An Investigation into Dynamics and Stability of a Powertrain with Half-Toroidal Type CVT

Transcription

1 An Investigation into Dynamis and Stability of a Powertrain with Half-Toroidal Type CVT Zhang N, Dutta-Roy T Mehatronis and Intelligent Systems Group, University of Tehnology, Sydney, Australia Copyright 004 SAE International ABSTRACT This paper presents a study on the dynamis and stability of a onventional powertrain with a Half Toroidal (HT) type Continuously Variable Unit (CVU). A mathemati system model of the powertrain is assembled from parametri finite elements that are formulated from lumped mass, torsional stiffness and damping and varying gear ratios. Simulations have been arried out to investigate the transient behaviour of the powertrain. The damping within the system has been varied to investigate its effet on the dynami stability of the powertrain. The obtained results show that transient responses of input and output rollers of the HT-CVU exist when luth hanges during vehile aeleration period starting from standstill ondition. Severe or even unstable responses of HT-CVU tae plae if the damping is insuffiient in HT-CVU and tyres not only in the initial aeleration period but also in later period after the luth hange. The response of the powertrain beomes stable again when a lo-up luth is applied even if the damping in tyres remains insuffiient. INTRODUCTION Continuously Variable Transmission (CVT) has reently beome an attrative and viable option for vehile power transmission in the automobile industry due to its inherent advantages lie smoothness of operation, easy drivability, powerful aeleration, large range of gear ratios and fuel effiieny. Until a few years ago (a deade ago), issues suh as drivability, ontrol and reliability had been a major drawba off all inds of CVT, limiting their use by ar manufaturers. Though the issues regarding reliability and ontrol have been aggressively dealt with, issues regarding drivability still need to be further improved. Osumi et al [1] have developed a dynami model for the half toroidal variator and hydrauli subsystems that are mated to a powertrain. The results obtained were analyzed for geared neutral ondition, launh and regime hange harateristis. The simulated results show transient for regime hange and orrelates well with the experimental data. ames [] has arried out dynami simulations for a Full Toroidal (FT) type infinitely variable transmission (IVT). The major thrust of his wor is on dynami simulation of the full toroidal variator integrated powertrain and hydrauli ontrol systems for regime hanges. The results of the simulation were experimentally validated by testing the full toroidal type IVT on a hassis dynamometer and demonstrating the quality of regime hange. ones et al [] developed a model for Perbury type FT-CVU and integrated it into a onventional powertrain model for simulation of the ontrol system and fuel onsumption studies. A modular approah has been used to model the CVT powertrain that onsists of individual models for the engine, transmission, vehile, ontrol system and a driver model. The results of the simulation had been obtained for the driving shedule, driver and powertrain ontrol signals and had been ompared with data obtained from testing on an experimental vehile equipped with a Perbury FT-CVT. This wor attempts to study the dynamis and stability of a powertrain integrated with a Half Toroidal (HT) type Continuously Variable Unit (CVU). The dynami model of the HT-CVU integrated powertrain is assembled from parametri

2 finite elements that are formulated using lumped mass and linearized torsional stiffness, damping. The inemati relationship between gear elements has been modeled using gear ratios. Simulations on transient response of the system have been arried out for hard aeleration and luth hange for onstant and multi throttle onditions. Engine harmonis have also been modeled for onstant and multi throttle onditions. Simulations have also been arried out with a soft part lie a lo up luth engaged at high speed range. A pieewise urved gear ratio map within a large speed range has been used for all the simulations. All simulations have been arried out for the vehile starting from stand still. The effet of damping in the tyres and damping within the HT-CVU on the stability of the powertrain has been investigated. It is reasonable to state that the tyre and HT-CVU damping has a signifiant effet on the dynami harateristis and the stability of the system. Simulations also show improved stability if a soft part is inluded between the engine and the HT-CVT of the powertrain for high speed ranges. HT-CVU is onstruted for a single roller. The speed ratios between the input toroid and roller ( R 1 ) and output toroid and roller ( R ) have been modeled using dummy elements ' and ' respetively. The dummy elements have zero inertia value. The ontat between the input toroid and roller and the output toroid and roller has been modeled by using linear ontat torsional stiffness (, ) and linear ontat torsional damping (, ).The gear ratios ( R 1, R ) are: ' R = R = ' 1, φ SIMPLIFIED PHYSICAL AND MATHEMATIC MODEL OF HT-CVU WORKING PRINCIPLE OF HT-CVU- The geometry of a HT-CVU is as shown in Fig 1. The HT- CVU onsists of an input half-toroidal one, rollers and output-half toroidal one. The input and output half-toroidal ones are in ontat with two rollers. Power is transmitted from engine to the input one through an input shaft. A loading am provides the required ontat load between the input toroid and the roller. The loading am an vary the ontat load aording to the torque transmitted by the input shaft. The diss transmit the power from the input toroid to the output toroid by shear ation of the fluid. The fluid in the CVU is an elastohydrodynami lubriant (EHL) in nature. The EHL momentarily turns into a solid under ertain load thus torque is transmitted between the rollers and the toroids. SIMPLIFIED MODEL OF HT-CVU The simplified physial model for the HT-CVU is shown in Fig. The HT-CVU has been disretized into lumped masses, linear spring and damper elements. Typially two rollers are used to transmit power between the input and output half-toroids. However, in this study it is assumed that the rollers are synhronized and lumped mass model for the 1 1 DUMMY ELEMENT 1 Fig 1:HT-CVU Shemati ' INPUT TOROID DUMMY ELEMENT ' ROLLER OUTPUT TOROID Fig : Disrete Mass, Stiffness and Damping Representation of HT-CVU The inertia, damping and stiffness element matries and element o-ordinate vetors for dynami analysis have been obtained by

3 ' ' using the Finite Element (FE) method and are given as: R = 1 ' e 0 R = 1 K e R1 0 R = 1 C e R1 R 1 R 1 e = (1) The gear ratios engaging gears, halftoroidal and rollers are determined as follows: ' ' 6B 7' R1 =, R =, R =, R 4 =, R 5 = 6A 7 8B 8A R = ' e 0 R = K e R 0 R = R C e 4 R R e = 4 () T eng INPUT TOROID ENGINE 1 ' DUMMY ELEMENT 1 ROLLER OUTPUT TOROID PLANETARY GEAR-SET B 8B 10 9 T WHEEL r These element dynami oeffiient matries vary when the speed ratio hanges and therefore, the HT-CVU is a parametri subsystem. The mathematial representation of the HT-CVU in equations (1) and () will be integrated into the dynami model of the ompleted powertrain. Due to la of experimental and published data the values of the ontat torsional stiffness (, ) for surfae ontat between the toroids and rollers have been assumed to be the same value for third luth lo-up taen from Zhang et al [4]. The values for ontat torsional damping (, ) have been taen from Boedo and Freyburger [5] and Patterson [6]. POWERTRAIN MODEL The powertrain onsists of an engine, a HT-CVU, luthes and planetary gear sets whih are used to expand the effetive gear ratios available for all operating onditions enountered during normal woring of the vehile, driveline, omponents inluding wheels and tyres. The HT-CVU is loated between an engine and luthes and planetary gear sets of a onventional powertrain. For torsional vibration analysis, the model of a ompleted powertrain an be represented by onneted lumped masses, torsional springs and damping elements and is shown in Figure. The differential is modelled with the independent o-ordinates of the pinion and the planetary gear train is modelled with the independent o-ordinates of the sun gear. The gear G1 is modelled with the independent o-ordinates of the drive pinion. ' DUMMY ELEMENT CLUTCH ' 7' GEAR G1 Fig :Lumped Mass Model of HT-CVU Integrated Powertrain EQUATIONS OF MOTION Finite element method is used to determine the equation of motion of the omplete powertrain. Nine elements represented in terms of mass moment of inertia oeffiient matrix, torsional stiffness oeffiient matrix and damping oeffiient matrix are obtained from individual element design speifiations and gear ratios. There are ten independent oordinates is the powertrain and the equations of motion of the system are obtained by assembling the mass, stiffness and damping oeffiient matries of the nine elements and is in the form: && + C & + K = T Where the global o-ordinate and global torque vetors are as follows: T = { } T= {T eng T r T r } T The equation of motion given above is a set of oupled seond order differential equation. The tehnique of order redution is used to redue the set of seond order differential equation to a set of oupled first order differential equation. The redued form of seond order oupled differential is given by: 7 7 8B 8B 8A A DIFFERENTIAL 9 9 T r WHEEL

4 . φ = [ ] nxn nx1 where, A nx1 0nxn [ A] 1 nx n nxn K = the system matrix 0nxn [ B ] nxn = [] 1 nxn nxn B T φ + [ ] nxn { } nx1 nxn [] [ ] [] 1 nxn nxn [ C] nxn nxn φ nx1 =. is the set of angular nx1 displaement and veloity vetors. φ nx1 =. I is.. is the set of angular nx1 veloity and aeleration vetors { T } is the global applied torque vetor. By inorporating a soft part lie a lo up luth/damper in the powertrain system, suspension of vibration for both the engine and the roller is made possible. This will have a positive effet on the ratio ontrol of the CVU and improve the stability of the powertrain system as a whole. NUMERICAL TRANSIENT ANALYSIS: For the transient analysis based on numerial simulations, the redued first order oupled differential equation an be approximately solved using various numerial methods. The Runge-Kutta 4 th Order method was used to solve the redued first order equation. The foring funtion for the transient analysis is engine torque (T eng ) and onstruted engine torque harmonis. A typial engine torque was taen for a given engine performane from its engine map as shown in Figure 4. FREE TORSIONAL VIBRATIONAL ANALYSIS Setting the value of global applied torque vetor in the state equation as zero, the equation an be solved as a standard eigenvalue problem whih would lead to free vibration analysis of the system. The detailed free vibration analysis for the powertrain has been reported by Dutta-Roy and Zhang [7] and the following findings have been obtained: 1. A hange in the gear ratios ( R 1, R ) of a toroidal CVU leads to a hange in the seond and fourth natural frequenies and their orresponding modal shapes in the low gear ratio (LGR) and high gear (HGR) onditions of the omplete powertrain system.. The hange in torsional ontat stiffness (, ) affets the third, fourth, fifth and sixth natural frequenies as well as the third, fourth and fifth natural frequenies in the LGR and HGR onditions respetively of the omplete powertrain system. There is a signifiant hange in the respetive modal shapes in both the LGR and the HGR onditions to a ertain value of ontat torsional stiffness. Further inrease in ontat torsional stiffness does not affet the modal shapes signifiantly. This indiates that the ontat torsional stiffness of the system is an important parameter in the half toroidal CVT. Fig 4: Engine Map Rolling frition and aerodynami drag are the two major soures of resistane (Tr) ating against the motion of the vehile. The rolling frition is given by: F rolling = f r W F where, rolling resistane fore f r is Rolling fritional is the rolling resistane oeffiient W is the weight of the vehile The rolling resistane oeffiient r f on a onrete surfae is given by [8]:

5 Vvehile.5 f r = f 0 +.4fs ( ) 100 where, f 0 is basi oeffiient f s is speed effet oeffiient V vehile is Linear vehile veloity in miles/hr The semi empirial formulation for the aerodynami drag is given by: The overall gear ratio (produt of R 1, R, and R ) between engine and driving propeller of the powertrain is shown in Figure 6. It should be pointed out that the speed ratio between input and output of an atual HT-CVT hanges smoothly over the whole of 0-19 seonds. The roughness at 1. seonds is due to the approximation of the HT-CVU speed ratio using two pieewise urved map. HT-CVT is the system that onsists of a HT-CVU and planetary gear sets and luthes. D A ρvvehile CDA = where, D A is aerodynami drag ρ is density of air CD is aerodynami drag oeffiient Vvehile is linear vehile veloity A is frontal area of vehile A three pieewise urved gear ratio map within a large speed range, shown in Figure 5, has been used for simulating the HT-CVU ratio hanges in all the simulations. Figure 5 shows that the HT- CVU ratio ontinuously hange from to 0.6 over the first 5.5 seonds of the aeleration period and at 5.5 seond, luth 1 of the transmission was hanged over to luth. Cluth 1 applies over a period of seonds and Cluth applies over a period of seonds. It should be pointed out that the speed ratio between input and output of an atual HT-CVU hanges smoothly over periods of seond and seond. The roughness at 1. seonds is due to the numerial approximation of the HT-CVU speed ratio using two pieewise urved map. Fig 6: Overall Gear Ratio Map of HT-CVT RESULTS AND DISCUSSIONS TRANSIENT RESPONSE DURING ENGINE ACCELERATION Transient responses of a omplete powertrain inorporated with a HT-CVT during vehile aeleration have been obtained numerially for various loading onditions. The simulation has been arried out for a vehile starting from stand-still to a speed of 150 Km per hour. This simulation is to understand the transient behavior of the powertrain aused mainly by the gear ratios hange and luth hange during vehile aeleration. The result for transient responses of onstant and multi throttle ondition is presented here. Constant Throttle Simulation Fig 5: HT-CVU Gear Ratio Map The simulation is arried out for a vehile starting from rest for a onstant throttle position (at 80% throttle position). The foring funtion (engine torque) is taen from the engine map for different engine speeds at 80% throttle ondition. The resistive torques on the wheels is a funtion of the linear vehile veloity and is numerially obtained at eah step.

6 Engine Output Torque and CVT Output Torque to Driveline As an be seen from Fig 7 during the initial 1. ses the CVT output torque to driveline steadily dereases from an initial high value with a steady inrease in the engine output torque. This leads to good aeleration harateristis of the vehile. The roughness at 1. ses is due to the numerial approximation of the HT- CVU speed ratio using two pieewise urved map. Between 1. to 5.5 ses the engine torque remains steady whereas the CVT output torque delines ontinuously. At 19 ses the overall gear ratio is 1 as shown in Fig 6. As it an be seen in Fig 7 engine torque and the CVT output torque are equal to eah other and this reflets the lo-up ondition. (5.5 ses) to 150 mph (19 ses) at the end of the simulation, the CVT torque to driveline ontinues to derease. The aeleration of the vehile also redues during this period. Fig 8: Vehile Speed and CVT Output Torque to Driveline for Constant Throttle (80%) Condition HT-CVU Elements Fig 7: Engine Output Torque and CVT Output Torque to Driveline for Constant Throttle (80%) Condition Vehile Speed and CVT Output Torque to Driveline Fig 8 shows the plot for vehile speed and CVT output torque to driveline at 80% onstant throttle ondition. High value of CVT output torque to driveline during the initial 1. ses leads to good aeleration of the vehile from rest position. The roughness at 1. ses is due to the numerial approximation of the HT- CVU speed ratio using two pieewise urved map. As the vehile speed inreases the CVT output torque to driveline steadily dereases as the overall gear ratio dereases ontinuously resulting in lower aeleration of the vehile. From 0 mph to 68 mph the vehile shows steady aeleration. At 68 mph (5.5 ses) luth hange ours. This leads to disturbane in the vehile veloity. These disturbanes would lead to an imperfet drive feel for the driver of the vehile. As the vehile speed rises from 68 mph The angular veloities of the roller and output toroid are plotted in Fig 9. As an be seen from the Fig 9 during the initial 1. ses (hard aeleration phase) the roller and output toroid angular veloities inrease. At 1. ses, due to the numerial approximation of the HT-CVU speed ratio using two pieewise urved map a roughness an be seen. Between 1. to 5.5 ses the roughness due to the numerial approximation gets damped out. At 5.5 ses the luth hange ours. As a result a transient an be seen in both the roller and output toroid angular veloities. The point of interest in this simulation is the transient in the roller during luth hange. As an be seen this would have an effet on the hydrauli ratio ontrol mehanism of the roller. In this simulation the hydraulis is not inluded, but the interation between the mehanial and hydrauli sub-assemblies of the roller would have a tremendous influene in the ratio ontrol of the HT- CVU. Another point of interest is the inrease of the angular veloity of the roller and output toroid between 1. to 5.5 ses. This reflets the possibility of instability reeping into the HT-CVU during the range of normal operation. The damping within the CVU and the overall damping of the system would definitely have an effet on the dynamis of the system. This point would be disussed in detail in later setions.

7 Fig 9: Angular Veloities of Roller and Output Toroid for Constant Throttle (80%) Condition Multi Throttle Simulation The simulation is arried out for vehile starting form rest and using a throttle map as shown in Fig 10. All other parameters for the simulation remain similar to onstant throttle ondition. The throttle is initially at 10% throttle setting. From rest to 1. ses the throttle is opened linearly to 0% throttle setting. This is done as initially the vehile is experiening hard aeleration. Thereafter the rate of throttle opening is slowed down and the throttle is at 40% throttle setting at 5.5 ses. Between 5.5 ses to 19 ses the throttle is opened from 40% throttle setting to full throttle. driveline steadily dereases from an initial high value due to ontinuous derease in gear ratio from 10 to with a steady inrease in the engine output torque. The throttle varies from 10% to 0% linearly. The roughness at 1. ses is due to the numerial approximation of the HT-CVU speed ratio using two pieewise urved map. Between 1. to 5.5 ses the engine torque remains steady whereas the CVT output torque delines ontinuously as the overall gear ratio dereases ontinuously from to 1.7. The engine torque inreases steadily between 5.5 ses to 19 ses with a steady derease of CVT output torque. At 19 ses the overall gear ratio is 1 as shown in Fig 6. As an be seen in Fig 11 engine torque and the CVT output torque are equal to eah other and this reflets the lo-up ondition. Between 5.5 ses and 19 ses the vehile is in ruise mode with slow aeleration and the throttle linearly inreases from 40% to full throttle setting. Fig 11: Engine Output Torque and CVT Output Torque for Multi Throttle Condition Vehile Speed and CVT Output Torque to Driveline Fig 10: Throttle Map Engine Output Torque and CVT Output Torque to Driveline As an be seen from Fig 11 during the initial 1. ses the CVT output torque to Fig 1 shows the plot for vehile veloity and CVT output torque to driveline for multi throttle ondition. High initial value of CVT output torque to driveline during initial 1. ses is the hard aeleration phase and leads to good aeleration of the vehile from rest position. From rest to 1. ses the throttle is linearly yarying from 10 to 0% throttle setting and the overall gear ratio ontinuously hanges from 10 to. The roughness at 1. ses is due to the numerial approximation of the HT-CVU speed ratio using two pieewise urved map. CVT output torque to driveline steadily dereases with inrease in vehile speed resulting in lower

8 aeleration of the vehile. From 17 mph (1. se) to 50 mph (5.5 ses) the vehile shows steady aeleration. The throttle varies linearly from 0% to 40% throttle setting and the over all gear ratio hanges ontinuously from to 1.7. damping within the system would have an effet on the stability of the system and this is disussed in detail in later setions. Fig 1: Vehile Veloity and CVT Output Torque to Driveline for Multi Throttle Condition At 50 mph (5.5 ses) luth hange ours. This leads to disturbane in the vehile veloity and CVT output torque. The disturbanes in both these parameters would lead to an unsatisfatory drive feel for the driver of the vehile. As the vehile speed rises from 50 mph (5.5 ses) to 15 mph (19 ses) at the end of the simulation yle, the CVT torque to driveline ontinues to derease as the overall ratio hanges from 1.7 to 1 and the throttle linearly varies from 40% to full throttle setting. The aeleration of the vehile also dereases during this period. HT-CVU Elements The roller and output toroid angular veloity are plotted in Fig 1. The roller and output toroid angular veloity inreases during the hard aeleration phase (0-1. ses). The roughness due to the numerial approximation of the HT-CVU speed ratio using two pieewise urved map a roughness an be seen at 1. ses. At 5.5 ses the luth hange ours. The transient due to luth hange an be seen learly in Fig 1. The point to note in the simulation is the transient of the roller during luth hange. Another point of interest is the inrease of angular veloities of the roller and output toroid. This indiates that instability ould reep into the system during normal woring ondition. The Fig 1: Angular Veloities of Roller and Output Toroid for Multi Throttle Condition TRANSIENT RESPONSE DUE TO ENGINE HARMONIC TORQUE The simulation of transient response of the powertrain to engine harmonis has been arried out for onstant and multi throttle onditions. The main fous of the results is to show modifiations in response when the engine harmonis are taen into aount. The engine harmonis is modeled up to the rd order [9]: T eng = T where, Torque mean + Σ(A m Sinm + B m Cosm ) T eng is Resultant Engine Output T mean is Constant Mean Tangential effort A m and B m are Constant Coeffiients of Mean Torque for m th Order m is Order Number is Cran Angle Constant Throttle Simulation HT-CVU Elements Fig 14 plots the angular veloity for the roller and the output toroid for onstant throttle (80%) ondition. As it an be seen from the figure the roller and output toroid angular veloity inreases during

9 the initial 1. ses (hard aeleration phase). The roughness at 1. ses is due to the numerial approximation of the HT- CVU speed ratio using two pieewise urved map. Between 1. to 5.5 ses the roughness is damped out, but the roller and toroid angular veloities ontinue to inrease. At 5.5 ses luth hange ours and the transients are seen. The point to note in the simulation is its omparison with Fig 9. Due the existene of engine harmonis, the foring torque is omposed of rih olletion of various foring frequenies. These frequenies are impressed on the response of both the roller and the output toroid and an be seen learly in Fig 14. The response for the roller and toroid under the same onditions but without harmonis in the engine as shown in Fig 9 do not show the impressed response when ompared to Fig 14. Multi Throttle Simulation The simulation is arried out for vehile starting form rest with engine harmonis modelled in. All other parameters for the simulation remain the same to that of multi throttle ondition simulation arried out for transient response during engine aeleration. HT-CVU Elements Fig 15 shows that the roller and output toroid angular veloity harateristis remain the same as disussed for Fig 1 but with the impression of the foring frequenies of the engine harmonis on the response of the roller and output toroid. The potential problems are the issues of resonane, transient during luth hange and instability as has been disussed in detail above. As these problems have been disussed in detail above, it will not be dealt with again. Fig 14: Angular Veloities of Roller and Output Toroid for Constant Throttle (80%) Condition with Engine Harmonis If the foring frequenies math the natural frequenies of the system it would lead to severe vibration. Another point of interest is the transient response of the roller at luth hange. The transient would effet on the hydrauli ratio ontrol mehanism. For the simulation the hydrauli subsystem has not been modeled, but the interation between the hydrauli and mehanial subsystems would have influene on the ratio ontrol of the HT- CVU. The problem is ompounded if any of the foring frequenies due to the engine harmonis is same or lose to the frequeny of transients for the ombined hydrauli-mehanial system. The results show the build up of roller and output toroid angular veloities between 1. to 5.5 ses and onsequently the possibility of instability within the system. Detailed disussion on instability is inluded in later setion. Fig 15: Angular Veloities of Roller and Output Toroid for Multi Throttle Condition with Engine Harmonis NOTE: The imperfet drive feel during luth hange as shown in Figs 7, 8, 11 and 1, remains when the engine harmonis is modeled. CVT output torque to the driveline shows the same harateristis as in the above mentioned figures with the modifiation of the foring frequenies from the engine harmonis being impressed on it. The results also indiate a possible ase for instability in the roller and output toroid during the normal running onditions of the powertrain. All the above results suggest the damping levels within the powertrain would have a signifiant impat on the dynamis of the powertrain. The effet of varying damping level within the HT-CVU and wheels has

10 been disussed in detail in the next setion. EFFECT OF DAMPING ON DYNAMICS OF HT-CVT POWERTRAINS Damping exists in several powertrain omponents suh as HT-CVU, engine, tyres and its value is usually not nown exatly. The numerial analysis has further been arried out for gaining understanding of the effet of damping on the dynamis of the HT-CVT powertrain. The effet of the damping level in several ey powertrain omponents on the stability of the omplete system operating under various onditions has been disussed. A lo-up luth is engaged in the powertrain during high vehile speed ranges and its effet on the dynami response of the powertrain has also been studied. The result of simulation with suffiient damping level is presented. No instability is seen in this ase. Thereafter the simulation is arried out with insuffiient level of damping in the tyres and HT-CVU. Instability in the input toroid, roller and output toroid is seen in this ase. The simulation is further arried out with a lo-up luth engaged during high speed ranges with insuffiient level of damping in tyres. These simulations will show the importane of having suffiient damping levels for all ranges of operation. The damping ratios for the global modes are also given for all ases. The damping ratio would hange throughout the operating range of the HT- CVT. Therefore it is neessary to have suffiient levels of damping within the system throughout the operating range of the HT-CVT. The numerial solver developed has been tested vigorously to rule out the possibility of instability aused by solver. The simulations are arried out for onstant throttle (80% throttle) ondition. Suffiient Damping in the Powertrain The simulation is arried out with suffiient damping level in the tyres and the HT-CVU. The damping ratio would hange throughout the operating range of the HT- CVT due to the hanges of its speed ratio. The damping ratio for the global modes at the extreme ends of HT-CVU gear ratio range is given in Table 1. The input toroid, roller and output toroid responses for onstant throttle ondition are given in Fig 16. The response shows stability of the HT-CVU throughout its operating range. HT-CVU GEAR RATIO GLOBAL DAMPING RATIO (%) ζ 1 ζ ζ ζ Table 1: Global damping ratios for suffiient damping level in tyres and HT-CVU Fig 16: Transient Responses for HT-CVU Components with Suffiient Damping level in Tyres and HT-CVU Insuffiient Damping in the Tyres This simulation is performed with insuffiient damping in the tyres while eeping the damping level in the HT-CVU the same as in the previous ase. The damping in the other shafts has also been varied. Table gives the damping ratio for the global modes at the extreme ends of HT-CVU gear ratio operation range. GLOBAL DAMPING RATIO (%) HT-CVU ζ 1 ζ ζ ζ 4 GEAR RATIO Table : Global damping ratios for insuffiient damping level in tyres As shown by Fig 17 the instability of the input toroid, roller and output toroid is seen. The response of the input toroid in Fig 17 is ompared with that in Fig 16. The input toroid response dampens out between 0-19 ses due to suffiient damping level in the tyres as shown in Fig 16. After luth hange at 5.5 ses the omparison indiates build up of input toroid response leading to instability of the input toroid during the high speed range (between 1-19 ses). The response of both the roller and the output toriod dampens out between 1. to 5.5 ses in Fig 16 when suffiient damping

11 level exists in the tyres. Also the response of the roller and output toroid dampens out between ses as shown in Fig 16. Comparison of the roller response in Fig 17 with that of Fig 16 during the time period ses shows that the response for the roller ontinues to build up in this period. After luth hange at 5.5 ses, at high speed range (16-19 ses) the response shows a build up after it had damped out initially. This build of roller angular veloity would lead to instability for the roller. simulation. The unstable response for the input toroid, roller and output toroid is seen in Fig 18. For the input toroid Figs 16 and 18 are ompared. The unstable response of the input toroid an be seen for the ondition of insuffiient damping levels in the HT- CVU and tyres as shown in Fig 18 during the high speed range of operation (10-19 ses). Comparison of Fig 16 and Fig 17 for the output toroid indiates inreased response for the output toroid during ses. In Fig 16 for suffiient damping level ondition the response of the output toroid dampens but this is not the ase in Fig 17. This build up indiates instability of the output toroid within the operational range of the HT-CVT. Fig 18: Transient Responses for HT-CVU Components with Insuffiient Damping Level in Tyre and HT-CVU In Fig 16 the response of the input toroid dampens out for all operating ranges of the HT-CVT. Fig 17: Transient Responses for HT-CVU Components with Insuffiient Damping level in tyres Insuffiient Damping in the HT-CVU and Tyres This simulation is performed with insuffiient damping in the HT-CVU and tyres. The damping in the other shafts has also been varied. Table gives the damping ratio for the global modes at the extreme ends of HT-CVU gear ratio operation range. GLOBAL DAMPING RATIO (%) HT-CVU ζ 1 ζ ζ ζ 4 GEAR RATIO Table : Global damping ratio for insuffiient damping level in HT-CVU and tyres Insuffiient damping level is present in the tyres and the HT-CVU for this As shown in Fig 16 for suffiient damping level in the tyre and HT-CVU the response of both the roller and the output toriod dampens out between 1. to 5.5 ses. Also the response of the roller and output toroid dampens out between ses as shown in Fig 16. Comparison of the roller response in Fig 18 with that of Fig 16 during the time period ses shows that the response of the roller ontinues to build up in this period. After luth hange at 5.5 ses, at high speed range (1-19 ses) the response shows unstable after it had damped out initially. Comparison of Fig 16 and Fig 18 for the output toroid indiates unstable response for the output toroid during ses. In Fig 16 for suffiient damping level ondition the response of the output toroid dampens but this is not the ase in Fig 18. This indiates instability of the output toroid within the operational range of the HT-CVT. Insuffiient Damping in the Tyres with Lo-Up Cluth Engaged at High Speed Range

12 The powertrain with a lo-up luth at high veloity range is simulated and the results are shown in Fig 19. The lo-up luth is engaged when the vehile is at 70 mph. The damping level in the tyres is insuffiient with suffiient damping in the HT-CVU is present. As it an be seen from Fig 19, the response of the input toroid during high speed range (1-19 ses) dampens out. During the high speed range the lo-up luth is engaged. In Fig 17 where the Lo-Up Cluth is not engaged the response builds up during 1-19 ses and instability results. Insuffiient damping in the tyre results in build up of roller and output toroid response between ses as an be seen in Figs 17,18 and 19. Also for the high speed range the input toroid and roller show instability in Figs 17 and 18. The damping level in the HT-CVU also affets the stability of the system as an be seen from Fig 18. As shown in Fig 18 if the damping level in HT-CVU is low the input toroid and roller response builds up during high speed range and gets ompounded by the fat that the damping level in the tyre is also low. If we ompare Fig 17 where only tyre damping level is low and Fig 18 where both tyre and HT-CVU damping levels are low, the instability in response for the input toroid and roller for the high speed range are higher in Fig 18 than for Fig 17. CONCLUSION Fig 19: Transient Responses for HT-CVU Components with Insuffiient Damping Level in Tyre and Lo-Up Cluth Engaged at High Speed Range Comparison of the roller response in Fig 19 with that of Fig 17 during the time period 1-19 ses shows that the roller response does not build up at high speed range (1-19 ses) whereas roller response in Fig 19 during the time period ses is the same as in Fig 17. During this period the roller response builds up. The response of the output toroid between ses remains the same for both Figs 17 and 19. Between ses the Lo-Up luth is not engaged. The output toroid response does build up in this period and ould result in instability in the system. COLLECTIVE NOTES/DISCUSSION The damping level in the tyre is of importane as an be seen by omparing Figs 17, 18 and 19 with Fig 16. If suffiient damping in the tyre is present as is the ase in Fig 16 the response of input toroid, roller and output toroid dampens out between ses. This paper presents a numerial investigation of the dynamis and stability of a powertrain that uses a HT- CVU in its transmission omponent. Detailed desription of the derivation of the mathematial model of the powertrain system that onsists of parametri elements due the ontinuous gear ratio hange was provided. Results of free torsional vibration analysis are briefly disussed with respet to various values of the dynami parameters of the HT-CVU. Substantial simulations have been arried out to investigate the transient behaviour of the powertrain and the effet of damping within the system on its dynami stability. The obtained results show that transient responses of input and output rollers of the HT-CVU exist when luth hanges during vehile aeleration period starting from stand-still ondition. Severe or even unstable responses of HT- CVU tae plae if the damping is insuffiient in HT-CVU and tyres not only in the initial aeleration period but also in later period after the luth hange. The response of the powertrain beomes stable again when a lo-up luth is applied even if the damping in tyres remains insuffiient. The presented wor also proves that the developed dynami model, assembled from a number of parametri finite elements in terms of mass, stiffness and damping matries, of a onventional powertrain onsisting of a HT-CVT is onvenient and suffiient for dynami analysis. The numerial simulations help researhers gain in-depth understanding of the dynami

13 behavior of a HT-CVT powertrain. In the presented wor the dynami parameters of HT-CVU were assumed as onstant values for the onveniene however in pratie, the value of these parameters may vary under different woring onditions of HT-CVU. Future wor will fous on the investigation into the influene of varying dynami harateristis of HT-CVU on the dynami behavior of the omplete powertrain. 9. Wilson W Ker. Pratial Solution of Torsional Vibration Problems. 196, Third ed. Chapman and Hall Ltd, London. REFERENCES 1. Osumi T, Kazuhio U, Nobumoto H, Saai M, Fuuma T. Transient Analysis of Geared Neutral Type Half Toroidal CVT. SAE Review 00,, ames IB. Dynami Charateristis of a Split-Power IVT Shool of Mehanial Engineering, University of Bath, ones RP, Kuriger IF, Hughes MTG, Holt M, Ironside M, Langley PA. Modelling and Simulation of an Automotive Powertrain Inorporating a Perbury Continuously Variable Transmission. In Proeedings of the International Conferene on Integrated Engine Transmission Systems, Avon, pp Zhang N, Crowther A, Liu DK, eyaumaran. A Finite Element Method for the Dynami Analysis of Automati Transmission Gear Shifting with a Four- Degree-of-Freedom Planetary Gearset Element. Proeedings of the Institution of Mehanial Engineers, Part D, ournal of Automobile Engineering 00, 17, Boedo S, Freyburger D. A Grease Filled Torsional Coupling for CVT Vehiles. SAE Tehnial Paper Series , Patterson M. The Full-Toroidal Variator in Theory and in Pratie. In Proeedings of the International Conferene on Continuously Variable Power Transmissions, Yoohama, 1996, pp Dutta-Roy T, Zhang N. Effet of Half- Toroidal Type CVU on the Dynamis of a Complete Powertrain: A Parametri Free Vibration Analysis. Proeedings of the Institution of Mehanial Engineers, Part D, ournal of Automobile Engineering 004; 18 (In Print). 8. Gillespie TD. Fundamentals of Vehile Dynamis. 199, Soiety of Automotive Engineers, Warrendale, Pa,