Računalniško simuliranje dinamike rotorjev Computer Simulation of the Dynamics of Rotors

Size: px
Start display at page:

Download "Računalniško simuliranje dinamike rotorjev Computer Simulation of the Dynamics of Rotors"

Transcription

1 STROJNIŠKI VESTNIK - JOURNAL OF MECHANICAL ENGINEERING, LJUBLJANA (42) 1996/ Računalniško simuliranje dinamike rotorjev Computer Simulation of the Dynamics of Rotors Robert Cokan, Miha Boltežar, Anton Kuhelj Univerza v Ljubljani, Fakulteta za strojništvo, 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 široko paleto specialnosti. Tudi strokovnjaki težko prepoznajo in povežejo vse vplive in pojave, ki se pojavljajo v praksi. V ta namen je bil razvit programski paket, ki omogoča obravnavo upogibnih nihanj linearnih sistemov in vrednotenje vpliva različnih parametrov na dinamiko rotorjev. Posebna pozornost je bila posvečena žiroskopskemu efektu in modeliranju drsnih ležajev ter njunemu vplivu na lastne frekvence in kritične hitrosti. Numerični del razvitega programskega paketa je zasnovan na metodi končnih elementov. Program je namenjen konstrukterjem za določitev dinamičnega obnašanja rotorja v začetni fazi razvoja. Abstract The dynamics of rotors has in recent decades developed into a wide range of specialties. Even for the experts it is difficult to recognize and draw a correlation among all phenomena that occur in practice. For that purpose a computer program has been developed that enables evaluation of bending vibrations of linear systems and assessment of influence of various parameters on rotordynamics. Special attention has been given to the gyroscopic effect and modeling of oilfilm bearings and its influence on eigenfrequencies and critical speeds. Numerical part of the developed software program is based on the finite element method. Program is intended to be used 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 wide široko področje problematike; od balansira- range of topics; from balancing of rigid ronja togih rotorjev do obravnave nestabilnosti tors to evaluation of instability due to nonzaradi nelinearnosti v sistemu. Zaradi kom- linearity of the system. Rotordynamics has pleksnosti posameznih sklopov se je veda v because of complexity of individual topics dezadnjih desetletjih razvila v disciplino z ve- veloped in a wide range of specialties. Even liko specialnostmi. Tudi strokovnjaki težko for the experts is difficult to evaluate whole prepoznajo in ovrednotijo celotno paleto pro- range of problems and draw the correlation blemov in povežejo vplive bistvenih parame- among significant parameters. The following trov. V nadaljevanju smo obravnavo ome- discussion has been restricted to the determijili na določanje lastnih frekvenc in kritičnih nation of eigenfrequencies and critical speeds hitrosti rotorjev. Glede na dejstvo, da ana- of rotors. Due to the fact that analytical solitične rešitve pokrivajo zelo ozko področje lutions are available only for a limited area problematike, je bil v preteklosti razvit niz of problems a range of discrete methods has diskretnih metod, ki omogočajo obravnavo been developed that make the evaluation of

2 STROJNIŠKI VESTNIK - JOURNAL OF MECHANICAL ENGINEERING, LJUBLJANA (42) 1996/ problematike dinamike rotorjev. Danes se rotordynamics possible. Nowadays two mev glavnem uporabljata dve metodi in sicer thods has been broadly used. These are first metoda končnih 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 splošni uporabnosti in razširjenosti Despite the fact that FEM is more and more metode končnih elementov se metoda preno- used the transfer matrix method is still used snih matrik še vedno uporablja za obravnavo for evaluation of some specific cases. nekaterih specifičnih primerov. Basic principles of the transfer matrix me- Sodobno zasnovo metode prenosnih ma- thod, based on the use of computer, can be trik, ki temelji na uporabi računalnika, lahko found in [1]. Somewhat modified formulanajdemo v [1]. Nekoliko spremenjeno formu- tion of the method that enables evaluation lacijo metode, ki omogoča popis žiroskopskega of gyroscopic effect and general description of efekta in splošen opis tirnice gibanja poljubne the path of the arbitrary point in the system točke v sistemu, lahko najdemo v [2] in [3]. can be found in [2] and [3]. The method was Metoda omogoča določitev vpliva deformabil- extended to include the influence of disk s flenosti diska na lastne frekvence in kritične hi- xibility on eigenfrequencies and critical speeds trosti [4], kot tudi obravnavo večlinijskih sis- [4] as well as evaluation of multi-line systems temov (rotorja s podporno konstrukcijo) [5]. (systems with support construction) [5]. Glede na izkušnje z metodo prenosnih ma- Considering our own experiences with the trik [6] se je izkazalo, da je omenjena metoda transfer matrix method [6] it has become evimanj primerna za obravnavo večlinijskih kon- dent that the use of the method is less suitastrukcij oz. rotorjev, kjer je potrebno po- ble, where besides the rotor itself the support leg osnovne linije modelirati tudi podporno construction has to be modeled as well. For konstrukcijo. V ta namen razvijamo v La- that purpose a computer program, based on boratoriju za dinamiko strojev in konstrukcij the FEM, has been developed in the Labora- (LADISK) na Fakulteti za strojništvo v Lju- tory for dynamics of machines and structures bljani programski paket, ki temelji na MKE, (LADISK) at the Faculty of mechanical enin bo v končni obliki omogočal obravnavo upo- gineering in Ljubljana that provides the megibnih nihanj rotorjev skupaj s podporno kon- ans for evaluation of various parameters and strukcijo, vrednotenje različnih parametrov in their influence on rotordynamics. The details njihov vpliv na dinamiko rotorjev. Podrob- of current development are shown in [7] and neje 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 široko upo- dynamics can be found in the literature. A rabnost metode končnih elementov pri obrav- scope of issues is covered and discussed using navi dinamike rotorjev. V [9] je podana the FEM in [9]. The eigenfrequencies [10] and obširna problematika na osnovi omenjene me- critical speeds [11] of the pre-twisted rotor, tode. Metoda omogoča tudi dolocčitev la- described by Timoshenko s equation, can be stnih frekvenc [10] in kritičnih hitrosti [11] determined using the FEM. For more accurate rotirajočega zvitega nosilca na osnovi Timo- description of the rotor the axisymmetric soshenkove enačbe. Za bolj točen popis rotorja lid finite elements can be used [12]. Considelahko uporabimo osno simetrične končne ele- ring coupled systems such as driving machinemente [12]. Pri pogonskih agregatih pride rotor the additional influences on rotordynazaradi zamika pogonske gredi motorja in ro- mics can be detected due to misalignment of torja 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 sufficient vadno zadostna, seveda pa metoda končnih but the FEM can also be used for evaluation elementov omogoča tudi popis nelinearnih ni- of non-linear vibrations [14]. Application of hanj [14]. Pri delu z metodo končnih elemen- the FEM, in contrast to the transfer matrix

3 STROJNIŠKI VESTNIK - JOURNAL OF MECHANICAL ENGINEERING, LJUBLJANA (42) 1996/ tov imamo, za razliko od metode prenosnih method where the size of transfer matrices is matrik, kjer je velikost prenosnih matrik kon- constant, implies the use of relatively large stantna, opravka z relativno velikimi matri- matrices. Matrix reduction is therefore rekami, zato je pogosto priporočljiva redukcija commended [12]. Considering the gyroscopic matrik [12]. Z upoštevanjem žiroskopskega effect the dynamic matrix becomes asymmeefekta postane dinamična matrika sistema ne- tric. Solving the eigenvalue problem implies simetrična. Za reševanja problema lastnih the use of time consuming general algorithms vrednosti moramo zato uporabljati splošne al- [15] and [16]. The problem can be avoided goritme [15] in [16], ki pa so časovno zahtevni. with some specific cases using the transfor- Pri obravnavi nekaterih specifičnih primerov mation of the dynamic matrix in symmetric se problemu lahko izognemo s transformacijo form [17]. dinamične matrike v simetrično [17]. 1. TEORETIČNO OZADJE 1. THEORETICAL BACKGROUND Za popis linearnih upogibnih nihanj, oz. The FEM has been used to describe fleza določitev matrične gibalne enačbe (1), je xural vibrations and to determine the matrix bila uporabljena metoda končnih elementov. equation of motion (1). Basic principles of the Osnove uporabljene metode, kot tudi načini method used as well as the mechanism to dedoločitve matrik posameznih elementov, so termine the matrices of distinct elements are splošno znane. Med drugim jih lahko najdemo well known. Among others they can be found v [9] in [18]. in [9] and [18]. 1.1 Problem lastnih vrednosti 1.1 The eigenvalue problem Probleme lastnih nihanj lahko splošno The problem of natural vibrations can be popišemo s homogeno gibalno enačbo v obliki: generally described by the homogeneous equation of motion in the form: M ẍ + B ẋ + K x = 0 (1) kjer so M masna, B vsota dušilne in where M, B and K are mass, the sum of žiroskopske matrike ter K togostna matrika. damping and gyroscopic matrices and stiff- Pomike in njihove časovne odvode označimo ness matrices respectively. Displacements and z x, ẋ in ẍ. Rešitev gibalne enačbe (1) vodi their time derivatives are denoted by x, ẋ do problema lastnih vrednosti. Kot rezul- and ẍ. Solution of the equation of motion tat dobimo lastne frekvence in pripadajoče la- (1) leads to the eigenvalue problem. As a restne oblike obravnavanega sistema. Ob za- sult one can obtain eigenfrequencies and cornemarljivem žiroskopskem efektu in dušenju, responding mode shapes of the system under pri B = 0, lahko zapišemo klasičen problem consideration. With negligible gyroscopic eflastnih 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 dinamično matriko sistema where A represents the dynamic matrix of the in I enotsko matriko. V primeru, ko sta ma- system and I is an identity matrix. In the sna in togostna matriki simetrični, lahko z case of symmetric mass in stiffness matrices dekompozicijo ene od matrik na spodnjo in the symmetricity of dynamic matrix A can be zgornjo trikotno matriko zagotovimo tudi si- guarantied through the decomposition of one metričnost dinamične matrike A. Takšen pro- of the matrices in lower and upper triangular

4 STROJNIŠKI VESTNIK - JOURNAL OF MECHANICAL ENGINEERING, LJUBLJANA (42) 1996/ blem je relativno enostavno rešljiv [15, 16, 19]. matrix. That kind of problems are relatively Tudi v splošnem primeru, kjer je B 0, in easy to solve [15, 16, 19]. so matrike lahko tudi nesimetrične, moramo In the general case, with B 0, and geneenačbo (1) zapisati v obliki (2). Množenje rally asymmetric matrices, the equation (1) enačbe (1) z M 1 pripelje, ob enakosti ẋ = ẋ, has to be written in the form of (2). Multiplido enačb: cation of the equation (1) by M 1 and using identity ẋ = ẋ leads to the equations: ẋ = I ẋ ẍ = M 1 K x M 1 B ẋ Z uvedbo Denoting z = { x ẋ } in and C = lahko tako transformirano gibalno enačbo zapišemo v obliki: [ 0 I M 1 K M 1 B ] (3) one can write the equation of motion in the form: ż Cz = 0 (4) Lastne vrednosti so v splošnem kompleksno Eigenvalues are generally complex conjukonjugirani pari in jih lahko zapišemo v obliki: gate pairs and can thus be written in the form: λ k = α k ± iω k (5) kjer ω k predstavlja k-to lastno frekvenco sis- where ω k represents the k-th eigenfrequency tema in α k koeficient rasti nihanj. Za ne- of the system and α k the coefficient governing gativne vrednosti α k se amplituda nihanj s the growth of vibrations. For negative values časom zmanjšuje, medtem ko se za pozitivne of α k vibrations decrease with time and for vrednosti povečuje. To pomeni, da je sistem positive values vibrations increase with time. pri pozitivnih vrednostih koeficienta rasti ni- That means that for positive values the syhanj nestabilen. stem becomes unstable. Podobno kot lastne vrednosti so tudi la- Similar as the eigenvalues the mode shapes stni vektorji v splošnem kompleksni. Lastna are generally complex. The natural vibrations nihanja celotnega modela lahko popišemo z of the whole model are described by the mogibanjem njegovih vozlišc. V uporabljenem tions at certain nodes. In the mathematical matematičnem modelu je gibanje sestavljeno model used these motions consist of displaiz pomikov, pravokotnih na os rotorja in od- cements perpendicular to the rotor axis and govarjajočih zasukov. Za numerično analizo the corresponding rotations. For the numezadošca zgolj obravnava pomikov, saj lahko rical analysis it is sufficient to evaluate only zasuke določimo iz upogibnice rotorja. displacements, because rotations can be cal- Pomike v poljubnem vozlišču lahko culated from the bent form of the rotor. zapišemo [9] z enačbo: Displacement of an arbitrary node can be written [9] by the equation: x ik = C k e α kt [r ik sin(ω k t + γ k ) + s ik cos(ω k t + γ k )] (6) kjer so: α k, ω k - realna in imaginarna kompo- where: α k, ω k - are the real and imaginary

5 STROJNIŠKI VESTNIK - JOURNAL OF MECHANICAL ENGINEERING, LJUBLJANA (42) 1996/ nenta lastne vrednosti, r ik, s ik - ustrezna re- parts of an eigenvalue, r ik, s ik - are the real alna in imaginarna komponenta lastnega vek- and imaginary parts an eigenvector and C k, γ k torja in C k, γ k - konstanti, ki ju določimo iz - are constants found from initial conditions. začetnih pogojev. When considering only natural vibrations Pri obravnavi lastnih nihanj lahko za- one can neglect the change in amplitude and nemarimo spremembo amplitude nihanj in put α k = 0. Further we can with no serious zapišemo α k = 0. Nadalje lahko zanema- restrictions put C k = 1 and γ k = 0. In this rimo konstanti, odvisni od začetnih pogojev way simplified equation has the form: in zapišemo C k = 1 in γ k = 0. Tako poenostavljeno enačbo lahko zapišemo v kompleksni obliki kot: z(t) = R e j(ωt+α ) + R e j(ωt α ) (7) Opazimo lahko, da je eliptična tirnica na It can be seen that the elliptical path in sliki 1 sestavljena kot vsota dveh rotirajočih Figure 1 consists of the sum of two rotating vektorjev, ki imata oba kotno hitrost ω. Vek- vectors, both having angular velocity ω. One tor dolžine R rotira sinhrono s smerjo rotacije vector of length R rotates in the positive dirotorja, medtem ko drugi vektor, dolžine R, rection while the other of length R rotates in rotira asinhrono. the negative direction. Im(z), x 2 R' ' R'' '' Re(z), x 1 To find the direction of rotation we intro- duce the rotation number Za določitev smeri opletanja vozlišca vpeljemo rotacijsko število Slika 1: Eliptična tirnica Figure 1: Elliptical path u = R R, (8) R + R ki določa smer opletanja obravnavanega vo- which indicates the direction of rotation of

6 STROJNIŠKI VESTNIK - JOURNAL OF MECHANICAL ENGINEERING, LJUBLJANA (42) 1996/ zlišca. Glede na rotacijsko število u je opleta- the node considered. Considering the rotanje lahko: tion number u the rotatioin can be: 1 u < 0 asinhrono opletanje 1 u < 0 asinhrono opletanje u = 0 gibanje vzdolž premice ali u = 0 gibanje vzdolž premice ali 0 < u 1 sinhrono opletanje 0 < u 1 sinhrono opletanje 1.2 Reševanje problema lastnih vrednosti 1.2 Solving the Eigenvalue Problem Za reševanje problema lastnih vrednosti To solve the eigenvalue problem a range of smo testirali in primerjali niz numeričnih me- numerical routines has been tested and comtod. Za simetrične matrike smo uporabili pared. For symmetric matrices the Househol- Hauseholderjevo metodo. Da bi obdržali si- der method has been used. To retain symmemetričnost dinamične matrike A, definirane v tric properties of the dynamic matrix A defienačbi (2), smo za masno matriko M upora- ned in equation (2), Cholesky decomposition bili dekompozicijo Choleskega. Za reševanje of mass matrix M has been used. Considesplošnega problema lastnih vrednosti, defini- ring the general eigenvalue problem defined by ranega z enačbo (4), program uporablja za equation (4) the HQR (Hessenberg) algorithm določitev lastnih vrednosti HQR (Hessenber- has been used to calculate eigenvalues while govo) metodo in inverzno iteracijo vektorjev inverse vector iteration has been applied to za določitev lastnih oblik. Podrobneje so ome- obtain eigenvectors. Details of all aforementinjene metode predstavljene v [15], [16] in [19]. oned methods can be found in references [15], [16] and [19]. 1.3 Teorija kratkega drsnega ležaja 1.3 Oil-film Bearing Osnove MKE, kot tudi načini določevanja The FEM as well as the mechanism to dematrik posameznih elementov, so splošno termine the matrices of distinct elements are znani [9] in [18]. V nadaljevanju je zaradi po- well known [9] and [18]. Because of the simembnega vpliva na dinamično obnašanje ce- gnificant influence on the dynamic behavior lotnega rotorja zato na kratko predstavljen le of entire rotor the model of the oil-film beauporabljeni model drsnega ležaja [9]. Primer ring is shortly discussed in the following. An takšnega ležaja je prikazan na sliki 2 example is shown in Figure 2. y y h 0 h R r J C B C J e J x C J C B z L Slika 2: Drsni ležaj Figure 2: Oil-film bearing Tlačne razmere v drsnem ležaju navadno The pressure conditions in the oil-film bepopišemo Reynoldsovo enačbo [9]. Za pri- aring are usually described by the Reynolds

7 STROJNIŠKI VESTNIK - JOURNAL OF MECHANICAL ENGINEERING, LJUBLJANA (42) 1996/ equation [9]. For the case of circular bearing and neglecting the change in pressure in cir- cumferential direction we can write the Reynolds equation: mer drsnega ležaja s krožno ležajno blazinico ima Reynoldsova enačba, ob zanemaritvi spremembe tlaka v obodni smeri, obliko: 2 p z 2 = 6η h 3 (ϕ, t) [e j(ω 2 γ) sin(ϕ γ) 2ė j cos(ϕ γ)] (9) kjer so: η - dinamična viskoznost maziva, Ω - where: η - is the dynamic viscosity of the lukotna hitrost gredi oz. čepa in p - tlak maziva. bricant, Ω - is the angular velocity of the shaft Govorimo o teoriji kratkega drsnega ležaja and p - is the pressure of the lubricant. oz. o linearnem modelu drsnega ležaja. The theory is known as short oil-film be- Za statično obremenjen horizontalno podprt aring theory or linear oil-film bearing model. rotor lahko togostne in dušilne koeficiente For a horizontally supported rotor with static določimo z odvajanjem komponent rezulti- load one can obtain stiffness and damping corajoče sile glede na pomike: efficients by differentiating component forces with respect to displacements: k ik = F i x k = γ ik F 0 δ, kjer sta γ ik in β ik brezdimenzijska togostna in dušilna koeficienta. d ik = F i ẋ k = β ik F 0 δω (10) where γ ik and β ik are dimensionless stiffness and damping coefficients respectively ik ik , Slika 3: Brezdimenzijski togostni koeficient Figure 3: Dimensionless stiffness and damping coefficients Na opisani način določena brezdimenzijska Coefficients obtained in the described way koeficienta sta prikazana na sliki 3 v odvi- are shown in Figure 3 as functions of the snosti od razmernika ekscentričnosti drsnega eccentricity ratio. Latest is defined by the ležaja. Le-ta je definiran z enačbo: equation: ɛ = e 3 R r 3 (11) Iz diagramov je razvidno, da sta togo- From the diagrams one can observe that stna in dušilna matrika drsnega ležaja nesi- the stiffness and damping matrices of the oilmetrični. Kot bo prikazano v nadaljevanju, film bearing are asymmetric. In the case of lahko neugodna kombinacija parametrov dr- unfavorable combination of parameters con-

8 STROJNIŠKI VESTNIK - JOURNAL OF MECHANICAL ENGINEERING, LJUBLJANA (42) 1996/ snega ležaja in celotnega rotorja privede tudi do nestabilnosti delovanja. cerning oil-film bearing and the whole rotor the aforementioned property can lead to the unstable operation as it will be shown in the following. 2. PROGRAMSKI PAKET 2. PROGRAM PACKAGE Za ovrednotenje vpliva različnih parame- To evaluate the influence of various paratrov na dinamiko rotorjev smo v Laboratoriju meters on rotordynamics the computer proza dinamiko strojev in konstrukcij (LADISK) gram, based on the finite element method, has razvili računalniški program, ki temelji na me- been developed in the Laboratory for dynatodi končnih elementov. Namen programa mics of machines and structures (LADISK). je zapolniti vrzeli komercialnih programov, ki The program is intended to fill the gaps večinoma (z redkimi izjemami) ne zadošcajo of general FEM programs, which (with rare posebnim zahtevam obravnave dinamike ro- exemptions) do not meet the special requiretorjev (žiroskopski efekt ter nesimetrične to- ments of rotordynamics (gyroscopic effect and gostne in dušilne matrike). asymmetric stiffness and damping matrices). Izdelani program vključuje numerično vre- The developed program incorporates nudnotenje matematičnega modela kot tudi merical processing of the mathematical model grafično pripravo podatkov in grafični pri- as well as graphical pre- and post-processing kaz dobljenih rezultatov. Groba programska of given numerical information. A rough proshema je prikazana na sliki 4. gram chart is shown in Figure 4. Delo z datotekami File management Zunanja obdelava External processing Disk Disk Matematièni model Mathematical model Nastavitve Options Tiskalnik Printer Grafièni modeler Graphic modeller Pomoè Help Elementi Elements Pregled rezultatov Results owerview Popravljanje Editing Izraèuni Calculations Lastne vrednosti Eigenvalues Kritiène hitrosti Critical speeds Dinamièni odziv Dynamic response Slika 4: Programska shema Figure 4: Program chart Zaradi poenostavitve dela s programom je To simplify handling of the program, a bil razvit grafični uporabniški vmesnik, ki graphics interface has been created that enaomogoča enostavno modeliranje in obdelavo bles elegant modeling and processing of data. podatkov. Vnos podatkov je omogočen s sis- Input of data is supported through the sytemom dialogov in menujev. Primer vnosa stem of dialogs and menus. An example of podatkov za gredni element je prikazan na data input is shown in Figure 5 for the beam sliki 5. Končni uporabnik tako ne potrebuje element. The user thus does not need any nobenega znanja o ukazih za modeliranje ma- knowledge about the syntax of a modeling tematičnega modela. Interpretacijo vnesenih language. Interpretation and translation of podatkov in pretvorbo v računalniku razu- input data to computer-understandable lan-

9 STROJNIŠKI VESTNIK - JOURNAL OF MECHANICAL ENGINEERING, LJUBLJANA (42) 1996/ mljiv jezik opravi sam program v ozadju, ne- guage is done in the background by the proodvisno od uporabnika. Kjer je bilo to mogoče gram itself. Where possible the checking of je vgrajeno preverjanje vnesenih podatkov, input information has been implemented in s čemer se izognemo nepotrebnim logičnim order to avoid any logical mistakes. As shown napakam pri vnosu podatkov. Kot je raz- in Figure 5 the program enables work with sevidno s slike 5, program omogoča hkratno veral models simultaneously. Transfer of data delo z več modeli. Prenos podatkov med between loaded models is made possible thromodeli je omogočen preko internega odla- ugh the program s internal clipboard. gališca podatkov. Slika 5: Uporabniški vmesnik - vnos podatkov Figure 5: User interface - data input Uporabnik lahko prilagodi program svo- The user can adjust the program to his rejim potrebam preko niza nastavitev. Možne quirements with a set of options. Appearance so nastavitve izgleda uporabniškega vmesnika of the user interface as well as some numerical kot tudi nekaterih numeričnih parametrov. parameters can be adapted to meet the user s Poleg interaktivnega dela s programom je requests. možna tudi zunanja paketna obdelava podat- Besides interactive work with the program, kov. Slednja je možna s programiranjem v external processing of data is also possible. poljubnem programskem jeziku, seveda pod The latest is possible by programming in arpogojem, da uporabnik pozna obliko zapisa bitrary computer language, assuming the file datotek in so mu na voljo potrebne numerične format used by the program and numerical rutine. Zunanja paketna obdelava podatkov routines are available to the user. That kind ima velikokrat prednost pred interaktivnim of work is sometimes preferred to interactive delom s programom vendar od uporabnika but it requires detailed knowledge about the zahteva dosti več znanja o samem programu program structure and general numerical anakot tudi o splošni numerični analizi. lysis. Za namen prikaza numeričnih rezultatov je To review the calculations a postprocessor bil narejen grafični post-procesor. Na sliki has been developed which enables graphic re-

10 STROJNIŠKI VESTNIK - JOURNAL OF MECHANICAL ENGINEERING, LJUBLJANA (42) 1996/ je predstavljen primer prikaza izračunanih presentation of results. An example is shown lastnih frekvenc in pripadajočih lastnih oblik in Figure 6 where calculated eigenfrequencies modela. and corresponding mode shapes are shown. Kot smo že omenili program omogoča As mentioned before the program supports tudi zunanjo obdelavo podatkov. Le-ta je further processing of numerical results. Latest omogočena tudi preko izvoza podatkov in re- has been made possible through the export of zultatov v ASCII obliki. Tako zapisane po- data in the ASCII file format. Results, writdatke lahko uporabljamo z večino komercial- ten in that form, can be accessed and further nih programskih paketov. Omogočen je iz- processed by almost any commercial compuvoz končnih rezultatov (lastne vrednosti in ter program. Export of results (eigenvalues kritične hitrosti) kot tudi posameznih matrik and critical speeds) as well as individual mav enačbi (1). trices in equation (1) is possible. Slika 6: Uporabniški vmesnik - prikaz rezultatov Figure 6: User interface - data output Program je napisan za PC kompatibilne The program is written for PC compatiračunalnike z operacijskim sistemom Win- ble computers with Windows 3.1 operating dows 3.1. Za smotrno delo s programom je systems. For better performance at least 486 zaželen vsaj 486 DX procesor. DX processor is preferred. Trenutno program omogoča določitev: At the present stage of development the - lastnih vrednosti in pripadajočih lastnih vek- program enables calculation of: torjev, - eigenvalues and eigenvectors, - kritičnih hitrosti, - critical speeds, - odziva modela v ustaljenem stanju zaradi - response due to dynamic force and zunanje harmonske dinamične obremenitve in - response to given kinematics of one or more - odziva modela zaradi predpisane kinematike arbitrary points of the system. ene ali več točk v sistemu. 3. DINAMIČNA ANALIZA MODELA VENTILATORJA 3. DYNAMIC ANALYSIS OF FAN MODEL

11 STROJNIŠKI VESTNIK - JOURNAL OF MECHANICAL ENGINEERING, LJUBLJANA (42) 1996/ Za analizo posameznih vplivov na dina- For the analysis of distinct influences on romiko rotorjev, predvsem žiroskopskega efekta tordynamics, mainly the gyroscopic effect and in vpliva drsnih ležajev, oz. za predstavitve the influence of the oil-film bearing, and to programskih zmogljivosti, smo oblikovali mo- demonstrate the program capabilities a ventidel ventilatorja. Osnovni model je prikazan lator model has been built. The basic model na sliki 7. Ventilator ima celotno maso 436 is shown in Figure 7. The ventilator has total kg in dolžino 1280 mm. Opazovano območje mass of 436 kg and length of 1280 mm. The obratovanja ventilatorja je od 0 do 1000 rad/s. range of operation in interest is from 500 to 1000 rad/s. Slika 7: Model ventilatorja Figure 7: Ventilator model Za vrednotenje posameznih vplivov na la- To identify distinct influences on eigenfrestne frekvence in kritične hitrosti smo primer- quencies and critical speeds, several variations jali nekaj variacij osnovnega modela. Osnovni of the basic model were considered. Geomegeometrijski in materialni podatki so enaki za tric and material properties are the same for vse obravnavane modele. Izvedba podprtja in all the tested models. Description of the suopis upoštevanih dodatnih parametrov je za pport conditions and additional model paravse modele zbran v tabeli 1. meters are given in Table 1. Tabela 1: Opis modelnih parametrov Table 1: Description of model parameters model model togi ležaji rigid bearings elastični ležaji elastic bearings drsni ležaj oil-film bearing strižne deformacije share deflection žiroskopski efekt gyroscopic effekt Matematični model sestavlja 13 elementov. The numerical model consists of 13 ele- Modeli z elastičnimi podporami in drsnimi ments. Models with elastic supports and ležaji (modeli 4-7) imajo tako 14*4=56 pro- oil-film bearings (models 4-7) have therefore stostnih stopenj. Modeli s togo izvedbo pod- 14*4=56 degrees of freedom. Rigidly supporprtja (modeli 1-3) imajo na mestu podprtja ted models (models 1-3) have at the support onemogočene pomike v vseh smereh in imajo point disabled motions in all directions and tako 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-

12 STROJNIŠKI VESTNIK - JOURNAL OF MECHANICAL ENGINEERING, LJUBLJANA (42) 1996/ sumed. Model 1 represents the basic model with no additional influences, while the eigen- values for model 2 were calculated considering shear deflection. First 4 eigenfrequencies are given in Table 2 for both models. podprtje rotorja. Model 1 predstavlja osnovni model rotorja brez dodatnih vplivov, medtem ko smo lastne frekvence modela 2 dolocili z upoštevanjem strižnih deformacij. Za oba modela so prve štiri lastne frekvence podane v tabeli 2. Tabela 2: Lastne frekvence modelov 1 in 2 Table 2: Eigenfrequencies of models 1 and 2 Model 1 Model 2 ω ω ω ω Zaradi strižnih deformacij lahko opazimo One can observe small changes in natural majhne spremembe lastnih frekvenc. Razlika frequencies due to shear deflection. The diffeleži v območju od -0.4% za prvo do -4.5% za rence lies in the range from -0.4% for the 1st četrto lastno frekvenco. Vpliv strižnih defor- to -4.5% for the 4th eigenfrequency. Due to macij je za obravnavani model majhen, saj the fact that the model is relatively slender imamo opravka z relativno vitkim rotorjem. the influence of shear deflection is small. For Za takšne rotorje je omenjeni vpliv majhen in such rotors the aforementioned influence is gega lahko zanemarimo, ne da bi s tem naredili nerally small and can be neglected with no sepreveliko napako. Vpliv strižnih deformacij je rious restrictions. The influence of shear debolj opazen pri kratkih rotorjih, katerih raz- flection is more noticeable with rotors where merje dolžine modela proti premeru gredi je the ratio of rotor length to the shaft diameter manjše. gets smaller. Za ovrednotenje žiroskopskega efekta smo Model 3 has been used to estimate the inuporabili model 3. Na sliki 8 je prikazana od- fluence of gyroscopic action. In Figure 8 the visnost lastnih frekvenc od krožne hitrosti ro- dependence of eigenfrequencies on rotor speed torja. Opazimo lahko, da se število lastnih is shown. One can observe that the number of frekvenc podvoji, če je krožna hitrost rotorja eigenfrequencies doubles with the shaft speed Ω 0. Oznake lastnih frekvenc na sliki 8 Ω 0 so the notation in Figure 8 has been smo zato ustrezno popravili. Opazimo lahko adjusted accordingly. Forward and backward sinhrono 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 inžiroskopskega efekta velik in ga nikakor ne gre fluence of gyroscopic action can not be nezanemariti. Zaključek velja splošno za rotorje glected. This is generally the case with rotors z rotacijskimi telesi, katerih masni vztrajno- having rotating bodies with mass moments of stni momenti so veliki v primerjavi z masnimi inertia relatively large compared to the one of vztrajnostnimi momenti same gredi modela. the shaft itself. Calculations have also been Opravili smo tudi primerjavo lastnih frekvenc done for the shaft of the model 3 alone. In za samo gred modela 3. V tem primeru lahko that case, the influence of gyroscopic action vpliv žiroskopskega efekta zanemarimo, saj je can be neglected since the changes of eigenrazlika lastnih frekvenc z in brez upoštevanja frequencies in the whole range of operation žiroskopskega efekta manjša od 1%. are less than 1%. Pri Ω = Ω S lahko opazimo zanimiv pojav. At Ω = Ω S an interesting phenomenon can Za vrednosti krožne frekvence Ω > Ω S je na- be observed. For Ω > Ω S the 3rd eigenfre-

13 STROJNIŠKI VESTNIK - JOURNAL OF MECHANICAL ENGINEERING, LJUBLJANA (42) 1996/ [rad/s] Tabela 3: Kritične hitrosti modela 3 Table 3: Critical speeds of model cr,1 cr,2 cr,3 cr,4 S [rad/s] Ω cr,1 = rad/s Ω cr,2 = rad/s Ω cr,3 = rad/s Ω cr,4 = rad/s Slika 8: Lastne frekvence modela 3 Figure 8: Eigenfrequencies of model 3 mreč tretja lastna frekvenca modela manjša quency is smaller than the 2nd. Both reod druge lastne frekvence. Obe frekvenci tain their direction of rotation; that is forward ohranita svojo smer opletanja; t.j. sinhrono whirl for the 2nd and backward whirl for the opletanje za 2. in asinhrono opletanje za 3. 3rd eigenfrequency. The phenomenon can be lastno frekvenco. Omenjeni pojav je opazen observed with rotors having rotating bodies pri modelih z velikimi rotacijskimi telesi, ka- with mass moments of inertia relatively large terih masni vztrajnostni momenti so veliki v compared to the one of the shaft itself. Four primerjavi z masnimi vztrajnostnimi momenti critical speeds, marked in Figure 8 and listed same gredi rotorja. V opazovanem območju in Table 3, can be found in the observed range lahko opazimo štiri kritične hitrosti, ki so of shaft speed. označene na sliki 8 in prikazane v tabeli Rotor z elastičnim podprtjem 3.2 Rotor with flexible bearings Za določitev vpliva elastičnosti podpor na To study the influence of bearing flexibility lastne frekvence in kritične hitrosti rotorja on eigenfrequencies and critical speeds models smo uporabili modela 4 in 5. Uporabili smo 4 and 5 have been considered. Bearings were poenostavljen model izotropnih podpor. Obe assumed isotropic. Both bearings have the podpori sta modelirani z enakimi vzmetnimi same stiffness coefficients. Model 4 has been konstantami. Model 4 smo uporabili za oceno used to assess the influence of bearing stiffness vpliva togosti ležajev na prve štiri lastne fre- to the first four eigenfrequencies. The ratios of kvence. Razmerje lastnih frekvenc modela 4 eigenfrequencies over reference frequencies are in referenčnih lastnih frekvenc je prikazano na shown in Figure 9. As the reference frequensliki 9. Kot referenčno 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 vacer omogoča modeliranje elastičnih podpor, lues for the stiffness coefficients of the beavendar pa so vrednosti za togost ležajev rings are difficult do determine. Usually they zelo težko določljive. Običajno jih moramo have to be determined experimentally or badoločiti s poskusi ali pa jih ocenimo na pod- sed on experiences. lagi izkušenj. Povečanje togosti ležajev vpliva na The increase of bearing stiffness results in povečanje lastnih frekvenc, ki limitirajo k vre- increased eigenfrequencies which in the limit dnostim za model s togimi podporami (model approach the values of the model with rigid 2). Zmanjšanje togosti ležajev pa se odrazi z bearings (model 2). On the other hand the nižjimi vrednostmi lastnih frekvenc, ki limi- decrease of bearing stiffness results in decrea-

14 STROJNIŠKI VESTNIK - JOURNAL OF MECHANICAL ENGINEERING, LJUBLJANA (42) 1996/ i Ri E+0 1E-3 1E-2 1E-1 1E+0 1E+1 1E+2 1E+4 1E+6 k [N/m] Slika 9: Lastne frekvence modela 4 Figure 9: Eigenfrequencies of model4 tirajo k vrednostim nepodprtega modela. Za sed eigenfrequencies which in the limit approzelo majhne togosti ležajev se prvi lastni fre- ach the values of the free rotor with respect kvenci približata vrednosti nič. Pri teh vre- to support conditions. For very small values dnostih celoten rotor niha kot togo telo. of bearing stiffness coefficients of bearings the first two eigenfrequencies approach zero. At that value the entire rotor vibrates as a rigid body. Na sliki 9 so lahko razvidna 3 ločena po- Three distinct areas can be seen in the Fidročja. V območju na začetku diagrama, kjer gure 9. In the initial part of the diagram je togost ležajev v območju 0 k 10 2 N/m where the bearing stiffness lies in the range, so spremembe lastnih frekvenc, v odvisnosti 0 k 10 2 N/m the changes of eigenfreod spremembe togosti podprtja, zelo majhne. quencies are small with regard to the change Za 3. in 4. lastno frekvenco je to območje of the bearing stiffness. The area is even wider še večje 0 k 1N/m. Podobno lahko for the 3rd and 4th eigenfrequency and amozaključimo za območje na desnem robu dia- unts to 0 k 1N/m. Similar can be said grama, kjer se vrednosti lastnih frekvenc uje- for the area at the right side of the diagram majo z vrednostmi togo podprtega modela. where the values correspond to the eigenfrequencies of rigidly supported rotor. V srednjem delu diagrama, kjer je togost In the central part of the diagram where ležajev v območju 0 k 10 2 N/m, pa so la- the bearing stiffness lies in the range the eistne frekvence zelo občutljive na spremembo genfrequencies are very sensitive to the change togosti ležajev. Že majhna sprememba to- of bearing stiffness. Even the small change of gosti lahko pomeni veliko spremembo lastnih the stiffness results as a large change of eifrekvenc. Problem pri modeliranju elastičnih genfrequencies. The problem of modeling the podpor je v dejstvu, da so togosti ležajev, ki elastic supports lies in the fact that the values se običajno pojavljajo v praksi, ravno v osre- that occur in the practice lie in the central dnjem območju diagrama. Izračunane lastne part of diagram. Calculated eigenfrequencies frekvence so tako zelo odvisne od izbranih vre- are therefore highly dependent on the choice dnosti za togost ležajev. of the values for the bearing stiffness. Vpliv žiroskopskega efekta na dinamiko ro- The influence of gyroscopic action to rotorjev pri modelu z elastičnimi podporami tordynamics for elastic supported rotor has

15 STROJNIŠKI VESTNIK - JOURNAL OF MECHANICAL ENGINEERING, LJUBLJANA (42) 1996/ [rad/s] S [rad/s] Tabela 4: Kritične hitrosti modela 5 Table 4: Critical speeds of model 5 Ω cr,1 = rad/s Ω cr,2 = rad/s Ω cr,3 = rad/s Ω cr,4 = rad/s Slika 10: Lastne frekvence modela 5 Figure 10: Eigenfrequencies of model 5 smo opazovali pri togosti obeh ležajev k = been estimated using the stiffness coefficients 10 7 N/m. Odvisnost prvih štirih lastnih fre- k = 10 7 N/m for both bearings. Dependency kvenc od krožne hitrosti rotorja je prikazana of the first four eigenfrequencies on the shaft na sliki 10. Opazimo lahko, da so vrednosti la- speed is shown in Figure 10. It can be noticed stnih frekvenc malo nižje kot so bile za primer that the first 4 eigenfrequencies are a bit lower togega podprtja. Vpliv žiroskopskega efekta than those of rigid support. The influence of je pričakovano manjši, saj je togost celotnega the gyroscopic action is as expected smaller rotorja manjša zaradi elastičnih podpor. since the stiffness of entire rotor is decreased Podobno kot za model s togimi podporami, due to elastic supports. lahko opazimo štiri kritične hitrosti, ki so Similar to the model with rigid bearings označene na sliki 10 in zbrane v tabeli 4. Za- four critical speeds, marked in Figure 10 and radi spremembe lastnih frekvenc pa so ustre- listed in Table 4, can be found. Since the eizno nižje tudi kritične hitrosti. genfrequencies have changed the critical speeds are correspondingly lower. 3.3 Rotor z drsnimi ležaji 3.3 Rotor with oil-film bearings Za ovrednotenje vpliva drsnih ležajev na Models 6 and 7 have been used to study dinamiko rotorjev smo obravnavali modela 6 the influence of the oil-film bearings on rotorin 7. Uporabljena je bila teorija kratkih dr- dynamics. Short oil-film bearing theory has snih ležajev, podrobneje predstavljena v po- been used as described in the chapter about glavju o rotorjih z drsnimi ležaji. Predposta- rotors with oil-film bearings. Dynamic viscovljena dinamična viskoznost maziva je η = sity of the lubricant was η = 0.01Ns/m 2 for 0.01Ns/m 2 za oba ležaja. Glede na statično both bearings. Considering the static load the obremenitev znaša tlak v ležajih 140 kpa za bearing pressures are 140 kpa and 1.25MPa levi in 1.25 MPa za desni ležaj. for left and right bearing respectively. Pri obravnavi drsnih ležajev, katerih raz- Considering the oil-film bearings with the merje dolžine in premera je manjše od 1, se ratio of the bearing length to the bearing divrednosti togostnih in dušilnih koeficientov, ameter smaller than 1 the values of stiffness dolocenih s teorijo kratkega drsnega ležaja, le and damping coefficients determined with the malo razlikujejo od realnih vrednosti. Napaka short bearing theory differ only slightly from se povecuje z narašcanjem razmernika ekscen- the real ones. The difference increases with tricnosti. Rezultati dobljeni pri zelo velikih the eccentricity ratio. Results obtained at vrednostih razmernika ekscentricnosti, ɛ 1, very high values of eccentricity ratio, ɛ 1, so zato dvomljivi. can therefore be considered as doubtful. Ker se razmernik ekscentričnosti hitro Since the eccentricity ratio decreases rapi-

16 STROJNIŠKI VESTNIK - JOURNAL OF MECHANICAL ENGINEERING, LJUBLJANA (42) 1996/ R L [rad/s] 2000 Slika 11: Razmernik ekscentričnosti ležajev Figure 11: Eccentricity ratio of bearings zmanjšuje s povečevanjem krožne hitrosti ro- dly with shaft speed W the range of the shaft torja Ω, je v nadaljevanju pri obravnavi speed rad/s is thus not evaluated in all the folizpušceno območje krožne hitrosti 0 Ω lowing. As can be seen in Figure 11 the values 200rad/s. Kot je prikazano na sliki 11 so of the eccentricity ratio are especially high for vrednosti razmernika ekscentričnosti v tem the right bearing. The omitted range can not območju velike predvsem za desni ležaj. be taken generally since the eccentricity ratio Izpuščenega območja seveda ne gre jemati for the specific case depends on geometric pasplošno, saj je razmernik ekscentričnosti za rameters of the bearings as well as the rotor konkreten primer odvisen od geometrijskih itself. parametrov ležaja kot tudi od rotorja samega. As expected the eccentricity ratio of both Po pričakovanjih se razmer- bearings decreases with rotor speed. In Figure nik ekscentričnosti obeh ležajev zmanjšuje s 11 ɛ L and ɛ R denotes the eccentricity ratio povečevanjem kotne hitrosti rotorja. Na sliki of left and right bearing respectively. Depen- 11 je z ɛ L označen razmernik ekscentričnosti dency of stiffness and damping coefficients on za levi, z ɛ R pa za desni ležaj. Odvisnost togo- rotor speed has been calculated after equation stnih in dušilnih koeficientov od kotne hitrosti (10) and are shown in Figure 12 for both berotorja smo določili z enačbama (10). Vre- arings. It can be noticed that all stiffness codnost koeficientov je za oba ležaja prikazana efficients for the right bearing converge toward na sliki 12. Opazimo lahko, da vsi togostni definite value. For the left bearing the increkoeficienti desnega ležaja konvergirajo k neki ase of off-diagonal stiffness coefficients with končni vrednosti. Pri levem ležaju pa lahko rotor speed can be noticed. The rotor repreopazimo povečevanje izvendiagonalnih togo- sents therefore highly cross-coupled system. stnih koeficientov s kotno hitrostjo rotorja. The vibration of cross-coupled systems can Rotor je zaradi tega močno vezan sistem. Kot became unstable as will be shown in the folbo razvidno v nadaljevanju, lahko nihanje ve- lowing. Such behavior can be explained with zanih sistemov postane tudi nestabilno. Dej- higher static load for the right bearing which stvo lahko razložimo z večjo statično obre- also results in increased bearing pressure. menitvijo desnega ležaja, ki ima za posledico večji tlak v ležaju. Dušilni koeficienti obeh ležajev, na sliki 12, Damping coefficients for both bearings, kažejo kvalitativno podobno vedenje in se mo- shown in Figure 12, show qualitatively similar notono zmanjšujejo s povečevanjem kotne hi- behavior for both bearings. They monotoni-

17 STROJNIŠKI VESTNIK - JOURNAL OF MECHANICAL ENGINEERING, LJUBLJANA (42) 1996/ kij [N/m] 8 *1E6 k12 *1E6 150 kij [N/m] k k k22 k k k21-50 k [rad/s] [rad/s] 2000 * 1E3 70 dij [Ns/m] d22 * 1E3 400 dij [Ns/m] d d11 0 d d12, d d12, d [rad/s] [rad/s] 2000 Levi le aj Left bearing Desni le aj Right bearing Slika 12: Togostni in dušilni koeficient ležajev Figure 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 been teoretični obravnavi dinamike rotorjev, pov- presented in the theoretical consideration of zroči žiroskopski efekt bodisi sinhrono ali pa rotordynamics the gyroscopic action causes eiasinhrono rotacijo rotorja. V večini 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 the velja za rotorje z drsnimi ležaji. Na sliki 13 je same for the entire rotor which does not stand prikazana 5. lastna oblika modela 7 pri kotni for the rotors with oil-film bearings. The Fihitrosti rotorja Ω = 500rad/s. Del rotorja ro- gure 13 shows the 5th mode shape of model tira sinhrono (prekinjene linije), medtem ko 7 at rotor speed Ω = 500rad/s. Parts of the deli rotorja, označeni s polno linijo, rotirajo rotor demonstrates a forward whirl (dashed asinhrono. Pojav je opazen le pri rotorjih, ki lines) while the rest of the rotor marked with so podprti z drsnimi ležaji (modela 6 in 7), solid lines is subjected to backward whirl. tako da si ga lahko razložimo z anizotropijo This phenomenon has been observed only ar in povezanostjo togostne in dušilne matrike rotors, supported by oil-film bearings and can drsnega ležaja. be explained through their anisotropy. Togostni koeficienti desnega ležaja modela Stiffness coefficients for the right bearing 7 so večji, kot so bili uporabljeni pri modelu of model 7 are relatively large compared to 5, medtem ko so togostni koeficienti za levi those used for model 5 while the values for ležaj primerljivi. Lastne frekvence modela z the left bearing are comparable. The eigendrsnimi ležaji, slika 14, so nekoliko nižje od frequencies, shown in Figure 14, are howe-

18 STROJNIŠKI VESTNIK - JOURNAL OF MECHANICAL ENGINEERING, LJUBLJANA (42) 1996/ Slika 13: Peta lastna oblika modela 7 pri Ω = 500rad/s Figure 13: Fifth mode shape of model 7 at Ω = 500rad/s lastnih frekvenc modela s homogenimi ela- ver somewhat lower than those of the model stičnimi podporami. Dejstvo lahko razložimo with homogeneous flexible bearings. This can s povezano togostno in dušilno matriko mo- be explained through the influence of crossdela 7. Prav tako je močno izražen vpliv coupling. Influence of gyroscopic action is also žiroskopskega efekta, saj se lastne frekvence clearly manifested since the eigenfrequencies močno spreminjajo s povečanjem krožne 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 no kompleksno konjugiranih parih (5). Pri mo- damping we have only imaginary component. delih, kjer nimamo dušenja, imajo lastne vre- Presence of damping in the model demonstradnosti le kompleksno komponento. V kolikor tes itself as the real component of the eigense v modelu pojavi dušenje, dobimo pri la- frequency. The Figure 15 shows the real comstnih vrednostih tudi realno komponento. Na ponents of the eigenvalues for the model with sliki 15 so prikazane realne komponente la- oil-film bearings. stnih vrednosti modela z drsnimi ležaji [rad/s] S [rad/s] S [rad/s] Slika 14: Lastne frekvence modela 7 Figure 14: Eigenfrequencies of model 7 Realni komponenti prvih dveh lastnih vrednosti sta pozitivni v celotnem opazovanem območju. Posledica tega je nestabilno obratovanje rotorja, saj amplituda nihanj s časom Slika 15: Diagram stabilnosti modela 7 Figure 15: Stability chart of model Real components of first two eigenvalues are positive while the others are positive in the whole range of interest. This can be the cause of instability of the system since the am-

TOPLJENEC ASOCIIRA LE V VODNI FAZI

TOPLJENEC ASOCIIRA LE V VODNI FAZI TOPLJENEC ASOCIIRA LE V VODNI FAZI V primeru asociacij molekul topljenca v vodni ali organski fazi eksperimentalno določeni navidezni porazdelitveni koeficient (P n ) v odvisnosti od koncentracije ni konstanten.

More information

Modal Analysis: What it is and is not Gerrit Visser

Modal Analysis: What it is and is not Gerrit Visser Modal Analysis: What it is and is not Gerrit Visser What is a Modal Analysis? What answers do we get out of it? How is it useful? What does it not tell us? In this article, we ll discuss where a modal

More information

SAMCEF For ROTORS. Chapter 1 : Physical Aspects of rotor dynamics. This document is the property of SAMTECH S.A. MEF A, Page 1

SAMCEF For ROTORS. Chapter 1 : Physical Aspects of rotor dynamics. This document is the property of SAMTECH S.A. MEF A, Page 1 SAMCEF For ROTORS Chapter 1 : Physical Aspects of rotor dynamics This document is the property of SAMTECH S.A. MEF 101-01-A, Page 1 Table of Contents rotor dynamics Introduction Rotating parts Gyroscopic

More information

PROJECT 2 DYNAMICS OF MACHINES 41514

PROJECT 2 DYNAMICS OF MACHINES 41514 PROJECT 2 DYNAMICS OF MACHINES 41514 Dynamics of Rotor-Bearing System Lateral Vibrations and Stability Threshold of Rotors Supported On Hydrodynamic Bearing and Ball Bearing. Ilmar Ferreira Santos, Prof.

More information

VIBRATION ANALYSIS OF TIE-ROD/TIE-BOLT ROTORS USING FEM

VIBRATION ANALYSIS OF TIE-ROD/TIE-BOLT ROTORS USING FEM VIBRATION ANALYSIS OF TIE-ROD/TIE-BOLT ROTORS USING FEM J. E. Jam, F. Meisami Composite Materials and Technology Center Tehran, IRAN jejaam@gmail.com N. G. Nia Iran Polymer & Petrochemical Institute, Tehran,

More information

Towards Rotordynamic Analysis with COMSOL Multiphysics

Towards Rotordynamic Analysis with COMSOL Multiphysics Towards Rotordynamic Analysis with COMSOL Multiphysics Martin Karlsson *1, and Jean-Claude Luneno 1 1 ÅF Sound & Vibration *Corresponding author: SE-169 99 Stockholm, martin.r.karlsson@afconsult.com Abstract:

More information

Program System for Machine Dynamics. Abstract. Version 5.0 November 2017

Program System for Machine Dynamics. Abstract. Version 5.0 November 2017 Program System for Machine Dynamics Abstract Version 5.0 November 2017 Ingenieur-Büro Klement Lerchenweg 2 D 65428 Rüsselsheim Phone +49/6142/55951 hd.klement@t-online.de What is MADYN? The program system

More information

Eigenvalues of Trusses and Beams Using the Accurate Element Method

Eigenvalues of Trusses and Beams Using the Accurate Element Method Eigenvalues of russes and Beams Using the Accurate Element Method Maty Blumenfeld Department of Strength of Materials Universitatea Politehnica Bucharest, Romania Paul Cizmas Department of Aerospace Engineering

More information

Baroklina nestabilnost

Baroklina nestabilnost Baroklina nestabilnost Navodila za projektno nalogo iz dinamične meteorologije 2012/2013 Januar 2013 Nedjeljka Zagar in Rahela Zabkar Naloga je zasnovana na dvoslojnem modelu baroklinega razvoja, napisana

More information

Rotor-dynamics Analysis Process

Rotor-dynamics Analysis Process 2001-36 Rotor-dynamics Analysis Process Mohammad A. Heidari, Ph.D. David L. Carlson Ted Yantis Technical Fellow Principal Engineer Principal Engineer The Boeing Company The Boeing Company The Boeing Company

More information

Structural Dynamics Lecture 4. Outline of Lecture 4. Multi-Degree-of-Freedom Systems. Formulation of Equations of Motions. Undamped Eigenvibrations.

Structural Dynamics Lecture 4. Outline of Lecture 4. Multi-Degree-of-Freedom Systems. Formulation of Equations of Motions. Undamped Eigenvibrations. Outline of Multi-Degree-of-Freedom Systems Formulation of Equations of Motions. Newton s 2 nd Law Applied to Free Masses. D Alembert s Principle. Basic Equations of Motion for Forced Vibrations of Linear

More information

Graphical User Interface (GUI) for Torsional Vibration Analysis of Rotor Systems Using Holzer and MatLab Techniques

Graphical User Interface (GUI) for Torsional Vibration Analysis of Rotor Systems Using Holzer and MatLab Techniques Basrah Journal for Engineering Sciences, vol. 14, no. 2, 2014 255 Graphical User Interface (GUI) for Torsional Vibration Analysis of Rotor Systems Using Holzer and MatLab Techniques Dr. Ameen Ahmed Nassar

More information

SIMETRIČNE KOMPONENTE

SIMETRIČNE KOMPONENTE Univerza v Ljubljani Fakulteta za elektrotehniko SIMETRIČNE KOMPONENTE Seminarska naloga pri predmetu Razdelilna in industrijska omrežja Poročilo izdelala: ELIZABETA STOJCHEVA Mentor: prof. dr. Grega Bizjak,

More information

This equation of motion may be solved either by differential equation method or by graphical method as discussed below:

This equation of motion may be solved either by differential equation method or by graphical method as discussed below: 2.15. Frequency of Under Damped Forced Vibrations Consider a system consisting of spring, mass and damper as shown in Fig. 22. Let the system is acted upon by an external periodic (i.e. simple harmonic)

More information

Theory and Practice of Rotor Dynamics Prof. Dr. Rajiv Tiwari Department of Mechanical Engineering Indian Institute of Technology Guwahati

Theory and Practice of Rotor Dynamics Prof. Dr. Rajiv Tiwari Department of Mechanical Engineering Indian Institute of Technology Guwahati Theory and Practice of Rotor Dynamics Prof. Dr. Rajiv Tiwari Department of Mechanical Engineering Indian Institute of Technology Guwahati Module - 2 Simpul Rotors Lecture - 2 Jeffcott Rotor Model In the

More information

Study of coupling between bending and torsional vibration of cracked rotor system supported by radial active magnetic bearings

Study of coupling between bending and torsional vibration of cracked rotor system supported by radial active magnetic bearings Applied and Computational Mechanics 1 (2007) 427-436 Study of coupling between bending and torsional vibration of cracked rotor system supported by radial active magnetic bearings P. Ferfecki a, * a Center

More information

Calculation of stress-strain dependence from tensile tests at high temperatures using final shapes of specimen s contours

Calculation of stress-strain dependence from tensile tests at high temperatures using final shapes of specimen s contours RMZ Materials and Geoenvironment, Vol. 59, No. 4, pp. 331 346, 2012 331 Calculation of stress-strain dependence from tensile tests at high temperatures using final shapes of specimen s contours Določitev

More information

Vibration Analysis of Hollow Profiled Shafts

Vibration Analysis of Hollow Profiled Shafts International Journal of Current Engineering and echnology ISSN 77-406 04 INPRESSCO. All Rights Reserved. Available at http://inpressco.com/category/ijcet Research Article Vibration Analysis of Hollow

More information

NX Nastran 10. Rotor Dynamics User s Guide

NX Nastran 10. Rotor Dynamics User s Guide NX Nastran 10 Rotor Dynamics User s Guide Proprietary & Restricted Rights Notice 2014 Siemens Product Lifecycle Management Software Inc. All Rights Reserved. This software and related documentation are

More information

A SIMULATION STUDY OF THE ROTOR VIBRATION IN A JOURNAL BEARING

A SIMULATION STUDY OF THE ROTOR VIBRATION IN A JOURNAL BEARING Engineering MECHANICS, Vol. 15, 2008, No. 6, p. 461 470 461 A SIMULATION STUDY OF THE ROTOR VIBRATION IN A JOURNAL BEARING JiříTůma*, Alena Bilošová**, Jiří Šimek***, Rudolf Svoboda*** The paper deals

More information

Dynamic Analysis of An 1150 MW Turbine Generator

Dynamic Analysis of An 1150 MW Turbine Generator Dyrobes Rotordynamics Software https://dyrobes.com 1 PWR2005-50142 Abract Dynamic Analysis of An 1150 MW Turbine Generator Edgar J. Gunter Fellow ASME RODYN Vibration Inc Charlottesville, Va. 22903 DrGunter@aol.com

More information

Computational Stiffness Method

Computational Stiffness Method Computational Stiffness Method Hand calculations are central in the classical stiffness method. In that approach, the stiffness matrix is established column-by-column by setting the degrees of freedom

More information

Shafts: Torsion of Circular Shafts Reading: Crandall, Dahl and Lardner 6.2, 6.3

Shafts: Torsion of Circular Shafts Reading: Crandall, Dahl and Lardner 6.2, 6.3 M9 Shafts: Torsion of Circular Shafts Reading: Crandall, Dahl and Lardner 6., 6.3 A shaft is a structural member which is long and slender and subject to a torque (moment) acting about its long axis. We

More information

Dinamska analiza nihalnega sistema

Dinamska analiza nihalnega sistema UNIVERZA V LJUBLJANI Fakulteta za strojništvo Dinamska analiza nihalnega sistema Magistrsko delo Magistrskega študijskega programa II. stopnje STROJNIŠTVO Aljaž Peternelj Ljubljana, junij 2018 UNIVERZA

More information

Attempt to prepare seasonal weather outlook for Slovenia

Attempt to prepare seasonal weather outlook for Slovenia Attempt to prepare seasonal weather outlook for Slovenia Main available sources (ECMWF, EUROSIP, IRI, CPC.NCEP.NOAA,..) Two parameters (T and RR anomally) Textual information ( Met Office like ) Issued

More information

Research Program Vibrations ENERGIFORSK Vibration Group

Research Program Vibrations ENERGIFORSK Vibration Group Vorlesungen Mechatronik im Wintersemester Research Program Vibrations ENERGIFORSK Vibration Group DIAM A Matrix Tool for Turbine and Generator Vibrations Detection, Investigation, Analysis, Mitigation

More information

Chapter 2: Rigid Bar Supported by Two Buckled Struts under Axial, Harmonic, Displacement Excitation..14

Chapter 2: Rigid Bar Supported by Two Buckled Struts under Axial, Harmonic, Displacement Excitation..14 Table of Contents Chapter 1: Research Objectives and Literature Review..1 1.1 Introduction...1 1.2 Literature Review......3 1.2.1 Describing Vibration......3 1.2.2 Vibration Isolation.....6 1.2.2.1 Overview.

More information

1820. Selection of torsional vibration damper based on the results of simulation

1820. Selection of torsional vibration damper based on the results of simulation 8. Selection of torsional vibration damper based on the results of simulation Tomasz Matyja, Bogusław Łazarz Silesian University of Technology, Faculty of Transport, Gliwice, Poland Corresponding author

More information

Acta Chim. Slov. 2003, 50,

Acta Chim. Slov. 2003, 50, 771 IMPACT OF STRUCTURED PACKING ON BUBBE COUMN MASS TRANSFER CHARACTERISTICS EVAUATION. Part 3. Sensitivity of ADM Volumetric Mass Transfer Coefficient evaluation Ana akota Faculty of Chemistry and Chemical

More information

ANALYSIS AND IDENTIFICATION IN ROTOR-BEARING SYSTEMS

ANALYSIS AND IDENTIFICATION IN ROTOR-BEARING SYSTEMS ANALYSIS AND IDENTIFICATION IN ROTOR-BEARING SYSTEMS A Lecture Notes Developed under the Curriculum Development Scheme of Quality Improvement Programme at IIT Guwahati Sponsored by All India Council of

More information

Structural System, Machines and Load Cases

Structural System, Machines and Load Cases Machine-Induced Vibrations Machine-Induced Vibrations In the following example the dynamic excitation of two rotating machines is analyzed. A time history analysis in the add-on module RF-DYNAM Pro - Forced

More information

Reševanje problemov in algoritmi

Reševanje problemov in algoritmi Reševanje problemov in algoritmi Vhod Algoritem Izhod Kaj bomo spoznali Zgodovina algoritmov. Primeri algoritmov. Algoritmi in programi. Kaj je algoritem? Algoritem je postopek, kako korak za korakom rešimo

More information

Outline. Structural Matrices. Giacomo Boffi. Introductory Remarks. Structural Matrices. Evaluation of Structural Matrices

Outline. Structural Matrices. Giacomo Boffi. Introductory Remarks. Structural Matrices. Evaluation of Structural Matrices Outline in MDOF Systems Dipartimento di Ingegneria Civile e Ambientale, Politecnico di Milano May 8, 014 Additional Today we will study the properties of structural matrices, that is the operators that

More information

DYNAMIC RESPONSE OF THIN-WALLED GIRDERS SUBJECTED TO COMBINED LOAD

DYNAMIC RESPONSE OF THIN-WALLED GIRDERS SUBJECTED TO COMBINED LOAD DYNAMIC RESPONSE OF THIN-WALLED GIRDERS SUBJECTED TO COMBINED LOAD P. WŁUKA, M. URBANIAK, T. KUBIAK Department of Strength of Materials, Lodz University of Technology, Stefanowskiego 1/15, 90-924 Łódź,

More information

Modeling and Control of Instabilities in Combustion Processes Modeliranje in upravljanje nestabilnosti v procesih zgorevanja

Modeling and Control of Instabilities in Combustion Processes Modeliranje in upravljanje nestabilnosti v procesih zgorevanja Izvirni znanstveni članek TEHNIKA - nestabilni termoakustični procesi zgorevanja Datum prejema: 30. julij 2014 ANALI PAZU 4/ 2014/ 1: 34-40 www.anali-pazu.si Modeling and Control of Instabilities in Combustion

More information

Structural Dynamics Lecture 2. Outline of Lecture 2. Single-Degree-of-Freedom Systems (cont.)

Structural Dynamics Lecture 2. Outline of Lecture 2. Single-Degree-of-Freedom Systems (cont.) Outline of Single-Degree-of-Freedom Systems (cont.) Linear Viscous Damped Eigenvibrations. Logarithmic decrement. Response to Harmonic and Periodic Loads. 1 Single-Degreee-of-Freedom Systems (cont.). Linear

More information

Stability Analysis of a Hydrodynamic Journal Bearing With Rotating Herringbone Grooves

Stability Analysis of a Hydrodynamic Journal Bearing With Rotating Herringbone Grooves G. H. Jang e-mail: ghjang@hanyang.ac.kr J. W. Yoon PREM, Department of Mechanical Engineering, Hanyang University, Seoul, 33-79, Korea Stability Analysis of a Hydrodynamic Journal Bearing With Rotating

More information

ENERGY AND MASS SPECTROSCOPY OF IONS AND NEUTRALS IN COLD PLASMA

ENERGY AND MASS SPECTROSCOPY OF IONS AND NEUTRALS IN COLD PLASMA UDK621.3:(53+54+621 +66), ISSN0352-9045 Informaclje MIDEM 3~(~UU8)4, Ljubljana ENERGY AND MASS SPECTROSCOPY OF IONS AND NEUTRALS IN COLD PLASMA Marijan Macek 1,2* Miha Cekada 2 1 University of Ljubljana,

More information

Finite element analysis of rotating structures

Finite element analysis of rotating structures Finite element analysis of rotating structures Dr. Louis Komzsik Chief Numerical Analyst Siemens PLM Software Why do rotor dynamics with FEM? Very complex structures with millions of degrees of freedom

More information

56 1 Upogib z osno silo

56 1 Upogib z osno silo 56 1 Upogib z osno silo PREGLEDNICA 1.5 (nadaljevanje): Upogibnice in notranje sile za nekatere nosilce d) Upogibnica prostoležečega nosilca obteženega s silo F Pomik in zasuk v polju 1: w 1 = F b x (L

More information

D : SOLID MECHANICS. Q. 1 Q. 9 carry one mark each.

D : SOLID MECHANICS. Q. 1 Q. 9 carry one mark each. GTE 2016 Q. 1 Q. 9 carry one mark each. D : SOLID MECHNICS Q.1 single degree of freedom vibrating system has mass of 5 kg, stiffness of 500 N/m and damping coefficient of 100 N-s/m. To make the system

More information

DYNAMIC ANALYSIS OF ROTOR-BEARING SYSTEM FOR FLEXIBLE BEARING SUPPORT CONDITION

DYNAMIC ANALYSIS OF ROTOR-BEARING SYSTEM FOR FLEXIBLE BEARING SUPPORT CONDITION International Journal of Mechanical Engineering and Technology (IJMET) Volume 8, Issue 7, July 2017, pp. 1785 1792, Article ID: IJMET_08_07_197 Available online at http://www.iaeme.com/ijmet/issues.asp?jtype=ijmet&vtype=8&itype=7

More information

General elastic beam with an elastic foundation

General elastic beam with an elastic foundation General elastic beam with an elastic foundation Figure 1 shows a beam-column on an elastic foundation. The beam is connected to a continuous series of foundation springs. The other end of the foundation

More information

Dynamics of Rotor Systems with Clearance and Weak Pedestals in Full Contact

Dynamics of Rotor Systems with Clearance and Weak Pedestals in Full Contact Paper ID No: 23 Dynamics of Rotor Systems with Clearance and Weak Pedestals in Full Contact Dr. Magnus Karlberg 1, Dr. Martin Karlsson 2, Prof. Lennart Karlsson 3 and Ass. Prof. Mats Näsström 4 1 Department

More information

OA07 ANNEX 4: SCOPE OF ACCREDITATION IN CALIBRATION

OA07 ANNEX 4: SCOPE OF ACCREDITATION IN CALIBRATION OA07 ANNEX 4: SCOPE OF ACCREDITATION IN CALIBRATION Table of contents 1 TECHNICAL FIELDS... 2 2 PRESENTING THE SCOPE OF A CALIBRATION LABOORATORY... 2 3 CONSIDERING CHANGES TO SCOPES... 6 4 CHANGES WITH

More information

Nonlinear Dynamic Analysis of a Hydrodynamic Journal Bearing Considering the Effect of a Rotating or Stationary Herringbone Groove

Nonlinear Dynamic Analysis of a Hydrodynamic Journal Bearing Considering the Effect of a Rotating or Stationary Herringbone Groove G. H. Jang e-mail: ghjang@hanyang.ac.kr J. W. Yoon PREM, Department of Mechanical Engineering, Hanyang University, Seoul, 133-791, Korea Nonlinear Dynamic Analysis of a Hydrodynamic Journal Bearing Considering

More information

Effects of Structural Forces on the Dynamic Performance of High Speed Rotating Impellers.

Effects of Structural Forces on the Dynamic Performance of High Speed Rotating Impellers. Effects of Structural Forces on the Dynamic Performance of High Speed Rotating Impellers. G Shenoy 1, B S Shenoy 1 and Raj C Thiagarajan 2 * 1 Dept. of Mechanical & Mfg. Engineering, Manipal Institute

More information

Abstract: Paper deals with subject of identification of aerostatic journal bearings dynamic properties with use of Rotor Kit Bently Nevada

Abstract: Paper deals with subject of identification of aerostatic journal bearings dynamic properties with use of Rotor Kit Bently Nevada IDENTIFICATION OF STIFFNESS AND DAMPING COEFFICIENTS OF AEROSTATIC JOURNAL BEARING Jan Kozánek Jiří Šimek Pavel Steinbauer Aleš Bílkovský Abstract: Paper deals with subject of identification of aerostatic

More information

Theory and Practice of Rotor Dynamics Prof. Rajiv Tiwari Department of Mechanical Engineering Indian Institute of Technology Guwahati

Theory and Practice of Rotor Dynamics Prof. Rajiv Tiwari Department of Mechanical Engineering Indian Institute of Technology Guwahati Theory and Practice of Rotor Dynamics Prof. Rajiv Tiwari Department of Mechanical Engineering Indian Institute of Technology Guwahati Module - 7 Instability in rotor systems Lecture - 4 Steam Whirl and

More information

Dynamics of assembled structures of rotor systems of aviation gas turbine engines of type two-rotor

Dynamics of assembled structures of rotor systems of aviation gas turbine engines of type two-rotor Dynamics of assembled structures of rotor systems of aviation gas turbine engines of type two-rotor Anatoly А. Pykhalov 1, Mikhail А. Dudaev 2, Mikhail Ye. Kolotnikov 3, Paul V. Makarov 4 1 Irkutsk State

More information

Critical Speed Analysis of Offset Jeffcott Rotor Using English and Metric Units

Critical Speed Analysis of Offset Jeffcott Rotor Using English and Metric Units Dyrobes Rotordynamics Software https://dyrobes.com Critical Speed Analysis of Offset Jeffcott Rotor Using English and Metric Units E. J. Gunter,PhD Fellow ASME February,2004 RODYN Vibration Inc. 1932 Arlington

More information

Dynamic Model of a Badminton Stroke

Dynamic Model of a Badminton Stroke ISEA 28 CONFERENCE Dynamic Model of a Badminton Stroke M. Kwan* and J. Rasmussen Department of Mechanical Engineering, Aalborg University, 922 Aalborg East, Denmark Phone: +45 994 9317 / Fax: +45 9815

More information

Assignment 6. Using the result for the Lagrangian for a double pendulum in Problem 1.22, we get

Assignment 6. Using the result for the Lagrangian for a double pendulum in Problem 1.22, we get Assignment 6 Goldstein 6.4 Obtain the normal modes of vibration for the double pendulum shown in Figure.4, assuming equal lengths, but not equal masses. Show that when the lower mass is small compared

More information

Q. 1 Q. 5 carry one mark each.

Q. 1 Q. 5 carry one mark each. General ptitude G Set-8 Q. 1 Q. 5 carry one mark each. Q.1 The chairman requested the aggrieved shareholders to him. () bare with () bore with (C) bear with (D) bare Q.2 Identify the correct spelling out

More information

Robust shaft design to compensate deformation in the hub press fitting and disk clamping process of 2.5 HDDs

Robust shaft design to compensate deformation in the hub press fitting and disk clamping process of 2.5 HDDs DOI 10.1007/s00542-016-2850-2 TECHNICAL PAPER Robust shaft design to compensate deformation in the hub press fitting and disk clamping process of 2.5 HDDs Bumcho Kim 1,2 Minho Lee 3 Gunhee Jang 3 Received:

More information

Perturbation of periodic equilibrium

Perturbation of periodic equilibrium Perturbation of periodic equilibrium by Arnaud Lazarus A spectral method to solve linear periodically time-varying systems 1 A few history Late 19 th century Emile Léonard Mathieu: Wave equation for an

More information

Electrical excitation and mechanical vibration of a piezoelectric cube

Electrical excitation and mechanical vibration of a piezoelectric cube Scientific original paper Journal of Microelectronics, Electronic Components and Materials Vol. 42, No. 3 (2012), 192 196 Electrical excitation and mechanical vibration of a piezoelectric cube Oumar Diallo

More information

Structural Matrices in MDOF Systems

Structural Matrices in MDOF Systems in MDOF Systems http://intranet.dica.polimi.it/people/boffi-giacomo Dipartimento di Ingegneria Civile Ambientale e Territoriale Politecnico di Milano April 9, 2016 Outline Additional Static Condensation

More information

Where and are the factored end moments of the column and >.

Where and are the factored end moments of the column and >. 11 LIMITATION OF THE SLENDERNESS RATIO----( ) 1-Nonsway (braced) frames: The ACI Code, Section 6.2.5 recommends the following limitations between short and long columns in braced (nonsway) frames: 1. The

More information

Structural Dynamics Lecture Eleven: Dynamic Response of MDOF Systems: (Chapter 11) By: H. Ahmadian

Structural Dynamics Lecture Eleven: Dynamic Response of MDOF Systems: (Chapter 11) By: H. Ahmadian Structural Dynamics Lecture Eleven: Dynamic Response of MDOF Systems: (Chapter 11) By: H. Ahmadian ahmadian@iust.ac.ir Dynamic Response of MDOF Systems: Mode-Superposition Method Mode-Superposition Method:

More information

Effect of an hourglass shaped sleeve on the performance of the fluid dynamic bearings of a HDD spindle motor

Effect of an hourglass shaped sleeve on the performance of the fluid dynamic bearings of a HDD spindle motor DOI 10.1007/s00542-014-2136-5 Technical Paper Effect of an hourglass shaped sleeve on the performance of the fluid dynamic bearings of a HDD spindle motor Jihoon Lee Minho Lee Gunhee Jang Received: 14

More information

Chapter 4 Analysis of a cantilever

Chapter 4 Analysis of a cantilever Chapter 4 Analysis of a cantilever Before a complex structure is studied performing a seismic analysis, the behaviour of simpler ones should be fully understood. To achieve this knowledge we will start

More information

Contact problems in rotor systems

Contact problems in rotor systems Contact problems in rotor systems Liudmila Banakh Mechanical Engineering Research Institute of RAS, Moscow, Russia E-mail: banl@inbox.ru (Received 18 July 2016; accepted 24 August 2016) Abstract. We consider

More information

NON-LINEAR ROTORDYNAMICS: COMPUTATIONAL STRATEGIES

NON-LINEAR ROTORDYNAMICS: COMPUTATIONAL STRATEGIES The 9th International Symposium on Transport Phenomena and Dynamics of Rotating Machinery Honolulu, Hawaii, February 1-14, NON-LINEAR ROTORDNAMICS: COMPUTATIONAL STRATEGIES Tom J. Chalko Head of Rotordynamic

More information

Advanced Vibrations. Elements of Analytical Dynamics. By: H. Ahmadian Lecture One

Advanced Vibrations. Elements of Analytical Dynamics. By: H. Ahmadian Lecture One Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian ahmadian@iust.ac.ir Elements of Analytical Dynamics Newton's laws were formulated for a single particle Can be extended to

More information

Centrifugal pumps (Agriculture) unbalance and shaft Dynamic analysis from the experimental data in a rotor system

Centrifugal pumps (Agriculture) unbalance and shaft Dynamic analysis from the experimental data in a rotor system Research Article International Journal of Current Engineering and Technology E-ISSN 77 416, P-ISSN 347-5161 14 INPRESSCO, All Rights Reserved Available at http://inpressco.com/category/ijcet Centrifugal

More information

PLEASURE VESSEL VIBRATION AND NOISE FINITE ELEMENT ANALYSIS

PLEASURE VESSEL VIBRATION AND NOISE FINITE ELEMENT ANALYSIS PLEASURE VESSEL VIBRATION AND NOISE FINITE ELEMENT ANALYSIS 1 Macchiavello, Sergio *, 2 Tonelli, Angelo 1 D Appolonia S.p.A., Italy, 2 Rina Services S.p.A., Italy KEYWORDS pleasure vessel, vibration analysis,

More information

Cveto Trampuž PRIMERJAVA ANALIZE VEČRAZSEŽNIH TABEL Z RAZLIČNIMI MODELI REGRESIJSKE ANALIZE DIHOTOMNIH SPREMENLJIVK

Cveto Trampuž PRIMERJAVA ANALIZE VEČRAZSEŽNIH TABEL Z RAZLIČNIMI MODELI REGRESIJSKE ANALIZE DIHOTOMNIH SPREMENLJIVK Cveto Trampuž PRIMERJAVA ANALIZE VEČRAZSEŽNIH TABEL Z RAZLIČNIMI MODELI REGRESIJSKE ANALIZE DIHOTOMNIH SPREMENLJIVK POVZETEK. Namen tega dela je prikazati osnove razlik, ki lahko nastanejo pri interpretaciji

More information

Dynamics of a rigid rotor in the elastic bearings

Dynamics of a rigid rotor in the elastic bearings Theoret. Appl. Mech., Vol.31, No.1, pp.73-83, Belgrade 2004 Dynamics of a rigid rotor in the elastic bearings Inga M. Arkhipova Abstract As a rule in the studies of a rigid rotor in the elastic bearings

More information

UNIT IV FLEXIBILTY AND STIFFNESS METHOD

UNIT IV FLEXIBILTY AND STIFFNESS METHOD SIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddharth Nagar, Narayanavanam Road 517583 QUESTION BANK (DESCRIPTIVE) Subject with Code : SA-II (13A01505) Year & Sem: III-B.Tech & I-Sem Course & Branch: B.Tech

More information

VIBRATION PROBLEMS IN ENGINEERING

VIBRATION PROBLEMS IN ENGINEERING VIBRATION PROBLEMS IN ENGINEERING FIFTH EDITION W. WEAVER, JR. Professor Emeritus of Structural Engineering The Late S. P. TIMOSHENKO Professor Emeritus of Engineering Mechanics The Late D. H. YOUNG Professor

More information

Dept.of Mechanical Engg, Defence Institute of Advanced Technology, Pune. India

Dept.of Mechanical Engg, Defence Institute of Advanced Technology, Pune. India Applied Mechanics and Materials Submitted: 2014-04-23 ISSN: 1662-7482, Vols. 592-594, pp 1084-1088 Revised: 2014-05-16 doi:10.4028/www.scientific.net/amm.592-594.1084 Accepted: 2014-05-19 2014 Trans Tech

More information

WORK SHEET FOR MEP311

WORK SHEET FOR MEP311 EXPERIMENT II-1A STUDY OF PRESSURE DISTRIBUTIONS IN LUBRICATING OIL FILMS USING MICHELL TILTING PAD APPARATUS OBJECTIVE To study generation of pressure profile along and across the thick fluid film (converging,

More information

Computational Simulation of Dynamic Response of Vehicle Tatra T815 and the Ground

Computational Simulation of Dynamic Response of Vehicle Tatra T815 and the Ground IOP Conference Series: Earth and Environmental Science PAPER OPEN ACCESS Computational Simulation of Dynamic Response of Vehicle Tatra T815 and the Ground To cite this article: Jozef Vlek and Veronika

More information

Nonlinear Rolling Element Bearings in MADYN 2000 Version 4.3

Nonlinear Rolling Element Bearings in MADYN 2000 Version 4.3 - 1 - Nonlinear Rolling Element Bearings in MADYN 2000 Version 4.3 In version 4.3 nonlinear rolling element bearings can be considered for transient analyses. The nonlinear forces are calculated with a

More information

On The Finite Element Modeling Of Turbo Machinery Rotors In Rotor Dynamic Analysis

On The Finite Element Modeling Of Turbo Machinery Rotors In Rotor Dynamic Analysis Proceedings of The Canadian Society for Mechanical Engineering International Congress 2018 CSME International Congress 2018 May 27-30, 2018, Toronto, On, Canada On The Finite Element Modeling Of Turbo

More information

3. Overview of MSC/NASTRAN

3. Overview of MSC/NASTRAN 3. Overview of MSC/NASTRAN MSC/NASTRAN is a general purpose finite element analysis program used in the field of static, dynamic, nonlinear, thermal, and optimization and is a FORTRAN program containing

More information

Parametrically Excited Vibration in Rolling Element Bearings

Parametrically Excited Vibration in Rolling Element Bearings Parametrically Ecited Vibration in Rolling Element Bearings R. Srinath ; A. Sarkar ; A. S. Sekhar 3,,3 Indian Institute of Technology Madras, India, 636 ABSTRACT A defect-free rolling element bearing has

More information

Determination of Dynamic Characteristics of the Frame Bearing Structures of the Vibrating Separating Machines

Determination of Dynamic Characteristics of the Frame Bearing Structures of the Vibrating Separating Machines IOP Conference Series: Materials Science and Engineering PAPER OPEN ACCESS Determination of Dynamic Characteristics of the Frame Bearing Structures of the Vibrating Separating Machines To cite this article:

More information

Use of Full Spectrum Cascade for Rotor Rub Identification

Use of Full Spectrum Cascade for Rotor Rub Identification Use of Full Spectrum Cascade for Rotor Rub Identification T. H. Patel 1, A. K. Darpe 2 Department of Mechanical Engineering, Indian Institute of Technology, Delhi 110016, India. 1 Research scholar, 2 Assistant

More information

Hakwoon Kim Gunhee Jang Sanghoon Lee. 1 Introduction

Hakwoon Kim Gunhee Jang Sanghoon Lee. 1 Introduction Microsyst Technol (2011) 17:749 759 DOI 10.1007/s00542-010-1188-4 TECHNICAL PAPER Complete determination of the dynamic coefficients of coupled journal and thrust bearings considering five degrees of freedom

More information

DYNAMICS OF MACHINERY 41514

DYNAMICS OF MACHINERY 41514 DYNAMICS OF MACHINERY 454 PROJECT : Theoretical and Experimental Modal Analysis and Validation of Mathematical Models in Multibody Dynamics Holistic Overview of the Project Steps & Their Conceptual Links

More information

Mechanics of Materials II. Chapter III. A review of the fundamental formulation of stress, strain, and deflection

Mechanics of Materials II. Chapter III. A review of the fundamental formulation of stress, strain, and deflection Mechanics of Materials II Chapter III A review of the fundamental formulation of stress, strain, and deflection Outline Introduction Assumtions and limitations Axial loading Torsion of circular shafts

More information

Numerical simulation of the coil spring and investigation the impact of tension and compression to the spring natural frequencies

Numerical simulation of the coil spring and investigation the impact of tension and compression to the spring natural frequencies Numerical simulation of the coil spring and investigation the impact of tension and compression to the spring natural frequencies F. D. Sorokin 1, Zhou Su 2 Bauman Moscow State Technical University, Moscow,

More information

CEE 271: Applied Mechanics II, Dynamics Lecture 25: Ch.17, Sec.4-5

CEE 271: Applied Mechanics II, Dynamics Lecture 25: Ch.17, Sec.4-5 1 / 36 CEE 271: Applied Mechanics II, Dynamics Lecture 25: Ch.17, Sec.4-5 Prof. Albert S. Kim Civil and Environmental Engineering, University of Hawaii at Manoa Date: 2 / 36 EQUATIONS OF MOTION: ROTATION

More information

Iskanje najcenejše poti v grafih preko polkolobarjev

Iskanje najcenejše poti v grafih preko polkolobarjev Univerza v Ljubljani Fakulteta za računalništvo in informatiko Veronika Horvat Iskanje najcenejše poti v grafih preko polkolobarjev DIPLOMSKO DELO VISOKOŠOLSKI STROKOVNI ŠTUDIJSKI PROGRAM PRVE STOPNJE

More information

Finite element method based analysis and modeling in rotordynamics

Finite element method based analysis and modeling in rotordynamics Finite element method based analysis and modeling in rotordynamics A thesis submitted to the Graduate School of the University of Cincinnati in partial fulfillment of the requirements for the degree of

More information

Multipla korelacija in regresija. Multipla regresija, multipla korelacija, statistično zaključevanje o multiplem R

Multipla korelacija in regresija. Multipla regresija, multipla korelacija, statistično zaključevanje o multiplem R Multipla koelacia in egesia Multipla egesia, multipla koelacia, statistično zaklučevane o multiplem Multipla egesia osnovni model in ačunane paametov Z multiplo egesio napoveduemo vednost kiteia (odvisne

More information

Mechatronics, Electrical Power, and Vehicular Technology

Mechatronics, Electrical Power, and Vehicular Technology Mechatronics, Electrical Power, and Vehicular Technology 05 (2014) 1-8 Mechatronics, Electrical Power, and Vehicular Technology e-issn:2088-6985 p-issn: 2087-3379 Accreditation Number: 432/Akred-LIPI/P2MI-LIPI/04/2012

More information

VIBRATION-BASED HEALTH MONITORING OF ROTATING SYSTEMS WITH GYROSCOPIC EFFECT

VIBRATION-BASED HEALTH MONITORING OF ROTATING SYSTEMS WITH GYROSCOPIC EFFECT VIBRATION-BASED HEALTH MONITORING OF ROTATING SYSTEMS WITH GYROSCOPIC EFFECT A Thesis presented to the Faculty of California Polytechnic State University, San Luis Obispo In Partial Fulfillment of the

More information

Distance reduction with the use of UDF and Mathematica. Redukcija dolžin z uporabo MS Excel ovih lastnih funkcij in programa Mathematica

Distance reduction with the use of UDF and Mathematica. Redukcija dolžin z uporabo MS Excel ovih lastnih funkcij in programa Mathematica RMZ Materials and Geoenvironment, Vol. 54, No. 2, pp. 265-286, 2007 265 Distance reduction with the use of UDF and Mathematica Redukcija dolžin z uporabo MS Excel ovih lastnih funkcij in programa Mathematica

More information

UČNI NAČRT PREDMETA / COURSE SYLLABUS Numerical linear algebra. Študijska smer Study field. Samost. delo Individ. work Klinične vaje work

UČNI NAČRT PREDMETA / COURSE SYLLABUS Numerical linear algebra. Študijska smer Study field. Samost. delo Individ. work Klinične vaje work Predmet: Course title: UČNI NAČRT PREDMETA / COURSE SYLLABUS Numerična linearna algebra Numerical linear algebra Študijski program in stopnja Study programme and level Univerzitetni študijski program Matematika

More information

1 Introduction. Minho Lee 1 Jihoon Lee 1 Gunhee Jang 1

1 Introduction. Minho Lee 1 Jihoon Lee 1 Gunhee Jang 1 DOI 10.1007/s005-015-5-5 TECHNICAL PAPER Stability analysis of a whirling rigid rotor supported by stationary grooved FDBs considering the five degrees of freedom of a general rotor bearing system Minho

More information

Nonlinear effects on the rotor driven by a motor with limited power

Nonlinear effects on the rotor driven by a motor with limited power Applied and Computational Mechanics 1 (007) 603-61 Nonlinear effects on the rotor driven by a motor with limited power L. Půst Institute of Thermomechanics, Academy of Sciences of CR, Dolejkova 5,18 00

More information

Imaginary. Axis. Real. Axis

Imaginary. Axis. Real. Axis Name ME6 Final. I certify that I upheld the Stanford Honor code during this exam Monday December 2, 25 3:3-6:3 p.m. ffl Print your name and sign the honor code statement ffl You may use your course notes,

More information

Evolucija dinamike Zemljine precesije

Evolucija dinamike Zemljine precesije Univerza v Ljubljani Fakulteta za matematiko in fiziko oddelek za fiziko Evolucija dinamike Zemljine precesije Avtor: Ivo Krajnik Ljubljana, 15. marec 2011 Povzetek Bistvo tega seminarja je v sklopu klasične

More information

UČNI NAČRT PREDMETA / COURSE SYLLABUS (leto / year 2017/18) Predmet: Analiza 3 Course title: Analysis 3. Študijska smer Study field ECTS

UČNI NAČRT PREDMETA / COURSE SYLLABUS (leto / year 2017/18) Predmet: Analiza 3 Course title: Analysis 3. Študijska smer Study field ECTS UČNI NAČRT PREDMETA / COURSE SYLLABUS (leto / year 2017/18) Predmet: Analiza 3 Course title: Analysis 3 Študijski program in stopnja Study programme and level Univerzitetni študijski program Matematika

More information

Aeroelastic effects of large blade deflections for wind turbines

Aeroelastic effects of large blade deflections for wind turbines Aeroelastic effects of large blade deflections for wind turbines Torben J. Larsen Anders M. Hansen Risoe, National Laboratory Risoe, National Laboratory P.O. Box 49, 4 Roskilde, Denmark P.O. Box 49, 4

More information

Stability analysis of a whirling disk-spindle system supported by FDBs with rotating grooves

Stability analysis of a whirling disk-spindle system supported by FDBs with rotating grooves Microsyst Technol (011) 17:787 797 DOI 10.1007/s0054-010-111-9 TECHNICAL PAPER Stability analysis of a whirling disk-spindle system supported by FDBs with rotating grooves Jihoon Lee Gunhee Jang Kyungmoon

More information

Review of Strain Energy Methods and Introduction to Stiffness Matrix Methods of Structural Analysis

Review of Strain Energy Methods and Introduction to Stiffness Matrix Methods of Structural Analysis uke University epartment of Civil and Environmental Engineering CEE 42L. Matrix Structural Analysis Henri P. Gavin Fall, 22 Review of Strain Energy Methods and Introduction to Stiffness Matrix Methods

More information