Engineering Tribology 2E Episode 7 doc

246 ENGINEERING TRIBOLOGY Misalignment has surprisingly little effect on the location of the reformation front or even the cavitation front compared to its effect on the pressure peak. This feature is probably due to the large film thickness in the cavitated regions of the hydrodynamic film which ensures a small relative change in film thickness with misalignment. Lubricant flow rates are also relatively unaffected by misalignment which renders unlikely the possibility of lubricant starvation with increasing shaft misalignment. The high values of maximum pressure, however, are undesirable. 5.6.4 VIBRATIONAL STABILITY IN JOURNAL BEARINGS As discussed already in Chapter 4, hydrodynamic bearings are prone to a vibrational instability known as ‘oil whirl’. Vibration characteristics of a hydrodynamic film can be modelled by a series of stiffness and damping coefficients. These coefficients can be computed from the solutions of the Reynolds equation. Vibration analysis of hydrodynamic bearings can be directed to the computation of the shaft trajectory in a vibrating bearing. This approach, however, involves a rigorous analysis of bearing instability and is a specialized task requiring extensive computing. A much simpler mode of analysis for practical engineering applications is discussed in this section. In this approach the limiting shaft speed at the onset of vibration is calculated using the Routh-Hurwitz criterion of stability. The criterion provides a conservative estimate of the shaft speed at which some level of sustained vibration occurs. It has often been found that at moderate shaft speeds, shaft vibration may occur but it is limited to a finite and safe amplitude. On the other hand at higher speeds, there is no limit to the amplitude of vibration and the shaft will oscillate in ever wider trajectories until it touches the bush which inevitably results in destruction of the bearing. In order to analyze shaft trajectories, the non-linear variation in stiffness and damping coefficients with shaft position must be included in the analysis. The advantage of the Routh-Hurwitz method is that only infinitesimal amplitudes of vibration are considered which allow the use of linearized stiffness and damping coefficients. The linearized Routh- Hurwitz analysis of bearing vibration and the computation method is described in the following sections. Determination of Stiffness and Damping Coefficients Stiffness and damping coefficients are obtained by including in the Reynolds equation the effect of small displacements and squeeze velocities. Stiffness and damping coefficients are calculated from the change in pressure integral, by dividing the changes by the displacement and squeeze velocity respectively. Magnitudes of displacements and squeeze velocities are held at small values in order to minimize inaccuracy due to non-linear variation of film forces. A cartesian coordinate system aligned with the direction of bearing load, shown in Figure 5.29, is established and values of stiffness and damping coefficients normal and co- directional with the load-line are then computed. Four stiffness coefficients relating to the range of possible bearing movements ‘K xx ’, ‘K yy ’, ‘K xy ’ and ‘K yx ’ and four damping coefficients ‘C xx ’, ‘C yy ’, ‘C xy ’ and ‘C yx ’ are required for vibration analysis. To find these coefficients the effect of small displacements on hydrodynamic pressure integral must be analyzed. Shaft displacements are modelled in the Reynolds equation in terms of their effect on dh/dx. It is convenient to use non-dimensional forms of shaft displacement in terms of the radial bearing clearance, i.e.: COMPUTATIONAL HYDRODYNAMICS 247 ∆x c = ∆x* (5.88) where: ∆x is the displacement of the shaft centre in the ‘x’ direction [m]; c is the radial clearance of the bearing [m]; ∆x* is the non-dimensional displacement. x y ω W FIGURE 5.29 Journal bearing coordinate configuration for vibration analysis. A similar relationship applies to ‘∆y’, the displacement in the ‘y’ direction. The equation for dh*/dx* is given in the following form according to basic geometrical principles: ∂h* ∂x* () = ∂h* ∂x* static + ∂ ∂x* [ ∆x*cos (x*) + ∆y*sin (x*) ] (5.89) where: x* refers to the film ordinate around the bearing; ∂h * ∂x * static is the variation in film thickness for the static case. The modified forms of ‘h*’ and ∂ 2 h * /∂x * 2 which are required for the Vogelpohl equation follow the scheme already described and are given by: h* = h* static + ∆x*cos (x*) + ∆y*sin (x*) (5.90) ∂ 2 h* ∂x* 2 () = ∂ 2 h* ∂x* 2 static + ∂ 2 ∂x* 2 [ ∆x*cos (x*) + ∆y*sin(x*) ] (5.91) The Vogelpohl equation (5.4) is then solved in terms of the modified forms of ‘h*’ and its derivatives, i.e. ∂h*/∂x*, ∂h*/∂y*, etc. Non-dimensional stiffness coefficients are defined as : 248 ENGINEERING TRIBOLOGY K* = Kc W (5.92) where: K* is the non-dimensional stiffness; K is the real stiffness (Note, in this section ‘K’ denotes the stiffness) [N/m]; c is the radial clearance of the bearing [m]; W is the bearing load [N]. This form of non-dimensionalization can be shown to be equivalent to: ∆W* ∆x*W* static K* = (5.93) Since ‘δx*’ is very small then: W* ≈ W static * In other words, non-dimensional stiffness coefficients are equal to the change in non- dimensional load divided by the product of non-dimensional displacement and static non- dimensional load. The change in load ‘∆W*’, is calculated from the total load found by integration of the hydrodynamic pressure field with the displacement parameters included, and the static load, i.e.: ∆W * = W * - W stat i c * (5.94) In exact terms, only the change in film force along the ‘x’ or ‘y’ axis is calculated not the change in the total load. For example, ‘K xx * ’ stiffness is calculated according to the following equation, i.e.: ∆W* x ∆x*W* K* xx = (5.95) where ‘ ∆W x * ’ is the load change in the ‘x’ direction, (i.e. first index denotes the axis along which the deflection occurs, while the second index denotes the axis of the force). Similarly stiffness ‘ K yx * ’ is given by: ∆W* y ∆x*W* K* yx = (5.96) where ‘ ∆W x * ’ is the load change in the ‘y’ direction. A similar convention applies for stiffnesses ‘ K yy * ’ and ‘ K xy * ’. Damping coefficients are found by adding appropriate squeeze terms to the Reynolds equation. A non-dimensional squeeze term is defined as: COMPUTATIONAL HYDRODYNAMICS 249 w cω w* = (5.97) where: w is the squeeze velocity [m/s]; c is the radial clearance of the bearing [m]; ω is the angular velocity of the shaft [rad/s]. and the non-dimensional form of the Reynolds equation with squeeze terms is given by: ∂ ∂x* ∂p* ∂x* h* 3 () + ( R L ) 2 ∂ ∂y* ∂p* ∂y* h* 3 () = ∂h* ∂x* + 2w* (5.98) The squeeze velocity is not constant around the hydrodynamic film but varies in a sinusoidal manner similar to the displacements. An expression for the dimensionless squeeze velocity at any position on the hydrodynamic film in terms of squeeze velocities along the ‘x’ and ‘y’ axes is given by: w* = w x *cos(x*) + w y *sin(x*) (5.99) The squeeze term ‘w*’ can be included in the parameter ‘G’ of the Vogelpohl equation, i.e.: = h* 1.5 ∂ 2 M v ∂x* 2 + ( R L ) 2 ∂ 2 M v ∂y* 2 = FM v + G FM v + ∂h* ∂x* + 2w* (5.100) Damping coefficients are computed in a similar manner to the stiffness coefficients, i.e. an arbitrary infinitesimal squeeze velocity is applied to cause a change in the pressure integral. The non-dimensional damping coefficient is defined in a similar manner to the non- dimensional stiffness coefficient, i.e.: cω W C* = C () (5.101) where: C* is the non-dimensional damping coefficient; C is the real damping coefficient [Ns/m]. Expressing (5.101) in terms of non-dimensional quantities gives the non-dimensional damping coefficient, i.e.: ∆W* w*W* C* = (5.102) 250 ENGINEERING TRIBOLOGY and a specific damping coefficient, e.g. ‘ C xx * ’ is calculated according to: ∆W* x w* x W* C* xx = (5.103) After determining all the necessary values of stiffness and damping coefficients the vibrational stability of a bearing can be evaluated. There are various theories of bearing vibrational analysis and the obtained stiffness and damping coefficients can be used in any of these methods. A very useful theory for vibrational analysis of a journal bearing was developed by Hori [7]. In this theory a simple disc of a mass ‘m’ mounted centrally on a shaft supported by two journal bearings is considered. The disc tends to vibrate in the ‘x’ and ‘y’ directions which are both normal to the shaft axis. The configuration is shown in Figure 5.30. m k 2 k 2 Bearing Bearing Combined shaft stiffness = k Oscillation Rotating mass e.g. turbine rotor FIGURE 5.30 Hori's model for journal bearing vibration analysis. There are two sources of disc deflection in this model; the shaft can bend and the two bearings are of finite stiffness which allows translation of the shaft. This system was analyzed by Newton's second law of motion to provide a series of equations relating the acceleration of the rotor in either the ‘x’ or the ‘y’ direction to the mass of the disc, shaft and bearing stiffnesses, and bearing damping coefficients. The description of this analysis can be found in [7]. The equations of motion of the disc can be solved to produce shaft trajectory but this is not often required since the most important information resulting from the analysis is the limiting shaft speed at the onset of bearing vibration. The limiting shaft speed is derived from the Routh-Hurwitz criterion which provides the following expression for the ‘threshold speed of self-excited vibration’ or the ‘critical frequency’ as it is often called: A 1 A 3 A 5 2 (A 1 2 + A 2 A 5 2 − A 1 A 4 A 5 )(A 5 +γA 1 ) ω* c 2 = (5.104) where: A 1 ,A 2 ,A 5 are the dimensionless stiffness and damping products; ω c * is the dimensionless bearing critical frequency. The bearing critical frequency is also given by: ω c ω* c = (g/c) 0.5 (5.105) COMPUTATIONAL HYDRODYNAMICS 251 where: ω c is the angular speed of the shaft [rad/s]; g is the acceleration due to gravity [m/s 2 ]; c is the radial clearance of the bearing [m]. and the ‘γ’ parameter is expressed by: W γ = kc (5.106) where; W is the weight on the shaft [N]; k is the stiffness of the shaft [N/m]. Since the ‘γ’ parameter is independent of bearing geometry is must be specified before commencing computing of a solution to equation (5.104). The ‘A’ terms relate to stiffness and damping coefficients in the following manner [7]: A 1 = K* xx C* yy − K* xy C* yx − K* yx C* xy + K* yy C* xx (5.107) A 2 = K* xx K* yy − K* xy K* yx (5.108) A 3 = C* xx C* yy − C* xy C* yx (5.109) A 4 = K* xx + K* yy (5.110) A 5 = C* xx + C* yy (5.111) The analysis is completed with the calculation of the non-dimensional critical frequency ‘ ω c * ’. Computer Program for the Analysis of Vibrational Stability in a Partial Arc Journal Bearing An example of a computer program ‘STABILITY’ for analysis of vibrational stability in a partial arc journal bearing is listed and described in the Appendix and its flow chart is shown in Figure 5.31. The program computes the limits of bearing vibrational stability. The Vogelpohl equation is solved by the same method described for the program ‘PARTIAL’. Although the program ‘STABILITY’ specifically refers to partial arc bearings a similar program could be developed for grooved bearings since the principles applied are the same. Example of the Analysis of Vibrational Stability in a Partial Arc Journal Bearing Comprehensive tables of a perfectly aligned bearing can be found in [7]. Of considerable practical interest, however, is the effect of shaft misalignment on bearing critical frequency. The computed results of the effect of shaft misalignment on critical frequency of a 120° partial arc bearing, L/D = 1, eccentricity ratio 0.7 and dimensionless exciter mass 0.1, are shown in Figure 5.32. A mesh density of 11 rows in both the ‘x*’ and ‘y*’ directions was applied in computation. 252 ENGINEERING TRIBOLOGY It can be seen from Figure 5.32 that there is a decline in critical frequency with increasing misalignment. However, at extreme values of misalignment the critical frequency rises as a result of the sharp increase in the principal stiffness coefficient ‘ K xx * ’. In practical bearing systems where misalignment is inevitable, operating the bearing at speeds very close to the critical speed as predicted from the perfectly aligned condition is not recommended. For example, if the value of radial clearance is 0.0002 [m] and g = 9.81 [m/s 2 ] then the conversion factor from non-dimensional to real frequency according to equation (5.105) is equal to: (g/c) 0.5 = (9.81/0.0002) 0.5 = 221.5 [Hz] The calculated difference between the minimum dimensionless critical speed for the bearing with a misalignment parameter of t = 0.2 and a perfectly aligned bearing is: ω c,misaligned * - ω c,aligned * = 2.2647 - 1.8591 = 0.4056 which makes the difference in the angular speed of the shaft about: 221.5 × 0.4056 = 89.6 [Hz] or [rad/s] Start Acquire bearing parameters Eccentricity ε L/D ratio Dimensionless exciter mass γ Misalignment parameter t Special settings of iteration parameters? No Yes Acquire parameters Set values of DX, DY, WX & WY Set initial values of arc position as bisecting minimum film thickness A Use preset values Set initial zero values of M(I,J), P(I,J) & switch function COMPUTATIONAL HYDRODYNAMICS 253 End A Solve Vogelpohl equation and attitude angle iterations by same method as in program Partial No Negative solution? Yes Assign SWITCH=1 to M(I,J) nodes Output: print KXX, KYX, KXY, KYY, CXX, CYX, CXY, CYY and dimensionless frequency limit 1 1 Except SWITCH(I,J)=1 nodes are excluded from the iteration Stabilization of cavitation front Store values of M(I,J) as MSAVE(I,J); Store load etc. Calculate new film thickness h and derivatives with # change imposed Call subroutine to solve Vogelpohl equation Calculate pressure field difference based on M(I,J)/W(I,J)^1.5 old − M(I,J)/W(I,J)^1.5 new 1 Call subroutine 2 to find load components 2 Calculate stiffness coefficients Call subroutine to solve Vogelpohl equation This loop is done 4 times: # = DX # = DY # = WX # = WY Calculate coefficients of Routh Hurwitz criterion Apply Routh Hurwitz criterion Print warning Take modulus of solution Subroutine 1 from program Partial Subroutine 2 from program Partial FIGURE 5.31 Flow chart of program for the analysis of vibration stability in partial arc bearings. 254 ENGINEERING TRIBOLOGY 1 3 4 0 0.1 0.2 0.3 0.4 0.5 0.6 Misalignment parameter t Dimensionless critical frequency 2 FIGURE 5.32 Effect of shaft misalignment on bearing dimensionless critical frequency. 5.7 SUMMARY Numerical analysis has allowed models of hydrodynamic lubrication to include closer approximations to the characteristics of real bearings than the original idealized analytical solutions. Some adaptations of numerical models presented in current research literature have been introduced in this chapter to illustrate the potential of computational methods. The strong influence of secondary effects such as lubricant heating and bearing deformation on load capacity is shown, together with possible methods of controlling the negative effects these have on bearing performance. The scope of numerical analysis is continually being extended. With increases in the speed of computing it may become possible to perform the simultaneous analysis of several different effects on bearing performance, e.g. the combined effect of heating, deformation and misalignment. The finite difference method used in numerical analysis is versatile and simple to apply, but is also relatively inaccurate. Newer methods of devising numerical equivalents of differential equations are being increasingly adopted. However, the fundamental principles of numerical analysis outlined in this chapter remain unaltered. REFERENCES 1 G. Vogelpohl, Beitrage zur Kenntnis der Gleitlagerreibung, Ver. Deutsch. Ing., Forschungsheft, 1937, pp. 386. 2 M.M. Reddi and T.Y. Chu, Finite Element Solution of the Steady-State Incompressible Lubrication Problem, Transactions ASME, Journal of Lubrication Technology, Vol. 92, 1970, pp. 495-503. 3 J.F. Booker and K.K. Huebner, Application of Finite Element Methods to Lubrication, an Engineering Approach, Transactions ASME, Journal of Lubrication Technology, Vol. 94, 1972, pp. 313-323. 4 A. Cameron, Principles of Lubrication, Chapter by M.R. Osborne on Computation of Reynolds' Equation, Longmans, London, 1966, pp. 426-439. 5 A.A. Raimondi and J. Boyd, A Solution for the Finite Journal Bearing and its Application to Analysis and Design, ASLE Transactions, Vol. 1, 1958, pp. 159-209. 6 A. Cameron, Principles of Lubrication, Longmans, London, 1966, pp. 305-340. 7 T. Someya (editor), Journal-Bearing Data-Book, Springer Verlag, Berlin, Heidelberg, 1989. COMPUTATIONAL HYDRODYNAMICS 255 8 A.J. Colynuck and J.B. Medley, Comparison of Two Finite Difference Methods for the Numerical Analysis of Thermohydrodynamic Lubrication, Tribology Transactions, Vol. 32, 1989, pp. 346-356. 9 C.M.Mc. Ettles, Transient Thermoelastic Effects in Fluid Film Bearings, Wear, Vol. 79, 1982, pp. 53-71. 10 J.H. Vohr, Prediction of the Operating Temperature of Thrust Bearings, Transactions ASME, Journal of Lubrication Technology, Vol. 103, 1981, pp. 97-106. 11 S.M. Rohde and K.P. Oh, A Thermoelastohydrodynamic Analysis of a Finite Slider Bearing, Transactions ASME, Journal of Lubrication Technology, Vol. 97, 1975, pp. 450-460. 12 H.G. Elrod, A Cavitation Algorithm, Transactions ASME, Journal of Lubrication Technology, Vol. 103, 1981, pp. 350-354. 13 B. Jakobsson and L. Floberg, The Finite Journal Bearing Considering Vaporization, Chalmers Tekniska Hoegskolas Madlinar, Vol. 190, 1957, pp. 1-116. 14 K.O. Olsson, Cavitation in Dynamically Loaded Journal Bearings, Chalmers University of Technology, 1965, Goteborg. 15 D. Dowson, A.A.S. Miranda and C.M. Taylor, Implementation of an Algorithm Enabling the Determination of Film Rupture and Reformation Boundaries in a Liquid Film Bearing, Proc. 10th Leeds-Lyon Symp. on Numerical and Experimental Methods in Tribology, Sept. 1983, editors: D. Dowson, C.M. Taylor, M. Godet and D. Berthe, Butterworths, 1984, pp. 60-70. [...]... shown in Table 6.1 TABLE 6.1 Viscosities of various gases at 20°C and 0.1 [MPa] pressure Gas Viscosity [Pas] Hydrogen 8.80 × 10-6 Helium 1.96 × 10-5 Nitrogen 1 .76 × 10-5 Oxygen 2.03 × 10 Carbon Dioxide 1. 47 × 10-5 Air 1.82 × 10-5 -5 276 ENGINEERING TRIBOLOGY Aerostatic bearings are used in the precision machine tool industry, metrology, computer peripheral devices and in dental drills, where the air used... 4 J.P O'Donoghue and W.B Rowe, Hydrostatic Bearing Design, Tribology, Vol 2, 1969, pp 25 -71 280 ENGINEERING TRIBOLOGY 5 M.E Mohsin, The Use of Controlled Restrictors for Compensating Hydrostatic Bearings, Advances in Machine Tool Design and Research, Pergamon Press, Oxford, 1962 6 W.A Gross, Fluid Film Lubrication, John Wiley and Sons, 1980 7 J.N Shinkle and K.G Hornung, Frictional Characteristics... for a conical hydrostatic bearing is given by: Hf = 2 π3 ηn2 [ R04 (R4 − R04) + hcosφ hr ] (6. 17) 266 ENGINEERING TRIBOLOGY As discussed for flat circular pad bearings, the allowance for recess effects in practical applications can be made by treating the recess area as a bearing load area and equation (6. 17) becomes: Hf = 2 π3 ηn2 R4 hcosφ The procedure described above can be applied in a similar manner... friction torque for flat 278 ENGINEERING TRIBOLOGY circular pad bearings can be calculated from equation (6.11) provided that laminar flow prevails EXAMPLE Calculate the load capacity of a 0.1 [m] radius flat circular aerostatic bearing with radius ratio of 2, and recess pressure of 1 [MPa] From Figure 6.9, K* = 0.63 The load capacity is then: W = π × 106 × 0.12 × 1 × 0.63 = 19 79 2 [N] Power Loss The... 6 .7 Flat circular pad bearing with orifice controlled flow The flow rate through an orifice of diameter ‘d’ is: Q= ( ( 1/2 πd2 ps − pr Cd 2 2ρ where: Q is the flow rate through the orifice [m3/s]; d is the diameter of the orifice [m]; ps is the lubricant supply pressure [Pa]; pr is the bearing recess pressure [Pa]; ρ is the lubricant density [kg/m3]; Cd is the discharge coefficient  274 ENGINEERING TRIBOLOGY. .. Frictional Characteristics of Liquid Hydrostatic Journal Bearings, Transactions ASME, Journal of Basic Engineering, Vol 87, 1965, pp 163-169 8 H Opitz, Pressure Pad Bearings, Conf Lubrication and Wear, Fundamentals and Application to Design, London, 19 67, Proc Inst Mech Engrs., Vol 182, Pt 3A, 19 67- 1968, pp 100-115 9 R Bassani and B Piccigallo, Hydrostatic Lubrication: Theory and Practice, Elsevier,... of controlling lubricant flow The capillary diameter and length are selected so that the required bearing stiffness is obtained The theory of capillary controlled stiffness is outlined below  272 ENGINEERING TRIBOLOGY The flow rate through a straight capillary is given by the Hagen-Poiseuille equation This equation applies to the tubes with high ‘l/d’ ratios, i.e l/d > 100 and a Reynolds number less... total power required is the sum of friction power and the pumping power, i.e.: H t = Hf + Hp 268 ENGINEERING TRIBOLOGY 1.0 ;;;;; ;; ;;;;; ;;;;; ;;;;; ;;;;; ;;;;; ;;; ;;;;; ; ;;;;; ;;;;; ;;;;; ;;;;; ;;;;; 20 ;;;; ;;;; ; ;;;; ;;;; ;;;; ; ;;;; ;; ;;;; ;;;; ;;;; ; ;;;; ; ;;;; ;;;; ;;;; ;;;; 0.9 C 0.8 10 B A A 0 .7 5 H 0.6 B B 2 0.5 1 0.4 0 0.2 0.4 0.6 0.8 1.0 C/B FIGURE 6.5 Design coefficients for square pad... minimum power Other parameters such as viscosity and bearing clearance can also be optimized and the details are given in [2] For example, the lubricant viscosity can be optimized by calculating 270 ENGINEERING TRIBOLOGY power losses and load capacity for a range of viscosities, while maintaining all the other parameters at required design levels The optimum clearance is obtained when the power ratio... Piccigallo, Hydrostatic Lubrication: Theory and Practice, Elsevier, Amsterdam, 1992 10 J.K Scharrer and R.I Hibs Jr., Flow Coefficients for the Orifice of a Hydrostatic Bearing, Tribology Transactions, Vol 33, 1990, pp 543-550 7 7.1 E L A S T O H Y D R O D Y N A M I C L U B R I C A T I O N INTRODUCTION Elastohydrodynamic lubrication can be defined as a form of hydrodynamic lubrication where the elastic . on Numerical and Experimental Methods in Tribology, Sept. 1983, editors: D. Dowson, C.M. Taylor, M. Godet and D. Berthe, Butterworths, 1984, pp. 60 -70 . 256 ENGINEERING TRIBOLOGY HYDROSTATIC 6 LUBRICATION 6.1. of Thermohydrodynamic Lubrication, Tribology Transactions, Vol. 32, 1989, pp. 346-356. 9 C.M.Mc. Ettles, Transient Thermoelastic Effects in Fluid Film Bearings, Wear, Vol. 79 , 1982, pp. 53 -71 . 10 J.H. Vohr, Prediction. 103, 1981, pp. 97- 106. 11 S.M. Rohde and K.P. Oh, A Thermoelastohydrodynamic Analysis of a Finite Slider Bearing, Transactions ASME, Journal of Lubrication Technology, Vol. 97, 1 975 , pp. 450-460. 12