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UBC Theses and Dissertations

Adaptive control performance indicators for internal combustion engines North, David Lawrence 1975

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ADAPTIVE CONTROL PERFORMANCE INDICATORS FOR INTERNAL COMBUSTION ENGINES by DAVID LAWRENCE NORTH B.A.Sc, U n i v e r s i t y of B r i t i s h Columbia Vancouver, B r i t i s h Columbia, 1973 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of Mechanical Engineering We accept t h i s t h e s i s as conforming t o the re q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1975 In presenting this thesis i n p a r t i a l fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library shall, make i t f r e e l y available f o r reference and studyo I further agree that permission.for extensive copying of t h i s thesis f o r sch o l a r l y purposes may be granted by the Head of my Department or by his representatives. I t i s understood that copying or p u b l i c a t i o n of t h i s thesis f o r 'financial gain s h a l l not be allowed without my wr i t t e n permission. Department of Mechariical Engineerirr The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada Date A p r i l 2nd, 1975 i i ABSTRACT The purpose of the i n v e s t i g a t i o n i s to study the dynamics of the i n t e r n a l combustion e n g i n e - v e h i c l e - d r i v e r system. S p e c i f i c a l l y , the system v a r i a b l e s , engine angular v e l o c i t y and engine angular a c c e l e r a t i o n are examined as p o t e n t i a l observers of the engine mean torque. Such an observer i s a requirement f o r the a p p l i c a t i o n of adaptive c o n t r o l t o i n t e r n a l combustion engines. This type of c o n t r o l system has shown promise i n p r o v i d i n g s o l u t i o n s to the present problems of f u e l economy, a i r p o l l u t i o n and performance. A n o n l i n e a r dynamic model o f the e n g i n e - v e h i c l e - d r i v e r system i s developed. This model i s l i n e a r i z e d and s i m p l i f i e d to provide expressions f o r the v a r i a b l e s of i n t e r e s t , the engine angular a c c e l e r a t i o n and v e l o c i t y . The v a l i d i t y of the s i m p l i f i e d model i s e s t a b l i s h e d by comparison w i t h r e s u l t s obtained from the computer s i m u l a t i o n of the n o n l i n e a r model. The agreement between the two models i s good. The s o l u t i o n s of the equivalent system model are analyzed to determine which i s the best observer f o r the mean torque. I t i s e s t a b l i s h e d that the steady s t a t e f o r c e d o s c i l l a - t o r y engine angular a c c e l e r a t i o n response provides the best observer. ACKNOWLEDGEMENT I would l i k e to thank Dr. T. N. Adams and Dr. R. E. McKechnie f o r t h e i r support and guidance throughout t h i s work. The c o n s t r u c t i v e c r i t i c i s m provided during the pr e p a r a t i o n of t h i s t h e s i s i s p a r t i c u l a r l y appreciated. I would a l s o "(Like to thank my w i f e f o r her patience and understanding i n t y p i n g the t h e s i s . i v TABLE OF CONTENTS SECTION PAGE 1 INTRODUCTION 1 (1.1) Purpose and Scope 1 (1.2) Work by Others 2 (1.3) Adaptive Control 4 (1.4) Application of Adaptive Control to Internal Combustion Engines 7 (1.5) Mean Torque Measurement 15 (1.6) Preview of Thesis 17 2 DYNAMIC ANALYSIS OF INTERNAL COMBUSTION ENGINE DRIVER VEHICLE SYSTEMS 18 (2.1) System Dynamics 18 (2.2) External Torques 20 3 AN EQUIVALENT LINEAR SYSTEM 34 (3.1) A Constant Speed System 34 (3.2) Equivalent Spring and Damper . 36 (3.3) Equations of Motion of Equivalent Linear System 38 4 SOLUTION OF THE EQUIVALENT LINEAR SYSTEM 39 (4.1) The Transient Response 39 (4.2) The Constant Response 40 (4.3) The O s c i l l a t o r y Response 41 (4.4) The Total Response a.* 42 V ' S E C T I O N  lE- A G E 5 SOLUTIONS OF A SPECIAL CASE . 43 (5.1) The Vehicle Considered , . 43 (5.2) The P a r t i c u l a r Case Considered 43 (5.3) The Transient Response 44 (5.4) The Constant Response 46 (5.5) The O s c i l l a t o r y Response 46 6 COMPUTER SOLUTION OF THE NONLINEAR SYSTEM 48 7 MEAN TORQUE INDICATORS 55 (7.1) The Transient Response 55 (7.2) The Constant Response . 57 (7.3) The O s c i l l a t o r y Response 57 8 CONCLUSIONS 60 9 RECOMMENDATIONS FOR FUTURE WORK 61 REFERENCES 62 APPENDICES I EQUATIONS OF MOTION OF AN INTERNAL COMBUSTION ENGINE VEHICLE SYSTEM 62 II SOLUTIONS OF THE EQUATIONS OF MOTION OF THE LINEAR EQUIVALENT SYSTEM 75 III PHYSICAL CONSTANTS USED IN THE ANALYSIS 80 IV COMPUTER PROGRAM 95 v i LIST OF TABLES TABLE PAGE I ADAPTIVE SPARK TIMING CONTROLLER TRUTH TABLE . . . . 10 II PHYSICAL CONSTANTS OF THE CORTINA 2000 c.c. SEDAN 28 III COMPARISON OF ANALYTIC AND SIMULATED RESPONSES TO A STEP CHANGE IN ENGINE OUTPUT TORQUE (80-23.5=56.5 (Nm) ) . . . 53 IV EVALUATION OF SYSTEM RESPONSE AS INDICATORS OF MEAN TORQUE 59 v i i LIST OF ILLUSTRATIONS FIGURE PAGE 1 S i n g l e L e v e l Extremum Seeking Adaptive C o n t r o l l e r 5 2 S t r u c t u r e of Co n t r o l s and Dynamics of the Engine V e h i c l e D r i v e r System . . . 8 3 S Conceptual Block Diagram of an Adaptive Spark Timing C o n t r o l l e r . . . 9 4 The E f f e c t of Spark Timing on the Mean Torque Output 12 5 Hypersurface I l l u s t r a t i n g the R e l a t i o n s h i p Between Engine V a r i a b l e s . ' 13 6 Lumped Parameter Engine V e h i c l e System 19 7 Comparison of the S i m p l i f i e d and Numerical Ser i e s Engine Torque Si g n a l s 22 8 Mean Torque of the C o r t i n a 2000 c.c. Engine . . . 24 9 Free Body Diagram I l l u s t r a t i n g the E x t e r n a l Forces on a V e h i c l e 30 10 Load Torque on the C o r t i n a Sedan . 32 v 11 Equivalent L i n e a r Engine V e h i c l e System 35 12 T h r o t t l e Command Input to the Computer Simul a t i o n Model 49 13 Mean Torque Response to the T h r o t t l e Command Input 50 14 Engine R o t a t i o n a l V e l o c i t y Response to the T h r o t t l e Command Input 51 15 Engine R o t a t i o n a l A c c e l e r a t i o n Response to the T h r o t t l e Command Input 52 v i i i FIGURE PAGE 16 Transient Cancellation 56 17 Free Body Diagram of Engine 69 18 Free Body Diagram of Clutch 69 19 Free Body Diagram of Transmission 71 20 Free Body Diagram of Driveshaft 71 21 Free Body Diagram of Rear End 72 22 Free Body Diagram of Tires 72 23 Free Body Diagram of Load 74 24 Flywheel and Crankshaft Layout of Cortina 2000. c.c. Engine 81 25 Connecting Rod, Piston Pin and Piston of Cortina 2000 c.c. Engine 82 26 Experimental Setup for Determining Clutch Spring and Damping Constants 88 27 Results of the Clutch Experiment 88 28 Clutch Plate and Springs 89 29 Driveshaft Dimensions . . . . . 91 30 Free Body Diagram of Vehicle 93 31 Front View of Cortina Sedan 93 i x NOTATION Symbol D e s c r i p t i o n Ap v e h i c l e p r o j e c t e d area B^-jBljB"^", ... F o u r i e r s i n torque s e r i e s c o e f f i c i e n t s C"2",Cl,C*2" ,... F o u r i e r cos torque s e r i e s c o e f f i c i e n t s DC equivalent l i n e a r c l u t c h damping c o e f f i c - i e n t s Dt t i r e damping c o e f f i c i e n t Dt' equivalent t i r e damping c o e f f i c i e n t at engine speed Je engine i n e r t i a Je<J e q u i v a l e n t system i n e r t i a JI load i n e r t i a vii' e q u i v a l e n t load i n e r t i a a t engine speed Kc c l u t c h s p r i n g constant Kf r o l l i n g r e s i s t a n c e drag c o e f f i c i e n t Ks d r i v e s h a f t s p r i n g constant Ks' e q u i v a l e n t d r i v e s h a f t s p r i n g constant at engine speed Kt t i r e s p r i n g constant Kt' equivalent t i r e s p r i n g constant at engine speed Kl r a t i o of the maximum to the mean torque per engine c y c l e Nf f r o n t wheel r e a c t i o n Symbol ; D e s c r i p t i o n Nf rear wheel r e a c t i o n R rear end r a t i o Rg gear r a t i o Rw dynamic wheel radius TdV engine mean torque output TdV^ optimum engine mean torque output Tb brake torque Tc n o n l i n e a r coulomb c l u t c h damping Te engine instantaneous torque output TI v e h i c l e load torque TI equivalent v e h i c l e load torque a t engine speed We engine angular v e l o c i t y Wengi i n i t i a l engine v e l o c i t y Wn n a t u r a l frequency Wnd damped n a t u r a l frequency W v e h i c l e weight |x| magnitude of v a r i a b l e x t time' creng engine constant a c c e l e r a t i o n 6 crank angle 0CX,0CX,$CX t r a n s i e n t motions of v a r i a b l e x 01 gradient angle 0S spark t i m i n g dSSCX,&SSCX,0SSCX steady s t a t e constant motions of v a r i a b l e Symbol • • • 0ssoxf0ssox,0$sox r rc ** T c D e s c r i p t i o n steady s t a t e o s c i l l a t o r y motions v a r i a b l e x pe r i o d of t r a n s i e n t o s c i l l a t i o n t r a n s i e n t time constant t h r o t t l e p o s i t i o n time of t r a n s i e n t damping r a t i o 1 1 INTRODUCTION (1.1) Purpose and Scope The c h i e f problems of the cu r r e n t i n t e r n a l combustion automotive engine are a i r p o l l u t i o n , f u e l economy and performance [ 3 > l ] , [ 2 ] , [ 3 ] , [ 4 ] , [ 5 ] , [ 6 ] / [ 7 ] . 1 A concept which has shown promise i n p r o v i d i n g a s o l u t i o n to these problems i s tha t of Adaptive C o n t r o l [ 8 ] , [ 9 ] , [ 1 0 ] . Using t h i s concept the present d i s t r i - butor advance u n i t and carburetor would be replaced by spark t i m i n g and a i r - f u e l r a t i o devices regulated by an adaptive c o n t r o l l e r . The p o t e n t i a l advantages of adaptive c o n t r o l i n c l u d e improved f u e l economy, reduced emissions, reduced maintenance and improved r e l i a b i l i t y [ 8 ] , [ 9 ] , [ 1 0 ] . An adaptive c o n t r o l scheme req u i r e s a s i g n a l which i n d i c a t e s system output or performance. I t has been shown by Adams [11] tha t f o r i n t e r n a l combustion engines the most s u i t a b l e s i g n a l i s the engine mean torque (Tdv)• The primary purpose of t h i s work i s to determine the best means of o b t a i n - ing t h i s torque s i g n a l . A secondary purpose i s to co n s t r u c t a model of the dynamics of an e n g i n e - v e h i c l e - d r i v e r system f o r l a t e r work i n determining the optimum adaptive c o n t r o l l e r c o n f i g - u r a t i o n . Numbers i n square brackets designate references at end of t h e s i s . 2 Any v a r i a b l e of the system-which acts as an observer of TQV w i l l be a f f e c t e d by disturbances a c t i n g on the system. Therefore, i t i s necessary t o study the dynamics o f the e n t i r e system to l e a r n how a p o t e n t i a l observer i s a f f e c t e d by the disturbances encountered under normal o p e r a t i n g c o n d i t i o n s . The dynamics of a lumped parameter engine v e h i c l e system are considered i n t h i s work. An i n i t i a l model i s developed and i t s equations of motion a>revs6l-vgdsb^us4muil>ation on the IBM 370 computer. This model i s l i n e a r i z e d and s i m p l i f i e d t o provide equations of motion which can be solved a n a l y t i c a l l y . The v a l i d i t y of the l i n e a r i z a t i o n and s i m p l i f i c a t i o n made i s checked by comparison of the simulated and a n a l y t i c s o l u t i o n s f o r a s p e c i a l case. The a n a l y t i c s o l u t i o n s are used to evaluate p o t e n t i a l observers f o r TQV • In connection w i t h t h i s work, models d e s c r i b i n g the engine mean torque (T(JV) and v e h i c l e load torque (T I ) are developed. Some p r e l i m i n a r y work on the adaptive c o n t r o l model and the human c o n t r o l model has a l s o been undertaken. This work w i l l be necessary i n fu t u r e to determine the optimum adaptive c o n t r o l l e r c o n f i g u r a t i o n . (1.2) Work by Others The recent work done i n the a p p l i c a t i o n of automatic c o n t r o l theory to i n t e r n a l combustion engines f a l l s i n t o three c a t e g o r i e s , d i g i t a l memory f u e l i n j e c t i o n c o n t r o l (DMC), c l o s e d 34 loop f u e l i n j e c t i o n c o n t r o l (CLC), and extremum seeking adaptive c o n t r o l s (AC) . (1.2.1) D i g i t a l Memory Fuel I n j e c t i o n C o n t r o l The DMC improves upon the conventional carburetor by- e l i m i n a t i n g the mechanical wear and aging problems [12], [13,]. The memory stores predetermined optimum f u e l i n j e c t i o n pulse times. These pulse times are addressed by in f o r m a t i o n d e r i v e d from engine speed and t h r o t t l e p o s i t i o n . The drawback of t h i s system i s the predetermined nature of the sto r e d i n f o r m a t i o n . For example, pulse times stored i n the memory are obtained from t e s t s on a standard engine under standard environmental c o n d i t i o n s . These pulse times are optimum f o r t h i s engine at the p a r t i c u l a r time they are determined. However, production t o l e r a n c e s i n engine manufactur- ing are not t i g h t enough to assure that the pulse times w i l l be optimum f o r every engine r o l l i n g o f f the assembly l i n e . An even more seriou s problem i s the wide range i n environmental c o n d i t i o n s a production engine w i l l experience during i t s l i f e . This range of environmental c o n d i t i o n s d e f i n i t e l y a f f e c t s the optimum pulse time [14]. (1.2.2) Closed Loop F u e l I n j e c t i o n C o n t r o l .The CLC senses the oxygen content of the exhaust gas stream and compares i t w i t h a reference value [14],[15]. I f the measured value d i f f e r s from the reference value, the c o n t r o l a d justs the f u e l i n j e c t i o n pulse time u n t i l the d i f f e r e n c e 4 disappears. This type of c o n t r o l was i n i t i a l l y developed to insure s a t i s f a c t o r y o p e r a t i o n of c a t a l y t i c converters by regu- l a t i n g the exhaust gas stream composition [14]. L a t e r i n v e s t - i g a t o r s [15] have extended the work to p r o v i d i n g optimum a i r - f u e l r a t i o s , but the workers have not shown how t h i s optimum may be i d e n t i f i e d . (1.2.3) Extremum-Seekinq Adaptive C o n t r o l AC a d j u s t s the spark t i m i n g and/or a i r - f u e l ratio:,, u n t i l a l o c a l optimum i n performance has been reached [ 8 ] , [ 9 ] , [10], Thus, AC d i f f e r s from conventional automotive engine c o n t r o l s and feedback c o n t r o l s i n t h a t i t searches f o r and maintains the optimum s e t p o i n t values of the v a r i a b l e s under c o n t r o l . These values maximize the engine performance. (1.3) Adaptive C o n t r o l F i g u r e 1 i s a block diagram r e p r e s e n t a t i o n of a s i n g l e l e v e l extremum seeking adaptive c o n t r o l system. The o b j e c t i v e of such a c o n t r o l system i s to i d e n t i f y and maintain the optimum s e t p o i n t X of the system. This s e t p o i n t w i l l maintain the out- put Z at i t s extremum ( i n t h i s case i t s maximum) value Z^» In t h i s simple system the r e l a t i o n s h i p between the input and the output (the system c h a r a c t e r i s t i c ) has been assumed to be given by: Z= Z # -X(X # -Xo) 2 (1.3-1) SYSTEM CHARACTERISTIC Xo 2X(X^Xo)K CONTROL LAW SLOPE DISCRIMINATOR -K •^-2MX*Xo) Figure 1: S i n g l e Level Extremum Seeking Adaptive C o n t r o l l e r Z=(Z#-X(X#-Xc)2) 6 The c o n d i t i o n which determines t h a t the optimum i s reached occurs when the slope of the system c h a r a c t e r i s t i c (AZ/AX) becomes zero. For example, i f the system i s i n i t i a l l y at some p o i n t Xo away from a n d the c o n t r o l makes a change to some new X, then a corresponding change i n Z w i l l r e s u l t . The change i n output Z i s then determined w i t h respect t o the change i n input X« The r e s u l t i n g s i g n a l i s used to d r i v e the system toward the optimum X where A7/AX approaches 0. The c o n t r o l law ( i . e . the law which governs the c o n t r o l response f o r a given e r r o r s i g n a l (AZ/AX) ) i n F i g u r e 1 i s a simple p r o p o r t i o n a l gain K- More s o p h i s t i c a t e d c o n t r o l laws may prove v a l u a b l e i n improving system s t a b i l i t y and response time. S t a b i l i t y r e f e r s to the tendency of the system to approach the optimum w i t h a decreasing amplitude of o s c i l l a t i o n . I f the system were to o s c i l l a t e w i t h i n c r e a s i n g amplitude around the optimum i t would be c l a s s e d as unstable. Response time r e f e r s to the time taken f o r the system to reach the optimum.for an i n i t i a l s e t p o i n t an a r b i t r a r y d i s t a n c e away from the optimum. For example, d e r i v a t i v e c o n t r o l ( i . e . c o n t r o l p r o p o r t i o n a l to the r a t e of change of er r o r ) may be a p p l i e d i n c o n j u n c t i o n w i t h the p r o p o r t i o n a l c o n t r o l to improve the system response time. A complete d i s c u s s i o n of the s t a b i l i t y and response time problems i s beyond the scope of t h i s work and the i n t e r e s t e d reader i s r e f e r r e d to [16],[17]. 7 An important feature of adaptive c o n t r o l i l l u s t r a t e d i n F i g u r e 1 i s tha t i f there i s only one maximum w i t h i n the oper a t i n g range then t h i s w i l l be the p o i n t the c o n t r o l w i l l seek to operate a t . A l l other p o i n t s w i l l produce a l o c a l gradient (AZ/AX) . Thus a c o n t r o l s i g n a l K(AZ/AX) w i l l e x i s t and d r i v e the system u n t i l AZ/^X becomes zero at a l o c a l optimum. Adaptive c o n t r o l can be e a s i l y extended to c o n t r o l two or more v a r i a b l e s . I n i t i a l small p e r t u r b a t i o n s are made i n the set p o i n t s of the v a r i a b l e s under c o n t r o l . The e f f e c t s of these p e r t u r b a t i o n s are then observed. The observations enable the c o n t r o l l e r to determine the d i r e c t i o n i n which the inputs are adjusted to achieve the "steepest ascent" toward t h e i r optimum s e t p o i n t s . (1.4) A p p l i c a t i o n of Adaptive C o n t r o l to I n t e r n a l Combustion Engines Figu r e 2 i l l u s t r a t e s the s t r u c t u r e of the c o n t r o l s and dynamics of the e n g i n e - v e h i c l e - d r i v e r system. The adaptive con- t r o l supervises the c o n t r o l o f s e v e r a l p l a n t v a r i a b l e s i n response to the d r i v e r c o n t r o l of f u e l r a t e . Figure 3 i s a conceptual i l l u s t r a t i o n of an adaptive spark t i m i n g c o n t r o l l e r f o r an i n t e r n a l combustion engine. The t r u t h t a b l e of the And-Nor l o g i c gate of t h i s c o n t r o l l e r i s presented i n Table I . The c o n t r o l senses the engine mean torque and spark t i m i n g . The two v a r i a b l e s are then compared by the LOAD DISTURBANCES DESIRED 6 ERROR VELOCITY HUMAN CONTROL MODEL <f>t=THROTTLE BRAKE MODEL H O DISTURBANCES S 3 l ENGINE TORQUE MODEL TI TAV ADAPTIVE CONTROL GEAR RATIO (RG) LOAD TORQUE MODEL SYSTEM DYNAMIC MODEL t i i DISTURBANCES Figure 2: St r u c t u r e of Controls and Dynamics of the Engine V e h i c l e D r i v e r System VEHICLE SPEED ENGINE SPEED 00 SIGNAL CONDITIONING AND-NOR COMPARATOR C O N T R O L L E R TAV CITAV d t | f AV 0s d0s d t DITHER SIGNAL ADVANCE CONTROL LAW RETARD CONTROL LAW Figure 3: Conceptual Block Diagram of an Adaptive Spark Timing Controller TABLE I : ADAPTIVE SPARK TIMING CONTROLLER TRUTH TABLE 1 Torque S i q n a l Spark Timinq L o q i c Gate Output 0- Decreasing 1- I n c r e a s i n g 0- Re t a r d i n g 1- Advancing 0- Retard the Spark 1- Advance the Spark 0 0 1 0 1 0 1 0 0 1 1 1 And-Nor l o g i c gate which a c t i v a t e s the appropriate c o n t r o l law. A d i t h e r s i g n a l , which i s merely a small p e r t u r b a t i o n a p p l i e d to the c o n t r o l s i g n a l , prevents the c o n t r o l from s t i c k i n g on a f a l s e optimum; For example, i f the c o n t r o l had i n i t i a l l y reached an optimum which because of disturbances has subsequently s h i f t e d , the system could not respond without a small d i t h e r to push the system o f f i t s o l d , now f a l s e optimum. Fi g u r e 4 i l l u s t r a t e s the r e l a t i o n s h i p between TdV and 8s c o r r e l a t e d f o r a l l loads and speeds. The r e l a t i o n s h i p d i s p l a y s the s i n g l e maximum torque which insures t h a t an adaptive c o n t r o l scheme w i l l achieve the optimum s e t p o i n t . F i g u r e 5 i l l u s t r a t e s the hypersurface r e l a t i o n s h i p between the four v a r i a b l e s TQV/ We,^T , and f o r a t y p i c a l i n t e r n a l combustion engine. The t h r o t t l e e f f e c t i s to s h i f t the three dimensional surface up and down without a f f e c t i n g i t s shape a p p r e c i a b l y . The surface d e s c r i b i n g a r e a l system i s more complex than t h a t of F i g u r e 5 because many more v a r i a b l e s are i n v o l v e d . One of the advantages of adaptive c o n t r o l i s t h a t i t r e q u i r e s no more than a q u a l i t a - t i v e understanding of t h i s hypersurface. S p e c i f i c a l l y , as long as only one optimum s e t p o i n t e x i s t s i n the o p e r a t i n g range t h i s i s a s u f f i c i e n t c o n d i t i o n to guarantee the r e a l i z a b i l i t y of an adaptive c o n t r o l l e r . The e f f e c t s of disturbances i n v a r i a b l e s i n f l u e n c i n g engine operation w i l l i n most circumstances a f f e c t the l o c a t i o n of the optimum spark t i m i n g and a i r - f u e l r a t i o . Thus adaptive - 30 -20 -10 0 +10 SPARK TIMING (0s) (DEGREES FROM OPTIMUM). F i g u r e 4: The E f f e c t of Spark Timing on the Mean Torque Output 13 F i g u r e 5: Hypersurface I l l u s t r a t i n g the R e l a t i o n s h i p Between Engine V a r i a b l e s c o n t r o l w i l l compensate f o r disturbances i n other u n c o n t r o l l e d v a r i a b l e s by continuously a d j u s t i n g the spark t i m i n g t o i t s l o c a l optimum. This compensation can be i l l u s t r a t e d by c o n s i d e r i n g the r e s u l t of a change i n t h r o t t l e p o s i t i o n . As a r e s u l t of the change i n t h r o t t l e p o s i t i o n a change i n engine v e l o c i t y (AWe) w i l l occur as shown i n Figure 5. Thus the l o c a t i o n of the optimum spark t i m i n g fls^will s h i f t byA0S^* The c o n t r o l l e r w i l l sense t h i s s h i f t by observing an increase i n the l o c a l g r a dient (ATdV/A0S) • This gradient w i l l d r i v e the system up to i t s new optimum. The adaptive c o n t r o l of i n t e r n a l combustion engines depends upon the f a c t t h a t adjustment of the v a r i a b l e s under c o n t r o l w i l l be on the order of only s e v e r a l c y c l e s . I d e a l l y , the c o n t r o l would adjust on a c y c l e to c y c l e b a s i s . This f a c t w i l l r e q u i r e the replacement of the c u r r e n t carburetor and d i s t r i b u t o r advance u n i t s by e l e c t r o n i c f u e l i n j e c t i o n and e l e c t r o n i c spark d i s t r i b u t i o n and t i m i n g devices. The present carburetor and d i s t r i b u t o r advance u n i t s have response times i n t-he order of many engine c y c l e s . These long responsestimes are caused by delays due to long vacuum and manifold connections The e l e c t r o n i c devices have the c a p a b i l i t y of r e a d j u s t i n g the a i r - f u e l r a t i o and spark t i m i n g on a c y c l e to c y c l e b a s i s thus a c h i e v i n g the d e s i r e d c o n t r o l l e r response. 15 The disturbances encountered by the automobile engine o r d i n a r i l y have long time constants ( i . e . the time taken f o r the d i s t u r b e d v a r i a b l e to move from i t s i n i t i a l value to w i t h i n 37% of i t s f i n a l v a l u e ) . For example, the disturbances to the o p t i - mum spark t i m i n g 6$ during warmup have time constants i n the order of minutes. Other disturbances are caused by a c c e l e r a t i o n (with time constants determined experimentally to be between 5 and 10 seconds i n normal urban d r i v i n g ) , aging and wear (time con- stants on the order of months) and environmental change (time constants on the order of hours).. The adaptive c o n t r o l l e r w i t h i t s a b i l i t y to ad j u s t the spark t i m i n g and a i r - f u e l r a t i o on a c y c l e to c y c l e b a s i s w i l l have l i t t l e problem i n compensating f o r these disturbances. Some disturbances encountered w i l l have very short time constants and may produce l a r g e changes i n the l o c a t i o n of the optimum s e t p o i n t . Such disturbances are caused by d r i v i n g onto i c e o r going up a steep h i l l . The system response time t o such disturbances i s determined by the c o n t r o l l e r time constant. C a r e f u l design of the c o n t r o l law w i l l minimize t h i s response time. (1.5) Mean Torque Measurement Several methods o f measuring the engine mean torque output are p o s s i b l e . One way i s to measure the t w i s t i n the crank or d r i v e s h a f t . This i n v o l v e s bonding s t r a i n gauge or v a r i a b l e r e l u c t a n c e transducers t o the s h a f t . Disadvantages of t h i s method i n c l u d e the space r e q u i r e d , the tr a n s m i s s i o n of the s i g n a l o f f the r o t a t i n g s h a f t , c o s t and r e l i a b i l i t y . L o c a t i o n of t h i s type of system downstream of the flywheel presents problems because of phase l a g and a t t e n u a t i o n . A second method of measuring output torque which i s used i n h e l i c o p t e r s c o n s i s t s of a h e l i c a l gear system. The output s h a f t of the power p l a n t as w e l l as d r i v i n g the l o a d a l s o engages a h e l i c a l gear connected to a h y d r a u l i c c y l i n d e r and c o n t r o l value. I f the output torque of the engine increases the h e l i c a l gear causes the c o n t r o l pressure to change, thereby r e p o s i t i o n i n g the f u e l c o n t r o l s p o o l . The disadvantages of t h i s system i n c l u d e frequency response, s i z e and c o s t . The frequency response problem i s the c h i e f disadvantage. In the i n t e r n a l combustion engine torque changes occur at a frequency of . 150 rad/sec or hig h e r . P r e l i m i n a r y a n a l y s i s shows t h a t a h e l i c a l gear system w i l l not respond to changes at t h i s frequency. A t h i r d p o s s i b i l i t y i s to measure the mean torque by observation of motion which i s dependent on the mean torque (Tav)• Such motions i n c l u d e the fly w h e e l a c c e l e r a t i o n and the r e l a t i v e angle across the c l u t c h ( i . e . the displacement of the dr i v e n p l a t e r e l a t i v e to the d r i v i n g p l a t e ) . These motions could be measured by using s u i t a b l e magnetic pickups above the fl y w h e e l s t e e t h o r notches on the c l u t c h p l a t e s . These s i g n a l s from the magnetic pickups could then be processed i n d i g i t a l 17 timing and l o g i c c i r c u i t s to produce the desired control signals. The t h i r d p o s s i b i l i t y i s the subject of t h i s work. (1.6) Preview of Thesis Section 2 presents the dynamics of the lumped parameter system. The engine torque model and the load torque model are also presented i n t h i s section. Section 3 presents the deriva- t i o n of an equivalent l i n e a r system. In Section 4 the equations of motion developed i n Section 3 are solved for the general case. Section 5 presents the solutions of Section 4 for a s p e c i a l case. In Section 6 the equations of motion from Section 2 are simu- lated by the IBM 370 computer for the same spe c i a l case as i n Section 4. A comparison of the a n a l y t i c and simulated solutions i s also made i n Section 6. Section 7 evaluates p o t e n t i a l observ- ers for TdV and Section 8 presents the conclusions of the t h e s i s . Section 9 presents recommendations for future work. 18 2 DYNAMIC ANALYSIS OF INTERNAL COMBUSTION ENGINE DRIVER VEHICLE SYSTEMS (2.1) System Dynamics Figure 6 i s a schematic representation of the chief dynamic elements and t h e i r arrangement i n an i d e a l i z e d lumped parameter engine vehicle system. The chief dynamic elements are; the engine i n e r t i a (Je) / the clutch spring (Kc) / the clutch damper (nonlinear coulomb damping,Tc)/ the transmission r a t i o (Rg) / the driveshaft spring (Ks) , the rear end r a t i o (R), the t i r e spring (Kt) / the t i r e damper (Dt) a n d the vehicle i n e r t i a (J|) . The system has six degrees of freedom given by; Q\ ,02, 03, 04, 05,06. Three external torques act on the system; the engine output torque (Te) / the load torque (TI) and the brake torque (Tb) • The dynamic analysis of the system results i n the following equations of motion: Te = J e 0 i + K c ( 0 i - 0 2 ) + T c (2.1-1) T3=Rg(Kc(0 i -02 )+Tc) (2.1-2) 02=Rg03 (2.1-3) T3 = T4=Ks(03- 04) (2.1-4) 04 = R05 (2.1-5) T5=RT4 (2.1-6) Figure 6 : Lumped Parameter Engine V e h i c l e System 20 T5 = Kt (05 -06 ) + D t ( 05 - 06) (2.1-7) J l 06+Tl+Tb=Kt(05-06) + Dt (05 -06 ) (2.1-8) For a d e t a i l e d a n a l y s i s and d e r i v a t i o n of the system governing equations i n c l u d i n g assumptions see Appendix I ; (2.2) E x t e r n a l Torques The three e x t e r n a l torques a c t i n g on the system are examined i n t h i s s e c t i o n . Assumptions are made which s i m p l i f y the expressions f o r these torques. These s i m p l i f i c a t i o n s are made so th a t the r e s u l t i n g model may be e a s i l y solved a n a l y t i - c a l l y . (2.2.1) Engine Output Torque (Te) I t i s shown i n [18] t h a t the instantaneous engine torque Te f o r a four stroke c y c l e engine can be expressed as a F o u r i e r s e r i e s ; Te = I + C_!_sin0 + Cisin0 + C3_sin30 + 2 2 2 2 Tav + Bi.cos0 + Bi cos0 + B3cos30 + -2 2 2 2 (2.2.1-1) Here $ i s the crank angle (0 = Wet) and TQV i s the engine mean 21 output torque. The c o e f f i c i e n t s CI,Cl,C3,•••/Bi,BI/B3/•••/ are 2" 2 2 2 given i n [18] f o r a s i n g l e c y l i n d e r four stroke c y c l e engine. The a n a l y s i s w i l l assume constant values f o r these torque c o e f f i c i e n t s . In a c t u a l engines t h i s i s not g e n e r a l l y the case as considerable c y c l e to c y c l e v a r i a t i o n occurs i n p r a c t i c e . In f u t u r e work the model being developed w i l l be used i n t e s t i n g adaptive c o n t r o l schemes. For the l a t e r work disturbances i n the c o e f f i c i e n t s can be introduced to simulate the v a r i a t i o n s when e v a l u a t i n g adaptive c o n t r o l schemes. For a four c y l i n d e r engine (the case of i n t e r e s t i n t h i s a n a l y s i s ) the instantaneous output torque i s given by summing the torques from i n d i v i d u a l c y l i n d e r s . The r e s u l t i n g torque expression i s : Te = l + 2C2sin20 + 4C4sin40 + • • • • + 8Bscos80 ( 2 . 2 . 1 - 2 ) Tav where terms to the eigh t h order have been i n c l u d e d . Terms higher than the ei g h t h order are much smaller and have much l e s s e f f e c t on the instantaneous engine torque. For the purpose of the a n a l y t i c i n v e s t i g a t i o n the expression (2.2.1-2) i s s i m p l i f i e d f u r t h e r to: ~-= l+Kls in(2We)t ( 2 . 2 . 1 - 3 ) lav where KI ^ITlQxlTQV^. 5 f o r four c y l i n d e r four s t r o k e c y c l e engines [19] andWe^ris the p e r i o d of the torque f l u c t u a t i o n . The e r r o r between the expression (2.2.1-2) and the expression (2.2.1-3) i s i l l u s t r a t e d i n F i g u r e 7. As shown, the F i g u r e 7: C o m p a r i s o n o f t h e S i m p l i f i e d a n d N u m e r i c a l S e r i e s E n g i n e T o r q u e S i g n a l s to approximation made i n equation (2,2.1-3) modifies the energy- input t i m i n g but maintains approximately the same energy input per c y c l e . The f a c t o r TdV must be i n v e s t i g a t e d to produce an a n a l y t i c expression f o r use i n the simulated model. Fi g u r e 8 i l l u s t r a t e s the r e l a t i o n s h i p between engine speed (We) , t h r o t t l e (<£t) and mean engine output torque (TdV) f o r the Ford C o r t i n a 2000 c.c. Engine (which w i l l be the engine to which the a n a l y s i s w i l l be a p p l i e d ) . R e c a l l t h a t F i g u r e 4 presents a sim- i l a r r e l a t i o n s h i p between spark t i m i n g (0S) and mean engine output torque (TdV) • Keeping the above r e l a t i o n s h i p s i n mind a f u n c t i o n a l . r e l a t i o n s h i p of the f o l l o w i n g form i s suggested: Tdv =Tav( We, <f>i, 0s, A/ F#, T, P, W,£) (2.2.1-4) In t h i s r e l a t i o n s h i p A/F i s the a i r - f u e l r a t i o , \jr i s the r e l a t i v e humidity, T i s the ambient a i r temperature, P i s the ambient a i r pressure,W i s the engine wear (age), and £ i s to account f o r other f a c t o r s . In t h i s a n a l y s i s a l l e f f e c t s other than those of speed (We)/ t h r o t t l e (<£t) and spark t i m i n g (Qs) w i l l be ignored. Equation (2.2.1-4) then reduces t o : Tav=Tav(We,<f>t,0s) (2.2.1-5) To evaluate (2.2.1-5) i t i s noted t h a t i n F i g u r e 8 f o r a f i x e d t h r o t t l e p o s i t i o n there e x i s t s a maximum torque (Tpvmox) which occurs a t a p a r t i c u l a r v e l o c i t y (Wê ) . Both 24 1 4 0 0 100 200 300 400 500 ENGINE SPEED (rad/sec) F i g u r e 8: Mean Torque of the C o r t i n a 2000 c.c. Engine of these v a r i a b l e s then are fun c t i o n s of t h r o t t l e (^t) alone, or: Tavmax = Tqvmax(<£t) ( 2 . 2 . 1 - 6 ) We^We*fyjt> ( 2 . 2 . 1 - 7 ) In the case of the C o r t i n a 2 0 0 0 c.c. engine they have been e v a l - uated by making p l o t s of TQVITICIXvs <f>\ and W e ^ v s ^ t . The f o l l o w i n g r e l a t i o n s h i p s r e s u l t : Tavmax =93.4 + 0.355x<£t (Nm) ( 2 . 2 . 1 - 8 ) We =3.86x<£t (rad/sec) ( 2 . 2 . 1 - 9 ) Assuming the r e l a t i o n s h i p between Tav and We to be p a r a b o l i c i n nature the f o l l o w i n g equation may be w r i t t e n : 2 Tav(*t,We)=Tavmax [«-CWe{^|#i}] ( 2 . 2 . 1 - 1 0 ) This leaves one remaining constant CWe w nich has to be evaluated from Figure 8. The r e s u l t of t h i s e v a l u a t i o n . i s : CWe=0.275 ( 2 . 2 . 1 - 1 1 ) The other f a c t o r , the spark t i m i n g , i s analyzed i n the same fa s h i o n as the t h r o t t l e . The f i r s t step i s determining the spark t i m i n g at which the maximum torque occurs. In t h i s a n a l y s i s i t i s t r e a t e d as a f u n c t i o n of speed alone: ( 2 . 2 . 1 - 1 2 ) From data f o r the C o r t i n a 2 0 0 0 c.c. engine, i t i s found t h a t : 0s#=(l.48x|O~2)We (Degrees BTDC) ( 2 . 2 . 1 - 1 3 ) With t h i s equation and remembering Fi g u r e 4, the f o l l o w i n g r e l a t i o n s h i p ' i s proposed: Tav(0s) = Tav(<^,We)[l-CiST(0s- 0 s * ) - C 2 S T ( 0 S - 0 S # ) 2 ] ( 2 . 2 . 1 - 1 4 ) w h e r e C I S T H . 2 4 6 X I 0 " 4 and C2ST =3.684X|0~3 From the preceding' paragraphs the engine mean torque (TdV) i s given by: Tav=Tavmax[l-CWe{^|-#-lf][l-CiST(0s-0s^-C2ST( 9s-6s*f] (2.2.1-15) ( 2 . 2 . 2 ) Load Torque (TI) As shown i n Figure 2 the load on a v e h i c l e r e s u l t s from three separate f o r c e s ; the aerodynamic drag f o r c e , the r o l l i n g r e s i s t a n c e f o r c e and the gradient f o r c e . An examina- t i o n of these three forces f o l l o w s . Ah expression f o r the load torque (TI) i s deriv e d from these f o r c e s . ( 2 . 2 . 2 . 1 ) Aerodynamic Drag Force (Df) The aerodynamic drag f o r c e may be c a l c u - l a t e d from the w e l l known r e l a t i o n s h i p : Df=Y-/oV2XCDoAp ( 2 . 2 . 2 . 1 - 1 ) In t h i s r e l a t i o n s h i p CDO i s the drag c o e f f i c i e n t taken as 0.45 ( t y p i c a l of modern streamlined v e h i c l e s ) , Ap i s the v e h i c l e p r o j e c t e d area (which i s i.53m f o r the C o r t i n a sedan) and p\l i s the dynamic pressure (where p i s the a i r ^ 2 4 v/ den s i t y (1.23 N sec /m ) and V i s the r e l a t i v e v e l o c i t y of the a i r ) . This r e s u l t s i n the f o l l o w i n g expression f o r the aero- dynamic drag: Df=0.424xv 2 (N) (2.2.2.1-2) where V i s i n m/sec. (2.2.2.2) R o l l i n g Resistance Force (Rf) As shown i n [20] the r o l l i n g r e s i s t a n c e fo r c e i s given by: Rf = KrXWt (2.2.2.2-1) where Kr i s an e m p i r i c a l constant g i v i n g the r o l l i n g r e s i s t a n c e due to mechanical f r i c t i o n , t i r e i n f l a t i o n and the aerodynamic and pumping e f f e c t s i n the t i r e s i n (N drag)/(N v e h i c l e w e i g h t ) . The constant i s given as: Kr=0,005 + 1.04X|Q3 + |.2IXV2 ( 2 2 2 2 _ 2 ) P P where P i s the t i r e i n f l a t i o n pressure i n Pascals and V i s the 5 v e h i c l e v e l o c i t y i n m/sec. C o n s u l t i n g Table I I , p=l.66x10 (Pa) and Kr becomes: Kr= 0.01126+ 7.3X|0~ 6 V 2 (2.2.2.2-3) 28 TABLE I I : PHYSICAL CONSTANTS OF THE CORTINA 200 c.c. SEDAN Quantity Symbol Value Units Engine Inertia Load In e r t i a Clutch Spring Driveshaft Spring T i r e Spring Clutch Damper Dynamic St a t i c Linearized Clutch Damping Ti r e Damper Vehicle Weight Frontal Area Ti r e I n f l a t i o n Pressure Dynamic Wheel Radius Rear End Ratio Gear Ratios 1st 2nd 3rd 4 th Je Jl Kc Ks Kt Tc Dc Dt Wt Ap P Rw R Rg 0.1.136 102 1500 7800 6950 ±3.2 +6.0 3.8 58.0 1100 1.53 1.66x10" 0.3 3.44 3.65 1.97 1.37 1.00 Nmsec^/rad Nmsec /rad Nm/rad Nm/rad Nm/rad Nm Nm Nmsec/rad Nmsec/rad N 2 m 2 Pa (N/m ) m Again c o n s u l t i n g Table I I and f i n d i n g Wt=11100 (N) (the v e h i c l e weight) the f i n a l r e s u l t f o r Rf can be given: Rf = KrXWt = l25 +0.0809X V 2 (N) (2.2.2.2-4) (2.2.2.3) Gradient Drag Force (Gf) The increase (or decrease) i n load e x p e r i - enced by a v e h i c l e c l i m b i n g (or descending) a gradient i s given by the f a m i l i a r expression: Gf = Wtxsin0i (2.2.2.3-1) where 0i i s the i n c l i n e angle. Therefore, the expression (2.2.2.3—1) can be w r i t t e n : , Gf » l l f 0 0 x s l n 6 i (N) $ 2 . 2 . 2 . 3-1) (2.2.2.4) Combination of the Load Force Figure 9 i s a f r e e body diagram i l l u s t r a t i n g the three forces a c t i n g on the v e h i c l e . Therefore, the t o t a l f o r c e a c t i n g a g a i n s t the v e h i c l e becomes: F| = Df+ Rf+Gf (2.2.2.4-1) which shows up as an e x t e r n a l torque (Tl) at the rear wheels: Tl=F|XRw (2.2.2.4-2) In the expression (2.2.2.4-2) Rw i s the wheel radius which i s given as 0.3 meters i n Table I I . As w e l l V can be replaced by: V = RWX 06 (2.2.2.4-3) 30 REAR WHEEL TI AERODYNAMIC ROLLING GRADIENT DRAG (Df) RESISTANCE(Rf) FORCE (Gf) VEHICLE Df Rf Nf Gf Wt Nr 'Figure 9: Free Body Diagram I l l u s t r a t i n g the Ex t e r n a l Forces on a V e h i c l e which r e s u l t s i n : TI = 37.5 + l . 5 2x iO~ 2 ( 0 6 ) 2 +3 . 32x iO 3 ( s i n 0 i ) (Nm) (2.2.2.4-4) The f i r s t two terms of (2.2.2.4-4) are p l o t t e d i n Figure 10 f o r a normal range of ope r a t i n g speeds. The e f f e c t of the t h i r d term i s i l l u s t r a t e d by the s h i f t i n g of the p l o t t e d curve upward as i n d i c a t e d i n Figu r e 10. The r e l a t i o n s h i p of (2.2.2.4-4) i s n o n l i n e a r . The a n a l y s i s can be s i m p l i f i e d by i n t r o d u c i n g the concept of an operatin g p o i n t (Tl^, 06^). That i s , a p o i n t at which the system i s i n e q u i l i b r i u m operation as shown i n Figu r e 10. I f a l i n e a r impedance of the form: r 7 _ d(potential) _ dTs° (2.2.2.4-5) d(flow) " d06° i s introduced where Ts i s the torque from the second term of (2.2.2.4-4) i The equation (2.2.2.4-5) may then be written;: Ts = l . 5 2 x i 0 " 2 0 6 2 « l . 52x iO * 2 ( 06 o ^3 .O4x iO " 2 06 o ( 06 - 06 0 ) (Nm) _0 (2.2.2.4-6) f o r small p e r t u r b a t i o n s (06—06) around, the op e r a t i n g p o i n t . The t h i r d term of (2.2.2.4-4) may be l i n e a r i z e d by re c o g n i z i n g t h a t 0 i w i l l g e n e r a l l y be small (<5®) and the r e - f o r e the small angle approximation can be made to o b t a i n : Tg = 3 . 3 2 x i O 3 ( 0 i ) (2.2.2.4-7) where 0I i s i n radi a n s . Therefore, the f i n a l expression f o r the load torque (TI) l i n e a r i z e d about an ope r a t i n g p o i n t (Tl^ t06^) i s :  3 3 TI=375 + i.52xiO- 2(06 o) + 3.O4x|O" 2 (06°)(06-06 o) + 3.32X|O 3(0i) ( 2 . 2 . 2 . 4 - 8 ) ( 2 . 2 . 3 ) Brake Torque (Tb) . The brake torque can be thought of as an increase i n the load torque which occurs as a r e s u l t of the d r i v e r ' s d e s i r e to slow down. The simplest model f o r the brake torque can be given by the expression: Tb=BCIxTbmax ( 2 . 2 . 3 - 1 ) -Here BCI i s a brake command input (percent of maximum p o s s i b l e brake a c t i o n ) . Tb represents the brake torque a p p l i e d which i s a f r a c t i o n of the maximum a v a i l a b l e brake torque, TbmOX. The expression ( 2 . 2 . 3 - 1 ) assumes the brake a c t i o n i s l i n e a r . 34 3 AN EQUIVALENT LINEAR SYSTEM The system described by equations (2.1-1) to (2.1-8) and F i g u r e 6 i s a s i x degree of freedom system which i s cumber- some to analyze a n a l y t i c a l l y . V a r i a b l e s downstream of 6 I w i l l experience a t t e n u a t i o n and phase s h i f t s i n the a f f e c t s caused by TQV. Thus i t i s d e s i r a b l e to focus primary a t t e n t i o n on 8\ and i t s r e l a t i o n s h i p to TdV. Therefore, by r e p l a c i n g the more complicated system of Figure 6 by the equivalent l i n e a r i z e d system of Figu r e 11 the a n a l y s i s can be co n s i d e r a b l y s i m p l i f i e d without e l i m i n a t i n g the important i n f o r m a t i o n . The v a l i d i t y of t h i s s i m p l i f i c a t i o n i s e s t a b l i s h e d l a t e r by computer s i m u l a t i o n of the system of Figure 6 and comparing the r e s u l t s w i t h those obtained from the a n a l y t i c i n v e s t i g a t i o n of Figu r e 11. (3.1) A Constant Speed System The f i r s t step i n the d e r i v a t i o n of the equi v a l e n t system of Figure 11 i s to e l i m i n a t e both the tra n s m i s s i o n and rear end r a t i o s . A l l components of the r e s u l t i n g system w i l l r o t a t e a t the same speed (the engine speed (We) ) . The e l i m i - n a t i o n of the tra n s m i s s i o n and rear end i s accomplished by making the f o l l o w i n g s u b s t i t u t i o n s : 06 = 06XRXRg (3.1-1) 0|, 0 ( , 01 06» 6̂ i 06 ENGINE EQUIVALENT SPRING DAMPER NETWORK EQUIVALENT LOAD F i g u r e 11: E q u i v a l e n t L i n e a r Engine V e h i c l e System 36 Tl' = - - ^ - r (3.1-2) R(Rg) J , ' = _ J ] _ _ (3.1-3) (R(Rg))2 Kt'= K t o (3.1-4) (R(Rg)) Dt'= T ~7^T I 2 ( 3 - 1 " 5 ) (R(Rg)) Ks ' = - ^ - r (3.1-6) Rg 2 '(-3.2) Equivalent Spring and Damper The next step i s to replace the spring-damper network between 01 and 06 by a s i n g l e equivalent s p r i n g (K6Q) and a s i n g l e equivalent damper (DeO) . This can be done by f i r s t r e c o g n i z i n g : T e - J e 0 i = T i ( 3 . 2 - 1 ) and Tl ' + J l W s T e ' s T i ( 3 . 2 - 2 ) where the components are a l l r o t a t i n g at the same speed. The displacement across the c l u t c h , d r i v e s h a f t and t i r e (01 — 06) i s : 37 ( f l , - 0 6 ' ) a - I i + - T i v | . . l l , o r (3.2-3) Kc Ks Kt (a,-06') = l L _ (3.2-4) Keq Using (3.2-3) and (3.2-4) the f o l l o w i n g r e s u l t i s obtained: Kaa^ KcKs'Kt' (3.2-5) • (KcKs' + Ks'Kt' + KcKt1) The f i r s t step i n combining the c l u t c h damper (Tc) and the equiv a l e n t t i r e damper (Dt') i s to replace the n o n l i n e a r c l u t c h damper w i t h an equi v a l e n t l i n e a r one. The method i s described i n [21] and the r e s u l t s are presented here. The equivalent l i n e a r damper (Dc) f o r the coulomb damped c l u t c h i s given by: p 4 X T p (3.2-6) U C 7TWX0O In t h i s equation Tc i s the coulomb damping torque which i s given i n Table I as ±3.2 (Nm) andWis the frequency of the torque s i g n a l (which i s twice the engine frequency (2WB) ) . Bo i s the amplitude of (Q\ — Q2) and i s given by: \ / ( 4TC V (3.2-7) Q .ToyKI v ' UTavKI/ *°""1<c-X . /W.\2 ' "\Wn/ In equation (3.2-7) Wn i s the n a t u r a l frequency of the c l u t c h which i s given by: W n ^ / f e q ~ (3-2-8) 38 where JeQ i s the equivalent system i n e r t i a . For t h i s two i n e r t i a system, JeQ i s given by.:: , JeJl ' q s Je+jT (3'2"9) Having a r r i v e d a t the eq u i v a l e n t l i n e a r c l u t c h damping the equivalent system damping can be found. The d i f f e r e n c e between the r o t a t i o n a l speeds of the engine and the load (al -66) must be d i s s i p a t e d i n the network between these two elements. Therefore: (0<-$6 , ) = J ^ + - £ j r ° r (3.2-10) (0,_0 6 ' ) = J T i _ (3.2-11) Oeq From equations (3.2-10) and (3.2-11) the expression f o r the equivalent system damping Deq i s : n n r , _ DcDt' D e q = Dc+Dt' ( 3 ' 2 " 1 2 ) (3.3) Equations of Motion of Equivalent L i n e a r System Having completed the above a n a l y s i s the equations of motion of Figure 11 can be w r i t t e n : Je0 i +Deq(0i-06 ,) + Keq(0i-06 ,)=Tav(l + KlXsin(2We)t)=Te (3.3-1) JI,6f6, + TI , = D e q ( 0 i - £ 6 , ) + Keq(0i-06') (3 .3-2) 39 4 SOLUTION OF THE EQUIVALENT LINEAR SYSTEM The solutions of equations (3.3-1) and (3.3-2) w i l l consist of three parts; the homogeneous (transient) part, and two p a r t i c u l a r parts. The p a r t i c u l a r parts are a constant solution and a sinusoidal solution. A detailed analysis of the solutions i s presented i n Appendix II and the results are given here. (4.1) The Transient Response The solution of the homogeneous equations of motion provides the transient response. For the system of Figure 11 the transient response i s given by: .In these -equations Q i s the damped natural frequency and P i s the negative exponent of decay. These two variables are given by: 0ci = 0~ p T (AcosqT +Bsinqr) (4.1-1) (4.1-2) p=£ Wn (4.1-3) (4.1-4) where £ i s the damping r a t i o and Wn i s the undamped n a t u r a l frequency. £ and Wn\ are given by: f - D e C » (4.1-5) fc Dcr W n *J Jeq Kea ( 4 - 1 - 6 ) Jeq where Deris the c r i t i c a l damping 'cote'ffi<c:rent 'given by: Dcr=2 v/jeqKeq ( 4 . 1 - 7 ) A and B are a r b i t r a r y constants which are determined from the system boundary c o n d i t i o n s . For example, f o r a step change i n mean torque- (ATdV) the boundary c o n d i t i o n s can be given by: a ATOV , rs\ ( 4 . 1 - 8 ) 0ci = — — T=0) Keq 0CI = O (T = 0 ) ( 4 . 1 - 9 ) T = © when t = t ' and t ' i s the time when the disturbance e x c i t i n g the t r a n s i e n t occurs. ( 4 . 2 ) The Constant Response The constant response i s given by the f o l l o w i n g equations: aen9 = 9sscî ssce'=To,v-T,lc' <«-2"l> Je+JI Weng s0ssci=0ssc6 , = Wengi+^aeng dr ( 4 . 2 - 2 ) 41 0eng = 0ssci = 0engi +J*WengdT (4.2-3) In the s p e c i a l case where aeng=0' the r e s u l t s (4.2-2) to (4.2-4) reduce to:- Weng = constant (4.2-5) 0eng=0ssci+Weng'(r) (4.2-6) * = K ^ j < 4 - 2 - 7 > (4.3) The O s c i l l a t o r y Response The o s c i l l a t o r y response of the system i s of the form: 0ssoi = KITavJ I ' g ^ je s l n ( 2Wet -¥ i ) ( 4 . 3 - i ) 0sso6=KITav lC6l+D6l s in(2Wet-^6') (4.3-2) E62 + F 6 2 In .these s o l u t i o n s Cl,C6,D I, D6,E I ,E6 and F l ,F6 arep^dep.endent-upon the system constants. For a more complete exp l a n a t i o n of t h i s s o l u t i o n see Appendix I I . The phase angle ¥ i s given by: ¥i = i n n- ' [ -& - ] ( 4 - 3 - 3 ) W-ton-'[•£&•] ( 4 - 3 - 4 ) 42 (4.4) The T o t a l Response The. t o t a l response of the system can be c a l c u l a t e d by summing a l g e b r a i c a l l y the responses given i n Sections (4.1) to (4.3). This i s p o s s i b l e because of the l i n e a r nature o f the system described by equations (3.3-1) and (3.3-2). For example the t o t a l s o l u t i o n f o r 01 i s given by summing equations (4.1-1)/ (4.2-3) and (4.3-1) t o y i e l d : 0i = * ~ p T (Acosqr +BsinqT) + 0engi + f Wengdr + ( 4 . 4 - 1 ) KITav / Q | 2 + P ' 2 s i n ( 2Wet -¥ i ) -V E r + F r 43 5 SOLUTIONS OF A SPECIAL CASE (5.1) The V e h i c l e Considered The a n a l y s i s of Sections 3 and 4 i s now a p p l i e d to a t y p i c a l small car system. The system s e l e c t e d was a Ford C o r t i n a automobile. The reason f o r t h i s choice i s t h a t t h i s v e h i c l e i s t y p i c a l of modern compact car design. Table I I presents the p h y s i c a l constants f o r the 2000 c.c. engine C o r t i n a Sedan. Appendix I I I presents the d e r i v a t i o n and determination of these constants. The a n a l y s i s can be e a s i l y extended to other v e h i c l e s and engines by s u b s t i t u t i o n of the appropriate p h y s i c a l constants and torque s i g n a l s , equation (2 . 2 .1-3), f o r those used i n t h i s a n a l y s i s . (5.2) The P a r t i c u l a r Case Considered For the equivalent system of Figu r e 11 and equations (3.3-1) and (3.3-2), the s o l u t i o n s (4.1-1) and (4.1-2), (4.2-1), (4.2-2), (4.2-3) and (4.2-4), (4.3-1) and (4.3.2) are determined f o r a p a r t i c u l a r case f o r l a t e r comparison purposes w i t h a com- puter s i m u l a t i o n of the system of Figure 6 and equations (2.1-1) to (2.1-8). The p a r t i c u l a r case i n v e s t i g a t e d i s an i n i t i a l engine speed f o r 280 rad/sec (~2800 rpm), i n t h i r d gear (Rg=1.37) producing a 23.5 Nm mean torque (-18.. 5% of maximum output t o r q u e ) . A 10% step change i n t h r o t t l e p o s i t i o n i s then made r e s u l t i n g i n a new mean output torque of 80 Nm (62% of maximum output torque). The system response to t h i s step change i s presented below. (5.3) The Transient Response C o n s u l t i n g Table I I and using equations (4.1-3), (4.1-4), (4.1-5), (4.1-6), (4.1-7) the f o l l o w i n g r e s u l t s are obtained: w n = J - £ ^ = 4 2 . 4 (rad/sec) ( 5 . 3 - 1 ) D c r 2^/JeqKeq Wnd = Wn^/l-C2 =42.2 (rod/sec) ( 5 . 3 - 3 ) Using these r e s u l t s i n equations (4.1-1) and (4..tl-2) the f o l l o w - i n g s o l u t i o n s can be w r i t t e n : 0ci = e " 3 ' 6 8 T ( A c o s 4 2.2T + Bsin42.2T) ( 5 . 3 - 4 ) 0c6,= ̂ " 3 ^ 8 T D(Acos42.2T+Bsin42.2T) ( 5 . 3 - 5 ) The constant D can be evaluated by the method i l l u s t r a t e d by equations ( A l l . 3 - 1 1 ) and ( A l l . 3 - 1 2 ) . The other two constants A and B are evaluated by c o n s i d e r a t i o n of the"boundary c o n d i t i o n s which are the same as those given by (4.1-8) and (4.1-9). The e v a l u a t i o n of these constants produces the f o l l o w - i n g set of r e s u l t s : 0ci = - f i »~ 3 - 6 8 T (2 .49X IO~ ' cos42.2T + 2 . 09X lO " 2 s i n 42.2T) (5.3-6) 0ci = ff"3-68T(4.l4 X | 0 2 C O S 4 2.2T - 3 . 6 0 X 1 0 , s i n 4 2.2T) ( 5 . 3 - 8 ) 0cs = e~368T(7.16x10"3sin42.2T +6 2 0 x I O ' 4 C O S 4 2 . 2 T ) X - I ( 5 . 3 - 9 ) 0C6,= ff-3-68T(3.O2xiO"",sin42.2r) ( 5 . 3 - 1 0 ) 0c6, = ̂ ~ 3 - 6 8 T (| ,28x iO , cos42.2T -| .O9s in42.2T) ( 5 . 3 - 1 1 ) These r e s u l t s (5.3-6) to (5.3-11) imply a time constant: r G STwn" S°' 2 7 1 4 ( S 6 C ) ( 5 The amplitude of the engine a n g u l a r . v e l o c i t y t r a n s i e n t i s : 0 c i s * - 3 . 6 8 T (9.80s in42.2T) ( 5 . 3 - 7 ) |0ci|= 9.80xr3-68T (5.3-13) and the engine angular a c c e l e r a t i o n t r a n s i e n t is,:- ]0cil*4l6 x ^ " 3 - 6 8 T (5.3-14) 46 ( 5 . 4 ) The Constant Response For the step change imposed on the system equation ( 4 . 2 - 1 ) can be evaluated to give: aeng =-^£p-12.0 (rod/sec 2 ) ( 5 . 4 - D Equation ( 4 . 2 - 2 ) can be evaluated assuming aeng i s constant for the f i r s t second a f t e r the step change i n TdV to produce: 0ssci =292 (rad/sec) ( 5 . 4 - 2 ) The assumption of constant angular acceleration aeng i s not s t r i c t l y true because Tic' w i l l increase due to the increase i n speed (WenQ) . This increase w i l l reduce the magnitude of aeng which thus does not remain constant. ( 5 . 5 ) The O s c i l l a t o r y Response For the p a r t i c u l a r case under consideration the value KITdV i s : KITav=200 (Nm) ( 5 . 5 - 1 ) a f t e r the step input. Substitution of the physical constants of Table II into equations ( 4 . 3 - 1 ) and ( 4 . 3 - 2 ) w i l l r e s u l t i n the following set of solutions: • 0ssoi = 4 .53x10 " 3 sin (5601 - 0.869°) ( 5 . 5 - 2 ) 0ssoi=KITav(2We)(2.2 6 5 x i O " 5 ) co s (2Wet -¥ i ) ( 5 . 5 - 3 ) = 2.54 cos(560t -0 .869°) 47 assoi = -K ITav (4We 2 ) (2 .265x|0 " 6 ) s l n (2Wet -^ i ) (5.5-4) = -I.43X 10 3 s in (560t - 0.869°) d s s o6 ' = 6 . 0 2 x|0 " 6 s i n ( 5 60 t - 9 0° ) (5.5-5) 0sso6'=KITav(2We)(3.OlxiO~ 8) cos(2Wet-¥e ' ) (5.5-6) = 3 .37x|0~ 3 c o s ( 5 6 0 t - 9 0 ° ) 6fsso6 ,=KITQv(4We 2)(3.0lxiO' 8)x-sin(2Wet~^6') (5.5-7) = - l . 8 8 s i n ( 5 6 0 t - 9 0 ° ) The s o l u t i o n (5.5-3) i m p l i e s a f i n a l e q u i l i b r i u m engine angular v e l o c i t y amplitude o f : ISssoil =2.54 (rod/sec) x (5.5-8) and a f i n a l e q u i l i b r i u m engine angular a c c e l e r a t i o n amplitude o f : Iflssoil =1420 ( rad/sec 2 ) (5.5-9) The phase d i f f e r e n c e between the change i n torque;and the o -5 response i s 0.0151 radians (0.869 ) or 5.39x10 sec at the o r i g i n a l o p e r a t i n g frequency (280 rad/sec)* 48 6 COMPUTER SOLUTION OF THE NONLINEAR SYSTEM The dynamics of the system of Figu r e 6 and equations (2.1-1) to (2.1-8) are simulated on the IBM 370 Computer us i n g the CSMP language. The mean torque (T<3V) and load torque (TI) models given by equations (2.2.1-15) and (2.2.2.4-4) res p e c t - i v e l y are inc l u d e d i n the s i m u l a t i o n . The instantaneous torque (Te) used i n the s i m u l a t i o n i s computed u s i n g expression (2.2.1-2) r a t h e r than expression (2.2.1-3). The computer programs used are presented i n Appendix IV. The s i m u l a t i o n i s c a r r i e d out f o r the same p a r t i c u l a r case as tha t given i n Section (5.2).to f a c i l i t a t e comparison between the simulated and a n a l y t i c r e s u l t s . F i g u r e 12 i l l u s - t r a t e s the t h r o t t l e input command step. Figure 13 i l l u s t r a t e s the response of mean torque (TdV) to the t h r o t t l e step i n p u t . F i g u r e 14 and Figu r e 15 i l l u s t r a t e r e s p e c t i v e l y the response of the engine r o t a t i o n a l v e l o c i t y and r o t a t i o n a l a c c e l e r a t i o n t o the t h r o t t l e i n p u t . The agreement between the a n a l y t i c and simulated models i s q u i t e good as i l l u s t r a t e d i n Table I I I . No r e l a t i v e e r r o r between the two s o l u t i o n s exceeds 10%. What e r r o r s do occur can be explained. The c a l c u l a t i o n s made i n c a l c u l a t i n g both the t r a n s i e n t response and the o s c i l l a t o r y response have -Her — 1 - r 4 - • — — - j — • " ~\i in U | ( T O H-OJ — i - "t • OS 0 1 1 M h i i E L 1 4 < 1 3 H 1 I 2, F i g u r e 12: T h r o t t l e Command Input t o the Computer S i m u l a t i o n Model -F i g u r e 13: Mean Torque Response to the T h r o t t l e Command Input o F i g u r e 14: Engine R o t a t i o n a l V e l o c i t y Response to the T h r o t t l e Command Input F i g u r e 15: Engine R o t a t i o n a l A c c e l e r a t i o n Response to the T h r o t t l e Command Input TABLE I I I : COMPARISON OF ANALYTIC AND SIMULATED RESPONSES TO A STEP CHANGE IN ENGINE OUTPUT TORQUE (80-23.5-56.5 (Nm) ) Item Response % E r r o r A n a l y t i c / Simulated i A n a l y t i c Simulated T r a n s i e n t Response (9d) Time Constant (1*0) 0.2714(sec) 0.27(sec) 0.5 Damped N a t u r a l Frequency (Wfldj 42.2(rad/sec) 42. .r0"( rad/sec) 0.5 V e l o c i t y Amplitude 9.80(rad/sec) 9.5(rad/sec) 3.0 A c c e l e r a t i o n Amplitude 416(rad/sec) 400(rad/sec) 4.0 Forced Steady State O s c i l l a t o r y Response (0SSOI) V e l o c i t y Amplitude 2.54(rad/sec) 2.40(rad/sec) 5.8 A c c e l e r a t i o n Amplitude 2 1420(rad/sec-) 2 1320(rad/sec>) 7.6 Forced Steady State Constant Response (0SSC1) V e l o c i t y a t T=l.0(sec) 292(rad/sec) 292.3(rad/sec) 2.5 ( i n i t i a l Velocity=280.0 (rad/sec)) assumed t h a t the frequency of the torque s i g n a l (2V\fe) has remained constant. As Figu r e 14 shows t h i s i s c l e a r l y not the case. This assumption then can account f o r the e r r o r between these two s o l u t i o n s . For example, consider the o s c i l l a t o r y angular v e l o c i t y amplitude as given by equation (5.5-8) t h i s was c a l c u l a t e d assuming We=280 rad/sec but as shown i n Fi g u r e 14 We ranges from 280 to 294 rad/sec. This v a r i a t i o n produces e r r o r s i n the c a l c u l a t e d response r e l a t i v e to the simulated response from 0 to 5%. Another e r r o r i n the a n a l y t i c r e s u l t s which has been pointed out p r e v i o u s l y i s the assumption t h a t aeng i s constant. This e r r o r e x p l a i n s the d i f f e r e n c e between the a n a l y t i c and simulated constant v e l o c i t y f o r T=1 sec. The a n a l y t i c model then, represents the dynamics of the simulated system c l o s e l y . The s i m u l a t i o n model can be used to evaluate proposed c o n t r o l laws f o r adaptive c o n t r o l s . The law would be programmed and the response observed over a t y p i c a l urban d r i v i n g c y c l e . A i r - f u e l r a t i o c o n t r o l w i l l r e q u i r e a f u r t h e r extension of the mean torque model given by equation (2.2.1-15). 55 7 MEAN TORQUE INDICATORS Now tha t the a n a l y s i s has been v e r i f i e d , the r e s u l t s can be used to i n v e s t i g a t e p o s s i b l e i n d i c a t o r s of the mean torque. In s p e c t i o n of the s o l u t i o n s (5.5-2) to (5.5-7) and (5.4-1) and (5.4-2) shows tha t they are a l l dependent on the mean torque (TQV) • The t r a n s i e n t s o l u t i o n (5.3-6) to (5.3-11) i s a l s o dependent on TdV for the s p e c i a l case of Section (5.2). A more d e t a i l e d a n a l y s i s of the p o t e n t i a l observers f o l l o w s . (7.1) The Transient Response The t r a n s i e n t response w i l l be e x c i t e d by disturbances other than those i n TdV thus making the boundary c o n d i t i o n s and hence the response independent of TdV* Another problem posed by using the t r a n s i e n t response as an i n d i c a t o r of torque i s i l l u s t r a t e d by Figur e 16. The f i g u r e shows an i n i t i a l t r a n s - i e n t followed by another t r a n s i e n t beginning i n the next c y c l e . Note th a t these t r a n s i e n t s have c a n c e l l e d each other out. The c a n c e l l a t i o n means t h a t no i n d i c a t i o n of; the wrong i n d i c a t i o n o f the change i n torque may r e s u l t from l o o k i n q at the t r a n s i e n t response. 56 INITIAL TRANSIENT LATER TRANSIENT T I M E F i g u r e 16: Transient C a n c e l l a t i o n 57 (7.2) The Constant Response The constant response could be used as an i n d i c a t o r of the mean torque. For example, measuring the gross v e h i c l e a c c e l - e r a t i o n or measuring the steady s t a t e constant p a r t of the r e l a - t i v e c l u t c h angle w i l l provide a s i g n a l of the constant response. The problem, w i t h u s i n g the gross v e h i c l e a c c e l e r a t i o n i s t h a t i t takes too long t o see the e f f e c t due to a c o n t r o l change because of phase s h i f t . With the c o n t r o l adjustment being on the order of every few c y c l e s t h i s l a g w i l l be unacceptable. Another serious problem a r i s e s because the constant response i s a l s o a f f e c t e d by disturbances i n loa d . The constant response w i l l not, t h e r e f o r e , always i n d i c a t e how the engine mean torque i s behaving because of the i n t e r f e r e n c e of these load torque, disturbances. (7.3) The O s c i l l a t o r y Response The steady s t a t e o s c i l l a t o r y response i s dependent on the r o t a t i o n a l frequency (We) as w e l l as the mean torque (TdV). However, from one c y c l e to the next l i t t l e change i n speed w i l l occur. As w e l l , the speed changes tend to be i n the same d i r e c t i o n as the changes i n mean torque, thus enhancing the i n d i c a t i o n of the change. The c o n t r o l would f i r s t observe the amplitude of the steady s t a t e o s c i l l a t o r y response, then a d j u s t the spark t i m i n g and then observe the 58 'steady s t a t e o s c i l l a t o r y amplitude of the f i r s t c y c l e a f t e r the adjustment. A comparison of the two amplitudes would then be made to determine the input to the c o n t r o l l e r of Figure 3. The t r a n s i e n t can be e f f e c t i v e l y separated from the steady s t a t e o s c i l l a t o r y response by a high pass f i l t e r . The t r a n s i e n t frequency i s always given by the damped n a t u r a l frequency which i n the case of the a n a l y s i s c a r r i e d out i n t h i s work i s 42.6 rad/sec (400 rpm) and by design i s kept w e l l below the lowest o p e r a t i n g frequency. The lowest frequency of the steady s t a t e o s c i l l a t o r y s i g n a l i s twice the engine i d l e frequency or 160 rad/sec (1500 rpm). Therefore, there i s normally a greater than f o u r f o l d s eparation of frequencies. The steady s t a t e o s c i l l a t o r y a c c e l e r a t i o n has a much l a r g e r amplitude than the v e l o c i t y s i g n a l as can be seen from i n s p e c t i o n of equations (5.5-3) and (5.5-4). For t h i s reason i t w i l l be a much e a s i e r s i g n a l to measure and changes i n TdV w i l l r e s u l t i n l a r g e r absolute changes i n a c c e l e r a t i o n amplitude than i n v e l o c i t y amplitude. The merits and drawbacks of a l l the responses as i n d i c a t o r s of TdV are summarized i n Table IV. TABLE IV: EVALUATION OF SYSTEM RESPONSE SOLUTIONS AS INDICATORS OF MEAN TORQUE S o l u t i o n A f f e c t e d by Disturbances Other than i n TdV Frequency (rad/sec) Amplitude Phase (rad) Transient 0ci 0CI Yes Wnd = 42.0 depends on disturbances depends on disturbance 0CI Constant aeng > Yes - depends on disturbance - O s c i l l a t o r y • 0SSOI KITav(2.27xiO" 5) 0SSOI > No 2We 2WeKlTav(2.27xlO~5) 0.0151 0S SO I 4We 2KITav(2.27xl0" 5) 60 8 CONCLUSIONS A s i m p l i f i e d l i n e a r model of the dynamics of the i n t e r n a l combustion engine v e h i c l e d r i v e r system has been dev- eloped. This model i s v e r i f i e d by comparison w i t h a computer s i m u l a t i o n model which includes the important n o n l i n e a r i t i e s . The l i n e a r model i s shown to be a good approximation of i t s simulated counterpart. Models d e s c r i b i n g the engine mean torque, are a l s o developed. The problems of measuring the mean torque are d i s - cussed. I t i s e s t a b l i s h e d t h a t measuring one of the engine v a r i a b l e s w i l l provide the best observer f o r the mean torque. Other v a r i a b l e s downstream of the engine are subject to phase s h i f t s and a t t e n u a t i o n . The s o l u t i o n s obtained f o r these v a r i a b l e s from the l i n e a r model have been used i n the evalua- t i o n and s e l e c t i o n of the best engine v a r i a b l e . I t i s shown tha t the t r a n s i e n t and constant s o l u t i o n s are e x c i t e d by disturbances i n the load torque as w e l l as by disturbances i n the engine mean torque. This f a c t r u l e s them out as p o t e n t i a l observers of the mean torque. The e v a l u a t i o n and s e l e c t i o n has determined t h a t the steady s t a t e o s c i l l a t o r y a c c e l e r a t i o n response provides the best i n d i c a t i o n of the mean torque. A system has b- " - - . 61 h RECOMMENDATIONS FOR FUTURE WORK Wuture work should be directed towards the develop- ment of a device for measuring the o s c i l l a t o r y acceleration. As well, the structure and quantitative determination of the adaptive c o n t r o l l e r should be pursued. To achieve the l a t t e r goal the simulation model developed i n this work should be extended to include the human, adaptive and feedback models. The simulation model would then be a useful tool i n the design, selection and optimization of adaptive control for i n t e r n a l combustion engines. 62 REFERENCES [1] Myers, P.S., Uyehara, O.A., Newhall, H.K., "The ABC's o f Engine Exhaust Emissions", Engineering Know How i n Engine Design - Part 19, Soci e t y of Automotive Engineers, SP-365, 1971. [2] Anonymous, " A i r P o l l u t i o n : The Problem and the R i s k s " , SAE J o u r n a l , Volume 76, No. 3, May, 1968, pp. 47-52. [3] Anonymous, "Fuel Economy and Emission C o n t r o l s " , U n i t e d States Environmental P r o t e c t i o n Agency O f f i c e of A i r and Water Programs Mobile Source P o l l u t i o n C o n t r o l Program, November, 1972. [4] Conta, L.D., "The C o n t r o l of Automotive Emissions", American S o c i e t y of Mechanical Engineers, No. 73-WA/DGP-3, November, 1973. [5% Stempel, R.C., and Ma r t i n s , S.W.., "Fuel Economy Trends and C a t a l y t i c Converters", S o c i e t y of Automotive Engineers, No. 740594, August, 1974. [6] LaPoint, C , "Factors A f f e c t i n g F u e l Economy", Automotive Engineering, Volume 81, No. 11, November, 1973, pp. 46-50. [7] Anonymous, "Present and Future Trends i n Auto F u e l Con- sumption", Automotive Engineering, Volume 81, No. 7, J u l y , 1973, pp. 48-50. [8] Schweitzer, Paul H., "Control of Exhaust P o l l u t i o n Through a Mixture Optimizer,', S o c i e t y of Automotive Engineers, No. 72025, January, 1972. [9] Schweitzer, Paul H., V o l z , C a r l , and Deluca, Frank, "Control System to Optimize Engine Power", S o c i e t y of Automotive Engineers, No. 7008.84, November, 1970. [10] Kehres, John K., and A l l a n , John J . , "Exhaust P o l l u t i o n M i n i m i z a t i o n i n Small Engines Using Adaptive D i g i t a l Con- t r o l " , S o c i e t y of Automotive Engineers, No. 730858, 1973. 63 [11] Adams, T.N., "Adaptive Control and Performance Indicators for Internal Combustion Engines", Private Communication, 1973. [12] Williams, M., "A D i g i t a l Memory Fuel Controller for Gas- ol i n e Engines", Society of Automotive Engineers, No. 720282, January, 1972. [13] Soltav, J.P., Senior, K.B., and Rowe, B.B., " D i g i t a l l y Programmed Engine F u e l l i n g Controlsy, Society of Auto- motive Engineers, No. 730128, January, 1973. [141] Rivard, J i G . , "Closed Loop Elec t r o n i c Fuel Injection Control of the Internal-Combustion Engine", Society of Automotive Engineers, No. 730005, January, 1973. [15] Hubbard, M., and Powell, J.D., "Closed Loop Control of Internal Combustion Engine Exhaust Emissions", Guidance and Control Laboratory, Department of Aeronautics and Astronautics, Stanford University, SUDAAR No. 473, February, 1974. [16] Westcott, J.H., An Exposition of Adaptive Control, London, England, Pergamon Press, 1961. [17] Kronauer, R.E., and Drew, P.G., "Design of Adaptive Feed- back Loop i n Parameter Perturbation Adaptive Controls", Theory of Self-Adaptive Control Systems, Proceedings of the Second IFAC Symposium on the Theory of Self-Adaptive Control Systems, Plenum Press, New York, 1966, pp. 299-308. [18] Taylor, C.V., The Internal Combustion Engine i n Theory and Practice, Volume l i t Combustion, Fuels, Materials, Design, Cambridge, Massachusetts, M.I.T. Press, 1968. [19] Yamamoto, Kenichi, Rotary Engine, Toyo Kogyo Lo. Ltd., October, 1971. [20] Hoerner, S.F., F l u i d Dynamic Drag, Sighard F. Hoerner, Midland Park, N.J., pp. 12-1 to 12-8. [21] Den Hartog, J.P., Mechanical Vibrations, McGraw H i l l Book Company, New York, N.Y., 1956, pp. 374-375. [22] ShigEey, J.E., Dynamic Analysis of Machines, McGraw H i l l Book Company, New York, N.Y., 1956 64 [23] Davisson, J.A., "Design and A p p l i c a t i o n of Commercial Type T i r e s " , S o c i e t y o f Automotive Engineers, No. 690001, January, 1969. [24] C o r t i n a Shop Manual, Service P u b l i c a t i o n s , Ford of B r i t a i n , May, 1972. [25] Baumeister, T., E d i t o r i n Chief, Standard Handbook f o r Mechanical Engineers, Seventh E d i t i o n , McGraw H i l l Book Company, 1966. APPENDIX I : EQUATIONS OF MOTION OF AN INTERNAL COMBUSTION ENGINE VEHICLE SYSTEM (A I . l ) Purpose The purpose of t h i s Appendix i s to d e r i v e the equa- t i o n s of motion ( 2 . 2 - 1 ) to ( 2 . 2 - 8 ) which apply to Figu r e 6. The important assumptions made i n making the lumped parameter system approximation are presented and t h e i r v a l i d i t y i n v e s t i - gated. (AI.2) Assumptions The assumptions made i n the lumped parameter model are; a l l i n e r t i a s other than the engine and load i n e r t i a s are neglected, a l l springs i n the system other than the t i r e s , d r i v e s h a f t or c l u t c h are assumed r i g i d and damping e f f e c t s other than those o f the c l u t c h and t i r e s are ignored. (AI.2.1) I n e r t i a Assumptions The components having the l a r g e s t i n e r t i a s next to the engine are the gears, c l u t c h p l a t e s and d r i v e s h a f t . The i n e r t i a of a t y p i c a l gear (Ig), a t y p i c a l c l u t c h p l a t e (Icp) and a t y p i c a l d r i v e s h a f t (Ids) are presented below: I g s — mg-rg* ( A I . 2 . 1 - 1 ) 66. where : mg=mass of gear=yt> ( i rTg 2 ) ! =1. 54 (Nsec 2/m) - 3 2 3 p ̂ d e n s i t y of steel=7.89x10 (Nsec /(m)cm ) r 9 -gear radius=5.0ci(;cm) t=gear thickness=2.5 (cm) .". Ig = 0.00194 (Nmsec 2/rad) (AI.2.1-2) Icp=-S-mC«rC2 ( A I . 2 . 1 - 3 ) 2 2 where:mC-mass of clutch=0.72 (Nsec /m) rc=clutch radius=10.79 (cm) .'. Icp=0.00419 (Nmsec 2/rad) (AI.2.1-4) Ids=-1-mds(ro2-ri2) ( A i . 2 . 1 - 5 ) 2 where:mds=mass of driveshaft=4.4 (Nsec /m) r 0 = o u t s i d e diameter of driveshaft=2 :.5 (cm) ri =inside diameter of d r i v e s h a f t = 2 . 3 5 (cm) .MdS= 0.00016 (Nmsec 2/rad) (AI.2.1-6) These i n e r t i a s are a l l a t l e a s t two orders of magnitude 2 smaller than the engine i n e r t i a of 0.14 (Nmsec /rad). Therefore, the i n e r t i a s of these components as w e l l as smaller components may be s a f e l y neglected. 67 (AI.2.2) Spring Assumptions The components having the lowest s p r i n g rates w i l l g e n e r a l l y be those of s m a l l e s t cross s e c t i o n a l area, g r e a t e s t length or made of the most f l e x i b l e m a t e r i a l (e.g. t i r e s ) . The t o r s i o n a l s p r i n g constant of s h a f t s i n the system i s i n v e r s e l y p r o p o r t i o n a l to t h e i r l e n gths. Therefore, s h a f t s such as the crankshaft, w i l l be e s s e n t i a l l y r i g i d when compared w i t h the d r i v e s h a f t . The s p r i n g constant f o r a s e c t i o n of crankshaft i s given by: K C S = ^ g ^ 0 5 (AI.2.2-1) 32 I 5 2 2 where: G=shear modulus of steel=8.45x10 (Nsec /(m)cm ) 2 g = a c c e l e r a t i o n of gravity=9.8 (m/sec ) <JCS=5.699 (cm) I =2.54 (cm) KcS=5.96xl0 5 (Nm/rad) (AI.2.2-2) This s p r i n g constant i s two orders of magnitude gr e a t e r than the d r i v e s h a f t s p r i n g constant o f : KS=7800 (Nm/rad) (AI.2.2-3) The assumption th a t a l l components of the system other than the t i r e , d r i v e s h a f t or c l u t c h are r i g i d i s thus e s t a b l i s h e d . (AI.2.3) Damping Assumptions Equation (5.3-2) i l l u s t r a t e s t h a t the damping r a t i o £ i s i n the range 0.1. Other forms of damping present i n the system i n c l u d e s t r u c t u r a l damping (£=0.01), j o u r n a l bearing 68 • f r i c t i o n (£=0.001) and gear f r i c t i o n (£=0.001)..The ef f e c t s of these damping mechanisms are ignored because t h e i r contribution to the motion of the system i s very small when compared with that of the clutch and t i r e . (AI . 2 . 4 ) : Limitations of Analysis Several factors have not been considered i n t h i s analy- s i s which may be of importance i n l a t e r work. Among these factors i s the engagement and disengagement analysis of the clutch, the ef f e c t of the d i f f e r e n t i a l upon the vehicle dynamics and the analysis of s l i p p i n g of the rear wheels. (AI .3) Equations of Motion Figure 6 may be used to draw free body diagrams of the system components to obtain the equations of motion. (AI. 3.1) The Engine Figure 15J i s a free body diagram of the engine i d e a l - ized as a pure i n e r t i a Je. A torque Te i s imposed on t h i s system and torque T l r e s u l t s . The analysis of t h i s system produces the following r e s u l t : T e - J e 0 e = Ti (Ai . 3 . 1 - 1 ) (AI . 3 . 2 ) The Clutch' -A • Figure 1 8 i s a free body diagram of the clutch i d e a l - ized as a p a r a l l e l spring (KC) and nonlinear coulomb damper (TO. 6 9 Tav Je T i F i g u r e lfZ: Free Body Diagram of Engine 01 02 Tl Kc T2 Tc F i g u r e ,-'18: Free Body Diagram of C l u t c h The torque i n T I i s equal to the torque out T2 and i s given by: Tl = K c ( 0 l - 0 2 ) + T c (AI.3.2-1) where Tc i s t n e value of the n o n l i n e a r coulomb damping. (AI.3,3) The Transmission The fr e e body diagram of Figure .19 i l l u s t r a t e s the tra n s m i s s i o n . The f o l l o w i n g equations r e s u l t from t h i s system: 0 2 = 0 3 X R g (AI.3.3-1) R g X T 2 = T3 (AI.3.3-2) where Rg i s the gear r a t i o . .(AI..3.4) The D r i v e s h a f t Figure 20 presents the f r e e body diagram of the d r i v e - s h a f t which i s i d e a l i z e d as a s p r i n g (KS) . The equations of motion of t h i s system are: . . T3 = T4 (AI.3.4-1) T3 = KS(03~04) (AI.3.4-2) (AI.3.5) The Rear End Figure 21 which i s much the same as Figure 19 i s the free body diagram of the rear end which i s an i d e a l transformer. The a n a l y s i s produces the f o l l o w i n g r e s u l t s : T5 = RXT4 (AI.3.5-1) 04=RX05 "(AI.3.5-2) (Al.3.6) The T i r e s Figure/22' i s the f r e e body diagram of the t i r e s . The t i r e s are i d e a l i z e d as a p a r a l l e l combination of a s p r i n g (Kt)• 71 T2 Rg T3 Figure 19: Free Body Diagram of Transmission Ks T4 Figure 2®.: Free Body Diagram of Driveshaft Figure ,;2I: Free Body Diagram of Rear 05 06 T5 Kt x - Dt~ Te Figure 22:• Free Body Diagram of Tires and a damper (Dt)• A n a l y s i s of the system shows: T5=T6 (AI.3.6-1) T5 = Kt (05- 06) + D t ( 0 5 " 0 6 ) (Al.3.6-2) (AI.3.7) The Load Figure 23 i s the f r e e body diagram of the load i d e a l - , i z e d as a pure i n e r t i a (J|)*-- The inputs to the system are the torque T 6 and TI- TI i s the load torque which i s a f u n c t i o n of speed. The i n v e s t i g a t i o n of t h i s system provides the f o l l o w i n g equation of motion: T I + J I 0 6 = T6 ' (AI.3.7-1) T6 J l TI F i g u r e 23: Free Body Diagram of Load APPENDIX I I : SOLUTIONS OF THE EQUATIONS OF MOTION OF THE LINEAR EQUIVALENT SYSTEM ( A I I . l ) Homogeneous S o l u t i o n The homogeneous equations of motion are: Je 0ci + Deq(0c i - 0ce') + Keq (0ci - 0ce')=O ( A I I . I - D jr0c6' + Deq(0c6'-0ci)+Keq(0c6'-0ci)=O ( A H . 1-2) The s o l u t i o n s of these equations are of the form: 0ci = 0aiXff S T ( A i i . 1 ^ 3 ) 0C6 ,s0a6'XtfST ( M I . 1-4) where S = —p+iq and p Is the negative of the exponent o f decay of the amplitude w h i l e Q i s the damped n a t u r a l frequency. S u b s t i - t u t i n g ( A H . 1 - 3 ) and ( A l l . 1-4) i n t o ( A l l . 1 - 1 ) and (Al l . 1 - 2 ) r e s u l t s i n : [jes 20oi+Deqs(0ai-0o6 ,) + Keq(0ai-0a6 ,)]*S T=O ( A H . 1 - 5 ) [j| ,s20a6 l+Deqs(0a6 ,-0ai)+Keq(0o6 ,-0ai) ]* S T=0 ( A H . 1 - 6 ) R e w r i t i n g ( A H . 1-5) and ( A L L . 1-6) produces the f o l l o w i n g r e s u l t s 0oi . Deos+Keo ( A H . 1 - 7 ) 006* Jes2+Deqs+Keq 006*_ Jl 's^Deqs+Keq ( A H . 1 - 8 ) 0a 1 Deqs+Keq Equating ( A H . 1-7) and ( A H . 1-8) r e s u l t s i n the frequency equa-- t i o n i J I o l l ' 2 ,s + Deqs +Keq=0 ( A H . 1 - 9 ) Je+J l ' Equation ( A H . 1 - 9 ) can be solved to produce the f o l l o w i n g r e s u l t - D e q ± y D e q 2 - 4 ( - ^ . ) K e q (AH.1-10) S = Je Jl ^ Je+JI , ; This r e s u l t can be evaluated knowing the numerical values of Je,JI* ,Deq and Keq. The f i n a l s o l u t i o n can be w r i t t e n i n the form: 0ci=tf~ p T ( A c o s q T + BsinqT) ( A I I . I - I I ) 0c6' = D e ~ p T ( A c o s q T +Bsinqr) ( A i i . 1 - 1 2 ) p_ Deq- (the r e a l p a r t of A H . 1 - 1 0 ) ( A H . 1 - 1 3 ) 2 J e q / Keq Peg 2 (the imaginary p a r t of A H . 1-10) V Jeq 4 J e q 2 ( A n . 1-14) The value of D can be c a l c u l a t e d from e i t h e r equation . ( A l l * 1 - 7 ) or ( A H . 1-8) and i s : D = — ' (AH.1-15) 001 A and B are a r b i t r a r y constants which are evaluated from the system boundafyneonditions. 77 ( A l l . 2 ) Constant S o l u t i o n The steady s t a t e constant equations of motion are: Je0ssci +Deq(0ssci - 0ssc6') + Keq(0ssci-0ssc6')= Tav ( A H . 2 - 1 ) jr0ssc6'+Deq(0ssc6 ,-0ssci) + Keq(0ssc6 ,-0ssci)=Tlc' ( A H . 2 - 2 ) .where TlC* i ; s the constant p a r t of the equivalent load torque... Now the. s o l u t i o n of these equations are,:. 0ssci = 0 s sc6 l = aeng ( A H . 2 - 3 ) 0ssci s0ssc6 ,=Wengi+ </*aengdT ( A H . 2 - 4 ) 0SSCI-0SSC6'= ^ ( A H . 2 - 5 ) where aeng i s the engine angular acceleration,Weng i i s the I n i t i a l engine angular v e l o c i t y and yfr i s the angular d e f l e c t i o n . S u b s t i t u t i n g the r e s u l t s ( A H . 2 - 3 ) to ( A H . 2 - 5 ) i n t o ( A H . 2-1) and ( A H . 2-2) produces; Tav-Tic' , a T T 9 ^ aeng = —;——71— ( A i i . 2 - 6 ) Je + J l and . _Tav Je(Tov-Tlc ') ( A H . 2 - 7 ) ^ " K e q Keq(Je+JI') In the s p e c i a l case where aeng=0 the r e s u l t ( A l l , 2-7,) reduces to: Tav ^=-7^ (AH.2-8) r Keq 78, (AH.3) O s c i l l a t o r y S o l u t i o n The steady s t a t e forced o s c i l l a t o r y equations of motion are: Je 0ssoi +Deq(0ssoi-0sso6') + Keq(0ssoi-0sso6')=Tavo ( A H . 3-1) ) JI ,0sso6'+Deq(0sso6 ,-0ssol)+Keq(0sso6 ,-0ssol)=-Tlo , ( A H . 3 - 2 In these equations Tlo' represents the f l u c t u a t i n g p a r t of the load torque and TdVO represents the f l u c t u a t i n g p a r t of the engine torque. These two f a c t o r s are given by:. Tlo' = C2 0SSO6' ( A l l . 3 - 3 ) and Tavo = KITavsin2Wet ( A I I . 3 - 4 ) Assuming the s o l u t i o n s of equations ( A H . 3-11) and (AH.3-2) are of the form: 0SSOI=|0SSOl l * + ? 2 W e t ( A l l . 3 - 5 ) 0SSO6'=|0SSO6'le + l 2 W e t ( A l l . 3 - 6 ) the f o l l o w i n g equations r e s u l t : [- Je4We 2 |0ssoi| +Deq2jWe(| 0ssoi|-| 0sso6'|) + Keq(|0ssoil - 10ssoe'l)] * l 2 W e t = KlTov e I 2 W E T ( A I I . 3 - 7 ) [-JI ,4We 2|0sso6l+Deq2iWe(|0ssp6l-|0ssoi|^ l0ssoi!)] ffi2Wet=-C2 2 i W e | 0 s s o 6 ' U l 2 W e t ( A i i . 3 - 8 ) These equations can be r e w r i t t e n to producer. [Keq + Deq2Wei-Je4We 2]l0ssoil-[Deq2Wei+Keq]|0sso6'|=KlTav (AH.3-9) 79 [Keq +(Deq + C2)2Wei-JI ,(4We2)]l0sso6 ,l-[Deq2Wei+Keq]|0ssoi|=O ( A H . 3 - 1 0 ) These equations can be solved by using Cramer's Rule: 0 S S O l l s K I T Q V (Deq2Wei + Keq) 0 (Keq+(Deq+C2)2Wei-Jl '4We 2) (Keq+Deq2Wei-Je4We 2HDeq2Wei + Keq) kDeq2Wei+Keq)(Keq + (Deq + C2)2Wei + Jl '4We 2) ( A H . 3 - 1 1 ) 0SSO6 = (Keq+Deq2Wei-Je,4We2) (Deq2Wei+Keq) KITav 0 (Keq+Deq2Wei-Je4We 2) (Deq2Wei+Keq) (Deq2Wei+Keq)(Keq+(Deq+C2)2Wei+Jl'4We 2) ( A I T . 3 - 1 2 ) Once the values of the p h y s i c a l constants and the operat i n g f r e - quency are known, equations ( A H . 3 - 1 1 ) and (All..3-12) can be evaluated. The f i n a l r e s u l t s w i l l be of the form: 0ssoi = A i sin(2Wet + ^ i ) ( A H . 3 - 1 3 ) 0sso6 = A6'sin(2Wet + ^6') ( A H . 3 - 1 4 ) where Al,A6',\/H and\fr6l are evaluated using equations ( A H . 3 - 1 1 ) and ( A H . 3 - 1 2 ) . 80 APPENDIX I I I : PHYSICAL CONSTANTS USED IN THE ANALYSIS ( A I I I . l ) Purpose The purpose of t h i s Appendix i s to present the sources and d e r i v a t i o n of the p h y s i c a l constants used i n the a n a l y s i s i n Sections 2,3,4, and 5. (AIII.2) Engine I n e r t i a (Je) Figure :24; i l l u s t r a t e s the flywheel and crankshaft l a y - out i n the C o r t i n a 2000 c.c. engine. Fi g u r e .25 i l l u s t r a t e s a connecting rod, p i s t o n p i n and p i s t o n from the same engine. Both Figures 24, and 25; are somewhat i d e a l i z e d and were constructed from data taken from [(24].. (AIII.2.1) Weight of P i s t o n The weight of t h e s p i s t o n i s determined by e v a l u a t i n g the f o l l o w i n g : W P = - ^ [ * < d 0 2 - d i 2 ) h l + » d 0 2 h 2 ] ( A I I I . 2 . 1 - 1 ) where the symbols are as shown i n Figures 2.4 and 26; except t h a t : -3 2 3 p =density of steel=7.89x10 (Nsec /(m)cm ) 2 g = a c c e l e r a t i o n of gravity=9.8 (m/sec ) .'. Wp =9.75 (N) (AIII.2.1^2) LARGE WEB SMALL WEB CRANKSHAFT FLYWHEEL SCALE FULL SIZE Mlw-J 1 s 1 o. o •o T •tlw I—tsw —[ Icj Icp :»1 a. -H !—tfw F i g u r e 24: Flywheel and Crankshaft Layout of C o r t i n a 2000 c.c. Engine 82 SiOiQM OF PISTON SCALE'.-^-FULL SIZE PISTON PIN Ipin " 1 CONNECTING ROD rduo r t 3i F i g u r e 25: Connecting Rod, P i s t o n P i n and P i s t o n of C o r t i n a 2000 c.c. Engine ,83 ( A I I I . 2 . 2 ) Weight of P i s t o n P i n The weight of the p i s t o n p i n i s given from: yog-Trlpindpin2 Wpin = =2.519 (N) ( A i n . 2 . 2 - 1 ) ( A I I I . 2 . . 3 ) Weight of the Connecting Rod The weight of the connecting rod i s given by: W c r ^ g t 3 - [ ^ D U ^ ^ ( A I I I . 2 . 3 - 1 ) ..Wcr= 5.297 (N) ( A I I I . 2 , 4 ) Weight of Crank J o u r n a l The weight of the crank j o u r n a l i s c a l c u l a t e d from: Wcj = -£p 7rdcj 2tcj = 5.0l (N) ( A I I I . 2 . 4 - 1 ) ( A I I I . 2 . 5 ) Weight of Crank P i n The weight of the crank p i n i s given by: Wcp=-£ptrdcp 2tcp = 4.171 (N) ( A I I I . 2 . 5 - 1 ) ( A I I I . 2 . 6 ) Weight of Small Web) The weight of the small web i s given by:; Wsw=/og-lsw-hswdsw=4.175 (N) ( A i n . 2 . 6 - 1 ) 84 ( A I I I . 2 . 7 ) Weight of Large Web The weight of the l a r g e web i s given by: Wl w= llw-dlw-hlw-/og = 6.089 (N) ( A i n . 2 . 7 - 1 ) ( A I I I . 2 . 8 ) Weight and I n e r t i a of Re c i p r o c a t i n g Mass D i v i d i n g the connecting rod mass so that one t h i r d i s r e c i p r o c a t i n g and two t h i r d s i s r o t a t i n g the t o t a l r e c i p r o c a t i n g weight becomes: Wrm = Wp + Wpin + - l - Wcr=l4.03 (N) ( A i n . 2 . 8 - 1 ) The moment of i n e r t i a of the r e c i p r o c a t i n g masses i s : Irm = - ^ ( S t r o k e x 0.5) = 2.12x10" 3 (Nmsec 2) ( A i n . 2 . 8 - 2 ) ( A I I I . 2 . 9 ) I n e r t i a of Rota t i n g P o r t i o n of the Connecting Rod The moment of i n e r t i a of the r o t a t i n g mass of the con- n e c t i n g rod i s : I r c r * W g c r ( S t r o k e ) 2 * 5 . 3 3 X I O - 4 (Nmsec 2) ( A i n . 2 . 9 - 1 ) 6g ( A I I I . 2 . 1 0 ) I n e r t i a of Crank J o u r n a l The moment of i n e r t i a of the crank j o u r n a l i s : I c j s ^ l £ c i f s 2 0 8 x I 0 - 4 (Nmsec 2 ) (A111.2.10-1) ( A I I I . 2 . 1 1 ) I n e r t i a of Crank P i n The moment of . i n e r t i a of the crank p i n w i t h respect to i t s own centre of mass i s : Icp(cm)= W 2 C ^ ' f C p =I.438X|0~4 (Nmsec 2) ( A H I . 2 . 1 1 - 1 ) 85 Using the p a r a l l e l a x i s theorem t h i s i n e r t i a i s t r a n s f e r r e d to the crank j o u r n a l a x i s : Icp/cj=Icp(cm)+ W c P r c p / c ' t 2 = 6 . 3 x i 0 - 4 (Nmsec2) ( A I I I . 2 . 1 1 - 2 ) 2g ( A I I I . 2 . 1 2 ) I n e r t i a of.Large Web The moment of i n e r t i a of the l a r g e web r e l a t i v e to i t s centre of mass i s : I lw (cm)=-J5f- ( Iw 2 +hlw 2 ) = 8 . 2 4 6 x i 0 ~ 4 (Nmsec 2) ( A I I I . 2 . 1 2 - 1 ) I2g The moment of i n e r t i a of the l a r g e web t r a n s f e r r e d to the crank j o u r n a l a x i s i s : I lw/cj=9. l4x iCT 4 (Nmsec 2) ( A I I I . 2 . 1 2 - 2 ) (AIII.2.13) I n e r t i a of Small Web The moment of i n e r t i a of the small web i s c a l c u l a t e d i n a s i m i l a r manner as t h a t of the l a r g e web. I t s value i s : I sw/cj=4.36xi0" 4 (Nmsec2) ( A i n . 2 . 1 3 - 1 ) (AIII.2.14) T o t a l I n e r t i a A s s o c i a t e d With One C y l i n d e r The t o t a l i n e r t i a a s s o c i a t e d w i t h one c y l i n d e r i s found from: Icyl = Irm + Ircr + Icj + Icp/CiJTi-Ilw/cj+Isw/cj = 4.84XlO" 3(Nmsec 2) (AIII.2.14-1) 8 6 (AIII.2.15) Weight and I n e r t i a of Flywheel The weight of the flywheel i s given as: />g«wdfw2tfw Wfw= r M =120 .9 (N) ( A I I I . 2 . i 5 - i ) 4 where: df W=28.0 (cm) (flywheel diameter) tfw=2 .54 (cm) (flywheel thickness) The moment of i n e r t i a of the flywheel i s : Wfw-rfw 2 Ifw= ' =1.21X10-' (Nmsec 2) (AI I I.2.1 5 - 2 ) 2 g (AIII.2.16) Weight and I n e r t i a of Shaft Between Crank and Flywheel The weight of the connecting s h a f t i s : Ws=/»g-irrs 2ls = 8.769 (N) ( A i n . 2 . 1 6 - 1 ) The moment of i n e r t i a of t h i s s h a f t i s given by: Ws» 2g Is= W s > r s 2 = 6.46XIQ- 4 (Nmsec2) ( A i n . 2 . 1 6 - 2 ) (AIII.2.17) The T o t a l I n e r t i a of The Engine The t o t a l i n e r t i a of the engine i s given by: Je=4Icyl + Is + Ifw=O.I4l (Nmsec 2 /rad) ( A i n . 2 . 1 7 - 1 ) (AIII.3) C l u t c h Spring (KC) and Damper (Tc) Constants The c l u t c h s p r i n g and damping constants were determined from experiments w i t h a small (16.5cm i n diameter) c l u t c h . The r e s u l t s are then e x t r a p o l a t e d to the C o r t i n a c l u t c h (21.5cm i n diameter). 87 Figure 26 i l l u s t r a t e s the experimental setup. F i g u r e 21-' presents the r e s u l t s of the experiment. The i n t e r c e p t of the s t r a i g h t l i n e i s the s t a t i c damping torque which i s : Tcs = 6.0 (Nm) ( A I I I.3-1) The dynamic damping torque can be c a l c u l a t e d knowing the r a t i o of the dynamic f r i c t i o n c o e f f i c i e n t to the s t a t i c f r i c t i o n c o e f f i - c i e n t f o r the c l u t c h p l a t e m a t e r i a l . This r a t i o i s found to be 0.53 f o r asbestos [25]. The dynamic damping torque i s then: Tcd = 3.2 (Nm) ( A I I I.3-2) The slope of the s t r a i g h t l i n e given i n Fi g u r e 2r7 gives the s p r i n g constant. Thus the s p r i n g constant i s given by: Kc=l030 (Nm/rad) ( A I I I.3-3) As a check, ' the s p r i n g constant i s c a l c u l a t e d . F i g u r e '-28 i l l u s - t r a t e s the dimensions and arrangement of- the c l u t c h p l a t e used i n the experiment. The s p r i n g constant of one s p r i n g of the c l u t c h p l a t e i s : K c | s £ r i = 2 . 3 2 x i 0 5 (N/m) ( A I I I.3-4) 10 2 where: G=shear modulus of s t e e l 5 ? 7 .92x10 (N/m ) A one Newton force a c t i n g on t h i s s p r i n g i s equi v a l e n t to a torque o f : I N e w t o n x - ^ = 3 . 5 x i O " 2 (Nm) ( A I I I.3-5) 2 Therefore, the s p r i n g constant Kci may be w r i t t e n : Kci = 284 .0 (Nm/rad) (A I I I.3-6) CLUTCH SUPPORTED IN VISE ECTION Figure 26: Experimental Setup f o r Determining C l u t c h Spring and Damping Constants I20-, 0.02 0.04 0.08 DEFLECTION (rod) o.io F,7.aure Xii - Experimental. Se£up^fpr Deterr ' - in i ; . F i g u r e 2-7: Results of Bthe C l u t c h Experiment c i u t c n Spring and Damping 8 9 CLUTCH SPRING D n = 5=no. of colls SPRING CIRCLE=dcs=7 (cm) F i g u r e 2 8 : C l u t c h P l a t e and Springs The t o t a l s p r i n g constant i s the sum of the i n d i v i d u a l s p r i n g constants o r : Kc = 4 x K c i = l l 4 0 (Nm/rad) ( A i n . 3 - 7 ) Comparison of the r e s u l t s i n d i c a t e s t h a t the.disagreement i s l e s s than 10%. S c a l i n g the s p r i n g constants l i n e a r l y w i t h c l u t c h , diameter produces the f o l l o w i n g r e s u l t : Kc=l500 (Nm/rad) ( A I I I . 3 - 8 ) The damping torque i s not a f f e c t e d by any change i n s i z e . ( A I I I . 4 ) Gear.Ratios (Rq) The gear r a t i o s are found i n [24]. They are: Gear Ra t i o 1st 3.65 2nd 1.97 3rd 1.37 4 th 1.00 ( A I I I.4-1) ( A I I I.5) D r i v e s h a f t Spring Constant (Ks) The d r i v e s h a f t s p r i n g constant i s c a l c u l a t e d u s i n g the dimensions i l l u s t r a t e d i n Figu r e 29. The r e s u l t i n g value i s : G-rrfdo^di4) / l k l ,. • — 3 2 1 = 7 8 0 0 (Nm/rad) ( A i n . 5 - 1 ) ( A I I I . 6 ) Rear End Rat i o (R) The rear end r a t i o i s given i n [24] as: R=3.44 ( A I I I.6-1) '91 l=!ength= 244 (cm) di=4.8 (cm) do=5.0 (cm) F i g u r e 29: D r i v e s h a f t Dimensions i 92 (AIII.7) T i r e Spring (Kt) and Damping (Dt) Constants The t i r e s p r i n g and damping constants are reported i n [23]. The average s p r i n g constant i s : Kt = 6950 (Nm/rad) ( A I I I . V - I ) and the average damping constant i s : Dt = 58.0 (Nmsec/rad) ( A M . 7 - 2 ) (AIII.8) Load I n e r t i a (J|) Figure 30 i s a fre e body diagram of the v e h i c l e and one of i t s wheels. The forc e F a c t i n g on the wheels can be w r i t t e n : F=/tNr ( A M . 8 - 1 ) . where Nr i s the p o r t i o n of the v e h i c l e weight over the rear wheels. The v e h i c l e weight i s given i n [24] as 11,100 (N). The weight d i s t r i b u t i o n i s 45% rear and 55% f r o n t . The c o e f f i c i e n t s of f r i c t i o n between the t i r e and a dry road i s found i n [25] to be ̂ |=0.8. Therefore: F=4995 (N) ( A M . 8 - 2 ) The maximum a c c e l e r a t i o n achieved by the v e h i c l e i s c a l c u l a t e d from: 3 amax=—^-=4.41 (m/sec) ( A M . 8 - 3 ) m At the rea r wheels, the maximum torque developed i s given by: Tmax=RwXF ( A M . 8 - 4 ) ' WHEEL Nr VEHICLE DRAG FORCE Figure 30: Free Body Diagram of V e h i c l e F i g u r e 31: Front View of C o r t i n a Sedan 94 where Rwis the dynamic wheel radius given i n [25] as: Rw = 0.2997 (m) ( A I I I.8-5) f o r the C o r t i n a t i r e s . The maximum torque i s r e l a t e d to the v e h i c l e i n e r t i a by: Tmax =JI amax , ( A I I I.8-6) where amax=- g7 1^= 14.71 (rad/sec2) Therefore, the v e h i c l e i n e r t i a i s given as-: J l = ^ m 0 x = 102 (Nmsec2/rad) ( A I H . S - 7 ) amax (AIII.9) V e h i c l e F r o n t a l Area (Ap) ' Figure M , i s a f r o n t view of the C o r t i n a sedan. The f r o n t a l area i s c a l c u l a t e d w i t h the a i d of t h i s F i g u r e and i s given by: Ap=(hi-h2)xtr = l .53 (m2) ( A I I I . 9 - I ) A I I I . 1 0 Engine Torque C o e f f i c i e n t s The engine torque c o e f f i c i e n t s used i n equation (2.2.1-2) are given i n References [18] as: C2=1.2 B2=-0.1 C4=0.2 B4=-0.02 C6=0.05 B6=-0.008 C8=0.01 B8=-0.005 9.5 APPENDIX IV: COMPUTER PROGRAM The purpose of t h i s Appendix i s to present the com- puter program used i n the simulation of Section 6 . The program i s stored i n the f i l e ICFILE (a l i s t i n g of t h i s f i l e appears i n t h i s Appendix) under the ID CNTL, The statements i n the INITIAL and TERMINAL sections may require r e v i s i o n from run to run. To.execute the program the following sequence of commands i s required: $SIG CNTL Password $R *CSMPTRAN SCARDS=ICFILE $R *FORTRAN SCARDS=-CSMP#7 $R *CSMPEXEC!H:-LOAD#+*CSMPLIB 5=-CSMP#5 S P R INT=DF I LE This sequence w i l l place the results of the simulation i n the f i l e DFILE. 1 MACRO Z Z = N O N L I N l A A C C , I , J , C A N V E L , R E L T O R ) 2 PROCEDURAL __. „ . 3 X X = D E L A Y ( I , J , R E L T C R ) 4 YY=AES(XX) 5 V,H = D E L A Y ( I , J . C A N V E L ) 6 V V = A R S m » 7 I F ( T I M E - P I . L E . O . O ) GO TC 107 d IF I V V - O . l ) 100 , 100 , 101 .,„ .. .. 9 100 ZZ = XX 10 I F I Y Y - 6 . 0 I 1 0 2 , 1 0 2 , 1 0 3 11 103 I F ( Z Z . G T . 6 . 0 ) G O T O 104 12 ZZ= -3 .2 13 GO TO 102 14 107 ZZ = A.ACC . __ _ . ... . _ 15 GO TO 102 16 104 11=2.2 17 GO TO 102 18 101 I F ( k k . G T . 3 . 0 ) GO TC 105 19 ZZ= -3 .2 20 . _ GO T O . 1 0 2 : - — - 21 105 ZZ=3.2 22 102 CONTINUE 23 ENDMAC 24 * T H E ABOVE MACRO DESCRIPT ION MODELS THE NONLINEAR CLUTCH DYNAMICS 25 MACRO S S = D L / Y ( A A B B , N , P A , A N V E L ) 26 PPQCEDLRAL „ . . . . . . .... . . _. - - 27 W=DELAY(M, PA, AKVEL ) 28 V = W 29 I F ( T I M E - P A . L E . O . O ) GO TC 108 30 SS=V*R*RG 31 GO TO 109 32 108 S S = AABB 32 109 CONTINUE 34 ENDMAC 35 MACRO TT=ANGLE<NN,ANGE,PK,N) 36 PROCEDURAL 37 C Q = O E L A Y ( N » P K t A N G E ) 38 111 P P = Q Q - N N * 2 . 0 * 3 . 1 4 1 6 39 I F I P P - 2 . 0 * 3 . 1 4 1 6 . L E . 0 . 0 ) GO TO 110 40 NN=NN+1 41 GO TO 111 42 110 T T = 2 . 0 * P P 43 ENDMAC 44 *THE ABOVE MACRO DESCRIPTION CALCULATES THE CRANKSHAFT ANGULAR 45 *D I SPLACEMENT 46 I N I T I A L 47 *TH I S SECTION DEFINES THE SYSTEM PARAMETERS AND IN IT IAL CONDITIONS 48 MEMORY NQNL I N 49 MEMORY DLAY 50 MfMORY ANGLE 51 PARAMETER J E = 0 . 1 4 , J L = 1 0 2 . 0 , K C = 1 5 0 0 . 0 , K T = 6 9 5 0 . 0 , K S = 7 8 0 0 . 0 , D T = 5 8 . 0 • . . . 5 2 R = 3 . 4 4 , P G = 1 . 3 7 , P 1 = 0 . 0 0 0 5 , P 2 = 0 . 0 0 1 , P 3 = 0 . 0 0 1 5 , P 4 = 0 .002 ,P5=0 .0025 . . . . 5 3 P6 = 0 . 0 0 3 , P 7 = 0 . 0 0 3 5 , P8 = 0 . 0 0 4 , P S = 0 . 0 0 4 5 , P 1 0 = 0 . C O 5,P11 = 0 .0055 ,P12 = 0 . 0 0 6 , . . . 54 P13 = 0. 0C6 5 ,P14 = 0.CO 1,PI 5 = 0 . 0 C 7 5 t P I 6 = 0 .008 , PI 7 = 0 . 0 0 8 5 , P I 8 = 0 . 0 0 9 , . . . 55 P 1 9 = 0 . 0 0 9 5 , P 2 0 = 0 . 0 1 , P 2 1 = 0 . 0 1 0 £ , P 2 2 = 0 . 0 1 1 , P 2 3 = 0 . 0 1 1 5 , P 2 4 = C . C 1 2 , . . . 56 P2 5=0. C 1 2 5 , P 2 6 = 0 . C 1 3 f P 2 7 = Q . 0 1 3 5 , P 2 8 = 0 . 0 1 4 , P 2 9 = 0 . 0 1 4 51P 3 0 = 0 . 0 1 5 , N2= 1, . . . 5 7 C 2 = 1 . 2 , C 4 = 0 . 2 , C 6 = 0 . 0 5 , C 8 = 0 . 0 1 , B 2 = - 0 . l , B 4 = - 0 . C2 , B 6 = - 0 . 0 0 8 , B 8 = - 0 . 0 0 5 58 INCON I C 1 = 2 8 0 . 0 , IC2 = 0 . 1 2 6 , I C 3 = 0 . 0 1 3 2 t I C 5 = 0 . 0 , 1 C 6 = 0 . 0 59 I C4= IC1 / (R *RG ) 60 DYNAMIC 61 *TH I S SECTION MODELS THE DYNAMICS OF THE SYSTEM 62 FPEC=DLAY( I C 1 , l i P l t S l T L C ) 63 TMAX=93.4+C.355*THRCT 64 +THROTTLE INPUT 65 T H R 0 T = 2 7 . 1 + 1 0 . 0 * S T E P ( 1 . 0 ) 66 S0THE0=3 .86 *THP0T 67 * AV EP AGE TOP QUE 63 ... . T A V - T ^ A X * l l . Q - Q . 2 7 5 * 1 ( ( FREQ/SQThEQ ) r U Q 1 * * 2 ) ) 69 S O T O R = A N G L E ( N 2 , S 0 T H E , P l f l ) 70 * INST ANT ANEOUS ENGINE TORQUE 71 S 0 T E = T A V * ( 1 . 0 + C 2 * 2 * S I h ( S Q T 0 R ) * C 4 * 4 * S I N ( 2 * S Q T U R ) + . . . . 7 2 B 2 * 2 * C O S ( S O T O R ) * - B 4 * 4 * C 0 S ( 2 * SOTORI+ C 6 * 6 * S I N ( 3 * S O T O R ) • . . 73 C 8 * 8 * S I N ( 4 * S O T O R ) * E 6 * 6 * C O S ( 3 * S 0 T O R ) * B 8 * 8 * C O S ( 4 *S0TCRJ ) 74 *LOAD TORQUE _. _ _ 75 S G T R = 3 7 . 5 + ( 0 . 0 1 5 2 * ( S 1 T L O * * 2 ) ) + S O T R D 76 *LOAD TORQUE DISTURBANCES 77 SOTPD=0.0 78 * SY STE M EQUATIONS OF MCTICN 79 S 2 T H E = S O T E / J E + S 0 T H E 2 * K C / J E - S 0 T D / J E - K C * S 0 T H E / J E QQ S I T H E r INTGRL l I C 1 » S2THE ) . .. . . 81 SOTHE= INTGRLJ IC2 , S1THE ) 82 S 0 T H E 2 = I S 0 T C * ( R G * * 2 ) + K C * ( R G * * 2 ) * S 0 T H E + K S * R G * S 0 T H E 4 ) / . . «3 ( K C * ( R G * * 2 ) + K S ) 84 SGTHE 3=S0THE 2/RG 85 S 0 T 4 = K S * ( S 0 T h E 3 - S 0 T H E 4 ) 86 S 0 T h E 4 = R * S 0 T h f 5 _ _ _ 87 S 1 T H E 5 = R * S 0 T 4 / D T + K T * S 0 T L 0 / D T + S l f L O - K T * S O T H E 5 / D T B8 S 0 T H E 5 = I N T G P L ( I C 3 . S 1 T H E 5 ) 89 S 2 T L 0 = ( D T * S l T F E 5 + K T * S 0 T h E 5 - S 0 T R - D T * S l T L 0 - K T * S 0 T L 0 ) / J L 90 S 1 T L 0 = I N T G R L ( I C 4 , S 2 T L 0 ) 91 S 0 T L 0 = I N T G R L ( I C 5 , S 1 T L 0 ) 92 S 1 T H E 2 = P E R I V ( I C 1 , S 0 T F E 2 ) 93 S1CL IT=S1THE -S1THE2 94 S 0 C L U T = S 0 T F E - S 0 T F E 2 95 S 0 T 0 = N C N L I N ( I C 6 , 1 , P 1 , S 1 C L U T , S 0 R R ) 96 S O R R = S 0 T E - S 0 T R / ( R * R G ) - J L * S 2 T L C / ( R * R G ) - J E * S 2 T H E 97 SO J E = J E * S 2 T F E 98 $OCS=KC*(SOCLUT) 99 NOSORT 100 *OUTPUT P R I N T E R CONTROL 101 T I f E I = T I M E I + D E L T 1 0 2 I F ( T I M E I - 2 . C * D E L T . E Q . 0 . 0 ) GO TO 1 1 7 1 0 3 GO TO 118 1 0 4 1 1 7 TI M E I r O . O 1 0 5 hR I T E ( 6 , 1 1 6 ) TIME , T A V , T H R O T , S 1 T H E , S 2 T H E 106 1 1 6 F 0 P M A T I 5 E 1 2 . 4 ) 1 0 7 1 1 8 CON T IM LE 108 SORT 1C9 T E R M I N A L 110 * T H I S S E C T I O N CONTROLS THE E X E C U T I C N - PHASE OF THE_ MCJDEL 111 T I M E R F I N T I M = 3 . 0 , P R D E L = 0 . 0 1 , C E L T = 0 . 0 0 0 5 112 METHOD RECT 1 1 3 END 114 STOP 115 ENDJOB 116 SFND

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