Graph-theoretic Modeling and Dynamic Simulation of an Automotive Torque Converter Joydeep M. Banerjee and John J. McPhee Department of Systems Design Engineering University of Waterloo, Waterloo, Ontario NL3G1, Canada e-mail: jbanerjee83@gmail.com) Abstract: A linear graph based model of an automotive torque converter is presented in this paper. The linear graph is used to represent the topology of the system and the connectivity of its components. The model accounts for the dynamic behavior of the hydraulic fluid and that of the mechanical components, i.e. the pump, the turbine and the stator. The governing equations in symbolic form are generated from the model in an automated fashion and are used to simulate the operation of the torque converter under specified input conditions. The model is capable of simulating both forward and backward flow modes. The simulation results are compared and validated using data available from existing literature. Keywords: Graph-theoretic Models, Computer Simulation, Symbolic Equations, Torque Converter, Powertrain. 1. INTRODUCTION Torque converters are used as coupling devices in automobile powertrains involving automatic transmissions. Efficient modeling of torque converters capturing various modes of operation is important for powertrain design and simulation, Hrovat and Tobler 1985), Ishihara and Emori 1966), Lucas and Rayner 1970). In this paper a linear graph model of a torque converter is presented and analyzed. The presented model accounts for the dynamic behavior of the mechanical components, i.e. the pump, the turbine and the stator, and that of the hydraulic fluid used in a torque converter. Conventional modeling methods require manual derivation of equations, which can be tedious and prone to errors, Hrovat and Tobler 1985). By employing linear graph theory to model the behavior of the torque converter, it is possible to generate the governing equations in an automated fashion. Also, using graph-theoretic methods makes it easier to extend the model to include other powertrain components from different physical domains. Our ultimate goal is to develop tools for automated generation of real time simulation codes for complete automotive systems. Currently, very efficient methods are available that can perform accurate simulations of multibody vehicular systems, Schmitke et al. 008). These methods are based on graph-theoretic modeling techniques. However, graph-theoretic models of certain key components of the transmission system are not available at this point. This is precisely the motivation behind the current research. Also, from the stand point of system analysis, a linear graph model of a torque converter would enable us to perform Financial support for this work has been provided by the Natural Sciences and Engineering Research Council of Canada NSERC), Toyota, and Maplesoft. graph-theoretic sensitivity analysis of the torque converter, Banerjee and McPhee 011). Fig. 1. Schematic diagram of a typical torque converter Figure 1 shows the schematic diagram of a typical torque converter. The mechanical components and the path followed by the hydraulic fluid during the forward flow mode are shown in the diagram. The operation of a torque converter can be described as that of a hydraulic pump driving a hydraulic turbine. The torque from the engine drives the pump which imparts energy to the hydraulic fluid in the system, the hydraulic fluid flows through the vaned construction of the turbine and makes it rotate thereby transmitting the torque coming from the engine towards the rest of the transmission system. From the standpoint of energy transmission, the pump converts mechanical energy into hydraulic energy and the turbine converts it back to the mechanical domain. A linear graph can be used to represent the flow of power across different domains of the system and account for the storage and losses incurred in the process.
. GRAPH-THEORETIC MODELING Graph-theoretical modeling represents a system using a set of nodes and connecting edges, McPhee 1996). A torsional inertia spring damper system and its corresponding linear graph are shown in Fig.. Fig.. Rotational inertial-spring-damper system In the graph shown above, the node g and a correspond to the frame of references fixed to the ground and the centre of mass of the rotating disk. The edges of the graph correspond to components of the system. The edges k 1 and c 4 represent the torsional spring and damper respectively, the edges I 3 and F 5 represent the rotational inertia and the torque applied on it and the edge r represent the revolute joint between the rotating disk and the ground. To describe the properties of a linear graph certain key terms need to be defined at this point. A path is a sequence of edges that connects two distinct nodes. A circuit is defined as the subgraph of a linear graph such that there are exactly two distinct paths between every pair of nodes in that subgraph. A tree is defined as a subgraph of a linear graph that contains all the nodes but doesn t contain any circuit. For example, the edge marked I 3 can be considered as a tree for the linear graph shown in Fig.. The portion of a linear graph that remains after the removal of a tree is known as a cotree. For the above selection of tree the cotree would include the edges k 1, r, c 4 and F 5. A linear graph can have more than one tree-cotree pairs, but for a particular selection of a tree there is always a unique cotree. The edges selected in a tree are known as branches and the edges that are selected in the cotree are known as chords. Physically, the edges represent mutually complementary sets of measurements. These measurements are the through variables τ and the across variables α. The identity of the through and across variables depend on the domain of the system,. For the mechanical domain, the through variables are the torques and forces acting on the system and the across variables are the displacements, velocities and accelerations of the system components. For hydraulic systems, the through variables are the volumetric flow rates and the across variables are the pressure differences between different points. The graph captures the interconnections of the through and across variables of different components. In other words, the topology of the system is captured by the structure of the linear graph representing it. The structure of a linear graph can be mathematically expressed by the two types of linear equations that can be generated from the graph itself. The equations are known as 1) The circuit equations, which is a set of linear relationship between the across variables and can be expressed in a matrix form as [B] {α} = 0 1) where B is a constant matrix known as the fundamental circuit matrix and α is the vector of across variables. ) The cutset equations, which is a set of linear relationship between the through variables and can be expressed in a matrix form as [A] {τ } = 0 ) where A is a constant matrix known as the fundamental cutset matrix and τ is the vector of through variables. The circuit and cutset equations alone cannot solve for the system behavior. For that, another set of equations are required to capture the connection between the through and across variables. These equations are known as terminal equations and they describe the nature of the components of the system. The general structure of a terminal equation can be expressed as a non-linear function of the through and across variables and the model parameters. f τ, τ, τ, α, α, α, b, t) = 0 3) By combining 1), ), and 3), we can form a set of equations sufficient in number to solve for the entire set of unknown through and across variables. However it is not always necessary to generate system equations in terms of all the unknowns. A method known as the branch-chord formulation can be used to reduce the number of generated equations. In this method a particular tree is selected and the edges are divided into branches and chords. Using 1) and ), it is possible to express the through variables of the branches in terms of the through variables of the chords and across variables of the chords in terms of across variables of the branches. This way the generated equations are formed in terms of a much smaller set of unknowns, i.e. the branch across variables, also known as branch co-ordinates, and the chord through variables. The number of unknowns can further be reduced by careful choice of branches and chords to include all the known across quantities in the tree and all the known through quantities in the cotree. In a torque converter mechanical and hydraulic components work in tandem. In the next section, a linear graph would be used to capture the topology of the system and to model the dynamic behavior of a torque converter. 3. MODELING A TORQUE CONVERTER The linear graph for the torque converter covers both mechanical and hydraulic domains. In the mechanical domain, the physical components are the pump, the turbine,
and the stator. To capture the motion of these components, a minimum of three nodes are required. On the other hand, in the hydraulic domain, the physical component is the hydraulic fluid which passes through the vanes, encountering different types of hydraulic events, such as transfer of momentum, losses and pressure drops. In this section, we will present the linear graph that captures the dynamics of the torque converter. 3.1 Cutset and circuit equations Edges 5, 15 and 3 are chosen as the branches for the mechanical domain, and edges 1,, 8, 9, 10, 11, 1, 18, 19, 0, 5, 6, 7, and 8 are selected as the branches for the hydraulic domain. The resulting cutset and circuit equations are given below. Mechanical domain circuit equations τ 3 + τ 4 + τ 5 + τ 6 τ 7 = 0 τ 13 + τ 14 + τ 15 + τ 16 τ 17 = 0 4) τ 1 + τ + τ 3 τ 4 = 0 Mechanical domain cutset equations ω 3 = ω 4 = ω 5 = ω 6 = ω 7 ω 13 = ω 14 = ω 15 = ω 16 = ω 17 5) ω 1 = ω = ω 3 = ω 4 Hydraulic domain circuit equations pi = 0 6) i = 1,, 8, 9, 10, 11, 1,18, 19, 0, 5, 6, 7, 8, 9 Hydraulic domain cutset equations Q 9 = Q [1,,8,9,10,11,1,18,19,0,5,6,7,8] 7) Fig. 3. Linear graph representing the torque converter The graph shown above has two non-connected subgraphs. These subgraphs correspond to the two domains of the system. The edges and nodes located inside the dashed circle belong to the mechanical domain and similarly the nodes and edges located outside the circle belong to the hydraulic domain. The symbols used to denote the through and across variables for different domains are given in the table below. Table 1. Through and Across variables Symbol Variable Domain Edges Q Through Hydraulic 1,, 8-1, 18-0, 5-9 p Across Hydraulic 1,, 8-1, 18-0, 5-9 ω Across Mechanical 3-7, 13-17, 1-4 τ Through Mechanical 3-7, 13-17, 1-4 The graph is also divided in three sections as illustrated in the figure. These three sections correspond to the three physical areas of the torque converter, i.e. the pump, the turbine and the stator. In the mechanical domain, the node g is the ground node. The nodes a, b, and c refer to the frame of reference fixed to the centers of mass of the pump, the stator, and the turbine. On the other hand, in the hydraulic domain, the nodes represent points on the path of the hydraulic fluid, where pressure and flow rate measurements are being obtained. The location of these points depend on the component they belong to. For example, the nodes that are connected by edges 1,, 8, 9, and 10 refer to points on the pump element. 3. Constitutive Equations Equations 4) - 7) don t convey any information about the functional dependence of the through variables on the across variables and vice versa. This critical piece of information is conveyed by a separate set of equations known as the constitutive equations. For every edge of the linear graph representing the system, constitutive or terminal equations describes the relationship between the associated through and across variables. Depending on the nature of this functional dependence, we can classify the edges to represent physical components like, inertia elements, energy dissipators or energy transducers. We will now present the terminal equations of the components in the context of the graph representing the torque converter. Inertia Elements For the mechanical portion of the graph, the edges 5, 15, and 3 correspond to the inertia of the pump, the turbine, and the stator respectively. The terminal equations for these edges are τ 5 = I p ω 5 τ 15 = I t ω 15 τ 3 = I s ω 3 ω 5 = ω p t) ω 15 = ω t t) ω 3 = ω s t) 8) where I p, I t, and I s are the moments of inertia of the pump, the turbine, and the stator respectively. For the hydraulic portion of the graph, the edges 1, 11, and 6 refer to the inertia of the fluid bodies present inside the vanes of the three components of the torque converter. The corresponding terminal equations are
p 1 = ρl p A Q 1, p 11 = ρl t A Q 11, p 6 = ρl s A Q 6 9) where ρ is the density of the hydraulic fluid, A is the cross-sectional area of the flow path and L p, L t, and L s are the lengths of the fluid paths inside the pump, the turbine and the stator respectively. Applied torques The edges 4 and 14 represent the external torques applied on to the shafts of the pump and the turbine elements. As a result these edges act as through variable drivers. The corresponding terminal equations are τ 4 = τ p t), τ 14 = τ t t) 10) Hydro-mechanical transducers In a torque converter, the pump element, forces the hydraulic fluid radially outward through its vanes. This imparts momentum to the fluid mass and creates a pressure difference along the path of the flow. Similar or opposite events occur at the turbine and the stator. The basic phenomenon can be described as one where the through variables of the hydraulic domain are being converted into the through variables of the mechanical domain and vice versa. This can be modeled by hydro-mechanical transducers. These transducers are multi-port components and have multiple edges associated with them. In the presented graph the following pairs of edges correspond to a total of six transducers used in the model. The pump : Edges { and 3} and {7 and 10} The turbine: Edges {1 and 13} and {17 and 0} The stator: Edges {1 and 6} and {4 and 9} The corresponding constitutive equations are τ 3 = g p Q and p = g p ω 3 τ 13 = g t Q 1 and p 1 = g t ω 13 τ 1 = g s Q 6 and p 6 = g s ω 1 τ 7 = ρs p Q 10 and p 10 = ρs p ω 7 τ 17 = ρs t Q 0 and p 0 = ρs t ω 17 τ 4 = ρs s Q 9 and p 9 = ρs s ω 4. 11) The transducer coefficients g p, g t, and g s are non-linear functions of the model parameters and the through and across variables of the graph. These quantities depend on the direction of fluid flow and thus are different for forward and backward flow modes.for the forward flow mode the following expressions are used. g p = ρ[r pω 5 + R p Q A tan α p R sω 3 R s Q 6 A tan α s] g t = ρ[r t ω 15 + R t Q 1 A tan α t R pω 5 R p Q A tan α p] g s = ρ[r sω 3 + R s Q 6 A tan α s R t ω 15 R t Q 1 A tan α t] 1) The coefficients S p, S t, and S s are design constants that depend on the vane profiles of the components of the torque converter and entry and exit vane angles. They have the dimension of area, and have an unit of m. The derivations of these equations follow from the momentum transfer equation for the flow of hydraulic fluid. The presented constitutive equations are based on derivations presented in Hrovat and Tobler 1985). Kinematic constraints Edges 6, 16, and represent the kinematic joints that connect the components of the torque converter to the ground. The pump and the turbine are connected to the automobile chassis through ideal revolute joints.consequently the terminal equations for the edges 6 and 16 are τ 6 = 0, τ 16 = 0 13) The stator is connected to the chassis using a one way clutch. It prevents the rotation of the stator when stator torque is positive but allows the stator to rotate freely when the stator torque becomes negative. This can be modeled by the following terminal equation. { ω = 0 and τ = τ s t) when τ s τ ε ω = ω s t) and τ = 0 when τ s < τ ε 14) where τ ε is a small quantity used to avoid numerical issues during simulation. Losses In this model, two types of losses are included. Both of these pertain to the hydraulic portion of the system. They are the shock losses, that arise due to non-ideal flow conditions at the junction of the components and the flow losses, that arise due to fluid friction. Although in reality these losses are distributed quantities, in this model, they are considered as lumped quantities. Edges 8, 18, and 7 refer to the flow losses. p 8 = ρc f Q 8 A 1 + tan α p ) ) signumq 8 ) p 18 = ρc f Q 18 A 1 + tan α t ) ) signumq 18 ) 15) p 7 = ρc f Q 7 A 1 + tan α s ) ) signumq 7 ) Edges 9, 19, and 8 represent the shock losses. p 9 = ρc s R p ω 5 ω 15 ) + Q 19 p 8 = ρc s p 19 = ρc s R s ω 3 ω 5 ) + Q 9 ) A tan α p tan β t ) A tan α s tan β p ) R t ω 15 ω 3 ) + Q 8 A tan α t tan β s ) ) ) 16) In the above equations, C f and C s are constants and are known as the flow and shock loss factors. The symbol R denotes radius, where as α and β are the exit and entrance angles of the vanes. The subscripts p, t, and s
refer to the pump, the turbine and the stator respectively. The signum function is incorporated to return the sign of the volumetric flow rate and make the expressions valid for backward flow modes as well. These expressions for the loss terms are derived using analytical and empirical formulations and are presented by Hrovat and Tobler 1985). 3.3 Generation of governing equations While it is true that 4) - 1) form a valid set of governing equations, it is not the most efficient method of simulation. To reduce the total number of unknowns and hence the number of equations to be solved, a branch chord formulation is used. The cutset and circuit equations are used to express the through variables of the branches in terms of the through variables of the chords and across variables of the chords in terms of across variables of the branches. Finally equation 8) to 1) are used to arrive at the final set of equations. For this particular selection of branches, as outlined in the previous section, the equations are generated in terms of the angular speeds of the components of the torque converter branch across variables for the mechanical domain), the torques applied on the components chord through variables for the mechanical domain) and the volumetric flow rate of hydraulic fluid chord through variable for the hydraulic domain). Since the applied torques are given functions of time, the state vector for this system is q = { ω p ω t ω s Q } T. 17) Since the kinematic joint at the stator is dependent on a dynamic criterion 14, the generated governing equations are different in different zones of simulation. To explain this phenomenon, it is important to describe the operation of a torque converter. For forward flow mode, during the acceleration phase, the stator torque is initially positive. The clutch holds the stator in place and forces the quantity ω s to be zero. After a while when stator torque becomes zero, the stator starts rotating. This particular event is known as the coupling point. If governing equations are generated using the cutset, circuit and constitutive equations pertaining to this particular phase of motion, the following set of equations are obtained. Where M. q +Gq + ξ Ψ = 0 with τ s = 0 18) I p 0 0 ρs p 0 I t 0 ρs t M = 0 0 I s ρs s 19) ρl p + L t + L s ) ρs p ρs t ρs s A 0 0 0 g p 0 0 0 g G = t 0 0 0 g 0) s g p g t g s 0 ξ = [ 0 0 0 P L ] T Ψ = [ τ p τ t τ s 0 ] T 1) In the above equation P L is the total pressure drop around the flow path due to friction and shock losses. For operation before the coupling point, the kinematic constraint equation given by 14) would be in effect. The resulting governing equation can be derived by setting ω s = 0 in 18) and assuming τ s t) 0. 3.4 Backward flow mode The torque converter can also work when the volumetric flow rate is negative, i.e. when the turbine has higher speed than that of the pump. This mode of operation corresponds to the action of engine braking and is an important aspect of the dynamics of a torque converter. To model the backward flow mode, we need to take note of the changes it brings to the equations we have presented so far. In the backward flow mode, the flow path becomes reversed. This makes the exit and entry angles of the vanes to switch places. Essentially the quantities α p, α t, and α s become β p, β t, β s and vice versa. The cutset and circuit equations are not affected by this change at all. In the event of Q < 0 they would simply change their signs and still remain valid. For the most part, the constitutive equations would remain the same, however for equations that are functions of the vane angles, the quantities α p,t,s need to swapped with the quantities β p,t,s e.g. the loss terms given by 15) and 16) and the transducer equations given by 11) and 1)). Groupings of the terms in the terminal equations also need to be modified. The modified expression for the transducer coefficient g p for the case of reverse flow mode is given below. g p = ρ[r pω 15 + R pq 1 A tan β t R sω 5 R sq A tan β p] ) 4. RESULTS The governing equations, 18), were solved numerically in Maple using Fehlberg fourth-fifth order Runge-Kutta method. Apart from the state variables, we also solve for the speed ratio ω r = ω t /ω p ) and the efficiency of the torque converter. The maximum function evaluation was set at maxfun = 10 5 with default error tolerance at ɛ = 10 5. To simulate the forward flow mode of the torque converter, a constant turbine torque of τ s = 500N m is applied. The pump torque is assumed to be a ramp function that increases from τ p = 00N m to τ p = 510N m.the values used for the model parameters and the initial conditions are given in Appendix A. We run the simulation through the coupling point and stop when the stator speed reaches a particular value ω s = 80 rad/s). The behavior of the torque converter is shown in Fig. 4. It clearly shows that under the action of fluid coupling, the turbine speeds up from rest and the difference between ω p and ω t decreases. After the coupling point is reached, the speed difference stop decreasing and the speed ratio saturates. This is a commonly observed phenomenon for torque converters, Hrovat and Tobler 1985).
Future work would be focused on the development of models of other hydraulic components for efficient modularized simulation of complicated hydraulic systems. Fig. 4. Variation of ω p and ω t with the change in ω r REFERENCES Abidi-Asl, H., Lashgarian-Azad, N., and McPhee, J.J. 011). Math-based modeling and parametric sensitivity analysis of torque converter performance characteristics. In Proceedings of the SAE World Congress 011, 11M- 0315. Banerjee, J.M. and McPhee, J.J. 011). Graph-theoretic sensitivity analysis of multibody systems. In J.C. Samin and P. Fisette eds.), Proceedings of the Multibody Dynamics 011 ECCOMAS Thematic Conference. Hrovat, D. and Tobler, W.E. 1985). Bond-graph modeling and computer simulation of automotive torque converters. Journal of the Franklin Institute, 319, 93 114. Ishihara, T. and Emori, R.I. 1966). Torque converter as a vibrator damper and its transient characterisitcs. In SAE Proceedings, 660368. Lucas, G.G. and Rayner, A. 1970). Torque converter design calculations. Automobile Engineering, 56 60. McPhee, J.J. 1996). On the use of linear graph theory in multibody system dynamics. Nonlinear Dynamics, 9, 73 90. Schmitke, C., Morency, K., and McPhee, J.J. 008). Using graph theory and symbolic computing to generate efficient models for multi-body vehicle dynamics. Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics, 4), 339 35. Appendix A. NUMERICAL VALUES Table A.1. Parameter values for simulation Fig. 5. Efficiency of torque transmission The efficiency of the torque converter as shown in Fig. 5 shows expected behavior, Abidi-Asl et al. 011). 5. CONCLUSION The central principle of a torque converter is the phenomenon of transduction of energy from one domain to another. This paper demonstrates the feasibility of modeling torque converter systems using graph-theoretic approaches, where conversion of energy between two domains are captured by multi-port hydro-mechanical transducers. The presented model can be easily extended to include other mechanical or hydraulic components. It is also easy to integrate it into existing graph-theoretic models of multibody vehicular systems. The components used in this model is easy to understand and has physical interpretations. It is therefore possible to upgrade this model in a modularized fashion. Furthermore, a graph-theoretic model would also allow further analysis if required. The approach of converting fluid dynamic formulations into terminal equations can also be applied to model systems like hydraulic drives, pumps and control valves. Name Description Value Unit ρ Density of the fluid 840 Kg/m 3 I p Inertia of pump 0.106 Kg m I t Inertia of turbine 0.89 Kg m I s Inertia of stator 0.05 Kg m R p Radius of pump 0.115 m R t Radius of turbine 0.068 m R s Radius of stator 0.088 m S p Design constant 0.0003 m S t Design constant 0.001 m S s Design constant 0.001 m α p Blade exit angle 0 Deg α t Blade exit angle -58 Deg α s Blade exit angle 65 Deg β p Blade entry angle -16 Deg β t Blade entry angle 48 Deg β s Blade entry angle -30 Deg C f Friction loss factor 0. N/A C sh Shock loss factor 1 N/A L total Length of fluid inertia 0.8 m A Flow area 0.01 m Table A.. Initial conditions for the simulation States Initial Values Unit Q 0.13 m 3 /s ω s 0 rad/s ω p 80 rad/s ω t 0 rad/s