International Journal of Advance Engineering and Research Development

Scientific Journal of Impact Factor (SJIF): 4.14 International Journal of Advance Engineering and Research Development Volume 3, Issue 3, March -2016 e-issn (O): 2348-4470 p-issn (P): 2348-6406 BIFURCATION ANALYSIS OF HYDRODYNAMIC PLAIN JOURNAL BEARING LUBRICATED WITH COUPLE STRESS FLUID Sanjeev Kumar Lambha 1, Rajiv Verma 2 and Vinod Kumar 2 1 Department of Mechanical Engineering, Faculty of Engineering & Technology, Gurukula Kangri Vishwavidyalaya Haridwar (Uttarakhand), India 2,3 Department of Mechanical Engineering, National Institute of Technology, Kurukshetra (Haryana), India Abstract The dynamic analysis of rotor bearing system supported by two couple stress fluid film journal bearings with non-linear suspension is studied in this paper. The hydrodynamic pressure is generated entirely by the motion of the journal and depends on the viscosity of the lubricated fluid. In this analysis the dynamic trajectory, the power spectra, the Poincare maps, and the bifurcation diagrams are used to analyse the behaviour of the rotor centre and bearing centre in horizontal and vertical directions under different operating conditions. The short journal bearing assumptions are used for the analysis. The model consists of non-linear rotor equations of motion, which are system of nonlinear differential equations. The generalized Reynold s equation is derived for the selected system of rotor bearing. A non dimensional parameter l has been used for couple stress fluid. The method used to solve the dynamic equations is Range Kutta by considering the effects of rotor speed and degree of rotor unbalance. The periodic, quasi-periodic, subharmonic and chaotic motion are demonstrated in this study. Results show that the dynamics of system depends on rotor speed and degree of rotor unbalance. By using the couple stress fluid as lubricant dynamic stability of rotor bearing system enhances considerably as compared to the system using Newtonian fluid as lubricant. Keywords: Journal bearing, dynamic analysis, stability, couple stress fluid, Bifurcation, non-linearity INTRODUCTION In the category of journal bearings, Hydrodynamic journal bearings are one of the examples which support the rotating shafts under a condition of high radial loads. A number of practical applications of these bearings are pumps, compressors, turbines etc. The substance used as lubricants may be Newtonian or non-newtonian in nature. In 1988, Oliver [1] reported that the addition of polymer particles increased the load-carrying capacity of the fluid and reduced the friction effects. Similarly, Spikes [2] showed that adding suitable additives to a base oil lubricant yielded a significant reduction in the frictional effects in elastohydrodynamic contacts and therefore minimized the surface damage to both members of the contact pair. However, in the complex lubricants described above, the viscosity changes with the strain rate, and hence the classic Newtonian linear shear stress strain rate relationship no longer holds. As a result, the rheological behaviour of non-newtonian fluids is generally described using some form of micro-continuum theory. In 1966, Stokes [3] proposed a simple micro-continuum theory to explain the behaviour of couple-stress fluids. The Stokes model provides a convenient means of investigating particle-size effects in couplestress fluids and has been successfully applied to analyse fluid phenomena in a wide variety of scientific and engineering applications. For example, in 1997, Lin [4] used the Stokes micro-continuum model to investigate the squeeze film characteristics of long partial journal bearings lubricated with a couple-stress fluid. The same author [5] also applied the model to explore the static and dynamic behaviour of pure squeeze films in short journal bearings lubricated with a couple-stress fluid. The results revealed that the use of a couple stress lubricants rather than a Newtonian fluid enhances the dynamic stiffness and damping characteristics of the bearing system and reduces the pumping power, leading to a reduction in the lubricant flow rate. In 1998, Das [6] used the Stokes model to investigate the problem of slider bearings lubricated with couple-stress fluids in a magnetic field. The results showed that both the maximum load-carrying capacity and the corresponding inlet outlet film thickness ratio were governed by the magnitude of the couple stress, the magnetic parameters and the bearing geometry. Lin [7,8] used Stokes microcontinuum theory to analyse the static and dynamic characteristics of externally pressurized circular step thrust bearings and to conduct a linear stability analysis of a rotor-bearing system lubricated with a couple-stress fluid. The analytical results revealed that rotor-bearing systems lubricated with a couple-stress fluid are more stable than those lubricated with a conventional Newtonian fluid. In 2001, Abdallah and Lotfi [9] applied the Stokes micro-continuum theory to conduct an inverse analysis of journal bearings lubricated with couple-stress fluids. In 2003, Hsu et al. [10] investigated the effects of surface roughness on the dynamic response of short journal bearings lubricated with a couple stress fluid and showed that the combined effects of the couple stress and the surface roughness improved the load-carrying capacity of the journal bearing system and reduced both the rotor attitude angle and the values of the friction factors. In a recent study, Lahmar [11] conducted an elastohydrodynamic analysis of double-layered journal bearings and demonstrated that the use of a couple-stress lubricant increased the load-carrying capacity and stability of the bearing system and reduced the friction effects and attitude angle of the rotor. There are some prominent literatures about non-linear dynamics of rotor-bearing systems. Ehrich [12] investigated bifurcations in a high-speed rotor- @IJAERD-2016, All rights Reserved 486

bearing system and found that the rotor response was characterized by a sub-harmonic vibration phenomenon. In 1978, Holmes et al. [13] investigated the steady-state motion of a rigid shaft supported by two short journal bearings. The results showed that at small values of eccentricity, the motion was asymptotically periodic and consisted of a small number of components, located principally at synchronous and half-synchronous frequencies. However, at higher eccentricities, the motion was complex and did not settle to a limit cycle, remaining instead in a state of aperiodic motion. In 1994, Brown et al. [14] applied short bearing theory to construct a simple model to describe the behaviour of a rigid, hydro dynamically supported journal and showed that the journal behaved chaotically when the rotating unbalance force exceeded the gravitational load. Adiletta et al. [15 17] performed a series of theoretical and experimental investigations about a rigid rotor supported in short bearings and showed that the rotor performed subharmonic, quasi-periodic and chaotic motions at critical values of the system parameters. Many deterministic physical systems exhibit complex aperiodic behaviour at certain parameter values. This behaviour typically has the form of a strange attractor, indicative of chaotic motion. When the motion of a system is chaotic, its response remains unpredictable for long periods of time, and the resulting aperiodic behaviour is extremely sensitive to the initial conditions. If a system falls into chaos, its behaviour is difficult to predict and control. Hence, a fundamental task when designing journal bearing systems is to identify the geometry and operating conditions which induce chaotic motion such that these conditions can be specifically avoided. In 2006, C. K. Chen et al. [18] proposed a non-linear model to study the dynamic behaviour of the rotor centre and bearing centre and explained that the dynamic behaviour of the system includes 2T-periodic, quasi-periodic and chaotic motions. In 2007, C. W. Chang-Jian et al. [19] investigated the dynamic analysis of a rotor supported by two couple stress fluid film journal bearings with non-linear suspension. It was concluded that rotor bearing system lubricated with couple stress fluid is more stable than that with the conventional Newtonian fluid. In 2007, C. K. Chen et al. [20] analyse an system of flexible rotor supported by two couple stress fluid film journal bearing with non-linear suspension system and studied the behaviour of system for the dynamic response of bearing centre and rotor centre. In 2013, Amamou et al. [21] used the numerical continuation on a long balanced hydrodynamic journal bearing and predict the branch of the journal equilibrium point, the Hopf bifurcation point and the emerging stable or unstable limit cycles. In this study we analyze the dynamic behaviour of a bearing system comprising a flexible rotor supported by two couple-stress fluid film journal bearings with non-linear suspension systems. Due to the non-linearity of the lubricant we assume the journal bearing be the short bearing approximation to get computation easily. The dynamic equations of the rotor centre and journal centre are solved using the Range Kutta method and the system response is illustrated by reference to bifurcation diagrams, Poincare maps and dynamic trajectory diagrams. Nomenclature c radial clearance, c = R-r c1 damping coefficient c2 viscous damping of rotor disk e eccentricity, e = X 2 + Y 2 f dimensionless parameter, f = mg/cks fe, fφ radial and tangential components of fluid film forces Fx, Fy fluid film force components in X direction and Y direction g acceleration due to gravity k1, k2 stiffness coefficients of springs supporting bearing housings ks shaft stiffness coefficient l characteristic length of additives, l =(ɳ/μ) 1/2 l* dimensionless couple-stress parameter, l* = l/c L bearing length m, m0 masses lumped at rotor mid-point and bearing housing mid-point Om rotor gravity center O1, O2, O3 geometric centers of bearing, rotor and journal p pressure distribution in fluid film R inner radius of bearing housing r radius of journal s rotational speed ratio, s =(ω 2 / ωn 2 ) s1 dimensionless parameter, s1 = (comcps 2 ) 1/2 X, Y, Z horizontal, vertical and axial coordinates x1, y1, x2, y2 X1/c, Y1/c, X2/c, Y2/c α dimensionless parameter, α = (k 2 c 2 )/(kscom) ρ mass eccentricity of rotor ɸ rotational angle (ɸ = ωt) rotational speed of rotor @IJAERD-2016, All rights Reserved 487

I. MATHEMATICAL MODELLING Figure 1 represents to a flexible rotor supported by two couple stress fluid film journal bearing with non-linear suspension systems. O m is the rotor gravitational centre, O 1 the geometrical centre of bearing attached, O 2 is geometrical centre of rotor, O 3 is the geometrical centre of journal; m 0 the mass of bearing housing, m the mass of rotor; k 1 and k 2 are the spring stiffness coefficients supported by two bearing housings, k s is the spring stiffness of the shaft; c 1 and c 2 are the damping coefficients of the supported structure and the rotor disk; the bearing housing inner radius is R, radius of shaft is r; ɸ is the rotational angle; the mass eccentricity of the rotor is ρ. Figure 1: A flexible rotor supported by two couple stress fluid film journal bearing and cross section of fluid film journal bearing To analyze this rotor bearing system a few assumptions are made: a) The rotor and bearing housing are rigid, b) The mass of rotor, torque of rotor disk, axial and torsional vibrations are assumed to be negligible. c) Theory of short bearing approximation is applied. d) A modified Reynolds equation is obtained to govern the couple stress fluid film forces. e) The revolving speed of rotor remains constant f) Unbalance of rotor located at mid-point of rotor g) Initially the rotor and bearing mass centres are located at the mid-points of rotor and bearing. h) The rotor, bearings and bearing housing supports are radially symmetric i) The damping at the mid-point of the rotor due to air resistance is viscous 1. MODIFIED REYNOLDS EQUATION According to Stokes micro-continuum theory the non-newtonian type of Reynolds equation used in the study is of the form p (g(h, l) ) + p h h (g(h, l) ) = 6μU + 12μ x x z z x where g(h, l)= h 3-12l 2 h+24l 3 tanh(h/2l), h c(1 + ε cos(γ φ(t))) = c(1 + ε cosθ). Therefore g(h, l)= c 3 (1+ ε cosθ) 3 12l 2 c(1+ ε cosθ) + 24l 3 tanh( x, (1) t h = -cε/r sin θ, t c(1+ε cosθ) 2l = cε cos θ + cεφ sin θ, x = Rθ, U=Rω, ε = e/c, h = ), (2) where l = (ɳ/ μ) 1/2, μ is the classical viscosity parameter, ɳ is a material constant for couple stress fluid. Thus p (g(h, l) ) + p (g(h, l) ) = 6μωcε sin θ + 12μ(cε cos θ + cεφ sin θ) (3) R θ R θ z z On the application of theory of short bearing approximation L/D<0.25 so p, therefore equation becomes, < p θ z 2 p/ z 2 6μωcε sin θ+12μ(cε cos θ + cεφ sin θ) = (4) g(h,l) The boundary conditions are: p/ z = 0 when z = 0 & p = 0, when z = ± L 2 3μ p = [(ω 2φ )ε sin θ 2ε cos g(h,l) θ](z2 -L 2 /4) (5) 2. FORCE EQUATIONS Forces acting on the centre of journal O 3(X 3, Y 3) are given by, F x= f ecosφ + f φsinφ = k s(x 2 X 3)/2, F y= f esinφ + f φcosφ = k s(y 2 Y 3)/2 (6) Where, f e and f φ are viscous damping forces in radial and tangential directions, and such that f e = -f r and f φ = -f t (7) f r= π L/2 0 L/2 π L/2 0 L/2 pr cos θ dzdθ, (8.1) f t = pr sin θ dzdθ (8.2) Therefore, f e = -μl 3 R/2c 2 π { 0 cosθ)3-12(l*) 2 (1+ ε cosθ) + 24(l*) 3 tanh(((1+ε cosθ))/2l*) )}dθ, (where l* = l/c) (9) @IJAERD-2016, All rights Reserved 488

f φ = -μl 3 R/2c 2 π { ([(ω-2φ )ε sin θ-2ε ċos θ]sinθ)/((1+ ε 0 cosθ)3-12(l*) 2 (1+ ε cosθ) + 24(l*) 3 tanh(((1+ε cosθ))/2l*) )}dθ, (where l* = l/c) (10) Suppose α 1 = -μl 3 R/2c 2, (11.1) π β 1 = { (sinθ cosθ)/((1+ ε 0 cosθ)3-12(l*) 2 (1+ ε cosθ) + 24(l*) 3 tanh(((1+ε cosθ))/2l*) )}dθ, (11.2) π γ 1 = { 0 (cos2 θ)/((1+ ε cosθ) 3-12(l*) 2 (1+ ε cosθ) + 24(l*) 3 tanh(((1+ε cosθ))/2l*) )}dθ, (11.3) π δ 1 = { 0 (sin2 θ)/((1+ ε cosθ) 3-12(l*) 2 (1+ ε cosθ) + 24(l*) 3 tanh(((1+ε cosθ))/2l*) )}dθ, (11.4) Such that f e = α 1{(ω 2φ )ε β 1-2ε γ 1}, f φ = α 1{(ω 2φ )ε δ 1-2ε β 1} (12) Therefore, F x = [α 1{(ω 2φ )ε β 1-2ε γ 1}cosφ] [α 1{(ω 2φ )ε δ 1-2ε β 1}sinφ], (13.1) F y = [α 1{(ω 2φ )ε β 1-2ε γ 1}sinφ] [α 1{(ω 2φ )ε δ 1-2ε β 1}cosφ], (13.2) & F x = ck s(x 2-x 1- εcosφ)/2, F y = ck s(y 2-y 1- εsinφ)/2 (14) Finally, ε = [β 1ck s{(x 2-x 1- εcosφ)sinφ - (y 2-y 1- εsinφ)cosφ} - δ 1ck s{(x 2-x 1- εcosφ)cosφ+(y 2-y 1- εsinφ)sinφ}]/(4α 1(γ 1δ 1 (β 1) 2 )), (15) φ = ω/2 - [γ 1ck s{(x 2-x 1- εcosφ)sinφ - (y 2-y 1- εsinφ)cosφ} - β 1ck s{(x 2-x 1- εcosφ)cosφ+(y 2-y 1- εsinφ)sinφ}]/(4α 1ε(γ 1δ 1 (β 1) 2 )) (16) In Non-Dimensional form ε = [β 1ck s{(x 2-x 1- εcosφ)sinφ - (y 2-y 1- εsinφ)cosφ} - δ 1ck s{(x 2-x 1- εcosφ)cosφ+(y 2-y 1- εsinφ)sinφ}]/(4α 1ω(γ 1δ 1 (β 1) 2 )) (17) φ = 1/2 - [γ 1ck s{(x 2-x 1- εcosφ)sinφ - (y 2-y 1- εsinφ)cosφ} - β 1ck s{(x 2-x 1- εcosφ)cosφ+(y 2-y 1- εsinφ)sinφ}]/(4α 1εω(γ 1δ 1 (β 1) 2 )) (18) 3. MASS EQUATIONS mx 2 + c 2X 2 + k s(x 2 X 3) = mρω 2 cosφ, (19.1) my 2 + c 2Y 2 + k s(y 2 Y 3) = mρω 2 sinφ - mg, (19.2) m 0X 1 + c 1X 1 + k 1X 1 + k 1X 1 3 = F x, (19.3) m 0Y 1 + c 1Y 1 + k 1Y 1 + k 1Y 1 3 = - m 0g +F y. (19.4) In non-dimensional form x 2 + (2ξ 2/s)x 2 + (1/s 2 )( x 2-x 1- εcosφ) = βcosφ, (20.1) y 2 + (2ξ 2/s)y 2 + (1/s 2 )( y 2-y 1- εcosφ) = βsinφ f/s 2, (20.2) x 1 + (2ξ 1/s 1)x 1 + (1/s 1 2 )x 1 + (α/s 2 )x 1 3 ½(1/C om)(1/s 2 )( x 2-x 1- εcosφ) = 0, (20.3) y 1 + (2ξ 1/s 1)y 1 + (1/s 1 2 )y 1 + (α/s 2 )y 1 3 ½(1/C om)(1/s 2 )( y 2-y 1- εcosφ) + (f/s 2 )= 0. (20.4) The systems of equations are solved to find solution by using the Range-Kutta scheme of numerical methods. RESULTS AND DISCUSSIONS The numerical analysis is carried out by using the Runge-Kutta method. The time steps for direct numerical integration is specified as pi/100 and the error tolerance is lower than 0.0001. In presenting the results, the time-series data relating to the first few revolutions of the rotor are deliberately excluded in order to ensure that analysed data corresponds to steadystate conditions. In the simulations, the system parameters are assigned the following values: ξ 1 = c 1/(2 k 1m 0 ) = 0.0525, ξ 2 = c 2/(2 k sm ) = 0.026, f = mg/ck s = 0.0972, c p = (k s/k 1)= 2.0, C om = (m 0/m) = 0.125, Ʌ = 0.12, β = ρ/c = 0.45, x 1(0) = 0.2, x 1 (0) = 0.00000001, y 1(0) = 0.4, y 1 (0) = 0.00000001, x 2(0) = 0.5, x 2 (0) = 0.00000001, y 2(0) = 0.4, y 2 (0) = 0.00000001 The dynamic responses of the rotor and bearing centers are evaluated by reference to bifurcation diagrams, dynamic trajectory diagrams and Poincare maps, respectively @IJAERD-2016, All rights Reserved 489

Figure 8: Trajectory diagram for s = 1.4 and l* = 0.1 Figure 9: Trajectory diagram for s = 1.8 and l* = 0.1 Figure 10: Trajectory diagram for s = 2.2 and l* = 0.1 Figure11: Bifurcation diagrams of bearing centre and rotor centre in horizontal and vertical direction for couple stress fluid (l* = 0.0) @IJAERD-2016, All rights Reserved 492

Figure12: Bifurcation diagrams of bearing centre and rotor centre in horizontal and vertical direction for couple stress fluid (l* = 0.1) With the help of trajectory diagrams & bifurcation diagrams it is very clear that for s=0.1 and l* = 0.0 both the bearing centre and the rotor centre exhibit a periodic motion. For s=0.1 and l* = 0.0 the bearing centre and the rotor centre exhibit the quasi-periodic motion. For s = 0.5-0.8 and l* = 0.0 the bearing centre exhibit and the rotor centre exhibit the chaotic motion. For s = 1.1 and l* = 0.1 the bearing centre exhibit and the rotor centre exhibit the quasi-periodic motion. For s = 1.4-2.0 and l* = 0.1 the bearing centre exhibit and the rotor centre exhibit the chaotic motion From s = 2.0 and l* = 0.1 the bearing centre exhibit and the rotor centre starts to become quasi-periodic II. CONCLUSION In this study the results are coated for two values of couple stress parameters l* = 0.0 & l* = 0.1. As the values of rotational speed ratio increases, the behaviour of the bearing centre and the rotor centre tends to be chaotic in nature first for both the cases. Then the centres start exhibiting quasi-static nature. Overall the results have shown that the stability of rotor bearing system depends upon the rotational speed of the rotor and its degree of unbalance. At higher values of rotational speed ratio the motion becomes quasi-periodic thus if we proceed further the motion tends to be periodic in nature. V. REFERENCES [1] D.R. Oliver, Load enhancement effects due to polymer thickening in a short model journal bearings, J Non- Newtonian Fluid Mech, volume 30, pp. 185 196, 1988 [2] H.A. Spikes, The behaviour of lubricants in contacts: current understanding and future possibilities, Proc Instn Mech Eng, Part J: J. Engg. Tribology, volume 28, pp. 3 15, 1994 [3] V.K. Stokes, Couple-stresses in fluids Physical Fluids, volume 9, pp. 1709 1715, 1966 [4] J.R. Lin, Squeeze film characteristics of long partial journal bearings lubricated with couple stress fluids Tribology International, volume 30(1), pp. 53 58, 1997 [5] J.R. Lin, Static and dynamic behaviour of pure squeeze films in couple stress fluid lubricated short journal bearings Proc Instn Mech Eng, volume 62(1), pp. 175 184, 1997 [6] N.C. Das, A study of optimum load-bearing capacity for slider bearings lubricated with couple stress fluids in magnetic field, Tribology International, volume 31(7), pp. 393 400, 1998 [7] J.R. Lin, Static and dynamic characteristics of externally pressurized circular step thrust bearings lubricated with couple stress fluids, Tribology International, volume 32, pp. 207 216, 1999 [8] J.R. Lin, Linear stability analysis of rotor-bearing system: couple stress fluid model, Computer Structure, volume 79, pp. 801 809, 2001 [9] A.E. Abdallah, HG. Lotfi, An inverse solution for finite journal bearings lubricated with couple stress fluids Tribology International, volume 34, pp. 107 118, 2001 [10] C.H. Hsu, J.R. Lin, H.L. Chiang, Combined effects of couple stresses and surface roughness on the lubrication of short journal bearings Ind Lubricat Tribol, volume 55(5), pp. 233-243, 2003 [11] M. Lahmar, Elastohydrodynamic analysis of double-layered journal bearings lubricated with couple-stress fluids Proc Instn Mech Eng, Part J: J Engg. Tribology, volume 219, pp. 145 171, 2005 [12] F.F. Ehrich, Some observations of chaotic vibration phenomena in high-speed rotordynamics ASME J Vibrat Acoust, volume 113, pp. 50 57, 1991 [13] A.G. Holmes, C.M. Ettles, L.W. Mayes, Aperiodic behavior of a rigid shaft in short journal bearings Int J Numer Mech Eng, volume 12, pp. 695 702, 1978 @IJAERD-2016, All rights Reserved 493

[14] R.D. Brown, P. Addison, A.H.C. Chan, Chaos in the unbalance response of journal bearings. Nonlinear Dyn, volume 5, pp. 421 432, 1994 [15] G. Adiletta, A.R. Guido, C. Rossi, Chaotic motions of a rigid rotor in short journal Bearings, Nonlinear Dynamics, volume 10, pp. 251 269, 1996 [16] G. Adiletta, A.R. Guido, C. Rossi, Nonlinear dynamics of a rigid unbalanced rotor in short bearings Part I: theoretical analysis, Nonlinear Dynamics, volume 14, pp. 57 87, 1997 [17] G. Adiletta, A.R. Guido, C. Rossi, Nonlinear dynamics of a rigid unbalanced rotor in short bearings. Part II: theoretical analysis Nonlinear Dynamics, volume 14, pp. 157 189, 1997 [18] C.K. Chen, C.W.C. Jian, Bifurcation and chaos analysis of a flexible rotor supported by turbulent long journal bearings, Chaos Solutions & Fractals, volume 34, pp. 1160-1179, 2006 [19] C.W. Chang-Jian, Non-linear numerical analysis of a flexible rotor equipped with squeeze couple stress fluid film bearings, Acta Mechanica Solida Sinica, volume 20(4), pp. 309-316, 2007 [20] C.K.Chen, C.W.C. Jian, Bifurcation analysis of a flexible rotor supported by couple stress fluid film bearings with non-linear suspension systems, Tribology International, volume 41, pp. 367-386, 2007 [21] A. Amamou, M. Chouchane, Nonlinear stability analysis of long hydrodynamic journal bearings using numerical continuation, Mech. And Machine Theory, volume 72, pp. 17-24, 2013 @IJAERD-2016, All rights Reserved 494