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ScienceDirect Procedia Engineering 150 (2016) 647 – 653

International Conference on Industrial Engineering, ICIE 2016

The Technique of an Interconnection Problem of the Hydrodynamic Lubrication Theory and the Nonlinear Dynamics for Mechanical Systems "An Elastic Crankshaft on Journal Bearings" Rozhdestvensky Yu.V., Khozeniuk N.A.*, Mylnikov A.A. South Ural State University, 76, Lenin Avenue, Chelyabinsk, 454080, The Russian Federation

Abstract In this paper a mathematical model and numerical methods for the modeling of the main bearing system of internal combustion engines are suggested. The novelty of the offered approach consists in 1) the modeling of the simultaneous influence of nonlinear damping properties of lubricant layers depending on the rheological properties of the modern lubricant layers, 2) the elastic links of the journals and the supports, 3) the skewnesses of the axis of mobile and immobile parts of bearings. These skewnesses are caused by external forces of different nature (dynamic forces or/and displacements, thermal factors). © by by Elsevier Ltd.Ltd. This is an open access article under the CC BY-NC-ND license © 2016 2016The TheAuthors. Authors.Published Published Elsevier (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICIE 2016. Peer-review under responsibility of the organizing committee of ICIE 2016

Keywords: main bearing; interconnected process; dynamic model; lubricant; non- newtonian properties; elastic compliance coefficient; crankshaft; crankcase; internal combustion engine

1. Introduction The performance assurance of a hydrodynamic tribosystem is one of the main problems in the creation and operational development of internal combustion engines design. Mainly it is caused by the complexity and interconnectivity of the processes and factors defining the reliability of the fluid friction units. For simulation of the hydrodynamic tribosystem it is necessary to take into account the following data: the geometry of the tribounits, the macro- and micro geometry of the friction surfaces, the velocity and loading parameters and the viscosity-

* Corresponding author. Tel.: +7-951-485-0950; fax: +7-351-267-9214. E-mail address: [email protected]

1877-7058 © 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICIE 2016

doi:10.1016/j.proeng.2016.07.061

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temperature and rheological properties of the liquid lubricant dividing the complex loaded surfaces of the friction. Together with these parameters considered in the autonomous tribounits, for the system of crankshaft main bearings of the internal combustion (IC) engines it is necessary to consider the cross-coupling effect of tribounits [1, 2] arising from elastic links of bearings and journals (by means of an engine crankcase and a crankshaft). The existing methods of modeling of the interconnected processes of lubricant flow in thin liquid layers and nonlinear dynamics for a system of "crankshaft - lubricating films - the crankcase” together are reduced to the solution of separate subtasks. One of the approaches consists in the decision of a dynamic problem for the system "an elastic shaft on resilient supports" without an intermediate lubricant layer. Such problem formulation is used for the definition of the dynamic loading and structural strength for a shaft and crankcase elements [1–6]. Its use allows us to model 3D behavior element. However, it is apparent that the nonlinear damping bearing properties significantly influence the dynamic behavior and structural strength of the shaft and crankcase elements. Still another approach is to present the performance of heavy-loaded hydrodynamic bearings as autonomous tribounits. In that case the mutual influence of support is excluded, and the approach purpose was the investigation of the process into the lubricant layer and development of calculation methods for various designs of bearings [8, 9, 10]. Such methods are most used for heavy-loaded hydrodynamic bearings. However, they don't allow us to estimate the influence of design and technological factors such as the different values of the elastic compliances of the crankshaft supports, misalignments of journals and main bearings and so on. The next approach is characterized by the accounting of elastic links between tribounits [12, 13], but considerable assumptions and approximate calculation methods of lubricating films of main bearings are used to obtain the results. The authors applied Holland`s method which didn't allow us to consider the structural features of the bearings and the non-Newtonian properties of modern lubricants. Prokopiev V., Vetrov M. [7] developed the approximating formulas based on Reynolds’ equation for the finite length bearing as well as Booker`s methods. It is shown that the accounting of the nonlinear damping properties of bearings significantly influences not only the dynamic behavior and durability of a shaft and a case, but also considerably changes the hydromechanical characteristics of the bearings (HMC). However, this technique doesn't allow us to consider a skewed axis of a shaft and the bearing, and also different values of the compliances supports of the crankshaft. The fullest model was offered by Zakharov [12, 14]. The technique considered the deviations of the necks of the crankshaft and the angular compliances of the crankshaft supports. But to obtain the results the authors were compelled to apply Holland` method, of which the limitations are given above. In spite of a considerable amount of works about the dynamics and lubrication of the system "elastic shaft rotating in the journal bearings" and the significance of its results, currently there are no general approaches to the solution of such tasks taking into account the angular misalignment of the axes of the mobile and immobile elements of tribounits and the elastic properties of the support simultaneously with features of the rheological behavior of lubricants and kinematic loading. 2. Theory The mathematical model of the dynamics of the interconnected system of the elastic multi-supporting crankshaft rotating in the hydrodynamics bearing is based on the mechanical model presented in Fig. 1. Crankshaft journals are interconnected through a resilient connection – IC engine crankshaft. The bushings of the main bearings are installed in the holes of the crankcase walls, and thus, the main bearings are interconnected via a flexible design of the crankcase. Identification procedure of the elastic properties of the shaft and linear K sO, ,xj , K sO, ,yj and angular K sI,,xj , K sI,,yj compliances is presented in work [15]. Influence of the lubricant layers dividing friction surfaces of the main bearing journals of a crankshaft and a crankcase of the IC engine is modeled by means of non-linear elastic elements with the compliances K LO,,xj W , K LO,,yj W , K LI ,,xj W , K LI ,, yj W . These compliances depend on the lubricant layers thickness which varies with time because the centers of the journals move on some trajectories under the loads F i ,( s ) , where i is number of the cranked port of shaft, s 1,.., 7 is number of the external force (Fig. 1).

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Fig. 1. Dynamic model of the crankshaft (a) and bearing (b)

Instant values of the compliances can be defined from the expressions

K LO,,xj W

X j W FXj W ; K LO,,yj W Y j W FYj W ; K LI ,,xj W D j W M Xj W ; K LI ,,yj W

E j W M Yj W (1)

where FXj W , FYj W , M Xj W , M Yj W are instant values of the forces and moments, acting to the j -th bearing;

X j W , Y j W , D j W , E j W are the position of the center of the j -th main bearing journal of the crankshaft. Elastic subtask. The mechanical model presented on Fig. 1 is a statically indeterminate 3D beam system on an elastic foundation. Forces and moments, acting to the j -th bearing FXj W , FYj W , M Xj W , M Yj W are defined by redundant problem solving. The solution of this problem is based on equation of five moments

[G ] ˜ ^M ` where

[G ]

^R`

(2) is

the

band

matrix

of

the

influence

coefficients

G ij G ij  K sO, ,xj 1, n , K sO, ,yj 1, n , K LO,,xj 1, n , K LI ,, yj , K sI,,xj , K sI,,yj , K LI ,, xj , K LI ,, yj ; ^M ` is the vector of unknown moments; ^R` is on the right part of Eq (2), each vector entry is the deflection under external forces and moments Ri Ri ( FXj 1, n ,( s ) , FYj 1, n,( s ) , K sO, ,xj 1, n , K sO, ,yj 1, n , K LO,,xj 1, n , K LI ,, yj , 'case , 'necks ) ; 'necks , 'case are misalignments of necks and main bearings. Usually these misalignments are correlated with technology and thermal factors. The technological misalignments are caused by a coaxiality tolerance of the holes into the crankcase and a run-out tolerance of the crankshaft journals. The technique of thermal state definition is presented in works [16, 17]. In the present work 'case value depends on the gas forces. Gas forces reduce the displacements of the centers of the holes of the crankcase walls because their direction is opposite to bearing forces. This fact wasn't considered by other researchers. The 3D finite elements (FE) model of the IC engine crankcase is used for definition of displacements of the centers of the bearings holes under the gas forces 'gaz case . For each calculation moment of the engine cycle the gas forces field is applied to the crankcase FE model and the displacements of the centers of the bearings holes 'gaz case are determined. Matrix 'gaz case 'gaz W for each moments of the ICE cycle as well as the influence coefficients are defined previously. The technique of the redundant problem taking into account linear and angular compliances of the crankshaft supports is in more detail presented in [18]. In the same article the expressions for determination of forces and moments operating on bearings FXj W , FYj W , M Xj W , M Yj W calculated on basis of moments vector ^M ` are presented.

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Dynamic subtask. The position of each central of crankshaft necks X j W , Y j W , D j W , E j W defined by the decision of the dynamic equation system (the subscript j isn't specified) mU J J

F (U , U , J , J ,W )  R (U , U , J , J ,W ) M F (U , U , J , J ,W )  M R (U , U , J , J ,W )  M G (U , U , J , J ,W )

(3)

where U { X , Y }, J {D , E } {J ˜ cos H , J ˜ sin H },U, J are vectors of linear and angular coordinates and accelerations of the journals respectively; J is the skewed angle between the neck and bearing axis; H is the angle between the skewed and coordinated planes; m, J are the journal inertial characteristics; F {F , F }, X

Y

M F {M X , M Y } are forces and moments acting on the bearing (they are defined by solving the system (2); R , M R are the resultant vectors of forces and moments received by integration of the Reynolds equation for the lubricant pressure; M G is the gyroscopic torque. Unlike autonomous bearing, the vectors F , M F for non-autonomous system of bearings depends on damping characteristics of all supports and thus coordinates and accelerations of all crankshaft journals centers. Hydrodynamics subtask. For the definition of a hydrodynamic reaction of each bearing lubricated by nonNewtonian liquid it is necessary to know a hydrodynamic pressure distribution in a thin lubricant layer p I , z ,W . This function is obtained by solving Reynolds’ differential equation [19-21]

M12 · wp º w w ª k 2 § « h ¨ M2  ¸U » wI «¬ M0 ¹ wI »¼ wz ©

ª k 2 § M12 · wp º « h ¨ M2  ¸U » M0 ¹ wz »¼ «¬ ©

§ M1 w ª «Z U h ¨1  wI «¬ © M0

·º w ¸ »  ( U h ). ¹ »¼ wW

(4)

where U is lubricant density; p ( p  pa )\ 2 / P0Z0 , h are a non-dimensional hydrodynamic pressure and film thickness respectively; p, pa are a dimensional hydrodynamic pressure and the atmospheric pressure value; \ , Z are relative values of a bearing clearance and angular velocity; z is a non-dimensional coordinate in the direction of a bearing width; I is an angular bearing coordinate; k is an index characterizing the degree of non-newtonian behavior of a lubricant;

Mk

y2

k * ³ y P dy , k 0,1, 2 ,

y1

y is a non-dimensional coordinate on the frictional surface normal; P * is a non-dimensional value of viscosity which depends of viscosity versus shear rate, pressure and temperature

P*

( I 2 )( k 1)/ 2 ˜ C1 ˜ e(C2 /(Te  C3 ))  E (Te )˜ p .

(5)

where I 2 (wV x wy )2  (wV z wy )2 is the second invariant of the shear rate; Te is the temperature of the lubricant layer; C1 , C2 , C3 are the constants, which are the empirical characteristics of the lubricant; E (Te ) is a piezoelectric coefficient of viscosity which depends on the temperature and chemical composition of the lubricants. The empirical procedure for determining the characteristics of a lubricant is proved in [23]. The dimensional function of the film thickness taken into account the skewness of axes is h(I , Z1 ,W )

h (I , Z1 )  e (W ) cos(I  G (W ))  Z 1

2 S (W ) ˜ cos(I  H (W )) B

(6)

Yu. V. Rozhdestvensky et al. / Procedia Engineering 150 (2016) 647 – 653

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where e W is the displacement of the journal mass centers relative to the bearing; G W is the angle taking into account the skewness of axes of the bearing and the journal; S (W ) is the distance between the geometric centers of the journal and the bearing at the ends of the tribounit; B is the width of the bearing, Z1  >  B 2; B 2@ ; h (I , Z1 ) is the bearing layer thickness with non-ideal geometry at the central journal position. At integration of the equation (4) the Stieber-Swift boundary conditions are used. The lubricant thermal state is based on the isothermal approach. It assumes that the calculated current temperature is the same at the all points of the lubricant film. This temperature is a highly inertial parameter and it is determined by solving the bearing heat balance equation for correcting of the lubricant viscosity. The calculating method for the autonomous bearing is in more detail presented in work [19]. An efficiency of the design of hydrodynamic tribosystems can be estimated by the calculation of the standard set of the hydromechanical characteristics (HMC) in order to predict the fatigue durability and wear resistance of bearings antifrictional layers, friction losses, thermal loading of tribosystems and to solve the problem of optimizing the design parameters [2]. The most important HMC for main bearings are: instant values of the minimum film thickness hmin W and the maximum hydrodynamic pressure pmax W of a lubricant layer, and also their extremes * * inf hmin , sup pmax and averages hmin , pmax magnitudes per cycle of loading; effective temperature of a lubricating

layer Te* , instant and average friction power losses N W , N * , lubricant flow rates Q W , Q* . 3. The offered method

Simulation of the interconnection process of a fluid flow in thin films and the nonlinear dynamics for mechanical systems “an elastic crankshaft on journal bearings” is based on integration of the three subsystems - the elastic subtask (2), the Dynamic subtask (3) and the thermo-hydrodynamics subtask (4)-(6). The system is integrated by an iterative method. The iteration convergence parameter is defined to forces operating on the main bearing. At the beginning it is supposed K LO,,xj W K LO,,yj W K LI ,,xj W K LI ,, yj W 0 , j 1, n . Then the redundant problem is solved and the forces acting on main bearings are defined. The trajectory of the crankshaft journals centers for each bearing are determined by means of consecutive solving the equations (3), (4)-(6) and new values of K LO,,xj W , K LO,,yj W , K LI ,,xj W , K LI ,, yj W are calculated. The finite differences method by the multigrid technique for the Reynolds equation was applied for the definition of the main journals movement trajectory. The method allows to consider a dependence of viscosity on temperature, pressure and shear rate. In addition the scheme of oil supplies in an each bearing layer, macrogeometrical characteristics of friction surfaces and other parameters are taken into account. Then next iteration is started and new values of forces acting on main bearing are defined. This algorithm is interrupted if the relative summary error between the "new" and "old" values of the main bearing loads differ by less than 5%. The linear and angular structural compliances of the crankcase using a 3D finite element model are defined at the initial stage. Simultaneously the displacements of the centers of the bearings holes 'gaz case W under the gas forces are determined for each time of the IC engine cycle. 4. The results

The offered method was applied to calculations of the main bearings HMC of the six-cylinder in-line diesel engine of the type 13/15. The full engine power mode is considered. Values of linear and angular compliances of bearings and displacements of the centers of the holes of crankcase walls 'gaz case are received using a 3D finite elements model of the crankcase. Some results of the calculations are presented in Table 1 and in Fig. 2, 3. Fig. 1, 2 illustrate an influence of the offered method of a main bearing load calculation on values of loads acting on the 4-th main bearing and a movement trajectory of the center of the 4-th crankshaft neck respectively. If the displacements of the centers of bearings from gas forces 'gaz case are considered, the maximum values of loads on 4th main bearing are 20% lower, than without center displacements 'gaz case

0 . Also the area of the maximum

652

Yu. V. Rozhdestvensky et al. / Procedia Engineering 150 (2016) 647 – 653 * bearing wear is changed (fig. 2), inf hmin and hmin increased by 29% and 24% respectively. Using the offered technique we receive more realistic values of the loads acting on the main bearings and HMC of tribounits.

Table 1. The hydromechanical characteristics of the main bearings of the six-cylinder in-line diesel engine Bearing number

inf hmin Pm

Dh deg.

* hmin Pm

sup pmax MPa

* pmax MPa

N* W

Q* l/s

T*

D1,5

0

%

ɋ

1

2,283

459,7

10,81

68,15

16,98

485,6

0,034

97,36

0,0

2

1,694

173,1

6,827

226,1

47,06

616,5

0,036

98,92

0,0

3

1,762

652,1

7,407

184,9

40,94

596,9

0,034

98,94

0,0

4

2,597

636,2

7,851

94,62

41,59

610,6

0,043

97,39

0,0

5

1,371

308,6

7,867

172

44,56

612

0,037

98,63

4,6

6

1,462

545,4

8,276

140,4

37,23

581,2

0,036

98,42

2,2

Fig. 2. Influence of support displacements from gas forces (a) on loadings of the 4-th main bearing, (b) on trajectory of the center of the 4-th crankshaft journal

The calculation results for crankshaft main bearings of the investigated engine taking into account displacements 'gaz case are shown in Table 1. The 5-th main bearing is the most loaded and at the same time the film thickness value hmin W less than hkritical for 4% of the engine cycle. 5. Conclusions

The offered method allows us to investigate the operability of the main bearings of a high-power diesel, considering at the same time the influence of such efficiency and operational factors as structural rigidity of a crankcase, its thermal and force deformations, technological misalignments caused by a coaxiality tolerance of the

Yu. V. Rozhdestvensky et al. / Procedia Engineering 150 (2016) 647 – 653

holes into the crankcases and a run-out tolerance of the crankshaft journals, design features of bearings (holes, grooves on friction surfaces), etc. Using the offered technique we receive more realistic values of the loads acting on the main bearings and the tribounit hydromechanical characteristics. Acknowlegements

This work was carried out with the scientific support of the Russian Fond of Fundamental Investigation (project ʋ 16-08-01020\16). References [1] V.A Rizhov, Development of diesels of new generation at the Kolomna plant, Dvigatelestroenie. 2 (2009) 18–20. [2] M.I. Raenko, V.A. Rizhov, Using of hierarchical system of models for an assessment of durability of engines parts, Dvigatelestroenie. 2 (2009) 21௅26. [3] B.S. Antropov, I.M. Sotskaya, M.J. Ananin, Operability of bearing shells of a crankshaft, Tractors and agricultural cars. 8 (2008) 44–45. [4] V.E. Fradin, Deformation method of calculation of loads of a crankshaft and the analysis of influence of clearances in bearings and rotation frequencies on tension in cheeks of a crankshaft, Dvigatelestroenie. 1 (1998) 16–19. [5] M.M. Abramishvili, V.K. Chistyakov, Dynamic intensity of a crankshaft and its support V-engine with eight cylinder working on external high-speed and load characteristics, Dvigatelestroenie. 1 (1990) 10–12. [6] A.N. Krasnokutsky, Ju.Ju. Trifonov, Calculation of a crankshaft durability according to continuous scheme, in: Proceeding of The collection of scientific papes on the engine-building problems, devoted to the 175 anniversary of Bauman University (MGTU). (2005) 96–102. [7] M.K. Vetrov, Development of a method of calculation of the parameters characterizing loading of bearings of multisupporting crankshaft of piston engines, Ph.D. diss., 1984. [8] E.A. Zadorojznaya, Improvement and expansion of a scope of a method of calculation of dynamics and hydromechanical characteristics of sliding support with floating rings, Ph.D. diss., 2002. [9] R. Li, Staticaly, Static and dynamic characteristics of radial bearings with the floating ring, ASME, J. of Tribol. Trans. 3 (1982) 64–70. [10] V.N. Prokopiev, A.K. Boyarshinova, E.A. Zadorojznaya, Characteristics of stability of movable elements of a rotor support of a turbocompressor of boosting system of transport cars engines, Herald of the Ural interregional office of Academy of transport. (2001) 180– 186. [11] A.N. Gots, V.V. Efros, Calculation of hydrodynamic characteristics of bearings on unsteady modes, Tractors and agricultural cars. 1 (2007) 41–43. [12] S.M. Zaharov, Ju.L. Tarsis, E.A. Shoroh, Joint calculation of a multisupporting crankshaft and sliding bearings, Messenger of mechanical engineering. 1 (1985) 5–7. [13] G.I. Semenov, Definition of efforts in crankshaft taking into account its deformations, Automotive industry. 4 (1975) 20–30. [14] S.M. Zaharov, V.I. Sirotenko, I.A. Zharov, Work modeling of tribosystem a crankshaft bearings - support of the block of cylinders of internal combustion engines, Friction and Wear. 16 (1995) 47–54. [15] A.A. Mylnikov, Experimental research of elastic characteristics of a crankshaft and block case of the four-cylinder in-line engine of ChN 13/15 type, Herald of SUSU, Series Mashinostroenie. 271 (2012) 218–222. [16] Y. Rozhdestvensky, N. Khozenjuk, A. Mylnikov, Modeling of a multisupporting cranckshaft tribo-system of internal combustion engine, In: Proceedings of the 15th Nordic Symposium on Tribology NORDTRIB 2012, Trondheim, Norway. (2012) 163. [17] Y. Rozhdestvensky, N. Khozeniuk, A. Mylnikov, I. Levanov, V. Romanov, Modeling of the Main Bearings of a Multi-Supporting Crankshaft of the Internal Combustion Engine, In: Proceeding of WTC2013. (2013). [18] A.A. Mylnikov, N.A. Khozeniuk, Technique of an assessment of loading of support of a crankshaft taking into account elastic properties of a crankcase of the engine, Almanac of modern science and education. 79 (2013) 127௅131. [19] V.N. Prokopiev, E.A. Zadorojznaya, V.G. Karavaev, Improvement of a calculation method of heavy loaded bearings of the sliding greased by non-newtonian oils, Problems of mechanical engineering and reliability. 1 (2010) 63–67. [20] R. Paranjpe, S. Tseregounis, M. Viola, Comparison between theoretical calculations and oil film thickness measurements using the total capacitance method for crankshaft bearings in a firing engine, J of Tribol Trans. 43 (2000) 345–356. [21] R. Zhang, H. Xueming, S. Yang, Perturbation solution of non-Newtonian lubrication with the convected Maxwell model, Trans of the ASME 2005. 127 (2005) 302–305. [22] V.N. Prokop'ev, E.A. Zadorozhnaya, V.G. Karavaev, I.G. Levanov, Improvement of the computation procedure for complex-loaded sleeve bearings lubricated with non-newtonian oils, Journal of Machinery Manufacture and Reliability. 39 (2010) 52௅55. [23] I.G. Levanov, Method of calculation of hydromechanical characteristics of heavy loaded bearings of sliding of the piston and rotor cars greased by non-newtonian oils, Herald of SUSU, Series Mashinostroenie. 18 (2011) 34–44. [24] I. Mukhortov, E. Zadorojznaya, I. Levanov, Reological Model of a boundary layer of lubricant, In: Proceeding of STLE Annual Meeting & Exhibition, Hilton Atlanta. (2011) 235–241.

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