- Email: [email protected]
Dynamic behaviour of elastic shaft supported by hydrodynamic bearings
Tribology Research: From Model Experiment to Industrial Problem G. Dalmaz et al. (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
DYNAMIC BEHAVIOUR OF ELASTIC SHAFT SUPPORTED BY HYDRODYNAMIC BEARINGS
Jean FRENE and Olivier BONNEAU Laboratoire de M6canique des Solides-UMR CNRS 6610, Universit6 de Poitiers, SP2MI, Bd Pierre et Marie Curie, T616port 2, BP 30179- 86960 Futuroscope - Chasseneuil Cedex High speed rotors present a lot of stability problems especially when the speed of rotation passes through a critical speed. The non-linear dynamic behaviour of the fluid bearings has, in that case, an important effect. This study presents several models of bearing dynamic behaviour taking into account the flexibility of the shaft: Squeeze film damper (SFD) behaviour is described. This element gives a lot of damping but its behaviour is totally non-linear. The coupling between the axial thrust bearing behaviour and the bending vibrations of the shaft is also studied.
1. I N T R O D U C T I O N In the turbomachinery field the designer has to model the shaft with high accuracy and has to take into account all the shaft surroundings. A lot of disciplines are concerned by this study: structural mechanics, acoustics, fluid mechanics, heat transfer, lubrication,... The difficulty, but also the challenge of the rotor study, comes from this diversity of subjects and from the choice of different models. Until a few years ago, each discipline was developing its own models: - very detailed shaft (finite element models) mounted in a simple support (with linear stiffness and damping coefficients) for structure mechanics approach; - very sophisticated bearing models (non-linear) but with simple shafts (rigid) in lubrication applications. This earlier work gives useful results for a first level of approach but these are inadequate when the mechanism works in severe conditions (near the critical speed for example). In such conditions it is necessary to analyze the coupled behaviour. The work presented here [1 to 5] is a synthesis of several studies done with coupled models. These models take into account shaft flexibility in fluid surroundings. Non linear shaft support behaviour will be specially emphasized. In the first part of the paper one describes bearing dynamic models (both linear and non-
linear models) and rotor dynamic models. The coupling between fluid element and the shaft is done by a non linear approach: the simulation of the non linear dynamic behaviour of the flexible rotor is a time step by step approach done in the shaft modal basis. In the second part of the paper some applications of this dynamic model are detailed: - Results obtained for a flexible shaft mounted in a squeeze film damper (SFD) have been chosen; this example is very interesting because of the high non linear behaviour of the fluid element. Numerical and experimental results are presented and the comparison shows the good accuracy of the numerical model. - The non-linear coupling between axial motion and the bending vibrations of the shaft is also presented. This coupled behaviour is due to the thrust bearing effects and to the shaft flexibility.
2. F L U I D
ELEMENTS
(FLUID
BEARINGS,
ANNULAR SEALS) 2.1. Presentation
Shafts are generally supported by two kinds of elements: ball bearings and (or) fluid bearings. The ball bearings give near-linear behaviour and this technology is not studied in this work. Fluid bearings have a complex behaviour; linearisation can be carried out but the behaviour is
sometimes quite non linear. Three kinds of fluid bearings are studied here: the traditional bearing, the squeeze film damper and the axial thrust bearing. Annular seals may also have an important effect on shaft behaviour [6 to 10]. The problems induced by these components are related to turbulent flow and inertia effect in the fluid which are due to the low fluid viscosity (in liquid hydrogen turbopump for example).
In the case of low viscosity fluids, for a large clearance or when rotation speed is high the flow could become turbulent. Then two turbulent viscosity coefficients are introduced. They are a function of turbulent parameters: local Reynolds number, axial and circumferential pressure gradient and roughness. Some turbulent flow models have been applied [6 to 9 ] to calculate turbulent viscosity coefficients.
.......----2.2. Fluid film modelling The characterization of fluid elements requires solution of the Navier-Stokes equations. This solution gives the values of the pressure field in the film. Two cases appear: - The film thickness is small compared to the other dimensions, thus this condition allows us to simplify the problem. An elliptic differential equation: the Reynolds equation is obtained. -It is impossible to neglect one dimension in relation to the others so it is necessary to solve the complete Navier-Stokes 3D systems. This approach is necessary for example in a labyrinth seal with very deep grooves. This aspect will not be described in this study. Arghir [10] developed numerical models adapted to this sort of seal geometry. 2.2.1. Lubrication equations We presem in this part the model adopted for journal bearing; the axial thrust bearing solution is similar [ 11 ] and we do not present it here. The assumption of a thin film thickness allows us to write the Reynolds equation as follows:
1 'E
h3
R2 8~ * /a 9
ha kxSO* 9
O)
= Xcos 0 +Ysin 0 + - - ~
- ~ ,u
kz ~
=
6;tl
200* Where: R and L are respectively the radius and the length of the bearing, h is the film thickness, P is the pressure field, co is the angular shaft velocity, ~t is the dynamic viscosity of the fluid, 0* and z are respectively the circumferential and axial parameter. All the different parameters are shown in Figure 1.
I
O"
Figure 1. Film characteristics. To solve the Reynolds equation, different boundary conditions should be applied: - Feed boundary conditions (feed pressure in holes, external pressure...) - Flow conditions, which characterize rupture in the film.: -Sommerfeld conditions: no rupture of the film (2 n film).
-Gtimbel solution: rupture when numerical pressure is negative (n film). -Reynolds conditions: the pressure and its derivative are nil when rupture of the film occurs. -Film reformation conditions: the pressure and its derivative are nil when rupture of the film occurs and the flow through the inactive part of the film is taking into account for the film reformation. Then the Reynolds equation is integrated using one of the following methods:
- Short bearing theory: the circumferential pressure variation effect is neglected. The fluid film force is formulated analytically and computing time is small. - Finite length bearing theory. In spite of its accuracy this solution implies heavy numerical calculations which are not suitable for the application discussed here. - Unidimensional theory. This solution, inspired from Rhode and Li [12], is based on an axial parabolic pressure assumption. A variational calculus implies a new equation of Reynolds where only one space variable 0 appears. The gain in time is important compared to the finite length model. These methods have been tested and their accuracies depend on the problem geometry. Integration of the pressure film gives fluid film forces on the rotor. Fx= fY2 ~2 ~~ P(0, z)cos 0Rd0dz
of the fluid load, Ax and Ay are small displacements near the static equilibrium and A:~ and AS' are small velocity variations. Fy (x o + Ax, Yo + Ay, A/~,A;r = Fy (Xo, Yo,0,0) + + Ax/CTy/+ Ay/--~)+
t,
-I-axC~ t~
Jo
A:~(~
o
t
A~,(---~)
)o
o
- Ay[-~--)-I- .... 0
Then a first order development is carried out: + Ax, Y0 + Ay, Ai, A~,,Aii, A~,) - Fx (0)~
?y(Xo+ Ax, Y0 + Ay, A:~,A~,,Aii, A~,)- F ( 0 ) J A:~ A~ (ff;} = -[aijIAA;}- [bijIA~} - [mijIA~}
Fy=~V2~ ~t P(0, z)sin 0Rd0dz where fx and fy are the additional loads created by
2.2.2. Dynamic behaviour The fluid element influence on rotor dynamics is characterized by the hydrodynamic forces generated by the pressure field. These forces are non-linear forces of the position and velocity of the shaft center in the bearing. In general the shaft is submitted to a combination of several forces: a static load (weight, belt tension,...) a dynamic load (unbalance, shaft vibration) The static load implies an equilibrium position (stable or unstable) and the dynamic load creates an orbital trajectory of the shaft. Depending on the conditions, two approaches can be carried out: a linear approach and a non-linear one. -
-
2.2.3. Linear theory The aim of this approach is to linearize the bearing behaviour around an equilibrium state. This theory is based on a small displacement hypothesis. Thus, close to a static equilibrium state, a first order development may be done. In these expressions F x and Fy are the components in the (.2, ~) reference
displacement and velocities, aij ' bij and mij are the stiffness, damping and added mass coefficients. The last coefficients (added mass) exist only for inertia flow. T w o numerical approaches may be used to calculate these coefficients: - a perturbation method[ 13] - a numerical differentiation [ 14 to 16]. These dynamic coefficients may be used to determine the linear stability of a rigid shaft supported by two identical bearings or to estimate the dynamic response of a shaft (to obtain critical speeds for example). However this model is false when the amplitude is large or when the equilibrium is unstable, it is necessary then to obtain a non linear solution. 2.2.4. Non linear model This approach is a numerical study of the equations of motion which are integrated by a step by step method: at each step, the Reynolds equation is solved to evaluate the film forces, then the fundamental principle of mechanics is integrated to obtain speeds and positions of the next step.
3. R O T O R M O D E L L I N G
obtain an acceptable precision. A change of variable is done in the following form: {8}= [g]{q} where
Two models could be considered, first the shaft is considered rigid and symmetrical (in geometry and in loading). It could be compared to a mass mounted in a fluid bearing. Then the equations of motion are simple (two degrees of freedom). This model is valid when the speed of rotation is far from a critical speed. This simple model is useful for a first approach but in this study a more advanced model is presented which concerns an elastic shaft. The rotor is modeled with typical beam finite elements including gyroscopic effects [1]. The differential system, with {8} the node displacement vector is the following:
[r is the matrix obtained with modal vectors. {gi }, and {q} is the modal vector of displacement. A new system is written where the bearing effect appears in the modal stiffness matrix. [m]{~} + [c]{cl}+ [k']{q} = {f} with : [m] :
[v/]t
[k'] = [gz]t[ K k ]
[c]= [v/]t[CIr {f} = [~]t {F}
To do a non linear calculus the linear bearing [Kbearing] should be subtracted from modal stiffness [k'] and should be introduced as a non linear effect. Then
effect Where [M]:is the mass matrix, [C] is the gyroscopic matrix, [K] is the stiffness matrix {Funb} are the unbalance forces, {Fr~} are the non linear bearing
[m]{~} + [c]{ft} + [k]{q} = {f} + {Fnl}
forces and {Fg~/ are the gravity forces. The damping
with [k] = [k'l- [I]/]t [kbearings I~/]
does not appear explicitly but it exists in the {Fr~} terms as a non-linear fluid damping. This system has 4(n+ 1) degrees of freedom (n is the number of nodes). The iterative non-linear calculation will be very expensive in computer time. To reduce the degrees of freedom a modal approach is used [ 17-18]. Two approaches are available to obtain modal base: -to take a free motion shaft but then it is important to conserve a lot of modes. -to take a judicious equivalent bearing stiffness. This stiffness is not the real one (because an uncentralised squeeze film damper has no linear stiffness). It is a symmetrical stiffness and its value leads to a representative modal base. Then this stiffness may be subtracted from the modal stiffness matrix. Then the undamped fundamental principle of mechanics on the rotor could be written:
[M]{r }q- ([Kr ] q-[Kbearings]){8} = 0 The solution has the following form: {8}=-{80 }ert and the first 6 modes are calculated. Lacroix [ 17] has shown that 6 modes are enough to
{Fnl} is the modal non linear force obtained, at each time step (see 4-1), by a modal basis change of the real non linear force. These real non-linear forces are calculated in the physical coordinates.
4. FLUID STRUCTURE COUPLING The coupling between fluid element and the shaft is done by a non-linear approach. The simulation of the non-linear dynamic behaviour of the flexible rotor is a step by step approach done in the modal basis. The flow chart is as follows: a) Beginning with initial values of modal positions and velocities. b) Calculus of external modal forces. c) Calculus of physical displacements and velocities in bearings (by modal basis change). d) Non linear bearings forces calculus (in the real basis). e) Computation of all forces in the modal basis (by modal basis change). f) Modal acceleration computation. g) Time integration (variable step Euler method). h) Shaft speed is incremented. i) The process begins again in b.
4.1. Flexible shaft on a Squeeze Film Damper 4.1.1. Squeeze Film Damper (SFD) Results obtained for a flexible shaft mounted in a squeeze film damper (SFD) have been chosen, this example is very interesting because of the high non-linear behaviour of this kind of fluid element. The basic idea for this kind of bearing is to support a ball bearing in a fluid bearing (figure 2). The rotation is ensured by the ball bearing and the oil film is squeezed between two non rotating rings (the external ring of the ball bearing is jammed in rotation). Two technologies for SFD are possible. A first kind of assembly consists in mounting, in parallel with the SFD centering springs [19-20]. The second technology is without centering springs then some stiffness problems appear: the creation of a significant pressure field (to support a static load) being dependent on a threshold of minimum perturbations. The results presented here are obtained only with this second technology.
\~o,
Bearing span: 0.8 rn, Rotor diameter 0.06 rn, Bearing length: 0.015m, Diameter: 0.09 rn, Radial clearance 0.05 mm. 4.1.3. Results A test rig was developed by Kassai at the National Institute of Applied Sciences [ 1 & 19]. The shaft displacements are measured by eddy current proximity probes. Numerical and experimental results are obtained for a linear rotation speed variation from 8500 rpm to 13500 rpm during 22 s. Numerical results obtained by short bearing theory and experimental results are shown in figure 4a and 4b (oil temperature = 25~ ~t=0.05 Pa.s) and in figure 5a and 5b (oil temperature = 60~ ~t=0.0135 Pa.s). These figures show the displacement amplitude of the shaft in the middle of the rotor (where the dynamic amplitude is maximum) versus rotational speed. Amplitude (ram)
O.4 t 0.3
0.2
0.1 I I I
Figure 2 - Scheme of a Squeeze Film Damper. 4.1.2. Shaft The shaft is a rotating flexible shaft supported at one end by a ball bearing and at the other end by an active squeeze film damper without a centralizing spring (figure 3).
o
I
9000
'
I
10000
'
I
11000
'
I
12000
'
I
13000
'
speed of rotation (rpm)
Figure 4a. Experimental results, ~t = 0.05 Pa.s.
Figure 3. Shaft representation. Results are obtained for a linear rotation speed variation from 7000 to 15000 rpm. The shaft and bearing characteristics are the following:
Figure 4b. Numerical results, ~t = 0.05 Pa.s.
Amplitude (mm)
The same conclusion has been obtained concerning the effect of the radial clearance (a high radial clearance is better closed to its critical speed and a low radial clearance is necessary for speeds different from the critical speed [3].
0.4u
5. ACTIVE SQUEEZE FILM DAMPER An active squeeze film damper can be chosen to present an optimal value of clearance or viscosity.
0 I
I
I
I
I
9000
10000
11000
12000
13000
speed of rotation (rpm)
Figure 5a. Experimental results, g = 0.0135 Pa.s.
5.1. Variable clearance
To perform the control, the active squeeze film damper has been modeled. The idea is to regulate the radial clearance by a parameter x corresponding to the position of a conical squeeze film damper in its housing (figure 6).
Amplitude (mm) 0Au
0.2
I 9000
'
I 10000
'
I 11000
'
I 12000
'
I 13000
' speed of rotation (rpm)
Figure 5 b. Numerical results, ~t = 0.0135 Pa.s. The comparison between experimental and numerical results is good. There is a critical speed around 11800 rpm for la=0.05 Pa.s and around 11500 rpm for ~t=0.0135 Pa.s. The experimental and numerical amplitudes are very similar. Comparison between figure 4a (respectively 4b) and 5a (respectively 5b) shows that the decrease of viscosity (which could be due to an increase in oil temperature) gives an increase of damping. In fact low viscosity leads to more displacement in the bearing and consequently higher energy dissipation may occur. The conclusion of these first results is the following: the influence of the squeeze film damper is very important and its effect is very different depending on the rotor speed. When the rotational speed is close to a critical speed, the squeeze film damper must dissipate a lot of energy, consequently the viscosity must be small. Conversely a large viscosity is better for speeds different to the critical speed (in that case the amplitude in the SFD is decreasing).
Figure 6. Active squeeze film damper. Some interesting work has been published [2223] on conical squeeze film dampers. The dynamic study is carried out with a simple shaft (symmetrical, centered, with one stiffness) and with a synchronous circular whirl. In our work the shaft is modelled with these 6 first modes and the trajectory can be of any shape. The numerical results are obtained with the same shaft as previously. The idea is to monitor the radial clearance by the rotational speed, the clearance evolution is a linear evolution from 0.05 mm to 0.1 mlrl.
Figure 7 presents the amplitude of the middle of the rotor and figure 8 gives the displacement in the squeeze film damper for 0.05 nma, 0.1 mm and for a linear evolution. It is remarkable (figure 7) that the shaft amplitude is filtered around the critical speed.
0.3
C=O05mm
0.2
C lin
7000Rpm
15000Rpn
Figure 7. Amplitude of the middle of the rotor for linear clearance variation.
0.03mI
[///~
0.02mr
tension required to obtain the ER effect, the total power required is low. The current never exceeds a few microamperes. Thus, ER fluids are able to provide a good interface between a mechanical device and an electric control system. Moreover, the speed of response could allow the realization of mechanical devices actively controlled by "electronic management". The idea is to feed the SFD with a negative ER Fluid and to apply an electric field between two electrodes. Figure 9 presents a scheme of this SFD. A negative ER Fluid was chosen for its Newtonian behaviour.
)
~:
0.01 mr
7000Rpm
15000Rpn"
Figure 8. Amplitude in the squeeze film damper for linear clearance variation. Figure 8 shows that when the speed is very different to the critical speed it is better to have a small clearance (to limit SFD amplitude). Near the critical speed, a large clearance in the squeeze film damper permits large displacements to dissipate energy. 5.2. Variable viscosity Electrorheological fluids, called ER fluids, are types of semi-conductor liquids [24 to 30]. They are concentrated suspensions of solid particles in an oil base liquid. Normally, they behave like newtonian oil, but when they are exposed to a strong electric field, they seem to "coagulate". An increase of the frictional forces of the fluid on the wall appears. So that "coagulation" induces a change in apparent viscosity of the fluid. This change is gradual, reversible and proportional to the applied electric field. It occurs very rapidly. The time lapse is very short, about 1 millisecond. In spite of the high
Figure 9. Electrorheological Squeeze Film Damper. Results obtained with this kind of active SFD [4] are close to results obtained with variable clearance, all the simulation shows that it could be possible to monitor the damping of the squeeze film damper.
6. COUPLING BY THRUST BEARING BETWEEN AXIAL VIBRATIONS AND BENDING VIBRATIONS. The influence of thrust bearing on the shaft dynamic behaviour, and more especially the effect on lateral shaft vibrations has been studied very little. Some work was done on the analysis of the dynamic characteristics of hydrodynamic thrust bearings and their action on axial motion [31 to 35]. The influence of thrust beating on lateral shaft vibrations has been studied by P.L. Jiang and L. Yu
10
[34]. They have shown that as the film thickness of the thrust bearing becomes smaller, then its action becomes more important and it may prevent the shaft from deflecting and, the first critical speed and the threshold speed increase. For the hydrodynamic thrust bearing case N. Mittwollen et al [35] have shown that both decrease in the total axial clearance and increase in the axial thrust can lead to a significant increase in the whirl amplitude at the first critical speed and in the threshold speed of instability. In these studies, influence of the hydrodynamic thrust bearing on the shaft dynamic behaviour has been assumed to be linear. Berger et al [11] have studied the effect of thrust bearing on a flexible shaft. The dynamic behaviour of the thrust has been assumed to be nonlinear. The authors have shown that a defect in the thrust bearing rotor can excite a critical speed. The aim of the work presented now is to study the non-linear influence of the thrust bearing on the dynamic behaviour of a flexible shaft. In particular, the coupling between the axial motion and the bending vibrations of the shaft is analyzed.
Rotor
.02
Stator
!
Zl
I
! ". ,,. ,,,
! !
'
t
Y0
Figure 10 : Thrust bearing scheme
6.1. Thrust bearing model The thin film equation is defined (Fig. 10) in the cylindrical reference (o0,r,t,z0) by: 8
Figure 11 shows the pressure field obtained in the thrust bearing, at a constant speed.
(h 3 Op) =
dr dh
dh
e rSH 8Hx - 12/a rU 2 -,--1 _ 12/a V2 dr 30
+ 6/~ hr
dU 2 dU 2 +6/~h + 6/a hU 2 + 12/~ rW 2 dr dO
Pad Pad 1
where p is the pressure field, h=H2-H1 is the film thickness, /~ is the dynamic viscosity and U2,V2,W 2 are the velocity components of the points ME of the rotor. The equation is solved by finite difference with the Gauss Seidel method. The boundary conditions used are as follows: Feed pressure in each groove (atmospheric pressure) and Reynolds boundary conditions.
Figure 11. Pressure field with thrust bearing The force components and the moment components are calculated by integration of the pressure field on the surface $2 of the rotor.
S2
$2
11
applied on the shaft. On the middle of the rotor, a static load is used in order to stabilize it. The rotational speed increases from 4000 rpm to 20000 rpm in 15 s.
6.2. Bearing and shaft models
In order to calculate the bearing forces, the Reynolds equation in the film should be solved by numerical methods. But the non-linear analysis of the elastic shaft implies heavy numerical calculation. In order to decrease the computing time one has opted to use the short bearing approximation. This assumption which allows simplification of the differential equation by neglecting the circumferential pressure gradient in the film. The force components are obtained by integration of the pressure field with the following boundary conditions: Feed pressure on both sides of the bearing (atmospheric pressure) and Giirnbel assumption (n film) i.e. negative pressures are set to zero. It should be noted that the use of f'mite length bearing solution leads to the same type of results. The elastic shaft is modeled as described in w using modal method of reduction.
h~
0.02m
0.76m
0.02m
Thrust
~adn.g.ll
Beating 1
"
0.06 rn
l
Bearing 2
Figure 12. Geometrical data of the shaft Bearing data is as follows: The length L=0.015 rn, the radius R=0.03 m, the radial clearance C=3.0E-05rn, the dynamic viscosity ~t=0.05 Pas. The thrust bearing has four pads. The specifications of each pad are the angle 13=80, the outer radius R2=0.04 m, the inner radius R~=0.02 rn, the difference between entry thickness and the exit thickness is 20 ~tm (l~-hr The axial clearance is defined as the distance hr between the rotor reference and thrust bearing center. (Figure 13) An axial dynamic force Fa excites the shaft at one end: Fa = (1000 + 800 sin(2n * N at)) ~ with Na the rotational shaft frequency.No unbalance force is
Pads
0
2
~[~
2~
I~
hs b~
="0
Figure 13. Film circumferential thickness in thrust bearing static Load Thrust I Fs Beating rl ~' [] [] Load o IT]:=__:I_____I__ .i_ _____.71~::~00,0 DynamiCFa y
Beating 1
l~
Dynamic Force StaticForce
6.3. Results
The shaft is mounted in two identical bearings and a thrust bearing supports axial load, at one end. The model of shaft is presented in figure 12.
Runner
Beating 2
I1+.11:~o. I1~.11=1000+ 800sin(2zN.t)
Figure 14. Rotating system The bending vibrations of the shaft are analyzed.. Figure 15 shows the trajectory amplitude of the shaft on the middle of the rotor. A critical speed appears at 14400 rpm. In the middle of the shaft, the peak amplitude is equal to 0.09 times the bearing radial clearance. Thus, an axial dynamic force can excite the lateral dynamic behaviour of the shaft through the thrust bearing. The frequency vibration corresponds to the first bending critical frequency.
7. CONCLUSION It is essential to couple the flexibility of the shaft with the non-linear behaviour of the fluid bearings when studying the dynamic behaviour of the rotor. Several bearing models are used: linear (with the small displacement assumption) and non-linear (in particular in the case of squeeze film damper). From the results obtained, the following conclusions can be drawn: 1. If the speed of rotation passes through a critical speed it is necessary to modelise the flexibility of the shaft.
12
o.o9
I
Shaft
Middle I
lilt
~176 ~176176
Il
~176 0.4
/ 0.6
0.8
1
1.2
\ 1.4
Speed of Rotation (rpm)
1.6
1.8
2 x 104
Figure 15. Lateral response of the shaft middle due to axial excitation of the shaft 2. The influence of the SFD is very different according to the rotor speed: when the speed is close to a critical speed, the SFD must dissipate a lot of energy, consequently the SFD radial clearance must be large or the viscosity of the lubricant must be small. Conversely a small radial clearance (or high viscosity) is better for speeds very different from the critical speed. 3. New designs of squeeze film damper have been studied (active SFD with variable clearance or variable viscosity) It could be possible to monitor the damping of this type of element. 4. Axial and lateral dynamic coupling by a thrust bearing has been demonstrated, its effect could be important.
REFERENCES
[1] O. Bonneau, A. Kassai, J. Fr~ne, J. Der Hagopian "Dynamic behaviour of an elastic rotor with squeeze film damper" EUROTRIB, Helsinki, Finlande, June 1989, Proceeding Vol.4, pp 145149. [2] O. Bonneau, J. Fr~ne "Study of a squeeze film damper with an axial controlled flow" IFTOM, Third international conference on Rotordynamic, Lyon, September 1990, Proceeding, pp 295-300. [3] O. Bonneau, J. Fr~ne "Numerical study of a flexible rotor mounted in an active squeeze film damper" IFTOM Fourth International Conference On Rotor Dynamics, Chicago, 1994 pp 327-331.
[4] B. Pecheux, O. Bonneau, J. Fr~ne "Investigation about Electro Rheological Squeeze Film Damper Applied to Active Control of Rotor Dynamic" International Journal of Rotating Machinery Gordon & Breach Science Publishers,1997, Vol. 3, pp53-60 [5] O. Bonneau, J. Franc "Influence of the Bearings Defects on the Dynamic Behaviour of on Elastic Shaft" International Journal of Rotating Machinery Gordon & Breach Science Publishers, 1996,Vol. 2 pp 281-287 [6] F. Simon, J. Franc. "Analysis for Incompressible Flow in Annular Pressure Seals", ASME Journal of tribology, 1991, vol. 114, pp.431-438, [7] F. Simon, J. Franc, "Rotordynamic coefficients for turbulent annular misaligned seals" Third International Symposium on transport Phenomena and Dynamics of Rotationg Machinery 1990, Hawaii, USA, PP 289-304 [8] V. Lucas, S. Danaila, O. Bonneau, J. Franc "Roughness Influence on turbulent flow through annular seals" Journal of Tribology ASME, 1994, vol 116, pp 321-329 [9] V. Lucas, O. Bonneau, J. Franc "Roughness Influence on the Turbulent Flow Through Annular Seals Including Inertia Effects" Journal of Tribology ASME, January 1996, vol. 118, pp 175-182 [ 10] M. Arghir J.Fr~ne "A Quasi-Two-Dimensional Method for the Rotordynamic analysis of Centred Labyrinth Liquid Seals" ASME, Journal of Engineering for Gas Turbines and Power, 1999, Vol. 121, pp 001-009. [11 ] S. Berger, O. Bonneau, J. Franc, "Influence of Axial Thrust Beating Defects on the Dynamic Behaviour of an Elastic Shaft", Proceedings of the 5tb International Tribology Conference Austrib'98, Brisbane, 6-9 December 1998, pp 389-394. [12] S.M. Rhodes, D.F. Li.,"A generalised short bearing theory" ASME Journal of Lubrication Technology, 1980, vol 102, pp278-282. [13] J.W. Lund, K.K. Thomsen "A calculation method and data for the dynamic coefficients of oil lubricated journal bearing" ASME, Topic in fluid film beating and rotor bearing system design and optimisation, 1978, pp 1-18. [ 14] J. Franc "R6gimes d'6coulement non laminaires en films minces, application aux paliers lisses" Th~se de doctorat d'6tat es Sciences, Universit6 Claude Bernard, Lyon 1974 [15] N. Abdul-Wahed "Comportement dynamique des paliers fluides. Etude lin6aire et non lin6aire"
13
Th~se de doctorat d'6tat, Universit6 Claude Bemard, Lyon, 1982. [16] B. Sternlicht "Elastic and damping properties of oil film journal bearing for application to unbalance vibration of calculation" ASME, J. Applied Mechanics., 1958, Vol 80, pp 141. [17] J. Lacroix "Comportement dynamique d'un rotor au passage des vitesses critiques" Thbse de Doctorat de I'INSA de Lyon, 1988. [18] M. Lalanne, G. Ferraris, "Rotordynamics Prediction in Engineering" J. Wiley, Chichester, 1990. [19] A. Kassai "Contribution ~ l'Etude Dynamique des Rotors Amortis" Th~se de Doctorat de I'INSA de Lyon, 1989. [20] C.H.T. Pan, J. Tonnesen "Excentric operation of the squeeze film damper" ASME, Journal of Lubrication Technology, 1978, Vol 100, pp 369377. [21 ] E.J. Hahn "Stability and unbalance response of centrally preloaded rotors mounted in journal and squeeze film beatings" ASME, Journal. of Lubrication Technology, 1979, Vol 101, pp 120128. [22] T.S.R. Murthy "Analysis of multi-scallop selfadjusting conical hydrodynamic bearings for high precision spindles" Tribology international, 1981, pp 147-150. [23] C. Mu., J..Darling., C.R. Burrows "An appraisal of a proposed active squeeze film damper" ASME, Journal of Tribology, 1991, vol. 113, pp 750-754. [24] A. W. Duff, "The viscosity of polarised dielectrics", Physic Revue, 1896,pp 4-23. [25] Y. Bj6mst/ihl, Snellman, Andrade, "The influence of an electic field on the viscosity of pure
liquid and colloidal suspensions", Koloid Z 78, pp258, 1937, 86, pp223 1939. [26] W. M. Winslow, "Induced fibration of suspensions, Journal of Applied Physics, vol. 20, December 1949, pp 1137-1140. [27] A. V. Lykov, R. G. Gorodl~, Z. P. Shulman, "Research into electrorheological effect", Engineering Physical Journal, 1968,15, 4, pp 589. [28] D. L. Klass, T. W. Martinek, "Electroviscous fluids I: Electrical properties", Journal of Applied Physics, January 1967, vol. 38, N~ pp 75-80. [29] A. Okawaga, Cox & S.G. Mason, 1974 J. Colloid Interfaces Sci. 47, 568. [30] J.E. Stangroom, "Electrorheological fluids", Phys. Technology, 1983, vol. 114, pp 290-296 [31] T. Someya and M. Fuhada, "Analysis and experimental Verification of Dynamic Characteristics of Oil Film Thrust Bearings", 1972, 15, pp 1004-1015, Bull. of JSME [32] M.C. Jeng and A.S. Szeri,"A thermohydrodynamic solution of pivoted thrust pads ", Trans. ASME, Journal of Tribology 1986, Vol. 108, pp. 195-218. [33] Q. Zhu, Y.B. Xie, L. Yu, "Axial Transient Forces of Thrust Bearing Rotor System in a Turboexpander", Proceedings of International Conference on Hydrodynamic Bearing-Rotor System Dynamics, 1990,China. [34] P.L. Jiang, L. Yu, "Effect of a Hydrodynamic Thrust Bearing on the Statics and Dynamics of a Rotor-Bearing system", Mechanics Research Communications 1998, Vol. 25, No. 2 pp.219-224. [35] N. Mittwollen, T. Hegel, J. Glienicke,"Effect of Hydrodynamic Thrust Bearings on Lateral Shaft Vibrations", ASME, Journal of Tribology October 1991, Vol. 113, pp.881-818.
DYNAMIC BEHAVIOUR OF ELASTIC SHAFT SUPPORTED BY HYDRODYNAMIC BEARINGS
Jean FRENE and Olivier BONNEAU Laboratoire de M6canique des Solides-UMR CNRS 6610, Universit6 de Poitiers, SP2MI, Bd Pierre et Marie Curie, T616port 2, BP 30179- 86960 Futuroscope - Chasseneuil Cedex High speed rotors present a lot of stability problems especially when the speed of rotation passes through a critical speed. The non-linear dynamic behaviour of the fluid bearings has, in that case, an important effect. This study presents several models of bearing dynamic behaviour taking into account the flexibility of the shaft: Squeeze film damper (SFD) behaviour is described. This element gives a lot of damping but its behaviour is totally non-linear. The coupling between the axial thrust bearing behaviour and the bending vibrations of the shaft is also studied.
1. I N T R O D U C T I O N In the turbomachinery field the designer has to model the shaft with high accuracy and has to take into account all the shaft surroundings. A lot of disciplines are concerned by this study: structural mechanics, acoustics, fluid mechanics, heat transfer, lubrication,... The difficulty, but also the challenge of the rotor study, comes from this diversity of subjects and from the choice of different models. Until a few years ago, each discipline was developing its own models: - very detailed shaft (finite element models) mounted in a simple support (with linear stiffness and damping coefficients) for structure mechanics approach; - very sophisticated bearing models (non-linear) but with simple shafts (rigid) in lubrication applications. This earlier work gives useful results for a first level of approach but these are inadequate when the mechanism works in severe conditions (near the critical speed for example). In such conditions it is necessary to analyze the coupled behaviour. The work presented here [1 to 5] is a synthesis of several studies done with coupled models. These models take into account shaft flexibility in fluid surroundings. Non linear shaft support behaviour will be specially emphasized. In the first part of the paper one describes bearing dynamic models (both linear and non-
linear models) and rotor dynamic models. The coupling between fluid element and the shaft is done by a non linear approach: the simulation of the non linear dynamic behaviour of the flexible rotor is a time step by step approach done in the shaft modal basis. In the second part of the paper some applications of this dynamic model are detailed: - Results obtained for a flexible shaft mounted in a squeeze film damper (SFD) have been chosen; this example is very interesting because of the high non linear behaviour of the fluid element. Numerical and experimental results are presented and the comparison shows the good accuracy of the numerical model. - The non-linear coupling between axial motion and the bending vibrations of the shaft is also presented. This coupled behaviour is due to the thrust bearing effects and to the shaft flexibility.
2. F L U I D
ELEMENTS
(FLUID
BEARINGS,
ANNULAR SEALS) 2.1. Presentation
Shafts are generally supported by two kinds of elements: ball bearings and (or) fluid bearings. The ball bearings give near-linear behaviour and this technology is not studied in this work. Fluid bearings have a complex behaviour; linearisation can be carried out but the behaviour is
sometimes quite non linear. Three kinds of fluid bearings are studied here: the traditional bearing, the squeeze film damper and the axial thrust bearing. Annular seals may also have an important effect on shaft behaviour [6 to 10]. The problems induced by these components are related to turbulent flow and inertia effect in the fluid which are due to the low fluid viscosity (in liquid hydrogen turbopump for example).
In the case of low viscosity fluids, for a large clearance or when rotation speed is high the flow could become turbulent. Then two turbulent viscosity coefficients are introduced. They are a function of turbulent parameters: local Reynolds number, axial and circumferential pressure gradient and roughness. Some turbulent flow models have been applied [6 to 9 ] to calculate turbulent viscosity coefficients.
.......----2.2. Fluid film modelling The characterization of fluid elements requires solution of the Navier-Stokes equations. This solution gives the values of the pressure field in the film. Two cases appear: - The film thickness is small compared to the other dimensions, thus this condition allows us to simplify the problem. An elliptic differential equation: the Reynolds equation is obtained. -It is impossible to neglect one dimension in relation to the others so it is necessary to solve the complete Navier-Stokes 3D systems. This approach is necessary for example in a labyrinth seal with very deep grooves. This aspect will not be described in this study. Arghir [10] developed numerical models adapted to this sort of seal geometry. 2.2.1. Lubrication equations We presem in this part the model adopted for journal bearing; the axial thrust bearing solution is similar [ 11 ] and we do not present it here. The assumption of a thin film thickness allows us to write the Reynolds equation as follows:
1 'E
h3
R2 8~ * /a 9
ha kxSO* 9
O)
= Xcos 0 +Ysin 0 + - - ~
- ~ ,u
kz ~
=
6;tl
200* Where: R and L are respectively the radius and the length of the bearing, h is the film thickness, P is the pressure field, co is the angular shaft velocity, ~t is the dynamic viscosity of the fluid, 0* and z are respectively the circumferential and axial parameter. All the different parameters are shown in Figure 1.
I
O"
Figure 1. Film characteristics. To solve the Reynolds equation, different boundary conditions should be applied: - Feed boundary conditions (feed pressure in holes, external pressure...) - Flow conditions, which characterize rupture in the film.: -Sommerfeld conditions: no rupture of the film (2 n film).
-Gtimbel solution: rupture when numerical pressure is negative (n film). -Reynolds conditions: the pressure and its derivative are nil when rupture of the film occurs. -Film reformation conditions: the pressure and its derivative are nil when rupture of the film occurs and the flow through the inactive part of the film is taking into account for the film reformation. Then the Reynolds equation is integrated using one of the following methods:
- Short bearing theory: the circumferential pressure variation effect is neglected. The fluid film force is formulated analytically and computing time is small. - Finite length bearing theory. In spite of its accuracy this solution implies heavy numerical calculations which are not suitable for the application discussed here. - Unidimensional theory. This solution, inspired from Rhode and Li [12], is based on an axial parabolic pressure assumption. A variational calculus implies a new equation of Reynolds where only one space variable 0 appears. The gain in time is important compared to the finite length model. These methods have been tested and their accuracies depend on the problem geometry. Integration of the pressure film gives fluid film forces on the rotor. Fx= fY2 ~2 ~~ P(0, z)cos 0Rd0dz
of the fluid load, Ax and Ay are small displacements near the static equilibrium and A:~ and AS' are small velocity variations. Fy (x o + Ax, Yo + Ay, A/~,A;r = Fy (Xo, Yo,0,0) + + Ax/CTy/+ Ay/--~)+
t,
-I-axC~ t~
Jo
A:~(~
o
t
A~,(---~)
)o
o
- Ay[-~--)-I- .... 0
Then a first order development is carried out: + Ax, Y0 + Ay, Ai, A~,,Aii, A~,) - Fx (0)~
?y(Xo+ Ax, Y0 + Ay, A:~,A~,,Aii, A~,)- F ( 0 ) J A:~ A~ (ff;} = -[aijIAA;}- [bijIA~} - [mijIA~}
Fy=~V2~ ~t P(0, z)sin 0Rd0dz where fx and fy are the additional loads created by
2.2.2. Dynamic behaviour The fluid element influence on rotor dynamics is characterized by the hydrodynamic forces generated by the pressure field. These forces are non-linear forces of the position and velocity of the shaft center in the bearing. In general the shaft is submitted to a combination of several forces: a static load (weight, belt tension,...) a dynamic load (unbalance, shaft vibration) The static load implies an equilibrium position (stable or unstable) and the dynamic load creates an orbital trajectory of the shaft. Depending on the conditions, two approaches can be carried out: a linear approach and a non-linear one. -
-
2.2.3. Linear theory The aim of this approach is to linearize the bearing behaviour around an equilibrium state. This theory is based on a small displacement hypothesis. Thus, close to a static equilibrium state, a first order development may be done. In these expressions F x and Fy are the components in the (.2, ~) reference
displacement and velocities, aij ' bij and mij are the stiffness, damping and added mass coefficients. The last coefficients (added mass) exist only for inertia flow. T w o numerical approaches may be used to calculate these coefficients: - a perturbation method[ 13] - a numerical differentiation [ 14 to 16]. These dynamic coefficients may be used to determine the linear stability of a rigid shaft supported by two identical bearings or to estimate the dynamic response of a shaft (to obtain critical speeds for example). However this model is false when the amplitude is large or when the equilibrium is unstable, it is necessary then to obtain a non linear solution. 2.2.4. Non linear model This approach is a numerical study of the equations of motion which are integrated by a step by step method: at each step, the Reynolds equation is solved to evaluate the film forces, then the fundamental principle of mechanics is integrated to obtain speeds and positions of the next step.
3. R O T O R M O D E L L I N G
obtain an acceptable precision. A change of variable is done in the following form: {8}= [g]{q} where
Two models could be considered, first the shaft is considered rigid and symmetrical (in geometry and in loading). It could be compared to a mass mounted in a fluid bearing. Then the equations of motion are simple (two degrees of freedom). This model is valid when the speed of rotation is far from a critical speed. This simple model is useful for a first approach but in this study a more advanced model is presented which concerns an elastic shaft. The rotor is modeled with typical beam finite elements including gyroscopic effects [1]. The differential system, with {8} the node displacement vector is the following:
[r is the matrix obtained with modal vectors. {gi }, and {q} is the modal vector of displacement. A new system is written where the bearing effect appears in the modal stiffness matrix. [m]{~} + [c]{cl}+ [k']{q} = {f} with : [m] :
[v/]t
[k'] = [gz]t[ K k ]
[c]= [v/]t[CIr {f} = [~]t {F}
To do a non linear calculus the linear bearing [Kbearing] should be subtracted from modal stiffness [k'] and should be introduced as a non linear effect. Then
effect Where [M]:is the mass matrix, [C] is the gyroscopic matrix, [K] is the stiffness matrix {Funb} are the unbalance forces, {Fr~} are the non linear bearing
[m]{~} + [c]{ft} + [k]{q} = {f} + {Fnl}
forces and {Fg~/ are the gravity forces. The damping
with [k] = [k'l- [I]/]t [kbearings I~/]
does not appear explicitly but it exists in the {Fr~} terms as a non-linear fluid damping. This system has 4(n+ 1) degrees of freedom (n is the number of nodes). The iterative non-linear calculation will be very expensive in computer time. To reduce the degrees of freedom a modal approach is used [ 17-18]. Two approaches are available to obtain modal base: -to take a free motion shaft but then it is important to conserve a lot of modes. -to take a judicious equivalent bearing stiffness. This stiffness is not the real one (because an uncentralised squeeze film damper has no linear stiffness). It is a symmetrical stiffness and its value leads to a representative modal base. Then this stiffness may be subtracted from the modal stiffness matrix. Then the undamped fundamental principle of mechanics on the rotor could be written:
[M]{r }q- ([Kr ] q-[Kbearings]){8} = 0 The solution has the following form: {8}=-{80 }ert and the first 6 modes are calculated. Lacroix [ 17] has shown that 6 modes are enough to
{Fnl} is the modal non linear force obtained, at each time step (see 4-1), by a modal basis change of the real non linear force. These real non-linear forces are calculated in the physical coordinates.
4. FLUID STRUCTURE COUPLING The coupling between fluid element and the shaft is done by a non-linear approach. The simulation of the non-linear dynamic behaviour of the flexible rotor is a step by step approach done in the modal basis. The flow chart is as follows: a) Beginning with initial values of modal positions and velocities. b) Calculus of external modal forces. c) Calculus of physical displacements and velocities in bearings (by modal basis change). d) Non linear bearings forces calculus (in the real basis). e) Computation of all forces in the modal basis (by modal basis change). f) Modal acceleration computation. g) Time integration (variable step Euler method). h) Shaft speed is incremented. i) The process begins again in b.
4.1. Flexible shaft on a Squeeze Film Damper 4.1.1. Squeeze Film Damper (SFD) Results obtained for a flexible shaft mounted in a squeeze film damper (SFD) have been chosen, this example is very interesting because of the high non-linear behaviour of this kind of fluid element. The basic idea for this kind of bearing is to support a ball bearing in a fluid bearing (figure 2). The rotation is ensured by the ball bearing and the oil film is squeezed between two non rotating rings (the external ring of the ball bearing is jammed in rotation). Two technologies for SFD are possible. A first kind of assembly consists in mounting, in parallel with the SFD centering springs [19-20]. The second technology is without centering springs then some stiffness problems appear: the creation of a significant pressure field (to support a static load) being dependent on a threshold of minimum perturbations. The results presented here are obtained only with this second technology.
\~o,
Bearing span: 0.8 rn, Rotor diameter 0.06 rn, Bearing length: 0.015m, Diameter: 0.09 rn, Radial clearance 0.05 mm. 4.1.3. Results A test rig was developed by Kassai at the National Institute of Applied Sciences [ 1 & 19]. The shaft displacements are measured by eddy current proximity probes. Numerical and experimental results are obtained for a linear rotation speed variation from 8500 rpm to 13500 rpm during 22 s. Numerical results obtained by short bearing theory and experimental results are shown in figure 4a and 4b (oil temperature = 25~ ~t=0.05 Pa.s) and in figure 5a and 5b (oil temperature = 60~ ~t=0.0135 Pa.s). These figures show the displacement amplitude of the shaft in the middle of the rotor (where the dynamic amplitude is maximum) versus rotational speed. Amplitude (ram)
O.4 t 0.3
0.2
0.1 I I I
Figure 2 - Scheme of a Squeeze Film Damper. 4.1.2. Shaft The shaft is a rotating flexible shaft supported at one end by a ball bearing and at the other end by an active squeeze film damper without a centralizing spring (figure 3).
o
I
9000
'
I
10000
'
I
11000
'
I
12000
'
I
13000
'
speed of rotation (rpm)
Figure 4a. Experimental results, ~t = 0.05 Pa.s.
Figure 3. Shaft representation. Results are obtained for a linear rotation speed variation from 7000 to 15000 rpm. The shaft and bearing characteristics are the following:
Figure 4b. Numerical results, ~t = 0.05 Pa.s.
Amplitude (mm)
The same conclusion has been obtained concerning the effect of the radial clearance (a high radial clearance is better closed to its critical speed and a low radial clearance is necessary for speeds different from the critical speed [3].
0.4u
5. ACTIVE SQUEEZE FILM DAMPER An active squeeze film damper can be chosen to present an optimal value of clearance or viscosity.
0 I
I
I
I
I
9000
10000
11000
12000
13000
speed of rotation (rpm)
Figure 5a. Experimental results, g = 0.0135 Pa.s.
5.1. Variable clearance
To perform the control, the active squeeze film damper has been modeled. The idea is to regulate the radial clearance by a parameter x corresponding to the position of a conical squeeze film damper in its housing (figure 6).
Amplitude (mm) 0Au
0.2
I 9000
'
I 10000
'
I 11000
'
I 12000
'
I 13000
' speed of rotation (rpm)
Figure 5 b. Numerical results, ~t = 0.0135 Pa.s. The comparison between experimental and numerical results is good. There is a critical speed around 11800 rpm for la=0.05 Pa.s and around 11500 rpm for ~t=0.0135 Pa.s. The experimental and numerical amplitudes are very similar. Comparison between figure 4a (respectively 4b) and 5a (respectively 5b) shows that the decrease of viscosity (which could be due to an increase in oil temperature) gives an increase of damping. In fact low viscosity leads to more displacement in the bearing and consequently higher energy dissipation may occur. The conclusion of these first results is the following: the influence of the squeeze film damper is very important and its effect is very different depending on the rotor speed. When the rotational speed is close to a critical speed, the squeeze film damper must dissipate a lot of energy, consequently the viscosity must be small. Conversely a large viscosity is better for speeds different to the critical speed (in that case the amplitude in the SFD is decreasing).
Figure 6. Active squeeze film damper. Some interesting work has been published [2223] on conical squeeze film dampers. The dynamic study is carried out with a simple shaft (symmetrical, centered, with one stiffness) and with a synchronous circular whirl. In our work the shaft is modelled with these 6 first modes and the trajectory can be of any shape. The numerical results are obtained with the same shaft as previously. The idea is to monitor the radial clearance by the rotational speed, the clearance evolution is a linear evolution from 0.05 mm to 0.1 mlrl.
Figure 7 presents the amplitude of the middle of the rotor and figure 8 gives the displacement in the squeeze film damper for 0.05 nma, 0.1 mm and for a linear evolution. It is remarkable (figure 7) that the shaft amplitude is filtered around the critical speed.
0.3
C=O05mm
0.2
C lin
7000Rpm
15000Rpn
Figure 7. Amplitude of the middle of the rotor for linear clearance variation.
0.03mI
[///~
0.02mr
tension required to obtain the ER effect, the total power required is low. The current never exceeds a few microamperes. Thus, ER fluids are able to provide a good interface between a mechanical device and an electric control system. Moreover, the speed of response could allow the realization of mechanical devices actively controlled by "electronic management". The idea is to feed the SFD with a negative ER Fluid and to apply an electric field between two electrodes. Figure 9 presents a scheme of this SFD. A negative ER Fluid was chosen for its Newtonian behaviour.
)
~:
0.01 mr
7000Rpm
15000Rpn"
Figure 8. Amplitude in the squeeze film damper for linear clearance variation. Figure 8 shows that when the speed is very different to the critical speed it is better to have a small clearance (to limit SFD amplitude). Near the critical speed, a large clearance in the squeeze film damper permits large displacements to dissipate energy. 5.2. Variable viscosity Electrorheological fluids, called ER fluids, are types of semi-conductor liquids [24 to 30]. They are concentrated suspensions of solid particles in an oil base liquid. Normally, they behave like newtonian oil, but when they are exposed to a strong electric field, they seem to "coagulate". An increase of the frictional forces of the fluid on the wall appears. So that "coagulation" induces a change in apparent viscosity of the fluid. This change is gradual, reversible and proportional to the applied electric field. It occurs very rapidly. The time lapse is very short, about 1 millisecond. In spite of the high
Figure 9. Electrorheological Squeeze Film Damper. Results obtained with this kind of active SFD [4] are close to results obtained with variable clearance, all the simulation shows that it could be possible to monitor the damping of the squeeze film damper.
6. COUPLING BY THRUST BEARING BETWEEN AXIAL VIBRATIONS AND BENDING VIBRATIONS. The influence of thrust bearing on the shaft dynamic behaviour, and more especially the effect on lateral shaft vibrations has been studied very little. Some work was done on the analysis of the dynamic characteristics of hydrodynamic thrust bearings and their action on axial motion [31 to 35]. The influence of thrust beating on lateral shaft vibrations has been studied by P.L. Jiang and L. Yu
10
[34]. They have shown that as the film thickness of the thrust bearing becomes smaller, then its action becomes more important and it may prevent the shaft from deflecting and, the first critical speed and the threshold speed increase. For the hydrodynamic thrust bearing case N. Mittwollen et al [35] have shown that both decrease in the total axial clearance and increase in the axial thrust can lead to a significant increase in the whirl amplitude at the first critical speed and in the threshold speed of instability. In these studies, influence of the hydrodynamic thrust bearing on the shaft dynamic behaviour has been assumed to be linear. Berger et al [11] have studied the effect of thrust bearing on a flexible shaft. The dynamic behaviour of the thrust has been assumed to be nonlinear. The authors have shown that a defect in the thrust bearing rotor can excite a critical speed. The aim of the work presented now is to study the non-linear influence of the thrust bearing on the dynamic behaviour of a flexible shaft. In particular, the coupling between the axial motion and the bending vibrations of the shaft is analyzed.
Rotor
.02
Stator
!
Zl
I
! ". ,,. ,,,
! !
'
t
Y0
Figure 10 : Thrust bearing scheme
6.1. Thrust bearing model The thin film equation is defined (Fig. 10) in the cylindrical reference (o0,r,t,z0) by: 8
Figure 11 shows the pressure field obtained in the thrust bearing, at a constant speed.
(h 3 Op) =
dr dh
dh
e rSH 8Hx - 12/a rU 2 -,--1 _ 12/a V2 dr 30
+ 6/~ hr
dU 2 dU 2 +6/~h + 6/a hU 2 + 12/~ rW 2 dr dO
Pad Pad 1
where p is the pressure field, h=H2-H1 is the film thickness, /~ is the dynamic viscosity and U2,V2,W 2 are the velocity components of the points ME of the rotor. The equation is solved by finite difference with the Gauss Seidel method. The boundary conditions used are as follows: Feed pressure in each groove (atmospheric pressure) and Reynolds boundary conditions.
Figure 11. Pressure field with thrust bearing The force components and the moment components are calculated by integration of the pressure field on the surface $2 of the rotor.
S2
$2
11
applied on the shaft. On the middle of the rotor, a static load is used in order to stabilize it. The rotational speed increases from 4000 rpm to 20000 rpm in 15 s.
6.2. Bearing and shaft models
In order to calculate the bearing forces, the Reynolds equation in the film should be solved by numerical methods. But the non-linear analysis of the elastic shaft implies heavy numerical calculation. In order to decrease the computing time one has opted to use the short bearing approximation. This assumption which allows simplification of the differential equation by neglecting the circumferential pressure gradient in the film. The force components are obtained by integration of the pressure field with the following boundary conditions: Feed pressure on both sides of the bearing (atmospheric pressure) and Giirnbel assumption (n film) i.e. negative pressures are set to zero. It should be noted that the use of f'mite length bearing solution leads to the same type of results. The elastic shaft is modeled as described in w using modal method of reduction.
h~
0.02m
0.76m
0.02m
Thrust
~adn.g.ll
Beating 1
"
0.06 rn
l
Bearing 2
Figure 12. Geometrical data of the shaft Bearing data is as follows: The length L=0.015 rn, the radius R=0.03 m, the radial clearance C=3.0E-05rn, the dynamic viscosity ~t=0.05 Pas. The thrust bearing has four pads. The specifications of each pad are the angle 13=80, the outer radius R2=0.04 m, the inner radius R~=0.02 rn, the difference between entry thickness and the exit thickness is 20 ~tm (l~-hr The axial clearance is defined as the distance hr between the rotor reference and thrust bearing center. (Figure 13) An axial dynamic force Fa excites the shaft at one end: Fa = (1000 + 800 sin(2n * N at)) ~ with Na the rotational shaft frequency.No unbalance force is
Pads
0
2
~[~
2~
I~
hs b~
="0
Figure 13. Film circumferential thickness in thrust bearing static Load Thrust I Fs Beating rl ~' [] [] Load o IT]:=__:I_____I__ .i_ _____.71~::~00,0 DynamiCFa y
Beating 1
l~
Dynamic Force StaticForce
6.3. Results
The shaft is mounted in two identical bearings and a thrust bearing supports axial load, at one end. The model of shaft is presented in figure 12.
Runner
Beating 2
I1+.11:~o. I1~.11=1000+ 800sin(2zN.t)
Figure 14. Rotating system The bending vibrations of the shaft are analyzed.. Figure 15 shows the trajectory amplitude of the shaft on the middle of the rotor. A critical speed appears at 14400 rpm. In the middle of the shaft, the peak amplitude is equal to 0.09 times the bearing radial clearance. Thus, an axial dynamic force can excite the lateral dynamic behaviour of the shaft through the thrust bearing. The frequency vibration corresponds to the first bending critical frequency.
7. CONCLUSION It is essential to couple the flexibility of the shaft with the non-linear behaviour of the fluid bearings when studying the dynamic behaviour of the rotor. Several bearing models are used: linear (with the small displacement assumption) and non-linear (in particular in the case of squeeze film damper). From the results obtained, the following conclusions can be drawn: 1. If the speed of rotation passes through a critical speed it is necessary to modelise the flexibility of the shaft.
12
o.o9
I
Shaft
Middle I
lilt
~176 ~176176
Il
~176 0.4
/ 0.6
0.8
1
1.2
\ 1.4
Speed of Rotation (rpm)
1.6
1.8
2 x 104
Figure 15. Lateral response of the shaft middle due to axial excitation of the shaft 2. The influence of the SFD is very different according to the rotor speed: when the speed is close to a critical speed, the SFD must dissipate a lot of energy, consequently the SFD radial clearance must be large or the viscosity of the lubricant must be small. Conversely a small radial clearance (or high viscosity) is better for speeds very different from the critical speed. 3. New designs of squeeze film damper have been studied (active SFD with variable clearance or variable viscosity) It could be possible to monitor the damping of this type of element. 4. Axial and lateral dynamic coupling by a thrust bearing has been demonstrated, its effect could be important.
REFERENCES
[1] O. Bonneau, A. Kassai, J. Fr~ne, J. Der Hagopian "Dynamic behaviour of an elastic rotor with squeeze film damper" EUROTRIB, Helsinki, Finlande, June 1989, Proceeding Vol.4, pp 145149. [2] O. Bonneau, J. Fr~ne "Study of a squeeze film damper with an axial controlled flow" IFTOM, Third international conference on Rotordynamic, Lyon, September 1990, Proceeding, pp 295-300. [3] O. Bonneau, J. Fr~ne "Numerical study of a flexible rotor mounted in an active squeeze film damper" IFTOM Fourth International Conference On Rotor Dynamics, Chicago, 1994 pp 327-331.
[4] B. Pecheux, O. Bonneau, J. Fr~ne "Investigation about Electro Rheological Squeeze Film Damper Applied to Active Control of Rotor Dynamic" International Journal of Rotating Machinery Gordon & Breach Science Publishers,1997, Vol. 3, pp53-60 [5] O. Bonneau, J. Franc "Influence of the Bearings Defects on the Dynamic Behaviour of on Elastic Shaft" International Journal of Rotating Machinery Gordon & Breach Science Publishers, 1996,Vol. 2 pp 281-287 [6] F. Simon, J. Franc. "Analysis for Incompressible Flow in Annular Pressure Seals", ASME Journal of tribology, 1991, vol. 114, pp.431-438, [7] F. Simon, J. Franc, "Rotordynamic coefficients for turbulent annular misaligned seals" Third International Symposium on transport Phenomena and Dynamics of Rotationg Machinery 1990, Hawaii, USA, PP 289-304 [8] V. Lucas, S. Danaila, O. Bonneau, J. Franc "Roughness Influence on turbulent flow through annular seals" Journal of Tribology ASME, 1994, vol 116, pp 321-329 [9] V. Lucas, O. Bonneau, J. Franc "Roughness Influence on the Turbulent Flow Through Annular Seals Including Inertia Effects" Journal of Tribology ASME, January 1996, vol. 118, pp 175-182 [ 10] M. Arghir J.Fr~ne "A Quasi-Two-Dimensional Method for the Rotordynamic analysis of Centred Labyrinth Liquid Seals" ASME, Journal of Engineering for Gas Turbines and Power, 1999, Vol. 121, pp 001-009. [11 ] S. Berger, O. Bonneau, J. Franc, "Influence of Axial Thrust Beating Defects on the Dynamic Behaviour of an Elastic Shaft", Proceedings of the 5tb International Tribology Conference Austrib'98, Brisbane, 6-9 December 1998, pp 389-394. [12] S.M. Rhodes, D.F. Li.,"A generalised short bearing theory" ASME Journal of Lubrication Technology, 1980, vol 102, pp278-282. [13] J.W. Lund, K.K. Thomsen "A calculation method and data for the dynamic coefficients of oil lubricated journal bearing" ASME, Topic in fluid film beating and rotor bearing system design and optimisation, 1978, pp 1-18. [ 14] J. Franc "R6gimes d'6coulement non laminaires en films minces, application aux paliers lisses" Th~se de doctorat d'6tat es Sciences, Universit6 Claude Bernard, Lyon 1974 [15] N. Abdul-Wahed "Comportement dynamique des paliers fluides. Etude lin6aire et non lin6aire"
13
Th~se de doctorat d'6tat, Universit6 Claude Bemard, Lyon, 1982. [16] B. Sternlicht "Elastic and damping properties of oil film journal bearing for application to unbalance vibration of calculation" ASME, J. Applied Mechanics., 1958, Vol 80, pp 141. [17] J. Lacroix "Comportement dynamique d'un rotor au passage des vitesses critiques" Thbse de Doctorat de I'INSA de Lyon, 1988. [18] M. Lalanne, G. Ferraris, "Rotordynamics Prediction in Engineering" J. Wiley, Chichester, 1990. [19] A. Kassai "Contribution ~ l'Etude Dynamique des Rotors Amortis" Th~se de Doctorat de I'INSA de Lyon, 1989. [20] C.H.T. Pan, J. Tonnesen "Excentric operation of the squeeze film damper" ASME, Journal of Lubrication Technology, 1978, Vol 100, pp 369377. [21 ] E.J. Hahn "Stability and unbalance response of centrally preloaded rotors mounted in journal and squeeze film beatings" ASME, Journal. of Lubrication Technology, 1979, Vol 101, pp 120128. [22] T.S.R. Murthy "Analysis of multi-scallop selfadjusting conical hydrodynamic bearings for high precision spindles" Tribology international, 1981, pp 147-150. [23] C. Mu., J..Darling., C.R. Burrows "An appraisal of a proposed active squeeze film damper" ASME, Journal of Tribology, 1991, vol. 113, pp 750-754. [24] A. W. Duff, "The viscosity of polarised dielectrics", Physic Revue, 1896,pp 4-23. [25] Y. Bj6mst/ihl, Snellman, Andrade, "The influence of an electic field on the viscosity of pure
liquid and colloidal suspensions", Koloid Z 78, pp258, 1937, 86, pp223 1939. [26] W. M. Winslow, "Induced fibration of suspensions, Journal of Applied Physics, vol. 20, December 1949, pp 1137-1140. [27] A. V. Lykov, R. G. Gorodl~, Z. P. Shulman, "Research into electrorheological effect", Engineering Physical Journal, 1968,15, 4, pp 589. [28] D. L. Klass, T. W. Martinek, "Electroviscous fluids I: Electrical properties", Journal of Applied Physics, January 1967, vol. 38, N~ pp 75-80. [29] A. Okawaga, Cox & S.G. Mason, 1974 J. Colloid Interfaces Sci. 47, 568. [30] J.E. Stangroom, "Electrorheological fluids", Phys. Technology, 1983, vol. 114, pp 290-296 [31] T. Someya and M. Fuhada, "Analysis and experimental Verification of Dynamic Characteristics of Oil Film Thrust Bearings", 1972, 15, pp 1004-1015, Bull. of JSME [32] M.C. Jeng and A.S. Szeri,"A thermohydrodynamic solution of pivoted thrust pads ", Trans. ASME, Journal of Tribology 1986, Vol. 108, pp. 195-218. [33] Q. Zhu, Y.B. Xie, L. Yu, "Axial Transient Forces of Thrust Bearing Rotor System in a Turboexpander", Proceedings of International Conference on Hydrodynamic Bearing-Rotor System Dynamics, 1990,China. [34] P.L. Jiang, L. Yu, "Effect of a Hydrodynamic Thrust Bearing on the Statics and Dynamics of a Rotor-Bearing system", Mechanics Research Communications 1998, Vol. 25, No. 2 pp.219-224. [35] N. Mittwollen, T. Hegel, J. Glienicke,"Effect of Hydrodynamic Thrust Bearings on Lateral Shaft Vibrations", ASME, Journal of Tribology October 1991, Vol. 113, pp.881-818.