Transmission Modeling and Simulation with MapleSim TM Introduction As automotive manufacturers strive to improve the fuel efficiency of their vehicles, they ve focused increasingly on transmission design as one of the key factors in the overall performance of the powertrain. Engineers are putting tremendous effort into determining exactly how power is lost and what can be done to reduce losses and improve overall efficiency. As a result, the transmission industry is now actively involved in optimizing existing designs and exploring new system architectures. These investigations are based on virtual prototypes of transmission systems used in computer simulations. Virtual prototyping can yield more efficient products at significantly reduced costs by allowing engineers to address design issues long before they invest in physical prototypes. Modeling and simulation environments like MapleSim have become critical tools in this process and are being increasingly adopted by powertrain and transmission manufacturers. Maplesoft has developed the MapleSim Driveline Component Library, a collection of components, transmission sub-assemblies, and full powertrain examples that highlight various driveline Figure 1: Powertrain modeling with MapleSim s Driveline Component Library applications. Built with guidance from major transmission manufacturers, the library covers all stages in the powertrain: from the engine through to the differential, wheels, and road loads (See Figure 1). The models created using the Driveline Component Library and other domain libraries of MapleSim are computationally very efficient which make them especially suitable for realtime simulations. One major advantage of the MapleSim Driveline Component Library is that the components are acausal. This allows users to simply connect components together in the required configuration without worrying about issues like torque/speed directions and load-flow. With issues like these resolved automatically by the underlying solver, users can produce complex transmission models more intuitively than with traditional signalflow simulation tools. Components in the Driveline Component Library can easily be switched from ideal (i.e. no losses) to lossy where power losses due to tooth-meshing, bearing friction and slip are accounted for. Incorporating empirical loss data allows users to make design decisions to reduce power losses and improve fuel efficiency based on the in-depth system-level insight provided by the MapleSim problem solving environment.
MapleSim Driveline Component Library Overview The library contains 31 components and 18 example models with additional examples available free online. The MapleSim Driveline Component Library components are grouped in the following categories (Figure 2): CVT and Torque Converter Engines and Dynamometers Loss Elements Differential Gears Vehicle and Tire Gear Selectors Simple Gear Sets Compound Gear Sets Figure 2: MapleSim s Driveline Component Library palette Multi-speed Actuation Components Clutches and Brakes (including One Way Clutch) Internal Combustion Engine Models Depending on the required level of detail, users can include one of the following models in a powertrain system model: Torque Driver Mean-value parameters of this component are shown in Figure 4. This component drives a rotational flange that is the crankshaft of the engine. The crankshaft can be connected to the remainder of the powertrain. Cycle-by-cycle The simple Torque driver model is an ideal starting point for firstapproximations. For a more detailed response, the Mean-value and Cycleby-cycle engine models allow users to customize their own engine parameters for greater fidelity. Mean Value Engine Model A mean-value model of an internal combustion engine provides the overall power/speed/torque output, without considering details like piston motion and ignition (See Figure 5). Typically, these models are used for engine control development, validation, and testing. The model incorporates standard equations for the mass flow rate through the throttle, pressure drop across the manifold, and indicated power from combustion of the fuel, accounting for volumetric and thermal efficiencies. More detail about this model can be found in the Maplesoft white paper, Mean-Value Internal Combustion Engine Model with MapleSim. Torque Driver Engine Model The Engine component of the Driveline Component Library is a simple torque driver model which provides a firstapproximation model for any type of internal combustion engine (spark ignition, diesel, etc.).in MapleSim, it is very easy to connect the drive shaft of the engine to an external mechanical system by simply drawing a connection between the rotational flange ports available on each component. For example, to attach the engine shaft to an inertial load, simply connect the ports of the components to each other as shown in Figure 3. The engine power characteristics are determined by a lookup table for power vs. engine speed data, or torque vs. engine speed data. Various options and Figure 3: Acausal connection: Engine and flywheel Figure 4: Parameters and Setting of the Engine Component Transmission Modeling and Simulation with MapleSim
Cycle-by-Cycle Engine Model This is a very detailed representation of the mechanical, gas-flow, and thermal elements of an internal combustion engine that considers the details throughout the 4-stroke cycle: gas intake, compression, ignition/combustion, and exhaust. As shown in Figure 6, this model includes piston, connecting rod, and crank assembly as multibody components, with a geared connection to the intake and exhaust valve trains, including the cams and the follower/ valve/spring assemblies. Within the combustion chamber, gas flow, compression, combustion, and exhaust Figure 5: Mean-value Engine Model are modeled using standard equations. Lost energy is assumed to generate heat in the cylinder wall and engine block. The model is intended to provide a comprehensive starting point and intuitive framework that can be used asis, but also allows users to add any detail required for a particular application. Figure 6: 3D visualization of the cycle-by-cycle engine model Engine/Transmission Coupling MapleSim includes standard Modelica clutch models, and the Driveline Component Library adds to this with extended and optimized clutches, brakes, and torque converters. Clutches As part of its standard component library, MapleSim provides two clutch models: a standard, controllable friction clutch and a one-way clutch. Both models provide the ability to include a lookup table of friction coefficients vs. relative speed between the friction pad and plate. Axial force on the plate(s) is provided by an input signal. Geometric considerations can also be entered, such as the effect of the inner and outer pad radii and number of pad/plate assemblies. The Driveline Component Library clutch models improve these core components by tracking loss power and adding additional layers of flexibility in specifying the clutch properties. Also, a dog clutch component is included (See Figure 7). function of the speed ratio. Directionality is determined by which port is receiving power. Both forward power flow (i.e. power flowing from pump to turbine) and backward flow (i.e. power flowing from turbine to pump) are possible. Torque Converter The Torque Converter component models a torque converter acting between the two connector ports (Pump and Turbine) as a Figure 7: Clutch and Brake components in the Driveline Component Library www.maplesoft.com
The torque converter is modeled using tables of measured data. The following characteristics are used: Torque Ratio vs Speed Ratio Load Capacity C vs Speed Ratio The required data can be stored and accessed within the component via lookup tables or as shown in Figure 8 can be provided from external calculations via real signal ports. The Torque Converter component supports two alternative formulations based on the following definitions of the load capacity: Figure 8: External data mode of the Torque Converter Component Backward flow, happens during deceleration of the vehicle where the vehicle kinetic energy is transmitted back through the transmission to the engine. In this situation, the turbine is pumping and the pump is acting as a turbine. Since torque converters are not designed to work optimally this way, the torque converter will have very different characteristics. This is accommodated in the lookup table by providing torque ratio and capacity data for >1, typically up to about 5. In the test model shown in Figure 9, the input (pump) torque increases and the transmission (turbine) torque closely matches it with a slight lag. However, due to inherent inefficiencies in the mechanism, the turbine speed is slightly less than the pump speed while the torque is driving the pump. Note that when the input torque drops at t = 15 s, the kinetic energy of the dynamometer changes the torque flow from forward to backward (i.e. turbine drives the pump), and the pump speed drops below the turbine speed. At low speeds, between t = 0 and 4 s, the turbine torque increases faster than the input torque. This is the torque multiplication effect. Figure 9: Torque converter test model Transmission Modeling and Simulation with MapleSim
Transmissions The transmissions section of the Driveline Component Library consists of simple and compound gear sets, related clutch packs, a continuously variable transmission, and example gearing configurations that use these components. The available component are shown in Figure 10. Planetary Gear The Planetary Gear (Figure 11) is made of three gears: the sun gear at the center of rotation; the planet that is mounted on a carrier arm that rotates around the sun and meshes with the sun; and a ring that meshes with the planet and rotates around the same axis as the sun. The base gear ratio is given by r_(r S) which is the ratio of the number of teeth in the ring gear to that of the sun gear. Dual-Ratio Planetary Gear The Dual-Ratio Planetary Gear (Figure 12) is based on the Planetary Gear component. It has a set of two co-rotating planet gears with different radii where the first planet meshes with the sun gear and the second planet meshes with the ring gear. Ring/ Planet(1) and Planet(2)/Sun gear ratios must be provided. Counter-rotating Planetary Gear The Counter-rotating Planetary Gear (Figure 13) configuration adds a Planet-Planet component to the basic planetary gear to represent two planets rotating around the sun and connected on the same carrier. The overall effect is to reverse the rotation of the ring relative to the sun. Ring/Small Sun and Ring/Large Sun gear ratios as well as the ratio of the two planets must be provided. Figure 11: Planetary Gear set Figure 12: Dual-ratio Planetary Gear set Figure 10: Gear components in the Driveline Component Library Figure 13: Counter-rotating Planetary Gear set www.maplesoft.com
Ravigneaux Gear Set The Ravigneaux configuration is a basic automatic transmission planetary assembly. As shown in Figure 14, it uses a large sun gear and a small sun gear, with two independent planetary gears connected to a rotating carrier. The inner and outer planets rotate independently of the carrier, but counter-rotate with a fixed gear ratio. The inner planet meshes with the small sun gear and the outer planet meshes with the large sun gear. The ring gear then meshes with the outer planet and couples the gear-train together. Figure 14: Ravigneaux Gear set The configuration is constructed using three Planet-Planet components and one PlanetRing component, connected as shown in Figure 15. Ring/Small Sun and Ring/Large Sun gear ratios must be provided. Selection of the required gear ratios is achieved by coupling or decoupling the mechanical ports with other shafts or with the gear housing, by means of clutch components. The Ravigneaux Actuation component can be used as shown in Figure 16 to easily create a 4-speed transmission. Simpson Gear Set The Simpson Gear consists of two planetary gears that have connected sun gears. As shown in Figure 17, the rear (driveshaft side) ring gear is connected to the front (engine side) carrier. Front Ring/Sun and Rear Ring/Sun gear ratios must be provided. The Simpson Actuation component can be used as shown in Figure 18 to easily create a 3-speed transmission. Figure 15: Internal structure of the Ravigneaux Gear Figure 16: Building a 4-speed transmission with the Ravigneaux Actuation component Figure 17: Simpson Gear set Transmission Modeling and Simulation with MapleSim
CR-CR Gear Set Figure 19 shows the CR-CR Gear component in the Driveline component library and a schematic view of the gear set configuration. This compound gear set is constructed from two planetary gears. The front carrier is connected to the rear ring and vice versa. Front Ring/ Sun and Rear Ring/Sun gear ratios must be provided. The CR-CR Actuation component can be used as shown in Figure 20 to easily create a 4-speed transmission. Lepelletier Gear Sets The two Lepelletier Actuation components (6-speed and 7-speed) can be used together with a Ravigneaux gear and a planetary gear to create 6-speed or 7-speed transmissions as shown in Figure 21. Figure 18: Building a 3-speed transmission with the Simpson Actuation component Continuously Variable Transmission A continuously variable transmission (CVT) allows a continuous range of gear ratios without the discrete gear changes necessary in other configurations (See Figure 22). Figure 20: Building a 4-speed transmission with the CR-CR Actuation component Figure 19: CR-CR Gear set Figure 22: CVT component (ideal mode) in the Driveline Component Library Figure 21: Building a 6-speed (top) and 7-speed (bottom) transmissions with the Lepelletier Actuation components www.maplesoft.com
Differential The differential provides a means of splitting power provided by an input shaft over two output shafts. Typically, this is used to balance the power to the left and right wheels on the drive axle, particularly when the vehicle is turning so the inner wheel needs to turn slower than the outer wheel. They are also used for other applications, such as power splitting between front and rear in four-wheel drive vehicles. The mechanical configuration can be seen in Figure 23, where the input shaft from the transmission is connected to a bevelled crown wheel, which in turn rotates two gears on a carrier, which then Figure 24: Internal structure of the Differential Gear mesh with gears that are connected to each of the two output shafts. As shown in Figure 24, the differential model uses three Planet-Planet components for the crown/carrier/ planet assembly, meshing with simple gears that drive the output shafts. The driveshaft pinion/crown gear ratio must be provided. Differential is the addition of a clutch component that connects the crown gear to one of the side gears. By activating the clutch, torque is transferred from the faster side to the slower side. Active Differential The Driveline Component Library also includes an active differential configuration known as the Active Limited Slip Differential (ALSD). As shown in Figure 25, the difference between the ALSD and the passive Figure 23: Differential component Figure 25: Internal structure of the Active Limited Slip Differential Gear Transmission Modeling and Simulation with MapleSim
Incorporating Losses The Driveline Component Library contains all the mechanical building blocks necessary to build a driveline system model to the required level of detail. To achieve improved levels of fidelity, the library also includes lossy versions of all the basic components, which allow the incorporation of loss information as component input signals. These signals can either be the results of calculations performed elsewhere in the model, or loss factors interpolated from can also be added using external Bearing Friction components as shown in Figure 28 for a Counter-rotating Planetary Gear. The bearing friction is expressed as a torque loss and is related only to the shaft speed. The non-ideal CVT component accounts for torque loss using the same efficiency approach as gears. This component can also include the belt slip. See Figure 29. Figure 26: Fundamental GUI option for all gears ideal = true/false tables of measured data. This approach means that you have full control over how the loss data is implemented, so that it best matches how you acquired the data from tests. Meshing Efficiency All gear components in the Driveline Component Library have the option to be switched between ideal and lossy modes (See Figure 26). Figure 27: Gear Mesh Leading/trailing edge contact When the modeling parameter ideal is set to false, the component icon changes and a red arrow appears. The list of component parameters also changes and more parameter options are provided. Meshing friction is expressed as a transmission efficiency which may be defined as a function of the gear angular velocity via data tables. As shown in Figure 27, forward (leading teeth edge contact) and backward (trailing teeth edge contact) efficiencies may be defined independently. Meshing efficiency data can be provided by: Figure 28: Adding bearing friction to gear sets Data files (*.csv, *.xls, or *.xlsx). Data files can either be on a hard-drive (external) or an attachment to the MapleSim file (internal). Array parameter via the user interface. Real signal input(s). Bearing Friction In compound gear sets (Planetary Gear, Dual-ratio Planetary Gear, Counterrotating Planetary Gear, Ravigneaux Gear, Simpson Gear, and CR-CR Gear), bearing damping can be added using the component options. Bearing friction Figure 29: Non-ideal CVT component in the Driveline Component Library www.maplesoft.com
Example Powertrain Models This library provides several example transmission models to illustrate how the driveline components can be used to build full powertrain system models: 3-speed Simpson 4-speed CR-CR 4-speed Dual Clutch 4-speed Ravigneaux 6-speed Lepelletier 7-speed Lepelletier Decomposed 6-speed Lepelletier Ideal Multi-Speed Gearbox ZF 4HP22 Transmission Figure 30: Full vehicle model in MapleSim ZF 5HP24 Transmission Case Study: ZF 4HP22 4-speed Automatic Transmission A full powertrain is modeled in MapleSim using the Driveline Component Library as well as other standard components as shown in Figure 30. This model includes a city driving speed profile which is provided via a lookup table containing the desired vehicle forward velocity over time. This example models a specific automatic transmission from a German manufacturer, ZF, coupled to the simple controlled torque engine model via a torque converter. This transmission model is based on a published gear schematic of the ZF 4HP22 that uses three connected planetary gears to provide four gear ratios (see Figure 31). Gear ratios are selected by engaging and disengaging clutches in a specific pattern. As shown in Figure 32, the clutch pattern is implemented using Ratio Selector components from the Driveline Component Library. The gear number is chosen by the Gear Shifter component of the Driveline Component Library which uses up-shift and down-shift engine RPM thresholds. The output from the Gear Shifter goes through a subsystem called Gear Override. To prevent engine stall in this subsystem, the gear is set to zero (neutral) whenever the engine RPM nears the idle speed. Sample simulation results are shown in Figure 33. Figure 31: ZF 4HP22 transmission configuration Transmission Modeling and Simulation with MapleSim
Figure 32: Implementing the Clutch table for ZF 4HP22 transmission Figure 33: Sample simulation results. Left: Desired (drive cycle) and actual vehicle speeds, Right: Engine RPM and selected gear number Real-time Performance Using MapleSim s API commands from Maple, the full vehicle model shown in Figure 30 can be simulated using fixed time-step solvers. Here, the Euler solver with a time step of 0.001 sec (typical of real-time hardware) is selected. The simulation is done on a 64-bit Windows 7 machine with Intel(R) Core(TM) Duo 2.40 GHz CPU. Figure 34 shows Maple commands for this example. These command extract and simplify the model equations and convert them to optimized c-code. The simulation results are obtained from a Maple procedure which includes the complied c-code. Figure 34: Running MapleSim simulation using API commands On average the simulation was done about 12 times faster than real-time (i.e. ~50 second of integration time for a 600-second simulation). Based on 15 consecutive runs the average simulation time was 50.2 seconds with standard deviation of 0.54. Conclusion The MapleSim Driveline Component Library is a comprehensive collection of components and models that provides the fundamental components needed to build and test complex transmission models. The acausal model representation allows you to drag and connect for rapid model development. The open structure of the components makes it very easy to modify them to suit your specific requirements. Custom transmission configurations can be created with very little effort. Furthermore, the open structure and intuitive energy-loss components allow the inclusion of power loss data configured to best reflect the way in which the loss data was acquired. MapleSim, with the addition of the MapleSim Driveline Component Library, provides the most flexible transmission modeling environment available, allowing you to mix the best of physical models and empirical data to maximize model fidelity, optimize your designs and improve overall vehicle fuel-efficiency. www.maplesoft.com
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