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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, University of British Columbia Vancouver, British Columbia, 1973 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in the Department of Mechanical Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA April, 1975 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall, make it freely available for reference and studyo I further agree that permission.for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for 'financial gain shall not be allowed without my written permission. Department of Mechariical Engineerirr The University of British Columbia Vancouver 8, Canada Date April 2nd, 1975 ii ABSTRACT The purpose of the investigation is to study the dynamics of the internal combustion engine-vehicle-driver system. Specifically, the system variables, engine angular velocity and engine angular acceleration are examined as potential observers of the engine mean torque. Such an observer is a requirement for the application of adaptive control to internal combustion engines. This type of control system has shown promise in providing solutions to the present problems of fuel economy, air pollution and performance. A nonlinear dynamic model of the engine-vehicle-driver system is developed. This model is linearized and simplified to provide expressions for the variables of interest, the engine angular acceleration and velocity. The validity of the simplified model is established by comparison with results obtained from the computer simulation of the nonlinear model. The agreement between the two models is good. The solutions of the equivalent system model are analyzed to determine which is the best observer for the mean torque. It is established that the steady state forced oscilla tory engine angular acceleration response provides the best observer. ACKNOWLEDGEMENT I would like to thank Dr. T. N. Adams and Dr. R. E. McKechnie for their support and guidance throughout this work. The constructive criticism provided during the preparation of this thesis is particularly appreciated. I would also "(Like to thank my wife for her patience and understanding in typing the thesis. iv 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 1(2.2) External Torques 20 3 AN EQUIVALENT LINEAR SYSTEM 34 (3.1) A Constant Speed System(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 3(4.2) The Constant Response 40 (4.3) The Oscillatory Response 1 (4.4) The Total Response a.* 42 V 'SECTION lE-AGE 5 SOLUTIONS OF A SPECIAL CASE . 43 (5.1) The Vehicle Considered , . 43 (5.2) The Particular Case Considered 43 (5.3) The Transient Response 44 (5.4) The Constant Response 6 (5.5) The Oscillatory Response 46 COMPUTER SOLUTION OF THE NONLINEAR SYSTEM 48 7 MEAN TORQUE INDICATORS 55 (7.1) The Transient Response 5(7.2) The Constant Response . 7 (7.3) The Oscillatory Response 58 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 vi 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 vii LIST OF ILLUSTRATIONS FIGURE PAGE 1 Single Level Extremum Seeking Adaptive Controller 5 2 Structure of Controls and Dynamics of the Engine Vehicle Driver System ... 8 3S Conceptual Block Diagram of an Adaptive Spark Timing Controller ... 9 4 The Effect of Spark Timing on the Mean Torque Output 12 5 Hypersurface Illustrating the Relationship Between Engine Variables . ' 13 6 Lumped Parameter Engine Vehicle System 19 7 Comparison of the Simplified and Numerical Series Engine Torque Signals 22 8 Mean Torque of the Cortina 2000 c.c. Engine ... 24 9 Free Body Diagram Illustrating the External Forces on a Vehicle 30 10 Load Torque on the Cortina Sedan . 32 v 11 Equivalent Linear Engine Vehicle System 35 12 Throttle Command Input to the Computer Simulation Model 49 13 Mean Torque Response to the Throttle Command Input 50 14 Engine Rotational Velocity Response to the Throttle Command Input 51 15 Engine Rotational Acceleration Response to the Throttle Command Input 52 viii FIGURE PAGE 16 Transient Cancellation 56 17 Free Body Diagram of Engine 69 18 Free Body Diagram of Clutch19 Free Body Diagram of Transmission 71 20 Free Body Diagram of Driveshaft21 Free Body Diagram of Rear End 72 22 Free Body Diagram of Tires23 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 Experiment28 Clutch Plate and Springs 89 29 Driveshaft Dimensions ..... 91 30 Free Body Diagram of Vehicle 93 31 Front View of Cortina Sedanix NOTATION Symbol Description Ap vehicle projected area B^-jBljB"^", ... Fourier sin torque series coefficients C"2",Cl,C*2" ,... Fourier cos torque series coefficients DC equivalent linear clutch damping coeffic ients Dt tire damping coefficient Dt' equivalent tire damping coefficient at engine speed Je engine inertia Je<J equivalent system inertia JI load inertia vii' equivalent load inertia at engine speed Kc clutch spring constant Kf rolling resistance drag coefficient Ks driveshaft spring constant Ks' equivalent driveshaft spring constant at engine speed Kt tire spring constant Kt' equivalent tire spring constant at engine speed Kl ratio of the maximum to the mean torque per engine cycle Nf front wheel reaction Symbol ; Description Nf rear wheel reaction R rear end ratio Rg gear ratio Rw dynamic wheel radius TdV engine mean torque output TdV^ optimum engine mean torque output Tb brake torque Tc nonlinear coulomb clutch damping Te engine instantaneous torque output TI vehicle load torque TI equivalent vehicle load torque at engine speed We engine angular velocity Wengi initial engine velocity Wn natural frequency Wnd damped natural frequency W vehicle weight |x| magnitude of variable x t time' creng engine constant acceleration 6 crank angle 0CX,0CX,$CX transient motions of variable x 01 gradient angle 0S spark timing dSSCX,&SSCX,0SSCX steady state constant motions of variable Symbol • • • 0ssoxf0ssox,0$sox r rc ** T c Description steady state oscillatory motions variable x period of transient oscillation transient time constant throttle position time of transient damping ratio 1 1 INTRODUCTION (1.1) Purpose and Scope The chief problems of the current internal combustion automotive engine are air pollution, fuel economy and performance [3>l],[2],[3],[4],[5],[6]/[7].1 A concept which has shown promise in providing a solution to these problems is that of Adaptive Control [8],[9],[10]. Using this concept the present distri butor advance unit and carburetor would be replaced by spark timing and air-fuel ratio devices regulated by an adaptive controller. The potential advantages of adaptive control include improved fuel economy, reduced emissions, reduced maintenance and improved reliability [8],[9],[10]. An adaptive control scheme requires a signal which indicates system output or performance. It has been shown by Adams [11] that for internal combustion engines the most suitable signal is the engine mean torque (Tdv)• The primary purpose of this work is to determine the best means of obtain ing this torque signal. A secondary purpose is to construct a model of the dynamics of an engine-vehicle-driver system for later work in determining the optimum adaptive controller config uration. Numbers in square brackets designate references at end of thesis. 2 Any variable of the system-which acts as an observer of TQV will be affected by disturbances acting on the system. Therefore, it is necessary to study the dynamics of the entire system to learn how a potential observer is affected by the disturbances encountered under normal operating conditions. The dynamics of a lumped parameter engine vehicle system are considered in this work. An initial model is developed and its equations of motion a>revs6l-vgdsb^us4muil>ation on the IBM 370 computer. This model is linearized and simplified to provide equations of motion which can be solved analytically. The validity of the linearization and simplification made is checked by comparison of the simulated and analytic solutions for a special case. The analytic solutions are used to evaluate potential observers for TQV • In connection with this work, models describing the engine mean torque (T(JV) and vehicle load torque (T I ) are developed. Some preliminary work on the adaptive control model and the human control model has also been undertaken. This work will be necessary in future to determine the optimum adaptive controller configuration. (1.2) Work by Others The recent work done in the application of automatic control theory to internal combustion engines falls into three categories, digital memory fuel injection control (DMC), closed 34 loop fuel injection control (CLC), and extremum seeking adaptive controls (AC) . (1.2.1) Digital Memory Fuel Injection Control The DMC improves upon the conventional carburetor by-eliminating the mechanical wear and aging problems [12], [13,]. The memory stores predetermined optimum fuel injection pulse times. These pulse times are addressed by information derived from engine speed and throttle position. The drawback of this system is the predetermined nature of the stored information. For example, pulse times stored in the memory are obtained from tests on a standard engine under standard environmental conditions. These pulse times are optimum for this engine at the particular time they are determined. However, production tolerances in engine manufactur ing are not tight enough to assure that the pulse times will be optimum for every engine rolling off the assembly line. An even more serious problem is the wide range in environmental conditions a production engine will experience during its life. This range of environmental conditions definitely affects the optimum pulse time [14]. (1.2.2) Closed Loop Fuel Injection Control .The CLC senses the oxygen content of the exhaust gas stream and compares it with a reference value [14],[15]. If the measured value differs from the reference value, the control adjusts the fuel injection pulse time until the difference 4 disappears. This type of control was initially developed to insure satisfactory operation of catalytic converters by regu lating the exhaust gas stream composition [14]. Later invest igators [15] have extended the work to providing optimum air-fuel ratios, but the workers have not shown how this optimum may be identified. (1.2.3) Extremum-Seekinq Adaptive Control AC adjusts the spark timing and/or air-fuel ratio:,, until a local optimum in performance has been reached [8],[9], [10], Thus, AC differs from conventional automotive engine controls and feedback controls in that it searches for and maintains the optimum setpoint values of the variables under control. These values maximize the engine performance. (1.3) Adaptive Control Figure 1 is a block diagram representation of a single level extremum seeking adaptive control system. The objective of such a control system is to identify and maintain the optimum setpoint X of the system. This setpoint will maintain the out put Z at its extremum (in this case its maximum) value Z^» In this simple system the relationship between the input and the output (the system characteristic) 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: Single Level Extremum Seeking Adaptive Controller Z=(Z#-X(X#-Xc)2) 6 The condition which determines that the optimum is reached occurs when the slope of the system characteristic (AZ/AX) becomes zero. For example, if the system is initially at some point Xo away from and the control makes a change to some new X, then a corresponding change in Z will result. The change in output Z is then determined with respect to the change in input X« The resulting signal is used to drive the system toward the optimum X where A7/AX approaches 0. The control law (i.e. the law which governs the control response for a given error signal (AZ/AX) ) in Figure 1 is a simple proportional gain K- More sophisticated control laws may prove valuable in improving system stability and response time. Stability refers to the tendency of the system to approach the optimum with a decreasing amplitude of oscillation. If the system were to oscillate with increasing amplitude around the optimum it would be classed as unstable. Response time refers to the time taken for the system to reach the optimum.for an initial setpoint an arbitrary distance away from the optimum. For example, derivative control (i.e. control proportional to the rate of change of error) may be applied in conjunction with the proportional control to improve the system response time. A complete discussion of the stability and response time problems is beyond the scope of this work and the interested reader is referred to [16],[17]. 7 An important feature of adaptive control illustrated in Figure 1 is that if there is only one maximum within the operating range then this will be the point the control will seek to operate at. All other points will produce a local gradient (AZ/AX) . Thus a control signal K(AZ/AX) will exist and drive the system until AZ/^X becomes zero at a local optimum. Adaptive control can be easily extended to control two or more variables. Initial small perturbations are made in the setpoints of the variables under control. The effects of these perturbations are then observed. The observations enable the controller to determine the direction in which the inputs are adjusted to achieve the "steepest ascent" toward their optimum setpoints. (1.4) Application of Adaptive Control to Internal Combustion  Engines Figure 2 illustrates the structure of the controls and dynamics of the engine-vehicle-driver system. The adaptive con trol supervises the control of several plant variables in response to the driver control of fuel rate. Figure 3 is a conceptual illustration of an adaptive spark timing controller for an internal combustion engine. The truth table of the And-Nor logic gate of this controller is presented in Table I. The control senses the engine mean torque and spark timing. The two variables are then compared by the LOAD DISTURBANCES DESIRED 6 ERROR VELOCITY HUMAN CONTROL MODEL <f>t=THROTTLE BRAKE MODEL HO DISTURBANCES S3l ENGINE TORQUE MODEL TI TAV ADAPTIVE CONTROL GEAR RATIO (RG) LOAD TORQUE MODEL SYSTEM DYNAMIC MODEL t i i DISTURBANCES Figure 2: Structure of Controls and Dynamics of the Engine Vehicle Driver System VEHICLE SPEED ENGINE SPEED 00 SIGNAL CONDITIONING AND-NOR COMPARATOR CONTROLLER TAV CITAV dt |f AV 0s d0s dt 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 Siqnal Spark Timinq Loqic Gate Output 0- Decreasing 1- Increasing 0- Retarding 1- Advancing 0- Retard the Spark 1- Advance the Spark 0 0 1 0 1 0 1 0 0 1 1 1 And-Nor logic gate which activates the appropriate control law. A dither signal, which is merely a small perturbation applied to the control signal, prevents the control from sticking on a false optimum; For example, if the control had initially reached an optimum which because of disturbances has subsequently shifted, the system could not respond without a small dither to push the system off its old, now false optimum. Figure 4 illustrates the relationship between TdV and 8s correlated for all loads and speeds. The relationship displays the single maximum torque which insures that an adaptive control scheme will achieve the optimum setpoint. Figure 5 illustrates the hypersurface relationship between the four variables TQV/ We,^T , and for a typical internal combustion engine. The throttle effect is to shift the three dimensional surface up and down without affecting its shape appreciably. The surface describing a real system is more complex than that of Figure 5 because many more variables are involved. One of the advantages of adaptive control is that it requires no more than a qualita tive understanding of this hypersurface. Specifically, as long as only one optimum setpoint exists in the operating range this is a sufficient condition to guarantee the realizability of an adaptive controller. The effects of disturbances in variables influencing engine operation will in most circumstances affect the location of the optimum spark timing and air-fuel ratio. Thus adaptive -30 -20 -10 0 +10 SPARK TIMING (0s) (DEGREES FROM OPTIMUM). Figure 4: The Effect of Spark Timing on the Mean Torque Output 13 Figure 5: Hypersurface Illustrating the Relationship Between Engine Variables control will compensate for disturbances in other uncontrolled variables by continuously adjusting the spark timing to its local optimum. This compensation can be illustrated by considering the result of a change in throttle position. As a result of the change in throttle position a change in engine velocity (AWe) will occur as shown in Figure 5. Thus the location of the optimum spark timing fls^will shift byA0S^* The controller will sense this shift by observing an increase in the local gradient (ATdV/A0S) • This gradient will drive the system up to its new optimum. The adaptive control of internal combustion engines depends upon the fact that adjustment of the variables under control will be on the order of only several cycles. Ideally, the control would adjust on a cycle to cycle basis. This fact will require the replacement of the current carburetor and distributor advance units by electronic fuel injection and electronic spark distribution and timing devices. The present carburetor and distributor advance units have response times in t-he order of many engine cycles. These long responsestimes are caused by delays due to long vacuum and manifold connections The electronic devices have the capability of readjusting the air-fuel ratio and spark timing on a cycle to cycle basis thus achieving the desired controller response. 15 The disturbances encountered by the automobile engine ordinarily have long time constants (i.e. the time taken for the disturbed variable to move from its initial value to within 37% of its final value). For example, the disturbances to the opti-mum spark timing 6$ during warmup have time constants in the order of minutes. Other disturbances are caused by acceleration (with time constants determined experimentally to be between 5 and 10 seconds in normal urban driving), aging and wear (time con stants on the order of months) and environmental change (time constants on the order of hours).. The adaptive controller with its ability to adjust the spark timing and air-fuel ratio on a cycle to cycle basis will have little problem in compensating for these disturbances. Some disturbances encountered will have very short time constants and may produce large changes in the location of the optimum setpoint. Such disturbances are caused by driving onto ice or going up a steep hill. The system response time to such disturbances is determined by the controller time constant. Careful design of the control law will minimize this response time. (1.5) Mean Torque Measurement Several methods of measuring the engine mean torque output are possible. One way is to measure the twist in the crank or driveshaft. This involves bonding strain gauge or variable reluctance transducers to the shaft. Disadvantages of this method include the space required, the transmission of the signal off the rotating shaft, cost and reliability. Location of this type of system downstream of the flywheel presents problems because of phase lag and attenuation. A second method of measuring output torque which is used in helicopters consists of a helical gear system. The output shaft of the power plant as well as driving the load also engages a helical gear connected to a hydraulic cylinder and control value. If the output torque of the engine increases the helical gear causes the control pressure to change, thereby repositioning the fuel control spool. The disadvantages of this system include frequency response, size and cost. The frequency response problem is the chief disadvantage. In the internal combustion engine torque changes occur at a frequency of . 150 rad/sec or higher. Preliminary analysis shows that a helical gear system will not respond to changes at this frequency. A third possibility is to measure the mean torque by observation of motion which is dependent on the mean torque (Tav)• Such motions include the flywheel acceleration and the relative angle across the clutch (i.e. the displacement of the driven plate relative to the driving plate). These motions could be measured by using suitable magnetic pickups above the flywheelsteeth or notches on the clutch plates. These signals from the magnetic pickups could then be processed in digital 17 timing and logic circuits to produce the desired control signals. The third possibility is the subject of this 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 in this section. Section 3 presents the deriva tion of an equivalent linear system. In Section 4 the equations of motion developed in Section 3 are solved for the general case. Section 5 presents the solutions of Section 4 for a special case. In Section 6 the equations of motion from Section 2 are simu lated by the IBM 370 computer for the same special case as in Section 4. A comparison of the analytic and simulated solutions is also made in Section 6. Section 7 evaluates potential observ ers for TdV and Section 8 presents the conclusions of the thesis. Section 9 presents recommendations for future work. 18 2 DYNAMIC ANALYSIS OF INTERNAL COMBUSTION ENGINE DRIVER VEHICLE SYSTEMS (2.1) System Dynamics Figure 6 is a schematic representation of the chief dynamic elements and their arrangement in an idealized lumped parameter engine vehicle system. The chief dynamic elements are; the engine inertia (Je) / the clutch spring (Kc) / the clutch damper (nonlinear coulomb damping,Tc)/ the transmission ratio (Rg) / the driveshaft spring (Ks) , the rear end ratio (R), the tire spring (Kt) / the tire damper (Dt) and the vehicle inertia (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 in the following equations of motion: Te = Je0i + Kc(0i-02)+Tc (2.1-1) T3=Rg(Kc(0i-02)+Tc) (2.1-202=Rg03 (2.1-3) T3 = T4=Ks(03- 04) (2.1-404 = R05 (2.1-5T5=RT4 (2.1-6) Figure 6: Lumped Parameter Engine Vehicle System 20 T5 = Kt(05-06) + Dt(05- 06) (2.1-7) Jl 06+Tl+Tb=Kt(05-06) + Dt(05-06) (2.1-8) For a detailed analysis and derivation of the system governing equations including assumptions see Appendix I; (2.2) External Torques The three external torques acting on the system are examined in this section. Assumptions are made which simplify the expressions for these torques. These simplifications are made so that the resulting model may be easily solved analyti cally. (2.2.1) Engine Output Torque (Te) It is shown in [18] that the instantaneous engine torque Te for a four stroke cycle engine can be expressed as a Fourier series; 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 $ is the crank angle (0 = Wet) and TQV is the engine mean 21 output torque. The coefficients CI,Cl,C3,•••/Bi,BI/B3/•••/ are 2" 2 2 2 given in [18] for a single cylinder four stroke cycle engine. The analysis will assume constant values for these torque coefficients. In actual engines this is not generally the case as considerable cycle to cycle variation occurs in practice. In future work the model being developed will be used in testing adaptive control schemes. For the later work disturbances in the coefficients can be introduced to simulate the variations when evaluating adaptive control schemes. For a four cylinder engine (the case of interest in this analysis) the instantaneous output torque is given by summing the torques from individual cylinders. The resulting torque expression is: Te = l + 2C2sin20 + 4C4sin40 + • • • • + 8Bscos80 (2.2.1-2) Tav where terms to the eighth order have been included. Terms higher than the eighth order are much smaller and have much less effect on the instantaneous engine torque. For the purpose of the analytic investigation the expression (2.2.1-2) is simplified further to: ~-=l+Klsin(2We)t (2.2.1-3) lav where KI ^ITlQxlTQV^. 5 for four cylinder four stroke cycle engines [19] andWe^ris the period of the torque fluctuation. The error between the expression (2.2.1-2) and the expression (2.2.1-3) is illustrated in Figure 7. As shown, the Figure 7: Comparison of the Simplified and Numerical Series Engine Torque Signals to approximation made in equation (2,2.1-3) modifies the energy-input timing but maintains approximately the same energy input per cycle. The factor TdV must be investigated to produce an analytic expression for use in the simulated model. Figure 8 illustrates the relationship between engine speed (We) , throttle (<£t) and mean engine output torque (TdV) for the Ford Cortina 2000 c.c. Engine (which will be the engine to which the analysis will be applied). Recall that Figure 4 presents a sim ilar relationship between spark timing (0S) and mean engine output torque (TdV) • Keeping the above relationships in mind a functional.relationship of the following form is suggested: Tdv =Tav( We, <f>i, 0s, A/ F#, T, P, W,£) (2.2.1-4) In this relationship A/F is the air-fuel ratio, \jr is the relative humidity, T is the ambient air temperature, P is the ambient air pressure,W is the engine wear (age), and £ is to account for other factors. In this analysis all effects other than those of speed (We)/ throttle (<£t) and spark timing (Qs) will be ignored. Equation (2.2.1-4) then reduces to: Tav=Tav(We,<f>t,0s) (2.2.1-5) To evaluate (2.2.1-5) it is noted that in Figure 8 for a fixed throttle position there exists a maximum torque (Tpvmox) which occurs at a particular velocity (We^) . Both 24 140 0 100 200 300 400 500 ENGINE SPEED (rad/sec) Figure 8: Mean Torque of the Cortina 2000 c.c. Engine of these variables then are functions of throttle (^t) alone, or: Tavmax = Tqvmax(<£t) (2.2.1-6) We^We*fyjt> (2.2.1-7In the case of the Cortina 2000 c.c. engine they have been eval uated by making plots of TQVITICIXvs <f>\ and We^vs^t. The following relationships result: Tavmax =93.4 + 0.355x<£t (Nm) (2.2.1-8) We =3.86x<£t (rad/sec) (2.2.1-9Assuming the relationship between Tav and We to be parabolic in nature the following equation may be written: 2 Tav(*t,We)=Tavmax [«-CWe{^|#i}] (2.2.1-10) This leaves one remaining constant CWewnich has to be evaluated from Figure 8. The result of this evaluation .is: CWe=0.275 (2.2.1-11) The other factor, the spark timing, is analyzed in the same fashion as the throttle. The first step is determining the spark timing at which the maximum torque occurs. In this analysis it is treated as a function of speed alone: (2.2.1-12) From data for the Cortina 2000 c.c. engine, it is found that: 0s#=(l.48x|O~2)We (Degrees BTDC) (2.2.1-13) With this equation and remembering Figure 4, the following relationship'is proposed: Tav(0s) = Tav(<^,We)[l-CiST(0s- 0s*)-C2ST(0S-0S#)2] (2.2.1-14) whereCISTH.246XI0"4 and C2ST =3.684X|0~3 From the preceding' paragraphs the engine mean torque (TdV) is 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 in Figure 2 the load on a vehicle results from three separate forces; the aerodynamic drag force, the rolling resistance force and the gradient force. An examina tion of these three forces follows. Ah expression for the load torque (TI) is derived from these forces. (2.2.2.1) Aerodynamic Drag Force (Df) The aerodynamic drag force may be calcu lated from the well known relationship: Df=Y-/oV2XCDoAp (2.2.2.1-1) In this relationship CDO is the drag coefficient taken as 0.45 (typical of modern streamlined vehicles), Ap is the vehicle projected area (which is i.53m for the Cortina sedan) and p\l is the dynamic pressure (where p is the air ^ 2 4 v/ density (1.23 N sec /m ) and V is the relative velocity of the air). This results in the following expression for the aero dynamic drag: Df=0.424xv2 (N) (2.2.2.1-2) where V is in m/sec. (2.2.2.2) Rolling Resistance Force (Rf) As shown in [20] the rolling resistance force is given by: Rf=KrXWt (2.2.2.2-1) where Kr is an empirical constant giving the rolling resistance due to mechanical friction, tire inflation and the aerodynamic and pumping effects in the tires in (N drag)/(N vehicle weight). The constant is given as: Kr=0,005 + 1.04X|Q3 + |.2IXV2 (2 2 2 2_2) P P where P is the tire inflation pressure in Pascals and V is the 5 vehicle velocity in m/sec. Consulting Table II, p=l.66x10 (Pa) and Kr becomes: Kr= 0.01126+ 7.3X|0~6V2 (2.2.2.2-3) 28 TABLE II: PHYSICAL CONSTANTS OF THE CORTINA 200 c.c. SEDAN Quantity Symbol Value Units Engine Inertia Load Inertia Clutch Spring Driveshaft Spring Tire Spring Clutch Damper Dynamic Static Linearized Clutch Damping Tire Damper Vehicle Weight Frontal Area Tire Inflation 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 consulting Table II and finding Wt=11100 (N) (the vehicle weight) the final result for Rf can be given: Rf = KrXWt = l25 +0.0809X V2 (N) (2.2.2.2-4) (2.2.2.3) Gradient Drag Force (Gf) The increase (or decrease) in load experi enced by a vehicle climbing (or descending) a gradient is given by the familiar expression: Gf = Wtxsin0i (2.2.2.3-1) where 0i is the incline angle. Therefore, the expression (2.2.2.3—1) can be written: , Gf »llf00xsln6i (N) $2.2.2.3-1) (2.2.2.4) Combination of the Load Force Figure 9 is a free body diagram illustrating the three forces acting on the vehicle. Therefore, the total force acting against the vehicle becomes: F| = Df+ Rf+Gf (2.2.2.4-1) which shows up as an external torque (Tl) at the rear wheels: Tl=F|XRw (2.2.2.4-2) In the expression (2.2.2.4-2) Rw is the wheel radius which is given as 0.3 meters in Table II. As well 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 Illustrating the External Forces on a Vehicle which results in: TI = 37.5 + l.52xiO~2(06)2+3.32xiO3(sin0i) (Nm) (2.2.2.4-4) The first two terms of (2.2.2.4-4) are plotted in Figure 10 for a normal range of operating speeds. The effect of the third term is illustrated by the shifting of the plotted curve upward as indicated in Figure 10. The relationship of (2.2.2.4-4) is nonlinear. The analysis can be simplified by introducing the concept of an operating point (Tl^, 06^). That is, a point at which the system is in equilibrium operation as shown in Figure 10. If a linear impedance of the form: r7_ d(potential) _ dTs° (2.2.2.4-5) d(flow) " d06° is introduced where Ts is 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.52xi0"2 062«l.52xiO*2(06o^3.O4xiO"206o(06-060) (Nm) _0 (2.2.2.4-6for small perturbations (06—06) around, the operating point. The third term of (2.2.2.4-4) may be linearized by recognizing that 0i will generally be small (<5®) and there fore the small angle approximation can be made to obtain: Tg = 3.32xiO3(0i) (2.2.2.4-7) where 0I is in radians. Therefore, the final expression for the load torque (TI) linearized about an operating point (Tl^t06^) is: 33 TI=375 + i.52xiO-2(06o) + 3.O4x|O"2(06°)(06-06o) + 3.32X|O3(0i) (2.2.2.4-8) (2.2.3) Brake Torque (Tb) . The brake torque can be thought of as an increase in the load torque which occurs as a result of the driver's desire to slow down. The simplest model for the brake torque can be given by the expression: Tb=BCIxTbmax (2.2.3-1) -Here BCI is a brake command input (percent of maximum possible brake action). Tb represents the brake torque applied which is a fraction of the maximum available brake torque, TbmOX. The expression (2.2.3-1) assumes the brake action is linear. 34 3 AN EQUIVALENT LINEAR SYSTEM The system described by equations (2.1-1) to (2.1-8) and Figure 6 is a six degree of freedom system which is cumber some to analyze analytically. Variables downstream of 6 I will experience attenuation and phase shifts in the affects caused by TQV. Thus it is desirable to focus primary attention on 8\ and its relationship to TdV. Therefore, by replacing the more complicated system of Figure 6 by the equivalent linearized system of Figure 11 the analysis can be considerably simplified without eliminating the important information. The validity of this simplification is established later by computer simulation of the system of Figure 6 and comparing the results with those obtained from the analytic investigation of Figure 11. (3.1) A Constant Speed System The first step in the derivation of the equivalent system of Figure 11 is to eliminate both the transmission and rear end ratios. All components of the resulting system will rotate at the same speed (the engine speed (We) ) . The elimi nation of the transmission and rear end is accomplished by making the following substitutions: 06 = 06XRXRg (3.1-1) 0|, 0(, 01 06» ^6 i 06 ENGINE EQUIVALENT SPRING DAMPER NETWORK EQUIVALENT LOAD Figure 11: Equivalent Linear Engine Vehicle System 36 Tl' = --^-r (3.1-2) R(Rg) J,'=_J]__ (3.1-3(R(Rg))2 Kt'= Kt o (3.1-4) (R(Rg)) Dt'= T~7^TI2 (3-1"5(R(Rg)) Ks'=-^-r (3.1-6) Rg2 '(-3.2) Equivalent Spring and Damper The next step is to replace the spring-damper network between 01 and 06 by a single equivalent spring (K6Q) and a single equivalent damper (DeO) . This can be done by first recognizing: Te-Je0i=Ti (3.2-1) and Tl' + JlWsTe'sTi (3.2-2) where the components are all rotating at the same speed. The displacement across the clutch, driveshaft and tire (01 — 06) is: 37 (fl,-06')a-Ii+-Tiv|..ll, or (3.2-3) Kc Ks Kt (a,-06') = lL_ (3.2-4Keq Using (3.2-3) and (3.2-4) the following result is obtained: Kaa^ KcKs'Kt' (3.2-5) • (KcKs' + Ks'Kt' + KcKt1) The first step in combining the clutch damper (Tc) and the equivalent tire damper (Dt') is to replace the nonlinear clutch damper with an equivalent linear one. The method is described in [21] and the results are presented here. The equivalent linear damper (Dc) for the coulomb damped clutch is given by: p 4XTp (3.2-6) UC 7TWX0O In this equation Tc is the coulomb damping torque which is given in Table I as ±3.2 (Nm) andWis the frequency of the torque signal (which is twice the engine frequency (2WB) ) . Bo is the amplitude of (Q\ — Q2) and is given by: \/ ( 4TC V (3.2-7) Q .ToyKI v ' UTavKI/ *°""1<c-X . /W.\2 '"\Wn/ In equation (3.2-7) Wn is the natural frequency of the clutch which is given by: Wn^/feq~ (3-2-8) 38 where JeQ is the equivalent system inertia. For this two inertia system, JeQ is given by.:: , JeJl' qs Je+jT (3'2"9) Having arrived at the equivalent linear clutch damping the equivalent system damping can be found. The difference between the rotational speeds of the engine and the load (al-66) must be dissipated in the network between these two elements. Therefore: (0<-$6,) = J^+-£jr °r (3.2-10) (0,_06')= JTi_ (3.2-11Oeq From equations (3.2-10) and (3.2-11) the expression for the equivalent system damping Deq is: nnr,_ DcDt' Deq= Dc+Dt' (3'2"12) (3.3) Equations of Motion of Equivalent Linear System Having completed the above analysis the equations of motion of Figure 11 can be written: Je0i +Deq(0i-06,) + Keq(0i-06,)=Tav(l + KlXsin(2We)t)=Te (3.3-1) JI,6f6, + TI, = Deq(0i-£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) will consist of three parts; the homogeneous (transient) part, and two particular parts. The particular parts are a constant solution and a sinusoidal solution. A detailed analysis of the solutions is presented in 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 is given by: .In these -equations Q is the damped natural frequency and P is the negative exponent of decay. These two variables are given by: 0ci = 0~pT(AcosqT +Bsinqr) (4.1-1) (4.1-2) p=£ Wn (4.1-3) (4.1-4) where £ is the damping ratio and Wn is the undamped natural frequency. £ and Wn\ are given by: f- DeC» (4.1-5) fc Dcr W n *J Jeq Kea(4-1-6Jeq where Deris the critical damping 'cote'ffi<c:rent 'given by: Dcr=2v/jeqKeq (4.1-7) A and B are arbitrary constants which are determined from the system boundary conditions. For example, for a step change in mean torque- (ATdV) the boundary conditions can be given by: a ATOV , rs\ (4.1-8) 0ci =—— T=0) Keq 0CI = O (T = 0) (4.1-9T=© when t=t' and t' is the time when the disturbance exciting the transient occurs. (4.2) The Constant Response The constant response is given by the following equations: aen9 = 9ssci^ssce'=To,v-T,lc' <«-2"l> Je+JI Wengs0ssci=0ssc6,= Wengi+^aeng dr (4.2-2) 41 0eng = 0ssci = 0engi +J*WengdT (4.2-3) In the special case where aeng=0' the results (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 Oscillatory Response The oscillatory response of the system is of the form: 0ssoi = KITavJ I'g^je sln(2Wet-¥i) (4.3-i) 0sso6=KITav lC6l+D6l sin(2Wet-^6') (4.3-2) E62 + F62 In .these solutions Cl,C6,D I, D6,E I ,E6 and Fl,F6 arep^dep.endent-upon the system constants. For a more complete explanation of this solution see Appendix II. The phase angle ¥ is given by: ¥i = inn-'[-&-] (4-3-3) W-ton-'[•£&•] (4-3-442 (4.4) The Total Response The. total response of the system can be calculated by summing algebraically the responses given in Sections (4.1) to (4.3). This is possible because of the linear nature of the system described by equations (3.3-1) and (3.3-2). For example the total solution for 01 is given by summing equations (4.1-1)/ (4.2-3) and (4.3-1) to yield: 0i = *~pT (Acosqr +BsinqT) + 0engi + f Wengdr + (4.4-1) KITav /Q|2 + P'2 sin(2Wet-¥i) -V Er+Fr 43 5 SOLUTIONS OF A SPECIAL CASE (5.1) The Vehicle Considered The analysis of Sections 3 and 4 is now applied to a typical small car system. The system selected was a Ford Cortina automobile. The reason for this choice is that this vehicle is typical of modern compact car design. Table II presents the physical constants for the 2000 c.c. engine Cortina Sedan. Appendix III presents the derivation and determination of these constants. The analysis can be easily extended to other vehicles and engines by substitution of the appropriate physical constants and torque signals, equation (2 . 2 .1-3), for those used in this analysis. (5.2) The Particular Case Considered For the equivalent system of Figure 11 and equations (3.3-1) and (3.3-2), the solutions (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 for a particular case for later comparison purposes with a com puter simulation of the system of Figure 6 and equations (2.1-1) to (2.1-8). The particular case investigated is an initial engine speed for 280 rad/sec (~2800 rpm), in third gear (Rg=1.37) producing a 23.5 Nm mean torque (-18.. 5% of maximum output torque). A 10% step change in throttle position is then made resulting in a new mean output torque of 80 Nm (62% of maximum output torque). The system response to this step change is presented below. (5.3) The Transient Response Consulting Table II and using equations (4.1-3), (4.1-4), (4.1-5), (4.1-6), (4.1-7) the following results are obtained: w n =J-£^=42.4 (rad/sec) (5.3-1) Dcr 2^/JeqKeq Wnd = Wn^/l-C2 =42.2 (rod/sec) (5.3-3) Using these results in equations (4.1-1) and (4..tl-2) the follow ing solutions can be written: 0ci = e"3'68T(Acos42.2T + Bsin42.2T) (5.3-4) 0c6,= ^"3^8TD(Acos42.2T+Bsin42.2T) (5.3-5) The constant D can be evaluated by the method illustrated by equations (All.3-11) and (All.3-12). The other two constants A and B are evaluated by consideration of the"boundary conditions which are the same as those given by (4.1-8) and (4.1-9). The evaluation of these constants produces the follow ing set of results: 0ci = -fi»~3-68T(2.49XIO~' cos42.2T + 2.09XlO"2sin42.2T) (5.3-6) 0ci = ff"3-68T(4.l4 X|02COS42.2T-3.60X10,sin42.2T) (5.3-8) 0cs = e~368T(7.16x10"3sin42.2T +6 20xIO'4COS42.2T)X-I (5.3-9) 0C6,= ff-3-68T(3.O2xiO"",sin42.2r) (5.3-10) 0c6, = ^~3-68T(|,28xiO,cos42.2T-|.O9sin42.2T) (5.3-11) These results (5.3-6) to (5.3-11) imply a time constant: rGSTwn"S°'2714 (S6C) (5 The amplitude of the engine angular.velocity transient is: 0cis* -3.68T (9.80sin42.2T) (5.3-7) |0ci|= 9.80xr3-68T (5.3-13) and the engine angular acceleration transient is,:-]0cil*4l6 x^"3-68T (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/sec2) (5.4-D Equation (4.2-2) can be evaluated assuming aeng is constant for the first second after the step change in TdV to produce: 0ssci =292 (rad/sec) (5.4-2) The assumption of constant angular acceleration aeng is not strictly true because Tic' will increase due to the increase in speed (WenQ) . This increase will reduce the magnitude of aeng which thus does not remain constant. (5.5) The Oscillatory Response For the particular case under consideration the value KITdV is: KITav=200 (Nm) (5.5-1) after the step input. Substitution of the physical constants of Table II into equations (4.3-1) and (4.3-2) will result in the following set of solutions: • 0ssoi = 4.53x10"3 sin (5601 - 0.869°) (5.5-2) 0ssoi=KITav(2We)(2.2 65xiO"5) cos(2Wet-¥i) (5.5-3) = 2.54 cos(560t -0.869°) 47 assoi = -KITav(4We2)(2.265x|0"6)sln(2Wet-^i) (5.5-4) = -I.43X 103sin(560t - 0.869°) dsso6'=6.02x|0"6sin(560t-90°) (5.5-5) 0sso6'=KITav(2We)(3.OlxiO~8) cos(2Wet-¥e') (5.5-6) = 3.37x|0~3 cos(560t-90°) 6fsso6,=KITQv(4We2)(3.0lxiO'8)x-sin(2Wet~^6') (5.5-7) = -l.88sin(560t-90°) The solution (5.5-3) implies a final equilibrium engine angular velocity amplitude of: ISssoil =2.54 (rod/sec) x (5.5-8) and a final equilibrium engine angular acceleration amplitude of: Iflssoil =1420 (rad/sec2) (5.5-9) The phase difference between the change in torque;and the o -5 response is 0.0151 radians (0.869 ) or 5.39x10 sec at the original operating frequency (280 rad/sec)* 48 6 COMPUTER SOLUTION OF THE NONLINEAR SYSTEM The dynamics of the system of Figure 6 and equations (2.1-1) to (2.1-8) are simulated on the IBM 370 Computer using 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) respect ively are included in the simulation. The instantaneous torque (Te) used in the simulation is computed using expression (2.2.1-2) rather than expression (2.2.1-3). The computer programs used are presented in Appendix IV. The simulation is carried out for the same particular case as that given in Section (5.2).to facilitate comparison between the simulated and analytic results. Figure 12 illus trates the throttle input command step. Figure 13 illustrates the response of mean torque (TdV) to the throttle step input. Figure 14 and Figure 15 illustrate respectively the response of the engine rotational velocity and rotational acceleration to the throttle input. The agreement between the analytic and simulated models is quite good as illustrated in Table III. No relative error between the two solutions exceeds 10%. What errors do occur can be explained. The calculations made in calculating both the transient response and the oscillatory response have -Her —1- r 4 - • — — -j — • " ~\i in U | (TO H-OJ —i -"t • OS 0 1 1 M h i i E L 1 4 < 1 3 H 1 I 2, Figure 12: Throttle Command Input to the Computer Simulation Model -Figure 13: Mean Torque Response to the Throttle Command Input o Figure 14: Engine Rotational Velocity Response to the Throttle Command Input Figure 15: Engine Rotational Acceleration Response to the Throttle Command Input TABLE III: COMPARISON OF ANALYTIC AND SIMULATED RESPONSES TO A STEP CHANGE IN ENGINE OUTPUT TORQUE (80-23.5-56.5 (Nm) ) Item Response % Error Analytic/ Simulated i Analytic Simulated Transient Response (9d) Time Constant (1*0) 0.2714(sec) 0.27(sec) 0.5 Damped Natural Frequency (Wfldj 42.2(rad/sec) 42. .r0"( rad/sec) 0.5 Velocity Amplitude 9.80(rad/sec) 9.5(rad/sec) 3.0 Acceleration Amplitude 416(rad/sec) 400(rad/sec) 4.0 Forced Steady State Oscillatory Response (0SSOI) Velocity Amplitude 2.54(rad/sec) 2.40(rad/sec) 5.8 Acceleration Amplitude 2 1420(rad/sec-) 2 1320(rad/sec>) 7.6 Forced Steady State Constant Response (0SSC1) Velocity at T=l.0(sec) 292(rad/sec) 292.3(rad/sec) 2.5 (initial Velocity=280.0 (rad/sec)) assumed that the frequency of the torque signal (2V\fe) has remained constant. As Figure 14 shows this is clearly not the case. This assumption then can account for the error between these two solutions. For example, consider the oscillatory angular velocity amplitude as given by equation (5.5-8) this was calculated assuming We=280 rad/sec but as shown in Figure 14 We ranges from 280 to 294 rad/sec. This variation produces errors in the calculated response relative to the simulated response from 0 to 5%. Another error in the analytic results which has been pointed out previously is the assumption that aeng is constant. This error explains the difference between the analytic and simulated constant velocity for T=1 sec. The analytic model then, represents the dynamics of the simulated system closely. The simulation model can be used to evaluate proposed control laws for adaptive controls. The law would be programmed and the response observed over a typical urban driving cycle. Air-fuel ratio control will require a further extension of the mean torque model given by equation (2.2.1-15). 55 7 MEAN TORQUE INDICATORS Now that the analysis has been verified, the results can be used to investigate possible indicators of the mean torque. Inspection of the solutions (5.5-2) to (5.5-7) and (5.4-1) and (5.4-2) shows that they are all dependent on the mean torque (TQV) • The transient solution (5.3-6) to (5.3-11) is also dependent on TdV for the special case of Section (5.2). A more detailed analysis of the potential observers follows. (7.1) The Transient Response The transient response will be excited by disturbances other than those in TdV thus making the boundary conditions and hence the response independent of TdV* Another problem posed by using the transient response as an indicator of torque is illustrated by Figure 16. The figure shows an initial trans ient followed by another transient beginning in the next cycle. Note that these transients have cancelled each other out. The cancellation means that no indication of; the wrong indication of the change in torque may result from lookinq at the transient response. 56 INITIAL TRANSIENT LATER TRANSIENT TIME Figure 16: Transient Cancellation 57 (7.2) The Constant Response The constant response could be used as an indicator of the mean torque. For example, measuring the gross vehicle accel eration or measuring the steady state constant part of the rela tive clutch angle will provide a signal of the constant response. The problem, with using the gross vehicle acceleration is that it takes too long to see the effect due to a control change because of phase shift. With the control adjustment being on the order of every few cycles this lag will be unacceptable. Another serious problem arises because the constant response is also affected by disturbances in load. The constant response will not, therefore, always indicate how the engine mean torque is behaving because of the interference of these load torque, disturbances. (7.3) The Oscillatory Response The steady state oscillatory response is dependent on the rotational frequency (We) as well as the mean torque (TdV). However, from one cycle to the next little change in speed will occur. As well, the speed changes tend to be in the same direction as the changes in mean torque, thus enhancing the indication of the change. The control would first observe the amplitude of the steady state oscillatory response, then adjust the spark timing and then observe the 58 'steady state oscillatory amplitude of the first cycle after the adjustment. A comparison of the two amplitudes would then be made to determine the input to the controller of Figure 3. The transient can be effectively separated from the steady state oscillatory response by a high pass filter. The transient frequency is always given by the damped natural frequency which in the case of the analysis carried out in this work is 42.6 rad/sec (400 rpm) and by design is kept well below the lowest operating frequency. The lowest frequency of the steady state oscillatory signal is twice the engine idle frequency or 160 rad/sec (1500 rpm). Therefore, there is normally a greater than fourfold separation of frequencies. The steady state oscillatory acceleration has a much larger amplitude than the velocity signal as can be seen from inspection of equations (5.5-3) and (5.5-4). For this reason it will be a much easier signal to measure and changes in TdV will result in larger absolute changes in acceleration amplitude than in velocity amplitude. The merits and drawbacks of all the responses as indicators of TdV are summarized in Table IV. TABLE IV: EVALUATION OF SYSTEM RESPONSE SOLUTIONS AS INDICATORS OF MEAN TORQUE Solution Affected by Disturbances Other than in 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 -Oscillatory • 0SSOI KITav(2.27xiO"5) 0SSOI > No 2We 2WeKlTav(2.27xlO~5) 0.0151 0S SO I 4We2KITav(2.27xl0"5) 60 8 CONCLUSIONS A simplified linear model of the dynamics of the internal combustion engine vehicle driver system has been dev eloped. This model is verified by comparison with a computer simulation model which includes the important nonlinearities. The linear model is shown to be a good approximation of its simulated counterpart. Models describing the engine mean torque, are also developed. The problems of measuring the mean torque are dis cussed. It is established that measuring one of the engine variables will provide the best observer for the mean torque. Other variables downstream of the engine are subject to phase shifts and attenuation. The solutions obtained for these variables from the linear model have been used in the evalua tion and selection of the best engine variable. It is shown that the transient and constant solutions are excited by disturbances in the load torque as well as by disturbances in the engine mean torque. This fact rules them out as potential observers of the mean torque. The evaluation and selection has determined that the steady state oscillatory acceleration response provides the best indication 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 oscillatory acceleration. As well, the structure and quantitative determination of the adaptive controller should be pursued. 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[10] Kehres, John K., and Allan, John J., "Exhaust Pollution Minimization in Small Engines Using Adaptive Digital Con trol", Society 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 Digital Memory Fuel Controller for Gas oline Engines", Society of Automotive Engineers, No. 720282, January, 1972. [13] Soltav, J.P., Senior, K.B., and Rowe, B.B., "Digitally Programmed Engine Fuelling Controlsy, Society of Auto motive Engineers, No. 730128, January, 1973. [141] Rivard, JiG., "Closed Loop Electronic 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 in 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 in Theory and  Practice, Volume lit 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., Fluid Dynamic Drag, Sighard F. Hoerner, Midland Park, N.J., pp. 12-1 to 12-8. [21] Den Hartog, J.P., Mechanical Vibrations, McGraw Hill Book Company, New York, N.Y., 1956, pp. 374-375. [22] ShigEey, J.E., Dynamic Analysis of Machines, McGraw Hill Book Company, New York, N.Y., 1956 64 [23] Davisson, J.A., "Design and Application of Commercial Type Tires", Society of Automotive Engineers, No. 690001, January, 1969. [24] Cortina Shop Manual, Service Publications, Ford of Britain, May, 1972. [25] Baumeister, T., Editor in Chief, Standard Handbook for  Mechanical Engineers, Seventh Edition, McGraw Hill Book Company, 1966. APPENDIX I: EQUATIONS OF MOTION OF AN INTERNAL COMBUSTION ENGINE VEHICLE SYSTEM (AI.l) Purpose The purpose of this Appendix is to derive the equa tions of motion (2.2-1) to (2.2-8) which apply to Figure 6. The important assumptions made in making the lumped parameter system approximation are presented and their validity investi gated. (AI.2) Assumptions The assumptions made in the lumped parameter model are; all inertias other than the engine and load inertias are neglected, all springs in the system other than the tires, driveshaft or clutch are assumed rigid and damping effects other than those of the clutch and tires are ignored. (AI.2.1) Inertia Assumptions The components having the largest inertias next to the engine are the gears, clutch plates and driveshaft. The inertia of a typical gear (Ig), a typical clutch plate (Icp) and a typical driveshaft (Ids) are presented below: Igs — mg-rg* (AI.2.1-1) 66. where : mg=mass of gear=yt> (irTg2)! =1. 54 (Nsec2/m) -3 2 3 p ^density of steel=7.89x10 (Nsec /(m)cm ) r9 -gear radius=5.0ci(;cm) t=gear thickness=2.5 (cm) .". Ig = 0.00194 (Nmsec2/rad) (AI.2.1-2) Icp=-S-mC«rC2 (AI.2.1-3) 2 2 where:mC-mass of clutch=0.72 (Nsec /m) rc=clutch radius=10.79 (cm) .'. Icp=0.00419 (Nmsec2/rad) (AI.2.1-4) Ids=-1-mds(ro2-ri2) (Ai.2.1-5) 2 where:mds=mass of driveshaft=4.4 (Nsec /m) r0=outside diameter of driveshaft=2:.5 (cm) ri =inside diameter of driveshaft=2.35 (cm) .MdS= 0.00016 (Nmsec2/rad) (AI.2.1-6) These inertias are all at least two orders of magnitude 2 smaller than the engine inertia of 0.14 (Nmsec /rad). Therefore, the inertias of these components as well as smaller components may be safely neglected. 67 (AI.2.2) Spring Assumptions The components having the lowest spring rates will generally be those of smallest cross sectional area, greatest length or made of the most flexible material (e.g. tires). The torsional spring constant of shafts in the system is inversely proportional to their lengths. Therefore, shafts such as the crankshaft, will be essentially rigid when compared with the driveshaft. The spring constant for a section of crankshaft is given by: KCS= ^g^05 (AI.2.2-1) 32 I 5 2 2 where: G=shear modulus of steel=8.45x10 (Nsec /(m)cm ) 2 g=acceleration of gravity=9.8 (m/sec ) <JCS=5.699 (cm) I =2.54 (cm) KcS=5.96xl05 (Nm/rad) (AI.2.2-2) This spring constant is two orders of magnitude greater than the driveshaft spring constant of: KS=7800 (Nm/rad) (AI.2.2-3) The assumption that all components of the system other than the tire, driveshaft or clutch are rigid is thus established. (AI.2.3) Damping Assumptions Equation (5.3-2) illustrates that the damping ratio £ is in the range 0.1. Other forms of damping present in the system include structural damping (£=0.01), journal bearing 68 • friction (£=0.001) and gear friction (£=0.001)..The effects of these damping mechanisms are ignored because their contribution to the motion of the system is very small when compared with that of the clutch and tire. (AI.2.4): Limitations of Analysis Several factors have not been considered in this analy sis which may be of importance in later work. Among these factors is the engagement and disengagement analysis of the clutch, the effect of the differential upon the vehicle dynamics and the analysis of slipping 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 is a free body diagram of the engine ideal ized as a pure inertia Je. A torque Te is imposed on this system and torque Tl results. The analysis of this system produces the following result: Te-Je0e = Ti (Ai.3.1-1) (AI.3.2) The Clutch' -A • Figure 18 is a free body diagram of the clutch ideal ized as a parallel spring (KC) and nonlinear coulomb damper (TO. 69 Tav Je Ti Figure lfZ: Free Body Diagram of Engine 01 02 Tl Kc T2 Tc Figure ,-'18: Free Body Diagram of Clutch The torque in TI is equal to the torque out T2 and is given by: Tl = Kc(0l-02)+Tc (AI.3.2-1) where Tc is tne value of the nonlinear coulomb damping. (AI.3,3) The Transmission The free body diagram of Figure .19 illustrates the transmission. The following equations result from this system: 02=03XRg (AI.3.3-1) RgXT2 = T3 (AI.3.3-2where Rg is the gear ratio. .(AI..3.4) The Driveshaft Figure 20 presents the free body diagram of the drive-shaft which is idealized as a spring (KS) . The equations of motion of this 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 is much the same as Figure 19 is the free body diagram of the rear end which is an ideal transformer. The analysis produces the following results: T5 = RXT4 (AI.3.5-1) 04=RX05 "(AI.3.5-2(Al.3.6) The Tires Figure/22' is the free body diagram of the tires. The tires are idealized as a parallel combination of a spring (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)• Analysis of the system shows: T5=T6 (AI.3.6-1) T5 = Kt(05- 06) + Dt(05"06) (Al.3.6-2(AI.3.7) The Load Figure 23 is the free body diagram of the load ideal-, ized as a pure inertia (J|)*-- The inputs to the system are the torque T6 and TI- TI is the load torque which is a function of speed. The investigation of this system provides the following equation of motion: TI+JI06 = T6 ' (AI.3.7-1) T6 Jl TI Figure 23: Free Body Diagram of Load APPENDIX II: SOLUTIONS OF THE EQUATIONS OF MOTION OF THE LINEAR EQUIVALENT SYSTEM (AII.l) Homogeneous Solution The homogeneous equations of motion are: Je 0ci + Deq(0c i - 0ce') + Keq (0ci - 0ce')=O (AII.I-D jr0c6' + Deq(0c6'-0ci)+Keq(0c6'-0ci)=O (AH. 1-2) The solutions of these equations are of the form: 0ci = 0aiXffST (Aii.1^3) 0C6,s0a6'XtfST (MI. 1-4) where S = —p+iq and p Is the negative of the exponent of decay of the amplitude while Q is the damped natural frequency. Substi tuting (AH.1-3) and (All. 1-4) into (All.1-1) and (All.1-2) results in: [jes20oi+Deqs(0ai-0o6,) + Keq(0ai-0a6,)]*ST=O (AH.1-5) [j|,s20a6l+Deqs(0a6,-0ai)+Keq(0o6,-0ai) ]*ST=0 (AH.1-6) Rewriting (AH. 1-5) and (ALL. 1-6) produces the following results 0oi . Deos+Keo (AH.1-7) 006* Jes2+Deqs+Keq 006*_ Jl's^Deqs+Keq (AH.1-8) 0a 1 Deqs+Keq Equating (AH. 1-7) and (AH. 1-8) results in the frequency equa--tioniJ Io ll' 2 ,s + Deqs +Keq=0 (AH.1-9) Je+Jl' Equation (AH.1-9) can be solved to produce the following result -Deq±yDeq2-4(-^.)Keq (AH.1-10) S = Je Jl ^ Je+JI,; This result can be evaluated knowing the numerical values of Je,JI* ,Deq and Keq. The final solution can be written in the form: 0ci=tf~pT(AcosqT + BsinqT) (AII.I-II) 0c6' = De~pT(AcosqT +Bsinqr) (Aii.1-12p_ Deq- (the real part of AH.1-10) (AH.1-13) 2Jeq / Keq Peg2 (the imaginary part of AH. 1-10) V Jeq 4Jeq2 (An. 1-14) The value of D can be calculated from either equation . (All*1-7) or (AH. 1-8) and is: D=—' (AH.1-15) 001 A and B are arbitrary constants which are evaluated from the system boundafyneonditions. 77 (All.2) Constant Solution The steady state constant equations of motion are: Je0ssci +Deq(0ssci - 0ssc6') + Keq(0ssci-0ssc6')= Tav (AH.2-1) jr0ssc6'+Deq(0ssc6,-0ssci) + Keq(0ssc6,-0ssci)=Tlc' (AH.2-2) .where TlC* i;s the constant part of the equivalent load torque... Now the. solution of these equations are,:. 0ssci = 0ssc6l=aeng (AH.2-3) 0sscis0ssc6,=Wengi+</*aengdT (AH.2-40SSCI-0SSC6'= ^ (AH.2-5) where aeng is the engine angular acceleration,Wengi is the Initial engine angular velocity and yfr is the angular deflection. Substituting the results (AH.2-3) to (AH.2-5) into (AH. 2-1) and (AH. 2-2) produces; Tav-Tic' ,aTT 9 ^ aeng = —;——71— (Aii.2-6) Je + J l and . _Tav Je(Tov-Tlc') (AH.2-7) ^"Keq Keq(Je+JI') In the special case where aeng=0 the result (All, 2-7,) reduces to: Tav ^=-7^ (AH.2-8) r Keq 78, (AH.3) Oscillatory Solution The steady state forced oscillatory equations of motion are: Je 0ssoi +Deq(0ssoi-0sso6') + Keq(0ssoi-0sso6')=Tavo (AH. 3-1) ) JI,0sso6'+Deq(0sso6,-0ssol)+Keq(0sso6,-0ssol)=-Tlo, (AH.3-2 In these equations Tlo' represents the fluctuating part of the load torque and TdVO represents the fluctuating part of the engine torque. These two factors are given by:. Tlo' = C2 0SSO6' (All.3-3) and Tavo = KITavsin2Wet (AII.3-4Assuming the solutions of equations (AH. 3-11) and (AH.3-2) are of the form: 0SSOI=|0SSOll*+?2Wet (All.3-5) 0SSO6'=|0SSO6'le+l2Wet (All.3-6) the following equations result: [- Je4We2|0ssoi| +Deq2jWe(| 0ssoi|-| 0sso6'|) + Keq(|0ssoil -10ssoe'l)] *l2Wet= KlTov eI2WET (AII.3-7) [-JI,4We2|0sso6l+Deq2iWe(|0ssp6l-|0ssoi|^ l0ssoi!)] ffi2Wet=-C2 2iWe|0sso6'Ul2Wet (Aii.3-8) These equations can be rewritten to producer. [Keq + Deq2Wei-Je4We2]l0ssoil-[Deq2Wei+Keq]|0sso6'|=KlTav (AH.3-9) 79 [Keq +(Deq + C2)2Wei-JI,(4We2)]l0sso6,l-[Deq2Wei+Keq]|0ssoi|=O (AH.3-10) These equations can be solved by using Cramer's Rule: 0SSOlls KITQV (Deq2Wei + Keq) 0 (Keq+(Deq+C2)2Wei-Jl'4We2) (Keq+Deq2Wei-Je4We2HDeq2Wei + Keq) kDeq2Wei+Keq)(Keq + (Deq + C2)2Wei + Jl'4We2) (AH.3-11) 0SSO6 = (Keq+Deq2Wei-Je,4We2) (Deq2Wei+Keq) KITav 0 (Keq+Deq2Wei-Je4We2) (Deq2Wei+Keq) (Deq2Wei+Keq)(Keq+(Deq+C2)2Wei+Jl'4We2) (AIT.3-12) Once the values of the physical constants and the operating fre quency are known, equations (AH.3-11) and (All..3-12) can be evaluated. The final results will be of the form: 0ssoi = Ai sin(2Wet + ^i) (AH.3-13) 0sso6 = A6'sin(2Wet + ^6') (AH.3-14where Al,A6',\/H and\fr6l are evaluated using equations (AH.3-11) and (AH.3-12) . 80 APPENDIX III: PHYSICAL CONSTANTS USED IN THE ANALYSIS (AIII.l) Purpose The purpose of this Appendix is to present the sources and derivation of the physical constants used in the analysis in Sections 2,3,4, and 5. (AIII.2) Engine Inertia (Je) Figure :24; illustrates the flywheel and crankshaft lay-out in the Cortina 2000 c.c. engine. Figure .25 illustrates a connecting rod, piston pin and piston from the same engine. Both Figures 24, and 25; are somewhat idealized and were constructed from data taken from [(24].. (AIII.2.1) Weight of Piston The weight of thespiston is determined by evaluating the following: WP=-^ [*<d02-di2)hl +»d02h2] (AIII.2.1-1) where the symbols are as shown in Figures 2.4 and 26; except that: -3 2 3 p =density of steel=7.89x10 (Nsec /(m)cm ) 2 g=acceleration 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 Figure 24: Flywheel and Crankshaft Layout of Cortina 2000 c.c. Engine 82 SiOiQM OF PISTON SCALE'.-^-FULL SIZE PISTON PIN Ipin "1 CONNECTING ROD rduo rt3i Figure 25: Connecting Rod, Piston Pin and Piston of Cortina 2000 c.c. Engine ,83 (AIII.2.2) Weight of Piston Pin The weight of the piston pin is given from: yog-Trlpindpin2 Wpin = =2.519 (N) (Ain.2.2-1) (AIII.2..3) Weight of the Connecting Rod The weight of the connecting rod is given by: Wcr^gt3-[ ^DU^^ (AIII.2.3-1) ..Wcr= 5.297 (N) (AIII.2,4) Weight of Crank Journal The weight of the crank journal is calculated from: Wcj = -£p 7rdcj2tcj = 5.0l (N) (AIII.2.4-1) (AIII.2.5) Weight of Crank Pin The weight of the crank pin is given by: Wcp=-£ptrdcp2tcp = 4.171 (N) (AIII.2.5-1) (AIII.2.6) Weight of Small Web) The weight of the small web is given by:; Wsw=/og-lsw-hswdsw=4.175 (N) (Ain.2.6-1) 84 (AIII.2.7) Weight of Large Web The weight of the large web is given by: Wl w= llw-dlw-hlw-/og = 6.089 (N) (Ain.2.7-1) (AIII.2.8) Weight and Inertia of Reciprocating Mass Dividing the connecting rod mass so that one third is reciprocating and two thirds is rotating the total reciprocating weight becomes: Wrm = Wp + Wpin + -l- Wcr=l4.03 (N) (Ain.2.8-1) The moment of inertia of the reciprocating masses is: Irm = -^(Strokex 0.5) = 2.12x10" 3 (Nmsec2) (Ain.2.8-2) (AIII.2.9) Inertia of Rotating Portion of the  Connecting Rod The moment of inertia of the rotating mass of the con necting rod is: Ircr* Wgcr(Stroke)2*5.33XIO-4 (Nmsec2) (Ain.2.9-1) 6g (AIII.2.10) Inertia of Crank Journal The moment of inertia of the crank journal is: Icjs^l£cifs208xI0-4 (Nmsec2) (A111.2.10-1) (AIII.2.11) Inertia of Crank Pin The moment of .inertia of the crank pin with respect to its own centre of mass is: Icp(cm)= W2C^'fCp =I.438X|0~4 (Nmsec2) (AHI.2.11-1) 85 Using the parallel axis theorem this inertia is transferred to the crank journal axis: Icp/cj=Icp(cm)+ WcP rcp/c't2=6.3xi0-4 (Nmsec2) (AIII.2.11-2) 2g (AIII.2.12) Inertia of.Large Web The moment of inertia of the large web relative to its centre of mass is: Ilw(cm)=-J5f- (Iw2+hlw2) = 8.246xi0~4 (Nmsec2) (AIII.2.12-1) I2g The moment of inertia of the large web transferred to the crank journal axis is: Ilw/cj=9.l4xiCT4 (Nmsec2) (AIII.2.12-2) (AIII.2.13) Inertia of Small Web The moment of inertia of the small web is calculated in a similar manner as that of the large web. Its value is: Isw/cj=4.36xi0"4 (Nmsec2) (Ain.2.13-1) (AIII.2.14) Total Inertia Associated With One Cylinder The total inertia associated with one cylinder is found from: Icyl = Irm + Ircr + Icj + Icp/CiJTi-Ilw/cj+Isw/cj = 4.84XlO"3(Nmsec2) (AIII.2.14-1) 86 (AIII.2.15) Weight and Inertia of Flywheel The weight of the flywheel is given as: />g«wdfw2tfw Wfw= r M =120.9 (N) (AIII.2.i5-i) 4 where: df W=28.0 (cm) (flywheel diameter) tfw=2.54 (cm) (flywheel thickness) The moment of inertia of the flywheel is: Wfw-rfw2 Ifw= ' =1.21X10-' (Nmsec2) (AIII.2.15-2) 2g (AIII.2.16) Weight and Inertia of Shaft Between  Crank and Flywheel The weight of the connecting shaft is: Ws=/»g-irrs2ls = 8.769 (N) (Ain.2.16-1) The moment of inertia of this shaft is given by: Ws» 2g Is= Ws>rs2 = 6.46XIQ-4 (Nmsec2) (Ain.2.16-2) (AIII.2.17) The Total Inertia of The Engine The total inertia of the engine is given by: Je=4Icyl + Is + Ifw=O.I4l (Nmsec2/rad) (Ain.2.17-1) (AIII.3) Clutch Spring (KC) and Damper (Tc) Constants The clutch spring and damping constants were determined from experiments with a small (16.5cm in diameter) clutch. The results are then extrapolated to the Cortina clutch (21.5cm in diameter). 87 Figure 26 illustrates the experimental setup. Figure 21-' presents the results of the experiment. The intercept of the straight line is the static damping torque which is: Tcs = 6.0 (Nm) (AIII.3-1) The dynamic damping torque can be calculated knowing the ratio of the dynamic friction coefficient to the static friction coeffi cient for the clutch plate material. This ratio is found to be 0.53 for asbestos [25]. The dynamic damping torque is then: Tcd = 3.2 (Nm) (AIII.3-2) The slope of the straight line given in Figure 2r7 gives the spring constant. Thus the spring constant is given by: Kc=l030 (Nm/rad) (AIII.3-3) As a check, ' the spring constant is calculated. Figure '-28 illus trates the dimensions and arrangement of- the clutch plate used in the experiment. The spring constant of one spring of the clutch plate is: Kc|s£ri=2.32xi05 (N/m) (AIII.3-4) 10 2 where: G=shear modulus of steel5?7 .92x10 (N/m ) A one Newton force acting on this spring is equivalent to a torque of: I Newtonx-^=3.5xiO"2 (Nm) (AIII.3-5) 2 Therefore, the spring constant Kci may be written: Kci = 284.0 (Nm/rad) (AIII.3-6) CLUTCH SUPPORTED IN VISE ECTION Figure 26: Experimental Setup for Determining Clutch Spring and Damping Constants I20-, 0.02 0.04 0.08 DEFLECTION (rod) o.io F,7.aure Xii - Experimental. Se£up^fpr Deterr'-ini;. Figure 2-7: Results of Bthe Clutch Experiment ciutcn Spring and Damping 89 CLUTCH SPRING D n = 5=no. of colls SPRING CIRCLE=dcs=7 (cm) Figure 28: Clutch Plate and Springs The total spring constant is the sum of the individual spring constants or: Kc = 4xKci = ll40 (Nm/rad) (Ain.3-7) Comparison of the results indicates that the.disagreement is less than 10%. Scaling the spring constants linearly with clutch, diameter produces the following result: Kc=l500 (Nm/rad) (AIII.3-8) The damping torque is not affected by any change in size. (AIII.4) Gear.Ratios (Rq) The gear ratios are found in [24]. They are: Gear Ratio 1st 3.65 2nd 1.97 3rd 1.37 4 th 1.00 (AIII.4-1) (AIII.5) Driveshaft Spring Constant (Ks) The driveshaft spring constant is calculated using the dimensions illustrated in Figure 29. The resulting value is: G-rrfdo^di4) /lkl ,. • —321 = 7800 (Nm/rad) (Ain.5-1) (AIII.6) Rear End Ratio (R) The rear end ratio is given in [24] as: R=3.44 (AIII.6-1) '91 l=!ength= 244 (cm) di=4.8 (cm) do=5.0 (cm) Figure 29: Driveshaft Dimensions i 92 (AIII.7) Tire Spring (Kt) and Damping (Dt) Constants The tire spring and damping constants are reported in [23]. The average spring constant is: Kt = 6950 (Nm/rad) (AIII.V-I) and the average damping constant is: Dt = 58.0 (Nmsec/rad) (AM.7-2) (AIII.8) Load Inertia (J|) Figure 30 is a free body diagram of the vehicle and one of its wheels. The force F acting on the wheels can be written: F=/tNr (AM.8-1). where Nr is the portion of the vehicle weight over the rear wheels. The vehicle weight is given in [24] as 11,100 (N). The weight distribution is 45% rear and 55% front. The coefficients of friction between the tire and a dry road is found in [25] to be ^|=0.8. Therefore: F=4995 (N) (AM.8-2) The maximum acceleration achieved by the vehicle is calculated from: 3 amax=—^-=4.41 (m/sec) (AM.8-3) m At the rear wheels, the maximum torque developed is given by: Tmax=RwXF (AM.8-4)' WHEEL Nr VEHICLE DRAG FORCE Figure 30: Free Body Diagram of Vehicle Figure 31: Front View of Cortina Sedan 94 where Rwis the dynamic wheel radius given in [25] as: Rw = 0.2997 (m) (AIII.8-5) for the Cortina tires. The maximum torque is related to the vehicle inertia by: Tmax =JI amax ,(AIII.8-6) where amax=-g71^= 14.71 (rad/sec2) Therefore, the vehicle inertia is given as-: Jl = ^m0x = 102 (Nmsec2/rad) (AIH.S-7) amax (AIII.9) Vehicle Frontal Area (Ap) ' Figure M,is a front view of the Cortina sedan. The frontal area is calculated with the aid of this Figure and is given by: Ap=(hi-h2)xtr = l .53 (m2) (AIII.9-I) AIII.10 Engine Torque Coefficients The engine torque coefficients used in equation (2.2.1-2) are given in 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 this Appendix is to present the com puter program used in the simulation of Section 6. The program is stored in the file ICFILE (a listing of this file appears in this Appendix) under the ID CNTL, The statements in the INITIAL and TERMINAL sections may require revision from run to run. To.execute the program the following sequence of commands is 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 will place the results of the simulation in the file DFILE. 1 MACRO ZZ=NONLINlAACC,I,J,CANVEL,RELTOR) 2 PROCEDURAL __. „ . 3 XX=DELAY(I,J,RELTCR) 4 YY=AES(XX) 5 V,H = DELAY(I, J.CANVEL) 6 VV=ARSm» 7 IF(TIME-PI.LE.O.O) GO TC 107 d IF IVV-O.l) 100, 100, 101 .,„ .. .. 9 100 ZZ = XX 10 IFIYY-6.0I 102,102,103 11 103 IF(ZZ.GT.6.0) GOTO 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 IF(kk.GT.3.0) GO TC 105 19 ZZ=-3.2 20 . _ GO TO.102 : - — -21 105 ZZ=3.2 22 102 CONTINUE 23 ENDMAC 24 *THE ABOVE MACRO DESCRIPTION MODELS THE NONLINEAR CLUTCH DYNAMICS 25 MACRO SS=DL/Y(AABB,N,PA,ANVEL) 26 PPQCEDLRAL „ . . . . . . .... . . _. - -27 W=DELAY(M, PA, AKVEL) 28 V = W 29 IF(TIME-PA.LE.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 CQ=OELAY(N»PKtANGE) 38 111 PP=QQ-NN*2.0*3.1416 39 IFIPP-2.0*3.1416.LE.0.0) GO TO 110 40 NN=NN+1 41 GO TO 111 42 110 TT=2.0*PP 43 ENDMAC 44 *THE ABOVE MACRO DESCRIPTION CALCULATES THE CRANKSHAFT ANGULAR 45 *DISPLACEMENT 46 INITIAL 47 *THIS SECTION DEFINES THE SYSTEM PARAMETERS AND INITIAL CONDITIONS 48 MEMORY NQNL I N 49 MEMORY DLAY 50 MfMORY ANGLE 51 PARAMETER JE=0.14,JL=102.0,KC=1500.0,KT=6950.0,KS=7800.0,DT=58.0•... 5 2 R=3.44,PG=1.37,P1=0.0005,P2=0.001,P3=0.0015,P4 = 0.002,P5=0.0025 .... 5 3 P6 = 0.003,P7=0.00 35, P8 = 0.004,PS=0.0045,P10=0.CO 5,P11 = 0.0055 ,P12 = 0. 006,. . . 54 P13 = 0. 0C6 5,P14 = 0.CO 1,PI 5 = 0.0C75tPI 6 = 0.008 , PI 7=0.0085,PI8=0.009, ... 55 P19=0.0095,P20=0.01,P21=0.010£,P22=0.011,P23=0.0115,P24=C.C12,... 56 P2 5=0. C125,P26=0.C13fP27=Q.0135,P28 = 0.014,P29=0.01451P30=0.015, N2= 1, ... 5 7 C2=1.2,C4 = 0.2,C6=0.05,C8=0. 01,B2=-0. l,B4=-0. C2 ,B6=-0.008,B8=-0.005 58 INCON IC1=280.0, IC2 = 0.126, IC3=0.0132tIC5=0.0,1C6=0.0 59 IC4=IC1/(R*RG) 60 DYNAMIC 61 *THIS SECTION MODELS THE DYNAMICS OF THE SYSTEM 62 FPEC=DLAY( IC1,liPltSlTLC) 63 TMAX=93.4+C.355*THRCT 64 +THROTTLE INPUT 65 THR0T=27.1+10.0*STEP(1.0) 66 S0THE0=3.86*THP0T 67 * AV EP AGE TOP QUE 63 ... . TAV-T^AX*ll.Q-Q.275*1 ( ( FREQ/SQThEQ )r UQ1**2)) 69 SOTOR=ANGLE(N2,S0THE,Plfl) 70 * INST ANT ANEOUS ENGINE TORQUE 71 S0TE=TAV*(1.0+C2*2*SIh(SQT0R)*C4*4*SIN(2*SQTUR)+... . 7 2 B2*2*COS(SOTOR)*-B4*4*C0S(2* SOTORI+ C6*6*SIN(3*SOTOR)•.. 73 C8*8*SIN(4*SOTOR)*E6*6*COS(3*S0TOR)*B8*8*COS( 4*S0TCRJ) 74 *LOAD TORQUE _. _ _ 75 SGTR=37.5+(0.0152*(S1TLO**2))+SOTRD 76 *LOAD TORQUE DISTURBANCES 77 SOTPD=0.0 78 * SY STE M EQUATIONS OF MCTICN 79 S2THE=SOTE/JE+S0THE2*KC/JE-S0TD/JE-KC*S0THE/JE QQ SITHEr INTGRLl IC1» S2THE ) . .. . . 81 SOTHE=INTGRLJIC2,S1THE) 82 S0THE2=IS0TC*(RG**2 )+KC*(RG**2)*S0THE + KS*RG*S0THE4)/.. «3 (KC*(RG**2)+KS) 84 SGTHE 3=S0THE 2/RG 85 S0T4=KS*(S0ThE3-S0THE4) 86 S0ThE4=R*S0Thf5 _ _ _ 87 S1THE5=R*S0T4/DT+KT*S0TL0/DT+SlfLO-KT*SOTHE5/DT B8 S0THE5=INTGPL(IC3.S1THE5) 89 S2TL0=(DT*SlTFE5+KT*S0ThE5-S0TR-DT*SlTL0-KT*S0TL0) /JL 90 S1TL0=INTGRL(IC4,S2TL0) 91 S0TL0=INTGRL(IC5,S1TL092 S1THE2=PERIV(IC1,S0TFE2) 93 S1CLIT=S1THE-S1THE2 94 S0CLUT=S0TFE-S0TFE95 S0T0=NCNLIN(IC6,1,P1,S1CLUT,S0RR) 96 SORR=S0TE-S0TR/(R*RG)-JL*S2TLC/(R*RG)-JE*S2THE 97 SO JE=JE*S2TFE 98 $OCS=KC*(SOCLUT) 99 NOSORT 100 *OUTPUT PRINTER CONTROL 101 TIfEI=TIMEI+DELT 102 IF(TIMEI-2.C*DELT.EQ.0.0) GO TO 117 103 GO TO 118 104 117 TI MEIrO.O 105 hR ITE(6,116) TIME ,TAV,THROT,S1THE,S2THE 106 116 F0PMATI5E12.4) 107 118 CON TIM LE 108 SORT 1C9 TERMINAL 110 *THIS SECTION CONTROLS THE EXECUTICN- PHASE OF THE_ MCJDEL 111 TIMER FINTIM=3.0,PRDEL=0.01,CELT=0.0005 112 METHOD RECT 113 END 114 STOP 115 ENDJOB 116 SFND 

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