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Simulation studies of vehicle-transmission interactions
Vol. 33, No. 3, pp. 143-153, 1996 Elsevier Science Ltd Copyright Q 1996 ISTVS Printed in Great Britain. All rights reserved 0022m4898/96 $15.00 + 0.00
Journal of Terramechanics,
PII: SOO22-4898(96)00015-8
SIMULATION
STUDIES OF VEHICLE-TRANSMISSION INTERACTIONS*
R. PAOLUZZ1.t G. RIGAMONTI
and L. G. ZAROTTIt
Summar-Within the frame of dynamic simulation techniques, the paper describes a vehicle locomotion model, inclusive of the internal power train and the soil-wheel interface. Two application areas are given as examples. The first refers to hydrostatic transmission sizing. The second treats the system response to fast transients due to external inputs or loads. Copyright 0 1996 ISTVS.
INTRODUCTION Dynamic simulation is gaining popularity as an additional tool available to designers to improve the characteristics of physical systems. The recognized benefits of the approach are financial (by saving one or more prototypes and by cheaper planning of experimental tests); and related to performance (better understanding of the sytem behaviour, study of special working conditions or faults and analysis of the system sensitivity to the design parameters)[l]. An additional benefit is less recognized (i.e. the ability of simulation to integrate contributions from different specialized fields). In actual fact, this is the case with vehicle locomotion, a complex problem which involves two main fields (i.e. soil-vehicle interaction and power transmission). It is easy to understand that they interact and influence each other, but experience shows that their development are almost independent. Consequently, the simulation environment seems to offer a promising opportunity of synthesis and synergy.
SYSTEM MODEL The best way to assemble the overall description of the system is to look at the vehicle firstly from the inside (the chain of power generation and transmission) and secondly from the outside (the body dynamics and the soil interaction). The basic scheme of the internal power chain is shown in Fig. 1, and includes the diesel engine, the hydrostatic transmission (pump and motor), the final reduction and two rear-driving wheels. The scheme is *Presented at the 4th Asia-Pacific Regional Conference of the International Society for Terrain-Vehicle Okinawa Heights, Ginowan, Okinawa, Japan, 20-21 November, 1995. tTo whom correspondence should be addressed. SCEMOTER-C.N.R., Istituto per le Maccine Movimento Terra e Veicoli Fuori-Strada del Consiglio delle Richerce, Via Canal Bianco, 28, 44044, Cassana, Ferrara, Italy. 143
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Nazionale
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EELS !
DIESEL Fig. 1. Schemeof the vehiclepower chain. not general, because it has the simplest hydrostatic transmission; however, it is a good starting point in view of other architectures (individual motors on wheels, more driving wheels, tracks instead of wheels etc.) because several building blocks are already available from the present model. Internal combustion engine
The engine is diesel, equipped with a speed limiter (governor) which may be continuously set between a minimum and a maximum value. The envelope of the engine operation is shown in Fig. 2. It is bounded by four curves: the two "natural" curves which correspond to the maximum and minimum (zero) injection rate, respectively, and the two w
maximum injection
0
J
0
maximum setting
minimum setting
SPEED zero injection Fig. 2. Mechanicalcharacteristicsof the diesel engine.
Simulation studies of vehicle-transmission interactions
145
"artificial" curves which correspond to the maximum and minimum governor setting. The maximum power is found in point A, and the maximum torque in point B. For a given governor set speed (e.g. point C on the maximum injection curve), the full engine characteristic is made by three parts: (a) the maximum injection curve until point C; (b) the governor curve from point C to point D, located on the zero injection curve; and (c) the zero injection curve from point D (however, there is no way to limit the speed along this curve). Within the engine operating envelope, the isolevels of the specific fuel consumption may also be plotted. (The consumption is computed, but not used in the following examples.) In the model, the dynamic properties of the engine come from the governor response. Without considering its actual implementation (mechanical, electronic etc.) the governor is modelled as a dynamic first order system which outputs a dimensionless injection variable. The system is not completely linear, because both rate of change and value of the injection variable are bounded; in particular, the value ranges from 0 to 1 (where 1 indicates full injection).
Hydrostatic transmission The scheme of the hydrostatic transmission is shown in Fig. 3. It implements the classical closed circuit layout [i.e. a variable displacement pump (1) and a fixed or variable displacement motor (2) connected by high and low pressure branches (main lines)]. The pump is driven by the engine at the same speed (although a speed ratio is possible), and the motor drives the final reduction gear. The main units are complemented by the standard boost circuit [pump (3), check valves (4) and (5), relief valve (6)], the pressure safety
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INTERNAL CONTROLS Fig. 4. Blockschemeof the displacementcontrol(pump or motor). block [cross relief valves (7) and (8)], and the bleeding block [shuttle valve (9) and relief valve (10)] which interacts with the boost block to control the low pressure line. For the sake of simplicity, the scheme does not include the fluid conditioning components (filters and heat exchanger). The valve characteristics are approximated in the steady state form, because detailed dynamics would be of little advantage. The flow and torque losses in the pump and motor (primary units) are modelled by nonlinear expressions which depend on pressure, speed and displacement, and are added to or subtracted from the ideal torque and flow. The model of losses (similar to those described in [2]) are also able to handle any reversed functional conditions i.e. when the role of pump and motor are exchanged, and the power flow changes from load to engine, or from load and engine to the cross valves. The dynamic properties of the hydrostatic transmission are focused on the compressibility of the connecting volumes, and the response of the primary units to displacement inputs. To this end, each unit is controlled by a first order dynamic system which collects inputs from the outside (the operator or an automatic supervisor) and the inside (typically, the automatic pressure limiter and the torque limiter). The block diagram of the system is shown in Fig. 4. Due to the internal end stops, the value of the dimensionless displacement variable ot is bounded (generally from + 1 to -1 in the pump, from a positive minimum, to + 1 in the motor). The rate of change of ot is also bounded, because it is strictly related with the flow consumption of the control system (this parameter is computed in the model, though not used in the following examples). The actual displacement variable and the signal requested by the internal controls are subtracted from the reference signal a~et. The resulting signal goes through the amplification stage 1 and the integration stage 3, whereas stages 2 and 4 work as limiters on the rate of change and value. Vehicle and wheels
The vehicle motion is supposed to be symmetrical (or two-dimensional) on a flat surface with positive, negative or null slope. The dynamic equilibrium is described in two parts. The first refers to the vehicle body under the action of internal and external forces (mass forces, wheel reactions and drawbar pull) and leads to two equations: the linear acceleration of the centre of gravity parallel and perpendicular to the direction of movement, and the angular acceleration around the centre of gravity. The basic geometric data of the vehicle are collected in Fig. 5. They locate, in particular, the centre of gravity G, and the point X, where the drawbar pull is applied. The second part of the dynamic equilibrium refers to the wheels, which are described by the model shown in Fig. 6. The force
Simulation studies of vehicle-transmissioninteractions
147
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Fig. 6. Model of the soil-wheelinteraction. exchanged between wheel and terrain has two components, Fv and FH. For a rigid terrain, the only effect of the vertical force is the wheel deformation d. The wheel reaction is treated with the Voigt Kelvin model i.e. as a spring and dumper combination[3]. Moreover, the motion resistance is supposed to cause a constant shift e of the vertical force in the direction of motion. The horizontal force is derived from the vertical one through the coefficient of traction (positive or negative), which depends on the wheel slip (the definition of slip is well known and is not recalled here). The qualitative relationship between the coefficient of traction and the wheel slip is shown in Fig. 6 (right) in the positive slip region, and is divided in three parts: a linear one form the origin to point 1, a second linear part from point 2 to point B (100% slip), and an intermediate nonlinear part from point 1 to point 2. The shape of the overall curve is mostly influenced by point A (i.e. position and value of the maximum coefficient). The stable (useful) operating range is from the origin to point A, whereas the range from point A to point B corresponds to an unstable operation, and plays a role only in some conditions. The same remarks apply to
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the negative slip region, where the traction characteristic is supposed to be reflected through the origin. The wheel has an additional interface with the internal power chain or braking system; this is described as a dynamic equilibrium equation which includes a driving or braking torque T.
APPLICATIONS The examples which follow deal with the hydrostatic transmission sizing, and the system response to fast transients. To restrict the number of case studies, all examples are based on the same engine and the same vehicle. The engine has a nominal torque of 220 Nm @ 2500 rpm (about 58 kW) and its governor is always set at the nominal speed. The vehicle has an overall mass of 7000 kg, the wheel base is 2000 mm, the wheel diameter is 500 mm, the centre of gravity is 1000 high and 800 mm before the rear axle. The drawbar pull is applied 1500 mm behind and 400 mm under the centre of gravity. The model has been developed and simulated by means of the Easy5x code[4].
Transmission sizing Sometimes, the technical literature suggests simplified (or first approximation) procedures which help the selection of the proper size of pump(s) and motor(s) for a specific application, and most of them include some data about the engine and the vehicle. The model described here moves to a second level approximation, which allows a more accurate provision of the almost steady state traction characteristics (where the dynamic effects are in general, negligible) and an attempt to optimize the overall performance. The performance of the simplest transmission scheme (variable displacement pump and fixed displacement motor) is shown in Fig. 7. Both pump and motor displacement are 100 cm3/rev, and the pump is equipped with pressure (300 bar) and torque (150 Nm) limiter. The final gear ratio is 35. Plot (a) and plot (b) result from computations made when the maximum traction coefficient is 0.8 at 20% slip. The overall transient (from maximum speed to full stop) lasts about 40 sec. Plot (a) is the drawbar pull characteristic and is divided into three parts: the first (high speed) has the pump at maximum displacement, the second (intermediate speed) has the pump at variable displacement relative to the torque limiter control, and the third (low speed) has the pump at variable displacement according to the pressure limiter control. At very low speed the curve drops due to the torque losses in the hydraulic motor. Plot (b) shows (not in scale) the output power of the whole vehicle. The slip of the driving wheels (not shown) closely follows the shape of plot (a), and its maximum is about 15%; therefore, a margin still exists because the maximum traction coefficient is not reached. (The performance is bounded by the maximum pressure.*) Needless to say, the parameters of the soil-wheel interface are variable and uncertain, and may cause unexpected results. An example is shown in plot (c), where the only difference is that the maximum (traction) coefficient is decreased from 0.8 to 0.6. Plot (a) is followed with a small gap until point A, where the coefficient enters the unstable region shown in Fig. 6, and the traction performance degenerates (and is no longer steady state). *It is interesting that in some applications the opposite happens, i.e. the maximum traction is used to bound the pressure within the transmission.
Simulation studies of vehicle-transmissioninteractions
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Further developments of the transmission circuit are made possible by a variable displacement motor. There are two ways to use the new degree of freedom: maintain the same displacement to extend the operating range, or reduce the displacement to save cost and size (with fewer benefits in terms of operating range). In any case one has to deal with the motor speed limits (at maximum or partial displacement), which do not allow the displacement to be decreased at will. The decision made here is to change the pump displacement from 100 to 70 cm3/rev, and the motor displacement from 100 to 80 cm3/rev. The pressure and torque limiter are still active and have the same setting points. The minimum motor displacement is set at 55% which keeps the speed slightly lower than its limit. At the same time, the final reduction ratio becomes 45 to assure approximately the same traction capability. In Fig. 8 two traction curves are shown. Plot (a) corresponds to the minimum motor displacement and plot (b) to the maximum. In between, an infinite number of intermediate curves define the operating envelope of the transmission. Compared with Fig. 7, the maximum speed increases and the high traction potential is similar. Plot (c) is the output power at minimum motor displacement and plot (d) at maximum displacement.
Dynamic transients The locomotion model is intrinsically dynamic, and this feature is fully exploited when the analysis of the time response is dominant. Two examples illustrate this application
150
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area. In the first, the hydrostatic transmission (sized as in Fig. 7) receives a request to change the pump displacement, from a very small value (+2.5% in terms of a dimensionless variable) to a high value ( + 80%), and then back to + 20%; both changes occur in 0.5 sec. In Fig. 9 the pressure levels are compared, as recorded in the main lines of the transmission. Plot (a) refers to the pump delivery line, and plot (b) to the pump suction line. Plot (a) shows a fast increase during the vehicle acceleration, which drops when the steady state condition is almost settled. The opposite happens during the vehicle deceleration (dynamic braking). The variations of the low pressure level (controlled by the boost and bleeding circuit) are due to the fluid compressibility effect during the fast displacement changes, and the mutual override of valve 6 and 10 (Fig. 3) when the high pressure shifts from one line to the other. In the second example, the transient is driven by a fast increase of the drawbar pull up to a value higher than the maximum vehicle capability, which causes the vehicle to stop. In the specific case, the external force reaches 35 kN in 20 ms. In Fig. 10 the system response is shown for two values of the maximum traction coefficient: in the first case the maximum coefficient is 0.8 (upper diagram), in the second it is 0.6 (lower diagram). Both diagrams have two plots: plot (a) gives the true vehicle speed, and plot (b) the ideal vehicle speed, i.e. the wheel rotational speed transformed through the effective wheel radius. (In practice, the true and ideal speed define the wheel slip.) In the first case, the applied pull is slightly higher than the maximum tractive capability of the vehicle (Fig. 7). Consequently the transmission reaches the pressure limiter setting, and from that point the vehicle
Simulation studies o f vehicle-transmission interactions
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slowly decelerates almost to a stop. At the end, the wheel slip is less than 20% because of the approximations used to model the external force around zero vehicle speed. In the second case, the pull is much higher than the maximum tractive capability of the vehicle; the maximum traction coefficient is reached and left, but large fluctuations take place between the stable and unstable region (Fig. 6), which resemble the so called "power hop". At the end, the wheel slip goes to 100%, and the vibration decreases. The "power hop" condition depends on several parameters (e.g. wheel inertia and stiffness) and its simulation is not easy. The results shown in Fig. l0 are to be considered as qualitative trends.
CONCLUSIONS The experience gained with the complete model of the vehicle system suggests the conclusion that, once proper tools are available, it is possible to build and handle the complexity of a reasonably realistic application (the previous examples depend on an average of 180 parameters). The building phase in particular takes advantage of simpler subsystems which are tuned separately before being assembled. The complexity is not an obstacle, but gives evidence to new problems or gives a fresh look at known problems.
152
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Simulation studies of vehicle-transmission interactions
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REFERENCES 1. Storey, S. F., Dynamic simulation cost benefits and design advantages: Proceedings of the 6th European ISTVS Conference, Vienna, Austria. International Society for Terrain-Vehicle Systems, 1994, I, 341-345. 2. Rydberg, K. E., On performance optimization and digital control of hydrostatic drives for vehicle applications. Linkoping Studies in Science and Technology, Dissertations, 1983, No. 99. 3. Kutzbach, H. D., Investigations on tractor tyres test stands and results: Proceedings of the 6th European ISTVS Conference, Vienna, Austria. International Society for Terrain Vehicle Systems, 1994, I, 219-237. 4. Harrison, J., Tollefson, J. and Gronewold, D., EASY5x Users Guide. Boeing Computer Services, Seattle, 1992.
Journal of Terramechanics,
PII: SOO22-4898(96)00015-8
SIMULATION
STUDIES OF VEHICLE-TRANSMISSION INTERACTIONS*
R. PAOLUZZ1.t G. RIGAMONTI
and L. G. ZAROTTIt
Summar-Within the frame of dynamic simulation techniques, the paper describes a vehicle locomotion model, inclusive of the internal power train and the soil-wheel interface. Two application areas are given as examples. The first refers to hydrostatic transmission sizing. The second treats the system response to fast transients due to external inputs or loads. Copyright 0 1996 ISTVS.
INTRODUCTION Dynamic simulation is gaining popularity as an additional tool available to designers to improve the characteristics of physical systems. The recognized benefits of the approach are financial (by saving one or more prototypes and by cheaper planning of experimental tests); and related to performance (better understanding of the sytem behaviour, study of special working conditions or faults and analysis of the system sensitivity to the design parameters)[l]. An additional benefit is less recognized (i.e. the ability of simulation to integrate contributions from different specialized fields). In actual fact, this is the case with vehicle locomotion, a complex problem which involves two main fields (i.e. soil-vehicle interaction and power transmission). It is easy to understand that they interact and influence each other, but experience shows that their development are almost independent. Consequently, the simulation environment seems to offer a promising opportunity of synthesis and synergy.
SYSTEM MODEL The best way to assemble the overall description of the system is to look at the vehicle firstly from the inside (the chain of power generation and transmission) and secondly from the outside (the body dynamics and the soil interaction). The basic scheme of the internal power chain is shown in Fig. 1, and includes the diesel engine, the hydrostatic transmission (pump and motor), the final reduction and two rear-driving wheels. The scheme is *Presented at the 4th Asia-Pacific Regional Conference of the International Society for Terrain-Vehicle Okinawa Heights, Ginowan, Okinawa, Japan, 20-21 November, 1995. tTo whom correspondence should be addressed. SCEMOTER-C.N.R., Istituto per le Maccine Movimento Terra e Veicoli Fuori-Strada del Consiglio delle Richerce, Via Canal Bianco, 28, 44044, Cassana, Ferrara, Italy. 143
Systems,
Nazionale
144
R. Paoluzziet al.
EELS !
DIESEL Fig. 1. Schemeof the vehiclepower chain. not general, because it has the simplest hydrostatic transmission; however, it is a good starting point in view of other architectures (individual motors on wheels, more driving wheels, tracks instead of wheels etc.) because several building blocks are already available from the present model. Internal combustion engine
The engine is diesel, equipped with a speed limiter (governor) which may be continuously set between a minimum and a maximum value. The envelope of the engine operation is shown in Fig. 2. It is bounded by four curves: the two "natural" curves which correspond to the maximum and minimum (zero) injection rate, respectively, and the two w
maximum injection
0
J
0
maximum setting
minimum setting
SPEED zero injection Fig. 2. Mechanicalcharacteristicsof the diesel engine.
Simulation studies of vehicle-transmission interactions
145
"artificial" curves which correspond to the maximum and minimum governor setting. The maximum power is found in point A, and the maximum torque in point B. For a given governor set speed (e.g. point C on the maximum injection curve), the full engine characteristic is made by three parts: (a) the maximum injection curve until point C; (b) the governor curve from point C to point D, located on the zero injection curve; and (c) the zero injection curve from point D (however, there is no way to limit the speed along this curve). Within the engine operating envelope, the isolevels of the specific fuel consumption may also be plotted. (The consumption is computed, but not used in the following examples.) In the model, the dynamic properties of the engine come from the governor response. Without considering its actual implementation (mechanical, electronic etc.) the governor is modelled as a dynamic first order system which outputs a dimensionless injection variable. The system is not completely linear, because both rate of change and value of the injection variable are bounded; in particular, the value ranges from 0 to 1 (where 1 indicates full injection).
Hydrostatic transmission The scheme of the hydrostatic transmission is shown in Fig. 3. It implements the classical closed circuit layout [i.e. a variable displacement pump (1) and a fixed or variable displacement motor (2) connected by high and low pressure branches (main lines)]. The pump is driven by the engine at the same speed (although a speed ratio is possible), and the motor drives the final reduction gear. The main units are complemented by the standard boost circuit [pump (3), check valves (4) and (5), relief valve (6)], the pressure safety
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146
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INTERNAL CONTROLS Fig. 4. Blockschemeof the displacementcontrol(pump or motor). block [cross relief valves (7) and (8)], and the bleeding block [shuttle valve (9) and relief valve (10)] which interacts with the boost block to control the low pressure line. For the sake of simplicity, the scheme does not include the fluid conditioning components (filters and heat exchanger). The valve characteristics are approximated in the steady state form, because detailed dynamics would be of little advantage. The flow and torque losses in the pump and motor (primary units) are modelled by nonlinear expressions which depend on pressure, speed and displacement, and are added to or subtracted from the ideal torque and flow. The model of losses (similar to those described in [2]) are also able to handle any reversed functional conditions i.e. when the role of pump and motor are exchanged, and the power flow changes from load to engine, or from load and engine to the cross valves. The dynamic properties of the hydrostatic transmission are focused on the compressibility of the connecting volumes, and the response of the primary units to displacement inputs. To this end, each unit is controlled by a first order dynamic system which collects inputs from the outside (the operator or an automatic supervisor) and the inside (typically, the automatic pressure limiter and the torque limiter). The block diagram of the system is shown in Fig. 4. Due to the internal end stops, the value of the dimensionless displacement variable ot is bounded (generally from + 1 to -1 in the pump, from a positive minimum, to + 1 in the motor). The rate of change of ot is also bounded, because it is strictly related with the flow consumption of the control system (this parameter is computed in the model, though not used in the following examples). The actual displacement variable and the signal requested by the internal controls are subtracted from the reference signal a~et. The resulting signal goes through the amplification stage 1 and the integration stage 3, whereas stages 2 and 4 work as limiters on the rate of change and value. Vehicle and wheels
The vehicle motion is supposed to be symmetrical (or two-dimensional) on a flat surface with positive, negative or null slope. The dynamic equilibrium is described in two parts. The first refers to the vehicle body under the action of internal and external forces (mass forces, wheel reactions and drawbar pull) and leads to two equations: the linear acceleration of the centre of gravity parallel and perpendicular to the direction of movement, and the angular acceleration around the centre of gravity. The basic geometric data of the vehicle are collected in Fig. 5. They locate, in particular, the centre of gravity G, and the point X, where the drawbar pull is applied. The second part of the dynamic equilibrium refers to the wheels, which are described by the model shown in Fig. 6. The force
Simulation studies of vehicle-transmissioninteractions
147
D1 L.
D
movement
,d, I
t .......i
r
I Fig. 5. Geometricaldata of the vehicle.
wheel
A
-
(-)
]
slip
ter Td
1
---N, e l ~ - -
FH
Fig. 6. Model of the soil-wheelinteraction. exchanged between wheel and terrain has two components, Fv and FH. For a rigid terrain, the only effect of the vertical force is the wheel deformation d. The wheel reaction is treated with the Voigt Kelvin model i.e. as a spring and dumper combination[3]. Moreover, the motion resistance is supposed to cause a constant shift e of the vertical force in the direction of motion. The horizontal force is derived from the vertical one through the coefficient of traction (positive or negative), which depends on the wheel slip (the definition of slip is well known and is not recalled here). The qualitative relationship between the coefficient of traction and the wheel slip is shown in Fig. 6 (right) in the positive slip region, and is divided in three parts: a linear one form the origin to point 1, a second linear part from point 2 to point B (100% slip), and an intermediate nonlinear part from point 1 to point 2. The shape of the overall curve is mostly influenced by point A (i.e. position and value of the maximum coefficient). The stable (useful) operating range is from the origin to point A, whereas the range from point A to point B corresponds to an unstable operation, and plays a role only in some conditions. The same remarks apply to
148
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the negative slip region, where the traction characteristic is supposed to be reflected through the origin. The wheel has an additional interface with the internal power chain or braking system; this is described as a dynamic equilibrium equation which includes a driving or braking torque T.
APPLICATIONS The examples which follow deal with the hydrostatic transmission sizing, and the system response to fast transients. To restrict the number of case studies, all examples are based on the same engine and the same vehicle. The engine has a nominal torque of 220 Nm @ 2500 rpm (about 58 kW) and its governor is always set at the nominal speed. The vehicle has an overall mass of 7000 kg, the wheel base is 2000 mm, the wheel diameter is 500 mm, the centre of gravity is 1000 high and 800 mm before the rear axle. The drawbar pull is applied 1500 mm behind and 400 mm under the centre of gravity. The model has been developed and simulated by means of the Easy5x code[4].
Transmission sizing Sometimes, the technical literature suggests simplified (or first approximation) procedures which help the selection of the proper size of pump(s) and motor(s) for a specific application, and most of them include some data about the engine and the vehicle. The model described here moves to a second level approximation, which allows a more accurate provision of the almost steady state traction characteristics (where the dynamic effects are in general, negligible) and an attempt to optimize the overall performance. The performance of the simplest transmission scheme (variable displacement pump and fixed displacement motor) is shown in Fig. 7. Both pump and motor displacement are 100 cm3/rev, and the pump is equipped with pressure (300 bar) and torque (150 Nm) limiter. The final gear ratio is 35. Plot (a) and plot (b) result from computations made when the maximum traction coefficient is 0.8 at 20% slip. The overall transient (from maximum speed to full stop) lasts about 40 sec. Plot (a) is the drawbar pull characteristic and is divided into three parts: the first (high speed) has the pump at maximum displacement, the second (intermediate speed) has the pump at variable displacement relative to the torque limiter control, and the third (low speed) has the pump at variable displacement according to the pressure limiter control. At very low speed the curve drops due to the torque losses in the hydraulic motor. Plot (b) shows (not in scale) the output power of the whole vehicle. The slip of the driving wheels (not shown) closely follows the shape of plot (a), and its maximum is about 15%; therefore, a margin still exists because the maximum traction coefficient is not reached. (The performance is bounded by the maximum pressure.*) Needless to say, the parameters of the soil-wheel interface are variable and uncertain, and may cause unexpected results. An example is shown in plot (c), where the only difference is that the maximum (traction) coefficient is decreased from 0.8 to 0.6. Plot (a) is followed with a small gap until point A, where the coefficient enters the unstable region shown in Fig. 6, and the traction performance degenerates (and is no longer steady state). *It is interesting that in some applications the opposite happens, i.e. the maximum traction is used to bound the pressure within the transmission.
Simulation studies of vehicle-transmissioninteractions
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9
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Further developments of the transmission circuit are made possible by a variable displacement motor. There are two ways to use the new degree of freedom: maintain the same displacement to extend the operating range, or reduce the displacement to save cost and size (with fewer benefits in terms of operating range). In any case one has to deal with the motor speed limits (at maximum or partial displacement), which do not allow the displacement to be decreased at will. The decision made here is to change the pump displacement from 100 to 70 cm3/rev, and the motor displacement from 100 to 80 cm3/rev. The pressure and torque limiter are still active and have the same setting points. The minimum motor displacement is set at 55% which keeps the speed slightly lower than its limit. At the same time, the final reduction ratio becomes 45 to assure approximately the same traction capability. In Fig. 8 two traction curves are shown. Plot (a) corresponds to the minimum motor displacement and plot (b) to the maximum. In between, an infinite number of intermediate curves define the operating envelope of the transmission. Compared with Fig. 7, the maximum speed increases and the high traction potential is similar. Plot (c) is the output power at minimum motor displacement and plot (d) at maximum displacement.
Dynamic transients The locomotion model is intrinsically dynamic, and this feature is fully exploited when the analysis of the time response is dominant. Two examples illustrate this application
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area. In the first, the hydrostatic transmission (sized as in Fig. 7) receives a request to change the pump displacement, from a very small value (+2.5% in terms of a dimensionless variable) to a high value ( + 80%), and then back to + 20%; both changes occur in 0.5 sec. In Fig. 9 the pressure levels are compared, as recorded in the main lines of the transmission. Plot (a) refers to the pump delivery line, and plot (b) to the pump suction line. Plot (a) shows a fast increase during the vehicle acceleration, which drops when the steady state condition is almost settled. The opposite happens during the vehicle deceleration (dynamic braking). The variations of the low pressure level (controlled by the boost and bleeding circuit) are due to the fluid compressibility effect during the fast displacement changes, and the mutual override of valve 6 and 10 (Fig. 3) when the high pressure shifts from one line to the other. In the second example, the transient is driven by a fast increase of the drawbar pull up to a value higher than the maximum vehicle capability, which causes the vehicle to stop. In the specific case, the external force reaches 35 kN in 20 ms. In Fig. 10 the system response is shown for two values of the maximum traction coefficient: in the first case the maximum coefficient is 0.8 (upper diagram), in the second it is 0.6 (lower diagram). Both diagrams have two plots: plot (a) gives the true vehicle speed, and plot (b) the ideal vehicle speed, i.e. the wheel rotational speed transformed through the effective wheel radius. (In practice, the true and ideal speed define the wheel slip.) In the first case, the applied pull is slightly higher than the maximum tractive capability of the vehicle (Fig. 7). Consequently the transmission reaches the pressure limiter setting, and from that point the vehicle
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slowly decelerates almost to a stop. At the end, the wheel slip is less than 20% because of the approximations used to model the external force around zero vehicle speed. In the second case, the pull is much higher than the maximum tractive capability of the vehicle; the maximum traction coefficient is reached and left, but large fluctuations take place between the stable and unstable region (Fig. 6), which resemble the so called "power hop". At the end, the wheel slip goes to 100%, and the vibration decreases. The "power hop" condition depends on several parameters (e.g. wheel inertia and stiffness) and its simulation is not easy. The results shown in Fig. l0 are to be considered as qualitative trends.
CONCLUSIONS The experience gained with the complete model of the vehicle system suggests the conclusion that, once proper tools are available, it is possible to build and handle the complexity of a reasonably realistic application (the previous examples depend on an average of 180 parameters). The building phase in particular takes advantage of simpler subsystems which are tuned separately before being assembled. The complexity is not an obstacle, but gives evidence to new problems or gives a fresh look at known problems.
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REFERENCES 1. Storey, S. F., Dynamic simulation cost benefits and design advantages: Proceedings of the 6th European ISTVS Conference, Vienna, Austria. International Society for Terrain-Vehicle Systems, 1994, I, 341-345. 2. Rydberg, K. E., On performance optimization and digital control of hydrostatic drives for vehicle applications. Linkoping Studies in Science and Technology, Dissertations, 1983, No. 99. 3. Kutzbach, H. D., Investigations on tractor tyres test stands and results: Proceedings of the 6th European ISTVS Conference, Vienna, Austria. International Society for Terrain Vehicle Systems, 1994, I, 219-237. 4. Harrison, J., Tollefson, J. and Gronewold, D., EASY5x Users Guide. Boeing Computer Services, Seattle, 1992.