Numerical Simulation of Nonlinear Oil Film Forces of Tilting-Pad Guide Bearing in Large Hydro-unit

A new numerical method is proposed for predicting the nonlinearity of tilting-pad guide bearing oilfilm force in the rotor-bearing system in a large hydro-unit. Nonlinear displacement and velocity of the journal center, as well as nonlinear tilting angles and angular velocities of the pads in non-stationary Reynolds equation are taken into account. This method is also suited for other small rotor-bearing system. As an example, the response due to a momentarily created unbalance is Calculated. The nonlinear motion patterns of the pad and journal whirling orbit are obtained. Finally, the nonlinear orbit is compared to the linear one that could be calculated from linear stiffness and damping coefficients. It is shown that there are important differences between those two orbits and that the nonlinear simulation is more accurate.


INTRODUCTION
With increasing size and performance of hydro-unit for power generation plants, knowledge of their behavior becomes more and more important.Basically, dynamic stability of the large hydro-unit is currently receiving a great deal of attention in literature since it influences the hydro-unit normal running directly.For example, in the Three Gorges Project, each hydro-unit's rated capacity reaches 700 MW, and its runner and guide bearing diameter are 10 and 3 m respectively.Especially, in order to determine design parameters (such as the number of guide bearings, radial clearance, supporting stiff- ness and journal's size, etc.), moreover, to evaluate the external exciting forces and deliver control tactics, it is indispensable to analyze the rotor system dynamic behavior accurately during the unit's design stage.Therefore, the dynamic behav- ior of rotor system is very important.
The large hydro-unit with a vertical arrangement is modeled in Fig. l(a).From dynamics point of view, rotor-bearing system of such machinery con- sists of shaft, generator rotor, water turbine, thrust bearings and seal rings, etc...For the sake of safety of the hydro-unit running, the critical speed, natural Corresponding author.Tel." (029)3268130.E-mail: e974511 @bull2.xjtu.edu.cn.frequency, and dynamic response of rotor system in lateral vibration would be important.All these respects have a close relation with the oil film dynamic behavior.The guide bearing, which serves as a support in radial direction, is not loaded static external force in radial direction in such vertical arrangement machinery, and the journal center should coincide with the bearing center in theory.It is well known that the stiffness and damping coefficients of the bearing perhaps are very small under previous conditions especially in the small pre-load bearing.So that, the displacement and velocity response of rotor system are perhaps comparable large as external excitation force is small.Sometime the Francis type hydraulic turbine would suffer a large lateral hydraulic excitation force when it is running on part load.This excitation force would make the center ofjournal whirl with a large amplitude.The linear theory is unsuitable for this situation.Consequently, we should look guide bearing film force as nonlinear function of journal center's displacement and velocity, pads' swing angles and angular velocities.

THEORETICAL ANALYSIS
Methods to Obtain Nonlinear Film Force (a) Approximate method (Furkawa et al., 1994;  Wang and Zhang, 1993).Reynolds equation is reduced by using long or short bearing assumption or using nonlinear coefficients that are defined as linear oil film model.
The database of non-stationary oil film force of bush segment has been set up and used to provide bush segment non-stationary oil film forces for assembling up the bearing nonlinear forces.
The first method reduced so much that it could not be satisfied with the need of the more accurate nonlinear analysis although its computation speed is fast.Both the accuracy and speed of the second method are better, but it only suits the fixed multilobe bearing at present.The accuracy and suitability of the third method are good, but it requires solving Reynolds equation many times and consumes large CPU-time in every time step.For this reason, the former simulations are only confined to the cylindrical journal bearing.However, in a long run, with the computer's hardware speed raising and software's functions improving, the shortcoming of numerical simulation will be dismissed gradually.The computer simulation process of nonlinear oil film force of tilting-pad guide bearing in large hydro-unit is presented in this paper.
The Rotor's Kinetic Equation Figure (b) is examined and the coordinate system used.It consists of a massless rigid shaft carrying a disk with mass (2m) at the middle of the span, and two identical tilting-pad bearings supported at both ends of the shaft respectively.The external lateral excitation force 2f(t) acts on the middle of the span, andf(t) -fx(t)[+fy(t)ff(t)is a known function of time.The equations of the journal center motion  can be described as: m2--Fx +fx (2.1) mj: -F +fy where, 2 and j are accelerations ofjournal center in x-and y-directions respectively.Fx, Fy are the film forces in x-and y-directions, which can be obtained by integration of the oil film pressure: where, p is film distributing pressure on the pad, L is pad length, R is pad radius, g) and z are coordinates in circumferential direction and axial direction, Fx and F are the functions of x, 2, y, , 5;, }i, i.e., Fx Fx (x, 2, y, , Si, i) Fy Fy (x, 2, y, J), 5i, }i). (2.3)

Reynolds Equation
In order to calculate the journal bearing force, the distribution pressure p on the pad has to be determined, while p must satisfy non-stationary Reynolds equation and Reynolds boundary condi- tion (2.4).Due to its complexity, the Reynolds equation is usually solved numerically.Here the finite element method (FEM) is used in this work.A large number of grid nodes is necessary to get an accurate solution.

(2.4)
Because the journal diameter is very large, journal speed is low, and radius clearance is small, the Reynolds number of fluid in the clearance is much lower than its critical value.So that, Eq. (2.4) is laminary flow equation assumed.Non-dimension non-stationary Reynolds equation can be deduced as (2.4b) and coordinate system is used as Fig.
(c): Since c _< 1, ff)min (10-3-10-4), min, 0 does not exceed 10 and is not too large in common, so that: The Pad's Kinetic Equation Due to each tilting-pad is allowed to pitch about its pivot point, the pads' motion equation can be gained as follows: Joi M 1,2,3,...,N), (2.5) where, J0 is pad moment of inertia, M; is the moment of the fluid film pressure upon the pad about pivot point.Because J0 can be quite small, Joi could be neglected, and (2.5) can be rewritten as: m 0 (i 1,2, 3,..., N). (2.6) The Process of Nonlinear Oil-film Force Computer Simulation Equation (2.6) can be used to determine the tilting- pad' swing angles.The purpose of nonlinear film force simulation is to find the nonlinear function in Eq. (2.3) in numerical form.The process of simu- lation is to resolve the simultaneous Eqs.(2.1), (2.4) and (2.6) in turn.
Selecting proper initial condition is one key step in the simulation process.It is assumed that the initial position of the journal center is always in its static equilibrium position.Thus, the initial film force (Fxo, F,o), the initial position of the jour- nal center (xo, Yo), and initial pad' angles 5i0 (i=1,2,3,...,N), can be obtained by solving the static Reynolds Eq. (2.4) and pad balanced Eq. (2.6).It should be pointed out that the journal center moving velocity and each pad's swing velocity are zero under these circumstances.Now, all the initial conditions of Eqs.(2.1) and (2.4) can be written as: Fx Fxo, Fy Fy o, (2.7) x--xo, Y.--Yo, 2--.9--0, 6i 6i0, 6i0 O. Equation (2.7) could express the initial conditions while the journal center equilibrium position is disturbed abruptly by an impact load, such as earthquake, magnetic pull failure or sudden partial mass failures.
The steps of simulation are presented as: (a) Obtain the initial conditions by resolving static Reynolds equation and equilibrium equations of pads.
According to the former simulation course, we have completed the nonlinear film force calculation program.The diagram of calculation procedure is shown in Fig. 2. A systematic study shows that it is necessary to set a convergence (IIAS/SII) criterion of 10 -4 for motion varies.criterion, numerical instabilities appear and con- vergence of the S is not achieved; using more severe convergence criteria provides no significant improvement in the final results, but the CPU-time greatly increase.

NUMERICAL EXAMPLES
In order to investigate the relationship between nonlinear film forces (Fx, Fy) and variables x, 2, y,f,(Si,i, the water turbine guide bearing and generator guide bearing of a 240 MW hydro-unit have been simulated respectively.The turbine guide bearing has 10 shoes (v= 1) and the generator guide bearing has 12 shoes (v= 119.4).They represent two kinds of typical tilting-pad bearings in large Francis hydro-unit.For the sake of the simplicity and convenience, we take a momentarily created unbalanced mass force as excitation force: f -fx +fyj, M Ecv: (sin cot [+ cos catf), (3.1) where, E is equivalent mass eccentricity, co is angle velocity of the journal.Substitution of Eq. (3.1)into (2.1), obtains (--ax / sin , Y---ay -+-cos -, where, The numerical time-integration of the motion of the journal center and calculation of the film force are performed according to Eqs. (3.2) and (2.4b) res- pectively.The parameter values of bearings are presented in Tables I and II.
Furthermore, a comparison of the nonlinear simulation with a linear one is carried out under the same operating conditions.In the linear simula- tion, the dynamic coefficients of bearing are calculated at the center of the bearing and listed in Table III.)min =2.12 10-4, )v l19.44,L/D 0.235, --0.5, M 945.9 t, #o 0.0045 kg.s/m2, a)= 68.18 rpm, D 1700mm, = 4.9, c=0.18mm.

RESULTS AND DISCUSSION
1. Pre-load (bv) influences the hydro-unit's behav- ior largely.In Figs.3(a) and 4(a), it is shown that the pre-load (bv) of bearing in- fluence on the swing angle and velocity of the pad largely.The generator guide bearing's pad swing angle-time (5-T) curve is obviously dif- ferent from that of the water turbine bearing's.
ZE-5 rad/d (a)   b)   are curves that show that the oil film force of single pad varies with time respectively.It is very interesting that the two curve forms are absolutely different for different preloads.
Figure 6 shows the nonlinear whirling orbit of journal center of turbine guide bearing (b,, 1).
FIGURE 6 The nonlinear whirling orbit of the water guide bearing (% 1).

So FIGURE 7
The resultant force versus time (S0---T) curve of the water guide bearing ('(or 1).
The resultant of film force of each bearing is also presented, in Figs. 7(v= 1) and 8 (bv= 119.4).It can be seen that their conver- gence trends are different.Difference between the results, which are calcu- lated by linear and nonlinear method respec- tively, is large.It should be noted that all the dynamic coefficients are equal to zero at the point of center of the turbine bearing (b 1) in theory.Therefore, even as the excitation force on the journal is small, the displacement and velocity of journal center will be comparatively large, and linearization may lead to errors.For generator guide bearing (b= 119.4), the linear and nonlinear whirling orbits are almost iden- tical perfectly when a small force is imposed on the journal.(see Figs. 9(a) and 9(b)).But, as the  force amplitude increases to a value big enough to make the radius of nonlinear whirling orbit reach to about 0.75, there is a distinct difference between these two.The linear result is wrong visibly (see Fig. 9(c)).Of course, the dynamic responses computed by nonlinear simulation appear to be more accurate and realistic.
However, the nonlinear simulations spend more CPU-time than the linear.
3. In the other point of view, when the external force is small, the linear and nonlinear whirling orbit are almost equal.When the external force is larger, the linear and nonlinear whirling orbit become larger, but the linear is larger than nonlinear.All these results proved that the nonlinear results are correct reasonably.

CONCLUSIONS
1. On the basis of the non-stationary Reynolds equation and nonlinear theory, the paper inves- tigates the nonlinear oil film forces of tiltingpad guide bearing in large hydro-unit.There is difference between the tilting-pad guide bear- ings' oil film forces that can be calculated by linear and nonlinear methods.It is important to analyze the bearing when pre-load is small (especially, to pre-load bv= l) by nonlinear method.
2. The pre-load (/:v 1) of the tilting-pad bearing is a crucial factor which influence the hydro-unit largely.3.At present, for the nonlinear film force theory is not perfect, in order to deal with the nonlinear film force of tilting guide bearing in actual engineering, the numerical simulation method is an effective way.
FIGURE l(a)The model of hydro-unit.
FIGURE l(b) The model of the rotor.
FIGURE l(c)The geometrical relation between the rotor and one pad.

FIGURE 2
FIGURE 2The diagram of calculation procedure.
FIGURE 3(a) One of the water turbine guide bearing's pad swing angle versus time curve ((Si-T).

FIGURE 5
FIGURE 5 Display of the result of simulation.

FIGURE 8
FIGURE 8The resultant force versus time (SO-T) curve of the generator guide bearing(v= 119.4).

FIGURE 9
FIGURE 9(c) The linear and nonlinear whirling orbit of the generator guide bearing (= 119.4) under same larger exter- nal force.
angle i-1,2, 3,..., N (N: total number of pads)E EN NE ER RG GY Y M MA AT TE ER RI IA AL LS S Materials Science & Engineering for Energy SystemsEconomic and environmental factors are creating ever greater pressures for the efficient generation, transmission and use of energy.Materials developments are crucial to progress in all these areas: to innovation in design; to extending lifetime and maintenance intervals; and to successful operation in more demanding environments.Drawing together the broad community with interests in these areas, Energy Materials addresses materials needs in future energy generation, transmission, utilisation, conservation and storage.The journal covers thermal generation and gas turbines; renewable power (wind, wave, tidal, hydro, solar and geothermal); fuel cells (low and high temperature); materials issues relevant to biomass and biotechnology; nuclear power generation (fission and fusion); hydrogen generation and storage in the context of the 'hydrogen economy'; and the transmission and storage of the energy produced.As well as publishing high-quality peer-reviewed research, Energy Materials promotes discussion of issues common to all sectors, through commissioned reviews and commentaries.The journal includes coverage of energy economics and policy, and broader social issues, since the political and legislative context influence research and investment decisions.S SU UB BS SC CR RI IP PT TI IO ON N I IN NF FO OR RM MA AT TI IO ON N Volume 1 (2006), 4 issues per yearPrint ISSN: 1748-9237 Online ISSN: 1748-9245 Individual rate: £76.00/US$141.00Institutional rate: £235.00/US$435.00Online-only institutional rate: £199.00/US$367.00For special IOM 3 member rates please email s su ub bs sc cr ri ip pt ti io on ns s@ @m ma an ne ey y. .cco o. .uuk k E ED DI IT TO OR RS S D Dr r F Fu uj ji io o A Ab be e NIMS, Japan D Dr r J Jo oh hn n H Ha al ld d, IPL-MPT, Technical University of Denmark, Denmark D Dr r R R V Vi is sw wa an na at th ha an n, EPRI, USA F Fo or r f fu ur rt th he er r i in nf fo or rm ma at ti io on n p pl le ea as se e c co on nt ta ac ct t: : Maney Publishing UK Tel: +44 (0)113 249 7481 Fax: +44 (0)113 248 6983 Email: subscriptions@maney.co.uk or Maney Publishing North America Tel (toll free): 866 297 5154 Fax: 617 354 6875 Email: maney@maneyusa.comFor further information or to subscribe online please visit w ww ww w. .mma an ne ey y. .cco o. .uuk k C CA AL LL L F FO OR R P PA AP PE ER RS S Contributions to the journal should be submitted online at http://ema.edmgr.comTo view the Notes for Contributors please visit: www.maney.co.uk/journals/notes/emaUpon publication in 2006, this journal will be available via the Ingenta Connect journals service.To view free sample content online visit: w ww ww w. .i in ng ge en nt ta ac co on nn ne ec ct t. .cco om m/ /c co on nt te en nt t/ /m ma an ne ey y