Dynamic behaviour of rotors supported by fluid-film bearings operated close to fluid-induced instability

Transcription

1 MATEC Web of Conferences 48, 43 (8 ICoEV 7 Dnamic behaviour of rotors supported b fluid-film bearings operated close to fluid-induced instabilit Luboš Smolík,, Miroslav Brtus,, Štěpán Dk,, and Michal Hajžman, Department of Mechanics, Universit of West Bohemia, Czech Republic Abstract. This paper is focused on an analsis of rotating sstems with fluid-film bearings, especiall on their nonlinear behaviour in the course of developing fluid-induced instabilit. The studied sstem consists of Jeffcott rotor supported b a fluid-film bearing characterised b the Renolds equation. The stead state response of the sstem is investigated b means of an approimate analtical solution of the Renolds equation while the transient response of the sstem is investigated using a comple numerical solution. Results suggest that the rotor eercise a bounded chaotic motion if it becomes unstable. If the fluid-induced instabilit further develops, the motion actuall becomes less chaotic and can be characterised as quasi-periodic. Introduction Fluid film bearings are widel used for supporting of various rotating machines because of their favourable properties, specificall low friction and wear and vibrationreducing capabilities. However, it has been known for more than 9 ears that a fluid-induced instabilit can develop in a poorl designed bearing []. The qualit of the bearing design can be epressed b Sommerfeld number S = ( p ψ /ηω, where p, ψare a rated load relative clearance of the bearing, η is a dnamic viscosit of a fluid film and ω is a rotor speed []. In general, bearings with low S i.e. not enough loaded bearings, bearings with high-speed rotors or bearings with highl viscous films are prone to the the fluid-induced instabilit. The instabilit, also known as oil whirl, was mathematicall described at the turn of the 5s and 6s [3, 4]. Both authors also epressed a stabilit criterion for a linearised rotor-bearing sstem which is based on the Routh-Hurwitz criterion. The occurrence of the fluid-induced instabilit and the processes accompaning it were studied eperimentall in the earl 8s [5 7]. The oil whirl starts when hdrodnamical (HD forces in the fluid film become higher than loading forces, which are predominantl caused b static loads. When this occurs, the HD forces start to push a journal a part of a shaft, which rests in a bearing in a forward circular motion around the bearing. The journal runs about the bearing with a period which in general lasts slightl longer than two revolutions. Therefore the oil whirl can be easil recognized b frequenc which is.4.49 to journal speed [6]. carlist@kme.zcu.cz mbrtus@kme.zcu.cz sdk@kme.zcu.cz mhajzman@kme.zcu.cz HD forces are determined b a pressure distribution of the fluid film, which is governed b a partial differential equation known as the Renolds equation (RE [8]. The eact analtical solution of RE has not been known until [9], however, approimate analtical solutions for infinitel short [, ] and infinitel long bearings [] have been known since the 5 s and later man researchers used perturbation methods in order to obtain approimate analtical solutions of full RE [, 3]. As the computing power of personal computers had been rising, robust numerical methods including finite differential [4], finite element [4, 5], finite volume [6] and element-based finite volume methods [7] were also emploed. Although all mentioned methods capture the phenomenon of oil whirl, there are relativel small number of studies which focus on nonlinear behaviours of rotors operated close to the instabilit threshold speed. A transient state which occurs in rotor-bearing sstem when a journal pass the threshold speed is studied in [3], where small amplitude perturbations are assumed in the bearing. The transient state of Jeffcott rotor supported b a gas bearing and accompaning bifurcations are studied in [8]. The main focus of this work is to perform an in-depth analsis of the transient state for Jeffcott rotor supported b journal bearing with consideration of all terms which are presented in RE. This ma lead to better understanding of the oil whirl instabilit, and hence to understand how to design rotor sstems which are operated in the unstable regime. The rest of the article is outlined as follows. Section presents a simple model on which computations of steadstate response were performed. Section 3 contains a brief description of an etended model for transient analsis. Model and simulation parameters, and results are discussed in sections 4 and 5, respectivel. Finall, section 6 draws some conclusions. The Authors, published b EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4. (

2 MATEC Web of Conferences 48, 43 (8 ICoEV 7 Model for stead-state analsis A general mathematical model of a simple rigid rotor which rotates at a constant angular velocit and is supported b a generic journal bearing is described in this section.. Equation of motion for a rigid rotor We assume the rigid rotor of mass m which rotates at a constant angular velocit ω and is subject to constant gravitational and out-of-balance forces f g and f u and to a vector of hdrodnamic forces f hd (see fig.. The equilibrium of acting and inertia forces can be written as [] M q = f g + f u + f hd, ( where M is a mass matri and q(t is an acceleration vector. If we further assume that the rotor does not perform aw and pitch movements then its motion is given b horizontal displacement and vertical displacement = (t and = (t and Eq. ( can be written in the form [ ][ẍ ] [ ] [ ] [ ] m cos = + U m ÿ m g st ω ω t F hd + sin ω t F hd, ( where g is the gravitational constant, U st is the static unbalance and F hd and F hd are horizontal and vertical components of vector f hd []. Alternativel, the motion of the rotor can be characterised b eccentricit e = e(t and angle to the eccentricit φ = φ(t, which are related to and as follows e = +, (3 φ = arctan ( arctan ( arctan ( > >, + π <, (4 + π > <. Time derivatives ė and φ of e and φ are obviousl r c r h ω ė = ẋ + ẏ +, (5 φ = ẏ ẋ +. (6 F hd tan z A U st f g f u ω t ϕ e hd F Figure. A rigid rotor supported b a journal bearing. s. Hdrodnamic forces in a journal bearing Hdrodnamic forces are obtained b integrating hdrodnamic pressure in an oil film over a surface of the bearing. Hdrodnamic pressure p = p(s, z, t is given b the Renolds equation [8] s ( h 3 µ p s + z ( h 3 µ p z = 6 ω r h h + s t, (7 where s and z are circumferential and aial coordinates, respectivel, h = h(s, z, t is a gap between the journal and the bearing and µ is dnamic viscosit of the oil film. The Renolds equation (7 can be solved under various assumptions. Let us assume the half Sommerfeld conditions [] and consider the following relations ( s h = c r e cos r φ and l, (8 where l is the bearing length. Then ial and tangential components of hdrodnamic forces F hd,sb and F hd,sb tan shown in Fig. can be epressed as [] F hd,sb = µ rl3 ω φ ε c ( r ε + π ( + ε ε ( ε.5, (9 ( F hd,sb tan = µ rl3 ω φ πε c r 4 ( ε + ε ε.5 ( ε, ( where ε = e /c r is the relative eccentricit. Equations (9 and ( approimate the analtical solution of Renolds equation (7 reasonabl accuratel if l < r holds []. If l is from interval r, 4 r, the components of hdrodnamic forces have to be corrected. Bastani proposed [] corrective functions in the form F hd = ( f ε 3 + f ε + f 3 ε + f 4 F hd,sb, ( Ftan hd = ( g ε + g ε + g 3 F hd,sb tan, ( where f i,g i are cubic or quartic polnomial functions which are dependant on l /r and are epressed in []. Hdrodnamic forces F hd and Fhd tan are derived in a rotated coordinate sstem shown in fig. and can be transformed to a fied coordinate sstem from equation of motion ( using relations F hd F hd.3 Solution strateg = F hd cos φ Fhd tan sin φ, (3 = F hd sin φ + Fhd tan cos φ. (4 Equation of motion ( can be formall rewritten as M q = f, (5 where f = f( q, q, t is a vector of all acting forces. Eq. (5 can be further transformed to the state-space; the resulting equation is then [ ] [ q q] q u = M, where u =. (6 f The equation of motion in this form can be integrated numericall b means of the Runge-Kutta method.

3 MATEC Web of Conferences 48, 43 (8 ICoEV Model for transient analsis An etension to the previous model allowing a transient analsis for ω const. is described in this section. 3. Equation of motion Let us assume that the angular speed of the rigid rotor from section. is time-dependant variable ϕ = ϕ(t. Since we want to be able to control ϕ, we introduce a controller and couple the controlle and the rotor with a rotational spring as shown in Fig.. The rotation of the rotor in such configuration is given b the equation I ϕ + d r ( ϕ Ω + k r (ϕ Φ = M f ric, (7 where I is a moment of inertia of the rotor to the ais of rotation, d r, k r are rotational damping and stiffness of the rotational spring, respectivel, Φ=Φ(t and Ω=Ω(t are arbitrar functions of controller s angular displacement and angular velocit and M f ric is a frictional moment []. Since frictional moment M f ric is of no concern because we do not analse losses due friction nor thermodnamics of the oil film, M f ric is further neglected. With respect to the fact that angular velocit of the rotor ϕ is not constant, equations of motion are m ÿ = U st ϕ cos ϕ + FEM F hd, (8 m z = m g + U ϕ sin ϕ + FEM Fz hd, (9 I ϕ + d r ϕ + k r ϕ = d r Ω+k r Φ, ( where the components of hdrodnamic forces FEM F hd and FEM F hd are obtained from finite-element-based numerical solution of the Renolds equation [5]. In the first step of the solution process, hdrodnamic pressure p is obtained from Renolds equation (7 under the assumptions tha i the bearing is full-flooded, ii there is constant ambient pressure p a at edges of the bearing, and c if hdrodnamic pressure p in an node of FE mesh drops below the predefined value p s, then p = p s is set in that particular node. In the second step, p is integrated over the bearing surface. This integral ields FEM F hd and FEM F hd [9]. Note that the approimate analtical solution of Eq. (7 which is described in section. is derived under the assumption that the angular speed of the journal is constant and therefore cannot be used for transient analsis. 3. Solution strateg The sstem of eqs. (8 has to be transformed to the state-space in order to be integrable b standard numerical solvers. The transformation was briefl described in section.3. 4 Model and simulation parameters Parameters of the sstem and their respective smbols for both stead-state and transient analsis can be found in tab.. The parameters are chosen so that the fluid-induced instabilit arises at ca. 3 rpm. The rotor was considered perfectl balanced in the case of the stead-state analsis. The simulations were performed in the speed interval n, 9 with the step 5 rpm. Time integration was performed in the interval t,.5 s with a variable time step. The results were then transformed to time series with the constant time step 4 s. Onl the time interval t.5,.5 s was subjected to subsequent post-processing. The transient analsis was performed in the speed interval n 8, 4 rpm with several values of constant angular acceleration and static unbalance U st. The selected values of U st correspond with balance qualit ges G, G6.3 and G6 per ISO 94. Time integration was performed with a variable time step t 5 7, 5 5 s and the results were then transformed to time series with the constant time step 4 s. Table. Parameters of analsed sstem. Items denoted with asterisk smbol ( are used onl during transient analsis. Parameter Smbol (unit Value rotor mass m (kg 3 moment of inertia I (kg m.3 rotor speed n (rpm variable gravitational constant g (m s 9.8 static unbalance G U st (g mm static unbalance G6.3 U st (g mm 45 static unbalance G6 U st (g mm bearing ius r (mm 3.69 bearing length l (mm ial clearance c r (mm.9 oil viscosit µ (Pa s.7 ambient pressure p a (bar saturation pressure p c (bar rotational stiffness k r ( kn m rotational damping d r (N m s controller Φ Ω k r, dr φ. φ,φ Figure. A rigid rotor coupled to a controller. rotor z oil film Table. Summar of performed simulations. Analsis: stead-state transient Static unbalance: g mm, 45, g mm Speed: const. const. const. angular acc. Min. speed: rpm 8 rpm Ma. speed: 9 rpm 4 rpm Simulation length:.5 s 5, 3, 6, s 3

4 MATEC Web of Conferences 48, 43 (8 ICoEV 7 a Relative eccentricit stead-state analsis transient analsis b Relative rotating displacement Figure 3. Results of brute force bifurcation. Depicted are a local etremes of stead-state responses and b values of relative rotating displacement ψ if relative rotating displacement χ is. Relative rotating displacements are defined in Eqs. ( and (. Fig. 3b can be also perceived as the intersection of rotating trajectories with nψ plane. The intersection is highlighted in fig. 4b. a b Fied coordinates Rotating coordinates Rel. displacement Rel. displacement ψ rpm χ rpm - - χ rpm - - χ rpm - - χ rpm - - χ c Spectra Rel. displacement Order Order Order Order Order Figure 4. Trajectories (obtained from stead-state analsis of the rotor at selected speeds are depicted in a fied coordinates and b rotating coordinates. Black dots show the position of the rotor at the beginning of each turn. There are also shown responses in the frequenc domain in fig. 4c. 5 Results and discussion The results of the aforementioned stead-state and transient analsis are presented and discussed in this section. 5. Stead-state analsis Fig. 3a shows the results of the stead-state response analsis, more precisel there are plotted local maima and minima of relative eccentricit ε. We assume that depicted trajectories are reasonabl close to trajectories in respective limit ccles. It is apparent that until ca. 35 rpm the sstem is working at its equilibrium. If the rotor speed further grows, the trajector will become elliptic as shown in fig. 4a. This elliptic motion is happening at a frequenc which is slightl lower than half of the rotor speed and might actuall be chaotic because significant broadband noise is present [9] 4

5 MATEC Web of Conferences 48, 43 (8 ICoEV 7 b.8 Relative eccentricit Relative eccentricit a.6.4 run-up s run-up 6 s run-up 3 s run-up 5 s stead-state analsis Ust =, runup s Ust =4, runup s. Ust =, runup s Ust =, stead-state Rel. displacement a Fied coordinates Figure 5. The results of transient analsis show that a the threshold speed for the fluid-induced instabilit is dependent on the angular acceleration of the sstem and b the static unbalance introduces new local etremes but almost does not influence the threshold speed rpm Rel. displacement ψ Rotating coordinates Order Order Order χ χ.5 χ.5 χ rpm.5 c Rel. displacement 75. rpm.5 χ Spectra 7.3 rpm.5 b rpm - - Order Order Figure 6. Trajectories (obtained from transient analsis of the balanced rotor at selected speeds through 4 turns are depicted in a fied coordinates and b rotating coordinates. Black dots show the position of the rotor at the beginning of each turn. There are also shown responses in the frequenc domain through 6 turns in fig. 6c. as can be seen in fig. 4c. It is safe to assume that around 35 rpm a threshold speed of the fluid-induced instabilit can be found. After 5 rpm mark is passed, the sstem comes into quasi-periodic motion with apparent response at the rotor speed and at even higher speed the motion is clearl limited b bearing clearance and the fluid-induced instabilit becomes full developed. We can stud the trajectories more rigorousl, if we transform them to a coordinate sstem (χ, ψ which rotates around z ais at the same speed as the rotor [9]. χ = χ(t and ψ = ψ(t are defined as follows χ = cos ϕ sin ϕ, ψ = sin ϕ + cos ϕ. ( ( We further refer to the transformed trajectories as rotating trajectories. Fig. 4b shows that there are three distinct tpes of the rotating trajectories in the zone delimited b the threshold speed of the instabilit and b the speed at which the instabilit is full-developed (we call this zone 5

6 MATEC Web of Conferences 48, 43 (8 ICoEV 7 the transient zone in the following tet. Firstl, there is almost circular trajector which shows properties of a bounded chaotic motion [9]. If the speed increases, the sstem comes into stable period doubling and tripling motion with less significant noise, possibl indicating quasiperiodic solution [3]. Finall, when the instabilit full develops, the rotating trajector is circular and the rotor covers roughl 54 in one ccle. All described phenomena are ehibited in the results of both stead-state and transient analsis with zero angular acceleration. However, all qualitative changes happen at higher speeds in the case of the transient analsis. 5. Transient analsis Fig. 5a shows how the threshold speed of the instabilit is influenced b angular acceleration of the rotor. Apparentl, if the angular acceleration is higher, the sstem will became unstable at higher speed. Fig. 5b shows that the influence of static unbalance on the threshold speed is rather small. Fig. 6 demonstrates there are three distinct tpes of rotating trajectories not onl if the rotor is spinning at constant speed but also if its speed is increasing. Note that trajectories and spectra depicted in fig. 6 are taken from the simulation of the perfectl balanced rotor with the lowest angular acceleration because this simulation offers the best frequenc resolution of frequenc analsis. Similar trajectories are to be found in all simulated cases. However, if the rotor is unbalanced, there is an additional snchronous (X component present in the response. 6 Conclusions This paper provided a brief et coherent theor for dnamic behaviour of rigid rotors supported b fluid-film bearings occurring in the so called transient zone delimited b the threshold speed of the fluid-induced instabilit and b the speed at which the instabilit is full-developed. There are three distinct tpes of the rotor motion in the transient zone. The first tpe appears as a bounded chaotic motion. Interestingl, later two tpes are less chaotic and tend to be quasi-periodic. This behaviour is presumabl caused b the fact that these motions are more limited b the geometr of the bearing. The later two tpes of motion are also characteristic b the response with snchronous (X components even if the rotor is perfectl balanced. These three tpes of motion can be find not onl if the rotor is analsed at constant speed but also if the angular speed is being constantl increased and if the rotor is unbalanced. Preliminar studies suggest that the presented conclusions are valid also for journals of fleible rotors which are supported b multiple journal bearings. However, to prove that such phenomena happen in general is left to future work. Acknowledgement This publication was supported b project 7-595S of Czech Science Foundation entitled Nonlinear dnamics of rotating sstems considering fluid film instabilities with the emphasis on local effects and b project SGS-6-38 of Universit of West Bohemia. The transient analsis was performed in the AVL Ecite software. The AVL Ecite software is available in the framework of the Universit Partnership Program of AVL List GmbH and its usage is greatl acknowledged. References [] B.L. Newkirk, H.D. Talor, General Electric Review, 8, (95 [] E. Krämer, Dnamics of Rotors and Foundations (Springer, Berlin, 993 [3] Y. Hori, Trans ASME Ser E, 6, (959 [4] A. Tondl, Some Problems of Rotor Dnamics (Chapman and Hall, London, 966 [5] M. Akkok, G. M. M. Ettles, ASLE Trans,, (98 [6] J.S. Mitchell, An Introduction to Machiner Analsis and Monitoring (Pennwell Publishing, Tulsa, 98 [7] D.E. Bentl, A. Musznska, NASA Conference Publication No. 5 (Teas A&M Univ., College Station, 98, 37-3 [8] O. Renolds, Proc R Soc Lond, 4, 9-3 (886 [9] D. Sfris, A. Chasalevris, Tribol Int, 55, ( [] G.B. DuBois, F.W. and Ocvirk, NACA Report No. 57 (NACA, 953, 99 6 [] Y. Bastani, M. de Queiroz, J Tribol, 3, ( [] F.K. Cho, M.J. Braun, Y. Hu, J Tribol, 3, (99 [3] L.-H. Yang, W.-M. Wang, S.-Q. Zhao, Y.-H. Sun, L. Yu, Appl Math Model, 38, (4 [4] L.R. Gero, C.M.M. Ettles, Tribol T, 9, 66-7 (986 [5] H.N. Chandrawat, R. Sinhasan, Wear, 9, (987 [6] A.T. Prata, R.T.S.S. Ferriera, D.E.B. Mile, M.G.D. Bortoli, International Compressor Engineering Conference (Purdue Universit, Purdue, 988, 34-4 [7] F.J. Profito, M. Giacopini, D.C. Zachariadis, D. Dini, Tribol Lett, 6, - (5 [8] J.Z. Zhang, W. Kang, Y. Liu, ASME J Comput Nonlinear Dn, 4, 7--9 (9 [9] A.D. Shaw, A.R. Champnes, M.I. Friswell, P Ro Soc Lond Ser-A, 47, (6 6