko razlozimo z vecjo staticno obre- also results in increased bearing pressure.menitvijo desnega lezaja, ki ima za posledicovecji tlak v lezaju.

Dusilni koeficienti obeh lezajev, na sliki 12, Damping coefficients for both bearings,kazejo kvalitativno podobno vedenje in se mo- shown in Figure 12, show qualitatively similarnotono zmanjsujejo s povecevanjem kotne hi- behavior for both bearings. They monotoni-


-10

-8

-6

-4

-2

0

2

4

6

8

10

0 500 1000 1500 2000

*1E6

k12

k11

k22

k21

W [rad/s]

kij [N/m]

-100

-50

0

50

100

150

0 500 1000 1500 2000

*1E6

k12

k11

k22

k21

W [rad/s]

kij [N/m]

-30

-20

-10

0

10

20

30

40

50

60

70

0 500 1000 1500 2000

* 1E3

d12, d21

d22

d11

W [rad/s]

dij [Ns/m]

-200

-100

0

100

200

300

400

0 500 1000 1500 2000

* 1E3

d12, d21

d22

d11

W [rad/s]

dij [Ns/m]

Levi leaj Left bearing Desni leaj Right bearing

Slika 12: Togostni in dusilni koeficient lezajevFigure 12: Stiffness and damping coefficients of bearings

trosti rotorja. cally decrease with rotor speed.Zanimiv pojav lahko opazimo pri lastnih An interesting observation can be made

oblikah za model 7. Kot je bilo prikazano pri from mode shapes of model 7. As it has beenteoreticni obravnavi dinamike rotorjev, pov- presented in the theoretical consideration ofzroci ziroskopski efekt bodisi sinhrono ali pa rotordynamics the gyroscopic action causes ei-asinhrono rotacijo rotorja. V vecini primerov ther forward or backward whirl of the rotor.je rotacija celotnega rotorja enaka, kar pa ne In most cases the direction of rotation is thevelja za rotorje z drsnimi lezaji. Na sliki 13 je same for the entire rotor which does not standprikazana 5. lastna oblika modela 7 pri kotni for the rotors with oil-film bearings. The Fi-hitrosti rotorja Ω = 500rad/s. Del rotorja ro- gure 13 shows the 5th mode shape of modeltira sinhrono (prekinjene linije), medtem ko 7 at rotor speed Ω = 500rad/s. Parts of thedeli rotorja, oznaceni s polno linijo, rotirajo rotor demonstrates a forward whirl (dashedasinhrono. Pojav je opazen le pri rotorjih, ki lines) while the rest of the rotor marked withso podprti z drsnimi lezaji (modela 6 in 7), solid lines is subjected to backward whirl.tako da si ga lahko razlozimo z anizotropijo This phenomenon has been observed only arin povezanostjo togostne in dusilne matrike rotors, supported by oil-film bearings and candrsnega lezaja. be explained through their anisotropy.

Togostni koeficienti desnega lezaja modela Stiffness coefficients for the right bearing7 so vecji, kot so bili uporabljeni pri modelu of model 7 are relatively large compared to5, medtem ko so togostni koeficienti za levi those used for model 5 while the values forlezaj primerljivi. Lastne frekvence modela z the left bearing are comparable. The eigen-drsnimi lezaji, slika 14, so nekoliko nizje od frequencies, shown in Figure 14, are howe-


Slika 13: Peta lastna oblika modela 7 pri Ω = 500rad/sFigure 13: Fifth mode shape of model 7 at Ω = 500rad/s

lastnih frekvenc modela s homogenimi ela- ver somewhat lower than those of the modelsticnimi podporami. Dejstvo lahko razlozimo with homogeneous flexible bearings. This cans povezano togostno in dusilno matriko mo- be explained through the influence of cross-dela 7. Prav tako je mocno izrazen vpliv coupling. Influence of gyroscopic action is alsoziroskopskega efekta, saj se lastne frekvence clearly manifested since the eigenfrequenciesmocno spreminjajo s povecanjem krozne hi- change a lot with increasing shaft speed.trosti rotorja. Eigenfrequencies appear always as complex

Lastne vrednosti se vedno pojavljajo v conjugate pairs (5). With models with nokompleksno konjugiranih parih (5). Pri mo- damping we have only imaginary component.delih, kjer nimamo dusenja, imajo lastne vre- Presence of damping in the model demonstra-dnosti le kompleksno komponento. V kolikor tes itself as the real component of the eigen-se v modelu pojavi dusenje, dobimo pri la- frequency. The Figure 15 shows the real com-stnih vrednostih tudi realno komponento. Na ponents of the eigenvalues for the model withsliki 15 so prikazane realne komponente la- oil-film bearings.stnih vrednosti modela z drsnimi lezaji.

0

50

100

150

200

250

0 500 1000 1500 2000

W [rad/s]

[rad/s]

WS

w01

w02

w03

w04

WS

w


-100

-75

-50

-25

0

25

50

75

100

0 500 1000 1500 2000

W [rad/s]

a

a1

a2

a3

a4

Slika 15: Diagram stabilnosti modela 7Figure 15: Stability chart of model

Realni komponenti prvih dveh lastnih vre- Real components of first two eigenvaluesdnosti sta pozitivni v celotnem opazovanem are positive while the others are positive inobmocju. Posledica tega je nestabilno obra- the whole range of interest. This can be thetovanje rotorja, saj amplituda nihanj s casom cause of instability of the system since the am-


narasca. Za zagotovitev stabilnosti obratova- plitude of vibration increases with time. Tonja bi bilo potrebno spremeniti geometrijske achieve stability of operation in such cases theparametre drsnih lezajev. bearing geometric parameters should be chan-

ged.

4. SKLEP 4. CONCLUSION

Namen prispevka je predstaviti problema- The purpose of the paper is to present thetiko dinamike rotorjev in ovrednotiti vpliv po- problems of rotordynamics an to estimate thesameznih parametrov na dinamicno vedenje influence of individual parameters on dynamicrotorja. Analiticne resitve s podrocja dina- behavior of rotor. Analytical solutions in themike rotorjev so zal omejene na zelo eno- domain of rotordynamics are unfortunately li-stavne modele. Pri obravnavi realnih siste- mited to elementary models. Considering rea-mov smo tako prisiljeni v uporabo diskretnih listic systems we are forced to use the discretemetod in numericno resevanje problemov. V methods and numerical solutions of problems.pricujocem clanku predstavljeni programski The program presented in this paper is in-paket naj bi zapolnil praznino med komercial- tended to fill the gaps of general FEM pro-nimi programi, ki temeljijo na metodi koncnih grams, which (with rare exemptions) do notelementov, in ne omogocajo obravnave neka- meet the special requirements of rotordyna-terih, za dinamiko rotorjev specificnih, po- mics (gyroscopic effect and asymmetric stiff-javov (npr. nesimetricna togostna matrika, ness and damping matrices). The programziroskopski efekt). Program je bil v celoti raz- has been entirely developed in the Labora-vit v Laboratoriju za dinamiko strojev in kon- tory for dynamics of machines and structuresstrukcij (LADISK) na Fakulteti za strojnistvo (LADISK) at the Faculty for mechanical engi-v Ljubljani in predstavlja uporabno orodje, neering in Ljubljana and represents an usefulnamenjeno konstrukterju v zacetni fazi ra- tool for engineers in the early stage of rotorzvoja rotorjev. design.

Razvoj taksnega programa predstavlja Development of such a program is anobsezen projekt, saj obravnavana tematika extensive project since the discussed issues co-pokriva siroko podrocje problematike. Po- ver wide range of problems. Besides the pro-leg problemov, neposredno povezanih z di- blems directly associated with rotordynamicsnamiko rotorjev, je v program vkljucen mo- the program incorporates a module for eva-dul za vrednotenje diskretnega modela z me- luation of a discrete model through the finitetodo koncnih elementov. Velik del program- element method. A big part of the programske kode je namenjen tudi numericnim me- code is also used by numerical methods fortodam za dolocanje lastnih vrednosti. Po- determination of eigenfrequencies. In addi-leg Hausholderjeve metode za resevanje sime- tion to the Householder method for evalua-tricnih problemov je obravnavi splosnih pro- tion of symmetric problems the Hessenbergblemov namenjena Hessenbergova metoda. method has been used for solving the general

Velika pozornost je bila posvecena ra- problems.zvoju uporabniskega vmesnika, ki uporabniku Special attention has been given to deve-omogoca enostavno modeliranje in delo s po- lopment of the user interface that enables easydatki. Zaradi splosnih smernic razvoja pro- modeling and data processing. Consideringgramske in strojne opreme program deluje the general tendencies of software and hard-na PC kompatibilnih racunalnikih v Windows ware development the program is intended tookolju. S tem je zagotovljena tudi osnova za be used on PC compatible computers withnadaljnji razvoj programa. Windows environment. That guaranties the

Odprta zasnova programske arhitekture in grounds for further development of the pro-uporaba objektnega programiranja zagota- gram.vljata enostavno nadgradnjo programa. Do- Opened outline of the program’s architec-


dajanje novih elementov zahteva le minimalne ture and the use of the object oriented pro-popravke v programu, ki ze zdaj obsega preko gramming assures easy further development30.000 programskih vrstic, obsezno knjiznico of the program. Adding new elements requi-z elementi uporabniskega vmesnika in obsirno res only minimal corrections in program thatpomoc. already contains more than 30.000 program

Metoda koncnih elementov, uporabljena za lines, extensive library with elements of thepopis diskretnega matematicnega modela, se user interface and thorough help.je pokazala kot zelo ucinkovita. Predstavljeni The finite element method, used to de-rezultati kazejo zelo dobro ujemanje s prever- scribe the discrete mathematical model, hasjenimi numericnimi in eksperimentalnimi re- proved itself as very effective. Presented re-zultati (Litostroj Ljubljana). sults show very good agreement with verified

Na podlagi predstavljenih rezultatov lahko numerical and experimental results (Litostrojzakljucimo, da vpliva ziroskopskega efekta pri Ljubljana).obravnavi dinamike rotorjev nikakor ne mo- On the grounds of presented results it canremo zanemariti. To velja se posebej za ro- be concluded that the influence of gyroscopictorje z rotirajocimi telesi, katerih glavni masni action can not be neglected when consideringvztrajnostni momenti so veliki v primerjavi z the rotordynamics. This is in particular validmasnimi vztrajnostnimi momenti gredi. Po- for the rotor having rotating bodies with massdobno velja tudi za vpliv drsnih lezajev, ki moments of inertia relatively large comparedlahko ob neugodnih konstrukcijskih parame- to the one of the shaft itself. The influencetrih povzrocijo celo nestabilnost modela. of the oil-film bearings to the rotordynamics

Programski paket ne pokriva celotnega has been also found important. Unfavorablepodrocja dinamike rotorjev. Vendar opra- construction parameters can even cause thevljena testiranja in zelo dobro ujemanje z refe- instability of the rotor.rencnimi rezultati predstavljajo, skupaj z od- The program does not cover the wholeprto zasnovano programsko arhitekturo, do- range of the rotordynamics phenomena. Per-bro osnovo za nadaljnje delo. formed tests and very good agreement with

reference results represent together with ope-ned outline of the program’s architecture goodbasic grounds for further development.


Literatura

[1] Leckie F.A.: Pestel E.C. Matrix Methods in Elastomechanics. McGraw-Hill, New York,1963.

[2] Liu S.-L. Lee A.-C., Kang Y. A modified transfer matrix method for linear rotor-bearingsystems. Journal of Applied Mechanics, 58:776–783, September 1991.

[3] Kang Y. Lee A.-C., Shih Y.-P. The analysis of linear rotor-bearing systems: A generaltransfer matrix method. Journal of Vibration and Acoustics, 115:490–497, October 1993.

[4] Wu F. Flowers G.T. A transfer matrix technique for evaluating the natural frequencies andcritical speeds of a rotor with multiple flexible disks. Journal of Vibration and Acoustics,114:242–248, April 1992.

[5] Athre K. Gupta K., Gupta K.D. Unbalance response of a dual rotor system: Theory andexperiment. Journal of Vibration and Acoustics, 115:427–435, October 1993.

[6] Kuhelj A. Cokan R., Boltezar M. Racunalniska simulacija nekaterih problemov dinamikerotorjev. Smarjeske Toplice,, del, 122-129 edition, Kuhljevi dnevi 93 zbornik 23.-24.9.1993:.

[7] Kuhelj A. Cokan R., Boltezar M. Technical report, 1995 v Proceedings of the IAHR WG1Meeting, Ljubljana, Slovenia, 5-7 September, G-4:1-14. Computer modeling of rotordyna-mics.

[8] Cokan R. Racunalniska simulacija dinamicnega obnasanja rotorja. Magistrska naloga,Fakulteta za strojnistvo, Ljubljana, 1996.

[9] Kramer E.:. Dynamics of Rotors and Foundations. Springer Verlag, Berlin, 1993.

[10] Keer L. M Chen W.-R. Transverse vibrations of a rotating twisted timoshenko beamunder axial loading. journal of vibration and acoustics. Journal of Vibration and Acoustics,115:285–294, July 1993.

[11] Keer L.M. Wijeyewickrema A.C. Critical speeds and buckling loads of a pre-twisted rotor.Journal of Sound and Vibration, 179(1):109–129, 1995.

[12] Rouch K.E. Stephenson R.W. Modeling rotating shafts using axisymmetric solid finiteelements with matrix reduction. Journal of Vibration and Acoustics, 115:484–489, October1993.

[13] Marangoni R.D. Xu M. Vibration analysis of a motor - flexible coupling - rotor systemsubjected to misalignment and unbalance, part i: Theoretical model and analysis. Journalof Sound and Vibration, 176(5):663–679, 1994.

[14] Lewandowski R. Non-linear free vibrations of beams by the finite element method andcontinuation methods. Journal of Sound and Vibration, 170(5):577–593, 1994.

[15] Wilkinson J.H. Algebraic Eigenvalue problem. Clarendon Press, Oxford, 1965.

[16] Goos G. and Hartmanis J. (eds.). Lecture Notes in Computer Science: Matrix EigensystemRoutines - EISPACK guide. Springer Verlag, 2nd edition, 1976.


[17] Kirkhope J. Wang W. New eigensolution and modal analysis for gyroscopc/rotor systems,part 1: Undamped systems. Journal of Sound and Vibration, 175(2):159–170, 1994.

[18] Sekulovic M. Metod konacnih elemenata. IRO Gradevinska knjiga, Beograd, 1988.

[19] Teukolsky S.A. Vetterling W.T. Press W.H., Flannery B.P. Numerical Recipes in Pascal.Cambridge Univesity Press, Cambridge, 1992.

Naslov avtorjev: Authors’ Adress:

mag. Robert Cokan, dipl. inz.dr. Miha Boltezar, dipl. inz.prof. dr. Anton Kuhelj, dipl. inz.Fakulteta za strojnistvoUniverze v LjubljaniAskerceva 61000 Ljubljana

Mag. Robert Cokan, Dipl. Ing.Dr. Miha Boltezar, Dipl. Ing.Prof. Dr. Anton Kuhelj, Dipl. Ing.Faculty of Mechanical EngineeringUniversity of LjubljaniAskerceva 61000 Ljubljana, Slovenia

Documents

Raˇcunalniˇsko simuliranje dinamike rotorjev Computer ...lab.fs.uni-lj.si/ladisk/data/pdf/clanek-dinamika_rotorjev.pdf · Dinamika rotorjev se je v zadnjih letih razvila vˇsiroko