Dynamic Modelling and Analysis of Vertical Machines

Transcription

1 LICENTIATE T H E SIS Dynamic Modelling and Analysis of Vertical Machines Erik Synnegård Computer Aided Design

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3 Dynamic Modelling and Analysis of Vertical Rotors Erik Synnegård Dept. of Engineering Science and Mathematics Luleå University of Technology Luleå, Sweden Supervisors: Jan-Olov Aidanpää and Rolf Gustavsson

4 Printed by Luleå University of Technology, Graphic Production 216 ISSN ISBN (print) ISBN (pdf) Luleå 216

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7 Abstract Since Sweden s hydropower plants are getting old and operating conditions have changed, there is a need for simulation tools to predict the functionality, durability and needs for maintenance. Also for breakdowns and major revisions, there is a benefit if dynamical problems could be predicted through simulations. The important parts in the modelling of hydropower machines are the interconnections (Bearings, Generator, Runner). Therefore, accurate and fast models are preferred to describe the interconnections and implement to the finite element model of the rotor. Since hydropower machines are oriented vertically the bearing coefficients needs to be described dynamically. This is due to the fact that vertical machines has no stationary operating point. In paper A simulations of a hydropower machine that suffered from resonance problems were performed. To reduce vibrations, a new type of viscoelastic supports was implemented between the supporting structure and bearing bracket. It was shown through simulations that this support would reduce the vibration levels and reduce the probability of resonance problems occurring when running at operating speed. In paper B the effect of cross-coupling bearing terms was investigated, commonly the cross-coupling terms are neglected since they are small in comparison to the radial components. This is shown to be true for some load cases, but if studying a vertical machine the load angle will change dynamically and for some load angles the cross-coupling terms cannot be neglected. Experiments on a vertical test rig with tilting pad bearings also agree well with the simulations performed. Since the tilting pad bearings in vertical machines introduce periodic coefficients in the bearing, studies in paper C shows that this periodic excitation can cause increased vibrations for certain operating conditions. v

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9 Contents Part I 1 Chapter 1 Thesis Introduction Overview of hydropower systems Introduction to rotordynamics Research question Chapter 2 Rotordynamics in hydro-electric machines Introduction Interconnections Chapter 3 Modelling Rotor Description Bearing model Numerical model Experimental setup Results Chapter 4 Conclusions and future work Conclusions Future work References 23 Part II 25 Paper A 27 Paper B 39 Paper C 61 vii

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11 Acknowledgments The research presented in this thesis was carried out as a part of Swedish Hydropower Centre - SVC. SVC has been established by the Swedish Energy Agency, Elforsk and Svenska Kraftnät together with Luleå University of Technology, KTH Royal Institute of Technology, Chalmers University of Technology and Uppsala University. Participating companies and industry associations are: Alstom Hydro Sweden, Andritz Hydro, E.ON Vattenkraft Sverige, Falu Energi & Vatten, Fortum Generation, Holmen Energi, Jämtkraft, Jönköping Energi, Karlstads Energi, Mälarenergi, Norconsult, Skellefteå Kraft, Sollefteåforsens, Statkraft Sverige, Sweco Energuide, Sweco Infrastructure, SveMin, Umeå Energi, Vattenfall Research and Development, Vattenfall Vattenkraft, Voith Hydro, WSP Sverige and ÅF Industry. I would like to thank my main supervisor professor Jan-Olov Aidanpää at Luleå university of technology for initiating this project and giving me the opportunity to do research and for his assistance and guidance. As well i would like to thank my assistant supervisor PhD. Rolf Gustavsson at Vattenfall R& D for his industrial expertise and help through this project. Finally i would also like to say thanks to my former colleagues at the division of solid mechanics and my current colleagues at computer aided design for making my time here enjoyable. A special thanks to my office mate Florian for taking his time to discuss technical problems as well as non work related issues. ix

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15 Chapter 1 Thesis Introduction 1.1 Overview of hydropower systems In the Swedish energy production, hydropower has been a core contributor for several decades. To date, it supplies approximately 65 TWh on a yearly basis and that is close to 4% of the total energy production in Sweden. With a large part of hydropower the energy system becomes robust, since hydropower is excellent at regulating the power output. However, most hydropower machines in Sweden were built in the middle of the twentieth century and many stations are in need of refurbishment. With increasing intermittent energy production such as wind power increases the amount regulating capabilities required. This unfortunately introduces unfavourable operating conditions for the hydropower machines e.g. changing blade and guide vane angles. Building better machines that can be operated in a wider range of operating conditions requires good models of the entire system. An important part of this system is the radial and thrust bearings of the system, which supports the rotor in radial and vertical direction. 3

16 4 Thesis Introduction 1.2 Introduction to rotordynamics Dynamics of rotating machines have been studied for more than one hundred years. In 1869 Rankine published a paper on whirling motions of a rotor [1]. Rankine concluded that rotating machines could never pass the first natural frequency since he neglected the imbalance of the system. In 1889 Gustav De Laval ran his steam turbine past the first critical speed and in 1919 Jeffcott [2] published the first paper where he confirmed the existence of stable overcritical operating speeds. Some limitations in Jeffcott s model is that the mass was represented as a point mass without inertia, this results in a natural frequency which is not depended on the rotating speed. A important part of rotordynamics is the influence of gyroscopic effects. Stodola [3] presented a paper in 1924 with a rigid disk with moment of inertia and mass on a flexible massless shaft. The gyroscopic terms in Stodola s model resulted in natural frequencies which are dependent on the rotating speed. This result is often visualised in Campbell diagram where the natural frequencies of the system is presented as function of the rotating speed. 1.3 Research question Since Sweden s hydropower plants are getting old and operating conditions have changed, there is a need for simulation tools to predict the functionality, durability and needs for maintenance. As well, for breakdowns and major revisions there is a benefit if dynamical problems could be predicted through simulations. This work is focused towards non-stationary events in hydropower machines, mainly start-ups and shut-downs. Since a large part of the hydropower machines in Sweden is supported by tilting-pad bearings, which change properties as function of rotating speed, load and position in the bearing, these effects should be included to model for example a start-up. Therefore the main focus in this thesis has been towards bearing modelling because of their importance to the stability of the machine. As well, fast and still accurate models needs to be evaluated to make simulations of non-stationary problems possible. Therefore, the research question is formulated as: Can fast simulation methods be used for solving non-stationary operating conditions in hydropower rotors with acceptable accuracy?

17 Chapter 2 Rotordynamics in hydro-electric machines 2.1 Introduction From the fundamental theories of rotordynamics it is possible to model the rotor of a hydropower machine. Complications occur when the interconnections that are part of the system are included. Extensive research has been conducted to develop theories on how bearings, generators and turbines affect the system. Almost all research that has been conducted has been focused on horizontal machines such as steam and gas turbines because of their importance in nuclear power and aviation industry. Rotordynamic studies on hydropower rotors is a relative new subject, if compared to steam and gas turbines. The main difference is that the rotor is orientated vertically and with low rotational speed, typically around 167 rpm. However, since the hydropower rotors have such large length and mass - around 1 m long and 3 tonnes of mass - dynamic problems can still occur. 5

18 6 Rotordynamics in hydro-electric machines Figure 2.1: Hydropower rotor with bearings, brackets, generator and runner [4] 2.2 Interconnections In hydropower, tilting pad bearings are commonly used for their good stability. The tilting pad thrust bearing was invented by Kingsbury and Mitchell. Mitchell also invented the tilting pad journal bearing and can be found in machines from as early as 1916 [5]. Lund [6] published his paper in 1964 where he introduced the pad assembly method and using Reynolds equation to solve the bearing coefficients in horizontal machines. The majority of the research on tilting pad bearings has been performed on horizontal machines. In these machines the dead weight of the rotor can be used to describe the force in the bearings, which allows a linear model to describe the dynamical behaviour. Glienicke et al [7] performed experiments on a horizontal rotor supported by tilting pad bearings. Childs et al [8] studied the difference between load on pad (LOP) and load between pad (LBP) for a 5 pad tilting pad bearing, see Figure 2.2.

19 2.2. Interconnections 7 (a) (b) Figure 2.2: Load between pad configuration (a) and load on pad configuration (b). Over the last 5 years bearing theory has evolved, T.Dimond et al [9] published a review of tilting pad bearing theory and the current state of the art now includes inertia effects, pad motions, thermal and mechanical deformations. However, including these effects in a rotordynamical analysis requires detailed models and would take to long time to simulate. In vertical machines the radial bearing load cannot be calculated using the dead weight of the rotor. The bearing load for a vertical machine is dependent on the eccentricity in the bearing which comes from unbalance loads e.g. mechanical, electromagnetic and fluid forces. Hence, in a vertical machine the bearing characteristics will change dynamically. To model this, Reynold s equation has been used to describe the bearing coefficients dynamically by M.Cha [1] and White et.al [11]. Using this approach, the bearing coefficients must be solved at each time-step, which slows down the simulation and makes parametric rotordynamical studies impractical. Therefore when evaluating a large parameter space simplified methods are preffered.

20 8 Rotordynamics in hydro-electric machines Nässelqvist et.al [12] proposed a method to describe the bearing coefficients dynamically as function of eccentricity and load angle. This method assumes that the shaft position in the bearing is static at each time-step. Hydropower generators is giving rise to unbalance magnetic pull (UMP), this force is dependent on the air gap between the rotor and the stator. Typically the force can be divided in two parts, radial force f r and tangential force f t. This force has a nonlinear behaviour as function of eccentricity in the generator. However, if the eccentricity is less then 1% of the air gap this behaviour can be considered linear [13]. In this region the UMP can be modelled as, f r = k R u r (2.1) f t = k T u t (2.2) where k R and k T is the magnetic stiffness in the radial and tangential direction. The UMP is also dependent on the whirling frequency, see Figure 2.3. Considering that there is a purely dynamic eccentricity, i.e whirling ratio Θ =1 the tangential force f t =. For this case, if considering low eccentricities the UMP can be modelled as, f M = K M u (2.3) where k R u x k M = k R,u= u y u θ (2.4) u φ

21 2.2. Interconnections Force [kn] whirl ratio [ ] Figure 2.3: Tangential and radial magnetic pull force as function of whirl ratio. A more realistic model would be to consider the generator shape. Since the shape of the generator can vary, (thermal expansion, tolerances etc.) this will affect the interaction between the generator and stator [14, 15]. Current research shows that each component in the hydropower machine can be linearized in some region. But if non-stationary analysis is of interest, for example start-ups, shut-downs or sudden change in operating conditions more detailed models are required to model the behaviour.

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23 Chapter 3 Modelling 3.1 Rotor Description In Figure 3.1 a finite element model (FEM) of a Jeffcott rotor can be seen. This model consists of 5 nodes for the rotor and 2 for the supporting bearing brackets. For more complex models, such as a hydropower rotor see Figure 3.2 d D Figure 3.1: 5 node finite element model of a Jeffcott rotor. 11

24 12 Modelling Exciter Rotor Runner UGB LGB TGB Figure 3.2: 52 node finite element model of a hydropower rotor. When performing non-stationary analysis the computational time can increase fast, and then it is important to optimize the system. One thing can be system reduction, where Guyan reduction [16] is most common reduction technique. However, a dynamical reduction method, for example the Improved Reduction System (IRS) [17] can also be used to reduce the number of degrees of freedom (DOF), without changing the lower eigenfrequencies. 3.2 Bearing model Since tilting pad bearings have different properties depending on where the load is applied, (LOP or LBP). Hence, for a vertical machine where the load angle will change for each time-step this results in periodic bearing coefficients with period time n Ωt for a synchronous whirl, where n is the number of pads and Ω is the rotating speed. Solving the bearing coefficients using Reynolds equation in Equation. 3.1 a finite difference method is used. 1 ( h 3 p ) + h 3 2 p R 2 θ θ x =6μΩ h 2 θ +12μ h (3.1) t From the calculated pressure distribution the load F (x, y, ẋ, ẏ) is derived by integration. Using a small perturbation the bearing coefficients can be derived, for example k xx. k xx = F x x = F x(x +Δx, y,, ) F x (x, y,, ) Δx (3.2) Similar process can be applied for k xy,k yy,c xy and c yy. Depending on grid size this process can become expensive numerically, and if this process is repeated for each time-step this would result in long simulation time.

25 3.2. Bearing model K ξξ K ηη 14 K ξξ LOP K ξξ LBP Stiffness N/m Kξξ N/m Eccentricity direction α [deg] eccentricity e/c b [%] Figure 3.3: Stiffness as function of α at 3% eccentricity in the coordinate system (ξ,η) (left) and stiffness LOP and LBP as function of eccentricity (right). Using a commercial bearing software [18] or the Reynolds equation described earlier, the bearing properties can be calculated for different load angles and eccentricities for a fixed load in the local coordinate system (ξ,η) see Figure 3.4. In Figure 3.3 the stiffness in the bearing is presented for such a calculation both as function of load angle and as function of eccentricity. Here, represent LOP and ±45 represent LBP. From Figure 3.3 it can be seen that the difference between LOP and LBP increases for higher eccentricities which will increase the periodic excitation of the machine. To model the bearing coefficients according to Nässelqvist s method [12], the stiffness and damping coefficients are calculated in a local coordinate system (ξ,η) for different eccentricities and load angles α, see Figure 3.4. Once the bearing coefficients as function of eccentricity is known, a polynomial fit is made to describe the coefficients as function of eccentricity k ij (e) for the extreme cases LOP and LBP. Since the load angle will change dynamically during operation, this parameter must be included.

26 14 Modelling η y k ξ k α O e x Figure 3.4: 4-pad bearing geometry, with coordinate system (ξ,η) rotated α degree s from the fixed coordinate system (x,y). If assuming a harmonic function between LOP and LBP the bearing coefficients as function of (e, α) can be describe as. k ij (α, e) = klop ij (e)+kij LBP (e) + klop ij (e) k LBP { ij (e) cos(4α) i = j (3.3) 2 2 sin(4α) i j c ij (α, e) = clop ij (e)+c LBP ij (e) + clop ij (e) c LBP { ij (e) cos(4α) i = j (3.4) 2 2 sin(4α) i j For each time-step the bearing coefficients are calculated and transformed to the fixed coordinate system (x, y) using the following transformation [19]. where [ T = K T = T T K B T (3.5) C T = T T C B T (3.6) cos(α) sin(α) ] sin(α) cos(α) (3.7) where K B and C B is the bearing stiffness and damping matrices in the local coordinate system (ξ,η). Using this method the stiffness and damping coefficients can be described for all eccentricities and load angles see Figure 3.5

27 3.2. Bearing model Kξξ N/m 3.6 Kξη N/m Eccentricity direction α [deg] Eccentricity direction α [deg] Kηξ N/m.8 Kηη N/m Eccentricity direction α [deg] Eccentricity direction α [deg] Figure 3.5: Stiffness as function of load angle at 3% eccentricity (e/c b ). (o) Represent simulations performed in the bearing program and solid line is Equation 3.3

28 16 Modelling 3.3 Numerical model From the rotor model, using Timoshenko beam elements the equation of motion can be written as: M q +(C +ΩG) q + Kq = (3.8) where M is the mass matrix, C is the damping matrix, G is the gyroscopic matrix, K is the stiffness matrix and Ω is the rotational speed. Interconnections can now be added to the equation of motion, the bearing properties which are calculated at each time-step as function of eccentricity e and load angle α. M q +(C + C T +ΩG) q +(K + K T )q = (3.9) If the bearings are connected to ground through a bearing bracket the bracket stiffness matrix H is also added. Depending on what load cases are studied, Unbalance (F ub ), Static (F s ) or Nonlinear (F nl ) these are added to the right hand side. M q +(C + C T +ΩG) q +(K + K T + H)q = F ub + F s + F nl (3.1) The equation of motion is rewritten using state-space formulation and solved using Runge-Kutta time integration. At each time step the bearing matrices are calculated according to equation 3.3 in the local coordinate system (ξ,η) and transformed to the fixed coordinate system (x, y). The equation of motion 3.1 is then updated and solved for the next time step. 3.4 Experimental setup To validate the bearing modelling a experimental rig is used, see Figure 3.6. With this rig, unbalance response studies can be performed. As well studying different operating speeds and start-ups. The measuring equipment consist of inductive displacement sensors at the bearings and strain gauges on the bearing bracket to measure bearing loads.

29 3.5. Results Results Figure 3.6: Experimental test rig In Figure 3.7 and 3.8 the displacement and force orbit in the bearing is presented for a vertical Jeffcott rotor, see Figure 3.1 with a 4-pad tilting pad bearing for pure unbalance response. As seen the 4-pad bearing produces a squared shape orbit in the fixed coordinate system, this is due to the periodicity of the bearing coefficients. Maximum force is achieved at the pad while maximum displacement is achieved between the pad. Recall from figure 3.3, here the difference between LOP and LBP increases for increased eccentricity, hence the periodic excitation in the bearing will increase for increased load. In Figure 3.8 the rotating speed is increased and hence the unbalance load is larger. Here the increased effect of the periodic excitation can be seen. FFT of the response, see Figure 3.9 show that frequency components are present at 3Ω and 5Ω in the stationary coordinate system. Since the 3Ω component is relatively large it could cause resonance problems if the machine is operating in the region Ω/ω n =.33 if the damping in the system is low.

30 18 Modelling.6 2 Displacement y [mm] Force y [N] Displacement x [mm] Force x [N] Figure 3.7: Experimental data (gray) and numerical simulation (black) of the bearing displacement and force orbit for unbalance magnitude m e = kgm and rotating speed Ω = 23 rpm.6 4 Displacement y [mm] Force y [N] Displacement x [mm] Force x [N] Figure 3.8: Experimental data (gray) and numerical simulation (black) of the bearing displacement and force orbit for unbalance magnitude m e = kgm and rotating speed Ω = 29 rpm

31 3.5. Results FFT Magnitude FFT Magnitude Frequency [Hz] (a) Frequency [Hz] (b) Figure 3.9: Fourier transformed force signal in the bearing from experiments (a) and from simulations (b). Black line is with rotating speed Ω = 23 rpm and red Ω = 29 rpm Simulations of a vertical Jeffcott rotor with a more slender shaft was performed to investigate this behaviour. Since there is less damping in the system and the eigenfrequency is lower a parametric study can be performed, see Figure 3.1. Here, the dashed lines represent Ω,3Ω,5Ω. For this case the operating speed was set in a range corresponding to Ω/ω 1 =(.18.38), where w 1 is the first natural frequency (w 1 = 1 rad/s). For each speed a total of 1 periods was recorded to see if the first natural frequency can be excited due to the bearing dynamics. From Figure 3.11 it is clear that resonance peaks are present when operating at Ω/ω 1 =.33. At this load the eccentricity in the bearing is around 4% of the bearing clearance, in Figure 3.3 it could be seen that the difference between LOP and LBP is large for this level of eccentricity and hence the periodic excitation is large.

32 2 Modelling Eigenfrequency [rad/s] 1 9 Ω/ω1=( ) Rotating speed [rpm] Figure 3.1: Campbell diagram for constant stiffness at 3% eccentricity Displacement [m] Ω/ω n [-] Figure 3.11: Displacement at the disk as function of rotating speed.

33 Chapter 4 Conclusions and future work 4.1 Conclusions From the different studies performed in this thesis, the dynamical behaviour of tilting pad bearings in vertical machines has been investigated. It is clear that the tilting pad bearings introduce periodic coefficients due to the difference between load on pad and load between pad. The simulation technique used in these studies, which is considerably faster if compared to solving Reynolds equation also agrees well with measurements. The 4-pad tilting pad bearing used mainly in this thesis also introduces higher frequency components to the response. These frequency components have been shown to excite the system for large bearing loads, due to the difference between load on pad and load between pad increases for increased load. 4.2 Future work The work presented in this thesis has been focused mainly on a small vertical machine with 4-pad tilting pad bearings. As a result, the future work will be to use the models from this case to a hydropower machine. Also, investigate the rotating speed dependency of the bearings and how the unbalance magnetic pull and turbine loads should be incorporated. 21

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35 References [1] W J Rankine. On the centrifugal force of rotating shafts. Engineer, 27: , [2] H. H. Jeffcott. The Literal Vibration of Loaded Shafts in the Neighborhood of a Whirling Speed. Philosophical Magazine, Series 6, 37(219):34 314, [3] A Stodolda. Dampf- und Gasturbinen. Springer Berlin Heidelberg, Berlin, Heidelberg, [4] M. Nässelqvist, R. Gustavsson, and J.-O. Aidanpää. Bearing Load Measurement in a Hydropower Unit Using Strain Gauges Installed Inside Pivot Pin. Experimental Mechanics, 52(4): , 212. [5] J E L Simmons and S D Advani. Michell and the development of tilting pad bearings. (iii):49 56, 191. [6] J. W. Lund. Spring and Damping Coefficients for the Tilting-Pad Journal Bearing. A S L E Transactions, 7(4): , [7] J Glienicke, D Han, and M Leonhard. Practical determination and use of bearing dynamic coefficients. (December):297 39, 198. [8] Dara W. Childs and Clint R. Carter. Rotordynamic Characteristics of a Five Pad, Rocker-Pivot, Tilting Pad Bearing in a Load-on-Pad Configuration; Comparisons to Predictions and Load-Between-Pad Results. Journal of Engineering for Gas Turbines and Power, 133(8):8253, 211. [9] Timothy Dimond, Amir Younan, and Paul Allaire. A review of tilting pad bearing theory. International Journal of Rotating Machinery, 211:1 23,

36 24 References [1] Matthew Cha and Sergei Glavatskih. Nonlinear dynamic behaviour of vertical and horizontal rotors in compliant liner tilting pad journal bearings: Some design considerations. Tribology International, 82: , 215. [11] Maurice F White, Erik Torbergsen, and Victor A Lumpkin. Rotordynamic analysis of a vertical pump with tilting-pad journal bearings. 27: , [12] Mattias Nässelqvist, Rolf Gustavsson, and Jan-Olov Aidanpää. Experimental and Numerical Simulation of Unbalance Response in Vertical Test Rig with Tilting-Pad Bearings. International Journal of Rotating Machinery, 214:1 1, 214. [13] L Lundström, Rolf Gustavsson, Jan-Olov Aidanpää, N Dahlbäck, and Mats Leijon. Influence on the stability of generator rotors due to radial and tangential magnetic pull force. I E T Electric Power Applications, 1(1):1 8, 27. [14] N. L P Lundström and Jan Olov Aidanpää. Dynamic consequences of electromagnetic pull due to deviations in generator shape. Journal of Sound and Vibration, 31(1-2):27 225, 27. [15] N. L P Lundström and Jan Olov Aidanpää. Whirling frequencies and amplitudes due to deviations of generator shape. International Journal of Non-Linear Mechanics, 43(9):933 94, 28. [16] R. J. GUYAN. Reduction of stiffness and mass matrices. AIAA Journal, 3(2):38 38, feb [17] Michael I. M.I. Friswell, S.D. Garvey, and J.E.T. Penny. Model reduction using dynamic and iterated IRS techniques. Journal of Sound and Vibration, 186(2): , [18] Rotordynamic-Seal Research. [19] D.; Plesha M.; Witt R.; Cook R.; Marlkus. Concepts and Applications of Finite Element Analysis. 22.

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39 Paper A Modelling of visco-elastic supports for hydropower applications Authors: Erik Synnegård, Rolf Gustavsson and Jan-Olov Aidanpää Reformatted version of paper originally published in: Proceedings of the 9th IFToMM International Conference on Rotor Dynamics,

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41 Modeling of visco-elastic supports for hydropower applications Erik Synnegård 1, Rolf Gustavsson 2, and Jan-Olov Aidanpää 1 1 Luleå University of Technology, Luleå, Sweden Department of Engineering Sciences and Mathematics, erik.synnegard@ltu.se, +46 () joa@ltu.se, +46 () Vattenfall R&D, rolf.gustavsson@vattenfall.com, +46 () Abstract. This paper presents a numerical model of how a visco-elastic support affects the dynamical response of a 42 MW Kaplan turbine that is experiencing resonance problems. The supports are placed between the bearing bracket and the supporting concrete structure. Since the supports are nonlinear, the nodal displacements are solved using a Runge-Kutta time integration method where the visco-elastic supports are implemented as a nonlinear force. To reduce calculation time the number of degrees of freedom of the rotor model is reduced using the Improved Reduction System. Excitation of the system is implemented as a stationary force in the runner with varying frequency. The resulting nodal displacement from the transient simulation is then compared to the system simulated without the supports to show how the machine dynamics are affected. The simulations show that the visco-elastic supports efficiently reduces the displacements in the lower vibration modes. The reduced vibration levels should decrease the probability of resonance problems occurring when running at operating speed. Keywords: Visco-elastic damper, Nonlinear, Hydropower, Transient 1 Introduction Hydro-electric power is an important source of energy in Sweden, but since the majority of the power stations where built in the middle of the twentieth century, most stations are old and in need of refurbishment. What makes it important is the ability to regulate the power output. Unfortunately this also introduces unfavourable operating conditions, e.g. operating the machine in regions where there can exist large excitation forces. In Fig.1, measurements of a failure in a Swedish hydropower rotor can be seen.

42 Fig. 1: Measurements from the hydro-power rotor going into resonance [1] The measurements were performed by [1], and it shows the machine going into resonance. The machine is now undergoing major refurbishments where one of the key features is the installation of eight visco-elastic supports to decrease the resonance levels when running at unfavourable operating conditions. This article focuses on the visco-elastic support elements placed between the bearing bracket and the supporting structure at the Upper Guide Bearing (UGB). Using visco-elastic materials as vibration absorbers for rotor-dynamics have been studied earlier by Lund [2] and Dutt [3]. In this article, simulations are performed to investigate the machine dynamics when incorporating this new element. Since the visco-elastic support is nonlinear, it is simulated as a force. The equation of motion is then solved for different excitation frequencies to investigate how the supports reduce vibrations. 2 Computational model The rotor model can be seen in Fig. 2. It consists of 52 conical elements and there are six important nodes for interconnections, three tilting pad journal bearings, generator, exciter and runner. Timoshenko beam elements are used to incorporate shear, rotary inertia and gyroscopic ef-

43 Exciter Rotor Runner UGB LGB TGB Fig. 2: Computational model fects, and the assembly of the matrices are performed according to [4]. The tilting pad journal bearings are simplified and assumed to have constant stiffness and damping coefficients, since at this stage the element of interest is the visco-elastic support. For small displacements, the Unbalanced magnetic pull from the generator and exciter can be considered as a linear function of the rotor displacement given by constant isotropic negative stiffness [5]. f M = K M u (1) The general machine data is summarized in Table 1. For the tilting pad bearings, the stiffness and damping properties are calculated at nominal speed Ω = 167 rpm at a bearing load of 5 kn using a commercial software [6]. Table 1: Properties of the hydropower rotor Symbol Item Exciter Rotor Runner m mass [kg] J d diametrical moment of inertia [kg.m 2 ] J p polar moment of inertia [kg.m 2 ] k M Magnetic stiffness [N/m] X UGB LGB TGB k xx, k yy Bearing stiffness [N/m] c xx, c yy Bearing damping[n.s/m] k bracket Bearing bracket stiffness [N/m] E Young s modulus [N/m 2 ] ν Poisson s ratio.3 ρ Density [kg/m 3 ] 781

44 2.1 Visco-elastic damping element The stiffness and damping of the visco-elastic support is described as k ext =22.9 kn/mm and c ext = 5 kn/(mm/s).2. Since the damping term is nonlinear, the problem is solved using a nonlinear force description to represent the damping term. In Fig. 3 (a) the characteristics for the viscoelastic damper are presented. From this we see that the force is high for low velocities, when compared with a regular linear damper. Since the displacements and velocities are relatively small, this behaviour will give a high restoring force. Visco-elastic damper characteristics 5 Force (kn) k 1 c 1 y 1 5 y 2 k 2 y Velocity (mm/s) k 3 k ext c ext Visco-elastic damper Linear damper (a) (b) Fig. 3: Damping force for the visco-elastic damper vs a linear damper (a) and the implementation of the visco-elastic supports (b) As described earlier, the dampers are placed between the bearing bracket and the supporting structure as shown in Fig. 3 (b). Here k 1 and c 1 represents the bearing properties, k 2 the bearing bracket, k 3 is the stator and k ext and c ext the visco-elastic support. From Lagrange energy equations, the system can then be modeled as, M q + C q + Kq = (2) where q = {x 1,y 1,x 2,y 2,x ext,y ext } T. The stiffness matrix K and damping matrix C that is given from Lagrange

45 energy equation can be seen below in eq But since the term c ext has a nonlinear dependency with regard to the velocity it is extracted from the matrix and modeled as a force. k 1xx k 1xy k 1xx k 1xy k 1yx k 1yy k 1yx k 1yy K = k 1xx k 1xy k 1xx + k 2 k 1xy k 2 k 1yx k 1yy k 1yx k 1yy + k 2 k 2 k 2 k 2 + k ext k 2 k 2 + k ext c 1xx c 1xy c 1xx c 1xy c 1yx c 1yy c 1yx c 1yy C = c 1xx c 1xy c 1xx c 1xy c 1yx c 1yy c 1yx c 1yy c ext c ext Hence by extracting the nonlinear damper coefficient, the equation of motion for the whole rotor can be described as. (3) (4) M q +(C + ΩG) q + Kq = f(t)+f nl ( q) (5) Where f( q) is the visco-elastic damper modeled as a force, and f(t) is the excitation force acting on the turbine and described as f(t) =F sin(ω t) (6) The force f(t) is modeled as a stationary force in the runner at different excitation frequencies ω, to verify at what frequencies resonance can occur and how the visco-elastic damper reduces vibrations.

46 2.2 Model Reduction When running simulations with parametric studies, simulation time becomes more important. To decrease the simulation time without changing the system properties, the rotor is reduced using the Improved Reduction System (IRS) method. The important nodes, e.g Exciter, Generator, Runner and Bearings are selected as Master nodes, while most shaft sections are set as Slave nodes. More information about this method is described in [7], and the transformation matrix is given by. T IRS = T s + SMT s M 1 R K R (7) The model is reduced from 52 nodes (28 DOF s) to 7 nodes (28 DOF s). Table 2 shows the difference in damped natural frequency and damping ratio for the reference case with 52 nodes and the reduced model with 7 nodes. Table 2: Damped natural frequency and damping ratio comparison. ω d ζ 2.3 Simulation method Mode 1 52 Nodes NodesIRS Mode 2 52 Nodes NodesIRS Mode 3 52 Nodes NodesIRS The equation of motion (5) is rewritten using a state-space formulation and solved using a Runge-Kutta time integration. The maximum response when reaching steady-state for different excitation frequencies is then calculated to retrieve the systems frequency response. 3 Results Figure 4-6 shows the maximum of the absolute displacement, U = x 2 + y 2 after reaching steady-state as a function of excitation frequency ω. The maximum response in the three bearings, generator and runner are investigated, and the reference response is compared to the response when the visco-elastic supports are included.

47 3 1 5 Response UGB Displacement (m) Excitation frequency ω (rad s 1 ) Reference With visco-elastic Fig. 4: Maximum response for the reference system (blue) and the system with the visco-elastic supports (red) as function of excitation frequency ω 1 5 Response LGB Displacement (m) Damped Natural Freq = 59.1 rad/s Excitation frequency ω (rad s 1 ) Reference With visco-elastic (a) (b) Fig. 5: Maximum response for the reference system (blue) and the system with the visco-elastic supports (red) as function of excitation frequency ω in (a) and the turbine mode shape for a damped natural frequency of 59.1 rad/s in (b).

48 In Fig. 4 the maximum of the absolute displacement in UGB are presented, here a decrease in vibrations can be observed when the viscoelastic supports are included. The results for the displacements in the Lower Generator Bearing (LGB), see Fig 5 also show some increased stability for the lower frequency responses, but for the higher turbine mode there is no improvement in the response. This is because the turbine mode shape have small displacements and velocities at the upper part of the shaft, where the visco-elastic dampers are located. This reduces the potential to absorb energy when large vibrations occur in the turbine. In the generator, which is close to UGB, there is a decrease in the resonance peaks for the lower frequencies, since these modes have a lot of displacement in the upper part of the shaft, see Fig.6 (a). To reduce the resonance peaks in the runner, see Fig. 6 (b), placing the visco-elastic supports at the UGB bracket has small effect. 1 4 Response Generator 1 3 Response Runner 2 Displacement (m) Displacement (m) Excitation frequency ω (rad s 1 ) Reference With visco-elastic (a) Excitation frequency ω (rad s 1 ) Reference With visco-elastic (b) Fig. 6: Maximum response for the reference system (blue) and the system with the visco-elastic supports (red) as function of excitation frequency ω in the generator (a) and runner (b)

49 4 Discussion The aim of this paper was to investigate if visco-elastic supports can be used to reduce vibrations in hydropower rotors. The results in Fig. 4-6, shows that the visco-elastic supports efficiently reduces large vibrations, if they are placed correctly. Since the mode shape associated with the generator, have relatively large displacements in UGB, the visco-elastic supports will absorb energy when vibrations occur in regions close to UGB. But for the turbine modes there are small displacements in UGB, and hence the supports cannot absorb energy, and the vibration levels will be unaffected. If resonance problems are frequent in the runner, a better solution would be placing the supports behind the LGB or TGB bracket where the displacement magnitude is larger for the turbine mode. Overall the supports look very promising for hydropower applications, here follows some key benefits. Large restoring forces for low displacements/velocities. Easy to install, no larger modifications needed since they are placed behind the bearing bracket. 5 Acknowledgement The research presented was carried out as a part of Swedish Hydropower Centre - SVC. SVC has been established by the Swedish Energy Agency, Elforsk and Svenska Kraftnät together with Luleå University of Technology, The Royal Institute of Technology, Chalmers University of Technology and Uppsala University.

50 Bibliography [1] Jan-Olov Aidanpää Mattias Nässelqvist, Rolf Gustavsson. Resonance problems in vertical hydropower unit after turbine upgrade. In 24th Symposium on Hydraulic Machinery and Systems, 212. [2] Jorgen W. Lund. The stability of an elastic rotor in journal bearings with flexible, damped supports. Journal of Applied Mechanics, 32(4): , URL [3] J.K. Dutt and B.C. Nakra. Stability of rotor systems with viscoelastic supports. Journal of Sound and Vibration, 153(1):89 96, ISSN 22-46X. doi: C. URL pii/2246x929629c. [4] M. I. Friswell. Dynamics of rotating machines. Cambridge University Press, Cambridge New York, 21. ISBN [5] Mattias Nässelqvist. Simulation and Characterization of Rotordynamic Properties for Vertical Machines. PhD thesis, Luleå University of Technology, December 211. [6] Rotordynamic-Seal Research. URL [7] M.I. Friswell, S.D. Garvey, and J.E.T. Penny. Model reduction using dynamic and iterated {IRS} techniques. Journal of Sound and Vibration, 186(2): , ISSN 22-46X. doi: org/1.16/jsvi URL science/article/pii/s2246x [8] R. J. Guyan. Reduction of stiffness and mass matrices. AIAA Journal, 3:38+, 1965.

51 Paper B Influence of cross-coupling stiffness in tilting pad journal bearings for vertical machines Authors: Erik Synnegård, Rolf Gustavsson and Jan-Olov Aidanpää Reformatted version of paper submitted to: Submitted to International Journal of Mechanical Sciences, October

52 4

53 Influence of cross-coupling stiffness in tilting pad journal bearings for vertical machines Erik Synnegård a,, Rolf Gustavsson b, Jan-Olov Aidanpää a a Luleå University of Technology, Luleå, Sweden b Vattenfall Research and Development, Älvkarleby, Sweden Abstract This paper evaluates how cross-coupling coefficients affects the dynamics of a vertical rotor with tilting pad journal bearings. For vertical machines, the bearing properties are dependent on the bearing load and direction. As a result, it generally requires that bearing properties are calculated at each time step using the governing fluid dynamic equations. This method gives a good representation of the bearing but computational time increases and can cause stability problems. In this study the bearing properties are instead modelled as a function of eccentricity and its direction. Hence, the bearing properties can be evaluated at each time step without solving Navier-Stokes or Reynold s equation. The main advantage of using this method is to decrease the computational time. The cross-coupling stiffness and damping coefficients are usually neglected since they are small compared to the radial stiffness and damping coefficients. In this paper, the simulated unbalanced response is compared to experimental results and it is seen that the crosscoupling stiffness for vertical machines can influence the dynamics. It is shown that the cross-coupling can be of the same order of magnitude as the radial stiffness component depending on the shaft angular position in the bearing. Including cross-coupling increases higher frequency components and the experiments show similar behaviour. Hence the cross-coupling stiffness and damping coefficients should be included when simulating vertical machines subjected to high loads or when the detailed dynamical behaviour is important to investigate. Keywords: Vertical rotor, journal bearing, cross-coupling stiffness 1. Introduction Tilting pad journal bearings (TPJB) are widely used as supports for hydroelectric power plants because of their good stability. During the last decades, the hydropower industry has been shifting from base electricity production to more regulating power. This introduces some negative effects such as increased wear and fatigue. Building new machines requires better and more efficient models for optimizing the design. When performing rotordynamic analysis with TPJB s, the challenge is to describe the stiffness and damping coefficients dynamically. As early as 1964, Lund [1] published a paper with stiffness and damping coefficients for TPJB s in horizontal machinery. For vertical machines, it is hard to find a dynamical model for the TPJB s, in comparison to a horizontal machine where there exists a stationary working point. In horizontal machines the dead weight of the rotor is usually sufficient to describe the force in the bearings and a linear model is normally used to describe the dynamic behaviour. Many studies have been done for horizontal machines, Glienicke et al. [2] performed experiments on a horizontal rotor supported by TPJB s. Studies Corresponding author addresses: erik.synnegard@ltu.se (Erik Synnegård), Rolf.Gustavsson@vattenfall.se (Rolf Gustavsson), joa@ltu.se (Jan-Olov Aidanpää) Preprint submitted to International Journal of Mechanical Sciences February 4, 216

54 on differences between load on pad (LOP) and load between pad (LBP) were performed by Childs [3]. In industry, most of the calculations on vertical machines are performed considering constant values for the bearing properties [4]. However, the stiffness and damping coefficients will depend on the shaft angular position in the bearing. Hence, since there is no static operating point the difference between LOP and LBP will introduce dynamic bearing coefficients. Similarities can be seen from cracked rotors which also introduce time dependent coefficients [5],[6]. To describe the bearing coefficients dynamically in vertical machines, Navier-Stokes or Reynold s equations needs to be solved. It is possible to describe the bearing dynamically by using the simplified Reynold s equation. Comparing with horizontal machines, there has not been as much studies performed on vertical machines with TPJB s. M. F. White et.al [7] investigated how radial clearance influenced the response using Reynold s equation to calculate the nonlinear bearing coefficients. M. Cha et.al [8] shows the difference between horizontal and vertical orientation of TPJB s using Comsol Multiphysics where Reynold s equation is solved. R. Cardinall et.al [9] modelled a hydropower rotor with TPJB s and used Reynold s equation to describe the coefficients. All of these studies require that Reynolds equation is solved at each time step to update the stiffness and damping coefficients. As a result, simulation time increases and performing parametric studies becomes impractical. As well, simulations of large hydropower systems with many degrees of freedom and large time periods requires a fast and good bearing description. This paper evaluates how cross-coupling stiffness and damping coefficients influence the response of a vertical rotor with two TPJB s where the general conclusion is that bearing cross-coupling terms are small and usually is neglected. A similar study has been performed by Nässelqvist [1] where he proposed a method to model the bearings using a bearing description as function of eccentricity and load angle. The method used in this paper is based on that model with some small modifications to investigate if cross-coupling coefficients are required to describe the dynamical behaviour of a vertical machine with tilting pad bearings. Using this method to model the bearing is much more time efficient then compared to using Reynolds equation. However, since this method considers the bearing dynamics quasi static at each time step some information is lost. 2. Method 2.1. Rotor description The test rig is modelled using 5 nodes with four degrees of freedom [x, y, θ, φ], see Figure 1. Each element is described using Timoshenko beam elements and consistent mass matrix, see [11] for beam and disc description. The rotor is supported by two radial tilting pad bearings connected through brackets to the surrounding structure. Specific parameters for the rotor is: shaft diameter d =49.84 mm, disk diameter D = 168 mm and length between the two bearing centre L = 5 mm. The rotor is connected to the motor through a jaw coupling and stinger to support the machine vertically. The general rotor data is presented in Table 1 and more information about the test rig can be found in [12]. This rotor configuration can be considered almost rigid, therefore the bearing dynamics can be isolated. In Figure 2 the Campbell diagram is presented, here the rotors eigenfrequencies are presented, with the first eigenfrequency at around 5Ω at Ω = 25 rpm. The Campbell diagram is calculated at 3% eccentricity using constant stiffness and damping in the bearing. This rotor design is selected since the only dynamic components of this machine are the bearings and unbalance loads. 2

55 Node 1 H H Node 2 d Node 3 D Node 4 Node 5 H H Figure 1: Test rig setup 6 Eigenfrequency [rad/s] Ω 1 3Ω Ω Rotating speed [rpm] Figure 2: Campbell diagram for the test rig considering the bearing coefficients fixed, here the dashed lines represent Ω, 3Ω and 5Ω 3

56 2.2. Bearing model Using a commercial software [? ], the bearing coefficients are calculated for different eccentricities and locations in the bearing. This program utilizes the Navier-Stokes equation to calculate the bearing coefficients. A polynomial fit is made to describe the stiffness and damping as a function of eccentricity e and angular position in the bearing α. The rotor and bearing properties can be seen in Table 1 and a sketch of the bearing used in the test rig is displayed below in Figure 3. η y k ξ R k α e O x Rp Cb θ θ p (a) (b) Figure 3: Sketch of the test rig tilting pad bearing with a eccentricity e in the ξ direction. Table 1: The test rig rotor and bearing properties. Rotor Symbol Description Item Value Unit d Shaft diameter mm D Disc diameter 168 mm e m Mass unbalance radius 7 mm H Bearing bracket stiffness 5 MN/m L Shaft length 5 mm Bearing Symbol Description Item Value Unit R Shaft radius mm R b Bearing radius mm R p Pad radius mm θ Pad angle 72 deg N No. of pads 4 - C b R b R Radial bearing clearance.125 mm C p R p R Radial pad clearance.155 mm m 1 C b /C p Preload factor.19 - x θ b /θ Pad pivot offset.6-4

57 4.2 x K ξξ K ηη 6 x 15 4 K ξη K ξη Stiffness N/m Stiffness N/m Eccentricity direction α [deg] Eccentricity direction α [deg] Figure 4: Stiffness as a function of load angle at 3 % eccentricity (e/c b ). The rotor-bearing setup is orientated in the fixed (x, y) coordinate system. When calculating the bearing properties, the local coordinate system (ξ,η) is used to evaluate the stiffness and damping for a constant eccentricity in the ξ-direction. The bearing stiffness and damping in the (ξ,η) coordinate system will vary with respect to α whichisdefinedastheanglebetweenthecoordinateaxisξ and the fixed coordinate axis x, see Figure 3. First the stiffness and damping coefficients as function of α [ 45, 45 ] are calculated. In Figure 4 the stiffness in the coordinate system (ξ,η) as function of α is presented. Here corresponds to the peg location and the interval investigated is [ ] from the peg location. In the bearing program the shaft angular position α and eccentricity e is considered fixed for each simulation. From Figure 4 it is clear that the radial stiffness k ξξ achieves maximum stiffness at the peg location. Also it is seen that the tangential stiffness k ηη achieves minimum stiffness at the peg location, and maximum stiffness between two pegs. However, the cross-coupling stiffness has maximum and minimum at π/8 from the peg location. The same is also true for the damping as function of the eccentricity direction and can be found in Appendix A Figure A.17. A closer look at Figure 4 shows that the cross-coupling stiffness is almost zero at the peg location. If this would be the only interesting region the terms can be neglected. However, by looking at the position where the cross-coupling reaches a maximum, it is seen that the cross-coupling stiffness is only a factor 5 less than the direct directions. When the angles α that achieves maxima and minima are known, stiffness and damping at these locations can be studied for different eccentricities. In Figure 5 the stiffness as function of eccentricity is shown together with a 4th order polynomial fit. 5