NON-LINEAR ROTORDYNAMICS: COMPUTATIONAL STRATEGIES
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1 The 9th International Symposium on Transport Phenomena and Dynamics of Rotating Machinery Honolulu, Hawaii, February 1-14, NON-LINEAR ROTORDNAMICS: COMPUTATIONAL STRATEGIES Tom J. Chalko Head of Rotordynamic Division, Scientific Eng. Research P/L, Melbourne, Australia, sci-e-research.com, ABSTRACT This article discusses strategies and algorithms for non-linear rotordynamic analysis from the point of view of their practical and effective engineering use. The algorithm implementation is demonstrated for the case of a cooling pump in a nuclear reactor. The pump has a vertical rotor supported in three hydrodynamic bearings, one contact seal and three other non-linear bearings, all in an elastic foundation. Unstable behavior of the machine and the development of limit cycle vibration following the loss of stability of the equilibrium position are demonstrated. Computational aspects as well as engineering results are discussed. INTRODUCTION Non-linear phenomena are quite common in rotating machinery. Among the most significant reasons for the non-linear phenomena in rotating machinery are: the fluid flow in hydrodynamic bearings (Fig 1), hydrostatic bearings, seals, impellers etc.. The fluid flow in the hydrodynamic suspension of the rotor provides not only a hydrodynamic force coupling between the rotor and foundation, but also facilitates a mechanism for the energy transfer from rotation to vibration. This mechanism of the energy transfer is the primary reason for the self-excited and limit cycle vibrations of rotors suspended hydrodynamically. non-linear elastic and energy dissipation properties of subsystems such as contact seals, rubber, polymer elements, elements with clearances such as partially worn ball bearings etc... linear, ordinary differential equations) that approximates the behavior of the non-linear system with sufficient accuracy in the vicinity of its equilibrium. This approach can be very effective, because it enables using all methods, techniques and algorithms of the well established linear vibration theory. The linearized approach has been successfully used to predict vibration performance of large turbine-generator sets (Chalko and Li [1]), analyze behavior of the system in terms of its alignment configuration (Li [], Krodkiewski and Sun [3]) and estimate alignment and the transverse static load in rotating machinery (Krodkiewski and Ding [4], Chalko and Li [5]). The main challenge of the linearization approach is determining the system equilibrium, especially for system with multiple non-linear bearings, where the system alignment effects are significant (Chalko and Li [1][6]). The main limitation of the linearization approach is that it has limited scope and applicability. In principle, this method can only be used if the Rotor-Bearing-Foundation (RBF) system equilibrium is stable and vibrations around such an equilibrium position are sufficiently small. Unfortunately these conditions are frequently not satisfied in certain classes of rotating machinery, such as in machines with vertical rotors supported in hydrodynamic bearings for example. TWO STRATEGIES There are essentially two strategies for vibration analysis of non-linear rotor-bearing-foundation systems that contain nonlinear sub-systems. 1. Linearization in the vicinity of a stable equilibrium position.. Direct solution of the non-linear differential equations of motion in the time domain. The essence of the linearization method is obtaining a reasonable linear system (a system that can be described by a set of Fig 1. Geometry of hydrodynamic bearing pad
2 Direct step by step solution of the non-linear differential equations of motion in the time domain is the alternative approach that doesn t impose limitations on the stability of the equilibrium position of the system or the magnitude of its dynamic deflections. In principle, this approach can be used for analysis of almost unlimited range of phenomena in rotating machinery, such as loss of stability of the static and dynamic equilibrium, self-excited fluid induced vibration and the development of limit cycles that are presented further on in this article. The main challenge in applying the direct step-by-step time domain solution approach in engineering practice arises due to high computational requirements of corresponding algorithms. The main computational load comes from the necessity of computing hydrodynamic forces (solving partial differential equations for the fluid flow) at every time step of integration. In addition, time domain solvers impose practical limits to the number of equations that can be simultaneously solved with satisfactory accuracy and hence impose practical limits on the number of degrees of freedom of the model. From the above, it becomes apparent that the practical implementation of the direct time domain approach in engineering practice depends on successful addressing of the two following issues: 1. Modelling. It is important to develop models in which the number of degrees of freedom is significantly reduced and that can satisfactorily represent the behavior of machinery in the given frequency range. Optimizing computations of hydrodynamic forces Application of the direct time domain approach will be demonstrated using a practical engineering example of a water pump used to circulate water coolant in nuclear reactors. MODELLING Every theoretical analysis and prediction is science and engineering relies of using sensible physical and mathematical models of Reality. Principles of modelling can be summarized in the following chart: REAL SSTEM Idealization PHSICAL MODEL Laws of physics OBSERVATION MATHEMATICAL MODEL Mathematics and computations RESULT? SOLUTION performance of the machine. If a simple model doesn t provide a satisfactory prediction of the machine performance, idealization process in the above chart is reviewed and the model is made more sophisticated. This section briefly describes the modelling strategy for the rotor-bearing-foundation (RBF) system of a centrifugal pump with a vertical shaft supported in 7 bearings: 3 of them hydrodynamic with 4 cylindrical pads each, 3 ball bearings and one contact seal. The presented model has been developed for the purpose of explaining the limit cycle development, predicting the transverse non-linear vibration response of the pump and establishing the influence of the pump parameters on its vibration levels. Models of the shaft and casing Elastic and inertia properties of the shaft and casing structures have been modelled using condensed [7] linear finite element models. The condensed model of the shaft contained 1 nodes and the condensed model of the casing contained 14 nodes. Each node of the reduced model had two degrees of freedom in the transverse directions x and y, so that the total 35 node shaftcasing model had 7 degrees of freedom Ball bearing Ball bearing Elastic coupling Ball bearing Contact seal 4-pad hydrodynamic bearing 4-pad hydrodynamic bearing 6 A good model of a non-linear rotor-bearing-foundation system should be capable of representing the range of phenomena of interest with satisfactory accuracy. It is important to stress that the complexity of the model depends on the range of phenomena to be represented and the accuracy requirements. The ultimate test for every model is a comparison of the predicted solution with a result of an experimental observation. In engineering practice it is sensible to aim for the simplest possible models that enable solving problems or predicting the pad hydrodynamic bearing Fig. Model of a pump with vertical rotor.
3 Hydrodynamic bearings The pump had 3 water lubricated hydrodynamic bearings as indicated in Fig. Each of them was cylindrical and had 4 fixed pads. The fluid flow between a pad and journal in a hydrodynamic bearing is governed by the Reynolds equation of the following form: µ 1 h 3 p R + µ h 3 p =6Ω h ϕ µ ϕ z µ z ϕ +1 h (1) t where, the film thickness is obtained from the geometry of the journal bearing (Fig 1) h = r + e cos (ϕ α c ) so that h t = e t cos (ϕ α c)+e α c t sin (ϕ α c). Reynolds equation is solved using the Optimized Finite Difference Method (Li and Chalko [8]) for specific geometry for each pad in each bearing. For each non-fixed pad (tilting, rolling, pivoting) the algorithm finds its equilibrium using an iterative method. The total instantaneous hydrodynamic force F kh (e, α,e v, α v,t) in bearing k is a function of the relative position (e, α) and velocity (e v, α v ) of the shaft with respect to the casing of the bearing and is computed as the integral (sum) of all pressure forces that act on all pads of the bearing. The finite difference grid and the example of the relative pressure distribution in all bearing pads in a 4 pad pressurized hydrodynamic bearing are shown in Fig 3. Finite series of analytical functions bearings and contact seals: F j = K j r kj + C j r cj () K j and C j above are matrices of coefficients of the connecting element j, r kj = r cj = ³ r xj r xj n k 1,r yj r yj n k 1 T and ³ ṙ xj ṙ xj nc 1, ṙ yj ṙ yj nc 1 T. rxj = x Sj x Fj and r yj = y Sj y Fj define relative positions of the shaft with respect to foundation/casing at the connecting element j along x and y axes respectively. This method enables convenient and flexible modelling of linear elastic elements (where n k =1and n c =1) as well as nonlinear elements (where n k > 1 and/or n c > 1). Anisotropy of stiffness and damping is conveniently controlled by matrices of coefficients K j and C j. An example of approximating the characteristics of a ball bearing F x (r xj r xj n k 1,,, ) for n k =3 is presented in Fig x Fig 4. Approximation of the elastic properties of a ball bearing. Horizontal axis - relative position of the shaft in [µm]. Vertical axis - bearing force in [N]. A ball bearing has near zero stiffness for shaft displacements smaller than bearing clearance. Fig 3. Finite difference grid and the relative pressure distribution in a fixed 4 pad pressurized cylindrical bearing. Shown only a half of the pressure distribution along the axial direction. have been used to approximate certain aspects of the Computational Fluid Dynamics algorithms and their solutions. Axial and radial symmetry in each bearing has been taken advantage of whenever possible. These steps enabled an acceleration of the hydrodynamic force calculation many times, without compromising the accuracy of hydrodynamic force calculation more than 3% in comparison to the standard Finite Difference solution of the Reynolds equation. Modelling of elastic bearings and contact seals The pump had 3 ball bearings, and one contact seal as indicated in Fig. The following analytical function has been used to model elastic non-linear and anisotropic elements such as ball Equations of motion Equations of motion of the rotor-bearing-foundation system can be written in the following form: ½ ¾ ½ ¾ MS ηs CS ηs + + M F η F C F ηf ½ ¾ KS ηs + = K F η F ½ ¾ F (ηs, η = F, η S, η F,t)+F S (t) (3) F (η S, η F, η S, η F,t)+F F (t) where η S =[x S,y S ] and η F =[x F,y F ] are vector of displacement of the shaft and foundation/casing respectively in transverse directions x and y, M S,C S,K S are mass, damping and stiffness matrices of the shaft and M F,C F,K F are mass, damping and stiffness matrices of the foundation. F S (t) and F F (t) denote external excitation forces on the shaft and foundation respectively.
4 Equations (3) are coupled by non-linear forces F (η S, η F, η S, η F,t) and hence are non-linear. Transformed to state-space u =(η S, η F, η S, η F ) equations (3) have the following form: du = f (t, u) (4) dt which is suitable for the numerical step by step time domain solution. Numerical algorithm Fehlberg 4-5 order Runge-Kutta method [9] has been adopted to solve (4) because it provided a good compromise between the accuracy and numerical efficiency for non-stiff equations typically encountered in rotordynamics. Fortran 95 code compiled and executed using Pentium III 85 MHz processor in Win operating system enabled computation of about 5 time steps per second for a system with 7 degrees of freedom containing 7 bearings, 3 of them hydrodynamic. This performance allows the non-linear rotordynamic computations to gain a status of a practical engineering tool in rotordynamic design and analysis. The code has been named TURBINE-PAK non-linear because it complements the previously developed code for linearized analysis of turbomachinery (Chalko and Li [1][6]). RESULTS The code computed the rotor-bearing-foundation (RBF) system response as a function of the system configuration (static load and alignment) bearings (types, geometry, lubricant properties etc.) shaft (geometry, pre-bending, material properties, temperature distribution etc.) foundation and casings dynamic properties (general condensed FEM superelement model) rotor speed excitation forces and unbalance The time step was chosen to correspond to 3 degrees of the shaft rotation, so there were 1 time steps per each rotation of the shaft. About 46 rotations were analyzed which corresponded to about 1.8 seconds of the pump operation at 14 rpm. Initial conditions were set to zero. Calculations took about 18 minutes on Pentium III 85 MHz computer. The rotor had an elastic coupling that coupled the motor with the shaft of the pump (see Fig ). It has been established that this coupling introduced an unbalance excitation to the shaft. For this reason, the model was excited by the unbalance located at the coupling as shown in Fig 5. After about rotations, the motion of the system became quasi-periodic. As the vertical equilibrium position of the shaft is not stable in the hydrodynamic suspension, the motion of the shaft in all 3 hydrodynamic bearings became dominated by hydrodynamic forces. The quasi-periodic motion is depicted in Fig 6. Development of the quasi-periodic limit cycle is shown in subsequent figures. At the bottom section of the shaft, supported in hydrodynamic bearings 5,6,7 hydrodynamic force excitation dominates and vibrations of the shaft have frequency of less than half of the frequency of rotation. The motor, supported in ball bearings is dynamically isolated from the pump by the elastic coupling. Shaft nodes between the elastic coupling and the seal 4 vibrate with essentially frequencies: the frequency of rotation that corresponds to the unbalance excitation and the frequency of the hydrodynamic excitation, which is slightly less than half of the frequency of rotation. Since these frequencies do not have a common multiple, the motion of this section of the shaft is non-periodic (Fig 7). Although the motion of the bottom section of the shaft is also non-periodic, the dominant frequency component there corresponds to the near half-speed hydrodynamic excitation (Fig 8, Fig 9). The relative motion of the shaft with respect to casing in the contact seal 4 is shown in Fig 1. The corresponding reaction is shown in Fig 1. The relative motion of the shaft with respect to casing in hydrodynamic bearings 5,6 and 7 are shown in Fig 11, Fig 14 and Fig 15 respectively. The corresponding reactions are shown in Fig13,Fig16andFig17. CONCLUSIONS Practical application of a fully non-linear rotordynamic analysis of a rotor-bearing-fondation system has been demonstrated using an example of a pump with hydrodynamically supported vertical rotor. It has been demonstrated that the non-linear approach in rotordynamics is not only theoretically possible, but can be effectively applied in engineering practice to predict the performance and dynamic behavior of rotor-bearing-foundation systems. The practical applicability of the non-linear approach to rotordynamics demonstrated in this article has been achieved by reducing the number of degrees of freedom (by using condensed FEM models) for the shaft and the foundation substructures and by optimizing the hydrodynamic force calculations Unbalance Max=.3E-3 Fig 5. Instantaneous position of the shaft (blue) and foundation/casing (red) during the quasi-periodic limit cycle vibration. Digits indicate bearings. Bearing 4 is a contact seal. Unbalance is located at the shaft node that corresponds to the elastic coupling.
5 .5 Motion of the shaft node Z Fig 6. Quasi-periodic motion of the shaft (red), seen from the bottom of the pump. Shown are trajectories in bearings. Every shaft rotation is coded in different color. Note that there are more than rotations per quasi-periodic cycle, that confirms the hydrodynamic excitation. Fig 8. Motion of the shaft node 13. Hydrodynamically excited half-speed twirl dominates. 1.E-5 1.E-5 8.E-6 6.E-6 4.E-6.E-6 -.E-6-4.E-6-6.E-6-8.E-6-1.E-5-1.E-5 Motion of the shaft node Motion of the shaft node 1 Fig 7. Motion of the shaft at node 7 (located between the contact seal 4 and the hydrodynamic bearing 5) is composed from frequencies that are not multiples of one another. Fig 9. Motion of the shaft node 1. Hydrodynamically excited half-speed twirl dominates.
6 1.E-5 1.E-5 8.E-6 6.E-6 4.E-6.E-6 -.E-6-4.E-6-6.E-6-8.E-6-1.E-5-1.E-5 Relative motion at bearing Reaction at bearing 4 Fig 1. Relative motion of the shaft with respect to casing at the contact seal 4. Fig 1. Reaction at the contact seal 4. The seal was modelled to have almost no stiffness, but significant viscous damping. Relative motion at bearing 5 Reaction at bearing 5 4.E-5 3.E E E E E E E-5 - Fig 11. Relative motion of the shaft with respect to the casing at hydrodynamic bearing 5 Fig 13. Reaction at hydrodynamic bearing 5 reflects its 4-pad design.
7 .6.5 Relative motion at bearing 6 4 Reaction at bearing Fig 14. Relative motion of the shaft with respect to the casing at hydrodynamic bearing 6 Fig 16. Reaction at hydrodynamic bearing 6 reflects its 4-pad design..6.4 Relative motion at bearing Reaction at bearing Fig 15. Relative motion of the shaft with respect to the casing at hydrodynamic bearing 7 Fig 17. Reaction at hydrodynamic bearing 7 reflects its 4-pad design.
8 Non-linear rotordynamic modelling and analysis has provided invaluable insight into the pump dynamics, enabling us to gain fundamental understanding of its non-linear behavior. A very important aspect of obtained results cannot actually be shown in this article - it is impossible to show here the animation of the computed non-linear vibration of the pump. Animation of computed behavior of the pump using a computer screen provides a very effective way of communicating computational results and therefore greatly assists engineers in gaining the understanding of non-linear phenomena in rotating machinery. ACKNOWLEDGMENTS The author is grateful to staff of Toshiba Corporation and Tokyo Electric Power Company in Japan for providing the data and discussing results of computations. REFERENCES [1] Chalko T.J., Li D.., Modelling Turbine Vibration in Terms of its Load Variation, International Journal of Rotating Machinery, 1995, Vol 1, No 3-4, pp [] Li D.., Dynamic Optimization of Multi-Bearing Rotors in Terms of System Configuration Parameters, PhD thesis, Department of Mechanical Engineering, University of Melbourne, Australia, 199. [3] J.M. Krodkiewski, L. Sun, Modelling of multi- bearing rotor system incorporating an active journal bearing, Journal of Sound and Vibration, Vol. 1, No. 1, p. 15-9, [4] J.M. Krodkiewski,J. Ding, Theory and experiment on a method for on-side identification of configuration of multi-bearing rotor system, Journal of Sound and Vibration, Vol. 164, No., p , [5] Chalko T.J., Li D.., Bearing Alignment Estimation in Rotating Machinery, Proceedings of 6-th International Symposium on Transport Phenomena and Dynamics of Rotating Machinery (ISROMAC-6) Honolulu, Hawaii, USA, Feb 3-8, 1996 [6] Chalko T.J., Li D.., Estimation of Alignment and Transverse Load in Multi-Bearing Rotor Systems International Journal of Rotating Machinery, 1996, Vol. 1, No. 7-8, pp [7] Krodkiewski J.M., Rotordynamics Lecture notes, The University of Melbourne, Dept of Mech Eng,, [8] Li D.., Chalko T.J., Hydrodynamic Characteristics of Combination Pad Journal Bearing, Proceedings of 6-th International Symposium on Transport Phenomena and Dynamics of Rotating Machinery (ISROMAC-6) Honolulu, Hawaii, USA, Feb 3-8, 1996 [9] Fehlberg E., Low-order classical Runge Kutta formulas with step size control, NASA TR R-315
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