, .4999 4‘.) -V “>4 1g “'1 t . . u_ . ‘ .315 . . 4‘ _I.I‘ 4 .‘r A 3c»! \ 9 up ABSTRACT A DESIGN OPTIMIZATION TECHNIQUE APPLIED To A SQUBEZE-FILM, GAS, JOURNAL BEARING By Carl Leander Strodtman Two different though related aspects of optimization of a com- pressible-fluid, squeeze-film journal bearing are treated. First, it is shown that the minimum clearance in the journal bearing can be maximized by the proper choice of the nominal clearance, the length- diameter ratio, and the excursion non-uniformity factor, for the case of fixed load force and volume. Second, it is shown that an optimum design can be selected by means of a merit function developed from the designer's value-judgement of the desired performance and cost. It is also shown that the merit function, multiplied by some suitable weighting function, can be used to select a maximax design. Although the merit function and the weighted merit function are ap- plied to the squeeze—film journal bearing, it is believed that they constitute a design procedure of much greater generality. A method of characterizing non-uniform driver excursion by means of its root-mean-square amplitude and a shape factor is developed. Carl Leander Strodtman An augmented, small-parameter solution of the squeeze-film partial differential equation including terms to the third power of the radial displacement is given. The results are compared to an alternating-direction, implicit, numerical method previously used. A study of sensitivity to parameter changes is presented by means of a response surface defined by the second derivatives, near the Optimum. Parameters not optimized are treated by employing sensitivity coefficients based on the first derivatives. Sensitivity is studied for both the clearance optimization and the merit optim- ization problems. A DESIGN OPTIMIZATION TECHNIQUE APPLIED TO A SQUEEZE-FILM, GAS, JOURNAL BEARING By Carl Leander Strodtman A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mechanical Engineering 1968 ACKNOWLEDGEMENTS The author wishes to thank his advisor, Dr. James V. Beck of the Michigan State University Mechanical Engineering Department, for his guidance and many helpful suggestions during the preparation of this thesis. In addition, the author expresses thanks to the Instrument Division of Lear Siegler, Incorporated, for their financial support, for their sponsorship of the work done on this thesis, and for the use of their facilities, in particular, the digital computer. Sincere thanks are given to Miss Anna D'Angelo and Miss Loretta Durkin for the typing of the manuscript and its numerous revisions. Finally, the author wishes to express his appreciation to his wife, Marion, and to his children for their interest, encouragement, and forebearance throughout this graduate program. ii TABLE OF CONTENTS Page LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . v LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . vi LIST OF APPENDICES . . . . . . . . . . . . . . . . . . . . . . viii LIST OF SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . ix CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . 1 1.1 Gas Bearings . ; . . . . . . . . . . . . . . . 1 1.2 Literature Review . . . . . . . . . . . . . . . 4 1.3 Design Optimization and Problem Statement . . . 6 CHAPTER 2 GOVERNING EQUATIONS . . . . . . . . . . . . . . . . 10 2.1 Gas Film Equation . . . . . . . . . . . . . . . 10 2.2 Load Support Capability . . . . . . . . . . . . 17 2.3 Non-Uniform Driver Excursion . . . . . . . . . 19 CHAPTER 3 OPTIMIZATION OF CLEARANCE . . . . . . . . . . . . . 24 3.1 Clearance Equations . . . . . . . . . . . . . . 24 3.2 Optimization of Small Parameter Equations . . . 28 3.3 Optimization of Clearance Using The Augmented, Small-Parameter Equation . . . . . . . . . . . 38 3.4 Sensitivity to Parameter Changes . . . . . . . 45 iii TABLE OF CONTENTS (Continued) Page CHAPTER 4 FIGURE-OF-MERIT . . . . . . . . . . . . . . . . . . 60 4.1 Merit Functions . . . . . . . . . . . . . . . 60 4.2 Application to the Squeeze-Film Journal Bearing . . . . . . . . . . . . . . . . . . . 74 4.3 Optimization of the Merit Function . . . . . . 82 4.4 Sensitivity to Parameter Changes . . . . . . . 85 CHAPTER 5 WEIGHTED FIGURE-OF-MERIT . . . . . . . . . . . . . 94 5.1 Weighted Merit Function . . . . . . . . . . . 94 5.2 Application to Squeeze-Film Bearing . . . . . 96 5.3 Stability Considerations . . . . . . . . . . . 104 CONCLUSIONS AND RECOMMENDATIONS . . . . . . . . . . . . . . . 108 LIST OF REFERENCES . . . . . . . . . . . . . . . . . . . . . 111 iv LIST OF TABLES Table Page 4.1 Functional Relationship Between the Merit Factors and the Independent Variables . . . . . . . . . . . . 81 4.2 Optimizing Parameters for the Merit Function. . . . . 85 4.3 Sensitivity Coefficients of the Merit Function (Weight Exponents Equal to Unity) . . . . . . . . . . . . . . 89 4.4 Effect of Weight Exponents on the Optimizing Parameters . . . . . . . . . . . . . . . . . . . 93 5.1 Comparison Between Design Optimized at Maximum Load (d= 1.0) and Design Optimized at the Weighted Optimum (d = 0.45) . . . . . . . . . . . . . . . . . . . . 102 8.1 Comparison of the Effect of the Number of Nodes on Load Support Calculations (€1r= 0.048, 62 = -0.8, = -1) . . .. . . . . . . . . . . . . . . . . . 127 LIST OF FIGURES Figure 2.1 Configuration of the squeeze-film journal bearing . 2.2 Transducer amplitudes for various ranges of the non-uniformity factor . 3.1 Ratio of maximum amplitude to rms amplitude as a function of the non-uniformity factor . 3.2 Clearance and radial displacement as functions of nominal clearance (small parameter equation) 3.3 Clearance as function of length-diameter ratio (small parameter equation) . . . . . . . . 3.4 Contours of constant clearance (small parameter equation) . . . . . . . . . . . .-. . . 3.5 Clearance and radial displacement as functions of nominal clearance (augmented, small-parameter equation) . . . . . . . . . . . . 3.6 Contours of constant clearance (augmented, small-parameter equation) . . . . . . . 3.7 First derivatives of clearance near the optimum (small parameter equation) . . . . . . . . 3.8 Second derivatives of clearance near the optimum (small parameter equation) 3.9 First derivatives of clearance near the optimum (augmented, small-parameter equation) 3.10 Second derivatives of clearance near the optimum (augmented, small-parameter equation) vi Page 14 21 29 32 33 35 43 44 48 50 52 S3 Figure 3. 11 LIST OF FIGURES (Continued) Optimum nominal clearance and optimum clearance as functions of load (augmented, small-parameter equation) . . . . . . . . . Effect of weighting exponent on representative functions 0 O I O O O O C O O C O 0 C O O O O Merit factors for the squeeze—film journal bearing Contours of equal merit for the merit function (augmented, small-parameter equation) First derivatives of the merit function near the optimum (augmented, small—parameter equation) . . Optimizing parameters as functions of load Performance of designs optimized at lesser loads when loaded at maximum load . Trajectory of optimums on contour maps. Merit values as function of load for optimum designs . Weighted merit as function of design Stability map adapted from reference [17] . . Load support as a function of mesh spacing in the axial direction Comparison between finite difference solution and augmented, small-parameter equation . vii Page 58 73 76 84 86 97 98 99 101 103 105 126 130 Appendix LIST OF APPENDICES Augmented, Small-Parameter Solution of the Squeeze-Film Equation . A.l Series Expansion . A.2 Solution for To and T1 A.3 Solution for T2 A.4 Solution for T3 Comparison of the Augmented, Small-Parameter Solution to the Numerical Solution viii Page 114 114 117 118 122 124 LIST OF SYMBOLS excursion amplitude at end of driver (microinches) excursion shape factor (A = b/a) excursion change at center of driver (microinches) minimum clearance (microinches) minimum clearance normalized by dhlr fraction of maximum load at which design is optimized f1(Z) evaluated at the boundaries shape function of the excursion function defined on page 27 variables in differential equation for T2 film thickness (microinch) nominal film thickness (microinches) variable in differential equation for T1 film thickness normalized by hO film thickness normalized by dhlr time average film thickness variables in differential equation for T3 a constant, K = (2n)1/3pa bearing length (inches) mesh spacing in Z direction merit function ix merit factor weighted merit function merit value of design optimized at a load equal to d, when loaded with s mesh spacing in a direction pressure (1bf per square inch) ambient pressure (1bf per square inch) pressure normalized by pa matrix of second derivatives radius of the bearing (inches) fraction of maximum load at which design is evaluated sensitivity coefficient (see page 46) time normalized by w‘ film characteristic, T = (PH)2 nEh-component of T in expansion (Appendix A) film characteristic, T' = (PH')2 weight exponent average total bearing load (lbf) average bearing load per unit length (1bf per inch) dimensionless average bearing load per unit length W 1 ZpaR generalized designation of a variable vector of AXi/Xi values longitudinal coordinate (inches) half length of the bearing (inches) 6h1(z) 6h 1m Ohlr half length of the driver (inches) longitudinal coordinate normalized by R length-diameter ratio of bearing (ZL = 3%) length-diameter ratio of driver “A A2 shape function, a = 1 + ;—-+ E—- dimensionless "mass” parameter (see page 106) amplitude of driver motion (microinches) maximum drive amplitude (microinches) root-mean-square drive amplitude (microinches) radial displacement (microinches) a reference drive amplitude normalized by h0 root-mean-square drive amplitude normalized by ho root-mean-square drive amplitude normalized by Ghlr eccentricity (radial displacement normalized by ho) eccentricity (radial displacement normalized by Ohlr) ratio of weight exponents (see page 64) angular coordinate gas viscosity (1bf - sec. per square inch) gas density (1bm per cubic inch) squeeze number (see page 11) also used in Chapter 5 for standard deviation of the load squeeze number (see page 40) time (seconds) weight function (see page 95) drive frequency (radians per second) a superscript asterisk denotes an optimum value xi CHAPTER 1 INTRODUCTION 1.1 GAS BEARINGS The use of a gas as a bearing lubricant has many advantages. There is an abundant supply of comparatively clean air available; gas bearings are very low friction devices having near infinite life; lubricating gases are comparatively free from adverse effects due to nuclear radiation; and for high temperature missile or low tempera- ture cryogenic applications, gas bearings are useful because the gas viscosity does not change much over wide temperature ranges. Gas bearings can be designed to have stiffnesses comparable to ball bearings and thus have wide applicability. Gas bearings have been used in textile spindles, machine tools, dental drills, jet engines, accelerometers, measuring instruments, and refrigerators. The first operational use of gas bearings was as gyroscope bearings for the German V-2 rockets. They are inherently well-suited in pre- cision instruments due to their low noise when rotating and to zero friction when used as a null device. Because of the increased demand of the aero-space industry for bearings embodying the above advan- tages the study of gas bearings has recently been brought to a high level. In the study of gas bearings to date, most of the work has re- lated to basic understanding of their operation and design. Little has been done on the field of design optimization. This thesis will study one type of gas bearing only recently developed, the squeeze— film bearing. A primary objective is to Show that the proper choice of parameters will maximize the minimum clearance in the bearing. A second objective is to develop a technique for design based on optimi— zing a series of functions developed from the designer's value judge- ment of the desired performance and cost. Film lubrication occurs when two closely spaced parallel or nearly parallel surfaces are completely separated by a lubricating fluid. In order for such a film to support a load, the pressure forces in the film must be such that their resultant produces a net force. There are three main types of film bearings recognized, (1) those which depend on an external source of lubricant to create a pressure field in the film (hydrostatic or externally pressurized bearings), (2) those which depend onrelative tangential motion of the bearing surfaces to create the pressure field (hydrodynamic or self-acting bearings), and (3) those which depend on relative normal motion of the bearing surfaces to create the pressure field (squeeze- film bearings). This paper will concern itself with the squeeze-film bearing using a compressible lubricant (gas), which should, more properly, be called the "steady-state, compressible-fluid squeeze- film bearing" to distinguish it from the transient bearing which is generated when two surfaces approach or recede from each other. It should also be distinguished from the incompressible fluid 3 squeeze-film bearing in that the compressibility of the lubricant plays a prime role in its operation. In the absence of tangential velocity or of external pressuri- zation, the squeeze-film gas bearing is able to support a load by virtue of a continuous high-frequency oscillation normal to the bearing surface. The squeeze-film bearing can be thought of as a self-pressuriz- ing bearing. Its advantages over the externally pressurized bearing include compactness, simplicity of construction, and ease in regula- tion. Further it seems to be relatively free of the stability problems associated with the externally pressurized bearing. The high-frequency oscillation can be sustained by electrical means through a suitable transducer, hence a suitable oscillator for driving the transducer is required. To conserve driving power, the transducer is operated at one of its mechanical resonant frequencies. The squeeze-film bearing, in its simplest form, is created be- tween two plates, one plate vibrating normal to the other. If the plates are closely spaced in relation to their lateral extent and if the vibration is at a relatively high frequency (both conditions are usual), then the effect, due to the finite viscosity of the gas, is to trap the gas between the plates at a superambient density, i.e., more gas can flow in when the plates are separated during the vibra- tion cycle than can flow out when the plates are close together. In the limit as a parameter, known as the squeeze number, goes to in- finity the trapped gas is completely isolated from the ambient and no external flow of gas occurs after an initial pump period. With the gas trapped between two closely Spaced plates, the effect of a non-linearity due to Boyle's law is to create a non-zero cyclic average pressure on the plates. Although described above in terms of a flat plate, the squeeze- film bearing can take many forms such as a round disk, journal, sphere or cone. 1.2 LITERATURE REVIEW ‘ Gross[l] reports that the Frenchman G. Hirn[2] in 1854 was ap- parently the first to mention that air might be used as a lubricant but that this type bearing was not discussed again until Kingsbury[3] built an air-lubricated journal bearing in 1897. The Englishman W. J. Harrison[4] in 1913 presented solutions for infinitely long gas-lubricated slider and journal bearings. It is only since about 1950 that the study of gas bearings has been noticeably accelerated. Taylor and Saffman[5] in 1957, stated that the Reiner effect[6] was probably due to either non-parallel plates or to normal vibra- tions rather than due to a non-Newtonian or visco-elastic property of air. They showed that it was possible to develop an average load-carrying pressure distribution in a compressible film by a re— lative normal motion of two surfaces. This is probably the first reference to a compressible squeeze-film bearing. One of the first analyses of squeeze effects for a gas bearing was that of Langlois[7], who considered the linearized problems for small periodic variations in the gap between infinitely long paral- lel plates and the gap between parallel disks. The case of periodic variation in the gap between parallel coaxial disks has also been considered experimentally and theoretically (numerical integration using a finite-difference scheme) by Salbu[8]. Malanoski and Pan[9] in the discussion of this paper, developed a mass content rule which allowed them to obtain the instantaneous film force and mean load- carrying capacity for large squeeze numbers. Their results were in excellent agreement with those of Salbu. A Pan[lO] was the first to publish an asymptotic method for large squeeze numbers which is applicable to arbitrary bearing shapes and arbitrary modes of oscillation. These asymptotic techniques have been applied to determine the load support capacity of a number of different bearing shapes such as (l) the infinitely long journal bearing by Pan, Malonoski, Broussard, and Burch[ll], the finite journal by Beck and Strodtman[12], the rotating sphere by Chiang, Malonoski, and Pan[l3], the non-rotating sphere by Beck and Strodtman[l4], and a variety of shapes by Pan and Broussard[15]. Other papers published include the following. A treatment of a high frequency instability for'the infinitely long squeeze-film journal bearing is given by Beck and Strodtman[16]. A treatment of the same form of instability in the finite journal bearing is by Nolan[l7]. An analysis has been developed by Elrod[18] of the ef- fect of low frequency vibration in the bearing. This analysis has been applied to spherical bearings by Pan and Chiang[19]. The asymptotic analysis for squeeze-film bearings including the effects of tangen— tial motion (the hybrid bearing) has been rederived on the basis of singular perturbation theory by DiPrima[20]. Pan and Chiang[Zl] de- rived the turbine torques on the supported load for the cylindrical journal bearing. The case of coaxial disks where one member is driven and the other is free to respond was treated by Beck, Hol- liday, and Strodtman[22]. There appears to be no publication of optimization techniques applied to squeeze-film bearings. 1.3. DESIGN OPTIMIZATION AND PROBLEM STATEMENT Although much has been written on optimization and on tech- niques for optimization, there does not seem to be any extensive field of literature on applications of these techniques to design problems, per se. Most of the applications use cost or profit as the objective function to be optimized. Sage[23] in his book "Optimum Systems Control" calls his objective function "a goal or a cost function", regardless of its nature. This is perhaps natural when one considers that cost is a common denominator in so much of man's activity. Other applications speak of maximizing range (of a rocket), minimizing the error in estimation (of position of an object), or minimizing the energy to achieve some end state. To attain an immediate goal it may be expedient to optimize the most important facet of a design. However, in the comprehensive treatment of a design problem there is not one, but many features, often conflicting, which must be considered for optimization. Yet the very word "optimum” means "the best". To resolve this conflict, it is proposed that a composite ob- jective function be created. This composite function will embody the designer's value judgements as to the importance of the various features entering into the design and the relationship between these features and the design variables or parameters. For want of a better name this composite fUnction will be called the figure-of- merit function or simply the merit function. An examination of recent literature failed to Show any previous work in this area. Starr[24] treated a similar problem in his quality function, although he used only a pure ratio between each feature and a standard feature. The fact that the design parameters can be used to define a merit function is one of the contributions of this thesis. Another contribution is made in studying the optimization of minimum clearance in the squeeze-film journal bearing of finite length when carrying a given load of fixed volume. The plan of this thesis is as follows: 1. The appropriate gas film equations for the squeeze-film bearing are stated and the equations for non-uniform driven excursion are derived. 2. The minimum clearance problem in the squeeze-film journal bearing is studied, the significant parameters are deter- mined, and it is shown that the minimum clearance can be optimized. 3. The figure-of-merit function for a squeeze-film journal bearing used as an accelerometer element is developed and optimized. 4. A figure-of—merit function weighted in terms of the ex- pected load is next developed. It is shown that this weighted merit function can also be optimized leading to what could be called an optimum of optimums or a maximax value. Throughout the study, whenever specific application was re- quired, the following practical design problem, called the demon- stration problem, was used. A single degree-of-freedom accelerometer is to be designed for an aerospace application. The accelerometer is to consist of a one . gram (.0022 lbm) proof-mass suspended in a squeeze-film journal bearing. The sensitive axis is along the axis of the journal bear- ing and sufficient restraints will be provided to withstand the maximum acceleration to be applied to the sensitive axis. Cross axis loading may be in any direction and may be any value up to thirty times the acceleration due to gravity. The proof-mass is to be completely contained within the squeeze-film transducer and is to consist of a cylinder of material with a density of .024 pounds per cubic inch. The squeeze-film bearing problem, extracted from this specification, is to support a .0022 pound load having a volume of .0925 cubic inches in a 30g acceleration field (maximum load equal to .066 pounds) in the radial direction. As a result of this study it was found that (l) the important parameters for maximizing the minimum clearance are the nominal clearance — drive amplitude ratio, the non-uniformity shape function of the excursion, and, to a lesser extent, the length-diameter ratio of the bearing; (2) a merit function based on the parameters above can be used to find an optimum combination of cost, radial displace- ment, length-diameter ratio, and drive amplitude; and (3) a weighted merit function can be used to show that a design optimized at less than the maximum load will not only work at all loads up to the maxi- mum anticipated, but will have a higher merit rating than the design optimized at the maximum load. CHAPTER 2 GOVERNING EQUATIONS 2.1 GAS FILM EQUATION The equation describing the fluid dynamics of laminar gas films is called the Reynolds equation. For a bearing in which there is no relative tangential motion between mating surfaces and for the cylin- drical coordinates z and 6 , Reynolds equation may be written[l] .12. 11.3.2.1». . .8; 11.325. (oh R2 30 p u 30 32 u at (2.1) where p is the density of the gas film at a general location 2 and 8 and at a general real time T ; h is the film thickness at (2,6,1); p is the pressure at (2,0,T); u is the gas viscosity; and R is the bearing radius. The assumption is generally made that the gas film is an iso- thermal, perfect gas, thus density is proportional to pressure and the viscosity may be treated as a constant. With this assumption then (2.1) becomes 1 a 313 = 3 (2h) E2- '3 [131136 wiEJh: 1] 1211 31' (2.2) 10 Equation (2.2) may be made the new variables where pa 15 the h is the o m is the Then (2.2) becomes .iL ae aifli [PH 36] + where .3. 32 11 dimensionless through substitution of 0313‘ zHN ambient pressure nominal film thickness squeeze frequency a dimensionless group called the squeeze number. The boundary conditions for use with (2.3) are + P(_ZL,0) $2.0) 38 [m3 a] = 04—13:: (2.3) 12 uwRZ Pahf; = I (2.4) QEIZ.T) 36 o (2.5) 12 where _L ZL'E‘R’ Equation (2.3) may be greatly simplified when G is large which is generally true in squeeze-film bearings. (Typically o is 1000 or greater.) C.H.T. Pan[lO] in 1966 first published an asymptotic solu- tion of (2.3) allowing 0 to go to infinity. With the assumption of infinite squeeze number (2.3) becomes 1332 as 32 373’ WIT—LP] =0 (2.6) where H is the time average of. H . Making the substitution T = (PH)2 reduces (2.6) to a HOT 3H a BET 3H _ a second order, linear, partial differential equation in T, inde- pendent of time. Since the asymptotic analysis used to derive (2.6) or (2.7) is valid only in the interior of the bearing, new boundary conditions where the bearing is exposed to the ambient are required. The boundary conditions at the ends of the bearing, (2.4), become (Pan[10]) 4.2,.) = :EIIEI :Z (2.8) 13 The derivative boundary conditions (2.5) are still valid but now take the form 31am) z flaw) 80 36 0 (2.9) For a journal of uniform radius constrained to have radial dis- placement only, the film thickness for the bearing,when the driving member moves sinusoidally, is given by h = ho - 6h1(z) sin wt - ah, cos a (2.10) where 6h1(z) is the amplitude of the excursion of the driving member at any 16cation along the z axis dhz is the displacement of the journal center above the bearing center measured in the plane defined by the journal center and 0 = 0. (See Figure 2.1) Equation (2.10) may be made dimensionless by dividing through by ho , and by using t = wT H = “1:?— = 1 - €1f1(Z) sin I: - £2 COS 0 . (2.11) o where c1 is some reference excursion f1(z) is a shape function for the excursion 82 is the eccentricity of the bearing 14 —A L‘“ __'__.. -. In” Figure 2.1 Configuration of the squeeze-film journal bearing From (2.11) then H = 1 - 52 cos 6 (2.12) and the boundary conditions (2.8) become _ _ 2 3 2 2 T(ZL,0) - (1 52 cos 6) + 2 clfl(zL) (2.13) 2 3 T( ZL’B) (I 92 cos 6) + 2 elf1 ( ZL) for the journal bearing. A solution to the asymptotic form of Reynolds equation has been approximated previously by two different methods, one numerical, the other analytical[12]. In the numerical approach, finite differences 15 were used in an alternating-direction, implicit method to generate a matrix of T values over the surface of the bearing. In the analyt- ical approximation, the series T 2 TO + €2T1 (2.14) was substituted into (2.7) and into the boundary conditions (2.9) and (2.13). A collection of like powers of €2_ gave two partial differ- ential equations, one in To , the other in To and T1. The solu- tion of these equations gave the answer 3 3 3 T.[ --:+ ,2] (2.15) l (uniform excursion). Later valid when 62 is small and ff(z) work produced the answer 2 1 cosh Z 2 T 2' l+§c§f2- 262c056[-%czf2c;.5h—-Z—+1+ie§f2] L (2.16) for non-uniform, but symmetrical, excursion where f = |f1(zL)| = IfII-ZL)‘ For this investigation it was concluded that (2.16) could be used only for preliminary investigations because of the restriction that 82 be small. The numerical method is valid for all 52 but re- quires a considerable amount of computer time for iterative solutions. 16 For these reasons the small parameter solution was expanded to in- clude two additional terms in the series T 2 TO + €2T1 + ESTZ + €3T3 (2.17) The details of this work are given in Appendix A together with a comparison in Appendix B between this augmented, small-parameter analysis and the numerical results. The conclusions reached are that the T-values found using the augmented, small-parameter solution agree with the numerical solutions for values of 52 as large as -.8, and that the former requires significantly less computer time. In all that follows then, the small parameter solution was used for prelim- inary investigations or to find starting values and the final solu- tion was made using the augmented, small-parameter equation. In the optimization procedure the solution of the partial differential equation (2.7) is required a large number of times. Without the augmented, small-parameter analysis, the total computer time may have been so large that the optimization procedure would not be practical. In summary then the film characteristic, T, is a function of the variables T = T(ho,6h1(z),6h2,pa,z,0,R,L) (2.18) in the dimensioned form or of T = T(€1,f1(Z),€2,Z,O,ZL) (2.19) in the dimensionless form. 17 2.2 LOAD SUPPORT CAPABILITY . A bearing at equilibrium can support an average load W1 , per unit length which is given by L 2n F2" -5 1 W1 = - 2TE' p cos 6 d6 dz dt 0 Jo —§ (2.20) In dimensionless terms (2.20) becomes r2.” r2". rZL W' _ w1 _ 1 _ 2p R — 8NZ P cos 0 d9 d2 dt a L Jo Jo J-ZL (2.21) The Sign of the force is chosen such that a displacement of the journal below the bearing center gives rise to a positive (upward) force. The pressure, P , in (2.21) is obtained from the solution of the gas film equation and since T = (PH)2, (2.22) Since H , given by (2.11), is the only factor of (2.21) containing 18 time and then in an especially simple manner, the integration of (2.21) over time may be performed explicity giving 12" IZL ;, T2 cos a de dZ w'=-i- , 4ZL [(1:62 cos 0)2 - eff?) é J J- o ZL (2.23) Again two methods of solution of (2.23) are available, numerical integration and a small parameter approximation. A two dimensional form of Simpson's rule was used for the numerical integration. _The small parameter solution, requiring now that cf as well as 52 be small led to the answer[12] n ( 3 tanh 2L) W' = - —-ezc 1 + -————-———- 2 2 1 2ZL (2.24) for uniform excursion and to IZL 2 cosh Z c c 2 w' = - 1- 1 2 2f1 * 3f cosh 2 dz 8 2L L J_ , (2.25) ZL for non—uniform but symmetrical excursion. In summary, the load support per unit length does not involve any variables other than those considered previously for the film 19 characteristic. In fact, the integration over the bearing area elim- inates the dependency on position so that in dimensioned form W l W' = = W' (h ,8h1(z),5h2,p ,R,L) ZpaR O a (2.26) and in dimensionless form w' = w'(:1.f1(2).e2.zL) (2.27) 2.3 NON-UNIFORM DRIVER EXCURSION The high frequency, sinusoidal displacement of the driving member, in general, will not be uniform along the axial length. As suggested by (2.11) the excursion may be represented by some refer- ence excursion multiplied by a non-uniformity shape function. An assumption, substantiated by some unpublished experiments by William G. Holliday[25] is that the amplitude of some of the simpler radial motions of a thin-walled cylinder can be expressed by a cosine function as am = a + b cos 97:5— (2.28) LC Where a and b are two parameters to be treated at greater length below and z is the half length of the driver. It is necessary LC to distinguish between the length of the driver and the length of the bearing since the two need not be equal. The length of the bearing is determined by the length over which the gas film extends. The driver excursion is defined over the length of the driver. 20 If a new variable,called the shape factor, is defined by the ratio mlc‘ (2.29) then (2.28) can be written oh, = a(l + A cos 22'2") (2.30) LC This accomplishes the separation of Ohl into a reference ampli- tude, a , and a shape function. If a is assumed to be a non-negative constant then a number of different drive shapes may be characterized by the magnitude of A as shown in Figure 2.2 , which depicts the range of displacement of one surface of the driving member. Equation (2.30) suffers from two deficiencies. First, the excursion, a , at the end is not a good reference, since as the in- put power changes both the end excursion and the shape factor, A , change, and second if the excursion at the end should be zero, Ohl is not defined. It was postulated that a spatial root-mean-square excursion value would make a better reference. Limited experiments by W. G. Holliday[25] did in fact show that a good degree of correlation existed between the rms excursion and the input power. 21 ___J£ 1 _—-—- _~‘~ 0 _J2_________. ___ HZ s‘~ ’4 CASEI A>O A "no- node" or ”barrel” mode ‘~ lob __. _ —T—-— ——_—-‘_ " L... CASE 3 'I 0) 2 Ohlm = a (A = 0) 3 Ohlm = a (-1 < A < 0) 4 Ohlm = a A = -l 5 6h1m = a for b :_-2a (-2 :_A < -1) dhlm = [bl -2a for b < -2a (A < -2) 6 ahlm = b (A = m) The first five cases may be expressed as Ohnn aF(A) where F(A) is defined as follows A > 0 F(A) = A + l -2 :_A :_0 F(A) = l A < —2 F(A) = |A + II But from (2.35) a - Ohlr a (3.8) Therefore 1m _ -% 5h1r - F(A) a (3.9) Equation (3.9) does not appear to cover case 6 since it is indeterminate as A + w ; however 6h . [- iim dhlm = 21m 2 1A+ll L = 2 (3.10) A400 11‘ A-_>°° 57+%+1]‘ 28 5h1m A values of interest. Differentiation of ( 3.9) with respect to A Figure 3.1 shows how the ratio varies in the range of fOr -2 §_A.§_0 gave the value of the maximum as 2.298 at A = -.% (A = - 1.2732). Putting (3.9) into the expression for clearance (3.2) gives _ F(A)6h1r c - hO - -——;?;——-+ th W, V, ho’ dhlr, A, pa, ZL) (3.11) as the expression for clearance to be maximized. 3.2 OPTIMIZATION OF SMALL PARAMETER EQUATION For the case when the gas film is the same length as the driver, the small parameter load support equation (2.25) may be integrated, using the non—uniform excursion expression (2.36), to obtain n 3 tanh ZL w' s - E-cfrsz 1 +--377;7§:—- (3.12) or in dimensioned terms w' _ _lL__ = _ 1.5h1r éhz 1 + 3.:EEE—EE. (3 13) - 2RL p 2 h3 2 a Z ' a o L Equation (3.13) may be solved for th and put into the clearance equation (3.11) yielding w h3 o 2 3 tanh ZL "RL Pa “WE * 737— L F(A) ah C = h .. 0 éhlr' (3.14) 29 nouonm »UAauomfl::n:oc 0:» mo :oauoqsm n ma monumead may ou ovsuflamad adafixda mo owumm H.m onsmfim ( NI no 0.. 5.3” E5“ 30 . ohfir 6h2 as the clearance equatlon for values of and -17—-<< 1, hg o (This implies that #2- must be near unity.) 0 Since constant volume was to be an equality constraint in the optimization it may be introduced by observing that the radius ap- pearing in (3.14) may be given by v 1/3 R = (21:2) (3.15) L and the length, L , by L = 2R zL (3.16) Then 3 Who F A c h,6hu,A,z - h - ~§l-mn, - 1 ° ' 1" ° 61 (211)1/3 pavz/a le/a ohfr [l o #21:“: L I (3.17) In order to gain some insight into the nature of the clearance function, it is instructive to study (3.17) by varying one parameter at a time; however to reduce the number of variables, divide (3.17) by dhlr and define the new variables air and ‘85 to get (3.18) 31 where 3 El : 5P2 = - W (Cir) 2 Phlr @é kg 3 tanh ZL "V ZL “—2??— L (3.19) h I = O Elr — 5h1r he K = (2“) Pa Thus it is possible to plot a dimensionless clearance in terms of a dimensionless nominal clearance, so that the results to follow, although specific for W and V , are general in nominal clearance and excursion. Figure 3.2 shows that for a fixed value of the load and the load volume, the clearance does have a definite maximum as a function of nominal clearance, and further that the maximum is a function of the non-uniformity of the excursion. Figure 3.2 also clearly shows the incompatibility of large nominal clearance and small eccentricity I since as Eir increases (large nominal clearance) so does 62 (large eccentricity). Figure 3.3 shows for a fixed value of non-uniform excursion that the optimum clearance is also a function of Z the length to L , diameter ratio of the bearing. The shape of the curves in Figure 3.3 suggest that Z may not be an important variable in determ- L ining the clearance. This is particularily true for €1r near the 32 ncowumsco nouoadnwm Hausmv oocnumoHo Hmcfisoc mo mcowuocam ma uaosoowammwv Heaven ecu oonnumoHu ~.m onsmflm III. as; on . 2 O. I N. o. o O Q #1 \ \ 9...... V \ \ \ . \\v\ \ \ \ \ \ \ \ \ \ K \T 3.32 \ V X‘ AOI(V.U \ \\\ \\ €8.55 V/ V) 6 hi. .o 1.6 1‘ \ II\\_ x m \ \ “wo \ x m. \. \ “ks \ 2 .\ Tm .. -(.m?l\\ 3.3 mv-\_.~ . nz.u~ao..> «cacao..3 3.3.6.6 5534...: .338 d _ a. 33 7 "‘-=--..‘:_«d,-I4 _________ / A - 12732 \IIMOOCLI Vh~l . . \ . //1 ‘ 1" \ "'\"2. VOL. - .0925 m3 55. if . ‘s / ‘W "5 N \ SHALL unmercn / \ counnou ols "' T ‘ ~~ ~ ._ I \ q 2, ~ \C." 3 |. 5.5 X /' ‘\ I ‘5 V \ 5 A - -l.2732¢ ' 8 .03. I... v . .0925 "‘3 oor'rco cunvcs 4:. >14 souo canvas 1.? 5:4 5 5 10 IN L2 L3 L4 L5 Figure 3.3 Clearance as function of length—diameter ratio (small parameter equation) 34 value of 14, the curve which yields the maximum clearance. For the dotted curves (sir > 14), the curvature becomes considerably more pronounced than for the solid curves (air §_l4) Figure 3.4 shows the contours of constant clearance plotted against the two most important variables, fixing ZL at 0.897. It is also possible to gain some insight into the maximum of (3.18) by the classical approach of setting the differential with respect to each of the three independent variables equal to zero. This procedure produces ' 3 h Z -1 8—CL = W(€1r) 3 Z ‘1/3 1 + :_t_a_n___li = 0 2 3 L KV/3 BZL L 2 a ZL (3.20) v 2 ac' - 1 3 W (Elr - o a 8' - - 3 tanh Z - L 2 a ZL (3.21) ' _ _. w ' 3 Stanhz '1 a— - apogwmoawy .glr , 'ax[1Tz—L .. . KV 2!. L (3.22) Performing the indicated differentiation of (3.22) with F(A) = l, which assumes that A lies between -2 and 0 , gives 36 0 3 ‘2 . .. ‘ __'________(clr) 3tanh 42L 3tanh 2!. A2‘4A"! 2(A+4)-0 2 1 2 n xv /32L /3 22 L(% + + :1) zzL n (3.23) ayn This equation is satisfied by A = - for any value of W, Z V, L, I or 81 That the value of A lies in the assumed range justifies r . the original assumption. Further, Beck and Strodtman[IZ] showed for small parameters that the load support of a finite journal bearing with uniform excursion is composed of two components, one due to interior excursion (non-linearity of the load support equation), the other due to the excursion at the boundary, (pressure pump-up). It was also shown that the second component was a function of the length diameter ratio, ZL , and for 2L 2 l was 1 1/2 times the first com- ponent. It has been shown that Shlm has a maximum at A = - %-. 1r This value of A is case 5 of Figure 2.2 with [bl < 2a thus 5h1m occurs at the boundary, enhancing the total load support. It is reasonable to expect a maximum value for the clearance when the load support capability is maximum. With the above value for A , since (3.20) is a function of A and ZL only, one can solve for 2 _ W *8 3 2, 2 a =-_- - n2 ZL tanh ZL 2 sech ZL “ (3.24) 37 Newton-Raphson iteration gives Z: = 0.89664 where here and elsewhere the asterisk will be used to denote optimum quantities. Using these values for ZL and A in (3.21) gives 26 % clr = 2.3501 w (3.25) For the assumed values of V = 0.0925 in3 and W = 0.066 lbs, oi; = 14.054 and c'* = 7.0719 That these values correspond to the maximum is evident from Figure 3.4 which is a contour map near the optimum. Further, in Section 3.4 it will be shown that the function defined by the second derivatives near the optimum is negative definite thus insuring that the optimum is a local maximum. To relate the above optimum values to dimensioned quantities it is necessary to know 5h1r . Typical values for the rms excursion range from five to fifteen microinches. The optimum nominal clear- ance, ho , will then be between 70 and 210 microinches and the min- imum clearance, c , will be between 35 and 105 micro-inches. The lower limit of 70 microinches for the nominal clearance is less than usually used in squeeze-film bearings. It is near the limit of nominal clearance which can be reliably fabricated and measured. 38 The upper limit of 210 microinches is well within typical dimensions encountered. A bearing with the lower limit of 35 microinches for the minimum clearance may not be satisfactory since a very small asperity or foreign particle might bridge the gap and destroy the bearing. It is also evident from (3.20) that if any other value of A is assumed for the excursion the corresponding maximums with respect I to ZL and c1 Spectively. It is also interesting to note that Z r only can be obtained from (3.24) and (3.21) re- L is independent of W and V thus only the optimum value of Fir changes as W and V are varied. The small parameter approach just described fails in one im- portant respect. The Optima predicted do not fulfill the small . . 6h , parameter restrictions; 1n particular 7;;- was requ1red to small 0 compared to unity. But at the optimum by using (3.21) and (3.19) one finds II I m| H which is not small compared to unity. 3.3 OPTIMIZATION OF CLEARANCE USING THE AUGMENTED, SMALL-PARAMETER EQUATION In the previous section, explicit solution of the clearance equation was analytically possible because a relatively simple ex- pression for dhz was available. However, a more accurate 39 solution is obviously needed since the clearance predicted by the small parameter equation did not meet the small parameter require- ment 5 . Equation (3.11) is still valid but th must be determined by a more accurate means. Two methods are available, (1) the finite difference solution and (2) the augmented, small-parameter equation. The latter was chosen because it uses less computer time with no significant difference in accuracy, in the range of interest for this problem. See Appendix B. The programs written to solve for the load support by either method result in a W' corresponding to given values of 51f: 52, A, Z To determine 82 corresponding to a given load it is ne- L . cessary to iterate on 82 until the given W' is obtained within some given error bound. To start the iteration, the small para- meter equation (3.12) was solved explicitly for £2 for the given load. Since for large loads or very small values of £1 , the small parameter equation tends to overstate the value of £2 , a check was necessary to insure that the resulting clearance was positive. If this condition was not satisfied, a value of 52 was selected which made the clearance slightly positive. The second value of 82 was always arbitrarily selected as 0.95 of the first. Linear inter- polation to the desired W' gave a third value. Quadratic inter- polation on the last three values was used for following steps to locate each new 52 . As each 82 was located the corresponding W' was computed and compared to the desired W', the operation terminating as soon as the error criterion was satisfied. 40 The load support calculations were normalized by dividing all film dimensions by hO , the nominal film thickness. But ho is one of the independent variables in the clearance equation. To get around this inconvenience, it is desirable to work either with dimensioned clearances or, to simplify the problem by reducing the number of variables, to find a new normalizing parameter. In the c previous section the use of c' = 5h was immediately evident 1r from (3.17). That the same normalization holds in general can be demonstrated from the gas film and load support equations. Define a new variable H' = 6N1 . Substitute this new 1r variable into (2.2) together with the other variables previously defined for P, Z, and t to get the analog of (2.3) in the form _3_ I3 23 1 I3 _a_p_ : I 30311.) as [PH 80:] I 32 [PH 82 O at (3.26) This is identical to (2.3) except that the squeeze number is redefined as v=.1_2_u_w_fi_ PaC5h1r)2 Since the squeeze number was assumed to be very large, this normali- zation makes it even larger as Ohlr < ho , and thus the asymptotic differential equation for the gas film characteristic is unchanged. 41 In the boundary conditions at the ends of the film T'=-== I (h) (3.27) thus the new boundary conditions are related to the old through the ratio of the normalizations. This means that the new T values (called T' = (PH')2) computed from (2.7) will differ from the old by a factor of h 2 ___9. 5h1r Consider the load support, W' , given by equation (2.21) with P = H, where H' = h = —h—0 ho dhlr ho dhlr therefore ho T% 61111‘ T% P " -_h_____— = '7; ° H and the dimensionless pressure used for computing the load support is unchanged proving that W' is the same using either normalization. 42 Thus the clearance equation (3.18) derived originally from the small parameter equation is valid for the more general solution as well. A steepest ascent method was used to find c'* , the optimum . . 1* d1mens1onless clearance, and the opt1m121ng parameters cl , Z A*, * L ’ numerically, for given values of W, the total load in pounds and V, the load volume in cubic inches. However, in developing this program it was instructive to also develop curves comparable to those pre- viously given for the small parameter treatment. Figure 3.5 compares the small parameter solution with the aug- mented solution. It is evident that after the optimum clearance is reached, further increases in the nominal clearance-drive ratio, Sir , result mainly in increasing c; , the eccentricity, With only a slow decrease in the clearance. It is also evident that the two solutions correspond closely so long as c; , which corresponds to 6h2 is small, and that the clearance curve in the more exact case has a flatter top. Although the solutions agree so far as clearance is concerned, the values of Elr corresponding to the maximums of the two curves are quite far apart. Figure 3.6 shows the contours of constant clearance for vary- ing A and sir with 2 fixed at 0.713. Again it appears that L the optimum value of 5h1mq/5h1r occurring at A = - %-, determines the A* value, certainly for the value of ZL selected. Other com- puter runs confirmed that, indeed A* = - g-for all values of ZL that were checked. Anewumsvo pouoanummIHHnEm .voucoamsmu oocmHmOHo Hmcwao: mo meoHuocsm mm ucosoomammfiv Heaven van oocmumoHu m.m munmfim . .I... .... On at on at. GNOO. I > 04000. n ’ n». .d unhN .7. 4 43 V on I o ./ \ a \ 81 o / swans: on] o -~ U I I) w. x / P 31k [10$] 0. 44 / \\ / / / 0"5 / c"- 7.07: l c... 1 / c’u'- 10.206 \/ 233372.13.” IO 20 30 40 50 00 3" Figure 3.6 Contours of constant clearance (augmented, small-parameter equation) 45 The optimization program was run for the demonstration design of W = 0.066 lbs and volume of 0.0925 in3 with the result c'* = 7.873, c1; = 18.205, A* = - 1.2732, 2: = 0.7131. Comparing these optimum values to those found previously from the small parameter equation, the nominal clearance, €1r , has in- creased 30 percent to 18.2. This represents a dimensioned nominal clearance of from 90 to 270 microinches depending on the drive amp- litude. The minimum clearance has increased slightly more than 10 percent giving clearances between 40 and 120 microinches. It appears that the small parameter solution is not too far from the more exact solution. The Optimum length-diameter ratio of 0.713 is not convenient and many applications of the squeeze-film bearing use a ZL between unity and 1.4. 3.4 SENSITIVITY TO PARAMETER CHANGES To gain some insight into the fashion in which the optimum dimensionless clearance changes with changes in the independent variables, make a Taylor series expansion around the Optimum giving N 32c * §;—§;—-AxiAx. + 0(Axi)3 1-1 j=l 1 j 3 IMZ 3 1 CI _._ CI*+2 1 ET ' * §£—-Ax. + . 3x. 1 1 l (3.28) where Axi and ij are the deviations from the optimum (x1 = €1r: x2 = ZL’ x3 = A)- 46 One of the definitions of sensitivity widely used in feedback control theory is the following[26] Sensitivity is normally used to express the ratio of the percentage variation in some specific system quantity such as gain, impedance, etc., to the percentage variation in one of the system parameters. The sensitivity function is defined as M d in M _ d M/M - Percentage change 1n M(due to change 1n xi) k d in x. - d x./x. Percentage change in x. 1 1 1 - - 1 where M is a transfer function and xi is a specified parameter. . . . . . ac' At an 1nter10r opt1mum, where x1 15 not constrained, 5;—- = 0 i leaving only thesecond derivatives and higher order terms, thus the definition given above is not applicable to this particular case. Following the ideas of Box [27] it is possible to fit a quadric surface in i dimensions to represent the response, c'-c'*, for small deviations near the optimum. Write (3.28) in the quadratic form c'-c'* = i Qx T (3.29) where Q is the matrix Of coefficients given by 47 F '1 32c' ; x 82c' ; x 82c' * axlaxl 1 1 axlax2 1 2 8x13x3 x1x3 Q = 32c' ; x 82c' ; x 82c' ; 3x28x1 2 1 8x23x2 2 2 3X23X3 2x3 32c' ; x 32c' ; x 32c' * 3X33X1 3 1 3X33X2 3 2 3X33X3 X3X3 L— ._J (3.30) and X1 x2 x3 x: II (—"'1 Axl sz Ax3 ] (3.31) The magnitude of the Off-diagonal terms in (3.30) indicates the degree of interaction between the variables. (Called the "factor dependence" by Box.) In the diagonalized form, the diagonal members of (3.30) give the curvature of the surface in each of the dimensions considered and are thus a measure of the sensitivity of the system to changes in the variables. Figure 3.7 shows the first derivatives near the optimum for the small parameter equation. The three curves in the tOp row I .53;. air but, by being plotted ; 611' versus 81 ZL and A respectively, have slopes which give the r, ’ second derivatives required by the first row of (3.30) (except for (Figure 3.7 (a), (b), (c))all are the required normalizing factor). 8p Acowudsvo nouosduam aamamu Esawumo on» use: oocmumo~o mo mo>wua>wuov umnflm n.m Ouswwm 3. 2: .3 THE C. .3 48 .3 .3 a: :5 49 This pattern is true for the other six curves also representing I I EE—- Z and §2—-IAI. Thus from Figure 3.7 (3.30) can be written BZL L BA for this particular problem as f-28.1 0 0 T Q = 0 -2.14 0 L 0 0 -24.84 (3.32) evaluated at (14.05, .897, -l.2732), the optimum. It can be shown from (3.20), (3.21), and (3.22) that the off—diagonal terms of (3.32) are identically zero at the optimum. Figure 3.8 shows the second derivatives of the dimensionless clearance as functions of the three independent variables. The curves are arranged so that the position of each corresponds to the location that the second derivative has in the Q matrix (3.30). Figure 3.8 contains more information than is required by (3.30), or by (3.32) for the specific problem, in that (3.30) requires evaluation of the derivatives at the Optimum only. Figure 3.8 shows the manner in which the second derivatives change away from the optimum. Note that only at the optimum are the pairs of mixed 2 I 32 I . . partial derivatives equal, i.e., 3—3—E——- = ——ZE—~T- This 15 not clraZL 3 Laelr unexpected, in that in moving away from the optimum the mixed partials are no longer being evaluated at the same points on the . 3 ac' surface, 1.e., SET—' 57—- 11‘ L in general, it is not the I* 811- * ZL + AZL 50 acowuazdo uouoswumm .HMEm. asawumo on» new: oonwuonU mo mo>wum>wuov vacuum w.m unawam N a... 0 s|~ .4». n0. . 0: , 9.. 0.. n0. 0 9.. 00.. 0.. on. a. ... .5. .o. .00. psr ““1 lJnN. . 0 aNfi O fiufl Hfi .¢_JN . Q .C‘Ubufi Au .oq ... .o. .s. \ N. v- —.o o .IIIII. «. o . sh... £3.“le u WW .1.» b... m" a .o. .a. .o. ow _- oc- / / 0 [I Our H8 :8 ll 2.2. / 350 .u-Q / .WQ a: 0 UN I W. p o 51 8 ac' SZ 36' ' same as * ZL For a maximum it is necessary that (3.30) be negative definite. This insures that the quadric surface is ellipsoidal and not hyper- bolic. The test for negative definiteness is that the principal minors of -IQI be positive. It is immediately evident that (3.32) is negative definite. It is further evident that no cross coupling exits, i.e., each variable has the same effect regardless of the value of the other variables. In the geometrical sense the quadric surface developed from (3.32) is of the form axf + bxg + cx§ = f(x1,x2,x3) (3.33) . . . 32f . . Since only second derivatives of the form 3&2-, (1 = 1,2,3) ex1st 2 i and the mixed derivatives §;§3§—-(i = 1,2,3; j = 1,2,3, i + j) do not i j occur. It is evident from Figure 3.7 and (3.32) that the nominal clearance-drive ratio, air, has the largest influence on the curva- ture of the quadric surface followed closely by the non-uniformity factor, A. The length-diameter ratio, ZL’ has very little influence. Thus it can be concluded that the two most important variables are ! €1r and A. Figures 3.9 and 3.10 contain the same type information as Figure 3.7 and 3.8 except that the clearance is now computed from 52 .cowpmsdo nouoawnmmaHHmam .voucoswsw. esswumo 0:» Hum: oocuumo.o mo mo>wud>wuov umuwm m.m ouswwm no.— .. .m— 8. O N- O O .0 O .O. .0 . \. .SHum / M NO. ~O. 2. c r. E «a 2.- ~r e- 5.. _.o 8.: 7\N O O JNJINIWW . U o I _. no . / U. / —. I. .. Z n. n... .o. —w .a. .0. \ —.- 0 0 Jun \ / .33 O p. p / / 4H .cowumavo HouoadHMQIHHMEm .voucoawsm. ESEHumo on“ you: oocwumo.u mo mo>wum>wuov vcooom o..m ondmwm u( can a...» 4 4m Hal 0. 0.. p 8.. —.— ..p 8.— p 8. O. .m— 8. p O. ... .3. .9. On 7 p- \ a A/ \\ n. 1' . O‘NU‘I! 0 fi% \ ._.... zap. . .. _ p h— p u a j Z _ .u. .o. .i. B u- u- a. 2'1 I! M3 my OJ”. .8 N a .0 .smIWQ a N bub . 00 L_. s. a. .o. .a. .0. 0.- u- ON- [I IlllrlllllLdo o o? sue farfloo #9..me «my —‘ n _ _ _ - _ 0 54 the augmented equation. The matrix of approximate second derivatives analogous to (3.32) is F n -lO.7 -l.20 -.157 Q = -1.16 -l.78 -.070 -.01 .018 -26.8 L . (3.34) evaluated at (18.2, .712, -l.2732), the optimum. The off-diagonal terms at q13, q23, q31, and q32 are essen- tially zero and could well be due to the inaccuracies in the finite difference procedure for computing the derivatives, to the finite error allowed in computing the eccentricity corresponding to a given load support and to the inability to determine the optimum exactly. Since no explicit solution is available for the clearance, c', it is not possible to show that these off—diagonal terms must be zero as was possible in the small parameter case. However since the clearance function is continuous and has continuous first derivatives, the order of differentiation at the optimum is immaterial. Thus the matrix (3.34) must be symmetrical. The terms at q23 and q32 are small and of opposite sign indicating a tendency to average to zero. The average of terms q13 and q31 is small compared to the diagonal terms. Thus q13 and q31 will be assumed to be zero. The off-diagonal terms at q12 and q21 are clearly not zero and demonstrate that an interaction occurs between air and ZL' By using the average of q12 and q21 terms (3.34) can be written as 110. c' - c'* = x -l.l The linear transformation leads to the diagonalized form c' - c'* = ‘ 1), the accompanying merit factor. The general form of the merit function for a design is then N M = TMXnEci) (i = 1,2,...1) (4.1) The argument for multiplying the merit factors is that this insures that if any one factor is zero, the product is zero also. (This may seem drastic, but what merit does a design have which is perfect in all respects except that it costs so much to produce there is no market? or one whose performance is so poor that the design fails its intended purpose?). Further, since each factor ranges from zero to one, the product does also, and additional factors may be appended still keeping the total product between zero and one. Also, one or more factors may be eliminated by set- ting the approPriate exponent equal to zero. In order to make the optimization of the merit function mean- ingful, it is desirable to define or specify each merit factor care- fully. The desired result is to have the merit function a strictly concave function in the vicinity of the optimum, thus assuring second derivatives with respect to all variables and a negative de- finite matrix of second derivatives. This is a necessary condition for the function to have a maximum on the interval. It is also 63 desired to have a merit function which has first derivatives with respect to all variables equal to zero on the open interval, thus insuring that the optimum is not at one limit. This is desired since any variable going to its limit means that that variable does not enter into the optimization and could just as well be eliminated by setting it equal to its limit value. Some requirements for attaining the desired properties in the merit function can be given for some of the simpler cases. It is assumed that the merit function will be single-valued, continuous and have continuous first derivatives in each of its variables. It is also assumed that the range of the merit function will be zero to unity and that the domain of the variables is closed and finite. Consider (4.1) written in the 10garithmic form N log M = 2 wn log Mn(xi) (4.2) n=1 then, assuming that an interior optimum exists, N a - . - 0.. SET-log M — Z wn SEE-log Mh(x1) - o (1 - 1,2, I) 1 n-l (4.3) It is evident from (4.3) that the individual values of the wn are not important in determining the optimum but rather only their ratio. 64 Thus in (4.3) wn can be replaced by ”n where n = (4.4) with the definition, n1 E 1. Consider now a particularly simple case of a merit function made up of two factors each of which is a function of the same independent variable on the interval (a,b), i.e., M M1 (x1) M2 (XI) (4.5) Then if on (a,b) 1 3M 1 8M1 1 3M2 __ = -— — + T] — —- = 0 (4'6) since n2 > 0 , M1 :_0 , M2 3_0 , (4.6) requires that the two first derivatives be of opposite sign at the optimum. The simplest pair of candidate functions which fulfill this requirement is two linear functions, one increasing and the other decreasing in XI. The other requirement for a maximum is that 2M 3M 3M 1 3142 2 1 32M2 -- I -— —-— —— -— -— 2 —- M 3x? M1 3x? n2 MIMZ 8x1 3x1 2' 2 M; 8x1 M2 axf (4.7) For the simplest pair of functions, the two linear functions, the two second derivatives vanish leaving 65 1 3M1 3M 3M 2 ( 2 2) < 0 (02 i 0) )+(n-1)1(_ MlMZ 3x1 8x1 2 EY' 3x1 (4.7a) Since the two first derivatives are of Opposite sign, the first term of (4.7a) will be negative. The sign of the second term is de- termined by (nz-l). If 0 < n2 :_1, then the sign of the second term will be negative also and clearly (4.7a) is negative. This is some- what more restrictive than needed as the final requirement is 3M1 3M2 | > (nz-l) 3M2 )2 (4.7b) M13X1 Mzaxll 2 Mzaxl In the event strict monotonicity is not required, both functions can be zero simultaneously over part of the interval even though one is an increasing function and the other decreasing. When this hap- pens, the product function will be zero over the entire interval. Another problem with non-strict monotonicity is that the functions may simultaneously be constant over part of the interval. Although the requirement of (4.6) will be satisfied, the second derivative (4.7) will be zero and not negative-definite as required for a max- imum. Since it is not desired to restrict the candidate functions to linear functions, any other pairs of function can be used provided the requirements of (4.6) and (4.7) are met. If considering more than two functions of a single variable, those which strictly increase may be grouped together and treated as a single, strictly increasing function. Those which strictly de- crease may be similarily grOuped. Since neither of the resulting pair need be linear, the tests of (4.6) and (4.7) must be met by the resulting merit function. 66 If now we consider a function of two variables n 2 M = M1 (x1,x2) M2 (x1,x2) (4.8) Then for an interior Optimum on the interval [a,b] 1 3M __ 1 3M1 + n 1 8M2 _ 0 ‘) ._.___ - .__.___ 2.__.___ _ M 8x1 M1 3x1 M2 3x1 (4.9) 1.211.131.41. 1.3.42.0 M 3X2 M1 3X2 n2 M; 3X2 J Write (4.9) in the form l.'3M 3 log M _ 3 a _ M 3x1 - 3x1 - EEI-log M1 + n2 SEY'lpg M2 - 0 (4.10) 1. 8M 8 log M 3 3 _ fi'axz 3x2 3x2 1°g M1 + n2 3x2 193 M2 ' O the 3 3x1 .1. 3x1 .1. 3x2 Try an iterative solution of (4.10) for x1 k+l iteration, k+l k k+1 k k+1 3 102 Ni 32 103 M1 103 Ml x1 + 6xl , x2 + 6x2 - 3x 3 2 1 X1 k 2 2 a log M 3 103 M 3 103 M +1 2 2 +1 2 k+1 103 ME ' _f§xl * ax? 6x§ * axlax2 x2 * k 2 2 k+1 - a log M1 8 log Ml k+1 a 103 M1 k+1 103 M1 3 + 6x2 + 3 3 1 x2 axg (x1 x2 1: a log M 32 103 M 32 103 M 10' M§*l - a 2 . 2 2 6x§+1 + 2 xg‘fl x2 3x2 axlax2 and x2 where, for k+l 32 log Mk + 1 Wfixgl+ooo (4.11) (4.12) (4.13) (4.14) 67 Putting (4.11), (4.12), (4.13) and (4.14) into (4.10) gives {- '3 F' "‘ r- ‘l 32 k 2 k 2 k 2 k Vx log MI n 3 log M 3 log M + n a log M 611'”! I 3x2 2 3x2 3x 3x 2 3x 3x I 1 1 1 2 1 2 2 k 2 k 2 k 2 k 9x2 3 log M1 . n 8 log M2 3 log Ml * n 3 log M2 dxk” axlax2 2 axlax2 Mg 2 3x7? 2 J (4.15) k+1 k k+1 k 3 log M1 3 log M1 3 log M2 3 log M where VX. = - -—-—--— + n2 ——-—— - —-—-—2- 1 8x. 3x. 8x. 3x. 1 1 1 1 For (4.15) to have a unique solution it is necessary that the square matrix, which will be called Q, be other than zero. Since (4.15) is to be iterated for a maximum, at which point the left side will be zero; it is necessary that Q be negative definite. This is to insure that a positive value of Gxi will decrease the value the left hand side of (4.15) when xi is less than the optimum. It is evident that Q is identical to the matrix of the quadratic form obtained by expanding (4.8) in a Taylor series about the optimum. It should also be noted that if the set of equations (4.15) is solved by Cramer's rule, the determinant of Q will be the denomin- ator in the solution. Thus if Q is near zero, small changes in Q will have a large effect on the answer. This then constitutes an "ill-conditioned" set of equations. The simplest candidate functions for M1 and M2 are linear 68 functions in x1 and x2. Assume M1 axl + bxz + e (4.16) M2 6X1 + dXz + f What requirements must be placed in the coefficients of equation (4.16) in order that it meet the requirements of (4.9) and that Q of (4.15) be negative definite? Using (4.9) one obtains 1 3M 3 C _ - 1 1 2 (4.17) 1 8M b d M ax2 M1 2 M2 J Evaluating Q from the square.matrix of (4.15) gives P m 2 2 + 2 2 M2 + dMZ a M2 nzc M1 ab 2 nzc 1 2 2 2 2 2 2 h-asz + n ch1 b M2 + nzd M1“d (4.18) 69 Since M1 and M2 are to be non-negative, (4.17) can be satisfied only if a and c are of opposite sign as well as b and d of opposite sign. This says that if M1 is an increasing linear function of xi then M2 must be a decreasing function of the same variable. Expanding the determinant of (4.18) leads to the require- ment (ad - bc)2 > o (4.19) Equation (4.19) says that Q, given by (4.18) will be negative definite for any values of the coefficients in (4.16) except those which make Q equal zero. However, so far, there is no restriction on the domain of x1 and x2 . Being assured that the merit function is negative definite, it is necessary to impose conditions that insure that at least a local optimum occurs on the Open in- tervals defined by x1 and x2 . Rolle's theorem hi one dimension states that a continuous function on a closed, finite interval which has a continuous first derivative will have its first derivative zero some place on the interior of the interval provided the values of the function at each end are equal to each other. By extending this to a plane and requiring all four of the combinations of bound- ary points to be equal,an interior optimum of M is assured. The requirements for an interior maximum of the merit function when made up of two linear factors Of the form given by (4.16) on the intervals 0 §_x1 §_l , 0 §_x2 :_l is then (1) ac < 0 (2) bd < 0 (3) (4) ad - bc + O 70 M(0,0) = M(l,0) = M(0,l) = M(1,1) = k If equation (4.2) is expanded in general for N merit factors with I variables, VXI VX2 VX where F (4.15) takes the general form r R E1 n 82 log Mk E 32 log M5 . 3' n 32 log M}; 6x?” n=1 n 3le n=1 n axlax2 n=1 n 8x131:I N N N k k E ”n 32 log M1,: 2 n“ 32 log Mn ..... Z ”n 32 log Mn “12‘” n=1 5X25X1 n=1 3x; n=1 3x23):I g 32 108 Mk 0000000000000000000000 g 32 log M: 6xk+l nn 3x 8x _ nn T I n=1 I 1 n-l I J J . k+1 k (4.20) N 3 log Mn 3 log Mn - Z nn 3x ' 5x. n=1 1 Equation (4.20) assumes that an interior optimum exists. An I-dimensional form of Rolle's theorem will be required to insure an interior optimum. Then, if the square matrix of (4.20) is negative definite, the optimum is at least a local maximum. It is very time consuming to apply the requirements of (4.20) to candidate functions but in general it appears that at least the following requirements must be met by either the individual merit factors or by the final merit function. 71 1. Each merit factor must be continuous, single-valued, and have continuous first derivatives. 2. Each merit factor must be dimensionless and be normalized to lie between zero and one. 3. If each merit factor is a linear combination of the varia- bles, then each variable must appear in at least two different fac- tors. Further if one appearance is in an increasing function of that variable (positive sensitivity coefficient) then the other must be in a decreasing function. 4. The weight exponents, wn , must be non-negative to pre- serve the zero to unity criterion. Each merit factor is to be a relationship describing one aspect of the design performance or requirement. It is also assumed that the different merit factors describe conflicting requirements on the same set of independent variables. Thus it is expected that each independent variable will not only appear in both increasing and decreasing functions, but that the resulting merit function will be near zero near the boundaries of the domain of interest for each variable. This does not guarantee that an interior optimum will exist but makes it plausible that one should exist. Since any merit factor going to zero yields a zero value for the merit function, considerable care must be exercised in choosing each merit factor. In effect, this says that no matter how well the other performance requirements are met, this particular one renders the design completely unacceptable. 72 Five representative functions which are suitable as merit factors are shown in Figure 4.1. For convenience, all are shown as monotonic decreasing functions of a single variable. Also shown is the effect of varying wn on each of the functions. It is apparent in all cases that wn < l keeps the function near unity over a larger portion of its domain than does wn = 1, thus de-emphasizing that factor. It should be noted that only the linear function, Figure 4.1(a), has both end points fixed (with the possible exception when wn = 0, in which case the right hand value will be defined as unity). Since the shape of each merit factor depends on the value judgement of the designer, it is not possible to give any rules for selecting each factor nor where to set the end points. This can only come from a thorough analysis of each application. In general though, lacking any better criteria, a good start might be with an approximate linear curve, such as shown in Figure 4.1(e). The weight exponent can then be used to alter the shape of the curve without changing the end points too drastically. It should be emphasized that the merit factors are measures of the design performance and requirements and as such will many times be expressed in terms of functions of the independent variables. This will be more apparent in the next section when specific appli- cation will be made to the squeeze-film journal bearing problem. 73 1 “M‘0 1 “m=o \ \ \ \ \ \ “\ Vinc'l \ \~k \\ \N \ \\ w"(1 Wu “ \ \ .1. \ \‘ u... \ \ w _1\ m:- \ ‘~ _- V~\ n' \\ \ Wn'l ~ MM?) \~ ‘\\\\‘ \ Wh>l\-\ .\ \‘I o \ ~— — 0 §‘”— — - 0 .2 .4 .e .e 10' o 2 3 4 :57 X I x S ' > (0) Mn 1 x O > o \\ \‘ ‘ o .5 1 L5 2 25 o .5 1 15 2 25 x x (0) Mn = e" no (0) Mn'C-x. no Wn=0 1 fi‘\ \\ \ \ 2 \\ \Wn<10-S a = 6837x10'3 a = o 2 ’ 9 1 o 0 The constants selected gave M2 = 0.98 at 0h2 = - 100 microinch and M2 = 0.5 at 6h2 = - 150 microinches (M2 = 0.024 at 0h2 = - 200 microinches). Also M2 = l for 6h2 3_- 82.8 microinches It can be shown from the small parameter equation that M2 is a monotonic increasing function of A and Ghlr and a monotonic 8M2 decreasing function of ho . Near the optimum §2_' is negative. L 79 The third merit factor was determined from the general shape of the package desired. If the length to diameter ratio becomes either too small or too large the instrument becomes an undesirable shape. It was decided that 0.7 §_ ZL §_1.S was the most desirable range. At the low end of the range the merit factor was chosen of the approximate linear form M3 = 1 - e-A3 (4.27) where 2 A3 = aZZL + alzL + a0 with a2 = 5.176 , a1 = -3.420 , a0 = 0 . The constants selected gave M3 = .02 at ZL = 0.7 and M3 = 0.954 at ZL = 1. At the high end of the range M3 = l for ZL < 1.3 decreasing to zero at Z = 1.5. See Figure 4.2(c). L The merit function, M3 , is a monotonic increasing function of ZL for ZL < 1 and a monotonic decreasing function for ZL > 1.3. There is a discontinuity in the first derivative at Z = 1.3, L however this was not removed as subsequent Optimization showed that 2: was always less than 1.3. The fourth item, minimum power, is related to the drive ampli- tude, 6h1r , in that limited, unpublished experiments by W. G. Holliday[25] at Lear Siegler, Incorporated showed that the rms value 80 of the excursion of the transducer could be directly correlated to the input power. The fourth merit factor was taken as a simple linear relation with M“ = 1 at dhlr = 0 and Mg = 0 at 6h1r = 20 microinch rms. See Figure 4.2(d). Ghlr M“ = 1 ' 20 (4.27a) The total merit function for the design was then taken as the product of the four individual merit factors in the form W W W W .. 1 2 3 '+ M - M1 M2 M3 M“ (4.28) where M1 = M1010! C(ho: 61111‘» A: ZL: W, V: pa)) M2 " M2(6112 (ho: 5h1r, A: ZL’ W, V, 133)) M3 ' M3 (21) M4 = M4 (51hr) Each of the merit functions is continuous, single-valued in the independent variables and, except at the high end of M3, all the merit factors have continuous first derivatives on the interval of interest. Although the merit factors are measures of different effects they are by no means independent in that they are various functions I of the independent variables of the squeeze-film problem. 81 Table 4.1 summarizes the type of functional relationship which exists between each merit function and the independent variables. In most cases the relationship was derived from the small parameter equation for the gas film. Although not linear functions, as dis- cussed earlier, each of the first three variables appears as an in- creasing function in at least one of the factors and as a decreasing function in one of the other factors. The exceptional case is the TABLE 4.1 FUNCTIONAL RELATIONSHIP BETWEEN THE MERIT FACTORS AND THE INDEPENDENT VARIABLES. M1 M2 M3 ML, ho Increasing Decreasing Not Applicable Not Applicable ZL Decreasing* Decreasing* Increasing* Not Applicable Ghlr Increasing* Increasing Not Applicable Decreasing A Increasing* Increasing Not Applicable Not Applicable * Local condition near the optimum. non-uniformity factor, A. The optimization, to be treated in the next section, was not, however, carried out with respect to A. The fact that M1 and M2 are both increasing functions of A results from the choice made for the value of A . There does not appear to be any significance to the fact that M1, for instance, is an increasing function of ho , 6h1r and A and a decreasing function of ZL . 82 4.3 OPTIMIZATION OF THE MERIT FUNCTION With the merit function as defined in the previous section, the ability to use 5h1r as a normalizing parameter is gone and all computations will be done on a dimensioned basis. A convenient di- mension for film thickness is the microinch which will be used ex— clusively for Ghlr , Ghz , and ho . Preliminary runs of the optimization program written to solve the merit problem showed that, as with clearance, the Optimum value of A is - However the value for A which has been physical- :Lh 1y demonstrated and which is closest to this value is A equal to -1. To reduce the number of variables, this value of A was used throughout the remainder of the work in preference to other less effective values of A which also have been demonstrated. This reduced the problem to the three independent variables (ho , Ghlr , and ZL) since again W, V, and p3 were assumed fixed in the demon- stration problem. Although introduction of the weighting exponents wn in (4.1) did, in effect bring in four new independent variables they generally were assumed equal to unity, although some limited experimentation was done varying them one at a time. It can be shown that the merit function, M , is essentially zero at each boundary of the independent variables thus insuring an in- terior optimum. For instance, Mn is zero when dhlr = 20 micro- inches , M2 is zero when dhlr = 0 microinches ; M1 is zero when h = 50 microinches; M2 is essentially zero when h0 = 1000 micro- 0 inches. 83 Figure 4.3 shows the merit contours computed using the aug- mented, small parameter gas film equation with ZL = Z: . Also shown on the figure are the numerically computed optima. The optimum merit value of 0.0860 shown on Figure 4.2 may seem rather small compared to the maximum possible of unity. The main contribution to the small merit value comes from M1 . This function was selected to have a value of unity for ho of 1000 microinches and c greater than 250 microinches. The gas film equation only al- lows h; = 175.6 microinches and c = 49.3 microinches, making M1 = 0.149. The other large contributor is Mg which linearly decreases from unity to zero as 6h1r goes from 0 to 20 microinches, thus at 6h? = 7.7 microinches, Mg = 0.615. The point here is that the numerical value of the merit function evaluated at the optimum is of no consequence. The values of the parameters which gave the optimum are important. The optimization routine was also performed using the small parameter equation to describe the gas film. The results are shown in Table 4.2 together with the results of the computation with the augmented, small-parameter equation. It is evident from Table 4.2 that, except for the optimum drive amplitude, the answers from the two different equations agree quite well. Since the small parameter equation required much less computer time than the augmented equation it was used to generate the start- ing vector for subsequent optimization procedure using the augmented, small-parameter equation. 84 :3 woe—.00 Houofidwwmo H Hdam . vopcofiuam :3 3:: . m m flue:— onu no.“ “$.35 :35 mo mason—So mé ouswfi 5.5.2.- 0‘. 08 SN 08 on. 09 on _ .2. , _ fl 4 _ .an wwo. u 3 . . .3. n3. .3 13 58.8.8.8 8.8. 5 no. 333 .8. 6.... n u oo. o. .‘ \\\\\\\\.~\\ \: Lao; u tum .22... cos.» n... ._..w \\ .22.: .h. u o: x g s S h\\\\\\\. \\k\\\ 9.. . w\\ .\\\\\\ MW“ \\ \ \ \\m\.\. g\ n\\“ \\ m A \Ms/gngm... \I\\\\ ON m, g 85 TABLE 4.2 OPTIMIZING PARAMETERS FOR THE MERIT FUNCTION. AUGMENTED, SMALL SMALL-PARAMETER PARAMETER EQUATION EQUATION h; microinches 167.5 175.6 ZE 1.117 1.097 Ghlr microinches 9.527 7.700 M* 0.07122 0.0860 c microinches 44.573 49.258 Ghz microinches —102.92 -110.16 4.4 SENSITIVITY TO PARAMETER CHANGES The first derivatives of the merit function are zero at the optimum, thus they cannot be used as sensitivity coefficients. But again a quadric surface can be used to represent the response sur- face. The second derivatives required for the matrix of the quadra- tic form can be found as the slopes of the first derivatives near the optimum, where the first derivatives are shown on Figure 4.4, calculated from the augmented small-parameter equation. Thus in analogy to (3.30) P 1.392 0.1055 Q = - 0.1058 0.3352 -O.S702 -0.0456 L. -0.5710 -0.0460 0.2898 I! 1.4 0.1 L-O. 0.1 -O.57 0.34 O 57 0 0.29 (4.29) 86 5:0wumsco nouoewummuHHmEm .woucoemsmv EdEwumo may Mme: :ofiuocsm payee may mo mo>wum>wuov umnwm v.v ouswwm laid . on“ this! 23“ on — _ oo— _ on a. — _. 0Q— — 8 o p _ no.— — no. a. .u. .3. .0. Ar NV 8.: ".1 \\1 \ulnilllv O \ O o....w.:l:w.% / .\ I 0 . 00. n. .sv .0. A .s. :o _r [Iii/1 .or 0 Int! 0 o .N .o / q .0. Au. .5. .6. V\\\MA. or _- Jill a? \ 0 i0 / 0030mm .. / 87 where _ 32M qi) axiaxj l J with _ Aho 02L A(5h1r) x " ho ' zL 6h1r The linear transformation F‘ H 1 0.072 -0.41 9 = 0 1 0 i 0 0 l L— .J (4.30) applied to (4.29) diagonalizes Q . The deviation of the merit value from the optimum merit value may then be written M - M* = 9 0 -0.333 0 L 0 0 -0.058 (4.31) where Aho 0.072 AZL 0.41 AGh1r ho 2L ' dhlr (4.32) 88 Equation (4.32) shows strong interaction between ho and Ohlr. The ridge of one-dimensional optima which divides Figure 4.3 is also evidence of ho - dhlr interaction. The presence of the ho - dhlr interaction was also evident during the numerical Optimization proce- dure. A steepest ascent method was used even though it was knwon that this method is inefficient for ridge systems[30]. The procedure did "zig-zag" across the ridge, eventually finding the optimum; but not without taking a great number of steps. Making the transforma- tion of variables (4.30) would improve the efficiency of the optimi- zation procedure; however, the transformation is not known until after the optimization has been performed. It is evident from (4.29) that the Optimum is relatively insensitive to the value of ZL . Since the merit function is a product of merit factors, it is possible to determine sensitivity coefficients, in the sense of the classical definition, for each of the merit factors. Using (4.3), write the sensitivity coefficient in the form x. M _ 1 3M _ a . _ Sx - 11 x. - xi 3x. log M (1 - l, 2, I) 1 1 1 (4.33) which is also N N x. 8M N M _ 3 _ _1_n _ Mn Sx - X1 z wn 3x log Mn - Z wn M 3x. ' X Wu Sx. 1 n-l 1 n=1 n 1 n=1 1 (4.34) At the Optimum, (4.34) is zero, for xi equal to h0 , ZL , or dhlr as shown in the second column of Table 4.3. (The small residue is due to the inability of the computer to find the exact 89 (WEIGHT EXPONENTS EQUAL TO UNITY) TABLE 4.3 SENSITIVITY COEFFICIENTS OF THE MERIT FUNCTION SMALL PARAMETER EQUATION xi 51‘. 8’2? S??? 851? Si“: 1 1 1 1 1 ho -0.0067173 1.0670 -1.0737 0 0 ZL 0.0023194 -0.0098701 -0.033910 0.045700 0 dhlr —0.00034617 0.19038 0.71582 0 -0.90967 A 0.44930 0.092708 0.35659 0 0 W -0.46322 -0.10541 —0.35782 0 0 V 0.30909 0.070334 0.23875 0 0 AUGMENTED,SMALL-PARAMETER EQUATION ho -.0061109 1.14999 1.15589 0 0 ZL -0.0093814 -0.017460 -0.085965 0.094039 0 5h1r -0.0057128 0.089228 0.53106 0 -0.62602 A 0.28457 0.038712 0.24586 0 0 W -0.31861 -0.053790 -0.26482 0 0 V 0.21176 0.035746 0.17600 0 0 90 Optimum). However, Table 4.3 shows how the individual merit factors are affected by the parameter changes. Thus a unit change in hO using the augmented equation, for instance, changes M1 and M2 by 1.15 units; whereas a unit change in Z changes M1 by 0.017 units, L M2 by 0.086 units, and M3 by 0.094 units. Thus, it is evident that ZL does not have as great an influence as ho or dhlr . Also shown in Table 4.3, are sensitivity coefficients with re- spect to the non-uniformity factor, A , the load, W , and the volume, V. These, of course, are not zero. Table 4.3 also gives the sensi- tivity coefficients of the merit factors for these parameters. By chain-rule differentiation 5141.213...st isfi.h°3bilahfi x. M 3x. M1 3c ax; c M1 3h 8x h 1 1 1 h 1 O o o c _ M1 C M] ho - Sc Sx. + Sh sx. h 1 o c 1 O (4.35) 8M2 = 5112 3M2 3(5h2) Xi = 8M2 Séhz Xi M2 3 (5112) 3X1 (5112 5112 Xi (4.36) This further reduces the sensitivity Of each factor to the product of parameter sensitivities. The merit factor M1 is an explicit func- tion of b0 and c ; M2 is an explicit function of 6h2 , thus three of the sensitivity coefficients in (4.35) and (4.36) are di- rectly obtainable. The factors 8: in (4.35) and Sihz in (4.36) i i 91 are obtainable from the clearance equation (3.11), either analytical- ly, using the small parameter equation, or numerically, using the augmented, small-parameter equation. The merit factors, M3 and Mg , are explicit functions Of Z L and dhlr, respectively, thus the sensitivity coefficients are obtained by direct differentiation. Typical values of the sensitivity coefficients for the small parameter equation at the optimum are 521 = 0.0002 c2 M1 e'(°°1°) = .04568 (4.37) h 0 5° = l- h + 3 6h = - 3.170 (4.38) h c o 2 0 712.5 M1 h M1 0 s = 4.39 ho (h - 50)2 ( ) O C 5:0 = 1 (4.40) 0 These give sfil = (.04568)(-3.l70) + 1.211 = 1.066 (4.41) O The second term of (4.41) shows that the explicit expression for ho in M1 has nearly ten times greater influence than the implicit expression for ho . Equations (4.37) and (4.39) are constants once the expression for M1 is selected. The parameter sensitivity 92 . . c . . coeff1c1ent Sx 1s different for each of the variables. This is i . M . also true 1n (4.36) where 86h 15 a constant for a given M2 but 2 the parameter sensitivity coefficient SCShz changes. Typical ex- 1 pressions for some of the simpler parameter sensitivity coefficients from the small parameter equation, in addition to (4.38) and (4.40), are c _ _ 1 F(A) ‘ $51111. C (T 0111]: + 2 5112) a (4.42) 5° = _ (4.43) _ 2 5 — -3— (4.44) 3112 86h1r > (4.45) 6h2 _ Sw - 1 3112 Sv =-£ 3 J Thus some insight into the relative importance of the various parameters can be obtained. Some limited experimentation was done to determine the effects of weighting exponent changes on the optimizing parameters. The results shown in Table 4.4, were obtained using the augmented, 93 TABLE 4.4 EFFECT OF WEIGHT EXPONENTS ON THE OPTIMIZING PARAMETERS. W1 w2 w3 wg M* h; Shir 2: 1 1 1 1 0.0860 175.6 7.700 1.097 0.5 1 1 1 0.2458 134.5 4.0 1.112 1 0.5 1 1 0.0903 182.1 7.328 1.101 1 1 0.5 1 0.0860 179.0 8.044 1.081 1 1 1 0.5 0.1202 216.2 11.96 1.092 small-parameter equation for the gas film. NO attempt was made to compute the partial derivatives with respect to the weight exponents. Rather, a significant change was made in one exponent at a time. It is probable in an application that changes would be Of a similar order of magnitude. alized through the appropriate merit factor. deemphasized the effect of ho. The effect of each exponent change can be visu- For instance "1 = .5 As a consequence the optimizing parameters shift to a smaller value of h; ; and with the smaller hO’ less drive is required to support the load, so Ohtr decreases. Another aspect of sensitivity, how the optimizing parameters change with load changes, is deferred to the next chapter. There a merit function weighted by the expected load to be applied will be developed. CHAPTER 5 WEIGHTED FIGURE-OF-MERIT 5.1 WEIGHTED MERIT FUNCTION The merit function developed in the previous chapter is capable Of further extension which may be of great value when one of the parameters, previously assumed constant, is permitted to vary. In particular, the load to be carried by the squeeze-film journal bearing has been treated as a constant. However, in many aerospace design problems, as in the demonstration problem, this load is the maximum at which the instrument is expected to perform. The actual load applied at any moment can be described by a probability function. A common practice is to design for the maximum load with the philosophy, "if it works at the maximum, it will surely work at all lesser loads." The argument with this philosophy is that it tends to create a heavy, bulky, more costly design than is necessary in order to have the reserve for the occasional maximum load. This is especially true if the merit function has been optimized for the maximum load. What is proposed is a figure-of—merit function, weighted in some suitable fashion by a probability function which describes the load. Such a weighted merit function may be created as follows: 94 95 l. Optimize the merit function of a number of competing designs by evaluating at different load values up to the maximum to be ex- pected. 2. Using the optimizing parameters found in step 1, evaluate each competing design at a number of values over the range of expected loads. Call these values Md,s where d is the fraction of the maximum load at which the design was optimized and s is the fractiOh of the maximum load at which the design is being evaluated. 3. Evaluate the integral Md = Md,s 0 ds (5.1) where 0 is the weighting function to be selected. It is then a simple matter to select the design having the maximum M6 value. This procedure should be approached with some caution, though, since it is possible to select a weight function which will so minimize the effect of large load values, that the design with the optimum ad will not function at the maximum load. This could be treated by constraining the optimum design in such a fashion to insure an acceptable level of performance at the maximum load. In any case, it is desirable to check at the maximum load to insure acceptable, 96 though no larger Optimum, performance at the maximum load. Although application of the weighted merit function is being made to the load, it should be remembered that other parameters could be used as well. 5.2 APPLICATION TO SQUEEZE—FILM BEARING The procedure described in the preceding section was applied to the squeeze—film journal problem, with Wfiax = .066 pounds and A = -1. Unless otherwise noted the weighting exponents, wn, in the merit function were all set equal to unity. The results of the first step, the determination of the Opti- mizing parameters for a number Of competing designs at loads, d, less than the maximum, is shown in Figure 5.1. It is immediately apparent that Z is not an important parameter and in the final program to L optimize M', Z was fixed at its first computed value and only ho L and 6h1 were used in the optimization program. r Figure 5.2 shows how the merit value, clearance and radial displacement vary when the maximum load is applied to each Of the competing designs. Evidently a design Optimized for a load of less than .3 of maximum cannot be used as the merit is rapidly approaching zero. This appears to be mainly due to the radial displacement approaching the undesirable region of 200 microinches. The path followed by the optimizing parameters is shown in Figure 5.3 (a) in the Ohlr-ho plane superimposed on contours Of constant merit and in Figure 5.3 (b) superimposed on constant clearance 97 .. \ J... ' v - .0025 IN‘ ‘— w.-w.-w.-w.-1 \Y as: .0 /“‘-\ ....... / \ / \\ 4‘ ‘ .1010” 2 50 \ 200 \‘§\ 0 .2 .4 .6 .0 LO 060 Figure 5.1 Optimizing parameters as functions of load 98 200 \ b .8 .2 100 GO .5... / \ 50 04.1 M4,! ' / w - .000 Les v - .0925 m' '.I":I"I'.I1 O .2 .4 .6 .0 L0 C Figure 5.2 Performance of designs optimized at lesser loads when loaded at maximum load 99 " ’ 7// J /’ // ,1/ //fl / / '3 in! O '93 ' 1 J .00 . ,1 /7 Nu mMMh‘W / ///// / zorrntnmnoaozxn -u °mo zoo zoo Mme.» 300 :50 400 (a) Merit contours " ,1 1 I to ,7” C' ‘5 '0 ‘5 communist :"————'——:;;:;::;' l I-—— ‘3 ‘0 H o 4...," Ad ‘.l. . r— . if: . ‘ muscronv or orrmuus M ”u 3:1 New!!!“ 4 : ‘000 zoo 250 300 350 400 “0 «launch (b) Clearance contours Figure 5.3 Trajectory Of optimums on contour maps 100 contours in the same plane. Figure 5.4 illustrates the results of the second step, the evaluation of each competing design over the load range. Again it can be seen that a design optimized at a design load of less than .3 is near failure at the maximum load. On the other hand, the merit rating of a design optimized at the maximum load changes very little over the load range. It is clear that if the design value is very small (d + O) the area under the curve goes to zero. It is not evident that there is a maximum area under one of the design curves, other than the one for d = l. The third step in the Optimization of the weighted merit function is to evaluate the integral given in (5.1). For this problem the weighting function was chosen to be the normal (Gaussian) distribution, ¢ = e (5.2) where o is the standard deviation of the load-frequency distribution. There is equal probability that a given load will be applied in any radial direction leading to the maximum load being i 30 g in the coordinate plane originally defined. Thus the mean value of the load-frequency distribution was taken as zero. 101 9383. gumo new 33 mo cowpogm we mead; «who: v.m 6.53m u. p I o I / I I T13 1 , a a... a ’8. w 8. / . . a 2 w . .. m— ». E. c. . .6: . n. ” ~ 2. - . / . . . _ ON F . . - . w 3. - - ’ Ill 8. 102 Numerical integration of (5.1) was performed for different values of sigma, with the results shown in Figure 5.5. The value of O = 0 corresponds to a weight of one, or all ordinates equally weighted. This has its maximum at d = .6. As sigma increases, the effect is to de-emphasize the larger load values and the maximum shifts to smaller d values. A sigma value Of three indicates that the maximum load would be applied approximately 0.3 percent of the time, thus strongly de-emphasizing the importance of the maximum load; in fact, any larger value for sigma says that the probability of this maximum load being applied is vanishingly small. Considering a reasonable value of sigma to be two, i.e., there is a 5 percent probability of the maximum load being applied at any instant, the Optimum weighted merit value corresponds to a design at d = 0.45. It is interesting in Table 5.1 to compare the Optimum parameters for this design with those for the design at d a 1. TABLE 5.1 COMPARISON BETWEEN DESIGN OPTIMIZED AT MAXIMUM LOAD (d = 1.0) AND DESIGN OPTIMIZED AT THE WEIGHTED OPTIMUM (d = 0.45) d 1.0 0.45 h: 175.6 microinches 230 microinches GhIr 7.7 microinches 9.1 microinches z: 1.094 1.098 M8,d 0.0860 0.1166 id 0.0232 0.0269 103 ca,-ufipnw5-¢v‘i1 W t .066 L88. vow .0925 m3 \10.0 \0-1 J ‘J\\\ // J J) / 5.110' \’0.2 2 \ 0.3 \; £704 ', O 0 2 4 .5 8 IO Figure 5.5 Weighted merit as function of design 104 Although the improvement in Md is only 16 percent, the improve- ment in the individual M; d is 36 percent. 3 The results, of course, will be different for a different set of merit functions and for a different problem. The squeeze-film journal bearing example has been carried out in considerable detail. The reason has been to illustrate the merit function and the weighted merit function technique on a concrete example. It should be evident that the merit function approach to a problem allows the selection of an optimum design after a designer analytically Specifies his pre— ferences; and, when applicable, the weighted merit function technique further sharpens the choice of Optimums. 5.3 STABILITY CONSIDERATIONS Since all compressible fluid bearings exhibit potential insta- bilities, this is a matter which must be investigated during the design of any gas bearing. Beck and Strodtman[lé] first investigated the stability of a journal bearing when no flow along the axis is permitted (infinite journal). Nolan[17] enlarged this investigation to include Z axis flow (finite journal). Although neither work is directly applicable except when A = 0 (uniform excursion), Nolan's results can be used to indicate whether stability might be a problem in the present case. Figure 5.6 is a combination of Figures 20 and 21 of Nolan's thesis. A design is considered stable if its parameters (8,62) fall below the appropriate 81 curve. The parameter 8 is the dimensionless "mass" parameter, 105 301.10 cunv: 2...! oonso canvas 2pm 0 .01 .02 .03 .04 .05 8 Figure 5.6 Stability map adapted from reference [17] 106 s = —— (5.3) In the demonstration problem, 8 has a value of .007 for a drive frequency of 80,000 Hertz, Z Of unity, and a nominal clearance L of 176 microinches. At the Optimum of Figure 4.3, h0 = 176 micro- inches, Ohl = 7.7 microinches, ZL = 1.097, A = —l, and the radial r displacement is 111 microinches. This corresponds to £2 = -0.6 and £1 = 0.0436. At 8 = 0.007, 81 = -0.6, the stability boundary of r Figure 5.6 is crossed at E1 approximately 0.3. If all the above parameters were to hold for A = 0 (where e1: 21 ) then, the Optimum r values give a design which is well removed from the stability boundary For values of A other than zero it can be argued that the equivalence between €1r and £1 at the stability boundary can be established by the value Of the static load support at the boundary. The basis for this argument lies in Figure 5.6, on the two curves for £1 = 0.3. One curve is for the infinite journal, the other for a finite journal, yet the stability boundaries practically coincide. A point such as C represents a static load support, W', of 0.27 whereas 0 represents a load support of 0.155. We can then bound the load support represented by the point E by interpreting 81 in two ways (1) as the maximum boundary excursion or (2) as the root-mean- square value of the excursion. Using interpretation (1) the static load support at point E for A = -l is()062 and for interpretation (2) it is 0.016. Similarily point F represents either 0.090 or 0.022. 107 The optima of Figure 4.3 give a load support of 0.018 at £2 = -0.63 and e = 0.0436. The point (0.007, - 0.63) falls well to 1r the left of the worst F (0.016, - 0.6) representing a load support between 0.022 and 0.090. The weighted Optimum design for sigma of two, i.e., for design load of 0.45 maximum load, had h0 = 230 microinches, and dhlr = 9.15 microinches, and gave 6h2 = 158 microinches when loaded at the maximum load equivalent to 52 = -0.69, Elr = 0.040. For the new value of h0 = 230 microinches, B has the new value of 0.0053. The load support required at the maximum load is 0.018 as before. Point G (0.01, -0.69) Of Figure 5.6 represents a load support either 0.024 or 0.11. The design value of (0.0053 - 0.69) is still on the stable side; however, under the worst interpretation of G as representing a load support of 0.024, the design is very near the stability boundary. It would appear that the locations of the stability boundaries for non-uniform excursion should be the subject for future investiga- tion in order to put this matter on a firm basis. CONCLUSIONS AND RECOMMENDATIONS One of the objectives of this thesis was to show that the proper choice at variables would maximize the minimum clearance in the bear- ing under the constraint of fixed load weight and load volume. A second objective was to develop a technique for design optimization. Both objectives have been met. In meeting the primary objectives, a number of other tasks were accomplished. A method of treating non-uniform driver excursion by means of its root-mean-square value and a non-uniformity factor was developed. Although, in this thesis, the non-uniform excursion was treated as a sinusoidal function in the axial direction, the treat- ment is general enough to accommodate other shape functions which may be encountered. The asymptotically-derived partial differential equation de— scribing the squeeze-film bearing was expanded in an asymptotic series of powers of the radial displacement. It was shown that sepa- ration of variables on the resulting series of partial differential equations could be accomplished. It was found that the resulting series solution including the third order term was of sufficient ac- curacy that a slower, alternating-direction, implicit, numerical solution of the squeeze-film differential equation was not required 108 109 for this problem. If desired, further improvement in accuracy is possible by following the pattern shown to be present in the series of partial differential equations, to develop higher order terms. It was found, using only first order terms in the small para- meter equation, that explicit optimization of the minimum clearance could be accomplished. A gradient method was used to optimize the clearance using the augmented, small-parameter equation. The optimum nominal clearance was found to be well within the range of values used in the squeeze-film bearings built to date. The optimum length- diameter ratio was considerably less than has usually been used. The Optimum non—uniformity factor appeared to be constant regardless Of whether the small parameter or the augmented equations were used. A study Of the response surface near the Optimum showed that, of the variables treated, the most important variables are the non- uniformity factor and the dimensionless nominal clearance-drive amplitude ratio. The length-diameter ratio did not appear to be an important variable. By varying the load on the bearing, it was found that the optimum nominal clearance-drive amplitude ratio varied in an especially simple fashion with the change on load. The concept of a figure-of-merit function as introduced in Chapter 4 should be applicable to a wide range of design problems. Its main forte is its ability to treat dissimilar quantities without 110 reducing each to a "cost" function. It was shown that a cost factor, a displacement factor, a length-diameter ratio factor, and a drive amplitude factor could be combined into a merit function. It was then possible to optimize the resulting merit function, which was done numerically by a gradient method. The merit function was fUrther improved by the introduction of the weighted merit function in Chapter 5. The weighted merit function showed that if one Of the constraints, such as load, was described by a probability function, the weighted merit function produced an optimum at less than maximum load. The design Optimized at this weighted optimum had a merit value 36 percent greater than that of the design Optimized at the maximum load. Among the problems left unanswered or at least which need additional work are the following: (1) A more complete set Of rules for the formation of the merit factors is needed. This was explored to some extent in Chapter 4 and some rules were given. (2) The treatment of stability of the journal bearing when the excursion is non-uniform is needed. Some arguments were advanced in Chapter 5 based on previous work which treated uniform excursion only [17]. (3) The effect Of certain manufacturing errors on the optimi- zation of clearance and on the merit function should be treated. Such errors include taper in the nominal clearance along the longi- tudinal axis, out-Of-roundness or taper in the nominal clearance on the angular direction, and non-uniform excursion on the angular direction. LI ST OF REFERENCES LIST OF REFERENCES Gross, W. A., Gas Film Lubrication, John Wiley and Sons, New York, 1962. Hirn, G., "Sur les principaux phénomenés qui présentent les frottementes médiats," Société industrielle de Mulhouse. Bulletin 26, 1854, pp. 188-277. Kingsbury, A., "Experiments with an Air Lubricated Journal," Journal of the American Society of Naval Engineers, Vol. 9, 1897, pp. 267-292. Harrison, W. J., "The Hydrodynamical Theory of Lubrication with Special Reference to Air as a Lubricant," Transactions of the Cambridge Philosophical Society, Vol. 22, 1913, pp. 39-54. Taylor, G. I., and Saffman, P. 6., "Effects Of Compressibility at Low Reynolds Numbers," Journal of the Aeronautical Sciences, Vol. 24, 1957, pp. 553-562. Reiner, M., "Researches on the Physics of Air Viscosity," Technical Report, Technion Research and Development Foundation, Haifa, Israel, 1956. Langlois, W. E., "Isothermal Squeeze Films," Quarterly of Applied Mathematics, Vol. 20, 1962, pp. 131-150. Salbu, E. O. J., "Compressible Squeeze Films and Squeeze Bearings," Journal of Basic Engineering, Transactions ASME, Series 0, Vol. 86, No. 2, June, 1964, pp. 355-364. Malanoski, S. B. and Pan, C. H. T., discussion of E. O. J. Salbu's paper ”Compressible Squeeze Films and Squeeze Bearings," Journal of Basic Engineering, Transactions ASME, Series D, Vol. 86, No. 2, June, 1964, pp. 364-366. Erratum, Journal of Basic Engineering, Transactions ASME, Series D, Vol. 86, No. 3, Sept., 1964, p. 638. Pan, C. H. T., "On Asymptotic Analysis Of Gaseous Squeeze-Film Bearings," Journal of Lubrication Technology, Transactions ASME, Series F, Vol. 89, No. 3, Julx,1967, pp. 245-253. 111 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 112 Pan, C. H. T., Malanoski, S. B., Broussard, P. H., Jr., and Burch, J. L., "Theory and Experiments Of Squeeze-Film Bearing, Part 1, Cylindrical Journal Bearings," Journal of Basic Engineering, Transactions ASME, Series 0, Vol. 88, NO. 1, March, 1966, pp. 191—198. Beck, J. V., and Strodtman, C. L., "Load Support of the Squeeze— Film Journal Bearing of Finite Length," Journal of Lubrication Technology, Transactions ASME, Series F, Vol. 90, No. 1, Jan., 1968, pp. 157-161. Chiang, T., Malanoski, S. B., Pan, C. H. T., "Spherical Squeeze- Film Hybrid Bearing with Small Steady-State Radial Displacement," Journal of Lubrication Technology, Transactions ASME, Series F, Vol. 89, No. 3, July, 1967, pp. 254-262. Beck, J. V. and Strodtman, C. L., "Load Support of Spherical Squeeze-Film Gas Bearings," Paper No. 68-LubS-3. Presented at the Lubrication Symposium, Las Vegas, Nev., June l7-20, 1968, of The American Society of Mechanical Engineers. Pan, C. H. T., and Broussard, P. H., Jr., "Squeeze-Film Lubrication," Paper No. 12, Gas Bearing Symposium on Design Methods and Applications, University of Southampton, Department of Mechanical Engineering, April, 1967. Beck, J. V. and Strodtman, C. L., "Stability of a Squeeze-Film Journal Bearing," Journal of Lubrication Technology, Transactions ASME, Series F, Vol. 89, No. 3, July, 1967, pp. 369-374. Nolan, J. E., Analytical Investigation of Stability of Squeeze- Film Journal Bearings, Ph.D. thesis, Michigan State University, Department of Mechanical Engineering, 1966. Elrod, H. 6., Jr., "A Differential Equation for Dynamic Operation of Squeeze-Film Bearings," Contribution A, Gas Bearing Symposium on Design Methods and Applications, Contributions and Discussions, University of Southampton, Department of Mechanical Engineering, April, 1967. Pan, C. H. T., and Chiang, T., "Dynamic Behaviour of Spherical Squeeze-Film Hybrid Bearing," Paper NO. 68-LubS—37. Presented at the Lubrication Symposium, Las Vegas, Nev., June 17-20, 1968, of The American Society of Mechanical Engineers. DiPrima, R. C., "Asymptotic Methods for an Infinitely Long Slider Squeeze-Film Bearing," Journal of Lubrication Technology, Transactions ASME, Series F, Vol. 90, NO. 1, Jan., 1968, pp. 173-183. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 113 Pan, C. H. T., and Chiang, T., "On Error Torques of Squeeze-Film Cylindrical Journal Bearings, Journal of Lubrication Technology, Transactions ASME, Series F, Vol. 90, No. 1, Jan., 1968, pp. 191-198. Beck, J. V., Holliday, W. G., Strodtman, C. L., "Experiment and Analysis of a Flat Disk Squeeze-Film Bearing Including the Effects of Supported Mass Motion," Paper No. 68-LubS-35. Presented to the Lubrication Symposium, Las Vegas, Nev., June 17-20, 1968, of The American Society of Mechanical Engineers. Sage, Andrew P., Optimum Systems Control, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1968. Starr, Martin Kenneth, Product Design and Decision Theory, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1968. Holliday, William G., Personal Communication, 1967. Kuo, Benjamin C., Automatic Control Systems, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1962. Box, G. E. P., "The Exploration and Exploitation of Response Surfaces: Some General Considerations and Examples," Biometrics, Vol. 10, 1954, pp. 16-60. Von Neumann, John and Morgenstern, Oskar, Theory of Games and Economic Behaviour, Princeton University Press, 1953. Luce, R. Duncan and Raiffa, Howard, Games and Decisions, John Wiley and Sons, 1957. Wilde, Douglass J. and Beightler, Charles S., Foundations of Optimization, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1967. APPENDIX A APPENDIX A AUGMENTED, SMALL-PARAMETER SOLUTION OF THE SQUEEZE-FILM EQUATION A.1 SERIES EXPANSION The squeeze-film equation, for the journal bearing of finite length, derived on the basis of infinite squeeze number is 0" Fun 011 a HOT 8H 53[§m'T55]*5z[232'Tsz] ' 0 (M) with the boundary conditions 3 'T(:Z ,0) = 11. (A.2) L H +Z ‘ L 91mm) 332.“) 00 — 30 0 (A.3) For the case considered in this thesis, where only displacement in the radial direction is permitted, H = 1 - 52 cos 0 - Elfl sin t . (A.4) Equation (A.1) becomes a 1-82 cos 0 3T . a 1-52 cos 0 8T _ 5‘61: 2 80'T5251“9]+az[_2 az '0 (A.5) 114 115 and the boundary conditions (A.2) become 2 3 2 2 T(:ZL,0) = (1-82 cos 0) + §'f1 (:ZL )81 (A.6) An approximation to T may be made by expanding T in a power series of 62, the eccentricity of the bearing, T = Z 52 Tn (A.7) n=o Putting (A.7) into (A.5) yields m +1 ) aZTn +1 3Tn +1 n n _ n - _ n 2 [:(52'62 cos 6 862 £2 Sin 0 80 2 62 cos 0 TH n=o + aZTn + (52-63 1 cos 0) -ng- ' 0 (A-3) Substitution of (A.7) into (A.6) gives the boundary conditions E E“ T +Z = l-e cos 0 2 + §,f2€2 (A 9) = 2 n ‘ L 2 2 1 ° where f = f1(iZL) The derivative boundary conditions (A.3) take the form CD 3 on n _ 3 on n _ ‘3'- ; CZ Tn(Z,O) E Z £2 Tn(z,fl) - 0 (A°10) 116 Expanding (A.8), (A.9) and (A.10) into the indicated series and collecting coefficients of like powers of 82 gives the series of partial differential equations and boundary conditions, n = 0 aZTO aZTO + = 0 A.11 302 322 C ) T (12 = 1 + 32 £262 (A.12) o L 2 1 3To(Z,O) aTo(Z,n) —3—0- = 3'5- : 0 (A.13) n = l 32To 8T0 02T 02T - cos 0 2 - sin 6 ——- - 2 T cos 6 + -——J-+ -——J- 30 o 302 322 aZT (A.14) - cos 0 f = 32 T1(:ZL) = - 2 cos 6 (A.15) EIIIZ»0) = 31112,") = 0 (A.16) 80 80 n = 2 32T1 3T1 32T2 82T2 -050 -sin —-2Tcose+ +— C 802 e -39 1 802 322 (A.17) 32T1 - cos 0 0 022 117 T2(:ZL) = c0520 = %~(l + cos 20) (A.18) 0T2(Z,O) 3T2(Z,fl) -50- - —5§— - 0 (A.19) n = 3 02T2 , 8T2 32T3 02T3 - cos 0 - Sin 0 -——-- 2 T2 cos 0 + + 392 30 382 322 (A.20) 2T - cos 0 = 322 T3(1ZL) = 0 (A.21) 8T3(Z,0) 3T3(Z,w) ae _ 86 - 0 (A.22) The same pattern that is evident in (A.14), (A.17), (A.20) continues for higher values Of n , and all boundary values are zero. A.2 SOLUTION FOR To AND T1 The solution of (A.11) is T = 1 . §.f2€2 (4.23) O 2 1 Putting this value of To into (A.14) gives ale 82T1 + 802 az2 - 2 TO cos 0 = 0 (A.24) 118 The variables may be separated in (A.24) by assuming a solution of the form T1 = h1(Z) cos 0 (A.25) giving h1 - h = 2 T (A.26) h1 = - 2 (A.27) :ZL The solution of (A.26) is cosh Z . (A.28) 1'11 = - 2 [To + (1-T0)——COS11 2L] giving T = - 2 cos 0 T + (l-T SEEELJE— (A 29) 1 o o cosh ZL ' A.3 SOLUTION FOR T2 Putting (A.25) for T1 into (A.17) gives the partial differ- ential equation for T2 , 2 2 2_IZ.+ 2.22.- (h + h") c0520 + hl sinze = 0 (A.30) 302 azZ 1 1 119 Assume a product solution of (A.30) of the form T2 = gO(Z) + g1(Z) cos 0 + g2(Z) cos 20 Putting (A.31) into (A.30) gives H H n hl H " h1 go - —3- + gl - g1 cos 0 + g2 - 4g2 - h1 - —5- co with the boundary conditions from (A.18) g1(:ZL) - 0 82(iZL) - % Since the terms in 1, cos 0, cos 20, --- cos n0 each coefficient of (A.32) must be zero thus leading ential equations '1‘g1 = O g3 - 4g2 - h1 - 2%. = 0 Equation (A.36) is 441402233, . (4.31) s 20 = 0 (A.32) (A.33) (4.34) (A.35) are orthogonal, to the differ- (A.36) (A.37) (A.38) (A.39) 120 which can be integrated twice to get cosh 2 go = - (1 - To) cosh Z + C1 Z + C2 L The boundary values (A.33) require that C1 = 0 and c2 = g- giving the solution go = - (I - To) EEEE—EZI+ 2' Equation (A.37) has the solution g1 = CleZ + Cze"Z therefore 81 = 0 Equation (A.38) can be written g3 - 4g2 = - 2 TO - 3(1 - To) cosh Z cosh Z L (A.40) (A.41) (A.42) (A.43) (A.44) 121 The solution of the homogenous part of (A.44) is -2Z g2H = cleZZ + C26 (A.45) A particular solution of (A.44) is T _ cosh Z o g2p - (I To) cosh ZL + -2' (A'46) The general solution with consideration of the boundary values (A.35) is then T _ cosh Z cosh 22 o g2 - (1 To) (cosh ZL - 2 cosh 22L)‘+'7T (A'47) The solution of the partial differential equation (A.30) for T2 is then _ §__ _ _ cosh Z T2 - 2 To (1 To) cosh ZL (A.48) cosh Z cosh 22 To + (1 - To) (cosh ZL - 2 cosh 2ZL )+'72 cos 26 It was hoped that this additional term would be enough to give answers of acceptable accuracy for large values of £2 ; however comparison to the numerical solution, though showing a significant improvement due to this term, was still not good enough. 122 A.4 SOLUTION FOR T3 Writing T2 = go(Z) + 82(2) cos 26 then (A.20) can be put in the form 32T3 82T3 +— 802 322 H + (Zgz - E3-- 2go - g3) cos 0 - gg-cos 30 = 0 2 2 (A.49) The variables in (A.49) may be separated by assuming a solution of the form T3 = ko(Z) + k1(Z) cos 0 + k2(Z) cos 20 + k3(Z) cos 30 (A.50) Putting (A.50) into (A.49) gives H k; +(k'1' - k1+ 2g2 - %2-- Zgo - gg)cos 0 + (k'z' - 4k2) cos 26 g +. (k3 - 9k3 - -§-)cos 30 = 0 (A.51) Putting (A.50) into the boundary values (A.21) gives ko(:zL) = k1(:ZL) = k2(:ZL) = k3(:ZL) = 0 (A.52) Again orthogonality of the series in cos n0 gives the dif- ferential equations k" = 0 (A.53) 123 n 82 k1 - k1 + 2g2 - 7 - 2go - g; = 0 (A.54) k2 - 4k2 = 0 (A.55) H 8'2 _ k3 - 91(3 - '2— - 0 (A.56) The only solutions of (A.53) and (A.55) which satisfy the boundary conditions are k = 0 (A.57) k2 = 0 (A.58) The solutions of (A.54) and (A.56) proceed in a straight forward fashion and eventually lead to 92 tanh Z cosh Z - 92 sinh Z + 12 (cosh Z - cosh Z ) k (1 T ) L L L 1 ‘ ‘ o 4 cosh ZL (A.59) and k = - 1 _ T cosh Z _ cosh 2Z + ll_cosh 32 3 l6 cosh z 5 cosh 22 80 cosh 3z L L L (A.60) The expression for T3 is now completely determined by substituting (A.57), (A.58). (A.59) and (A.60) into (4.50) Additional terms could be found in the same manner but sub- sequent tests showed that terms including those to 53 gave suffi- cient accuracy. APPENDIX B APPENDIX B COMPARISON OF THE AUGMENTED, SMALL-PARAMETER SOLUTION TO THE NUMERICAL SOLUTION In order to show that the augmented equation had sufficient terms to be an acceptable solution, the load support computed by using T from the finite-difference method was compared to the load support computed by using T from the augmented, small—parameter 3 solution with the £2 term. In both cases, the integration of the load support equation was performed numerically using Simpson's rule. It was known that an error existed in the direct numerical method due to the finite grid size used. Since the answers computed by this method were to be used as a standard, the error must be re- duced to a minimum. The numerical method approximated the deriva- tives in the squeeze-film equation by central differences, of second order accuracy, thus it was expected that the error in T would be proportional to the square of the mesh spacing in both the Z and 0 directions. If M and N are the number of nodes in the Z and 0 directions respectively, then m = —l—- and n = —l—- are M-1 N—l the respective mesh spacings. One method of improving the accuracy would be to increase the number of nodes in the two directions, this however, would certainly increase the computer time needed for a solution and, if carried to extremes, might well increase the truncation and roundoff errors to 124 125 unacceptable levels. The computations were to be performed on an IBM System/360-50, which in single precision, carries six hexadecimal digits (equivalent to approximately 7.2 decimal digits) thus roundoff error could rather quickly reach unacceptable levels. An investigation was made to determine if an extrapolation to zero mesh size, based on the assumed error being 0(m2) and 0(n2), would be effective. It was assumed that the domain of interest for the independent variables was 0<€ _<_0.5 A sampling was made using mesh sizes ranging from 9 x 9 to 17 x 41 at the extremes of the above domain. It was quickly apparent that the effect of mesh size was greatest for small values of £1 , for large absolute values of £2 , for A = - l , and for large values of ZL . It also appeared that the N mesh spacing (0 di- rection) was more critical than the M spacing (2 direction). The worst combination of values which was found is shown in Table B.l, and plotted versus n in Figure B.l. It appears from Figure 3.1 that the points lie on a straight line (N = 9 appears to be a slight exception) and that the error is 0(n2). It also appears that an answer extrapolated from the 9 x 9 W"1103 126 (9x9) 19x") Figure B.l Load support as a function of mesh spacing in the axial direction Ntl? ‘\ ‘\ \ \‘L ‘\ ‘\ Cu'.04. \ ‘ a --I \ _ .3 ' -.. “ \. \\ 10 N0. (At 05-00004) 0 so 40 so I'lm‘ 127 TABLE B.1 COMPARISON OF THE EFFECT OF THE NUMBER OF NODES ON LOAD SUPPORT CALCULATIONS. (clr=o.048, £2 = - 0.8, A = - 1) M m2 N n2 w' AT zL = .5 w' AT zL = 1,7 9 .0156 9 .0156 .025626 -.086639 9 .0156 17 .00391 .062890 .010319 .9 .0156 25 .00174 .069058 .025852 9 .0156 33 .000977 .071179 .031138 9 .0156 41 .000625 .072168 .033602 17 .00391 17 .00391 .062717 .0092033 17 .00391 25 .00174 .068907 .024931 17 .00391 33 .000977 .071035 .030282 17 .00391 41 .000625 .072029 .032778 grids (W' = 0.04253) is closer to the answer extrapolated from 9 x 17 and 9 x 24 (W' = 0.03831) than is the single computation at 9 x 41 (W' = 0.033602). It appears that extrapolation from two coarse grids gives a better answer than that from one fine grid. Using grid size as a measure of computer time, the 9 x 9 and 9 x 17 combination should be nearly twice as fast as a single 9 x 41 grid. In all other cases checked the slope of the extrapolation curve was less than that of Figure 3.1 indicating that the accuracy of the answer extrapolated from the coarse grids was proportionately better. At the other extreme of large values of 61 and small absolute values of 82 it was found that the slope of the curve of W' versus 112 was essentially zero but that W' plotted versus 1112 now had a significant slope. Thus in one case extrapolation on n2 improves the answer; in the other case extrapolation on m2 is 128 necessary. To cover both cases then, extrapolation is necessary in the m2 - n2 plane rather than linear extrapolation on either m2 or n2 although the greatest improvement is made by n2 extrapola- tion. The final result was that the computation was performed three times with meshes Of 9 x 17, 9 x 25, and 17 x 17 respectively, with extrapolation Of a plane to zero mesh size in both the m2 and n2 directions. A short investigation was also made to determine if the Simpson's rule integration for W' was introducing any measureable error. This was done by computing the T values for a given mesh (say 17 x 17) then using quadratic interpolation to generate the missing values in a mesh with twice the number of intervals (33 x 33). Each set of values was integrated and the final answers com- pared. There was no significant difference between the two computa- tions. Consider the case when T in (2.23) is given by T = T[1 + 0(AO)2 + 0(AZ)2] (B.1) A where T is the exact value. Then T1/2 = T13 [I + 0(A0)2 + 0012?] (B.2) All other terms in the integrand of (2.23) can be expressed exactly. Since the error in Simpson's rule is of the order'of the fourth power of the mesh spacing, this error is evidently insignificant compared to the 0(m2) or 0(n2) of the T computation itself. An error analysis performed on the finite difference equations 129 for small absolute values of 82 showed that the magnitude of the error is approximately 2 2 m 5n n z E _ + — 8.3 1 1max 2[12 6 :1 ( ) bearing out the observation that the n interval (0 direction) is more important than the m interval (Z direction). Finally, the finite difference computation for W' was compared to the augmented, small-parameter solution. The results of this comparison are shown in Figure B.2 for the worst case. (Worst in that the augmented equations are exact in 51 , so that the maximum error should be associated with small 81 and large absolute values of 82.) In the range Of interest Of 0.7 §_Z :_1.5 L the maximum difference for the augmented equation including terms to 83 was at ZL = 0.7 and amounted to less than 10%. The error for the augmented equation to including terms to 8% however was an unacceptable 28%. 3 It was concluded that the augmented equation with terms to 82 gave answers which would be more thsn accurate enough for the present investigation. Further, when only load support is needed in any future work the augmented equation will suffice. The computer time for the finite-difference equation is approximately 20 seconds per load support computation compared to less than 4 seconds for the augmented equation when using the IBM System/360-50. 130 0 7 \ w': 10' ‘\ s \\ \ \\ ‘\ \ ‘\ a ‘\ mm: surnames \ ( unwoo ‘ \ \‘ mm c’ TERM we»: no suALL "saunas soumow 3 mm a: nun ear I .048 AI-l 03 I - .0 2 7 9 I 1 1.3 I s Figure B.2 Comparison between finite difference solution and augmented, small-parameter equation. S S! nrcwxcnw TATE UNIV. LIBRARIES JIHIJJJIJH “IWWIJJJIJIJJIHHIWIWJIIJIJHI 312 3006498425