Friction and Lubrication in Mechanical Design Episode 1 Part 9 ppt

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 b

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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

Figure 6.12 (a) Synchronous whirl for balanced rotor at 1750 rpm (b)

Eccentricity-time plot for unbalanced rotor at 1750 rpm

Design of Fluid Film Bearings

(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

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

Design of Fluid Film Bearings 189

select the bearing parameters, which meet their objective with a minimum

The bearing graphs represent a plot of temperature rise, Af, versus

average viscosity for a bearing with a known characteristic number,

K = ( R / C ) 2 N , 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

[ 2 4 ] and are shown in Figs 6.19-6.22 for average pressure values of 100,

Design of Fluid Film Bearings 191

500, 1000, and 2000 psi, respectively The length-to-diameter ratios LID

considered are 0.25, 0.50, and 1.0

The graphs for the lubricants represent the change of average viscosity

with temperature rise for any particular initial temperature Figures 6.23- 6.25 represent such plots for SAE 10, 20, and 30 oils, respectively These graphs give a convenient means of analysis, as well as the design of bearings,

as explained in the following section

Analysis Procedure

For a bearing with a given geometry, load, and speed, a characteristic

number, K = ( R / C ) 2 N , can be readily calculated As can be seen from

Eq (6.8), this number represents the Sommerfeld number for a particular

value of viscosity and average pressure That is:

K = S ( F )

Design of Fluid Film Bearings I93

Figure 6.25 SAE 30 oil chart

The bearing graph (which represents the relationship between viscosity, p, versus temperature rise, At, for the particular value of K, LID, and P), can be readily plotted on a transparent sheet by interpolation from Figs 6.9-6.12 Given the type of oil and its inlet temperature, the oil graph (which represents average viscosity versus temperature rise for the oil), is also plotted on the same sheet from Figs 6.23-6.25 Intersection of the two curves as can be seen

in the example illustrated in Fig 6.26, gives the temperature rise in the bear- ing and the corresponding average viscosity, p The Sommerfeld number for the bearing is then calculated from S = K p / P

Consequently, all the behavioral characteristics of the bearing can be read from Figs 6.3-6.8 or calculated from the given bearing performance equations, which 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

Figure 6.26 Bearing design chart: application for clearance selection

at 10,000rpm is symmetrically supported by two bearings, each of length

1 .O in The lubricating oil is SAE No 10 with an inlet temprature of 150°F The objective is to select a value for the radial clearance, C, which minimizes both the oil flow and temperature rise Because these are conflicting objec- tives, a weighting factor, k, has to be specified to describe their relative importance for a particular bearing application The design criterion can therefore be formulated as:

Find C , which minimizes U = At + kQ (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 calcu- lated as:

Design of Fluid Film Bearings 195

L

LD 1 x 2 - 5oOpsi and - D = 0.5

p=-= w 2000/2

Arbitrary values for the design parameter, C are assumed and the corre-

sponding bearing parameter, k, is calculated in each case For example, if C

is selected equal to 0.006in., the corresponding parameter is

= (+= (&) * (107000 =4.63 7 )x 106

The bearing performance curve, corresponding to this value of k for

P = 5OOpsi and LID = 0.5, can be interpolated from Fig 6.21 as plotted

in Fig 6.26 The oil characteristic curve for SAE 10 for the 150°F inlet temperature is also traced from Fig 6.23 as shown in Fig 6.2 The inter- section of the two curves yields the following values for the temperature rise and average viscosity:

The merit value is calculated from Eq (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 Numerical Results for Bearing Design

U

Q

C CL (reyn) C At (“F) (in.3/sec) k = 2 k = 5 k = 7 0.003 1.22 x 10-6 0.0454 38 2.81 43.62 52.05 57.67 0.006 1.53 x 10-6 0.0142 15 5.81 26.6 44 55.6 0.012 1.61 x 10-6 0.00375 8 12.00 32 68 92

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O.OO0 0.002 0.004 0.006 0.008 0.010 0.012 0.014

Clearance, C

Figure 6.27 Selection of optimum clearance for the difference objectives

The optimum clearances can be deduced from the figure for the different values of the weighting factor k as:

k = 2: C* = 0.005in

k = 5: C* = 0.006in

k = 7: C* = 0.007in

6.2.2 Automated Design System

This section presents an automated system for the selection of the main design parameters to optimize the performance of the hydrodynamic bear- ing In spite of the wealth of literature on the analysis of these bearings, the selection of design parameters in the past has relied heavily on empirical guides Empiricism was necessary because of complexity of the interaction between the different parameters which govern the behavior of such bear- ings The analytical relationships describing the bearing performance are

Design of Fluid Film Bearings 197

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 optimally satisfy the designer’s objec-

tive within the constraints imposed on the design A full journal bearing to

operating at a constant speed and supporting a known constant load is considered The procedure is extended to cover the selection of an optimum bearing for applications where the load and speed may vary from time to time within given bounds

System Parameters

The main independent parameters for the problem under consideration are

(D, L, C), p, and ( W , N ) These parameters, as grouped, describe the bear-

ing geometry, oil characteristics, and load specifications, respectively In formulating the problem, it will be assumed that D, N , W are given inputs

for the bearing design The design parameters are therefore LID, C , p The constraints on the design are:

The first five of these inequality constraints represent the limit on the oil film thickness, temperature rise, maximum allowable pressure, minimum oil visc- osity, and bearing length These limits are dictated by the quality of machin- ing, the characteristics of the material-lubricant pair, and the available space The sixth constraint describes a condition for bearing stability, as described in Fig 6.8

The Governing Equations

The equations governing the behavior of the bearing in this study are devel- oped by curve fitting from Raimondi and Boyd’s numerical solution to Reynolds’ equation, Eq (6.1) These equations, which are given in Section 6.1.3, allow the calculation of the temperature rise, minimum oil film thick- ness, maximum oil film pressure, oil flow, frictional loss, and so forth The

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curve-fitted equations for the stability analysis by Lund and Saibel, given in Section 6.1.4, provide a simplified mathematical relationship for the onset of instability constraint

Design Criterion

The selection of an optimum solution requires the development of a design criterion, which accurately describes the designer’s objective The topogra- phy of this criterion and its interaction with the boundaries of the design domain (constraint surfaces), have a significant effect on the efficiency and success of the search In the problem of bearing design, many decision criterion can be envisioned Some of these are: minimizing the maximum temperature rise of the bearing, minimizing the quantity of oil flow required for adequate lubrication, minimizing the frictional loss, and so forth The objective may also be composed of a multitude of the previously mentioned factors, and weighing their relative importance requires skilled judgement by the individual designer

Search Method

In formulating the problem for automated design, the following factors are considered in developing a search strategy: (I) the nature of the objective function, (2) the design domain and behavior of the constraints, (3) the

sensitivity of the objective function to the individual changes in the decision parameters, and (4) the inclusion of a preset criterion for search effectiveness and convergence

A block diagram describing the search is shown in Fig (6.28) Arbitrary values of the design parameters within their given constraints are the entry point to the system These values need not satisfy the functional constraints The first phase of the search deals with guidance of the entry point into the feasible region In this phase, incremental viscosity changes, of the order of

10%, and clearance changes, on the order of 0.001 in per inch radius,

proved to be adequate When the stability constraint is violated, a feasible point may be located by dropping the length-to-diameter ratio to its lower limit, and simultaneously halving the viscosity and the clearance To avoid looping in this phase, a counter can be set to limit the number of iterations

If a feasible point can not be successfully located, the designer can readjust the entry point according to the experience gained from the performed computations When a feasible point in the design domain is located, the gradient search is initiated according to: