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Energy-Efficient Trajectories Rotors Supported on Radial Fluid-Film Bearings
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ScienceDirect ScienceDirect Procedia Engineering 00 (2017)000–000
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Procedia Engineering 00 (2017)000–000
www.elsevier.com/locate/procedia www.elsevier.com/locate/procedia
Procedia Engineering 206 (2017) 527–532
International Conference on Industrial Engineering, ICIE 2017 International Conference on Industrial Engineering, ICIE 2017
Energy-Efficient Trajectories Rotors Supported on Radial FluidEnergy-Efficient Trajectories Supported on Radial FluidFilm Rotors Bearings Film Bearings S.V. Majorov, L.A. Savin, A.Y. Koltsov* S.V. Majorov, L.A. Savin, A.Y. Koltsov* Orel State Univesity, 29, Naugorskoe sh, Orel 302020,Russia Orel State Univesity, 29, Naugorskoe sh, Orel 302020,Russia
Abstract Abstract
The problem of rotor machines modification requires development of new approaches to the design of specific nodes. As bearings are machines consideredmodification to be the most important elements of approaches imply The problem of rotor requires development of rotor new machines, approachessuch to the design ofcanspecific optimization application. The present paper considers the approach for enhancing energy efficiency of rotor nodes. As bearings are considered to be the most important elements of rotor machines, such approaches can imply machines by means of generating optimal trajectories using the example of fluid-film bearings. The authors provide optimization application. The present paper considers the approach for enhancing energy efficiency of rotor the meansby to means analyzeofthe obtainedoptimal trajectories types. The also covers the approach for setting the criteria of machines generating trajectories usingpaper the example of fluid-film bearings. The authors provide rotor's trajectories energy efficiency. Besides, the paper the results conducted analysis the means to analyze the obtained trajectories types. Thepresents paper also covers of thethe approach for numeric setting the criteriathat of led to the hypothesis aboutefficiency. the existence of nonthe trivial rotor energy-efficient rotor's trajectories energy Besides, paper presents the resultsorbital of the trajectories. conducted numeric analysis that © 2017 The Authors. Published by Elsevier B.V. led2017 to the about the of non trivial rotor energy-efficient orbital trajectories. © Thehypothesis Authors. Published byexistence Elsevier Ltd. Peer-review under responsibility of the scientific committee of the International Conference on Industrial Engineering. © 2017 The under Authors. Published by B.V.committee Peer-review responsibility of Elsevier thefluid-film scientific of the International Conference Industrial Engineering Keywords: energy-efficient; rotor system; bearing; rotordynamics; hydrodynamic lubrication on theory. Peer-review under responsibility of the scientific committee of the International Conference on Industrial Engineering. Keywords: energy-efficient; rotor system; fluid-film bearing; rotordynamics; hydrodynamic lubrication theory.
1. Introduction 1. Introduction The rotor with fluid film bearings is active-dissipative system in which the complex of hydrodynamic and heat transfer processes, and also of axial and lateral vibrations realized. Viscosity, surface wettability, The rotor with fluid film different bearings types is active-dissipative system in whichis the complex of hydrodynamic and heat compressibility, transitions, multiphase, properties, technologysurface factors, thermal, transfer processes,phase and also differentturbulence, types of axial and lateralrheological vibrations is realized. Viscosity, wettability, elastic, inertial processes acting important role multiphase, for load-carrying capability of a fluid film [1-6]. Providing the compressibility, phase transitions, turbulence, rheological properties, technology factors, thermal, sufficient bearing capability in rotor systemsrole withforbearings of liquid friction itofisaalways connected a taskthe of elastic, inertial processes acting important load-carrying capability fluid film [1-6]. with Providing orbital stability of capability movementinofrotor a rotor [7-12]. At bearings the sameof time often the questions connected withwith an efficiency sufficient bearing systems with liquid friction it is always connected a task of operating evaluation of such systems the At point viewtime energy pass into connected the background. in orbital stability of movement of a rotorfrom [7-12]. theofsame oftencosts the questions with anHowever efficiency operating evaluation of such systems from the point of view energy costs pass into the background. However in
* Corresponding author. Tel.: +7-910-309-8250; fax: +7-486-241-9849. E-mail address:author. [email protected] * Corresponding Tel.: +7-910-309-8250; fax: +7-486-241-9849.
E-mail address: [email protected] 1877-7058 © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the International Conference on Industrial Engineering . 1877-7058 © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the International Conference on Industrial Engineering .
1877-7058 © 2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the scientific committee of the International Conference on Industrial Engineering. 10.1016/j.proeng.2017.10.511
528 2
S.V. Majorov et al. / Procedia Engineering 206 (2017) 527–532 Majorov S.V., Savin L.A., Koltsov A.Y. / Procedia Engineering 00 (2017) 000–000
some cases, what is especially characteristic of autonomous systems the question of energy efficiency rises especially sharply. In this work the hypothesis about existence of rotor energy-efficient orbital trajectories is made, and also on the basis of a computing experiment the proof of her justice is provided. 2. Fluid-film bearing modeling Main complexity of rotor-bearing modeling is calculation of fluid film bearing reaction.(Fig. 1). Fluid film bearing design model represented Fig. 1.
Fig. 1.Fluid film bearing design model (D – bearing diameter; L – length bearing)
The majority of works by calculation of fluid film bearing reaction is based on the solution of the Reynolds Eq. (1). And here, determination of the pressure distribution is a complex boundary problem based on the Reynolds equation. Using the operators of the partial first derivatives in the forms of ∂/∂x and ∂/∂z, this equation reads: h3 p h3 p 6 Uh 12 V , x K x x z K z z x
(1)
where x and z are the circumferential and axial coordinates of bearing’s sweep, respectively; p represents the pressure inside the bearing; μ and ρ are the viscosity and density of the fluid accordingly; h is the radial gap function; Kx and Kz are the turbulence coefficients in the directions of turbulence, namely in the x and z directions, correspondingly, that determine the effective viscosity when the flow is turbulent considering the Reynolds stress. In expression Eq. (1), U and V are the values of the circumferential and radial velocities of the surface points of the shaft. The form of Eq. (1) is not exclusive and can vary depending on the different types and operational conditions of the bearing, more detail on the forms of Eq. (1) can be found in [1-5].However there are works [6] in which modeling is based on solution Navier-Stokes equation(NS equation). Expressions for terms in Eq.(1) can be find at following references [1,3,5].The radial gap function can be expressed by following term: 3. Fluid-film bearing reactions The pressure field p(x,z) findings from Eq (1) is basis for determination of the hydrodynamic forces operating on a rotor shaft namely reactions of RX and RY of a lubricant layer in the X and Y directions and Mfr friction moment. These terms can be determined by the following expressions LD LD 2x 2x p x,z cos dxdz x,z sin dxdz ; RX D D 0 0 0 0
(2)
LD LD 2x 2x RY p x,z sin dxdz x,z cos dxdz ; D D 0 0 0 0
(3)
S.V. Majorov et al. /A.Y. Procedia Engineering 206 (2017) 527–532 Majorov S.V., Savin L.A., Koltsov / Procedia Engineering 00 (2017) 000–000
M fr
D LD x,z dxdz , 20 0
5293
(4)
where (x,z) – shear stress field, that can obtained from pressure distribution by following relationship:
Kx
x ,z
h
h p x,z . 2 x
(5)
4. Energy-efficient of rotor system The basic trend in rotor trajectories is an attribute of an unbalance rotor, which dynamics is parametric optimization, to which a big performs forced oscillations caused by the centrifugal number of papers lately are dedicated [13 – 20]. The main controversial problem of parametric optimization is the determination of the quality criteria for a specific rotor system. In the present study the criteria is suggested which is based on the effective use of the supplied energy. The main function of any rotor system is the energy transfer from the drive to the execution device, and it is obvious that this energy characteristic can be expressed as kinetic energy of rotational motion: Teff
J 2 , 2
(6)
where J – inertia momentum with respect to rotation axis; – rotor's angular speed. During the revolution of the rotor in fluid-film bearings the dissipation of energy will have two distinctive components: 1) dissipation of energy due to revolution, in other words, the action of viscous forces along the rot ; 2) dissipation of energy due to oscillations, in other words, the action of the forces on angular displacement Adis disp trajectories considered above of the rotor Adis and these values can be evaluated as follows:
rot Adis
*
M fr d ;
(7)
0
disp Adis R ds ,
(8)
s
where Mfr –friction momentum of a fluid-film bearing; R –reaction force vector in fluid-film bearing; ds – * singular vector tangent to the trajectory (s) of shaft’s center; – angle of a rotor displacement during the time on trajectory. So, for every closed trajectory the following energy efficient Eeff
rot disp Adis Adis Teff
(9)
The parametric optimization with a suggested criteria is indeed a laborious task in terms of calculation. One of the alternatives to it is active control of the fluid-film bearing’s parameters. The variable parameters are here changed so that the trajectory with the lowest Eeff is obtained from a number of possible.
S.V. Majorov et al. / Procedia Engineering 206 (2017) 527–532 Majorov S.V., Savin L.A., Koltsov A.Y. / Procedia Engineering 00 (2017) 000–000
530 4
5. Results In present paper the bearing with the following geometric and operating parameters is investigated: D=50 mm; L=50 mm; h0=100 µm; =500 rad/s µ=10-3 Pa s; =1000 kg/m3. Let's set possible elliptic trajectories of a rotor by following relationships:
A X t h0 AX cos t t cos t X sin t t sin t X c 2
(10)
A Y t h0 AX cos t t sin t X sin t t cos t Yc 2
(11)
where t – angular frequency on elliptic orbit; t – trajectory rotation angle from X axis; AX – major ellipse axis related to nominal radial clearance; Xc and Yc – coordinate of elliptical orbit center related to nominal radial clearance. And set following ranges of kinematic parameters for elliptical trajectories: 1. Ax = [0.1 0.2 0.3]; 2. Xc= [0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9]; 3. Yc= [0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9]; 4. t=[0 30 60 90 120 150]; 5) t=[/2 2]. Energy efficiency indicators for various values of the ranged parameters have been calculated. It should be noted that not all combinations of the ranged parameters are kinematic possible, that is at some combinations the trajectory goes beyond a radial gap, for such cases calculation of an indicator of energy efficiency wasn't carried out and its value wasn't considered in the analysis. In tables 1-3 global minima of an indicator of energy efficiency at a variation of coordinates of the centers of elliptic trajectories. Table 1. Global minims of energy-efficiency (t=/2) Ax=0.1
Ax=0.2
Eeff
Xc
Yc
Ax=0.3
Eeff
Xc
Yc
Eeff
Xc
Yc
0
0.000162
0
0.8
0.015798
0
0.8
0.040355
0
0.7
30
0.00597
0.3
0
0.023505
0.3
0
0.065238
0.2
0
60
9.9710-5
0.6
0.1
6.7410-4
0.4
0
5.4010-3
0.4
0
90
0.000143
0.8
0
0.001235
0.5
0.4
0.003305
0.4
0.3
120
0.000179
0.2
0.4
0.002351
0.2
0.4
0.000645
0.2
0.4
0
0.4
150
8.9510
-5
0.1
0.5
6.8510
-4
0
0.4
5.4110
-3
Table 2. Global minims of energy-efficiency (t=)
Ax=0.1;
Ax=0.2
Ax=0.3
Eeff
Xc
Yc
Eeff
Xc
Yc
Eeff
Xc
Yc
0
0.021522
0
0
0.021522
0
0
0.200904
0
0
30
0.021295
0.2
0
0.021295
0.2
0
0.200885
0
0
0
0
0.5
0
2.0610
0.018841
0.3
0.2
0.018912
0.2
0.3
60
2.0610
90 120 150
-2
1.8810
-2
0
0.3
-2
-1
0.5
0
2.0110
0.018841
0.3
0.2
0.200904
0
0
0.018912
0.2
0.3
0.200885
0
0
0
0
1.8810
-2
0
0.3
2.0110
-1
S.V. Majorov et al. / Procedia Engineering 206 (2017) 527–532 Majorov S.V., Savin L.A., Koltsov A.Y. / Procedia Engineering 00 (2017) 000–000
531 5
Table 3. Global minims of energy-efficiency (t=2)
Ax=0.1
Ax=0.2
Ax=0.3
Eeff
Xc
Yc
Eeff
Xc
Yc
Eeff
Xc
Yc
0
0.028028
0
0
0.113523
0
0
0.265208
0
0
30
0.028028
0
0
0.113523
0
0
0.265206
0
0
60
2.8410-2
0.1
0
1.1410-1
0
0
2.6510-1
0
0
90
0.028028
0
0
0.113523
0
0
0.265208
0
0
120
0.028028
0
0
0.113523
0
0
0.265206
0
0
150
2.8010-2
0
0
1.2710-1
0.2
0.1
2.6510-1
0
0
From the analysis of the given results it is visible that global minima of energy efficient other than trivial decisions take place that confirms a hypothesis of a possibility of finding of optimum trajectories in energy efficiency parameter in the field of kinematic parameters. 6. Conclusions It is shown what among a set of possible elliptic trajectories for rotor supported on fluid-film bearings friction can be finded optimum by offered energy efficiency criterion. It is necessary to mark that the global minimum is achievable not only in trivial points (Xc=Yc=0). These show especially for subharmonic oscillations (t=/2) in case of «oil whirl» effect, and also in case of oscillations with synchronous frequency to the rotor speed (t=) in case of unbalance load. These ranges of oscillations are prevailing on the main operation modes of rotor with fluid-film bearings that allows to use the criterion in optimization problems of rotor-bearing systems as objective function. Acknowledgment This work was partly supported by an RSF grant, project 16-16-00186 «Planning of optimal energy efficient trajecotries of rotors in mechatronic modules in complex rheology media». References [1] O.V. Solomin, Dynamic characteristics of hydrostatodynamic supports under two-phase lubrication conditions, Izvestya vyzov, Machinostroenye. 1 (2006) 14–23. [2] Y.A. Ravikovych, Construction and design of journal bearings of engine aircraft, MAI, Moskow, 1995. [3] N.P. Artemenko, Hydrostatic supports of high-speed rotors, Osnova, Kharkiv, 1992. [4] V.A. Maksimov, G.S. Batkis, Trybology of fluid-film bearings and seals of high-speed machines, FEN, Kazan, 1998. [5] Y. Hory, Hydrodynamic lubrication, Springer Verlag, Tokyo, 2006. [6] H. Liu, H. Xu, P. Ellison, Z. Jin, Application of computational fluid dynamics and fluid–structure interaction method to the lubrication study of a rotor–bearing system, Tribology Letters. 38(3) (2010) 325–336. [7] T. Yamamoto, Y. Ishida, Linear and nonlinear rotordynamics. A modern treatment with applications, John Willey&Sons, New York, 2001. [8] M.L. Adams, Rotating machinery vibration: from analysis to troubleshooting, Marcel Dekker Inc., New York, 2001. [9] E. Kramer, Dynamics of rotors and foundations, Springer Verlag, Berlin, 1993. [10] D. Childs, Turbomachineryrotordynamics: phenomena, modeling and analysis, John Willey&Sons, New York, 1993. [11] V.G. Lukanenko, Vibrations of high-speed rotors supported on hydrostatic bearings and methods decreasing of machine vibrationsах, Samarsky NC RAN, Samara, 2001. [12] E. Logan, Jr, Handbook of turbomachinery, Marcel Dekker Inc., New York, 1995. [13] A.O. Pugachev, M. Deckner, Shape optimization of a labyrinth seal: leakage minimization and sensitivity of rotordynamic coefficients, IMechE Ninth International Conference on Vibrations in Rotating Machinery, University of Exeter, UK, 2008, pp. 849–859. [14] L. Gorasso, L. Wang, Journal bearing optimization using nonsorted genetic algorithm and artificial bee colony algorithm, Advances in Mechanical Engineering. (2014) 1–18. [15] A.O. Pugachev, A.V. Sheremetyev, V.V. Tykhomirov, Gradient-based optimization of a turboprop rotor system with constraints on stresses and natural frequencies, 6th AIAA Multidisciplinary Design Optimization Specialist Conference, Orlando, FL, USA, 2010, AIAA 20103006. doi:10.2514/6.2010-3006.
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[16] R. Helfrich, N. Wagner, Application of optimization methods in rotor dynamics, Proceedings of the 9th IFToMM International Conference on Rotor Dynamics. Volume 21 of the series Mechanisms and Machine Science, pp. 1715–1725. [17] E.P. Petrov, Advanced analysis and optimization of nonlinear resonance vibrations in gas-turbine structures with friction and gaps, Proceedings of IUTAM Symposium on Emerging Trends in Rotor Dynamics, IIT Delhi, IUTAM bookseries, Vol. 25, pp. 297–308. [18] J.S. Rao, Optimized life using frequency and time domain approaches, Proceedings of IUTAM Symposium on Emerging Trends in Rotor Dynamics, IIT Delhi, IUTAM bookseries, Vol. 25, pp. 13–26. [19] I.F. Santos, Trends in controllable oil film bearings, Proceedings of IUTAM Symposium on Emerging Trends in Rotor Dynamics, IIT Delhi, IUTAM bookseries, Vol. 25, pp. 185–200. [20] L. Savin, D. Shutin, Control of rotor motion using active fluid-film bearings, Proceedings of the 9th IFToMM International Conference on Rotor Dynamics, of the series Mechanisms and Machine Science, Vol.21, pp. 1115–1122.
ScienceDirect ScienceDirect Procedia Engineering 00 (2017)000–000
Available online at www.sciencedirect.com
ScienceDirect
Procedia Engineering 00 (2017)000–000
www.elsevier.com/locate/procedia www.elsevier.com/locate/procedia
Procedia Engineering 206 (2017) 527–532
International Conference on Industrial Engineering, ICIE 2017 International Conference on Industrial Engineering, ICIE 2017
Energy-Efficient Trajectories Rotors Supported on Radial FluidEnergy-Efficient Trajectories Supported on Radial FluidFilm Rotors Bearings Film Bearings S.V. Majorov, L.A. Savin, A.Y. Koltsov* S.V. Majorov, L.A. Savin, A.Y. Koltsov* Orel State Univesity, 29, Naugorskoe sh, Orel 302020,Russia Orel State Univesity, 29, Naugorskoe sh, Orel 302020,Russia
Abstract Abstract
The problem of rotor machines modification requires development of new approaches to the design of specific nodes. As bearings are machines consideredmodification to be the most important elements of approaches imply The problem of rotor requires development of rotor new machines, approachessuch to the design ofcanspecific optimization application. The present paper considers the approach for enhancing energy efficiency of rotor nodes. As bearings are considered to be the most important elements of rotor machines, such approaches can imply machines by means of generating optimal trajectories using the example of fluid-film bearings. The authors provide optimization application. The present paper considers the approach for enhancing energy efficiency of rotor the meansby to means analyzeofthe obtainedoptimal trajectories types. The also covers the approach for setting the criteria of machines generating trajectories usingpaper the example of fluid-film bearings. The authors provide rotor's trajectories energy efficiency. Besides, the paper the results conducted analysis the means to analyze the obtained trajectories types. Thepresents paper also covers of thethe approach for numeric setting the criteriathat of led to the hypothesis aboutefficiency. the existence of nonthe trivial rotor energy-efficient rotor's trajectories energy Besides, paper presents the resultsorbital of the trajectories. conducted numeric analysis that © 2017 The Authors. Published by Elsevier B.V. led2017 to the about the of non trivial rotor energy-efficient orbital trajectories. © Thehypothesis Authors. Published byexistence Elsevier Ltd. Peer-review under responsibility of the scientific committee of the International Conference on Industrial Engineering. © 2017 The under Authors. Published by B.V.committee Peer-review responsibility of Elsevier thefluid-film scientific of the International Conference Industrial Engineering Keywords: energy-efficient; rotor system; bearing; rotordynamics; hydrodynamic lubrication on theory. Peer-review under responsibility of the scientific committee of the International Conference on Industrial Engineering. Keywords: energy-efficient; rotor system; fluid-film bearing; rotordynamics; hydrodynamic lubrication theory.
1. Introduction 1. Introduction The rotor with fluid film bearings is active-dissipative system in which the complex of hydrodynamic and heat transfer processes, and also of axial and lateral vibrations realized. Viscosity, surface wettability, The rotor with fluid film different bearings types is active-dissipative system in whichis the complex of hydrodynamic and heat compressibility, transitions, multiphase, properties, technologysurface factors, thermal, transfer processes,phase and also differentturbulence, types of axial and lateralrheological vibrations is realized. Viscosity, wettability, elastic, inertial processes acting important role multiphase, for load-carrying capability of a fluid film [1-6]. Providing the compressibility, phase transitions, turbulence, rheological properties, technology factors, thermal, sufficient bearing capability in rotor systemsrole withforbearings of liquid friction itofisaalways connected a taskthe of elastic, inertial processes acting important load-carrying capability fluid film [1-6]. with Providing orbital stability of capability movementinofrotor a rotor [7-12]. At bearings the sameof time often the questions connected withwith an efficiency sufficient bearing systems with liquid friction it is always connected a task of operating evaluation of such systems the At point viewtime energy pass into connected the background. in orbital stability of movement of a rotorfrom [7-12]. theofsame oftencosts the questions with anHowever efficiency operating evaluation of such systems from the point of view energy costs pass into the background. However in
* Corresponding author. Tel.: +7-910-309-8250; fax: +7-486-241-9849. E-mail address:author. [email protected] * Corresponding Tel.: +7-910-309-8250; fax: +7-486-241-9849.
E-mail address: [email protected] 1877-7058 © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the International Conference on Industrial Engineering . 1877-7058 © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the International Conference on Industrial Engineering .
1877-7058 © 2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the scientific committee of the International Conference on Industrial Engineering. 10.1016/j.proeng.2017.10.511
528 2
S.V. Majorov et al. / Procedia Engineering 206 (2017) 527–532 Majorov S.V., Savin L.A., Koltsov A.Y. / Procedia Engineering 00 (2017) 000–000
some cases, what is especially characteristic of autonomous systems the question of energy efficiency rises especially sharply. In this work the hypothesis about existence of rotor energy-efficient orbital trajectories is made, and also on the basis of a computing experiment the proof of her justice is provided. 2. Fluid-film bearing modeling Main complexity of rotor-bearing modeling is calculation of fluid film bearing reaction.(Fig. 1). Fluid film bearing design model represented Fig. 1.
Fig. 1.Fluid film bearing design model (D – bearing diameter; L – length bearing)
The majority of works by calculation of fluid film bearing reaction is based on the solution of the Reynolds Eq. (1). And here, determination of the pressure distribution is a complex boundary problem based on the Reynolds equation. Using the operators of the partial first derivatives in the forms of ∂/∂x and ∂/∂z, this equation reads: h3 p h3 p 6 Uh 12 V , x K x x z K z z x
(1)
where x and z are the circumferential and axial coordinates of bearing’s sweep, respectively; p represents the pressure inside the bearing; μ and ρ are the viscosity and density of the fluid accordingly; h is the radial gap function; Kx and Kz are the turbulence coefficients in the directions of turbulence, namely in the x and z directions, correspondingly, that determine the effective viscosity when the flow is turbulent considering the Reynolds stress. In expression Eq. (1), U and V are the values of the circumferential and radial velocities of the surface points of the shaft. The form of Eq. (1) is not exclusive and can vary depending on the different types and operational conditions of the bearing, more detail on the forms of Eq. (1) can be found in [1-5].However there are works [6] in which modeling is based on solution Navier-Stokes equation(NS equation). Expressions for terms in Eq.(1) can be find at following references [1,3,5].The radial gap function can be expressed by following term: 3. Fluid-film bearing reactions The pressure field p(x,z) findings from Eq (1) is basis for determination of the hydrodynamic forces operating on a rotor shaft namely reactions of RX and RY of a lubricant layer in the X and Y directions and Mfr friction moment. These terms can be determined by the following expressions LD LD 2x 2x p x,z cos dxdz x,z sin dxdz ; RX D D 0 0 0 0
(2)
LD LD 2x 2x RY p x,z sin dxdz x,z cos dxdz ; D D 0 0 0 0
(3)
S.V. Majorov et al. /A.Y. Procedia Engineering 206 (2017) 527–532 Majorov S.V., Savin L.A., Koltsov / Procedia Engineering 00 (2017) 000–000
M fr
D LD x,z dxdz , 20 0
5293
(4)
where (x,z) – shear stress field, that can obtained from pressure distribution by following relationship:
Kx
x ,z
h
h p x,z . 2 x
(5)
4. Energy-efficient of rotor system The basic trend in rotor trajectories is an attribute of an unbalance rotor, which dynamics is parametric optimization, to which a big performs forced oscillations caused by the centrifugal number of papers lately are dedicated [13 – 20]. The main controversial problem of parametric optimization is the determination of the quality criteria for a specific rotor system. In the present study the criteria is suggested which is based on the effective use of the supplied energy. The main function of any rotor system is the energy transfer from the drive to the execution device, and it is obvious that this energy characteristic can be expressed as kinetic energy of rotational motion: Teff
J 2 , 2
(6)
where J – inertia momentum with respect to rotation axis; – rotor's angular speed. During the revolution of the rotor in fluid-film bearings the dissipation of energy will have two distinctive components: 1) dissipation of energy due to revolution, in other words, the action of viscous forces along the rot ; 2) dissipation of energy due to oscillations, in other words, the action of the forces on angular displacement Adis disp trajectories considered above of the rotor Adis and these values can be evaluated as follows:
rot Adis
*
M fr d ;
(7)
0
disp Adis R ds ,
(8)
s
where Mfr –friction momentum of a fluid-film bearing; R –reaction force vector in fluid-film bearing; ds – * singular vector tangent to the trajectory (s) of shaft’s center; – angle of a rotor displacement during the time on trajectory. So, for every closed trajectory the following energy efficient Eeff
rot disp Adis Adis Teff
(9)
The parametric optimization with a suggested criteria is indeed a laborious task in terms of calculation. One of the alternatives to it is active control of the fluid-film bearing’s parameters. The variable parameters are here changed so that the trajectory with the lowest Eeff is obtained from a number of possible.
S.V. Majorov et al. / Procedia Engineering 206 (2017) 527–532 Majorov S.V., Savin L.A., Koltsov A.Y. / Procedia Engineering 00 (2017) 000–000
530 4
5. Results In present paper the bearing with the following geometric and operating parameters is investigated: D=50 mm; L=50 mm; h0=100 µm; =500 rad/s µ=10-3 Pa s; =1000 kg/m3. Let's set possible elliptic trajectories of a rotor by following relationships:
A X t h0 AX cos t t cos t X sin t t sin t X c 2
(10)
A Y t h0 AX cos t t sin t X sin t t cos t Yc 2
(11)
where t – angular frequency on elliptic orbit; t – trajectory rotation angle from X axis; AX – major ellipse axis related to nominal radial clearance; Xc and Yc – coordinate of elliptical orbit center related to nominal radial clearance. And set following ranges of kinematic parameters for elliptical trajectories: 1. Ax = [0.1 0.2 0.3]; 2. Xc= [0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9]; 3. Yc= [0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9]; 4. t=[0 30 60 90 120 150]; 5) t=[/2 2]. Energy efficiency indicators for various values of the ranged parameters have been calculated. It should be noted that not all combinations of the ranged parameters are kinematic possible, that is at some combinations the trajectory goes beyond a radial gap, for such cases calculation of an indicator of energy efficiency wasn't carried out and its value wasn't considered in the analysis. In tables 1-3 global minima of an indicator of energy efficiency at a variation of coordinates of the centers of elliptic trajectories. Table 1. Global minims of energy-efficiency (t=/2) Ax=0.1
Ax=0.2
Eeff
Xc
Yc
Ax=0.3
Eeff
Xc
Yc
Eeff
Xc
Yc
0
0.000162
0
0.8
0.015798
0
0.8
0.040355
0
0.7
30
0.00597
0.3
0
0.023505
0.3
0
0.065238
0.2
0
60
9.9710-5
0.6
0.1
6.7410-4
0.4
0
5.4010-3
0.4
0
90
0.000143
0.8
0
0.001235
0.5
0.4
0.003305
0.4
0.3
120
0.000179
0.2
0.4
0.002351
0.2
0.4
0.000645
0.2
0.4
0
0.4
150
8.9510
-5
0.1
0.5
6.8510
-4
0
0.4
5.4110
-3
Table 2. Global minims of energy-efficiency (t=)
Ax=0.1;
Ax=0.2
Ax=0.3
Eeff
Xc
Yc
Eeff
Xc
Yc
Eeff
Xc
Yc
0
0.021522
0
0
0.021522
0
0
0.200904
0
0
30
0.021295
0.2
0
0.021295
0.2
0
0.200885
0
0
0
0
0.5
0
2.0610
0.018841
0.3
0.2
0.018912
0.2
0.3
60
2.0610
90 120 150
-2
1.8810
-2
0
0.3
-2
-1
0.5
0
2.0110
0.018841
0.3
0.2
0.200904
0
0
0.018912
0.2
0.3
0.200885
0
0
0
0
1.8810
-2
0
0.3
2.0110
-1
S.V. Majorov et al. / Procedia Engineering 206 (2017) 527–532 Majorov S.V., Savin L.A., Koltsov A.Y. / Procedia Engineering 00 (2017) 000–000
531 5
Table 3. Global minims of energy-efficiency (t=2)
Ax=0.1
Ax=0.2
Ax=0.3
Eeff
Xc
Yc
Eeff
Xc
Yc
Eeff
Xc
Yc
0
0.028028
0
0
0.113523
0
0
0.265208
0
0
30
0.028028
0
0
0.113523
0
0
0.265206
0
0
60
2.8410-2
0.1
0
1.1410-1
0
0
2.6510-1
0
0
90
0.028028
0
0
0.113523
0
0
0.265208
0
0
120
0.028028
0
0
0.113523
0
0
0.265206
0
0
150
2.8010-2
0
0
1.2710-1
0.2
0.1
2.6510-1
0
0
From the analysis of the given results it is visible that global minima of energy efficient other than trivial decisions take place that confirms a hypothesis of a possibility of finding of optimum trajectories in energy efficiency parameter in the field of kinematic parameters. 6. Conclusions It is shown what among a set of possible elliptic trajectories for rotor supported on fluid-film bearings friction can be finded optimum by offered energy efficiency criterion. It is necessary to mark that the global minimum is achievable not only in trivial points (Xc=Yc=0). These show especially for subharmonic oscillations (t=/2) in case of «oil whirl» effect, and also in case of oscillations with synchronous frequency to the rotor speed (t=) in case of unbalance load. These ranges of oscillations are prevailing on the main operation modes of rotor with fluid-film bearings that allows to use the criterion in optimization problems of rotor-bearing systems as objective function. Acknowledgment This work was partly supported by an RSF grant, project 16-16-00186 «Planning of optimal energy efficient trajecotries of rotors in mechatronic modules in complex rheology media». References [1] O.V. Solomin, Dynamic characteristics of hydrostatodynamic supports under two-phase lubrication conditions, Izvestya vyzov, Machinostroenye. 1 (2006) 14–23. [2] Y.A. Ravikovych, Construction and design of journal bearings of engine aircraft, MAI, Moskow, 1995. [3] N.P. Artemenko, Hydrostatic supports of high-speed rotors, Osnova, Kharkiv, 1992. [4] V.A. Maksimov, G.S. Batkis, Trybology of fluid-film bearings and seals of high-speed machines, FEN, Kazan, 1998. [5] Y. Hory, Hydrodynamic lubrication, Springer Verlag, Tokyo, 2006. [6] H. Liu, H. Xu, P. Ellison, Z. Jin, Application of computational fluid dynamics and fluid–structure interaction method to the lubrication study of a rotor–bearing system, Tribology Letters. 38(3) (2010) 325–336. [7] T. Yamamoto, Y. Ishida, Linear and nonlinear rotordynamics. A modern treatment with applications, John Willey&Sons, New York, 2001. [8] M.L. Adams, Rotating machinery vibration: from analysis to troubleshooting, Marcel Dekker Inc., New York, 2001. [9] E. Kramer, Dynamics of rotors and foundations, Springer Verlag, Berlin, 1993. [10] D. Childs, Turbomachineryrotordynamics: phenomena, modeling and analysis, John Willey&Sons, New York, 1993. [11] V.G. Lukanenko, Vibrations of high-speed rotors supported on hydrostatic bearings and methods decreasing of machine vibrationsах, Samarsky NC RAN, Samara, 2001. [12] E. Logan, Jr, Handbook of turbomachinery, Marcel Dekker Inc., New York, 1995. [13] A.O. Pugachev, M. Deckner, Shape optimization of a labyrinth seal: leakage minimization and sensitivity of rotordynamic coefficients, IMechE Ninth International Conference on Vibrations in Rotating Machinery, University of Exeter, UK, 2008, pp. 849–859. [14] L. Gorasso, L. Wang, Journal bearing optimization using nonsorted genetic algorithm and artificial bee colony algorithm, Advances in Mechanical Engineering. (2014) 1–18. [15] A.O. Pugachev, A.V. Sheremetyev, V.V. Tykhomirov, Gradient-based optimization of a turboprop rotor system with constraints on stresses and natural frequencies, 6th AIAA Multidisciplinary Design Optimization Specialist Conference, Orlando, FL, USA, 2010, AIAA 20103006. doi:10.2514/6.2010-3006.
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