Engineering Tribology Episode 2 Part 2 pdf

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) TEAM LRN 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. TEAM LRN 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 TEAM LRN 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. TEAM LRN 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. TEAM LRN 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. TEAM LRN 256 ENGINEERING TRIBOLOGY TEAM LRN HYDROSTATIC 6 LUBRICATION 6.1 INTRODUCTION In hydrostatic lubrication the bearing surfaces are fully separated by a lubricating film of liquid or gas forced between the surfaces by an external pressure. The pressure is generated by an external pump instead of by viscous drag as is the case with hydrodynamic lubrication. As long as a continuous supply of pressurized lubricant is maintained, a complete film is present even at zero sliding speed. Hydrostatic films usually have a considerable thickness reaching 100 [µm] and therefore prevent contact between the asperities of even the roughest surfaces. This ensures a complete absence of sticking friction. Furthermore, the friction generated by viscous shear of the lubricant decreases to zero at zero sliding speed. Hydrostatic bearings can support very large masses and allow them to be moved from their stationary positions with the use of minimal force. These extraordinary features of zero static friction and high load capacity were applied, for example, in the 5.08 [m] diameter Mount Palomar telescope and in many radar installations. With other types of bearing, starting friction is inevitable and can cause distortion and damage to large structures. This problem is critical to the design of large telescopes which rely on extreme accuracy of telescope positioning. Hydrostatic bearings have a wide range of characteristics and need to be carefully controlled for optimum effect. The following questions summarize the potential problems that an engineer or tribologist might confront. If it is possible to generate films similar to hydrodynamic films, how can these films be controlled and produced when needed? What are the practical applications of this type of lubrication? What are the critical design parameters of hydrostatic bearings? What is the bearing stiffness and how can it be controlled? The engineer or tribologist should know how to find the answer to all these questions. According to available records, the first hydrostatic bearing was invented in 1851 by Girard [1,2] who employed a bearing fed by high pressure water for a system of railway propulsion. Since then there have been a number of patents and publications dealing with different design aspects and incorporating various features. Some of these designs introduced genuine improvements but the majority merely introduced complexity rather than simplicity and are destined to be forgotten. As well as the true hydrostatic bearing, hybrid bearings have also been developed. These are hydrodynamic bearings assisted by an externally pressurized lubricant supply. TEAM LRN 258 ENGINEERING TRIBOLOGY In this chapter, the mechanism of film generation in hydrostatic bearings together with methods of calculating basic bearing operational and design parameters are discussed. Commonly used methods of controlling the bearing stiffness are also outlined. 6.2 HYDROSTATIC BEARING ANALYSIS The analysis of hydrostatic bearings is much simpler than the analysis of hydrodynamic bearings. It is greatly simplified by the condition that the surfaces of these bearings are parallel. Flat Circular Hydrostatic Pad Bearings Consider, as an example, a flat circular hydrostatic pad bearing with a central recess as shown in Figure 6.1 [2]. Pressure profile p y h R R 0 p = p r p = p r p = 0 r dr Recess ω h r FIGURE 6.1 Flat circular pad bearing with a central recess. · Pressure distribution The pressure distribution can be calculated by considering the lubricant flow in a bearing. For a bearing supplied with lubricant under pressure, the flow rate given by equation (4.18) becomes: q x = − h 3 12η ∂p ∂x Since the bearing is circular, the flow through the elemental ring at radius ‘r’ is: Q = − h 3 12η ∂p ∂r 2πr (6.1) rearranging and integrating yields (surfaces are parallel, i.e. h ≠ f(r)): p = − 6ηQ πh 3 dr r ⌠ ⌡ = − 6ηQ πh 3 lnr + C (6.2) TEAM LRN HYDROSTATIC LUBRICATION 259 Boundary conditions from Figure 6.1 are: p = 0 at r = R Substituting into equation (6.2), yields the constant ‘C’: C = 6ηQ πh 3 lnR (6.3) Hence the pressure distribution for this type of bearing in terms of lubricant flow, bearing geometry and lubricant viscosity is given by: p = 6ηQ πh 3 ln ( ( R r (6.4) · Lubricant Flow By rearranging equation (6.4), the lubricant flow, i.e. the minimum amount of lubricant required from the pump to maintain film thickness ‘h’ in a bearing, is obtained: Q = πh 3 p 6η 1 ln(R/r) Since at r = R 0 , p = p r then: Q = πh 3 p r 6η 1 ln(R/R 0 ) (6.5) where: p r is the recess pressure [Pa]; h is the lubricant film thickness [m]; η is the lubricant dynamic viscosity [Pas]; R is the outer radius of the bearing [m]; R 0 is the radius of the recess [m]; Q is the lubricant flow [m 3 /s]. It can be seen that by merely substituting for flow (eq. 6.5), the pressure distribution (eq. 6.4) is expressed only in terms of the recess pressure and bearing geometry, i.e.: p = p r ln(R/R 0 ) ln(R/r) (6.6) · Load Capacity The total load supported by the bearing can be obtained by integrating the pressure distribution over the specific bearing area: TEAM LRN [...]... bearing (6.19): ( ( p h3 πd2 ps − pr 1 /2 Cd = r B 2 2ρ η Rearranging: pr = ( ( 1 /2 πd2 η ps − pr Cd 3 2h B 2 Substituting into the expression for load (6.18) gives: W= ( ( 1 /2 Aπd2 ηA ps − pr Cd 2 2h3 B The bearing stiffness with an orifice is found by differentiating with respect to film thickness ‘h’: λ=− ( ( 1 /2 dW 3Aπd2 ηA ps − pr = Cd 2h4 B dh 2 Hence: λ= 3W h (6 .26 ) which is the same expression... gives: T =⌠ 2 R0 2 R ⌠ η U r2 dθ dr +⌠ ⌠ η U r2 dθdr ⌡ ⌡ hr ⌡ ⌡ h 0 R 0 0 0 Expressing ‘U’ in terms of revolutions per second: U = 2 rn where: n is the speed of the bearing [rev/s] TEAM LRN 26 2 ENGINEERING TRIBOLOGY and substituting gives: T =⌠ 2 2 R0 R ⌠ η 2 π n r3 dθ dr +⌠ ⌠ η 2 πn r3 dθdr ⌡ ⌡ hr ⌡ ⌡ h 0 R 0 0 0 Assuming constant viscosity and velocity and integrating yields: T= 2 η n 4 2 η n 4... to: p pr R W = pr πR 02 + 2 ⌠ prdr R ⌡ dr 0 dθ Substituting for pressure (eq 6.6): R y dx = dr dy = rdθ r x (( 1 ⌠ rln R dr W = pr πR 02 + 2 pr r ln(R/R0) ⌡ 0 R R R0 (6.7) Integrating by parts and substituting yields: (( r2 1 r2 R ln W = pr π R0 + 2 πpr + 4 r ln(R/ R0) 2 2 R R0 After simplifying, the expression for the total load that the bearing can support is: W= [ pr π R2 − R 02 2 ln(R/ R0) [ (6.8)... Ht = p = W W W (6 .20 ) Substituting for load (6.18) and flow (6.19), the ratio of pumping power to load expressed in terms of non-dimensional coefficients becomes: Hp = W pr ( ( pr h3 B W η Wh3 B = 2 4 π R η A2 (π R2 pr A )2 Thus: Hp Wh3 = 2 4 H W πRη (6 .21 ) The values of the ‘H’ coefficient for flat circular and square pad bearings can be found from Figures 6.4 and 6.5 respectively [2] · High Speed Recessed... 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 TEAM LRN 27 4 ENGINEERING TRIBOLOGY Equating flow through the orifice to the total lubricant flow through the bearing (6.19): ( ( p h3 πd2... LRN HYDROSTATIC LUBRICATION 26 7 1.0 20 0.9 10 0.8 5 0.7 H 0.6 B B 2 A 0.5 1 ;;;;; ;;;;; ;;;;; ;;;;; ;;;;; ;;;;; ;;; ;;;;; ;;;;; ;; ;;;;; ;;;;; ;;;;; ;;;;; 0.4 ;;;; ;;;; ; ;;;; ; ;;;; ;; ;;;; ;;;; ;;;; ;;;; ;;;; ;;;; ;; ;;;; ;;;; ;;;; ;;;; 0.5 0.3 R0 0 .2 0.1 0 .2 R 1 A 1.5 0.1 2 3 5 10 ∞ 0 R/R0 FIGURE 6.4 Design coefficients for flat circular pad bearings [2] W = Apr A = B2 pr A Q= pr h3 B η where: A... times a non-dimensional scale factor W = Apr A = π R2 pr A Q= (6.18) pr h3 B η (6.19) where: A is the total pad area [m2]; A and B are non-dimensional load and flow coefficients defined as: [ A= R2 − R 02 1 2 2 R ln(R/ R0) B= [ π 1 6 ln(R/ R0) Non-dimensional coefficients A and B are usually plotted against bearing dimensions as shown in Figure 6.4 [2] The bearing analysis and design is now greatly simplified... surfaces for a conical hydrostatic bearing is given by: Hf = 2 π3 η n2 [ R04 (R4 − R04) + h cos φ hr ] (6.17) TEAM LRN 26 6 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 η n2 R4 h cos φ The procedure described above can be applied... i.e.: H t = Hf + Hp TEAM LRN 26 8 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 bearings; C is the width of the recess [2] Pumping power ‘Hp’ is... ln(R/ r) ln(R/ R0) TEAM LRN HYDROSTATIC LUBRICATION 26 5 · Load Capacity The total load capacity that the bearing can support is found by integration of pressure over the bearing area: W =⌠ 2 2 R0 R ⌠ pr r dθ dr + ⌠ ⌠ prdθ dr cos φ ⌡ ⌡ cosφ ⌡⌡ R 0 0 0 0 Substituting for ‘p’ and integrating gives the bearing load capacity ‘W’: W= [ pr π R2 − R 02 2 cos φ ln(R/ R0) [ (6.15) · Friction Torque The friction . = ⌠ ⌡ 0 2 ⌠ ⌡ 0 R 0 η r 2 dθdr + U h r ⌠ ⌡ 0 2 ⌠ ⌡ R 0 η r 2 dθdr U h R Expressing ‘U’ in terms of revolutions per second: U = 2 rn where: n is the speed of the bearing [rev/s]. TEAM LRN 26 2 ENGINEERING. 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. TEAM LRN 25 4 ENGINEERING TRIBOLOGY 1 3 4 0. integrals: W = ⌠ ⌡ 0 2 ⌠ ⌡ 0 R 0 p r rdθdr + ⌠ ⌡ 0 2 ⌠ ⌡ R 0 R prdθdr W = p r πR 0 2 + 2 ⌠ ⌡ R 0 prdr R p dr dθ R y R 0 r x dx = dr dy = rdθ p r W = p r πR 0 2 + 2 p r ⌠ ⌡ R 0 rln R Substituting