180 Chapter 6 disturbed from that position, the whirl orbit will gradually decay to the equilibrium point. The minimum condition is reached, for this case, at a speed of 5000rpm (Fig. 6.10). Any increase in the speed beyond this value would produce limit cycle orbits (Fig. 6.1 l), with increasing amplitudes. Finally, at a speed of 6200rpm, the orbit becomes large enough to consume all the bearing clearance, producing contact between the shaft and the sleeve (neglecting the effect of large amplitudes and near-wall operation on the dynamic characteristics of the film). Three different whirl conditions were found to occur, also, for the unbalanced rotor as illustrated in Figs 6.12-6.14 for an unbalance mr = 0.0001 lb-sec2 (0.0000455kg-sec2) and To = 0. At low speeds, the unbalance produces synchronous whirl with relatively high maximum eccen- tricity, as shown in Fig. 6.12. The maximum eccentricity of the orbit decreases with increasing speed until a minimum condition is reached at a speed of 4900 rpm corresponding to this unbalance. Higher rotor speeds beyond the minimum condition begin to produce nonsynchronous whirl with increasing maximum eccentricities (Fig. 6.13). Finally, at a speed of 6 100 rpm, the orbit continues to increase until contact with the sleeve occurs (Fig. 6.14). A summary plot for these different orbit conditions as affected by the magnitude of unbalance is given in Fig. 6.15a. Isoeccentricity ratio lines are plotted from the steady-state orbits to illustrate the effect of speed and unbalance on the type and magnitude of the rotor vibration. A similar plot is given in Fig. 6.15b for the peak eccentricity occurring during the rotor operation. These eccentricities generally occur during the transient phase before steady-state orbits are attained. Of particular interest is the minimum peak eccentricity locus shown in broken lines in Fig. 6.15a. Also of interest is the sleeve contact curve. Although this curve is obtained with simplifying assumptions, it serves to illustrate the expected trend for the upper speed limit of rotor operation. Both conditions impose a reduction on the corresponding speed as the magnitude of unbalance increases. It is also interesting to note that there appears to be practically a constant speed range of approximately 1200 rpm between the minimum peak eccentricity condition and the sleeve contact conditions. The following results illustrate the influence of some of the main para- meters on the rotor whilr. Figure 6.16 illustrates the effect of increasing the rotor weight on the whirl. The results show that increasing the rotor weight from 45.5kg to 142 kg increases the speed for the instability threshold. It also significantly reduces the whirl amplitude. Changes in the whirl conditions can be seen in Fig. 6.17, when the bearing clearance is changed from 0.0063in. to 0.01 in. (0.016 to Design of Fluid Film Bearings 181 0.03 , , I I -0.03 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 XI0 7 ,,,,,,,,,,,,*,,,,,,f 6- 5- E- 04- E: - z: u3- 5- f: : 2- 1- 0.56 0.57 0.58 0.59 0.60 Eccentricity Ratio, E Figure 6.12 (a) Synchronous whirl for balanced rotor at 1750 rpm. (b) Eccentricity-time plot for unbalanced rotor at 1750 rpm. 182 Chapter 6 -0.2 1 s 0 < 0.0 0.1 0.2 0.3 0.4 0.5 Ecoontrlcity Ratio, E Figure 6.13 (a) Nonsynchronous whirl for balanced rotor at 5500 rpm. (b) Eccentricity-time plot for unbalanced rotor at 5500 rpm. Design of Fluid Film Bearings 1.01 . . , . . . . , . *. .I. - - - 1 0.5 -1.0 -0.5 0.0 0.5 1 .o xlo 15 IR f '0 10 L a 5 183 0 0.0 0.2 0.4 0.6 0.8 1 .o Eccentricity Ratio, 6 Figure 6.14 (a) Whirl of unbalanced rotor at sleeve contact condition (6100 rpm). (b) Eccentricity-time plot for unbalanced rotor at 6100 rpm. 184 Chapter 6 .7 Y n n n n c) c) A A n n n n c) Y n " .8 Y Y Y 3,- .c n 1 I I I I I O.Oo0 0.001 0.m 0.m 0.004 0.005 0.006 0.007 WhS) kw l~l~l~l~l'l-l O.oo00 O.ooo5 0.0010 0.0015 O.Oa20 0.0025 O.Oo30 5ooo P Y 0 0 v) 8 (b) 0 O.Oo0 0.001 0.002 0.003 0.004 0.005 0.008 0.007 mr (Ibm-8q ~-I-I.I-I-I-I O.OMx1 O.oo(# 0.0010 0.0015 0.0020 0.0025 O.Oo30 kg-8* Figure 6.1 5 (b) Spectrum of transient peak eccentricity for unbalanced rotor. (a) Spectrum of steady-state peak eccentricity for unbalanced rotor. Design of Fluid Film Bearings 185 w 0.4 - w= 142 ko 0.2 - - W - 46.6 kg nu = 0.000227 kgd 0 SbadyS1.1.EfmntrMyfor8.kncrdRotor 0.0 1.1.1.1.1.1. 0 1000 2000 3000 4000 5000 6000 7000 Figure 6.16 (a) Effect of rotor weight on steady-state peak eccentricity for dif- ferent unbalanced magnitudes. (b) Effects of rotor weight on amplitude of whirl (mr = 0.000227 kg-sec2). 186 Chapter 6 1 .o 0.8 0.6 w 0.4 0.2 0.0 Speed (vm) Figure 6.1 7 (a) Effect of clearance on steady-state peak eccentricity for different unbalanced magnitudes. (b) Effect of clearance on amplitude of whirl (mv = 0.000227 kg-sec2). Design of Fluid Film Bearings 187 0.0254 cm). Figure 6.17a shows that increasing the clearance causes a reduc- tion in the instability threshold. Figure 6.17b on the other hand, shows little effect on the actual whirl orbit amplitude due to the clearance change with 0.000227 kg-sec2 unbalance. Two opposite effects of changing the average film temperature are shown in Fig. 6.18. In the first example, with W = lOOlb (45.5 kg), C = 0.0063 in. (0.0 16 cm), and mr = 0.005 lb-sec2 (0.000227 kg-sec2), increasing the average film temperature from 373°C to 94°C resulted in a considerable reduction in the threshold speed, as well as an increase in the whirl amplitude (Fig. 6.18a). On the other hand, the second example, W = 1000 lbf = 455 kg and C = 0.016 cm, shows that considerable reduc- tions in the whirl amplitude resulted from the same increase in the average film temperature (Fig. 6.18b). The case of a rigid rotor on an isoviscous film considered in this illus- tration provided a relatively simple model to approximately investigate the effect of rotor unbalance and film properties on the rotor whirl. The developed response spectrum shown in Fig. 6.15a gives a complete view of the nature of the rotor whirl as affected by the speed and the unbalance magnitude. Of particular interest is the existence of a rotational speed for any particular unbalance where the peak eccentricity is minimal. Nonsynchronous whirl, with increasing amplitudes and eventual instability or rotor sleeve contact, occurs as the speed is increased beyond that condi- tion. It should be noted here that results associated with large whirl ampli- tudes and those near bearing walls represent qualitative trends rather than accurate evaluation of the whirl in view of the assumptions made. Investigation of the influence of system parameters on whirl for the considered cases showed, as expected, that improved rotor performance can be attained by increasing the load and reducing the clearance. Increasing the average film temperature showed that an increase or a reduc- tion in the whirl amplitude may occur depending on the particular system parameters. Although a relatively simple model is used in this study, the technique can be readily adapted to the analysis of more complex rotor systems and film properties. 6.2 DESIGN SYSTEMS 6.2.1 This is an illustration of graph-aided design for journal bearings. The graphs are constructed in such a manner as to enable the designers to Procedure Based on Design Graphs 188 1 .o 0.8 0.6 w 0.4 0.2 0.0 Chapter 6 Figure 6.18 (b) W = 455 kg. Effect of average film temperature on rotor whirl: (a) W = 45.5 kg; Design of Fluid Film Bearings 189 select the bearing parameters, which meet their objective with a minimum of calculations. In constructing the graphs, the main parameters influencing the bearing behavior were divided into two groups. The first deals with the bearing geometry (L, D, R, C), load W, and speed N. The second deals with the oil, and its temperature-viscosity char- acteristics. Because many types of oil can be used in the same bearing, the basic approach in the design graphs given here is to construct separate graphs for the different bearings and oils. The bearing graphs represent a plot of temperature rise, Af, versus average viscosity for a bearing with a known characteristic number, K = (R/C)2N, length-to-diameter ratio, t/D, and average pressure P. They are constructed by assuming the average viscosity, calculating the Sommerfeld number and the corresponding A T. Such plots are based on the numerical results of Raymondi and Boyd [24] and are shown in Figs 6.19-6.22 for average pressure values of 100, 9E-6 8E-6 7E-6 6E-6 5E-6 n 4E-6 f a 3E-6 3. 2E-6 1 E-6 P=lOOpsi ___ Ud I 1.0 Ud 0.5 Ud = 0.25 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 AT (OF) Figure 6.19 Bearing chart for P = 1oOpsi. [...]... (6 .15 ) for the given weighting factor, k The process is repeated for different selections of the clearance (0.003 in and 0.0 12 in are tried in this example) The results are listed in Table 6 .1 and plotted in Fig 6.27 Table 6 .1 C 0.003 0.006 0. 012 Numerical Results for Bearing Design CL (reyn) C 1. 22 x 10 -6 1. 53 x 10 -6 1. 61 x 10 -6 0.0454 0. 014 2 0.00375 U Q At (“F) (in. 3/sec) k = 2 38 15 8 2. 81 5. 81 12.00... are based on the curve fitting of these figures illustrative Example The use of the bearing design graphs is illustrated by the following example It is assumed that a shaft 2 in in diameter, carrying a radial load of 2000 lb 19 4 Chapter 6 9x1o8 8x1O8 7x1O8 6x10" 5x10" n * 4x10" B 3x106 2x1o8 1OS 0 20 40 60 80 10 0 12 0 14 0 16 0 18 0 200 220 240 260 280 A (OF) T Figure 6.26 Bearing design chart: application...P = SO0 p 1 Udml.O - ud 10 .5 - - - Ud m 0.2s 9E-6 8E-6 7E-6 6E-6 5E-6 a 3E-6 2E-6 1E-6 0 20 4 0 60 80 10 0 12 0 14 0 16 0 18 0 200 220 240 260 280 18 0 200 220 240 260 280 AT (OF) Figure 6.20 Bearing chart for P = 500 psi 9E-6 8E-6 7E-6 6E-6 5E-6 2E-6 1E-6 0 20 40 60 80 10 0 12 0 14 0 16 0 AT (OF) Figure 6- 21 Bearing chart for P = 1OOOpsi  19 1 Design of Fluid Film Bearings 9E-6 7E-6 6E-6 5E-6 8E-6... viscosity and average pressure That is: K =S(F) 9E-6 8E-6 SAE 10 7E-6 - t, = 40 17 0 O F 6E-6 5E-6 4E-6 n r) 5 3E-6 e Y =L 2E-6 1E-6 0 20 40 60 80 10 0 12 0 14 0 AT Figure 6.23 16 0 18 0 200 220 240 260 280 18 0 200 220 240 260 280 (OF) SAE 10 oil chart 9E-6 8E-6 7E-6 6E-6 5E-6 4E-6 n (D t 3E-6 U 3 2E-6 1E-6 0 20 40 60 80 10 0 12 0 14 0 AT Figure 6.24 SAE 20 oil chart 16 0 (OF) I93 Design of Fluid Film Bearings 9E-6... a feasible point in the design domain is located, the gradient search is initiated according to: 19 9 Design of Fluid Film Bearings I + 1 Inputs W, R,N C P , U d (within side consUaints) I StaltGradiiSearch I violated 1 Wmin Feasible Region satisfedory Point Locatrn (a) Figure 6.28 (a) Search method flow diagram 1 (b) Search method flow diagram 2 200 Chapter 6 P,+I = Pn - where n,, nc, and nL = scale... describing the bearing performance are Design o Fluid Film Bearings f 19 7 generally based on Reynolds’ equation and are, in most cases, numerical solutions of the equation with certain assumptions and approximations In this section, the curve-fitted numerical solutions, given in Sections 6 .1. 3 and 6 .1. 4, are utilized in a design system that rationally selects the significant parameters of a bearing to... determined from: n, = =I nc7= = order of 10 3 nL = = order of 10 s l :1 1 41 The control of the step size is exercised by including a provision for changing A,, in the computational logic One way to accomplish this is to require a specified percentage change in one of the parameters and to set upper limits on the incremental changes in the other parameters, thus offering a safeguard against the overshooting... Examples Two common bearing applications are considered to illustrate the design procedure Designof Bearingsfor Constant Load and Speed Condition The inputs are taken as: W = 2000, 10 00,500,2501b, respectively N = 16 .66,33.33,83.33, 16 666,250,333.33 rps, respectively D = 2in The constraints are: hmin= 5 x 10 - in , = 300°F = maximum allowable temperature , tmin Pm,, = 30,000 lb /in? = maximum allowable... bearing application The design criterion can therefore be formulated as: Find C , which minimizes U = At + k Q (6 .15 ) where At = temperature rise (OF) Q = oil flow (in. 3/sec) Values of k = 2, 5, and 7 are considered to illustrate the influence of the weighting factor on the final design The average pressure and the length-to-diameter ratio are first calculated as: 19 5 Design of Fluid Film Bearings... 1x2 L - 5oOpsi and - = 0.5 D Arbitrary values for the design parameter, C are assumed and the corresponding bearing parameter, k , is calculated in each case For example, if C is selected equal to 0.00 6in. , the corresponding parameter is = (+= (&) * (=4.63 ) 10 7000 7 10 6 x The bearing performance curve, corresponding to this value of k for P = 5OOpsi and LID = 0.5, can be interpolated from Fig 6.21 . radial load of 2000 lb 19 4 Chapter 6 9x1 o8 8x1 O8 7x1 O8 6x1 0" 5x1 0" 4x10" 3x106 n * B 2x1 o8 1 OS 0 20 40 60 80 10 0 12 0 14 0 16 0 18 0 200 220 240 260. 57.67 0.006 1. 53 x 10 -6 0. 014 2 15 5. 81 26.6 44 55.6 0. 012 1. 61 x 10 -6 0.00375 8 12 .00 32 68 92 19 6 Chapter 6 O.OO0 0.002 0.004 0.006 0.008 0. 010 0. 012 0. 014 Clearance,. 5E-6 2E-6 1 E-6 0 20 40 60 80 10 0 12 0 14 0 16 0 18 0 200 220 240 260 280 AT (OF) Figure 6- 21 Bearing chart for P = 1OOOpsi. Design of Fluid Film Bearings 19 1 9E-6 8E-6