'1':- I . . COMPUTER AIDED OPTIMIZATION OF: . NONLINEAR SERVOMECHANISMS' _ _ EMPLOYING A DIRECTED SEARCH OF MULTIRARAMETER g COMPONENT LIBRARIES AND STATISTICAL TDLERANCING‘ ' f £f i : ‘ Thesis for the Degree of Ph. D.- ‘ MICHIGAN STATE UNIVERSITY BRUCE ALLEN CHUBB I 1969 will Ir.” Y ' L I B R A R Mid‘iigan State University This is to certify that the thesis entitled Computer Aided Optimization of Nonlinear Servomechanisms Employing A Directed Search of Multiparameter Component Libraries And Statistical Tolerancing. presented by Bruce Allen Chubb has been accepted towards fulfillment of the requirements for 1&— degree in _él_'}_L_ 0-169 DIRE Technic libraries ti minimum (101; function to PTODability {Omance Sp Part nmber Cmoment p nah“'Afacturi Considered COmpa- Search Pro has COnsid Conileuter I: ABSTRACT COMPUTER AIDED OPTIMIZATION OF NONLINEAR SERVOMECHANISMS EMPLOYING A DIRECTED SEARCH OF MULTIPARAMETER COMPONENT LIBRARIES AND STATISTICAL TOLERANCING by Bruce Allen Chubb Techniques are developed to automatically select from computerized libraries the components that satisfy a given system specification at minimum dollar cost. This is accomplished by defining an object function to be the system cost which in turn is a function of the probability that the design will be successful in meeting the per- formance specification. Starting with an initial set of component part numbers, the total system cost is minimized by iterating the component part numbers using a directed search technique. The manufacturing tolerances associated with the component parameters are considered in calculating the probability of success. Comparisons are made between the Monte Carlo and the directed search procedure which illustrate that the directed search technique has considerable advantage. Several examples demonstrate that such a computer program can result in considerable cost savings. The techniques are developed around an instrument servomechanism as a specific example. Four component libraries are established to list the part characteristics for the followup, amplifier, Rotor-generator, a assigned performar and time response coulomb friction, the effect of fini performance. Equ: bacxlash without a step inputs, and I motor-generator, and geartrain. Combinations of up to eight pre— assigned performance specifications in the areas of damping, accuracy, and time response are considered. The nonlinear effects of backlash, coulomb friction, and amplifier saturation are considered as well as the effect of finite geartrain stiffness in evaluating the system performance. Equations are derived for calculating, l) the allowable backlash without a limit cycle, 2) the nonlinear overshoot for large step inputs, and 3) the effective bandwidth for sinusoidal inputs. NO DIRECTED S in pg [I COMPUTER AIDED OPTIMIZATION OF NONLINEAR SERVOMECHANISMS EMPLOYING A DIRECTED SEARCH OF MULTIPARAMETER COMPONENT LIBRARIES AND STATISTICAL TOLERANCING By Bruce Allen Chubb A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Electrical Engineering 1969 gjé/ H 6'13: 67 TO MY MOTHER AND FATHER ii The author in can of the Electr Michigan State Un tions during the due to my associa especially Mr. G. and thorough read. ciated. In addition, nent of the InstrI Sponsorship of thc facilities, in pa' Thanks are a DUIkln for their I numerous magic“S I Final ly the } ACKNOWLEGEMENTS The author wishes to thank his advisor, Dr. H. E. Koenig, chair- man of the Electrical Engineering and System Science Department, Michigan State University, for his guidance and many helpful sugges- tions during the preparation of this thesis. Special thanks are also due to my associates at Lear Siegler, Inc., Instrument Division, especially Mr. G. Garvelink and Dr. R. Wierenga. Their assistance and thorough reading of the complete manuscript are gratefully appre- ciated. In addition, the author expresses sincere thanks to the manage- ment of the Instrument Division for their financial support, for their sponsorship of the work done on this thesis, and for the use of their facilities, in particular the analog and digital computer. Without their support this work would not have been done. Thanks are also given to Miss Anna D'Angelo and Miss Loretta Durkin for their patience in typing the complete manuscript and its numerous revisions. Finally the author wishes to express his appreciation to his wife, Janet, for her interest and encouragement throughout this graduate program. iii LIST OF TABLES . LIST OF FIGURES LIST OF APPENDICI 1. . DEVELOPMENT 33 CaICUIaI 3.4 ObIBCt . 3 5 Design IXTRODUCT IO.\' 1.1 Statemer 1.2 Example 1-3 Survey c 1-4 SCOpe of 2-1 BaSlCA 2'2 FOI‘IITula 2'3 System 2.4 Variabi DEIELORNENT 3.1 Basic A TABLE OF CONTENTS LIST OF TABLES . LIST OF FIGURES LIST OF APPENDICES . 1. INTRODUCTION . 1.1 Statement of Problem 1.2 Example System 1.3 Survey of Present Techniques 1.4 Scope of Investigation 2. DEVELOPMENT OF ANALYSIS PROGRAM 2.1 Basic Approach 2.2 Formulation of System State Equations . 2.3 System Specifications and Design Equations 2.4 Variability Analysis Techniques . 3. DEVELOPMENT OF COMPUTER OPTIMIZATION DESIGN PROCEDURE 3.1 Basic Approach 3.2 Generation of Object Functions 3.3 Calculation of Rejection Ratio 3.4 Object Function Derivatives . 3.5 Design Program Strategy . iv Page vi viii 10 15 27 32 32 35 38 46 53 4. EI'LWPLE DESI 4.1 Compone 4.2 First I 4.3 Second 5. CONCLUSIONS LIST OF REFERENC TABLE OF CONTENTS (cont.) Page 4. EXAMPLE DESIGN PROBLEMS . . . . . . . . . . . . . . . . . 68 4.1 Component Libraries and Search Matrices . . . . . . 68 4.2 First Design Example . . . . . . . . . . . . . . . . 74 4.3 Second Design Example . . . . . . . . . . . . . . . 85 5. CONCLUSIONS AND RECOMMENDATIONS . . . . . . . . . . . . . 98 LIST OF REFERENCES . . . . . . . . . . . . . . . . . . . . . 148 Table 11 System par a. Component L3 State equa 14 System Spe 35 Component 2-6 Location 0 4-1 Followup 1 4-2 Amplifier I4 Geartrain 4.3 FOlth'up 5 I6 A”Plifier 4.7 Motolu-gene ~I.8 Geartl‘ain I example {10 Best deSig: design I 4.11 DITECted St fOr fir: LIZ Directed Se fOr fit: 4&3 Local minim LIST OF TABLES System parameter definitions for state model . Component parameter definitions for state model State equations for calculating nonlinear overshoot System specifications Component parameter definitions for library Location of F functions Followup library data Amplifier library data . Motor-generator library data . Geartrain library data . Followup search matrix . Amplifier search matrix Motor-generator search matrix Geartrain search matrix Intermediate Monte Carlo printout for first design . example Best design obtained using Monte Carlo for first . design example Directed search with initial guess underdesigned . for first design example Directed search with initial guess overdesigned for first design example Local minimums obtained for first design example . vi Page 12 12 23 24 26 27 69 7O 71 72 75 76 77 78 8O 81 82 84 85 fade L14 Best desi 4&5 4J6 417 418 AZ RS C1 C2 first Intermedi desigr Best desi seconc Directed for se Directed for s: Directed local Local miI Best des SeCOH Nonlinea diSpl Nonlinea diSp1 NonliREa di5p1 NOnlinea saIUr Table 4. 14 .15 .16 .17 .18 .19 .20 .21 LIST OF TABLES (cont.) Best design obtained using directed search for . first design example Intermediate Monte Carlo printout for second . design example Best design obtained using Monte Carlo for . second design example Directed search with initial guess overdesigned for second design example Directed search with initial guess underdesigned . for second design example Directed search resulting in an unsatisfactory . local minimum Local minimums obtained for second design example Best design obtained using directed search for . second design example Nonlinear overshoot calculations for 0.1 radian displacement Nonlinear overshoot calculations for 0.2 radian displacement Nonlinear overshoot calculations for 0.35 radian . displacement Nonlinear bandwidth calculations with friction and . saturation Nonlinear bandwidth calculations for zero friction . cas e vii Page 86 88 89 90 91 93 96 95 132 133 134 142 143 Rpme 12 13 i1 12 id Al A2 A5 A.4 As is A1 A2 is 14 A6 Schematic servome A simplifi Nonlinear servome Nonlinear Phase-plan computer a 09518“ prc Comparisor Nonlinear sen'omd Backlash-fi BaCk lash d EffeCtive‘ Backlash-g System res System reg Analog CoI Nonlinear Phase‘Pla: SYStem Te, TyPiCal s- Figure 1.1 LIST OF FIGURES Schematic diagram of motor-generator instrument . servomechanism A simplified design flow diagram Nonlinear state model diagram for an instrument . servomechanism Nonlinear function definitions Phase—plane diagram showing piecewise linear regions Computer aided design program flow chart Design program simplified logic diagram . Comparison of Monte Carlo and directed search . Nonlinear state model diagram for an instrument . servomechanism Backlash-friction diagram . Backlash describing function Effective gain of friction vs. frequency Backlash-friction ratio vs. frequency . System response curves for various friction values System response on phase-plane diagram Analog computer diagram used for transient analysis . Nonlinear overshoot logic flow diagram Phase-plane interpolation diagram . System response on phase-plane diagram Typical system response curves viii Page 11 13 22 33 63 94 103 102 108 111 113 114 117 118 126 129 135 136 thre C1 Nonlinear C.2 Effective zero-tI C.3 Analog cor Figure C.1 C.2 C.3 LIST OF FIGURES (cont.) Nonlinear bandwidth flow diagram Effective bandwidth as a function of input zero-to-peak amplitude Analog computer diagram used for bandwidth analysis . ix Page 141 145 146 Appendix A C LIST OF APPENDICES Derivation of Backlash-Friction Slope Equation . Derivation of Nonlinear Overshoot Equation . Derivation of Nonlinear Bandwidth Equation . Page 101 116 137 1.1 STATEME.‘ The sysI manufacturing nation and cc 3) A 56 b) A be thes C) A Se tion The basic pro satisfies the AUtOmate. multitude Of ( the problem 01 amplifier COnf thns . If one Should be appr and designated standard reSi51 1. INTRODUCTION 1.1 STATEMENT OF THE PROBLEM The system engineer Operating within the framework of a typical manufacturing organization operates from the following basic infor- mation and constraints: a) A set of customer specifications to be met, b) A basic system configuration to be used in realizing these specifications, c) A set of standard components that fit into this configura- tion. The basic problem is to determine the collection of components that satisfies the given specification at minimum total dollar cost. Automated techniques for selecting the optimum set of components for a system are necessitated by today's competitive market and the multitude of candidate components available. As an example, consider the problem of selecting an optimum set of components for a fixed amplifier configuration to meet a given set of customer specifica- tions. If one extrapolates data from a 1964 survey [1], today there should be approximately 60,000 semiconductor devices manufactured and designated by part number. If one adds to this, the number of standard resistors, capacitors, transformers, etc., it becomes obvious that manual tecl coaconent select The same 5' there the config pcnents are avai of course, be 5 analysis prograx for any candidaI priate part hum} retrieve the da, various perme; techniques, pro. QUIETS in 3Ut0rr, Problem, 1'2 EXANPLE ST The techni Ilaily the Same tetnniqUes devg The examp], de ' VICE, electrc a A Rd SeartI‘ain t101'] . that manual techniques cannot come close to yielding an optimum component selection. The same situation exists in every area of system engineering where the configuration is ”fixed" and a multitude of candidate com- ponents are available. The characteristics of these components can, of course, be stored in computer libraries by part numbers and an analysis program can be written to systematically analyze the system for any candidate set of components by merely inserting the appro- priate part numbers. Such computer programs are structured so as to retrieve the data for each particular component, proceed with the various performance calculations and display the results to the de- signer for each set of part numbers manually selected. The goal of this study is to go one step further and deveIOp techniques, procedures, and programs for the effective use of com- puters in automating the solution to the above class of design problem. 1.2 EXAMPLE SYSTEM The techniques presented are developed around an instrument servomechanism as a specific example. The design problem is essen- tially the same as that discussed in references [2,3]; however, the techniques developed are believed to be much improved. The example instrument servomechanism consists of a follow-up device, electronic amplifier, drive motor with feedback generator, and geartrain. A pictorial diagram showing a fixed system configura- tion using these components is shown in Figure 1.1. Excnanor \ohage It ‘15 a: Up to elght liShEd to li a) Fol b) Am; C) I101 d) Ge.- Eve“ thOugh wall» the manly: 25 The 0p Satisfies t total COStH Amplifier output voltage (.1 In at Excitation Control . voltage, transformer (CT) 6'18 0m Control gamma“), 1 it . 3‘" ,1, ransmn er (CX) I I / o“ i” I \ I I, ‘c ’ ' ' - - 7 I, S 'III‘ ii Generator 1 I 0 * Gear Imt‘ . . train OUlpUt voltage q Jena“ 1 I l I Generator l Amplifier input voltage e, Error voltage 9.. voltage to Figure 1.1. Schematic diagram of motor-generator instrument servomechanism. It is assumed that the design of this configuration must meet up to eight preassigned specifications in the areas of damping, accuracy, and time response. Four component libraries are estab- lished to list the part characteristics as follows: a) Follow-up — 25 part numbers b) Amplifier - 50 part numbers c) Motor-generator - 25 part numbers d) Geartrain - 25 part numbers Even though the size of each demonstration library was purposely kept small, the number of theoretical possible candidate systems is large, namely: 25 X 50 X 25 X 25 = 781,250. The optimum collection of components is defined as "the one that satisfies the given specification in a manner resulting in minimum total cost". Three an origin in the calculating t mm for a n problems repr mechanism des 1.3 SURVEY C Literal] and describe more comprehI Host of this meter values scalar funct Paramet 81‘s up Three c fouwine t} a) Rat] Al‘ te gi Si me Si ‘10 Du 4 Three ancillary problems considered in the thesis have their origin in the fact that design equations did not heretofore exist for calculating the allowable backlash, large step overshoot, and band- width for a nonlinear instrument system. Solutions to these three problems represent significant advancements in the field of servo- mechanism design, and are presented as Appendices. 1.3 SURVEY OF PRESENT TECHNIQUES Literally thousands of articles have been published which list and describe work that has been done in optimization. A few of the more comprehensive publications are listed as references [4,5,6]. Most of this work is concerned with finding that set of n para- meter values, X1, X2, ---, Xn, which maximizes (or minimizes) a given scalar function F(X1, X2, ---, Xn), subject to constraints on these parameters which limit their range to realizable values. Three of the most popular techniques are centered about the following three basic approaches: a) Random Experimentation Although crude forms of this method are as old as design technology itself, the best early formal documentation as given in 1958 [7] uses repeated solution of the system de- sign equations with random selections of the input para- meters generated through Monte Carlo methods. In its simple form, a large number of computer solutions are re- quired to achieve good results. When for reasons of com— puter costs, only a few runs can be justified, a partitioned b) or s In g some the be 5 Stet furI putt tecl par‘ llUIll' 10c in exa met Dir The th mod StI Seq res tri Sel- b) e) or stratified form [8] usually provides better efficiency. In general, improved results are obtained most often if some form of strategy or learning can be employed to adjust the frequency distributions representing the parameters to be selected. Steepest Ascent This method was introduced by R. R. Brown in 1957 [9] and further improved in 1959 [10]. Today there are many com- puter programs available for general use which employ this technique. The Steepest Ascent methods calculate the partial derivatives aF/axl, aF/8X2,---, aF/exn usually numerically, and then proceed along the gradient until a maximum is obtained. Since the result represents only a local or relative maximum, various starting points are used in an attempt to find the global maximum. GREAT [11] is one example of a highly effective program that is based on this method. Direct Search The Direct Search technique is attributed to Hook and Jeeves who presented the unconstrained case in 1961 [12]. This was modified in 1965 by Weisman and Wood [13] to include con- straints. In direct search, the minimum is found by the sequential examination of a finite set of trial values. The result of each trial is compared with the best previous trial and the new value accepted if an improvement is ob- served. This series of exploratory moves, in which each var: the whiI mow p10 pea Agl 1.4 SCOPE 0? A logic is show in tion 100ps, tion (9°8- C ponent SBIec comPonent Se If eacp the Solutim Components meter and u: direct SEar SolutiOn is minal compo Ponent Part Pollent S . variable is individually adjusted, is used to determine the "best" direction for a successful move. This move,in which all parameters are changed, is called the "pattern move". Each pattern move is followed by a sequence of ex- ploratory moves to revise the pattern. The sequence is re- peated until the scalar function can be increased no further. A good application of this type of algorithm is LOOK[14]. 1.4 SCOPE OF INVESTIGATION A logic flow diagram representing an effective design procedure is shown in Figure 1.2. As indicated, there are two design itera- tion loops. One loop concerns changes in the basic system configura- tion (e.g. component interconnection) and the other changes in com- ponent selection. Only the problems associated with automating the component selection are considered here. If each component could be represented by a single parameter, the solution would be quite straightforward. One could arrange the components in the library in ascending order of its single para- meter and use a modified form of either the steepest ascent or direct search method to find the optimum. However, the general solution is far more difficult, since libraries consist of multiter- minal components, and several parameters are required to describe each component. These parameters are associated only with the com- ponent part number, and there is no natural ordering between com- ponents. COIN START ASSUME INITIAL CONFIGURATION ASSUME INITIAL COMPONENTS ALTER CONFIGURATION _ ANALYZE T SYSTEM ALTER COMPONENTS NO BUILD AND SHIP Figure 1.2. A simplified design flow diagram. The pro‘ N-Cube [15]. certain meth at each poir our particul example, if using a big! to solve th The pr is concerne fomulatior of the Syst tions and ; This effor The 5 design 10c neccssary comPonent nus effo. SECt 0Red prOg Press“ 5 The problem is analogous to A. M. Gleason's Search in the N-Cube [15]. Gleason states "It is entirely clear that there is no certain method of finding the maximum, short of computing the function at each point of the set in question". This exhaustive search for our particular design problem, however, is out of the question. For example, if we consider 30 seconds to be required for each analysis using a high speed machine, it would take 6510 hours of computer time to solve the problem in question. There must be a better way! The problem can be divided into three aspects. The first aspect is concerned with developing an effective analysis program, including formulation of the necessary nonlinear state equations, codification of the system specifications, developing the required design equa- tions and a method of handling the component parameter tolerances. This effort is presented in Section 2. The second task is to incorporate the analysis program in the design loop by adding an optimization procedure. To this end it is necessary to formulate the object function to be minimized, set up component libraries, and to formulate the optimization strategy. This effort is presented in Section 3. Section 4 of this thesis presents the application of the devel- oped program to typical hardware design problems. Section 5 then presents the conclusions of the study and provides suggested guide- lines for future work. 2. DEVELOPMENT OF ANALYSIS PROGRAM 2.1 BASIC APPROACH The analysis section is the starting point of any computer-aided or automated design program. Optimization, in the design context, is derived from an efficient use of iterative analysis techniques. Devoid of a good analysis capability, the designer has nothing. Its presence provides a powerful tool in itself. In this case, however, it is simply a means to an end - Automated Design. "But what are the requirements for an effective analysis pro- gram?" First, and primary, is the fact that it must accurately re- present the hardware. This requires a significantly detailed model, including often overlooked nonlinearities, and a realistic consider- ation of component tolerance effects. This means that the program- mer is faced with the solution of nonlinear differential equations, and that system parameters, instead of being constants, must be treated as random variables. Second the outputs of the analysis program must have a one-to-one correspondence with the list of system specifications. That is, if the customer specifies overshoot, re- sponse time, accuracy, etc., then the program must have the capabil- ity of calculating the performance characteristics in this form. Third and last, since the analysis is to be repeated many times in an iterative fashion, the solution time should be a minimum. 10 2.2 FORMULATION OF SYSTEM STATE EQUATIONS A most effective method of obtaining the response of a system is by using the state variable model [16]. Much work has been done in the effective application of this approach to the analysis of phys- ical systems [17,18]. Many aspects of the particular problem consid- ered here are presented in reference [2]. However, in the interest of continuity a limited development is repeated here. The example system under consideration consists of a followup, amplifier, servomotor with an integral mounted feedback generator, geartrain, and load. The load is made up of inertia and coulomb friction. Experience had demonstrated [2] that geartrain resilience, along with the nonlinearities of gear backlash, amplifier saturation, and coulomb friction, must be considered. The state model diagram of the system is shown in Figure 2.1 and the system and component parameters are defined in Tables 2.1 and 2.2,respective1y. Four state variables are required to define the system. These are the outputs of the 4 integrators of Figure 2.1 and correspond to motor velocity, motor position, load velocity, and load position. It should be noted that the motor velocity and posi- tion have been reflected to equivalent values at the load (i.e. the hardware values are simply those given by Figure 2.1 times the gear ratio, N). The amplifier saturation is represented by an equiv- alent torque saturation (i.e. the torque level is set at a value equal to that of the amplifier voltage level times the product of the motor torque gain and gear ratio). 11 .amwcmnuoeo>uem ucoazuumcfl an new soammwv Hobos cumum newsflacoz .H.~ onsmwm :mqnxuam fl m_.z 33¢ .pzuzuojama /.II coho: 3.53.12. 2.3% E004u> £05.02 OPEN—mun. logo—cu 030:. :00 zoc<¢3h.—_uod> 93 V V 3 so 533.2»... Soul / #352. 624.800 Table 2. ’— Symbol 12 Table 2.1. System parameter definitions for state model. Symbol 2:52:12: Name Units BM N28In Reflected motor damping oz-in/rad/sec JL J! + Jg Load inertia oz-in/rad/sec2 JM NZJn Reflected motor inertia oz-in/rad/sec2 KG KgKagKmNz Generator damping coefficient oz-in/rad/sec KT foameN System torque constant oz-in/rad TL Ti + T3 Load friction oz-in TSAT KmEsat System torque saturation oz-in Table 2.2. Component parameter definitions for state model. Symbol Definition Units B Geartrain backlash radians B. Motor viscous damping oz-in/rad/sec Esat Amplifier saturation level Ivolts J8 Geartrain inertia oz-in/rad/sec2 J1 Load inertia oz-in/rad/sec2 JIll Motor-generator inertia oz-in/rad/sec2 Kaf Amplifier gain to followup volts/volt K38 Amplifier gain to generator volts/volt Kf Followup gain volts/rad Kg Generator gain volts/rad/sec K. Motor torque gain oz-in/volt Ks Geartrain spring constant oz-in/rad N Gear ratio T8 Geartrain friction oz-in T‘ Load coulomb friction oz-in The nonlin friction (N2) . #q &_0CK DIA Ill-0L)-—O- CL HQ. u 13 The nonlinear functions representing backlash (N1), coulomb friction (N2), and saturation (N3) are defined in Figure 2.2. BLOCK DIAGRAM EOUATIONSIWHEREZ 890 9'9/l9Il "I ,,... Wile-444119.40 (Du-9 l—v | I __r_..T '°'I°-'9LI2° I ' r,-o "i ”'8 m l9.'9,I sB BACKLASH N . 2 Tt IITLlpgn 9L ‘l’L _ tor 0L! 0 T 5,, ——v I ¢ r, --rr ”Tl. tor 9L IO COULDMB FRICTION N3 sure-I _ _ - \ Tm. KT(9| 8L) K. 0.. . \ torlfil9.-0Ll-K.0.I5T." “fwl. L)_K.O4u-. f Tm A Tm. Tut .°n[K'(9| -9L,-K.éll _/ m] attired-madam SATURATION Figure 2.2. Nonlinear function definitions. The nonlinear state model for the system can be obtained di- rectly from Figure 2.1 as: Fe F- 51 o o o I-é F—l-(T -r M JM M JM m r) 9M 1 o o 0 6M 0 i - + dt . ' - 1 9L 0 0 o 0 6L J—-(T 4f) L e o _ L4 5 o o 1 o J cal... B 3 (2.1) The cor 1 for a: II N, :1 for 1 dt E Where f is generatol. am and in a like 14 The corresponding linear approximation is obtained by setting 2 ll 1 l for zero backlash; N2 = 0 for zero coulomb friction; and N, = l for no amplifier saturation. P6 1 r- “1ch) _:S_ 0 (KS-KTIj r'é 1 Fir-T M JM JM JM M . JM 6“ 1 o o 0 6M 0 d a? s + 61 K5 K3 ' e 0 — 0 - — e 0 L JL JL L 9L 0 o 1 0 6L 0 (2.2) In the special case when the geartrain stiffness is considered in- finite (i.e. Ks = 00) the linear state model becomes (2.3) where fT is the total effective viscous damping from the feedback generator and motor. This is f =3 +K (2.4) and in a like manner JT is the total system inertia given by J = J + J (2.5) 9.3 [satiation (- where g and normally defi under conside 2-3 SYSTEM 5 The firs understanding sl'Stem must 5 equati0ns tha t0 the speCif of both the a Fer a C0 plication, th the ComPUt er 15 Equation (2.3) may be written also in the convenient form (2.6) where c and m are the damping ratio and natural frequency as N normally defined for a second order system. For the particular case under consideration c = —— (2.7) 24% ’KT m = — (2.8) N JT 2.3 SYSTEM SPECIFICATIONS AND DESIGN EQUATIONS The first step in realizing a design is to establish a thorough understanding of the set of performance specifications that the system must satisfy. The second step required is to develop a set of equations that enable one to evaluate a potential design in relation to the specifications. This section is devoted to the accomplishment of both the above tasks. For a computer program to be effective in design, it must cover a somewhat general set of specifications. Then, for any given ap- plication, the user may choose the particular desired set and instruct the computer to ignore the others. A set of eight specifications is selected for Rwy are reP commercial 5 hmtrument E Michigan. The eig the correspo design. 1) Sta fie mea th sys inc nul fri ing ant Ste 16 selected for the example program developed as part of this study. They are representative of those listed in numerous military and commercial specifications for such systems as manufactured by the Instrument Division of Lear Siegler, Incorporated, Grand Rapids, Michigan. The eight specifications are now discussed one at a time, with the corresponding design equations used to evaluate a proposed design. 1) Static Accuracy Static accuracy is unquestionably the most often speci- fied requirement for any instrument servo. It is simply a measure of the magnitude of the error that can exist be- tween the command input and the indicated output of the system under static conditions. Contributions to this error include followup tracking error, amplifier and generator null offsets, motor starting voltage, and gearing and load friction. By taking each of these error sources and divid- ing by the corresponding dc gain back to the error angle, and summing, the following equation is derived for the static accuracy (8A). E K E E T +T£ €A=ef+KRn+lCKn+KKS 4'Ic KN f af f af f af fKaf m (2.9) 3) wl' ii In; 2) 3) 17 where 6f = followup tracking accuracy (rad) an = amplifier output null voltage (volts) Egn = generator output null voltage (volts) E5 = motor-generator no-load starting voltage (volts) and all other notation is defined in Table 2.2. Resolution Resolution is a measure of the total dead-zone in an instrument servomechanism. It therefore represents the maximum amount that the input can be displaced without no- ting any motion at the output. This deadzone results from. the fact that a certain amount of error must be built-up to overcome the motor starting voltage and coulomb frictions. Thus, the total deadzone or resolution (ER) is given by ES T +T2 e = 2 —— + (2.10) R KfKaf KfKameN Velocity Lag Velocity lag is a measure of the servo's accuracy under constant velocity conditions. It is defined as the steady- state positional difference between the command input and the indicated output with the input rotating at a constant velocity. Since the resulting lag error is a function of the input velocity, the latter also must be specified. 4) 5) The eqi whe to as Fol tha fri Spe the the Whe the Dam 4) 5) 18 The velocity lag (2L) may be calculated using the equation (see reference [2]) N2 Bm+K Ka Km) = 0 5L ———LB—K K K N ein + EA (2.11) f af m where éin is the input velocity at which the lag error is to be measured or calculated and e A in the static accuracy as defined by (2.9). Followup Rate Followup rate is a measure of the maximum velocity that the servo is capable of producing. If there were no friction loading, it would be simply the motor no-load speed divided by the gear ratio. However to account for the load, one can calculate the followup rate (éL) using , the equation: (see reference [2]) 6I. = TI” 1 -_I%'T— (2'12) where the symbols are as defined in Table 2.2 except for the additional ones which are a). ll motor no-load speed (rad/sec) v-l ll motor stall torque (oz-in) Damping Ratio Damping ratio is the most often used measure of system 19 stability. This is unfortunate since its definition applies only for a linear 2nd order system. However if one makes this linear approximation, then the damping ratio equation may be obtained directly in terms of the component parameters by substituting the definitions of Table 2.1 into Equa- tion (2.7). Thus: N2 B +K K Km) "L.1L£§L_____._. (2.13) — 2 2‘//RfKameN(N Jm+Jg+J£) 6) Null Oscillation C Null oscillations are small amplitude steady state oscillations (limit cycles) that exist about a null and are a result of backlash being present in the geartrain. A typical specification states that "no such oscillation shall exist." In Reference [2], it was established that the amount of backlash that a given design can tolerate without such a limit cycle is proportional to the amount of coulomb friction on the load side of the backlash. In this study, we shall derive the equation for the proportionality con- stant (derivation in Appendix A) thereby obtaining the equation for the allowable backlash as follows: B(allowable) - M I N 3-7- IKT'JszIz’ “1"")2 nEN1(w)+.4 JOINZOD) ("1(w)Ks_JMw2)2+(fTw)2 L o_._.< omz v zoEwt / 22 \ \S zo_._._._._m0n_ n 20.0%. \\ \\ 202.05.... 0... 6259:: .6 ezmzuofiama \1 m3 MES. Bio 816 ezmzuofiams 35:2. 83\OQ >._._UOJ w> 852252 a L hoozmmm>o 23 Table 2.3. State equations for calculating nonlinear overshoot. Region 1 (Negative torque saturation) . TsAr‘TL - 7.3.5.111 60m 00(0) 0 T 2 0 I“ u .. wt - I. 9 o (c) "0(0) - “£44151 15"-“- o (o) . 6°“) .IIIE‘W'T'J/I" ‘ o 2‘“ ° ‘n‘u Region 2 (Unsaturated) l’ " r’ 1. ,' 60(2) In“; s ,/r,2-1I 60(0) 9 «IN? 90(0) - 3% _ N“ .. \/‘_2:l ) t O T 10" Jc’d o c -é(o)-u t-I/42-1II(O). L O“ )J L 0 N( I O Jf‘fl“ . I7C2'l i- H A [- TL ‘ F' ‘i u _ _ / 2_ ' _ 2 _ o a.NI‘ C 1’00“” mM 00(0) I J1. _‘"(‘ - ‘2‘! ) t O I r 2““ ' ‘ '1 r {NONI-(co Jtz-1)O(O)- L I- o u o 7777-1] . T”N(‘ . E l' T ' 'r , ‘ . ’ tom T:- - afloom - «5300(0) I 00(0) I 0 I - g -u t - to a” o e N O r T TL "‘ ° u - _.':_ e (0) - i — u I..°m. L 00(0) 9 N110(0) r131 . L o [1. . LIL, . U 5.“) 60(0) a: - chm - 5.0.10) 0 - I H .."I‘ meFE’zo-fi? 5(0) 1" the. has 0' V 0.“) 0,10) - it {:- ° "om ' i,— I: U Region 3 (Positive torque saturation) .TSNI'-TI. --TSAT-TL ‘°(‘) 60(0) 9 T o ,. ."‘vru‘ _ , , '5 I” ’ '7 ‘ISA "I. I (0) ' EMT'ILIZN ’0“) Jfigsfl—LILIE + 00(0) 9 0 tun" ’ Simnarize number ; IIOD is ; Specifyix meance SEYVQ is “Tidy. tl COUlomb 24 However, because of system nonlinearities; namely, satura- tion and coulomb friction, the actual system bandwidth is a function of the amplitude of the input sinusoid. The neces- sary procedure for including this nonlinear effect is de- veloped as part of this study. The development is included as Appendix C and is based on the use of describing func- tion approximations to obtain effective values for w and N C. The eight system specifications that have now been described are summarized in Table 2.4. This table lists the name, symbol, and number assigned to each specification, tells whether the specifica- tion is an upper or lower bound, and the units used. In addition to specifying any desired combination of the above described eight per- formance requirements, the user must also define the load that the servo is to drive. For the example program developed as part of this study, the load is represented by an inertia (J2) and a nonlinear coulomb friction (T2). Table 2.4. System specifications- Name Symbol Boundary Units Static accuracy 81 upper degrees Resolution 82 upper degrees Velocity lag S3 upper degrees Follow-up rate Sh lower deg/sec Damping ratio 55 lower - Allowable backlash 86 upper minutes Overshoot 87 upper degrees Bandwidth S8 lower hertz 25 The analysis problem can be now defined mathematically by let- ting S, Y, and X be vectors,defined in general as: s = [31, 52, sk] Y = [Y1, Y2, °'°, Yk] x = [x1, x2, xn] (2.19) where k = number of performance specifications n = number of component parameters S. = numerical value for the ith specification as defined in Table 2.4 (l §_i :_k) Y1 = system performance function corresponding to 1th specification (1 §_i §_k) X3. = numerical value for jth component parameter (lijin) Thus one can write in general that r-Y1‘q PFiixia x2, x3,---, xnI-1 Y2 = F2(x1, x2, x3,---. xn) .YkJ LFk(x1, x2, x3.°°', an (2.20) It is only necessary, at this time, that the X vector contain the elements as required to calculate the system performance function vector Y . However, it is convenient to include the component costs as part of the X vector [even though they will not show up expli- citly in (2.20)] since they are required to calculate the optimization function that is introduced later. practice for our particular example, k = 8 and vector is defined in Table 2.5. 26 fined in Table 2.6. Following this n = 23, where the X Likewise, the F functions are de- Table 2.5. Component parameter definitions for library. COMP VAR PARAMETER NAME SYMBOL UNITS F xl Cost Cf dollars 0 L X2 Gain Kf volts/rad L 0 X3 Accuracy of minutes W U P A X“ Cost C dollars M a P x Gain to Followup volts/volt L 5 af 1 x6 Gain to Generator K volts/volt r “3 I 7 Output Saturation Level Esat volts E R 8 Output Null Voltage Ean volts ' G X9 Cost Cm dollars E Xlo Stall Torque Ts oz-in M N ' O E X11 No-load Speed 6. rpm T R X Inertia J gm-cm2 0 A 12 m R T X13 Starting Voltage Es volts 3 Xlu Generator Gain Kg volt571000 rpm X15 Generator Null E8n millivolts G X15 Cost GB dollars E . _2 A X17 Inertia Jg gm cm R X Stiffness K ‘oz-in/rad T 19 s R x19 Friction T8 oz-in ? X20 Backlash B minutes N X21 Gear Ratio N - 3 x22 Inertia J; gm-cm2 A T oz-in D X23 Friction t 27 Table 2.6. Location of' F Functions. Function Location F| Equation (2.9) F2 Equation (2.10) F3 Equation (2.11) F4 Equation (2.12) F5 Equation (2.13) F6 Actual backlash (X20) F7 Table 2.3 with logic from Figure 8.3 F8 Equation (2.18) with ”N and c replaced by effective values as defined in Appendix C Thus (2.20) can be used to calculate the system performance vector (Y) given any component vector (X). By programming this equation as presented, one obtains the desired analysis program ex- cept fer one deficiency. That is, due to manufacturing tolerances, the X vector varies from unit to unit, and we are interested not in a particular value of Y ’but what spread or limits to expect. For this reason, the next section is devoted to selecting a suitable method for determining this tolerance spread. 2.4 VARIABILITY ANALYSIS TECHNIQUES Variability Analysis refers to the methods used to determine the ability of a system to continue to give specified performance while its component parts change value within specified limits. One met the specifie analysis. A a worst-caSe program the spect to eaC parameters t "worst- case" For mul method becom puter, due t Even if the unrealistic same system throughout t ante require The app design probl iStic PiCtur dUCIIOn. St Shelihart i n I in the early the Statistic artICles and 28 One method of insuring that a given system design meets all of the specified performance critera is to use some form of worst-case analysis. An example of this type of procedure is MANDEX which is a worst-case circuit analysis computer program [19]. Using this program the first derivative of all the output variables with re- spect to each of the input parameters is used to set each of the parameters to their "worst-case" tolerance extreme, so that a "worst-case" condition exists at each of the circuit outputs. For multivariable systems, the application of the worst-case method becomes very time consuming, even when using a high speed com- puter, due to the multitude of possibilities that must be considered. Even if the worst-case stackup can be found, the resulting design is unrealistic since it assumes that everything is at worst-case on the same system at the same time. Using this criteria consistently throughout the whole design invariably results in component toler- ance requirements that are so tight the cost is prohibitive. The resulting system is greatly overdesigned. The application of statistical tolerance theory to iterative design problems overcomes this difficulty and provides a most real- istic picture of the control system behavior to be expected in pro- duction. Statistical tolerance theory was first introduced by Shewhart in his book "Economic Control of Quality of Manufactured Products" [20]. Following this, S. S. Wilks of Princeton University in the early 1940's published two papers [21], [22] that developed the statistical foundation for tolerance theory. However these articles and those that followed [23], [24], [25] up until as late as 1963 conceI | problem of as: chanical part: only to the 5. tion of the 0 system perforr parameters an The Monte Car nay be applie sented and th by Mark and L The Mont under investi Asystem IS 5 randomly from 93th componer Then each pa: in a tabulati °UIPUt varia: can be calcu.‘ The Home about the met. tinns of the I Partial deri'l 29 as 1963 concerned themselves almost universally with the design problem of assigning tolerances to the physical dimensioning of me- chanical parts. From a systems point of view, this case applies only to the situation where the system function is a linear combina- tion of the component parameters. In general, and for this example, system performance is a complex nonlinear function of the component parameters and the simple root-sum-square technique is not adequate. The Monte Carlo and Moment methods deveIOped in the last few years may be applied to handle this problem. Both techniques are pre- sented and the merits of each are compared by D. G. Mark [26] and by Mark and L. H. Stember [27]. The Monte Carlo technique assumes that each component parameter under investigation can be represented by a frequency distribution. A system is simulated mathematically by choosing each parameter value randomly from its frequency distribution. After parameter values for each component in the system are selected, a solution is obtained. Then each parameter value is again chosen as before and another solu- tion is obtained. This sequence is repeated many times, resulting in a tabulation of data representing the distributions of the desired output variables. From this, the resulting mean and 3 sigma values can be calculated. The Moment technique makes use of an expansion of the function about the mean parameters using a Taylor series. The higher order terms of the series are usually neglected. This requires taking the partial derivative of each performance variable with respect to each component parameter. Assuming that the component performance 30 parameters are independent and noting that the aYi/BXj = 0 if Xj is a component cost, the mean value of Yi is given by the equation Yi(mean) = Fi[:X1(mean), X2(mean), ooo, Xn(mean)] (2.21) and the standard deviation of Yi is approximated by the equation: er. 2 er. 2 ar. 2 w I Ia N H +---+IIo I—‘I 1 x1 1 x2 3x2 xn axn (2.22) where i = l, 2, --°, k and the partial derivatives are evaluated while all other parameters are held at their mean value. Since the higher order derivatives are neglected, the Moment method prediction is considered less accurate than the Monte Carlo method, but still adequate for most purposes. The Moment method has the advantage that it provides information that is extremely useful to the designer in pinpointing sensitive areas and reducing this sensitivity to parameter variability. Because of this latter advan- tage and the fact that satisfactory results can be obtained with a lesser number of computer runs, the Moment method is used here. As can be seen from (2.22), the use of the Moment method re- quires that we calculate the partial derivatives of each system per- formance functiOn with respect to each component parameter. The matrix of these partials is the Jacobian 31 r 1 er1 3Y1 3Y1 ax1 3x2 axn _ 3(Y1, Y2, ---, Yk) _ . 3Y2 av2 ... 3Y2 3(x1, x2, ---, xn) ’ ax1 3x2 x ark ark ... ark x1 3X2 aXn (2.23) Approximation of these partials is easily obtained numer- ically by programming (2.20) and using a subroutine to make the fol- lowing steps: 1) 2) 3) Set all the Xi's equal to their mean value (Xi), and the calculated Y vector is taken to be the mean value Y. X is replaced by (X1 + AXl) and the corresponding 1 value of Y }is calculated with all other X's at their mean value. From this, we obtain the first column of the Jacobian matrix using for i = 1, 2, ..., k and j = 1 Step 2 is repeated for each Xj for J = l, 2, ---, n thereby obtaining the complete Jacobian matrix. 3.1 BASIC Use 01 vious sect method, ne computer. the compm toward Op' cOl'ii'e‘onent system . Fig 111 a (1).]. fOr all data Ca 3. DEVELOPMENT OF COMPUTER OPTIMIZATION DESIGN PROCEDURE 3.1 BASIC APPROACH Use of the computer-aided design procedure developed in the pre- vious section, although many times more effective than any manual method, nevertheless represents only a passive use of the digital computer. That is, the engineer makes all the design decisions and the computer only serves as a fast calculator. The next logical step toward optimized design is to use the computer to determine how the components should be varied to converge on the desired minimum cost system. Figure 3.1 illustrates in general, how a computer could be used in a dynamic sense. The prerequisite to design is to input the data for all components. This is accomplished by loading in the component data cards pre-punched in a prescribed format. This need be done only the first time and thereafter only if that data is to be changed, e.gq,updated. These data are then stored by part number in an easily retrievable form on magnetic disk and are referred to as the "component libraries!‘ In order to provide the mainline design program with a ‘guide as to part number selection, some ordered array of these is desired. This is accomplished by using a "search matrix library," the precise working of which is explained later. Thus, immediately 32 ---——-—q I l l I r Figure 3.1. COMPONENT LIBRARY I I Immnmmue PARAMETERS TO COMPUTE PARTIALS 33 INPUT l. Specs. 2. Labor Cost 3. nitial I___C9mp cuts... COMPONENT DATA FROM LIBRARIES BY PART NUMBER ~ CALCULATE (mumUfiE COMPONENT SEARCH MATRICES — . JACOBIAN MATRIX l.TOTAL COST 2.REICHONIIIK) 3.COMPONENTS INPUT NEW COMPONENT SELECTION / DETERMINE NEXT COMPONENT SELECTION t PRINT 1. COMPONENTS 2. PERFORMANCE Computer aided design program flow chart. 34 after generation of the component libraries, the computer calculates the component search matrices and stores these in a second block of data-u-the search matrix library. Now the program is ready to be used. The designer inputs the system specifications, fixed produc- tion labor cost, and any initial set of components of his choice. The latter item could be made a random selection if desired. In either event, the computer retrieves the component data from libra- ries and proceeds to calculate the system performance. The component parameters are then perturbated one at a time and the partials of each system performance function with respect to each component para- meter are determined. Once this is completed the partials are stored in the form of a Jacobian matrix. The calculated performance limits are then compared to the specification limits. The fraction of the units produced that statistically fall outside of the specification limits is then calculated as the "rejection ratio." From this rejection ratio, the fixed labor cost, and the summation of the parts cost, the total cost is calculated. A printout is then made so that the user can follow the steps that the computer makes. Following this, some method must be employed to determine if cost is a minimum. If it is, then a final printout can be made. If it is not, then an option is shown as to how one wants to optimize. This can be accomplished by the user reading in another set of part numbers or the computer automatically can select a set in the manner described in Section 3.5 using the search matrix library. This procedure is repeated in an iterative manner until the optimum design is reached. 35 There are many associated details that are not shown in Figure 3.1. This diagram, however, gives the general outline of the pro- cedure. 3.2 GENERATION OF OBJECT FUNCTIONS The first question that must be answered in an optimization problem is, "What is to be optimized and what is optimum?” Often, this is not a trivial problem in itself since there are many separ- ate and usually conflicting factors; i.e., minimum cost, maximum accuracy, small volume, best response, etc. These factors may be considered simultaneously by defining a scalar F of the form k 2 F = _2 Ai(Yi-Di) (3.1) 1=l where: F = object function to be minimized k = number of desired properties Ai = weight factor selected to give the ith property the desired priority Y1 = current value of ith property Di = desired value for ith property A serious difficulty inherent in this approach, however, consists A ., A such that 1’ 2’ k scaling between the various terms is properly considered in order in finding a set of weighting factors A to maintain sensitivity and obtain good convergence. Considering fur: quir maxi: by 1: How give Thu; Tot C05- 36 properties such as accuracy, weight, cost and response, these weight selections often become subjective in nature. It is proposed in this thesis that an entirely different object fUnction shall be used. It is founded on the competitive philosophy that the manufacturer wants a design that fulfills the customer re- quirements at minimum overall cost. With this result, he can either maximize his chances of competing or if his sale price is "fixed" he maximizes his profits. Using this minimum cost philosophy, an appropriate object function can be generated in the following manner. The total cost to build a given number of systems is represented by the equation Total _ Number [Labor + Component] [1 + Overhead] Cost - Built Cost Costs Ratio (3.2) However, the number that must be built for a given contract is given by Number = Number Required (3 3) Built 1 _ Rejection ' [: Ratio Thus, we have for the total cost Total _ Number Required Labor + Component 1 + Overhead Cost - 1 _ Rejection Cost Costs Ratio [: Ratio :] (3.4) 37 Since the number of required units and (1 + overhead ratio) are product terms which are not functions of the components, one obtains the same cost minimizing set of components using the function Labor Component Cost + Costs COSt - Rejection (3'5) 1 - . Ratio Equation (3.5) is the object function used in this thesis for what is defined later as "the fine search mode." When it is at a minimum, the desired optimum set of components has been defined- However, one problem may exist in the early portion of the iteration cycle. That is, the design can be so far away from specification that, for all practical purposes, the rejection ratio is unity, the denominator of (3.5) goes to zero, resulting in infinite cost. As long as this occurs, (3.5) has no practical value. In fact, one loses all sensitivity in calculating partials, and there is no way of telling if one design is better than another. For this reason, a "coarse search mode" is defined. Its corresponding object function is: k - _ 2 F — . A1111”i si) (3.6) i=1 where F = object function to be minimized k = number of specifications to be met A1 = weight factor for ith specification . . . .th . . . R. = rejection ratio for i speCification de t1“. 38 Yi = calculated system performance 3 sigma limit corresponding to ith specification .th . . . . . Si = l speCification limit It should be further noted that Y1 = Yi-3aYi if Si is a lower limit, and i Yi+3oYi if Si is an upper limit. '-< ll Since Equation (3.6) is used only in the coarse search mode, selec- tion of the weight factors is not too critical. For this study, Ai was set at l/Si2 except for the case when Si equals zero and then Ai was arbitrarily set equal to unity. In the coarse search mode, cost is neglected in an attempt to determine the performance such that the rejection ratio becomes less than unity. The incorporation of the Ri term in (3.6) greatly aids in the accomplishment of this condition. First it nulls each term in the summation which represents an overdesigned condition (i.e. Ri=0) and secondly it applies a linearily increasing weight on the others according to their significance. Once each of the Ri's is driven less than unity, the cost becomes finite, and the optimization process is switched from the coarse to the fine search where (3.5) is used as the object function. 3.3 CALCULATION OF REJECTION RATIO Let Sl, ---, Sk be the k specification limits for a given design, e.g., static accuracy, overshoot, etc. There corresponds h I h‘h 39 then, R random variables Y1, -°-, Y that represent the actual k performance to be expected. Since these are a function of the n component equations, one can write as before that FYI‘ PFl‘M’°°”)%)fi Y2 = ‘2 (xv Km) 9.. w. (x1: x.) - m Looking at small perturbations F 1 F‘ 7 F' n AYI 311 312 "' a1n Axl AY2 = a21 a22 °'° agn sz AYk akl ak2 ~-- akn AXn (3.8) L .4 L ..4 L- .J BY. where the k_x n matrix has the general element aij = 52$- j therefore is identical to the Jacobian matrix (J) as defined in and it (2.23). The joint density of the Y's is given by: - [(Y-?)MY‘1(Y-Y)T J e Y1,Y2,...Yk(y1,y2,---yk) = (2")k/2‘\v/q3§;[ f (3.9) where and‘ Sinc and cova Sin< fali cat 40 where: (H?) = [(yl-yl).(y2-§2)» (’1'ka and the (k x k) covariance matrix MY is T MY = JMXJ (3.10) Since the component performance parameters are assumed independent and ox = 0 if xi is a component cost, one can write the component i covariance matrix Mx as r H o 2 O --- 0 x1 = o o 2 --- o Mx X2 0.. 2 _ 0 0 0‘xn d (3.11) Since the total rejection ratio R is the probability of a design falling outside of the specification, and assuming that the specifi- cation limits are constant, it is given by (L12 'L22 (Lkz R = 1 ' "' le’YZ’...’Yk(yl’y2’...’yk)dy1dy2 ... dyk J J J L11 L21 Lkl (3.12) 41 where: L. = -m 11 .th . . . for the i spec1fication an upper bound . = S. 12 i L. = S. 11 1 th for the i specification a lower bound Li2 - w and fyl, Y2» ..., YkCYl, yz. -'-. yk) ls given by (3 9). In order to evaluate R using (3.12), one must evaluate the multiple integral of dimension k where k = 8 for the example in this study. This can be accomplished using numerical techniques [28], and [29], however, the process is very time consuming. In the in- terest of minimizing computer time, three alternate procedures are considered. First one could use the upper bound on R which is simply II II M” 73 H0 H1 II MW 7U A H R(upper bound) 1 1 i 1 = 1 otherwise (3.13) where R1 is the individual rejection ratio corresponding to the ith specification and is calculated as rLi2 Y-UY 2 l_ i - 2 CY. R=1—_—l—— e ldy i 2nd 2 (3.14) Y. 1J L i1 42 where the limits of the integral are as defined for (3.12). Equation (3.14) can be evaluated by using the standard error function _ 2 -u2 ERF(Z) - W e du (3_15) o for the upper limit case Si-uYi R. = 0.5 1 — ERF for S. > u 1 V3 0Y 1 - Yi i “Yi'si = 0.5 l + ERF for S. < u ' /E 0Y. 1 Y1 1 (3.16) and fbr the lower limit case 51'“Yi R. = 0.5 l + ERF for S. :_u i /§'OY 1 Y1 i uYi-Si = 0.5 l - ERF for S. < u /2 CY 1 Y1 i (3.17) A second possibility for approximating the total rejection ratio R is to use the lower bound given by 3.1 43 R(lower bound) = Rj (3.18) where: R. < R. for all 1 < i < k J " 1 " " Since (3.13) represents an overdesigned case and (3.18) an underdesigned case, it would be wise to have available an approxi- mation that lies between these extremes. A quantity which has this property is k R(independent) = l - I_I'(l-Ri) (3-19) i=1 which is equal to the true R for the case when the Y's are in- dependent. For the example program, the user is given the opportunity of selecting either the R(upper bound) or R(independent) approximations. The R(lower bound), although readily available, is eliminated as a choice since it is never on the safe side. One difficulty remains since (3.16) cannot be used to calculate the rejection ratio for the null oscillation specification. This specification that no null oscillation shall exist is converted by the computer to a specification limit on the actual backlash. This limit is not a constant but a random variable computed as described in Appendix A. Therefore, the rejection ratio must be computed by examing two frequency distributions, namely that of the allowable backlash and that of the actual backlash of the geartrain being con- sidered. Thus a separate subroutine was written to calculate R6 the derivation of which is explained in the remainder of this section. 44 For this derivation only, the random variable Y is used to represent the actual backlash and S the allowable backlash. Since both are assumed to be normally distributed their density functions are defined as - 1(Y'uy)2 2 o g1(y) = ;;7é===§ e Y (3.20) ZNOY - i (“5)2 2 o g2(s) = 1 e S (3.21) and the corresponding rejection ratio is given by the probability that Y > S as m y R6 = PCY > 5) = g(y.5) dsdy (3.22) .00 .00 and since Y and S are independent 8(y.5) s g1(y)g2(5) (3 23) By using (3.23 in (3.22) and substituting in for g1(y) and g2(s) using (3.20) and (3.21), and simplifying, (3.22) can be written as r 1 y-“Y 2 ry 1 S"‘s 2 - — - — dsdy 2 o 2 o R _ l e Y 1 e S 6 - 2 2 21rdY -\/2nos 4-00 J-oo (3.24) 4S Letting A(y) = -———————- e ds (3.25) where A(y) can be evaluated by using the standard error function as before, one obtains y-u s .A = 0 5 ].+ ERF for > (y) (:gf;§;) Y —-“s (us-Y ) f = 0.5 1 - ERF or y < u s /§.°s (3.26) and since one is interested only in the region inside the 3-sigma limits, R6 is evaluated as {YMAX _ 2 _ (y ”Y) 20 2 1 Y R6 = -———————- A(y) e dy “ /21roY2~ JYMIN (3.27) where A(y) is evaluated using (3.26) and YMAX and YMIN are taken to be “Y + 30 and “Y - 30 respectively. Y Y 46 3.4 OBJECT FUNCTION DERIVATIVES It is of necessity that the partial derivatives of the object function be calculated in the steepest ascent method of optimization. If these derivatives were somehow known for the direct search tech- nique, it would be of advantage since one could then conduct explor— atory moves in descending order of importance. In our case, it would be a major task to perturbate each of the component parameters again and calculate the resulting change in the object function to obtain the partial derivatives. It is shown, however, that these can be obtained directly from the Jacobian matrix which is already available from the tolerance calculations, namely, Equation (2.23). This is accomplished in the following manner as derived first for the fine search and then for the coarse search. The object function used in fine search, Equation (3.5), can be written as C(X) = [x+f(X)][1-R(X)]'1 (3.28) where X = component parameter vector [:X1, X2, °°', Xn‘] C(X) = total system cost K = labor cost R(X) = rejection ratio f(X) = 2 component cost Taking the partial derivative of C with respect to Xi 3%E. = §§§§2.(1-R(x))-1 + (K+f(X))(l-R(X))-2 5§E- (3.29) 47 and expanding to include all Xi _a__C_ _a..c_ .332. . _.__1__ a; _a_f_ a_f 3X1 ’ 3X2 ’ ’ 3Xn l-R(X) 3X1 ’ 3X2 ’ ’ 3 mm [8R , 8R , .91 (3.30) (1_R(x))2 ax1 ax2 axn The latter vector (BR/3X) can be obtained by making use of the Jacobian defined by (2.23). Thus F -1 3Y1 ... aY1 3X 3X 1 11 an ER ... 3R aR aR .. an t . 9 ’ ’ .9 a .o o 0 3X1 8X2 3X BY1 3Y2 BYk 3X1 3X L.- “J (3.31) Substituting (3.31) into (3.30) one obtains the desired matrix equation for the fine search cost derivative vector as ac ac ... ac = 1 3f af ... af 3 1 ’ 3X2 ’ ’ axn l-R(X) 3X1 ’ 3X2 ’ ’ axk r- n 3Y1 aY1 Si—' '3?— 1 II + K+f!X! 2 33R ’ 33R , , 3:3 E E (1'R(x)) 1‘ 2 k aY aY _k . . . —k 3X1 3X I— n J (3.32) 48 where: 3f 1 if X1 is a component cost -——- = (3.33) i 0 otherwise and the vector 8 BY ’ 3 2 ’ ’ Yk R 3R 3R 1 is referred to as the "rejection ratio derivative vector" and given the notation aR/aY. The calculation of the aR/aY vector, as required for the fine search mode, depends on the particular equation used in approx— imating the rejectiOn ratio R [i.e., (3.13) or (3.19)]. Consider first the case where R is approximated by the upper bound. Since in the fine search mode < 1 "MW 56 i one has R(upper bound) = R1 + R2 + --- + Rk (3.34) and since Rj is a function of Yi~ only for i = j aRmLer bound) = _i_ for i = 1, 2, ..., k 3Y1- 3Y1. (3.35) and only the partials of the individual rejection ratios are required. This is also shown to be the case when R is approximated by using the case where the Y's are assumed independent as given by 49 R(independent) = l - (l-R1)(l-R2) "° (l-Rk) (3.36) Again using the fact that Ri is only a function of Y1 , one obtains 3R(independent) 3R1 k = ___. II - . 3.37 aYi BYi (1 RJ) ( ) i=1 3+1 The task remaining, then, is to obtain expressions for aRi/aYi . For the case where the Specification limit is a constant, the magnitude of aRi/BYi is given by the Yi» density function evaluated at the point yi = Si and the sign of aRj/BYi depends on whether Si is an upper or a lower bound. That is S.-u 2 -_1_ 1 Y1 2 0 3R1 :1 Yi 5— = —-——-e (3.38) i» -‘/2noY2 where .th . . . . . S1 = i spec1fication limit “Y = mean value of Y1 distribution 1 CY = standard deviation of Y1 distribution 1. and the + sign is taken if Si is an upper limit and the - sign is taken if Si is a lower limit. 50 For the case where the specification limit is not a constant but a random variable (e.g., 86), the corresponding equation is gy— = ’g1(2)g,_(2)dz (3.39) where the density functions g1(.) and' g2(.) are defined by (3.20) and (3.21) respectively. The solution to (3.39) is approx- imated in the example design program by using numerical integration over the region from “Y.-30Y. . . i l i i In summary, Equation (3.32) gives the required partial de- rivatives of cost with respect to each component parameter. The necessary elements of the rejection ratio derivative vector are Obtained using either (3.35) or (3.37) and with the aRi/BYi entries furnished by either (3.38) or (3.39) as the requirements dictate. A similar development is presented now for the coarse search mode. The object function used for coarse search is of the form [see (3.6)] F(x) ' A1R1(x) [Yl(x)'51]2 + A2R2(X)[Y2(x)-8232 + ... + AkRk(x)[Yk(x)_Sk]2 (3.40) where for the case of the example program the value used for $6 is taken to be its mean value. 51 Taking the partial derivative of F with respect to X ark 2 aRk * " ZAkRk(Yk'Sk) axl“ “ Ak(Yk‘Sk) ax"; (3.41) Thus, in total vector form: T F ‘ "M1 M1 M1 ‘ A1R1(Y1‘51) ax1 ax2 "' ax r- fl SF SF aF 3Y2 ayz 3Y2 ax ’ ax ’ '°°’ ax = 2 A2R2(Y2 S2 ax ax "' ax 1 2 n 1 2 L J : : : : Y -s ark ark ark AkRk( k k) ax1 3x2 ax n i- .4 b .J -‘T C 2 'aRI aR1 aR 7 A1(Y1'S1) ax1 ax2 axn a + A (y -3 )2 3E£_ 353. ..Eg 2 2 2 ax1 ax2 axn _ k( k R J __ax1 ax2 axn 3 (3.42) 52 Using the further relationship that: P '1 F' 'fi r' '- BR BR BR BR BR BR BY BY BY BX BX BX BY BY BY BX BX BX 1 n 1 2 k 1 2 n 3R2 BR ... 3R2 = BR2 BR2 ... BR2 3Y2 3Y2 ... BY2 BX1 BX2 BXn BY1 3Y2 BYk BX1 3X2 BXn BRk BRk BRk BRk BRk ... BRk BYk BYk ... BYk BX BX BX BY BY BY BX BX BX b 1 2 n_) L 1 2 k 4 L 1 2 n _ (3.43) BRi and noting that BY—'= 0 for all i + j, and substituting (3.43) in- j. to (3.42), one obtains P FTP aanl avl avl avl ] A1(Y1"51)R1 " (Y1'31)3'v_1' W1- '55 i'x: 23R av av av a? ' 'aixp' ' "" 3%:- ' 2 A2("2‘32)Rz * (Yz'sz) Ra 7331—2 f '_2' I 2 n 2 I 2 BX“ ‘ank av av av k k k Y-SR+Y-S -—- — —-"°— _‘1( k k) k (k k) 3ka 3": axa axnd (3.44) Equation (3.44) gives the desired partial derivatives of the coarse search object function with respect to each component para- meter in the system. Again, like (3.32),it is in terms of the al- ready available Jacobian matrix and no further parameter perturba- tions are required. 53 3.5 DESIGN PROGRAM STRATEGY The design program developed as part of this study has two basic operating options -analysis and directed search. When operating with the analysis option, the four component part numbers required for each analysis may be either read in from cards or selected at random by the program. In either case, as many consecutive runs are made as requested and a final printout is provided summarizing the best de- sign obtained. Thus the engineer can make a rapid evaluation of a selected number of designs of his choosing, or, he can perfbrm Monte Carlo runs by letting the computer select the part numbers at random. With the directed search option, the computer program uses the object derivatives in connection with search matrices to direct the next component selection in an attempt to reduce the object function. This process is repeated in an iterative fashion until a local mini- mum is obtained. Since there is no guarantee that this condition is the absolute minimum, numerous starting points are employed and the one with the lowest cost is assumed to be the best design. The starting points for each search may be specified by the user or otherwise selected at random by the program. The generation of the search matrices is a prerequisite to a directed search. A separate search matrix is used along with each couponent library and their generation automatically follows each library update. These matrices consist of an ordered array of the component part numbers defined by 54 Q. = S21 5‘22 522 ‘1' . sm1 sz '°° 5mg J (3.45) where 2 = the number of parameters used to describe the .th 1 component m = the number of part numbers for ith component stored in the library = a component part number for l §_n §.m and n3' 1 3_j :_2 Each column of ESE corresponds to a particular parameter of the ith component and the entries of that column consist of all the ith component part numbers arranged in ascending order of the mean value of that parameter. That is, let the jth column of Si correspond to the kth component parameter of the X vector. Then slj, st, ---, smj are chosen such that kuSU) i xk(521') i Xk(53j) 1'” 5- xk(5mj) (3.46) where xk(§nj) signifies the mean value of the component parameter Xk for the part number stored in location snj 55 In order to explain the strategy used by the design program to conduct a search, the following definitions are established: search = minimization process which begins with the initial set of part numbers and ends once a local minimum is found. basegpoint = set of part numbers for which the object function is less than that calculated for any previous set of part numbers in a given search. sub—search that part of a search which takes place be- tween successive base points. exploratory move a set of part numbers which are at least tentatively being considered for a system performance analysis. failure = an exploratory move which is analyzed and the object function obtained is greater than (or equal to) that of the base point. success . = an exploratory move which is analyzed and the object function obtained is less than that of the base point. local minimum = the object function corresponding to the base point which remains once all the exploratory moves analyzed in a given sub-search result in failure. Thus a search is made up of many sub-searches and each of the latter are in turn made up of numerous exploratory moves. Each exploratory move consists of changing one component part number while keeping the others fixed at the base point. Once an exploratory move 56 results in "success," the move is defined as a new base point and a new sub-search is started. This process is repeated until all the exploratory moves of a sub-search are exhausted and no success is found. minimum. The base point for this last sub-search defines the local The following ten steps describe the general pattern of the program's search strategy: 1) 2) 3) 4) The object function being minimized is SCALAR [defined as F in Equation (3.6)] while in the coarse search mode and COST [Equation (3.5)] while in the fine search mode. The program is in the coarse search mode as long as the total rejection ratio [Equation (3.13) or (3.19)] is equal to unity, once less than unity the program switches to the fine search mode. Each time a lower object function is found, the corresponding part numbers are stored as a new base point. At each new base point, calculations are made to establish the object function derivative vector using Equation (3.44) for the coarse search mode and (3.32) for the fine search mode. Priority and direction vectors are established as the bases for making exploratory moves. The priority vector (IPAR) consists of a re—ordering of the component parameter numbers (i.e., subscripts of the X parameter vector) such that 3 object 3_ 3 object :_--- 3_ 3 object (3.47) 8xIPAR1 3xIPAR2 axIPARm 5) 6) 57 where 1n, the dimension of IPAR, equals the number of component parameters excluding the load (e.g., with the vector defined in Table 2.5, m = 21). The direction vector (IDEX) is defined by 3 object BXII II ‘3 object BXII IDEX for 1 :_II :_m (3.48) Thus IDEXII +1 if the IIth parameter should be increased -1 if the IIth parameter should be decreased in order to achieve a reduction in the object function. A "sub-search progress number," denoted by the symbol II, is used by the program as the subscript for the IPAR and IDEX vectors. It is initialized equal to unity (i.e., II = l) at the beginning of each sub-search and incremented under pro- gram control as the sub-search progresses. As II is increased from 1 to m, IPAR corresponds to the component II parameter numbers having decreasing sensitivity values with respect to the object function. Likewise, IDEX IPARII corresponds to the desired direction the IPARII parameter is to be changed. Each exploratory move is initiated by calling a subroutine, named SEARCH, to select the new part number which is to be investigated. This is accomplished using the statement: CALL SEARCH[IDEX IPN IBOUND] IPAR ,IPAR 9 II JJJ II’ 58 where IDEXIPARII = direction IPARII parameter is to be changed IPARII = parameter number for change being con- sidered IPNJJJ = present part number on entering the subroutine and on return it is the new part number to be used IBOUND = 0 unless present part number is already at the boundary and cannot be changed further, then it is set to 1 by the sub- routine and for this example, the JJJ subscript is established from Table 2.5 as JJ = l for l §.IPARII §_3 = 2 for 4 f-IPARII §_8 = 3 for 9 :IPARII §_15 = 4 for 16 :IPARII §_21 The SEARCH subroutine takes the IPARII entry which corre- sponds to the subscript of the X vector and seeks the corresponding column of the appropriate search matrix. This column is then searched until the currently used part number is found (IPNJ Once this occurs the subroutine incre- JJ)' ments either down or up one location depending on whether IDEX is +1 or -1 and replaces the old part number with the new one found. If the old part number happens to be on a 7) 59 boundary such that a new part number cannot be obtained, the subroutine sets IBOUND to l and returns with the old part number. If this occurs, no further minimization can be obtained considering the IPARII parameter, therefore one returns the part numbers to the base point and increments to the next most significant parameter by increasing the sub-search progress number (II) by l and step 6 is repeated. For each new component selected by SEARCH a library sub- routine, named LIBR, is called to retrieve the corresponding parameter data. This is accomplished by the statement CALL LIBR[IPN XMAX,XMIN] JJJ’ where IPNJJJ= part number for which data is desired XMAX = a vector containing the mean +3 sigma values for the total X parameter vector XMIN = a vector containing the mean -3 sigma values for the total X parameter vector The LIBR subroutine takes the part number (IPNJJJ) and searches the appropriate component library, stored off-line on magnetic disk, until the part number is located. Once located its associated parameter data is read back and inserted in the proper locations of the XMAX and XMIN vector. Thus by calling the LIBR subroutine with a part number, one is able to automatically update the 3 sigma limits for the X's corresponding to that part leaving the others unchanged. 60 8) After the new data is obtained for the exploratory move, the program checks for the existence of two conditions before the system performance is evaluated. The first is used to control the extent that the program explores changes based on a given parameter before it moves on to the next para- meter. This is accomplished by calculating a normalized distance (DIST) according to XMINi DIST = m; for IDEXi > 0 XMINS. = —_i_ for IDEXi < o (3.49) where 1 = IPARII XMAXS = a vector containing the mean +3 sigma values for the total X parameter vector for the base point. XMINS = a vector containing the mean -3 sigma values for the total X parameter vector for the base point. This normalized distance is then compared to a program input parameter XNN. For XNN > 1, one is assured that the XIPARII random variable has been varied so that its frequency distribution inside the 3 sigma limits lies outside the distribution for the corresponding base point parameter. Thus by selecting the value of XNN, the program user can control the extent to which exploratory moves are made. A value of XNN = 1.5 was found to give satisfactory results 61 and is used for the examples presented in this thesis. By making XNN larger one explores more possibilities at the expense of increased computer time. Thus for DIST > XNN the program returns the part numbers to the base point, in- crements to the next most significant parameter by incrementing the sub-search progress number by l, and returns to step 6 above by calling SEARCH. If DIST :_XNN, the pro- gram continues to make the second check. This second check consists of calculating the estimated change in the object function based on its first derivative vector using the equation Aobject = i "MB 51—99199; [XNOM. - XNOMS.] (3.50) 1 3X1 1 1 where XNOM and XNOMS are the mean component parameter vectors corresponding respectively to the exploratory part number vector and the base point part number vector. Since the i = IPARII term in (3.50) is negative, one knows that if Aobject turns out to be positive, the summation of the changes caused by the parameters in IPNJJJ other than IPARII have resulted in an estimated increase in the object function. Since an increase in Aobject is undesirable, one returns to step 6 above, when Aobject >0 and calls SEARCH keeping the same sub-search progress number (II). If Aobject :9, a complete system performance analysis is made using the exploratory move part numbers. 62 9) If the exploratory move turns out to be "a success" (i.e., the object function is reduced) one returns to step 2 above and the process is repeated. If it is "a failure" (i.e., the object function is not reduced) one returns to step 6 and the next exploratory move is investigated. 10) The optimization procedure terminates once all the explor- atory moves made from a given base point are completed "without success." This base point defines the local minimum. Figure 3.2 is a simplified logic flow diagram for the total de- sign program. For simplicity sake, only the logic fundamental to the directed search option is included. The path used to update the component libraries, and to calculate and store the search matrices is shown by the single dashed line. The linkage between the design program and the component and search matrix libraries via the above subroutines is illustrated with the double dashed lines. In order to describe the operation of the program, the following additional program logic variables must be defined: MODE a l for the first analysis using the initial part numbers for each search 2 for all following analyses in coarse search mode 3 for all following analyses in fine search mode 4 for final analysis of each search ICOUNT = number of analyses that have been conducted as part of each search ISER = search number O‘ M E) 2% 3 E” 4.___— 5....--_____ uu “Longtime _ @ CW!“ Luau ‘Tlld‘l I "z: -_ % n:_-s.-L-.r1-o m" ‘ I _ ~4 -______—)-__ FT}!— m» soc-mun I l I CILCMYI PAIHALS [TIT—3 —rfiagy*** Figure 3.2. Design program simplified logic diagram. 64 NUMRUN = specified number of searches to be made in an attempt to find the best design IMAX = a maximum allowable iterations per seardh A design begins by the user inputting the system specifications, labor cost, and the required program logic after which the program initializes numerous parameters as shown in Figure 3.2. The program then branches according to whether the input is to be from cards or selected at random. Once the part numbers are obtained, the program is at point 20 and the desired component data is retrieved from the component library using the LIBR subroutine. Since MODE was initi- alized = l, the program branches to point 200 and the system performance is calculated, the component parameters are then per- turbated one at a time, and the system performance is re-evaluated. This process is repeated, using the steps described in Section 2.4, until all the entries in the Jacobian have been calculated after which the program calculates the rejection ratio, scalar, and cost (pro- viding the later is finite). At this point, a decision is reached whether or not to make an intermediate printout. In either event, ICOUNT is incremented by l and a check is made to see if it equals the maximum allowable value. Assuming the answer to be no, the program goes to the mode direction 'block. Since MODE = 1, it branches to point 1 and sets MODE = 2 (coarse search), branches to 41 and checks to see if the rejection jpercentage is less than 100. Assuming the answer to be no, the pro- gram stays in the coarse search mode, sets OBJECT = SCALAR and compares OBJECT to SAVE. If OBJECT < SAVE, as it is the first 65 time and thereafter anytime a lower object has been found, the last analysis is considered to be "a success" and the program branches to point 54 in order to store the data as a new "base point." For the cases when OBJECT > SAVE, the last analysis is considered to be "a failure" and the program continues on with the search using the old base point via point 58. At point 54, the new base point is established by storing the part numbers in a vector named IPS, the object function as SAVE, and the mean i 3 sigma component parameter vectors as vectors XMAXS and XMINS respectively. Following the establishment of each new base point, the elements of the gg-vector are calculated using (3.35) or (3.37) along with the object derivative vector which in the coarse search mode equals the scalar derivative given by (3.44). The priority and direction vectors IPAR and IDEX are then established as defined by (3.47) and (3.48). After this, the sub-search progress number (II) is initialized to unity and the program is at point 57 of Figure 3.2 ready to begin a sub-search by making exploratory moves to look for a smaller object function. At point 57, the program checks to see if aobject/BXIPAR =0. II If it is zero for a given progress number II, the program declares the base point to be a local minimum. This is accomplished by branching to point 63, setting MODE = 4 and returning to point 20 to terminate this search by repeating the best run obtained. 66 Consider now the case where for any given progress number II, the derivative is not zero. The program branches to point 58 and the SEARCH subroutine is called to select a new part number. Upon return from SEARCH the parameter IBOUND is examined to determine if one is at a boundary condition. If the answer is yes, no further minimization can be obtained considering the IPARII parameter, there- fore the program branches to point 59, returns the part numbers to the base point and increments to the next most significant parameter by increasing the progress number II by 1. If II is then greater than 21, the program has exhausted all possible parameters and there- fore branches to 63 declaring the base point as the local minimum and terminates the search. If II :_21, the program returns to point 57 and the exploratory moves are continued with the next least signif- icant parameter. Returning to point 40 and assuming that the new part number selected by the SEARCH subroutine is not on a boundary, the program branches to point 20 and the new component data is retrieved from the library and the program branches to point 64. At point 64, the program strategy checks for the existence of two conditions before an analysis is made. The first is to compute DIST as given by (3.49) and compare it to the program input parameter XNN. If DIST > XNN, the program branches to point 59, the part numbers are returned to the base point and the sub-search progress number is incremented to explore based on the next parameter. If DIST < XNN, the program branches to point 67 for the second check by calculating Aobject (3.50). If Aobject :_0, the program branches 67 to point 200 and an analysis is made, while if it is not, it branches to 58 and the SEARCH subroutine is called to make another part number selection. The above described coarse search procedure is repeated until R becomes less than 100% at which time the program branches to point 32 and sets MODE = 3 (for fine search) and OBJECT = COST and continues on as before at point 54, the only difference being that the object derivative vector is taken as the cost derivative (3.32). The program continues in the fine search mode until either the object derivative is equal to zero or all m parameters (this case m = 21) have been considered and the search is terminated via point 63. Execution of this termination is obtained by setting MODE = 4 and repeating the system analysis using the part numbers resulting in the lowest cost for this search (i.e., the base point), the latter being defined as a local minimum. Since this time MODE = 4, the program branches to point 43 and ISER is compared to NUMRUN. As long as the number of searches is less than the number requested, the program branches to point 73 where the cost is compared to the best local minimum. If the new cost is less, the vector IPBEST is updated with the new part numbers and BEST is updated with the corresponding cost. In any event, ISER is incremented by l and a new search is started. This process is repeated until ISER = NUMRUN whereupon the vector IPN is set equal to IPBEST and the analysis is repeated for the best local minimum. Upon exiting at point 43 (ISER > NUMRUN), the final printout is obtained at point 36. The results obtained using the above described program are described in the next section. 4. EXAMPLE DESIGN PROBLEMS 4.1 COMPONENT LIBRARIES AND SEARCH MATRICES The components selected to make up the libraries for this study, chosen so as to provide a broad base of design, are typical of those used throughout the servomechanism industry. The actual component parameter values used are listed in Tables 4.1 through 4.4 which con- sist of the "component libraries." Referring to these tables, the design problem is simply explained as "picking the one part from each table such that when combined in a system, they meet a given specifi- cation at minimum dollar cost.” Each column of the library data is labeled with the appropriate X-vector notation; i.e., X1, X2, ..., X21 each of which is assumed to be a random variable with a normal distribution defined for each component by the mean i 3 sigma limits given by the MAX and MIN values shown. The variables Xi for i = l, 4, 9, l6, and 21, which are the individual component costs and the gear ratio and have no manufacturing tolerance, are still treated as "random variables" but having zero variance; i.e., XMAXi = XMINi. It should be noted that many of the numerical units of measure for the variables are purposely included 2, etc.). as an inconsistent set (e.g., min of arc, rpm, oz-in, gm-cm This is done to place them in one-to-one correspondence with what is normally given in vendor catalogs and component specification sheets. 68 69 Table 4.1. Followup library data. X VECTOR NOTATION X1 X2 X3 PART NO. COST FOLLOWUP GAIN ACCURACY DOLLARS (VOLTS/RADI (MIN 0F ARC) 453234 MIN. Max—am... 1001 300.00 23.6000 21.4000 1.0 0.0 1002 24.00 12.7000 10.3000 10.0 0.0 1003 35.00 24.8000 20.2000 7.0 0.0 1004 200.00 0.5050 0.4950 30.0 0.0 1005 600.00 0.5025 0.4975 10.0 0.0 1006 28.00 24.8000 20.2000 15.0 0.0 1007 40.00 12.1000 10.9000 3.0 0.0 1008 36.00 12.1000 10.9000 7.0 0.0 1009 22.00 12.7000 10.3000 15.0 0.0 1013 30.00 0.5050 0.4950 120.0 0.0 1011 95.00 11.7000 11.3000 2.0 0.0 1012 90.00 0.5050 0.4950 60.0 0.0 1013 300.00 0.5050 0.4950 15.0 0.0 1014 60.00 24.800C 20.2000 3.0 0.0 1015 16.00 12.7CCC 10.3000 30.0 0.0 1016 30.00 25.9000 19.1000 10.0 0.0 1017 260.00 11.7000 11.3000 1.0 0.0 1018 150.00 23.6000 21.4000 2.0 0.0 1019 20.00 27.0000 18.0000 30.0 0.0 1020 28.00 0.5150 0.5050 180.0 0.0 1021 26.00 5.5000 4.5000 10.0 0.0 1022 30.00 5.2500 4.7500 5.0 0.0 1023 20.00 5.5000 4.5000 15.0 0.0 1024 28.00 5.2500 4.7500 7.0 0.0 1025 18.00 5.5000 4.5000 30.0 0.0 Table 4.2. 70 Amplifier library data. X1mCOM1NOUU10N x“ x5 x6 I x7 x8 PART N04 COST AMPLIFIER GAIN AMPLIFIER CAIN SAT. LEVEL OUTPUT NULL DOLLARS TU FOLLOHUP TU GENERATOR IVOLTSI (VOLTSI IVOLTS/VHLTI IVOLTS/VDLTI MAX. MIN. MAX. MIN. MAX. MIN; MAX. "IN; 2001 20.00 12. 8. 12. 8. 20.0 16.0 1.00 0.0 2002 75.00 55. 45. 110. 90. 25.0 17.0 0.50 0.0 2003 85.00 1150. 850. 1150. 850. 28.0 20.0 2.00 0.0 2004 200.00 55. 45. 11000. 9000. 28.0 20.0 3.00 0.0 2005 140.00 11. 9. 1100. 900. 25.0 17.0 1.00 0.0 2006 250.00 105. 95. 10500. 9500. 28.0 20.0 2.00 0.0 2007 170.00 5250. 4750. 10. 9. 26.0 18.0 1.00 0.0 2008 107.00 550. 450. 55. 45. 25.0 17.0 1.00 0.0 2009 220.00 5500. 4500. 11000. 9000. 28.0 24.0 2.00 0.0 2010 150.00 11500. 8500. 575. 425. 28.0 20.0 2.00 0.0 2011 160.00 55. 45. 5500. 4500. 26.0 18.0 1.00 0.0 2012 90.00 105. 95. 53. 48. 25.0 17.0 1.00 0.0 2013 185.00 1050. 950. 5250. 4750. 28.0 20.0 1.00 0.0 2014 180.00 105. 95.‘ 5250. 4750. 26.0 20.0 2.00 0.0 2015 50.00 55. 45. 11. 9. 23.0 17.0 2.00 0.0 2016 75.00 11. 9. 110. 90. 23.0 17.0 0.50 0.0 2017 175.00 5500. 4500. 110. 90. 28.0 20.0 1.00 0.0 2018 220.00 1100. 900. 11000. 0000. 28.0 24.0 2.00 0.0 2019 180.00 11000. 9000. 11. 9. 26.0 18.0 1.00 0.0 2020 170.00 11000. 9000. 1100. 900. 28.0 24.0 2.00 0.0 2021 200.00 550. 450. 11000. 9000. 28.0 20.0 3.00 0.0 2022 170.00 1050. 950. 105. 95. 26.0 18.0 1.00 0.0 2023 250.00 10. 9. 10500. 9500. 26.0 18.0 2.00 0.0 2024 60.00 110. 90. 11. 9. 23.0 17.0 2.00 0.0 2025 180.00 525. 475. 525. 475. 26.0 18.0 1.00 0.0 2026 142.00 55. 45. 1100. 900. 26.0 18.0 1.00 0.0 2027 170.00 5500. 4500. 550. 450. 28.0 20.0 1.00 0.0 2028 130000 11000 900. 55. ‘05. 2600 3800 1000 0.0 2029 170.00 550. 450. 5500. 4500. 28.0 20.0 2.00 0.0 2030 170.00 11000. 9000. 5500. 4500. 28.0 24.0 3.00 0.0 2031 185.00 11000. 9000. 55. 45. 28.0 20.0 1.00 0.0 2032 130.00 110. 00. 1100. 900. 26.0 18.0 2.00 0.0 2033 165.00 5500. 4500. 1100. 900. 28.0 20.0 2.00 0.0 2034 70.00 53. 48. 53. 48. 23.0 17.0 2.00 0.0 2035 130.00 53. 48. 525. 5. 25.0 17.0 1.00 0.0 2036 165.00 5500. 4500. 55. 45. 26.0 18.0 1.00 0.0 2037 110.00 550. 450. 1100. 900. 28.0 20.0 3.00 0.0 2038 105.00 5‘0. “50. 310 0. 2500 1700 1000 0.0 2039 160.00 11. 9. 5500. 4500. 26.0 18.0 3.00 0.0 2040 110.00 550. 450. 110. 90. 26.0 18.0 1.00 0.0 2041 170.00 5500. 4500. 5500. 4500. 28.0 24.0 1.00 0.0 2042 160.00 11000. 9000. 110. 90. 28.0 20.0 2.00 0.0 2043 85.00 11. 8. 575. 425. 25.0 17.0 2.00 0.0 2044 100.00 110. 90. 550. 450. 26.0 18.0 2.00 0.0 2045 160.00 1050. 950. 10. 9. 25.0 17.0 1.00 0.0 2046 70000 310. 90. 81°. 90. 25.0 1700 1000 0.0 2047 30.00 12. 8. 60. 40. 23.0 17.0 1.00 0.0 2048 130.00 1100. 900. 550. 450. 28.0 20.0 2.00 0.0 2049 300.00 10500. 9500. 10500. 9500.‘ 32.0 24.0 2.00 0.0 2050 250.00 306. 294. 306. 294. 26.0 20.0 2.00 0.0 71 06.6. 6.6 6.0 6.6 0.6 66.0 om.~ 66—.0 oc~.o .uom6 .0606 upwo.o 0660.0 60.66 mNnm oo.6~ 6.0 6.6 0.0 0.6 mm.“ om.~ 666.. 066.“ .oooom .6666N 0666.0 6006." 66.66 «~66 66.60 6.0 uo.m_ 60660.0 onoo~.o 06.0 co." 666.0 066.6 .0066 .006» uo-.o 6066.0 06.66 m~nm 66.6: 0.6 6.0 6.6 c.o o.o oo._ 666.6 066.: .6666 .ooco uUm~.u oosm.o 06.66 «Nam oo.o~ 0.0 6.0 6.0 c.o 6.0 66.. 066.0 ooc._ .6066 .ooaud 6666.0 6066.0 06.66 -nm oo.~r c.o co.- 66666.0 66066.0 o.o o~.~ pom.~ oos.— .oooo .ooo- Coo—.0 oo6~.c on.m- nmnm co.m~ c.o 0.0 6.6 0.0 66.0 co.“ 066.0 666.0 .6606 .~om> 6666.6 06-.o 66.66 6666 oo.m~ 0.6 6.6 6.0 0.0 06.6 on.“ 660.0 oo~.o .6066 .soms 6666.6 «066.6 06.6 6~om oo.m~ 0.6 0.0 6.0 6.0 06.0 on.“ 060.6 006.0 .6606 .6666 ~¢~o.u mo~_.o oo.~6 sanm oo.m~ 0.0 00.66 oco-.o 66666.0 66.0 oo.~ 066.6 com._ .6066 .ooo_H uu-.6 umm~.o 66.66 6~om oo.6~ o.o o.c 6.0 0.0 66.0 66._ o~o.o oc_.o .0666 .0066 Cos—.0 cco~.o 66.66 6666 06.6w 0.0 00.66 66666.6 00666.0 66.0 om.~ 006.0 cc..— .0666 .0000" goo—.6 6066.0 oo.~m «dam on.: 0.0 00:: ammocd m-66.c 06.0 064 coo.~ 006.6 .0066 .0006. 6.66.6.0 0066.0 00.2: 62.6 00.6w 0.0 06.6 6m~oo.o o~mno.o 06.0 o~.~ 666.6 606.0 .0066 .0066 coo—.0 0066.0 00.6» Ndnm 00.66 0.0 00.6 0666—.6 ooo-.o o.o 0n.~. 666.0 666.0 .6066 .0066 6666.6 0666.6 06.66 -cm oo.6~ 0.0 00.3 0.36.6.0 66066.0 64.0 00.0 coo.~ 000.6 .oomo .0006 6663...... 9.65.0 00.66 San oo.6~ o.o oo.- 06666.0 chm—6.0 06.0 o~.~ 666.0 666.0 .6066 .6060 6666.6 oom~.c 06.60 66nm on.o~ 0.0 06.6. c¢m~m.o 65666.6 66.0 66.0 060.6 066.6 .6066 .0006 6600.0 6666.0 00.66 room oo.6~ 0.0 6.6 0.6 0.0 66.6 om._ o6o.o c4..o .0066 .ooc- 66-.o 0066.6 oo.o~ Foam oo.- 0.0 00.6. 0206.0 2.36.0 06.0 00.0 com; 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Also, with respect to the motor data, the effective values of stall torque (TS) and no-load speed (ém) are functions of the voltage capability of the driving amplifier. In order to account for this effect, the values used in the program are obtained from the rated condition given in the library by the equations sat 6m(effect1ve) = 6m(rated) BC (4.1) Esat Ts(effective) = Ts(rated) E (4.2) c where Ec is the rated control voltage of the motor and Esat is the amplifier saturation level, both of which are included as part of the library data. Although (4.1) and (4.2) are not exact [see reference (2)] they are considered adequate for this study. In ad- dition to the above, the motor torque gain (Km) and damping coeffi- cient (Bm) are terms that are used by the systems engineer and are required as part of this study; however, they are normally not pro- vided directly by the vendor. They must be calculated from what is normally provided; no-load speed (ém), stall torque (TS), and rated control voltage (EC). The equations used are Ts(rated) Km = ——_E_——___ (4.3) c Ts(rated) B =———— (4.4) m ém(rated) 74 The search matrices are generated immediately after the library data is stored in the computer system. These search matrices are shown as Tables 4.5 through 4.8 and consist of the component part numbers arranged in an ordered array as previously explained in Section 3.5. The computer program developed as part of this study has been tested on several design problems,each employing different specification sets. Each one has met with about equal success; however, the two that have received the most comprehensive study are presented here as examples and comprise the rest of this section. 4.2 FIRST DESIGN EXAMPLE In order to demonstrate the application of the program in its most comprehensive form, a customer requirement is assumed which makes use of all eight specifications. The particular set is: 1. Static accuracy = 1.0 degrees 2. Resolution = 0.5 degrees 3. Velocity lag for 90 deg/sec input = 5 degrees 4. Followup rate 90 deg/sec S. Damping ratio 0.3 6. Null oscillation = none allowed 7. Overshoot for 10 deg step = 2.0 degrees 8. Bandwidth for 2.0 deg peak sinusoid = S hertz The assumed labor cost is $200.00. First to be considered are the results obtained by using the above specification set and the program operating in the Monte Carlo mode. 75 Table 4.5. Followup search matrix. COST KéF THETA 1015 1010 1017 1025 1005 1001 1023 1004 1018 1019 1012 1011 1009 1013 1014 1002 1020 1007 1021 1024 1022 1006 1021 1024 1024 1025 1003 1020 1022 1008 1022 1023 1021 1010 1015 1005 1016 1007 1002 1003 1011 1016 1008 1009 1013 1007 1017 1006 1014 1008 1023 1012 1002 1009 1011 1014 1015 1013 1006 1025 1004 1003 1019 1017 1018 1004 1013 1016 1012 1001 1001 1010 1005 1019 1020 ' a. 3:. 76 Table 4.6. Amplifier search matrix. COST K-AF K-AG SAT NULL 2001 2001 2001 2001 2002 2047 2043 2045 2047 2016 2015 2047 2019 2034 2027 2024 2023 2024 2015 2036 2046 2039 2015 2024 2008 2034 2016 2038 2016 2028 2016 2005 2007 2046 2022 2002 2015 2012 2035 2012 2003 2011 2047 2005 2019 2043 2026 2008 2012 2001 2012 2034 2036 2002 2047 2044 2004 2034 2045 2013 2038 2002 2028 2043 2007 2008 2035 2031 2008 2031 2040 2032 2040 2038 2025 2037 2044 2022 2007 2046 2028 2006 2017 2028 2045 2048 2046 2002 2026 2038 2035 2014 2042 2022 2005 2032 2012 2016 2011 2040 2005 2024 2046 2040 2011 2026 2050 2035 2023 2041 2010 2029 2050 2019 2035 2011 2025 2027 2036 2017 2045 2021 2025 2044 2026 2042 2040 2048 2025 2015 2039 2038 2044 2032 2029 2036 2037 2010 2039 2014 2033 2008 2043 2014 2003 2030 2028 2026 2050 2050 2007 2013 2003 2031 2048 2029 2003 2020 2029 2024 2027 2048 2005 2027 2006 2022 2045 2037 2013 2023 2041 2022 2033 2048 2044 2020 2018 2032 2006 2010 2017 2007 2030 2003 2020 2014 2027 2029 2042 2049 2025 2041 2014 2021 2018 2019 2009 2013 2010 2009 2031 2036 2011 2037 2042 2013 2017 2041 2004 2043 2021 2033 2039 2017 2034 2004 2031 2049 2033 2033 2018 2030 2006 2030 2032 2009 2049 2023 2041 2030 2050 2042 2021 2020 2021 2006 2020 2018 2018 2039 2023 2010 2009 2009 2037 2049 2019 2004 2049 2004 77 Table 4.7. Motor-generator search matrix. COST T-S T-M-D J-M ESTART K-G NULL 3007 3025 3003 3C15 3011 3024 3022 3002 3003 3005 3C19 3022 3022 3001 3015 3019 3012 3017 3021 3007 3018 3016 3017 3004 3001 3004 3017 3007 3021 3018 3008 3018 3008 3021 3015 3024 3004 3011 30C7 3020 3001 3025 3003 3001 3023 3025 3001 3015 3021 3022 3008 3013 3022 3010 3025 3024 3004 3009 3010 3012 3006 3019 3019 3001 3005 3015 3009 3015 3018 3017 3025 3014 3006 3023 3023 3012 3012 3023 3023 3018 3003 3005 3023 3011 3008 3020 3019 3011 3018 3005 3005 3014 3007 3017 3005 3017 3011 3006 3006 3011 3002 3021 3016 3003 3003 3011 3002 3025 3008 3019 3020 3002 3010 3015 3009 3014 3012 3002 3020 3017 3016 3022 3002 3009 3016 3009 3012 3012 3014 3004 3025 3008 3014 3018 3022 3007 3016 3007 3004 3023 3019 3021 3021 3020 3014 3006 3010 3005 3013 3016 3006 3013 3009 3008 3009 3010 3020 3024 3024 3014 3004 3013 3006 3001 3013 3003 3010 3013 3020 3024 3024 3010 3002 3013 3016 78 Table 4.8. Geartrain search matrix. COST J-G K-S T-G B N 4014 4014 4014 4014 4001 4001 4009 4020 4017 4020 4012 4022 4020 4022 4021 4021 4021 4020 4013 4002 4023 4023 4004 4014 4023 4009 4002 4009 4003 4002 4006 4013 4009 4013 4017 4023 4018 4001 4005 4002 4015 4009 4025 4023 4001 4024 4008 4019 4007 4018 4013 4022 4019 4024 4002 4019 4010 4001 4022 4021 4010 4024 4012 4017 4011 4015 4024 4015 4004 4018 4024 4013 4005 4021 4025 4004 4005 4005 4011 4011 4020 4019 4002 4018 4016 4006 4016 4005 4016 4004 4022 4005 4024 4015 4007 4007 4019 4007 4022 4003 4023 4003 4015 4004 4007 4011 4020 4011 4017 4025 4018 4010 4010 4006 4008 4017 4003 4006 4018 4017 4021 4010 4006 4025 4014 4025 4001 4003 4015 4007 4013 4012 4004 4016 4011 4012 4025 4010 4003 4008 4019 4016 4006 4016 4012 4012 4008 4008 4009 4008 79 This is accomplished by instructing the computer to make multiple analysis runs selecting the component part numbers at random. Table 4.9 illustrates a typical program output by showing the first 50 lines of intermediate printout. Each line represents an analysis run and lists the cost (3.5), scalar (3.6), total reject (3.19), the four component part numbers used, and the individual specification rejec- tion percentages [Ri using (3.16) or (3.17) for i = l, --', 8 exclu- ding R6 which is given by (3.27)] arranged in order as R1, R2, °°-, R8. For this example, the total percentage rejection was calcu- lated using the assumption that the Y's are independent [i.e., Equation (3.19)]. As shown, 43 out of the 50 runs illustrated have 100% rejection and therefore an infinite cost (shown as **** when cost :_1. x 106 dollars) while run number 20 with a cost of $663.28 and a percent rejection of 37.88 is the best of the 50. A total of 467 Monte Carlo runs where made (about 1.5 hours of computer time on an IBM 360 model 50). The lowest cost unit found was $374.02, with a zero percent rejection, using part numbers 1008, 2015, 3008, and 4006. Table 4.10 is the final output sheet obtained summarizing this design. The results obtained using the program in the direct search mode now are illustrated in detail for two searches. The first, shown in Table 4.11, is a case where the initial guess fails completely to meet 4 out of the 8 required specifications, thus resulting in an infinite cost. Twenty-seven iterations are required by the program to minimize the scalar object function to the point where the cost becomes finite and the program switches from the course to the fine search mode. It should be noted that for this run and most subsequent computer runs, 8O Table 4.9. Intermediate Monte Carlo printout for first de51gn example. 1 PERCENI COMPONENTS SELECTED 089888.1N01V10UAL SPECIFICAVIUN IEJECVIONSOOOOOOOO £051 SCALAR REJECI FOUP AND NOCIN 6818 $7811; RES 146 DAMP NULL 0968 84 0 090000000 6.041E901 100.00 1011 2047U3005 4005 100.00 100.06 10 . O. . . . 000000000 8.508f901 100.00 1022 2033 3025 4020 0.0 0.0 0.0 100.00 100.00 100.00 0.0 99999999. 1.856E9OZ 100.00 1023 2047 3017 4007 100.00 100.00 100.00 0.0 0.0 0.0 100.00 9.0900000 2.3617904 100.00 1003 2021 3002 4012 0.00 0.0 100.00 0.0 0.0 0.0 100.00 09.99.99. 3.780E901 100.00 1009 2(16 3010 4010 51.34 99.91 100.00 0.0 0.0 0.0 100.00 000000000 3.7831901 100.00 1019 2028 3017 4014 0.0 0.0 0.0 100.00 100.00 100.00 0.0 99000000. 3.4571901 100.00 1009 2040 3018 4001 0.0 0.0 0.0 0.0 100.00 0.0 100.00 0.0 99999999‘ 1.047(006 100.00 1022 2006 3004 4016 99.70 0.00 100.03 100.00 0.0 0.0 0.0 100.00 99.99.99. 1.409(903 100.00 1014 2039 3024 4021 33.91 100.00 0.0 0.0 100.00 0.0 100.00 0.0 00000000. 3.4516901 100.00 1002 2045 3015 4022 0.0 0.0 0.0 0.0 100.00 63.50 99.98 0.0 09900090. 4.944(000 100.00 1002 2028 3018 4010 0.0 0.0 0.0 95.13 100.00 100.00 0.0 0.0 90999099. 1.636t902 100.00 1022 2044 3016 4001 93.82 92.35 100.00 0.0 0.0 0.0 0.0 100.00 600066690 5.751(901 100.00 1002 2001 3020 4021 99.89 100.00 59.41 0.0 31.34 0.0 50.46 100.00 009909000 8.530(902 100.00 1018 2009 3009 4002 0.0 0.0 0.0 0.0 0.0 100.00 0.0 0.00 9.0999909 8.4386900 100.00 1006 2042 3016 4006 0.0 0.0 0.0 0.0 100.00 100.00 100.00 0.00 999999999 1.1866904 100.00 1017 2014 3002 4018 86.81 0.00 100.00 0.0 0.0 0.0 0.0 100.00 99.90.99. 5.040(900 100.00 1006 2012 3005 4009 0.0 0.0 0.0 0.0 100.00 99.73 93.34 0.0 37928.44 2.2666901 98.81 1019 2048 3002 4016 0.0 0.0 0.00 0.0 61.08 0.0 86.48 0.9990999 1.2221905 100.00 1008 2004 3008 4005 99.70 67.82 100.00 0.0 0.0 0.0 100.00 663.28 5.156E9OO 37.88 1021 2046 3017 4007 0.0 37.88 0.0 0.00 0.0 0.0 0.0 09999900. 2.327f902 100.00 1023 2034 3013 4012 0.65 95.52 100.09 0.0 0.0 0.0 100.00 99.09.99. 1.221E901 100.00 1001 2031 3015 4017 0.0 0.0 0.0 100.00 100.00 0.0 0.0 00000000. 6.974F9OU 100.00 1019 2016 3024 4004 7.18 100.00 0.0 100.00 0.0 100.00 0.00 999999999 2.1741003 100.00 1005 2003 3014 4019 92.41 99.94 100.00 0.0 0.0 0.0 100.00 099900... 8.1106003 100.00 1003 2011 3004 4021 96.05 0.00 100.00 0.0 0.0 0.0 100.00 999909... 1.0016901 100.00 1015 2040 3013 4010 0.0 0.0 39.38 0.00 3.48 0.0 100.00 060060000 9.3246900 100.00 1022 2050 3023 4007 0.0 0.0 88.29 0.00 0.0 0.0 100.00 099.000.. 1.464E901 100.00 1022 2007 3003 4022 0.0 0.0 0.0 100.00 100.00 100.00 0.00 099999090 8.316%904 100.00 1016 2004 3009 4022 98.98 2.17 100.00 0.0 0.0 0.0 100.00 099000000 5.791E001 100.00 1010 2021 3018 4014 99.93 100.00 0.00 100.00 0.0 100.00 0.0 9.9900009 1.2276901 100.00 1018 2011 3020 4024 91.68 0.00 99.99 0.00 0.0 0.0 100.00 3918.51 2.7466901 83.74 1021 2008 3010 4001 0.0 0.0 0.0 0.13 0.0 83.72 0.00 000000000 3.317(900 100.00 1025 2012 3005 4006 0.00 45.55 99.63 0.0 0.0 0.0 100.00 090000000 8.918E901 100.00 1010 2009 3001 4020 66.39 0.0 0.0 100.00 0.0 100.00 0.0 90099090. 1.4151904 100.00 1012 2032 3016 4001 99.97 100.00 100.00 0.0 0.0 0.0 100.00 9.0900090 2.3071904 100.00 1017 2018 3016 4016 51.81 0.0 100.00 0.0 0.0 0.0 100.00 09999099. 2.190f905 100.00 1007 2023 3011 4021 99.93 100.00 100.00 0.0 0.0 0.0 100.00 183335.81 1.321E901 99.67 1018 2040 3009 4010 0.0 0.0 0.0 0.0 98.43 0.0 0.06 009909090 5.194E900 100.00 1006 2040 3022 4007 0.0 0.0 0.0 100.00 100.00 0.0 0.0 0.9990900 1.4081901 100.00 1003 2027 3014 4012 0.0 0.0 0.0 0.0 100.00 0.00 89.73 2185.52 2.1766901 69.71 1021 2033 3008 4021 0.0 0.0 0.0 0.0 69.71 0.00 0.0 90000000. 1.0206901 100.00 1022 2017 3007 4011 0.0 0.0 0.0 100.00 99.70 0.00 0.0 00000000. 4.4781901 100.00 1018 2029 3019 4022 0.0 0.0 0.0 100.00 1.12 100.00 0.0 00000099. 2.846(902 100.00 1015 2012 3024 4022 0.00 0.0 0.0. 100.00 100.00 100.00 0.0 09999990. 6.4491900 100.00 1020 2019 3005 4003 85.41 0.0 0.03 100.00 21.71 0.26 0.00 129559.62 4.2461900 99.56 1008 2035 3017 4008 0.0 0.00 0.0 9 0.0 0.0 0.0 0.0 00990009. 1.4361902 100.00 1017 2040 3024 4024 0.0 0.0 0.0 100.00 100.00 100.00 0.0 90090909. 1.0031901 100.00 1017 2017 3018 4007 0.0 0.0 0.0 100.00 100.00 0.0 0.0 09999909. 1.186(00 100.00 1017 2014 3002 4018 86.81 0.00 100.00 0.0 0.0 0.0 100.00 Table 4.10. 81 Best design obtained using Monte Carlo for first design example. JANUARY 209 AUTOMATED DESIGN RESEARCH PROGRAM 1969 9’99DEFINITION OF LOA0999‘ FRICTION (OZ-1N1 9999PART 1008 MAX MIN INERTIA IGM-CMSQRI 9.000E902 7.000E902 0.000E-01~ 4.000E-01 NUMBERS OF COMPONENTS SELECTED9999 FOLLOHUP AMPLIFIER MOTOR-GEN GEAR TRAIN 2015 3000 4006 9‘99PERFORMANCE9999 MAXIMUM MINIMUM SPEC LIMIT PCT REJ 4.100E904 3.020E004. - TOTAL INERTIA IGM-CMSQRI 6.141E402 4.0326002 TOROUE CONSTANT IOl-IN/RAOI 1.523E901 1.031E001 DAMPING COEFFICIENT IOl~IN~SEC1 5.210E000 4.226E400 NATURAL FREQUENCY IHERTZI 4.764E‘01 2.1936-01 1.000 0.0 STATIC ACCURACY IDEGI 4.649E-01 2.647E-01 0.500 0.00 RESOLUTION IDEGI 2.973E900 2.242E900 5.000 0.0 LAG FOR 90. DEG/SEC RAMP IDEGI 1.639E002 1.122E902 90.000 0.00 FOLLOHUP RATE IDEGISECI 4.2026-01 3.269e-01 0.300 0.00 OAMPING RATIO 4.500E901 2.250E401 SEE BELOH 0.0 BACKLASM IMINI 1.712E400 1.107E000 2.000 0.00 OVERSHOOT FOR 10. DEC STEP IDEGI 6.4706900 5.243E000 5.000 0.00 BANOHIDTH FOR 2. DEG SINE INERTZI ALLOUABLE BACKLASH SPECIFICATION (MINI ‘MAXIMUM 8 3.0616902 MINIMUM I 9.7426901 99"COST SUMMARY9999 0.00 PCT REJECTION (UPPER BOUND! 0.00 PCT REJECTION IINDEPENOENTI 0.00 PCT REJECTION ILOUER BOUND) 200.00 LABOR COST 174.00 PARTS COST ' 374.02 TOTAL COST IUSING R-INDEPENDENTI E SIGNIFIES CONVENTIONAL POHER-OF-TEN NOTATION 82 . 11.1!1l 0 0.0 0.0 00.~ 00.0 0.0 0.0 00.0 00.0 _n~00 m~0m m_uw n_0~ wo.~ ~o+wcwm._ 05.n0m «h n 0.0 0.0 o0.~ 00.0 0.0 0.0 00.0 00.0 n~00 maom maum m~o~ wo.~ ~c+wqwm.~ 05.n0m 00 n 0.0 0.0 .0.0 00.0 0.0 0.0 00.0 00.0 n—oc m~0m «nu~ mac" “0.0 ~0000wm.~ "0.0mm mm m 0.0 0.0 00.— 00.0 0.0 0.0 00.0 00.0 Mao; muom meow muod wc.~ ~o+0>~0.~ no.mom _m n 0.0 0.0 0.0 0.0 0040 0.0 00.0 00.0 0000 m—om ~00~ m~o~ 00.0 bo+wm00.p 0n.00m 00 m 0.0 0.0 0.0 0.0 00.0 0.0 00.0 00.0 0000 -0m Noon 0‘0" 00.0 co+uemm.~ 00.npm on m 0.0 0.0 00.~m 00.0w 00.0 0.0 0.0 00.0 0000 -0m -0~ m~0~ 00.~0 000000~.m w~.mmn~ mm m 0.0 0.0 00.~m on.0~ 00.0 0.0 0.0 0.0 0000 -0m ~_c~ ooo— m0.~0 00+w~mm.m 00.0mo~ on N 0.0 0.0 00.~m cm.0~ 00.0 0.0 0.0 0.0 0000 -om -0~ ~00. m0.~0 00.0000.m -.000a >~ ~ 0.0 0.0 00.00u 00.00“ 00.0 0.0 0.0 00.0 0000 -0m -0~ 0.0” 00.000 009wowm.n 99.9.9999 ma ~ 00.00~ 0.0 0.0 0.0 00.0 00.00— 00.00— 00.000 0000 wuom Ndum 0000 00.ru~ «n+mwnr.o 9.9099090 :~ ~ 00.00— 0.0 0.0 0.0 00.0 00.00— 00.00~ 00.00~ 0000 naom -0~ cucd o .no~ Noowmoo.~ 9.0.0999: - ~ 00.00— 0.0 0.0 0.0 00.0 00.00. 00.00— 00.00“ 0000 muum Nucn n.0— co.oc~ Roomsmn.~ 9.90.909. A. ~ 00.00— 0.0 0.0 0.0 00.0 00.00_ 00.00— 00.000 0000 poem -0~ mac" 00.00" woowmwm.~ 99.999590 0 ~ 00.00— 0.0 0.0 0.0 o~.0 00.00— 00.000 00.00~ 0000 mwom -0~ wdou 00.no~ ~u+wom~.n 999999909 m N 00.00u 0.0 0.0 0.0 -.o oo.00~ 00.00— 00.00~ 0000 -0m w—ow m_o~ 00.00" ~o+woo¢.m 00.9.9999 p ~ 00.00— 0.0 0.0 0.0 00.0 00.00— 00.00— 00.00~ 0000 0~om -0~ -u~ oc.tc_ ~m+ummn.¢ 00.0099.» 0 ~ 00.00u 0.0 0.0 0.0 00.0 00.00— 00.00— 00.00— 0000 000m -0~ N—o— 00.nc~ ~n+m0>m.m «9.00999» 0 ~ 00.00u 0.0 0.0 0.0 00.0 00.00" 00.00" 00.00~ 0000 000m -0m -u~ oo.ou~ Noowow~.o 99099905» m w 00.00u 0.0 0.0 0.0 00.0 oo.oo~ co.00— 00.00» 0000 000m ngm -o~ 00.0.. mo.-on.~ 999.9999; ~ 00.n0~ 0.0 0.0 0.0 00.0 00.00— 00.00~ 00.00~ 0000 000m -¢~ -n~ 00.n0~ n+mwon.~ 900099.90 d w 020: aw>0 44:2 040 20m u—qum mucaam 0:00 apac 2000; 014 azcu puusaa xcaqum hmau .0: 0 O'Dooouomzoupumswx zo_p—0z_699099§ Qwhumawm mFszcazou pzwuaw 20¢ o I 1 909600999 a 000:0: x0a0.0 0.0 0000 0000 mm0~ 0000 >0.n 00+m-m.0 «p.000 NM 0 0.0 0.0 0.0 0.0 00.0 0.0 0.0 0.0 000: m00n mm0~ 0000 00.0 009w~0m.~ 00.000 0n n 0.0 0.0 0.0 0.0 00.0 0.0 0.0 .0.0 0000 0000 mm0~ ~000 00.0 0090n~0.> 00.~mc on m 0.0 . 0.0 0.0 00.0 00.0 0.0 0.0 0.0 0000 000M 0m0~ 0000 00.0 oo+w¢~o.~ 00.009 0~ n 0.0 0.0 0.0 00.0 00.0 0.0 0.0 0.0 0000 n00m 0~0~ 0000 00.0 0090000.0 00.x00 00 m 0.0 0.0 0.0 00.0 00.0 0.0 0.0 0.0 0000 000m 0~0~ 0000 00.0 009w¢0n.o 00.000 ~ 0.D 0.0 0.0 00.0. 00.0 0.0 rb.0 0.0 0000 m00m 0~0~ M000 00.0 0090000.o 00.0%H1. 0Il w 0200 uw>0 0002 0:40 004000 000 m0« 00p00209999999 00000000 002020at00 0200awa 20¢ 0 999999999 ~ «00:03 zuqum 200009999999999 .o0memxo sawmow amn0m How 00:00movuo>o mmmsm 0000000 0003 condom 00000000 .N0.v 000.0 85 Table 4.13. Local minimums obtained for first design example. Component Part Numbers Re'ection 1 $634.99 19.37% 1022 2048 3002 4015 1 $475.00 0.0 % 1003 2026 3015 4011 1 $380.61 0.42% 1025 2003 3003 4020 A 2 $346.43 2.72% 1015 2015 3007 4006 3 5 $340.70 1.09% 1015 2015 3015 4013 Based on the results listed in Table 4.13, the system obtained using part numbers 1015, 2015, 3015, and 4013 is assumed to be the best design at a cost of $340.70 per unit. The final computer print- out sheet summarizing this combination is shown as Table 4.14. 4,3 SECOND DESIGN EXAMPLE The specification set for the second example is chosen such that the computer solution time per analysis is minimized thereby enabling more example runs per dollar. This is accomplished by considering a customer requirement to consist only of the first five specifications: 1. Static accuracy = 0.35 degrees 2. Resolution = 0.3 degrees 3. Velocity lag for 300 deg/sec input = 5 degrees 4. Followup rate 300 deg/sec 5. Damping ratio 0.5 Since the last three specifications are not included, the cal- culation of Y Y7, and Y as well as R 6’ 8 6’ R7, R8 and S6 can be bypassed Table 4.14. 86 Best design obtained using directed. search for first design example. MAXIMUM 1.902E003 00769E§02 2.7575900 2.2305001 0.022E-01 000955-01 1.365E000 3.775E002 ‘O‘OOE-O‘ 4.5008+01 1.2fi06900 205996001 AUTOMATED DESIGN RESEARCH PROGRAM JANUARY 20. I969 ¢*'*DEFINITIDN OF LOAD‘¥‘* MAX MIN INERTIA (GM-CMSORI 9.000E002 7.000E+OZ FRICTION (OZ-(NI 8.000E-01 4.000E-01 ..‘RPART NUMBERS OF COMPONENTS SELECTED‘t‘O FULLOHUP AMPLIFIER MOTOR-GEN GEAR TRAIN 1015 2015 3015 4013 'fitfiPERFORMANCE‘tii MINIMUM SPEC LIMIT PCT REJ 1.502E003 TOTAL INERTIA (GM-CMSORI 3.4135902 'TOROUE CONSTANT (OZ-IN/RADI 2.1175900 DAMPING COEFFICIENT (OZ-IN-SECI 1.032E901 NATURAL FREQUENCY (MERTZI 2.526E-01 1.000 0.00 STATIC ACCURACY (DEGI 2.612E-01 0.500 0.00 RESOLUTION (OEGI 7.619E‘01 5.000 0.0 LAG FOR 90. DEG/SEC RAMP (DEG! 2.57IE+02 90.000 0.0 FOLLOHUP RATE (DEG/SEC! 3.30QE-01 0.300 0.00 DAMPING RATIO 2.250E+01 SEE BELOH 10.85 BACKLASH (MINI 6.622E-01 2.000 0.0 OVERSHODT FOR 10. DEG STEP (DEGI 2.135E901 5.000 0.0 BANDHIDTH FOR 2. DEG SINE (HERTlI ALLDNABLE BACKLASM SPECIFICATION (MINI MAXIMUM 8 6.946E001‘ 3.158E901 MINIMUM - "P‘COST SUMMARY“‘¢ 1.09 PCT REJECTIDN (UPPER BOUNDI 1.09 PCT REJECTION (INDEPENDENTI 1.08 PCT REJECTION (LOHER BOUND) 200.00 LABOR COST 137.00 PARTS COST 360.70 TOTAL COST (USING R-INDEPENDENTI E SIGNIFIES CONVENTIONAL POHER-OF-TEN NOTATION 87 (i.e., set equal to zero). With this alteration to the program, the computer time is reduced from approximately 18.0 to 0.3 seconds per solution -- a factor of 60. For this specification, two sets of Monte Carlo data were obtained each comprising 4000 runs. The first 50 lines of data obtained from the first set is shown as Table 4.15. As can be seen, only 4 of the L 50 have a finite cost, the best being $517.00. It should be noted that the individual rejections for the last 3 specifications are zero since no specification exists. The total rejection for this example 3" is calculated based on the upper bound approximation; i.e., Equation (3.13). Out of the total 8000 Monte Carlo runs made, which took about 40 minutes of computer time, the lowest cost design was found to be $375.00 obtained using part numbers 1006, 2003, 3002, and 4014 with a percentage rejection of zero. A summary of this combination is shown in Table 4.16. The results obtained using the program in the direct search mode now are illustrated in detail for three searches. The first, shown in Table 4.17, illustrates the condition where the initial guess at first hand looks like a "reasonable design"; i.e., the rejection is only 0.77%. However, after 74 iterations in the direct searCh mode, the cost has been reduced from the original design value of $555.30 to only $374.27 -- a savings of $181.03 per unit! The computer run time was less than one minute! Table 4.18 illustrates the opposite condition where the initial selection of part numbers yields a system that fails completely to Table 4.15. 88 Intermediate Monte Carlo printout for second design example. RUN RERCENI CCMPDNENTS SELECTED 0000699|NOIVIDUA1 SPECIFICATION REJECTIONSOOOOOOUU “1' R a MC! Fnuv 1m» MOGEN can t I n 0 999999999 9.7026001 100.00 1025 2003 3010 6019 6. . . . . . . . 2 909999900 6.3396000 100.00 1019 2031 3000 6010 11.62 0.0 0.0 100.00 100.00 0.0 0.0 0.0 3 999999999 7.7320000 100.00 1015 2022 3016 6022 10.59 0.0 0.0 0.0 100.00 0.0 0.0 0.0 6 999999999 3.627(002 100.00 1020 2006 3021 6010 100.00 100.00 100.00 0.00 90.99 0.0 0.0 0.0 5 99000999. 2.5131003 100.00 1006 2026 3007 6012 100.00 100.00 100.00 100.00 0.0 0.0 0.0 0.0 6 000000000 6.1676001 100.00 1013 2021 3017 6026 100.00 100.00 0.00 0.00 100.00 0.0 0.0 0.0 7 99999.99. 1.337F007 100.00 1005 2005 3000 6023 100.00 100.00 100.00 0.71 0.0 0.0 0.0 0.0 0 999999099 2.2560003 100.00 1006 2066 3006 6000 0.15 0.0 100.00 100.00 0.0 0.0 0.0 0.0 9 000000600 3.9156000 100.00 1015 2007 3011 6023 12.26 0.0 0.0 20.32 100.00 0.0 0.0 0.0 10 900909099 6.7226900 100.00 1016 2022 3006 6010 0.0 0.0 0.0 100.00 0.0 0.0 0.0 0.0 11 99.9.9909 2.7305003 100.00 1000 2016 3005 6026 99.60 75.61 100.00 61.01 0.0 0.0 0.0 0.0 12 602.00 7.0566000 -0.0 1003 2020 3006 6021 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 13 999999999 5.063E600 100.00 1019 2060 3007 6019 16.66 0.0 0.0 0.00 100.00 0.0 0.0 0.0 16‘999999999 1.6675001 100.00 1013 2033 3026 6023 0.02 0.0 0.0 0.0 100.00 0.0 0.0. 0.0 15 999999999 6.671E000 100.00 1016 2010 3010 6010 0.0 0.0 0.0 0.00 100.00 0.0 0.0 0.0 16 999999990 5.2307000 100.00 1006 2020 3001 6010 09.66 56.05 100.00 100.00 0.03 0.0 0.0 0.0 17 999999999 5.027E000 100.00 1025 2007 3023 6026 13.60 0.0 0.0 32.57 100.00 0.0 0.0 0.0 10 999999999 9.9706001 100.00 1021 2002 3002 6022 100.00 100.00 100.00 0.00 0.00 0.0 0.0 0.0 19 999999999 1.0066900 100.00 1020 2006 3003 6007 99.95 100.00 100.00 100.00 0.0 0.0 0.0 0.0 20 999090900 3.032E906 100.00 1026 2023 3012 6010 99.95 100.00 100.00 13.66 0.0 0.0 0.0 0.0 21 0.9999999 2.7036001 100.00 1010 2007 3011 6025 90.02 0.0 53.26 100.00 100.00 0.0 0.0 0.0 22 099999999 5.675E000 100.00 1026 2065 3016 6009 0.0 0.0 0.0 0.00 100.00 0.0 0.0 0.0 23 999999990 1.0155003 100.00 1021 2035 3003 6010 100.00 100.00 99.09 95.11 0.32 0.0 0.0 0.0 26 000099000 6.622E000 100.00 1015 2062 3007 6009 11.95 0.0 0.0 0.00 100.00 0.0 0.0 0.0 25 999999999 5.0736000 100.00 1001 2026 3017 6011 0.0 0.0 0.0 100.00 100.00 0.0 0.0 0.0 26 999999909 5.6526000 100.00 1006 2025 3015 6016 0.00 0.0 0.0 100.00 100.00 0.0 0.0 0.0 27 909999999 6.3120000 100.00 1017 2065 3006 6006 0.0 0.0 0.0 100.00 100.00 0.0 0.0 0.0 20 “00000" 5.6026000 100.00 1006 2030 3021 6012 10.60 0.0 0.0 100.00 100.00 0.0 0.0 0.0 29 one..." 5. 5606.00 99.99 1016 2010 3016 6011 0.0 0.0 0.0 99.99 0.01 0.0 0.0 0.0 30 9.9999900 6.3775000 100.00 1001 2027 3005 6020 0.0 0.0 0.0 0.00 100.00 0.0 0.0 0.0 31 909.990.. 0.6611000 100.00 1007 2050 3011 6003 0.0 0.0 100.00 100.00 0.0 0.0 0.0 0.0 32 999990900 6.177(006 100.00 1002 2016 3009 6006 99.73 2.95'100.00 0.00 0.0 0.0 0.0 0.0 33 9.0999900 0.2606900 100.00 1013 2020 3015 6002 99.00 100.00 0.0 0.00 100.00 0.0 0.0 0.0 36 999999999 7.196F000 100.00 1017 2007 3017 6000 0.0 0.0 0.0 100.00 100.00 0.0 0.0 0.0 35 999000090 7.0305000 100.00 1019 2019 3016 6020 11.76 0.0 0.0 0.00 100.00 0.0 0.0 0.0 36 “999“” 6.2886001 100.00 1020 2031 3017 6025 99.06 0.0 0.00 100.00 100.00 0.0 0.0 0.0 37 one..." 6.0006600 100.00 1019 2010 3019 6025 12.96 0.0 0.0 100.00 100.00 0.0 0.0 0.0 30 099990990 7.255t900 100.00 1001 2029 3021 6013 0.0 0.0 0.0 0.00 100.00 0.0 0.0 0.0 39 000000600 9.670E006 100.00 1005 2001 3020 6010 100.00 100.00 100.00 62.65 0.00 0.0 0.0 0.0 60 999990000 2.2625605 100.00 1005 2002 3016 6012 100.00 100.00 100.00 100.00 0.0 0.0 0.0 0.0 61 9.9909909 6.1201000 100.00 1003 2030 3002 6010 0.0 0.0 0.0 100.00 100.00 0.0 0.0 0.0 62 900000000 7.722190) 100.00 1015 2026 3006 6007 99.96 90.30 100.00 100.00 0.0 0.0 0.0 0.0 63 999999999 5.057F000 100.00 1023 2062 3006 6015 0.00 0.0 0.0 0.00 100.00 0.0 0.0 0.0 66 099099099 1.7750006 100.00 1002 2006 3016 6017 99.05 97.16 100.00 99.90 0.0 0.0 0.0 0.0 65 999990999 5.580(002 100.00 1010 2026 3006 6006 93.90 3.06 100.00 57.07 0.0 0.0 0.0 0.0 66 ""9”" 2.6661602 100.00 1011 2036 3016 6000 36.70 96.13 100.00 100.00 0.0 0.0 0.0 0.0 67 550.00 6.6366000 -0.0 1003 2020 3009 6026 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 60 0009"” 7.2936001 100.00 1010 2067 3020 6013 100.00 100.00 99.99 2.60 6.37 0.0 0.0 0.0 ‘9 $17600 16296‘901 “0.0 1019 2°30 5°02 9°25 00° 00° 00° 00° 00° 00° 00° 00° 50‘990099909 0.307E000 100.00 1012 2017 3000 6020 96.17 0.00 0.0 0.00 100.00 0.0 0.0 0.0 [[77 89 Table 4.16. Best design obtained using Monte Carlo for second design example. MAXIMUM 6.071E603 8.9536603 6.169E601 6.865E601 201656-01 2.776E600 9.702E902 1.592E000 0.0 0.0 0 0 MINIM 2.928 6.027 2.717 6.503 2.685 1.366 1.633 6.192 8.818 .0 .0 0 00° 0.00 0.00 0.00 200.00 175.00 375.00 E AUTOMATED DESIGN RESEARCH PROGRAM JANUARY 15. 1969 **9‘OFFINITION OF LOAO‘R“ MAX MIN INERTIA (GM-CMSORI 9.0006602 7.000E002 FRICTION (Ul-IN1 8.000E-01 6.000E-01 9‘99PART NUMBERS OF COMPONENTS SELECTEDR‘9’ FOLLOHUP AMPLIFIER MOTOR-GEN GEAR TRAIN 1006 2003 3002 6016 ttttPERFORRANCE¢996 UM spec LINII PCT REJ F003 TOTAL INERTIA IGN-CNSORI £903 TORQUE CONSTANT (DZ-INIRADT 5.01 OANPINC COEFFICIENI IOl-IN-SECI E001 NATURAL FREQUENCY (HERT21 E-OZ (.350 0.00 STATIC ACCURACY (DEG) E-OZ 0.300 0.0 RFSOLUIION IOLGI E900 5.000 0.0 LAG FOR 300. DEG/SEC RAMP (DEG! 6.02 300.000 0.00 FOLLOHUP RATE (DEG/SEC! E-01 0.500 0.00 DAMPING RATIO .......... 0.0 BACKLASH ININI .......... 0.0 OVERSHODT FOR 0. DEG STEP (OEGI .......... 0.0 BANDHIDTH FOR 0. DEG SINE (HERTII 666tc051 SUMMARY.... PCT REJECTION (UPPER aOUNOI PCT REJECTIUN IINOFPENOFNII PCT REJECTION ILONER BOUNDI LABOR COST PARTS COST TOTAL COST (USING R-UPPFR BOUNDI SIGNIFIFS CONVENTIONAL PONER-OF-TEN NOTATION - .m! ‘- .-_ ...- .; 90 « 6.0 0.0 0.0 60.6 60.0 o«.~ o.c oo.o «~o« ~oon moow cuu~ ~«.~ oo.wmp6.o h~.«hn ho n 0.0 o.o 6.0 00.0 no.0 o«.~ o.o oo.o «~o« ~oon moo~ ooo~ ~«.~ cp.mmpo.o -.«~m «s n o.o 0.6 0.0 00.0 oo.o o.o 0.0 oo.o «~o« ~oom o«c~ ooo~ 00.0 oo+w««~.m oo.«~« on n 0.0 0.6 0.0 60.6 00.0 6.0 o.o oo.o «~o« ~oon mmo~ ooo~ oc.o oo+wo«h.m ~n.o«« «« m 0.0 o.o 0.0 0.0 0.0 0.0 0.0 00.0 «~o« ~00n enoN ooo~ co.o ~o.womm.~ uo.«m« ~«, n 0.6 0.0 6.0 0.0 0.0 0~.o o.o oo.o «~o« ~oom ~«o~ ooo~ o~.o ~oow~oo.~ ou.«m« on n 0.0 0.0 0.0 00.0 oo.o m~.o o.o ~o.o «~o« ~oon mmcw ooo~ s~.o oo.wooo.« op.«o« - n 0.0 0.0 0.0 00.0 00.0 00.0 0.0 00.0 «~o« ~oon m~o~ Noo~ no.0 oo.womo.« mm.oc« o~ m 0.0 0.0 0.0 oo.o 0.0 -.o 0.0 .oo.o «~o« ~oon omo~ ~oc~ -.o oo+w~mm.« 0c.onm - m 0.6 0.0 0.0 ~o.o 0.0 0.0 0.0 oo.o «~o« ~oom omow ooo~ ~0.o oc.w-o.« o~.n«m o n 0.0 0.0 o.o oo.o 06.6 0.0 . 0.0 00.0 oooc ~oom omu~ ooo~ 00.0 oo.womo.n ~o.««m p n 0.0 0.0 0.0 ~o.o 0.0 c.o 0.0 00.0 o~o« Noon omo~ ooo~ No.0 oo.w~mh.« -.~«m n n 0.0 0.0 0.0 00.6 00.0 o«.o 0.0 00.0 n~o« ~oom omo~ ooo~ m«.o oo.womo.« no.0mm ~ ~ 0.0 0.9 o.o no.0 00.0 n~.o o.o «o.o m~o« o~ummbmo~ cuu~ nb.o oo.m~m«.« bm.an ~ in «a ¢w>o 44:2 6149 w»<¢3u 945 mm: u_>¢»m apam zmcox 61¢ 6:00 rumsma aqgcum hmnu .02 c 66666.6.mzo~bqum¢ zo-~Q2~6696666 cwbuwawm m»2wzoaxou pzwuama 23¢ o P 3 666666666 ~ awaxaz runawm z_¢wm6666666666 .o~mamxo cm~mov vcooom pom wocm~movuo>o mmosm ~m~uwc~ np~3 :onmom cowoopflo .n~.v o~nmb 91 99.9..... ~ ¢w610¢ 1066mm z~0w69...999999 « 0.0 0.0 0.0 00.0 00.0 6«.~ 0.0 00.0 «~0« ~66m n60~ ooo~ ~«.~ ooownho.o p~.«~n m- n 0.6 0.0 0.0 60.0 00.0 6«.~ 6.0 60.0 «~0« ~60n nou~ 066~ ~«.~ ooomnhc.« h~.«pm ~6~ n 60.0 0.0 0.0 00.0 00.0 0.0 6.0 00.0 «~0« ~00n 6«0~ 060~ 60.6 60.m««~.m 06.«~« «6~ n 0.0 0.0 0.0 00.0 06.0 0.0 6.0 00.0 «~0« ~00n o«0~ ~06~ 00.6 ooommo~.m 06.o~« ~6~ n 0.6 0.0 0.0 ~0.~ 66.0 0.0 0.0 60.0 «~0« Noon 6«0~ 660~ ~6.~ 66+w-m.n 6~.«~« 6h~ n 0.6 0.6 0.0 o~.~ 00.0 6.0 0.0 0.0 «~0« ~60n 6«0~ o~6~ o~.~ 60.mooo.m «o.¢~« -~ n 0.6 0.0 0.0 00.0 06.0 6.0 0.6 0.0 «~0« ~00n c«0~ 606~ 66.0 ooowom«.m 06.m~« ~o~ n 0.0 0.0 0.0 00.0 00.0 0.0 6.0 0.0 «~0« ~00n o«0~ ~06~ 66.6 66.6«06.m 06.~n« cm~ n 0.0 0.0 0.0 00.0 00.0 0.0 0.0 0.0 «~0« ~00m c«0~ -o~ 00.0 ooowoo~.m uo.~.n« o«~ n 0.6 0.0 0.0 00.0 00.0 0.0 6.0 6.0 «~0« ~0om nm0~ -0~ 06.0 06.6«o«.o ~o.~mm om~ n 0.6 0.0 0.0 0.0 0.0 0.0 0.0 0.0 «~0« ~60n 6m0~ -6~ 6.6 ~oowomo.~ 06.-n nm~ n 0.0 0.0 0.0 0.0 0.0 ~0.0 6.0 0.0 «~0« ~60n ~«6~ -6~ ~o.o ~o.w006.~ mo.-n -~ .6 0.6 6.0 0.0 00.0 06.0 60.0 6.0 0.6 «~0« ~06m n~0~ -0~ 06.6 coowo«m.m ~6.~mm -~ n 0.6 6.0 6.0 00.0 .00.0 60.0 6.0 0.6 0~0« ~06n n~0~ -6~ 66.6 ooowo-.m ~6.««n 6- n 0.6 0.6 0.0 00.0 ~o.0 0.0 0.0 0.0 0~0« -0n m~0~ -6~ ~6.6 66.0«m~.« «6.mom o- n 0.. 0.6 0.0 0.0 ~n.o~ 60.0 6.0 0.0 n~0« -6m m~0~ -0~ ~m.6~ 669m~«~.« mm.mo~ oo~ n 0.6 0.6 0.0 00.6 -.~n on.o~ 0~.- 6.6 m~0« -6m o«0~ -6~ no.«m 66+mm-.m om.omo~ 06 ~ 6.6 0.0 0.0 ~n.o 0~.«m oo.o~ h~.- 66.0 «~0« -0m o«6~ -6~ 06.60 06+w-~.n 0m.~o~6n mm ~ 6.0 0.6 0.0 09.0 o~.«m no.0~. n«.- 60.0 «~0« -0n 0«0~ h06~ 00.60~ 60+uh0~.m 99....99. ~« ~ 0.6 0.6 0.0 00.nn o«.oo no.h~ ~N.«~ 60.0 «~0« noon o«0~ ~66~ 00.66~ oo+m~o~.m ....9.... 6« N 0.6 6.6 0.0 ch.«o o~.om o~.0 ~n.no 60.0 «~0« nmom o«0~ ~06~ 06.66~ 66+womm.m 9....99.. mm ~ 0.6 0.0 0.0 00.06~ no.0 0.0 00.0 0.0 «~0« -om o«0~ h66~ 06.66~ oo+w«mm.« 9.9.9.99. «m N 0.6 6.0 0.0 60.60~ no.0 0.0 00.00~ 66.66 «~0« -0n ««6~ -o~ 06.oo~ 66+u~o«.« 9.999.... m~ ~ 0.6 0.6 0.0 66.60~ 66.6 0.0 no.- ~0.o «~0« -0m o«6~ -o~ 00.66~ ooouomm.m 9.......9 o~ ~ 0.0 0.0 0.0 60.60~ 6.0 6.0 oo.~o 00.6 -0« -0n o«0~ -o~ 06.no~ ooom-o.6 ........9 - ~ 6.6 6.0 6.0 06.00~ 0.6 6.6 66.~0 6~.« -0« -0n o«0~ «~0~ 06.60~ oo+mw~o.6 .99....9. 6 ~ 0.6 6.0 0.0 00.06~ 0.0 6.0 «n.~0 m«.6~ -0« -6n o«6~ -6~ 06.66~ oo+m~oo.6 99....... m N 0.0 6.6 0.6 66.06~ 0.0 66.00~ 60.06~ 00.06~ -0« -6m o«0~ moo~ 60.60~ ~69m~«o.m 9.99..... « ~ 6.6 0.6 6.6 oo.~ 60.6 66.00~ 06.60~ 06.00~ -0« -0n o«0~ moo~ 06.nc~ Noowh«~.~ 9....999. m u 0.0 D.[.b no.0 00.0 60.2: no .DDH 00.2: NNO... m~om m«pN%oc~ 06.00~ oowwwhé 0am....nu — w 6266 ¢w>0 44:: 0:46 whczau 060 mwa u~p~6z~..9.... owhowawm wh2w20atou pzwuuwe 20¢ 0 P t .o~memxo cm~mo6 vacuum you wonu~mocuovcs mmmsm ~m~u~6~ gu~3 nonmom wouuon~6 .w~.v oHDQH 92 meet 3 out of the 5 specifications. It takes the program 55 itera- tions to obtain a finite cost and switch to fine search and then 147 more to reach a cost of $374.27 which is the same local minimum. The third search is shown in Table 4.19 where this time the initial parts result in a design which fails completely to meet 5 out of the 6 specifications. After 55 iterations, the program has ’.'!1.'z£" reduced the scalar from 59,610,000 to 3.396 and only one specifica- tion remains a complete failure; however, this point turns out to be a local minimum and no further reduction is obtained. Figure 4.1 illustrates the convergence of the directed searches as compared to the results obtained from the two Monte Carlo runs by showing a plot of improved system cost versus iteration number. The two directed searches reach the minimum in 74 and 202 iterations respectively while the Monte Carlo runs employ 4000 solutions each and neither has found the minimum. A total of 15 searches were made and the local minimums found and their frequencies are summarized in Table 4.20. Based on the results listed in Table 4.20, the system obtained using part num- bers 1009, 2003, 3002, and 4014 is assumed to be the best design. The final computer printout sheet summarizing this combination is shown as Table 4.21. 93 6.6 0.6 0.0. NN6~ 00.66~ Chaoor.n ......... ow~ 0.6 0.0 6.0 60.0 00.00~ 00.6 m~.~« 00.«~ h~0« n~0m ooo~ -6~ 06.66~ 66.060m.m ......... mm 0.6 0.0 0.0 06.0 oo.0o~ 00.0 n~.~« 66.«~ -o« m~6m «~0~ -o~ 06.66~ oo+mo0m.n ......... 0~ 6.6 6.6 6.0 60.0 oo.00~ 60.0 m~.~« mo.0~ h~0« n~6m «~0~ «~o~ 06.6o~ 66.0«n«.m ......... 0~ 0.6 6.0 0.0 «0.00 00.00~ 66.0 00.00 «0.om -o« -6n «~6~ «~o~ 60.60~ oo+woeh.m ......... - 0.6 0.6 0.0 6.0 00.66~ 00.06~ 00.00 ~0.00 >~6« -6m «~0« «~0~ 66.60~ ~6+mm«o.o ......... o~ 0.6 6.0 0.0 60.0 oo.00~ 60.06~ no.00 06.00 -0« m~om «~6~ «~o~ oo.oo~ «6+mhmo.m ......... 0~ 0.6 0.6 6.0 60.0 oo.00~ 60.66~ 06.60~ m0.00 p~6« n~6n «~o~ o~6~ 06.66~ co+¢o-.m ......... «~ 0.6 0.6. 0.0 00.0 oo.00~ 66.60~ oo.00~ m0.00 -o« n~6n «~6~ m~0~ 00.66~ ooommm«.m ......... m~ 6.6 0.6 0.0 6.0 00.00~ oo.00~ 00.00~ 60.00 -0« moon «~0~ m~6~ 66.60~ oo+woho.o ......... «— o.6 0.6 6.0 0.0 06.00~ 00.00~ 06.00~ 60.00 -03 -6n «~0N m~6~ 06.60~ ooowo-.0 .......9. - 0.6 0.0 6.0 0.6 00.00~ 60.06~ 66.oo~ 00.00 h~0« moon «~6~ m~6~ 66.66~ ~o+m«06.~ ......... 6~ 6.6 6.0 0.0 0.0 oo.0o~ 66.06~ 60.06~ 60.00 h~0« Noon «~0N m~6~ 66.66~ ~6+w-m.~ ......... 6 0.6 0.6 0.0 .0.0 n~.oh 66.06~ 66.06~ 00.00 -0« o~0m «~o~ m~6~ 66.66~ >6.m0~0.~ ......... s 0.6 0.6 0.6 0.6 60.00~ oo.oo~ 66.06~ ~0.00 h~6« «66m «~0N m~o~ 06.66~ ~o+m6>6.~ ......... m 0.6 0.6 0.6 0.0 06.00~ 66.66~ 66.66~ «0.00 -0« 006m «~0~ m~o~ 66.66~ s6+mm~6.« ......... m 0.6 0.6 6.0 0.6 oo.00~ 66.06~ 60.60~ m0.00 -6« «~om «~0~ n~o~ 60.66~ s0+mo~o.« .......9. ~ 6.6 6.6 .6 0.6 66.06~ 66.06~.66.oo~ ~0.00 -0« 6~6n «~0~ n~o~ 66.66~ bo+m~00. . .PIII quad mama 443a ”and unqm=u mm; «mm u—p4Hu muuu auooa eta 0660 pomowm aco<0m pmno .62 ..9.....mzo~huwsw¢ 26~p<0~u~uwxm a<=o~>~62~9...... owhowawm mFszoazou hzwuamm 26¢ 00.011 00.oam aqua u—unfl uu.«_ H—an wnum «com IDONdNNNNNNNNNNNNNNN’N" ...9.9... m ammtoz 2004mm z~0wm.......... .E:E~:~E ~moo~ xhouomMmfiummcs cm :a mcwu~3mou condom oopoohfin .m~.v o~nmh S94 .noumom oouoonao 6:0 o~uuu 00:02 mo oomwummaou umnzaz zcnhiumhu 0000~ 000~ . 00a . mzsm coo. h< amp $374.27 thus proving the latter to be the absolute minimum. It should be pointed out that the above procedure is not prac- tical for all situations as readily apparent if one considers the $340.70 value obtained for the first design example (see Table 4.14). This time there are only 4,249 combinations which satisfy (4.5), however, with the 18 seconds required per solution a proof would take 21 hours of computer time. 5. CONCLUSIONS AND RECOMMENDATIONS The objective of this thesis is to deve10p techniques to automatically select a collection of components that satisfy a given system specification at minimum dollar cost. This objective has been realized. The techniques developed are sufficiently general to provide an effective design tool especially when there are significant pro- duction quantities and a large number of standard components available. The application of the developed technique is demonstrated by establishing libraries for electromechanical components and writing a computer program to automatically design instrument servomechanisms. The results of this effort are most rewarding. For example starting with an initial design that satisfied a customer specification at a cost of $475.00 per unit, the computer program in 23 minutes produces a modified set of part numbers that meet the same specification at a cost of $340.70 per unit. This amounts to a cost savings of $134.30 per unit, representing a 28.2% cost reduction. In a similar manner, starting with an initial design which failed completely to meet 4 out of 8 specifications, the program brings the design from the "infinite cost condition" down to the same minimum in only 60 iterations. 98 99 The optimization techniques developed here, like most others, present no guarantee that the result obtained is an absolute minimum. A method is presented, however, that is practical in some cases for testing whether or not a local minimum is indeed the absolute minimum. This is accomplished by analyzing only a subset of the total number of possibilities which, for some cases, can be reasonably small. The test is made for one of the design examples presented which establishes that the minimum found by the program is truly the absolute minimum. Comparisons are made between designs using purely Monte Carlo techniques and directed search. These demonstrate that the latter offers a decided advantage in faster convergence to the lowest cost design. The exact convergence time, of course, depends a great deal on the closeness of the initial guess. In meeting the primary objective, a number of other tasks are accomplished in the area of calculating nonlinear servomechanism performance. An equation is derived fer calculating the allowable Inicklash in a servomechanism without it displaying null oscillations. Ikrretofore this could only be "calculated" by iterating a direct Silnulation of the nonlinear state equations until the critical point "815 found. This could literally take several hours of computer time to (abtain when considering nominal values. The thought of including the: component tolerance effects was, therefore, out of the question. Wit [1.the equation derived in this study, the solution is obtained, witihl tolerances, in 2 seconds of computer time. 100 Advantages similar to the above are obtained also in the area of computing the overshoot and bandwidth for the nonlinear servo- mechanism. An obvious next step for future work would be to extend the development presented here to include such design specifications as weight, volume, power consumption, and reliability. This would pro- vide the system engineer with an even more effective total design capability. For applications where the number of parts stored in a component library becomes extremely large, one should consider the use of a "working library" selected by the system engineer from a large "standard library." With this approach, an experienced user could eliminate, on an "a priori" basis, many components undesirable for a given application. Automated procedures would need to be developed to aid in sorting out the components with the desired features. The search technique as developed in this thesis could be {possibly fUrther improved by extending the exploratory move strategy ‘to include simultaneous multiple component changes. However, the zadded complexity of the control logic required might very well out- weigh any advantages gained. APPENDIX A DERIVATION OF BACKLASH-FRICTION SLOPE EQUATION It is well known that backlash can cause small amplitude oscil- lations about null. In 1947, A. Tustin [30] presented a graphical method for analyzing the stability of systems with backlash. His work was then extended by others [31], [32], [33], and [34]. However, each of these assume zero load after the backlash (i.e., the output stops instantly each time the motor reverses). For this condition, the backlash is "represented" by a simple hysteresis nonlinearity. However, it can be shown that, for the second order system with hysteresis, the sufficient condition for asymptotic stability is only that the damping ratio (c) be greater than one half [35]. But in actual practice [2] and [36], many systems oscillate with c > 0.5, which can be attributed to the fact that the hysteresis characteristic does not represent the physical facts. References [37], [38], [39], and [40] attempt to circumvent the problem by considering a load which (zontinues to move after the motor reverses. Each, however, is Iaased on the assumption that the gearing has infinite stiffness (i.e., the impact is perfectly inelastic). However, J. Liversidge [41] in 1951 demonstrated with hardware that low gear train stiffness greatly aggravated the null oscillation problem. Later efforts [42], [43], and [44] demonstrate, at least via simulation, that for an accurate model, gear stiffness must be considered along with backlash. 101 102 C. H. Thomas [45] considered this as early as 1954 but made the assumption that there was zero damping at the load. Thus, the gen- eral problem, except for direct simulation, has remained unsolved. In the last few years, there has been considerable discussion [46], [47], [48], [49], and [50] about extending the application of describing functions to handle two separated nonlinearities in a system. The validity of such an approach, of course, increases as the amount of integration effect between them is increased. Since at least one pure integration exists in every path between the backlash and friction blocks of Figure A.1, it was decided to follow this 4» approach. The null oscillation problem is formulated in the following manner. Consider the 2-space of backlash and friction and some point R that represents the numerical values of each for a partic- ular system (see Figure A.2). From Reference [2] one knows that there Backlash + line A Friction Figure A.2 Backlash-friction diagram 103 .am~:mnooao>uom ucoashumc~ :6 now swummwo ~oooa oumum Hmocfl~=oz .~.< onow~m 20:03: 080.. :00 1948040 0. + .326 ham-.3333 c» £46 (:95: 353.12. :_u_ 61:»... 9.3] I 20533 A A— Jh a E» n o . a I '9. + . Es / a .55: 624223 .3 an Esau) EOPOI ENC! . {El 104 exists a straight line (A) through the backlash-friction space that is strictly a function of the system linear parameters. If R lies above this line, the system has a limit cycle while if R lies be- low this line, the system is asymptotically stable. Our problem is to find an equation for calculating the slope (m) of this line in terms of the system linear parameters. The purpose of the remaining portion of this Appendix is to accomplish this task. Consider again the state model diagram shown in Figure A.1. Since presently only the null oscillation problem is being considered, the saturation block may be ignored (i.e., assume a gain of unity for N3). By inserting for the backlash and coulomb friction blocks their effective gains N1 and N2, the autonomous state model can be written as F‘ d! K - q q 6 r_ f:_ NlKS 0 N1 5 r_é M JM JM JM M 0M 1 O 0 O 6M _d.. ._. dt N K N N K . 6 0 1 s _ _3_ _ ‘; s 9L L JI. JL L 6L 0 0 1 0 6L 1... J L. J L. .J (A.1) The corresponding characteristic polynomial is then given by .x- ‘ ‘2' I Inn-'51 '7 1'-“ .| 105 f N f, +J KOJNK f K+NNK .NK 0(1) . -xu , 31+ 33 13 , er M21 3 L 1 s 12 * TN} 5 2 1 s A ‘ K} 1 s M L ,Pl. MJI. NI. (A.2) The describing function gains N1 and N2 are given as func- tions of their respective input zero-to-peak values, E1 and E2 , by the equations _ 2 .-1 B B B 2 g N1(51) - 1-; sm (Eff)+7§f\ll-(251) f” E13-2 = 0 otherwise, (A.3) 4TL N2(E2) = EEE- for all E2 (A.4) with the corresponding range and domains git-4:15.00 01E2<°° (A.5) 0 < N1 < l w :_NZ > 0 Since N1 and N2 are both real and frequency independent, one can separate out terms in (A.2) and write the characteristic equa- tion in the form PACK) + N1(E1)F1(l) + N2(52)F2(A) + N1(E1)N2(Ez)F3(A) = 0 (A.6) u‘- - 106 where Fq(l) = JMJLA“ + fTJL13 F3(A) = KSA = 3 2 F2(A) JMA + fTA = 2 F1(1) JTKSA + fTKSA + KTKS (A.7) Substituting jw for 1 and defining the real and imaginary parts of each coefficient Fn(jw) = (JMJqu) + j(-fTJLw3) E Pg + ij F3(jw) = j(sz) 5 P3 + an 1:20..) = (41.2) + j(-JMw3) 2 P2 + jQz w = (ww + mo) -.1 + m. (A.8) Separating the real and imaginary parts of the characteristic equation pu(w) + N1(E1)p1(w) + N2(E2)P2(w) + N1(E1)N2(E2)P3(w) = O (A.9) Q.(w) + N1(E1)Q1(w) + N2(EZ)Q2(w) + N1(E1)N2(EZ)Q3(w) = o (A.10) Equations (A.9) and (A.10) provide two of the necessary conditions for a limit cycle in terms of three unknowns N N2, and w. Solving 1’ simultaneously the two equations for N1 and N2 in terms of w and substituting in for the P's and Q's,one obtains equation (A.11) for N1 and (A.12) for N2, in terms of w and the linear system parameters 107 "1(a) - -UJ’HL".J“JTK')“N(PT2"-J"‘TK')HZJ °fiIJK’L‘s"’K’T‘s)”~'(FTZ‘wa‘r‘sI“J("[55:49:52] LJMZJL“6°FT2JL”~J ZEW-JTKNJ (A.11) "2 (U) . - [Juzuz-JukrofTZJ . ME Juzuz—Jgrrofirzl 2-4fT EJsz-IJ‘Z-JLfTKTJ (A , 1 2) T "1 Equations (A.11) and (A.12) will be referred to as the frequency relationships as they give the required describing function values as a function of frequency and the linear parameters. The third necessary condition for a limit cycle is obtained from the fact that the derivation of (A.11) and (A.12) did not place any restrictions on the amplitude requirement that must exist between E1 and E2, thus N1 and N2. From the state model diagram (Figure A.1)one»can derive a transfer function relating el and e2 as 2 61(A) = :1. JMA +£TA+KT (A.13) e (A) A 2 JMA +fTA+N1Ks Letting A = jw and by taking the magnitude of both sides and E1 E323 be the peak values of e1(jm) and e2(jw), El. = l. (KT-JMmZ)2+(fTw)2 (A.14) E2 “ (les'wa2)2+(fTw)2 Solutions are now required for El and E2 in terms of N1 mic! N; . This is difficult to accomplish for E1 . Figure A.3 shcnvs a plot of’ N1 vs. 2E1/B with and without the third term of -t~ MD :o~uo::m wc~nfiuomoo nmm~xomm .m.< onsm~m alum .0 ~¢ ..m I—N 108 . _. EL 1: ,_ 20.h340m E :o~uo~um mo :Hmm 03000.3... .v.< 0.5me 8332: 522.02... 0w. 0: 00. 00 00 0m 06 8 9 On 0w b 020—5080 5% 2:0. Euoz¢.\ 04¢}..-n0 000.. My. 62.2.78 0000.. .9. .9..1¢.o..3...20.:aou on» -273 on . . 3.... 333...: «10.... 000» . as «so I... 0000...... . .5522... sup»; 0. T... .C 0 88‘38 02.4 5200002; ~ l1 IOVU/NI-ZOI NOIIOIUJ JO NWO 3M133i13 112 where N1(w) and N2(w) are defined as in (A.11) and (A.12). Figure A.5 shows a plot of the bracketed quantity of (A.20) as a function of w . In general the problem can be solved by using a simple optimization subroutine. The one selected is SGREAT which is the subroutine version of the program described in Reference [11]. Figure A.6 illustrates transient response curves for a typical system as obtained via a direct digital computer simulation of the nonlinear state Equation (2.1). The system parameters used are identical to those listed in Figure A.4 except that coulomb friction (TL) was varied for the various curves. Looking at Figure A.5 one seestflun:for low values of friction the system displays a limit cycle. As the friction is increased, the only noticeable change is a slight decrease in the limit cycle frequency (mo) until finally one goes beyond the critical value and the limit cycle suddenly disap- pears. Qualitively the results agree exactly with what one would expect from the theory that has been developed (e.g., Figure A.4). Quantitatively, however, Figure A,6 shows that the critical value of friction is between 0.1 and 0.15 oz-in, say 0.125. The backlash value used in the simulation was 10 minutes of arc. Thus the cor- responding slope of the backlash-friction curve is calculated as B 10 m = TL(critical) = .07125' = 80 minutes/oz-in as compared to the value of 49.55 determined using (A.20). The dif- ference is, of course, mainly attributed to the describing function approximation made in the derivation of (A.20). It can be shown, lumwever, that neglecting the harmonics is on the safe side. That 400»- ) u 0 O l T ZOO-11- BACKLASH-FRICTION RATIO (MlNUTES/OZ-IN 113 '00“ MINIMUM POINT m. .49 55 MINUTES T ‘ OZ-IN I I -a I (no - 7| RAD/SEC L - - IA”. . . 013 20 40 60 do 100 150 FREQUENCY (RAD/SEC) Figure A.5 Backlash—friction ratio vs. frequency 5 . .Ir #1.... i...§.~....._; ...f A. 114 .15 OZ-IN TL. .10 n- ._..u . mmmmmnu a nun. .uwuu mu nuuummuuwu NHHHHHHHANH..W ...aa......... ..m.mm~_~_uu_ .05 8.. 8... 8... .6203 B 6252. . 5L5 TL 3 .025 A 4 a a III. 0 ~. ¢ "0.. - NILLISCCWBIXIO' I v .3... IIIC v System response curves for various friction values Figure A.6. 115 is, if the harmonics had been included, the values of N1 and N2 would be somewhat higher. This would mean that the real damping effect would be greater and the backlash effect less than calculated in our derivation. Thus the amount of true allowable backlash is always greater than that calculated by (A.20), and the method used is always on the safe side. APPENDIX B DERIVATION 0F NONLINEAR OVERSHOOT EQUATION The purpose of this appendix is to derive the necessary equations for calculating the system overshoot for large step inputs. One way this could be accomplished would be a direct numerical integration solution of the nonlinear state model given by Equation (2.1). However, this direct simulation procedure is time consuming on a digital computer. In order to minimize the solution time, the non- linear state equations are solved explicitly using piecewise linear techniques. For simplicity sake, the system is considered to have zero backlash and infinite stiffness. This approximation is justi- fied when considering response to large inputs since these only add small oscillations about the nominal solution. The various solutions required are visualized best by looking at the system response on the phase-plane as shown in Figure 8.1. The three response curves shown were generated on an analog computer using the simulation shown in Figure B.2. Each curve is for a dif- ferent initial displacement which was selected so as to demonstrate the three different types of mode switching that can take place up to the first overshoot. Since one is concerned with the response up to the first overshoot, only the portion of the space below the abscissa is of interest. This can be divided up into three regions, in region one the drive torque is negative and saturated, and in 116 117 pl I- Emumw~0 ocm~muommnm :0 omcommon soumxm ...m 88mm. 825.32 2.... 20.523: 3.532 N 29:22.3 4 .2069... 20.32 I W>rpo_.mooma¢ \..2..... IL on. on. 3. o.~. n... on. 32454506 #2028335 3......6 3.. 3 . u :3 330%.”... WW. 4» 20.5.5. oh 273 69. 2:» moo .23 9:5 $3273 on.» . :. ..3\ 0% O... owm ..z_INO n N u an :_UOJN> . uwmuzTNo nNn u .0. omdeimoz o<¢\z_INO wnn. . p0. 118 1M .m~m»~mcm uco~mcmuu pow 06m: Ewumm~0 Hopsmsoo mo~m=< .m.m ousmflm o. .- NI no».. .2 0» 0 oo. 6 SSTWW. x - R.) 092...:B g 09+ 6 I Z... >o¢ + n P e ... 2, A E... 00. 2.32. 3.22.6 pzuzuofiama «:2 >3. . 23 moo... o. 000.. "330 2.» 2.36. ... 2.; n In®lldf 0 5.60.5; 9x4-» 660. A— . 2.80m ..ou2_\>o. . >oo.+ m2: m_x2 O—I H >4 N II I E 2: Substituting each of the above into (8.8) and simplifying A1‘2 Alt Azt 80“) " Z11[80(0) t Agile t Z21[:00(0) + ARIEL}: - [A2211 * 11212] "9" for z + l (3.17) and -th ZIZQ -mNt = Z12 [00(0) ' 51]“ " [90(0) ' 21% ' ...Nz ]8 * [”14” ” 212]??? N N for 1;= l (8.18) .1“; 1.1 I 123 where the constituent idempotent matrices are calculated as 111 (C- a] 02-1 ) w 2 N N AZU-P -1 wN(-c- 4:2-1) —’ Z = -——————— = 11 92-h -2w Jr??? N (8.19) 15?” fl 2_ 2 wN[c+ c I) “N z -+ «11+ $72.11) 21 - AI'AZ - 2U) C2_1 N and the constituent nilpotent as - _ 2 LON LON 212 = [P + wNu] = 1 (13.20) “’N c=l Substituting (8.20), and (8.15) into (8.12) and simplifying r 0 { . T I 0(t) 1.1;. g2-18(0)+w290-._L - 0 ~( ~/ J. N a“ J. -.N(..¢.—2.1). = e ZMN C -l . ¢.?_) TL 0 a) -e an -w 1 - c-l 6(0)+ L o J _ o N( o 11.ka , M1734)"l [‘ . TL ‘ r’ m -wN(c - ¢c2—1)60(0) -wNZ 00(0) + 3— 0 + T e-wN(C - (2'1 ) t 2 7T? 5 (0) + w (c + \/c2-1 )9 (0) - TL “N V/C TL 0 N o J - £5- — 1. 'I'mN(c 1)J .KTJ (for c + l) (3.21) 124 i'" and F- '1 P‘ T fl 1 .1 ' L - 2 . 90(t) 37r- - wN90(0) - wN 60(0) f 90(0) {— 0 -w t ~11: t = te N + e N , e (t) 50(0) + wNB (0) - TL 00(0) - :5 IL + ° + L ° Fr!“ . H 1 fr . for c a 1 (3.22) A __ “t ———.. Even though (8.21) is valid for g < 1, it does involve complex arith- metic. A better form for this case is obtained directly from (8.21) by expressing the exponentials in terms of sine and cosine and simpli- fying resulting in ‘ 1' com 60(0) 7:};— - :5°(0) - «Noam o - e-“th 4 cos I-t t 9 1 sin ‘/l-c2 t 9 u" 1-c2 . u" > TL 80“” 1.I. C TI- °°m 60(0) - q “’11 . zoom) - 1? q J for: c < l (8.23) The necessary state equations for the unsaturated region have thus been derived. These are (8.21) for c > 1 , (8.22) for c = l, and (8.23) for c < l. 8.3 SATURATED REGION NUMBER THREE The only difference between saturated region number 1 and number 3 is that the driving torque is reversed. Therefore, the 125 state equation for this region is obtained by simply reversing the sign of TSAT in (8.10). Thus for region 3 we have ' P . T *T P P TSAT'TL 60(1)) 60(0) - —§%1——E 1 o 1 -—1§;——- 1 -2 w _ o ‘MN‘1 t+ 'éo(°) ’ (TSAT’TLII/BM TSAT'TL e (0) 1 60(0) ’ (TSAT’TLIIIBM . L60“) L ZLMwN L BM Lo ZCMUN F I (8.24) 1 8.4 SWITCHING MODE LOGIC The remaining task is to tie together the state equations for ~- the various regions in order to arrive at the value for the system overshoot. The procedure used is summarized by the flow chart of Figure 8.3, the derivation of which can be explained in the following manner . The first step is to examine whether the servo is initially saturated by comparing 60(0) to TSATY/KT' Assuming that the initial displacement is large enough to cause saturation, the next step is to solve for the intersection of the trajectory and the first saturation boundary. The boundary line defining this saturation is given by T 2; - SAT - J 90 (13.25) “’11 6b1 ’ KT 1 o - o . . (atting the (TSAT Tl.L/BM term of Equation (8 10) be defined as s , which is the servo followup rate, and then substituting (8.10) into (13.25): 126 7:5 1. USING 02 ‘o.(1.)usmo 0.10 NO SET 1-At “ J13 (Kaif'infl’ - M . __LALCILLAIL_ MATRIX cocmcu MArmx COEFFICIENTS ron 0.21 CALCULATE FOR 3.23 MAmx co r1 1 Iron 3.22 5 1 SET STORE 1- 1 .At 0_- o, u-m) (>1 % {-1 CALCULAI'E CALCULATE 0. (nusme 3.21 0.1uusmo 11.23 CALCULATE 1111115190 0.22 _.f£\. no 0.101121113110071 .0 .."1’0 9 (mom '5’ +11 .gteouwoAav) usmc 0.32 ° usmo 0.30 CALCULATE m MATRI X COEFFICIENTS FOR I. 24 I CALCULATE 1, USING 0.33 ‘ 1, usmc 11.3. 8.1.3) USING 8.24 0.11.1031“ 0.2. SET 0.10vsnsnoon .- 9.1..) Figure 8.3. Nonlinear overshoot logic flow diagram. 127 —eo(0)-eS ~2cMth . é o(0)+és T— e - est + 8 (0) + -—2-—— mm 0 Cwm TSA 2; , , -2cw 2c , = —- - —-g- 6 (O)+6 e Mm Nt ___£_ 6 KTT wN o s wN s (8.26) g and collecting terms i 2: . 2: t1 2.. 7:£-- 2 1 e (0)+e e MwN - 9 t1 N CMwN 5 . . T 2; . -§El—— 90(0)+es + 9 0(0) - —§AI-- -—3-9 = o MwN Kr wN S (0.27) Equation (8.27) can then be solved for the time at the boundary (t1) by iteration. A good initial guess is e0(0)-TSA'D/KT tl(est.) = . - (8.28) 6 S Once t1 is found it is substituted into (8.10) to obtain the desired boundary conditions which are the initial conditions for :region number 2. These two steps are illustrated as the first oper- ation in Figure 8.3 for the saturated case. Region number 2 is then entered, either after the above calcu- lations for the saturated case, or directly for the unsaturated case. 128 Since region number 2 is one of indecision, that is the next boundary is not known, one simply continues to solve for 50(t) and 60(t) incrementing time in steps of At until one of the two boundaries is crossed. However, the particular state equation to be evaluated in region number 2 depends on the value of c . Thus the next step is to evaluate c and, depending on its value, calcu- late the matrix coefficients which are independent of time as re- quired for either (8.21), (3.22), or (3.23). One is now at point 5 of Figure 8.3 with t = 0. The previous values of the state vari- ables are stored as 005 and depending on the value of c either (8.21), (8.22), or (8.23) is evaluated. It should be noted that the latter is a relatively simple calculation since the matrix coeffi- cients are independent of t and have already been evaluated. After point 9, one simply checks to see if either of the two possible boundary conditions have been exceeded, i.e. if 60(t) > O or if 60(t) < ebz where 6b2 in the corresponding values for the second saturation line given by T 2; SAT ' 9b2 = - fi-Tfeo (3.29) If neither of the above inequalities is satisfied then an appropriate At is calculated, t is incremented and one returns to point 5 and 'the process is repeated until an exit is obtained at either point 10 <3r 11. Consider first that the exit is via 11. The corresponding situ- :Ition is illustrated in Figure 8.4, i.e. (90(t) lies in region 129 VELOCITY 6° (RAD/SEC) DISPLACEMENT 6 o ( RADI ANS) SATURATION BOUNDARY INTERSECTION POINT 00 (t2) STRAIGHT INTERPOUTION eon-At) Figure 8.4. Phase-plane interpolation diagram. number 3 while (90(t-At) lies in region 2. By using a straight interpolation line between these two points, the desired intersection point is readily calculated using the derived interpolation equation F. , ‘1 r' . “ 60(t2) 1 wN60(t-At)-wNeo(t-At)-wNMTSAT/KT 2C8M+wN h-E)O(t2)_J _3CgMeo(t-At)-2Cgeo(t-At)'wNTSAT/KT J (8.30) where M is the slope of the interpolation line given by éo(t) - éo(t-At) M = (3.31) 60(t) - 80(t-At) 130 In a similar manner, if one exceeds the éo = O boundary (i.e. exit via point 10) the intersection point, which this time cor- responds to the desired overshoot, may be evaluated using é (t-At) 90(overshoot) = M - 80(t-At) (3.32) where M is as given in (8.31). If the intersection is on the éo = 0 line, one is finished; if not, then one proceeds into region number 3 using the new initial conditions as given by (8.30). Region number 3 is also one of indeci- sion reguarding the next boundary condition. However one can readily derive which boundary applies in the following manner. If one re- mained in region number 3 80(t) would equal zero at some time say t3. This value can be solved forby using the first equation of (8.24) yielding B 6 (0) M 0 2n 1 - -————-—- (3.33) ZCMmN TSAT+TL t3 and using this value in the second equation one readily obtains 60(t3). If this value is less than -TSAT/KT , the assumption of remaining in region number 3 is valid and one ends up at point 12 of Figure 8.3 with 90(overshoot) = - 60(t3). If not then the trajectory must enter region number 2 for the second time. An equation giving the appropriate time in region 3 can be obtained in the same manner as (8.27) by letting ass = (TSAT+TL)/BM and using (8.24) instead of (8.10). 131 2C . . -2§ w t . __g,_ 2 1 6 (O)+6SS e M N 3 + asst3 wN CMwN o . . T 2c , + 2 1 90(0)-eSS + e (0) + EAT + g 955 = o _ CMwN o T “’N n (3.34) Once the time in region number 3 (t3) is obtained by sloving (8.34) it is substituted into (8.24) and the initial conditions are obtained for region number 2, t is set to At and one returns to point 5. The procedure is thus repeated as explained before except this time a termination is reached via point 10 and Equation (8.32). Tables 8.1, 8.2, and 8.3 show sample outputs of a computer program that was written to perform the above described procedure. The three tabulations correspond to the three trajectories shown in Figure 8.1. These trajectories are repeated as Figure 8.5 with the corresponding calculated data points from the digital computer program supperimposed. Figure 8.6 illustrates the same results except on the displacement vs. time plot instead of the phase-plane. 132 Nonlinear overshoot calculations for 0.1 radian displacement. Table 8.1. AUIOHATED DESIGN RESEARCH PROBLEM NONLINEAR OVERSHOOT CALCULATIONS DECEMBER 231 1968 INPUI PARAMETERS T-SAT THETA-O J-T B-N K-G IOZ‘INI (RAD! IOZ-IN-SECZIIOZ-IN-SECIIOl-IN-SECI T-L K-I 102-IN) IOZ-IN/RADI 1.0000101 1.53.0+03 4.0000101 1.0000-01 3.6600+OO 1.5000100 3.1500+01 CALCULATED PARAMETERS zETA-n [ETA-G ZETA OMEGA-N THETA-00T-ss (FAD/SEC! (RAD/SEC) 5.0010-02 2.5010-01 3.0010-01 2.0.90.01 4.2.70.00 CALCULATED RESPONSE 350100 1-2 BOUNDARY CONDITIONS T105 T-Cuess THETA-DOT IHETA-DOT THETA THETA-B (SEC) 15301 1310/sec1 NORMALIZED 13A01 10A01 0.1086 0.0101 -0.5150-01 -4.1560-02 5.2040-02 5.2040-02 START REGION 2 A‘ y 11. fill E OVERSHOOI' 1.407E-02 RADIANS IIHE THETA’OOT THEIA-DOI THETA IHEIA-B ISEC1 IRAD/SECI NORMALIZED (RAD) (RAD) 0.0 -8.5150-01 -4.156D-02 5.2040-02 -1.0460-02 0.0029 -8.699D-01 -4.2460'02 4.9550-02 -1.0010-02 0.0058 -8.849D-01 -4.319D-02 4.6960-02 '9.647D-03 0.0088 -8.963D-01 -4.3750-02 4.4300-02 -9.368D-03 0.0119 -9.04lD-01 -4.413D-02 4.1570-02 -9.1770*03 0.0149 -9.0830-Cl ~4.4340-02 3.8780-02 -9.0760-03 0.0181 -9.0870‘01 -4.4360-02 3.5940-02 -9.0660-03 0.0212 -9.053D-01 -4.419D-02 3.3070-02'-9.147D-03 0.0244 -8.983D-01 -4.3850-02 3.0170-02 -9.3190-03 0.0277 -8.8750-01 -4.3320-02 2.7250‘02 -9.5820-03 0.0310 -8.7300-01 -4.262D-02 2.4340-02 -9.936D-03 0.0344 -8.5480-01 '4.173D-02 2.1430-02 -1.0380-02 0.0378 -8.3300-01 -4.0660-02 1.8530-02 -1.0910-02 0.0413 -8.0760-01 -3.94ZD-02 1.5670-02 -1.1530-02 0.0449 -7.7850-01 -3.8OCD-OZ 1.2830-02 -1.224D-02 0.0485 -7.4580-01 -3.6400-02 1.0050-02 -1.304D-02 0.0523 -7.09SD-01 -3.463D-CZ 7.3190-03 -l.393D-02 0.0561 -6.69SD-01 -3.2680-02 4.6550‘03 -1.49OD-02 000601 ’602590‘01 -300550'02 200680-03 -105970-02 0.0643 -5.784D-01 -2.8240-02 -4.2950-04 -1.7130-02 0.0686 -5.2710-01 -2.5730-02 -2.8230-03 -1.8380-02 0.0732 -4.7170-01 ‘2.3020-02 -5.094D-03 -1.973D-02 0.0780 ~4.1200-01 -2.0110-02 -7.2220-03 -2.1190-02 0.0831 -3.4760-01 -1.697D-02 -9.1780-03 -2.2760-02 0.0887 -2.783D-01 -1.3580-02 -1.U920-02 '2.4460-02 0.0948 -2.033D-01 -9.924D-03 -1.24OD-02 -2.629D-02 0.1017 -1.22OD-01 -5.957D-O3 -1.3510-02 -2.8270-02 0.1096 -3.3710-02 -1.6460-03 -1.4120-02 -3.043D-02 0.1190 6.1680-02 3.0110-03 -1.3980-02 -3.2760-02 133 ’ Table 8.2. Nonlinear overshoot calculations for 0.2 radian displacement. AUTOMATED DESIGN RESEARCH PROBLEM NONLINEAR UVERSHUOT CALCULATIONS DECEMfiER 23. 1968 INPUT PARAMETERS T-L K-T T-SAT THETA-O J-T 8-M K-G (OZ-1N1 (Ol'IN/RAOI (OZ-(NI (RADI (OZ-IN-SECZI(Ol-IN-SECI(Ol-IN-SEC1 1.6000901 1.5560003 4.8000001 2.0000-01 3.6600000 7.5000000 3.7500001 CALCULATED PARAMETERS lTTA-G ZETA OMEGA-N THETA-DOT-SS (RAD/SECI '(RADISECI 5.0010-02 2.5010-01 3.0010-01 2.0490001 4.2670000 lETA-M CALCULATED RESPONSE REGION [’2 BOUNDARY CONDITIONS TIME T-GUESS THETA-DOT THETA’OUT THETA THETA-B (SECI (SEC) (RAD/SEC) NORMALIZEO (RADI (RADI START REGION 2 TIME THETA-OOT THETA-DOT THETA THETA-D (SECI (RAD/SECI NORMALIZED (RAD) (RADI 0.0 “1.3600900 -6.6400-02 6.4460-02 1.9580-03 0.0022 -1.3720000 -6.6950'02 6.1510-02 2.2350-03 0.0043 -1.3800+OO '6.7360-02 5.8530-02 2.4420-03 0.0065 -1.386DOOO ‘6.7640-02 5.5500-02 2.5780-03 0.0109 '1.3880§00 '6.7760-02 4.9370-02 2.6400-03 0.0132 -1.3850000 -6.7610-02 4.6270-02 2.5660-03 0.0154 -1.379D+00 -6.7330-02 4.3160-02 2.4230-03 0.0177 -I.3710000 -6.6900-02 4.0050-02 2.2120-03 0.0200 '1.3590000 -6.6350'02 3.6940‘02 1.9330-03 0.0223 “1.3450000 -6.5660'02 3.3830-02 1.5880'03 0.0246 “1.3280000 -6.4840-02 3.0740—02 1.1780-03 0.0269 -1.309D+00 -¢.3890-02 2.7660-02 7.0310'04 0.0293 '1.2870000 -6.2810-02 2.4600-02 1.6460-04 0.0316 ’1.2620§00 -6.1610-02 2.1570-02 '4.3650-04 0.0340 '1.2350000 ‘6.0280-02 1.8570-32 '1.099D-03 0.0365 -1.2050000 ~5.884D—02 1.5610‘02 ”1.3230-03 000389 “1.1730900 -507270-02 102680-02 '206070‘03 0.0414 '1.139D+00 -5.5580-02 9.8050'03 '3.4500-03 0.0439 -1.1020900 -S.3780-02 6.9740—03 -4.353D-03 0.0465 -1.0620*00 -5.1860-02 4.1970-03 '5.313D-03 0.0491 -1.0210*C0 -4.982D-02 1.4780-03 -6.3320'03 0.0518 -9.7650r01 -4.7670-02 “1.1770-03 -T.409D-03 0.0545 -9.3OOD—01 -4.54CD-02 -3.7640—03 -8.5450-03 0.0573 -8.8110-01 -4.3CID-02 -6.2760-03 -9.7380-03 0.0601 -8.2990-01 -4.0510-02 -8.7070-03 -I.O99D-02 0.0630 -7.7620-01 -3.7890-02 ‘1.1050-02 -1.23OD-02 0.0660 '7.2010-01 -3.5150-02 -1.33OD-02 -1.367D-02 0.0691 -6.6150-01 -3.2290-02 -1.544D-02 '1.5100-02 REGION 2-3 BOUNDARY CONDITIONS THETA-DOT THETA-DOT THETA (RAD/SECI NORMALIZED (RAD. -6.894D-01 -3.365D-02.-1.4420-02 050100 3-2 00000101 CONDITIONS 1103 T-GUESS TutTA-oor THETA-DOT THETA THETA-a 15301 (SEC) IRAD/SECI NORMALIZEO 10A01 10101 0.0259 0.0190 -2.1360-01 -1.0430-02 -z.0040-02 -2.004o-02 START REGION 2 TIME THETA-DOT THETA-DOT ~TMETA THETA-B (SECI (RAD/SECI NORMALIZED (RADI (RADI 0.0 -2.136D-01 -1.043D-02 -2.6040‘02 -2.6040-02 0.0027 '1.6520-01 -8.0660-03 -Z.6550-02 -2.7220-02 0.0056 -1.1580-01 -5.6530-03 ‘2.6950-02 -2.8420-02 0.0086 -6.5310-02 -3.188D-03 '2.7220’02 -Z.9660-02 0.0117 '1.3850-02 -6.7600-04 -2.7350-02 ~3.0910-02 0.0151 3.8410-02 1.8750-03 ‘207310‘02 -3.2190-02 OVERSHOOT' 2.734E-02 RADIANS Table 8.3. 134 Nonlinear overshoot calculations for 0.35 radian displacement. AUTOMATED DESIGN RESEARCH PROBLEM NONLINEAR OVERSHOOT CALCULATIONS DECEMBER 23. T-L K-T (OZ-IN) (OZ-IN/RAD) 106000201 105360903 1968 INPUT PARAMETERS T-SAT (OZ-(NI 4.8000901 THETA-O (RAD) 3.5000-01 J-T CALCULATED PARAMETERS K-G (OZ-IN-SECZI(Ol-IN-SEC)(Ol-IM-SEC) 3.6600000 7.5000000 3.7500001 lETA-H [ETA-G ZETA OMEGA-N THETA-OOT-SS (RAD/SEC) (RAD/SEC, 5.0010-02 2.5010-01 3.0010-01 2.0490401 4.2670’00 CALCULATED RESPONSE REGION 1'2 BOUNDARY CONDITIONS T106 T-GUFSS THETA-DOT THETA-DOT THETA THETA-e (SECI (SEC) (RAD/SECI NORMALIZED (RAD. (RAD) 0.2737 0.0747 “1.8320900 *8.9420-02 7.5970-02 7.5970-02 START REGION 2 TIME THETA-DOT THETA-OOT THETA THETA-B (SEC) (RAD/SEC) NORMALIZEO (RAD) (RAOI 0.0 '1.8320*00 -8.9420-02 7.5970‘02 1.3470-02 0.0017 '1.839D¢00 ‘8.9760-02 7.2930-02 1.365D-02 0.0033 '1.8440900 '9.0000-02 6.9870-02 1.3770'02 0.0050 '1.847D+00 '9.014D-02 6.6790-02 1.3830-02 0.0067 '1.84TD§00 '9.0160-02 6.3700-02 1.3840-02 0.0083 “1.8450000 -9.0080-02 6.0590-02 1.3000‘02 0.0100 '1.84ZD+00 -8.989D-02 5.7480-02 1.3710-02 000111 -108360900 '809600‘02 504360-02 103560-02 0.0134 ’1.8280+00 ’8.9210-02 5.1240-02 1.337D‘O2 0.0151 '1.3170000 '8.8710-02 4.8120‘02 1.3120-02 0.0169 ‘1.8050+OO '8.8120'02 4.5010'02 1.2820-02 0.0186 -I.79ID+OO -8.74ZO-02 4.1910-02 1.2470‘02 0.0203 ‘1.775D*00 -8.663D-02 3.8820-02 1.2080-02 0.0221 ”1.7570000 '8.5750-02 3.5750‘02 1.1640-02 0.0238 '1.7370§00 ’8.477D-02 3.2690-02 1.1150'02 0.0256 ‘1.7140900 ‘0.369D-02 2.9660-02 1.0610'02 0.0291 '1.665D+00 -8.127D-02 2.3660'02 9.3970‘03 0.0309 '1.6370+00 '7.9920-02 2.0700‘02 0.7240-03 0.0327 '1.6080+00 -7.849D-02 1.7760-02 0.0070'03 0.0345 ‘105770900 ‘TQOQTO‘OZ 100870-02 702470-03 0.0364 '1.544D§00 ”7.5370'02 1.2000-02 6.4440'03 000382 “105090900 “703680-02 901740-03 505980-03 0.0401 ‘1.4730+00 -7.1900-02 6.3860-03 4.7110'03 0.0420 ’1.4350*00 “7.0040-02 3.6400-03 3.7020-03 REGION 2‘3 BOUNDARY CONDITIONS THEIA‘DOT THETA-DOT THETA (RAD/SEC) NURMALIZED (RAOI ‘1.4380§00 '7.CI90-02 3.0540-03 OVERSHOOT'I 4.936E-02 RADIANS #3:" Jm: 1.. mfi%ififi1 .‘ 135 .swnmmfiw ocmHm1ommnm :o emcommmu Eoumxm .m.m onzmflm 9.1. o .1; run. 0...... mu. o.~. n... om. Mo. 8259:. . .m. 02w... 83.05 3...} no . mu.» S 0833. 3.8.... 2.13 0.. a» .3 .m o... 2.13 we . 2.» €633 «umm1z_1NO mm.» 0 I. CNN—442mg.— uum .2700 3... .6 000-273 a ...n .. .0. 352.18 0?... 5. no... .35 . 25- .IO DISPLACEMENT (RADIANS) -.05d 136 KT =1530 02-10mm K. = 37.5 02-10-550 3. =7.5 02-10-5130 Jr a 3.66 oz-IN-SEC‘- Tm -43 02-10 TL -16 02—10 “0'20-49 RAD/SEC Cg 3.25 gms.05 HEAVY LINE , ‘/ UNIT SATURATED DEAD ZONE WIDTH -0UE T0 FRICTION LIGHT LINE UNIT UN SATURATED 3:211be {3330 E .2003 “-m' -- .8 I. I4) 1!er .. “ME ‘7 I 1 (SEC) Figure 8.6. Typical system response curves. l .fi, APPENDIX C DERIVATION OF NONLINEAR BANDWIDTH EQUATION The purpose of this appendix is to derive the necessary equa- tions for calculating the system bandwidth. Often, this is "accom- plished" by using the linear closed loop transfer function, with 5 replaced by jw, and calculating the frequency at which leo(j0)/ei(jm)| is down 3 db. However in the real world, one rarely realizes this value. This discrepancy is mainly a result of amplifier saturation, which in turn causes the actual bandwidth to be a function of input level. The true affects of saturation, as well as coulomb friction, backlash and finite stiffness, could be ac- counted for by direct simulation of the nonlinear state equations presented in Section 2.2. However, including this simulation in an iteration loop (as necessary to find w = wB such that the response is '3 db) is very time consuming. This is due to the fact that for each iteration one must wait for steady-state conditions before an evaluation can be made. In order to minimize the solution time, an algebraic equation for the bandwidth frequency is derived including the effects of saturation and coulomb friction. This is accomplished by using describing function approximations. For simplicity sake, the system is considered to have zero backlash and infinite stiffness. This approximation is justified since the displacements are considered to be large compared to any deflections that may exist in the gear train. 137 138 The bandwidth frequency for the linear second order system is found by taking the closed loop transfer function 2 00(5) wN = 2 2 (C.1) 91(5) 5 +2chs+0N replacing s by jm and setting the magnitude equal to 0.707 for (02 N 2 = 0.707 (0.2) k "”321 * jLZCwNwB) (“N and solving for the bandwidth frequency (08) LOB = 1111N-\/l-2C2+«\/2-4524'451+ (C-S) This equation can be extended to the desired nonlinear case using describing functions, by replacing w and c by their correspond- N ing effective values wN' and c' given by KfKa'Km wN' = -—f3———— (C.4) T +K K ' K +02 (2' BM 0a] a (C.S) 2 l/KfKa'KMJT 139 where Ka' and N2 are the effective gain values for the amplifier and coulomb friction elements as given by their respective describing fUnctions. For the friction element N__j1_.__ 2- . (0.6) 060(peak) and letting N3 be the describing function for the amplifier satu- ration 2Ka sin1201 N3 = —;—- w + 2 for Ei(peak) Z-EsatKa (C.7) = Ka otherwise where Esat w = arc Sln KaEi(peak) (0.8) Thus N3 is a function of its respective input Ei(peak) and like- wise N2 is a function of 00(peak). One can obtain Ei(peak) for any w from the transfer function 2 - 1 I 51(5) K S +2"’ML “N s (09) = ’7 12 0i(s) f s +2c'wN's+wN where 8 +N . = M 2 (0.10) 140 by replacing s with jw and solving for Ei(peak) 10‘I + (ZEML'wN'w)2 E (peak) = K 0.(peak) i f 2-22 2 (wN' w ) + (ZC'wN'w) 1 (C.11) Since the output amplitude is 0.707 of the input amplitude .umg 00(peak) = 0.707 wei(peak) (0.12) and thus 4TL N2 = 0.707 nwei(peak) (C'IS) The necessary relationships have now been derived and “8 given by “’3 = wN1‘/1_2C12+_‘/2_4C12+4C14 (C.14) is calculated using the iteration scheme illustrated in Figure C.1. This procedure consists of one iteration inside another. The outer loop is used to adjust w until it equals w as given by (C.14) B to the desired accuracy (008) , while the inner loop adjusts, for each value of w , Ka' so that it equals its describing function value (N3) as given by (C.7) to the desired accuracy (GKa). Tables C.1 and C.2 show sample outputs from a computer program that was written to perform the above described procedure. The only difference between the two tables is that Table C.2 is for the zero friction case. Each table consists of a tabulation of the system 141 ( START ) INITIALIZE W'U. (LINEAR) K0 ' Ko CALCULATE N, USING 0.13 DETERMINE NEW 0,, USING ITERI SU0- ROUTINE DETER MINE NEW In USING ITER 2 SUBROUTINE CALCULATE l‘51 ;' USING 0.5 w.‘ USING 0.4 C ‘ USING 0.10 IL (PEAK) USING C.|I CALCULATE 01 USING ca 0. USING 0.7 Figure C.1. C ALCUL ATE III. USING 0.14 Nonlinear bandwidth flow diagram. 142 Table C.1. Nonlinear bandwidth calculations with friction and amplifier saturation. AUTOMATED DESIGN RESEARCH PROBLEM NONLINEAR BANDHIDTH CALCULATIONS JANUARY 15. 1969 INPUT PARAMETERS K-F K-A K-M K-G (VOLT/RAD) (VIV) (Cl-IN/V) (OZ-IN-SEC) 1.460E001 1.420E+02 1.350E900 5.000E+01 B-M J-T T-L E-SAT DELTA Ol-IN-SECIIOZ-IN-SECZI (OZ-(NI (VOLTS) 3.000EOOO 1.500E-01 1.000E000 2.050E001 1.000E-O3 CALCULATED LIAEAR PARAMETERS H-N ZETA H-B THETA-SAT (RAD/SEC) (RAD/SEC) (DEG) 1.366E*02 1.293EO0C 6.174E§01 3.731E*00 NONLINEAR BANDHIDTH VS THFTA-IN CALCULATIONS THETA H-B N-3 N-2 (DEG) (RAD/SECI (V/VI (OZ-IN‘SECI 5.0005-01 5.665E001 1.420E902 3.638E000 1.000Ef00 5.919E001 1.420E'02 1.742E*00 1.500E+00 6.004E+CI 1.420E002 1.145EO00 2.000E*00 6.046E+01 1.420E‘02 8.5?85-01 2.500E’00 6.072E+01 1.420E*02 6.7935’01 3.000E000 60089E+01 104205+02 SCOQSE-OI 3.SOOE+00 6.1015001 1.420E+02 4.829E-01 4.000E+OO 6.129E001 1.360E002 4.2075-01 4.500E+00 6.278E401 3.928E+01 3.645E’01 5.000E000 5.9555+01 2.899E001 3.464E‘01 5.500E+00 5.665E+01 2.382E’01 3.311E-01 6.000E+00 5.4115001 2.047E+01 3.178E“01 6.500E000 5.185E+01 1.805E401 3.061E'01 7.000E900 4.9825901 1.620E*01 2.9575-01 7.500Ef00 4.803E+01 1.474E4C1 2.8655-01 8.000E+00 4.635E+01 1.351E+(1 2.781E‘01 8.500FTOO 4.482E+01 1.249EOC1 2.706E-01 9.000E+00 4.349E+01 1.166E001 2.6465'01 QQDOOETOO “OZZOETOI SOOQIE+C1 205816-01 1.000E001 4.101E+01 1.026E+01 2.522E-01 100506901 309915901 9.684E+C0 20467E‘01 1.100E’01 3.887E*01 9.175E*00 2.416E-01 1.1505+01 3.79IE’C1 B.720£*00 2.369E-01 1.2OOE*OI 3.700E+01 3.312E000 2.325F'01 1.250E001 3.615E‘01 7.941E+CO 2.2835-01 1.300E401 3.5325901 7.594E+CO 2.24OF-01 1.350C901 3.4575*01 7.292E*C0 2.206E‘01 1.400E+01 3.386F001 7.013E*00 2.173F-01 1.450E+01 3.318E001 6.756E+C0 2.142E-01 1.5OOE+01 3.253E+01 6.518E000 2.112E'01 1.5505901 3.192E901 6.298E*00 2.084E-01 1.600E+01 3.138E*01 6.IIOE+00 2.057E'01 1.650Et01 3.080E*01 5.912E000 2.03IE'01 1.7005*01 3.024E901 5.7275+00 2.0075-01 1.750E*01 2.971Et01 5.5545900 1.984E-01 1.8005+01 2.9205+01 5.393E400 1.9615'01 lK-WL'JC . '.“. .'-_- . “’0‘ 143 Table C.2. Nonlinear bandwidth calculations for zero friction case. AUTOMATED DESIGN RESLARCH PROBLEM NONLINEAR BANDHIDTH CALCULATIONS JANUARY IS, 1969 INPUT PARAMETERS K-F K-A K-M K-G (VOLT/RAD) IV/VI (OZ-IN/VI IDZ-IN-SECI 1.460E001 1.4205602 1.350EOOO 5.000E001 B-M J-T T-L E-SAT DELTA DZ-IN-SECIIDZ-IN-SECZI IDZ-INI IVOLTSI 3.000E000 1.500E-OI 0.0 2.C50E+OI I.OOOE-03 CALCULATED LINEAR PARAMETERS H-N ZETA H-B THETA-SAT (RAD/SEC) IRAD/SECI IDEGI I.36bE+02 I.293t+OC 6.176E001 3.73IE000 NDNLINEAR BANDHIDTH VS THETA-IN CALCULATIONS THETA H'B N-3 N-Z IDEGI IPAD/SECI IV/VI IDl-IN-SECI 5.000E-0I 6.174EOOI 194205002 0.0 1.000E000 6.174EOOI 1.420E9C2 0.0 1.500E+00 6.174EOOI. I.§20E002 0.0 2.000E*00 6.174E*DI I.420E*C2 0.0 2.500E*00 6.174E+OI 1.620E+C2 0.0 3.000E+00 6.176E001 1.420E+(2 0.0 3.SOOE+OO 6.174FOOI 1.420E0C2 0.0 [900006.00 602016.01 ‘0347EPCZ 000 4.500E+00 b.326E001 3.449E*CI 0.0 5.000E000 5.996E901 2.663E001 0.0 5.500(000 5.706EOOI 2.2256+CI 0.0 6.000E+OO 5.4505001 1.9296001 0.0 6.500E000 5.222E+OI 1.710E+Ol 0.0 7.000E+DO 5.019EOOI 1.539EOOI 0.0 7.500E*00 4.842EOOI 1.407E0CI 0.0 8.000E000 4.6756601 1.294EOCI 0.0 8.500E900 '4.524E+OI l.I99E+OI 0.0 9.000E000 4.384E+CI 1.118EOCI 0.0 9.500E+OO 4.256FOCI I.048E+OI 0.0 1.000E+01 4.137FOCl 9.865E+00 0.0 IoOSDEOOI 4.0?7EOOI 9.323F+00 0.0 1.1005001 3.924F+OI 8.843E000 0.0 1.150E+01 3.827E*OI 8.608E+00 0.0 I.ZOOE+OI 3.73TE+CI 8.023EOCO 0.0 1.250FOOI 3.652EOCI 7.h72E+00 0.0 1.300E*OI 3.572E’C1 7.353E900 0.0 I.3bOE*OI 3.500E001 7.076E000 0.0 1.400EOOI 3.426EOOI 6.800E900 0.0 1.4506401 3.356E+OI 6.5A6E900 0.0 1.500EtOI 3.29OEOCI 6.312EOOO 0.0 1.550E901 3.232EOC1 6.115E900 0.0 [.600EODI 3.172F+CI 5.9I4EOOO 0.0 I.bSOE+OI 3.110E901 5.712E000 0.0 loTOOE+Ol 3.060EOOI 5.551E*00 0.0 I.750E+OI 3.0IIF+CI 5.399E900 0.0 1.800E001 2.956E*OI 5.233E+OO 0.0 - .AIL 144 bandwidth as a function of peak magnitude of the input sine wave. There is little difference between the two tabulations thereby demonstrating that amplifier saturation is the dominate nonlinear— ity except for small input levels. At small input levels (less than about 4 degrees) the zero friction case is equivalent to the linear case (no saturation) and the bandwidth equals the linear value of 61.74 rad/sec. For the case with friction the bandwidth falls off for both high and low values of input amplitude, as would be expected. The results obtained are plotted as Figure C.2. An analog computer was used to check the validity of the describing function approximations used to represent the two nonlinearities. Figure C.3 documents the simulation used. The analog computer was operated by setting potentiometers 5 and 6 to the corresponding values of w and 91(peak) as tabulated B by the digital program and making Lissajous diagrams in predrawn boxes with the height (output) equal to 0.707 times the base (input). As demonstrated by Figure C.4 the analog computer re- sponses are almost exactly tangent to all sides, thereby demon- strating that the describing function assumption provides an effective model of the system. :5 ’2'“ 145 , . rfifiLE .1.I . . m . .owzpwfimfiw.xwomtousoao~ “Shaw mo cowuocsm a ma :uvwzvcmn o>Hpoommm .N.u oaswfim 33:32:42. 6 O. 0. Q. N. O. 0 o t N O 1 q d # J A. d [1 - n6 uCDoE 20:14:35 eon—(ad 2.53 09.093 ”P253 2.0NO 00 SN was. 2323 m4 .. ...— on O? «3395273 n. .o . .:. IV on 933452.- No .n a an .. 339:. 2.73 .on u ex :73 0.. u w 952.53 dog» 5. Leo «552:: 22.96 2 z. .3 .9...» Ob (an/om 'n aouanonu mamas" 146 EF“. .mflmxdmcm subwzvcmn pom vow: Emummflo Housmeoo wofimc< .m.u onsmflm o. 33. u. the. .moa. . _ n Anmv . . u o. . 00. Q0 0000 «Imxl b 2.32. .6 «2:3. C ...zu3m044am3 m_x< x 28.5“ a 5m» h... :6. . e a o. . _ a 2.2.3. ..o» 256. #523333 2.2 > . 4 L >.o. .+. a p . Po. 0. ..m m 3.3.36 omo. . _. .. ® >00 7... 147 slope-0.707 r””'i 1°01 ”’,, ///”/ I .707 A// .r._ J/._ 1 5 ’ I _ ,/’//7 ei ; 61 (peak) I 2 degrees ‘ wi I 60.46 rad/sec #_",.—’ Oi (peak) I 4 degrees w I 61.29 rad/sec .707 .707 01 (peak) I 6 degrees “1 I 54.11 rad/sec 01 (peak) I 10 degrees ”1 I 41.01 rad/sec .707 .707 61 (peak) 0 14 degrees «1 I 33.86 rad/sec 01 (peak) I 18 degrees m1 I 29.20 rad/sec Figure C.4. Lissajous diagrams obtained from analog simulation. 10. ll. 12. 13. LIST OF REFERENCES Booz, Allen Applied Research, Inc. Semiconductor Information Storage and Retrieval System. May, 1964. Report No. 185-5T56-2. Chubb, B. A. Modern Analytical Design of Instrument Servo- mechanisms. Addison-Wesley, 1967. Hannom, T. J. B. and Kaskey, G. Digital Computer As a Design_ Tool. IEEE. International Convention Record, volume 13, Part 6 (Symposium on Automatic Control, Systems Sciences, Cybernetics, Human Factors) 1965 - pp 27-38. Wilde, D. J. Optimum Seeking Methods. Prentice-Hall, 1964. 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