Dec 29, 2003 - developed by Sayers and Riley (1996) and will henceforth be ...... Steve Riley of MSC software is recognized for his work on modeling the ...
International Journal of Heavy Vehicle Systems December 29, 2003
Generating Proper Dynamic Models for Truck Mobility and Handling Loucas S. Louca, D. Geoff Rideout, Jeffrey L. Stein, and Gregory M. Hulbert The University of Michigan Department of Mechanical Engineering, Automated Modeling Laboratory G029 W.E. Lay Automotive Lab Ann Arbor, MI 48109-2121 Email: {cyl, drideout, stein, and hulbert}@umich.edu ABSTRACT In previous work, a “proper model” was defined as the model of minimal complexity, with physically meaningful parameters, which accurately predicts dynamic system outputs. Proper models can be generated by an energy-based model reduction methodology that removes unnecessary complexity from models (linear or nonlinear) without altering the physical interpretation of the remaining parameters and variables. Energy-based model reduction allows design in a search space of reduced dimension, and in general improves computational efficiency. The current work demonstrates the effectiveness and utility of an energy-based model reduction algorithm for vehicle system modeling applications. Model reduction is performed for two different vehicle modeling applications. The first case study focuses on the vehicle dynamics model of a military heavy-duty tractor semitrailer. The full model is developed using a classical multibody system approach and accurate reduced models are then generated for a lane change maneuver. The second case study develops, validates, and reduces an integrated vehicle system model of a single-unit medium-size commercial truck composed of engine, drivetrain, and vehicle dynamic subsystems. The systematically reduced model accurately predicts vehicle forward speed/acceleration and engine behavior during full-throttle acceleration and braking with a twofold increase in computation efficiency. The reduced models generated by the energy-based methodology retain predictive quality, are useful for studying trade-offs involved in redesigning components and control strategies for improved vehicle performance, and are less computationally intensive. Keywords: Vehicle dynamics, energy metrics, model reduction, automated modeling, proper models, integrated vehicle systems 1. INTRODUCTION Vehicle design is a costly process that starts with a comprehensive analysis of the vehicle system in order to determine the desired characteristics of subsystems. Detailed design of subsystems and components follows, and the process concludes with building and testing of prototypes. Simulation-based design, in which a virtual vehicle is created and tested for a variety of conditions, can reduce development time and cost and result in prototypes that are closer to the final product. Barriers continue to exist, however, that prevent the full potential of simulation-based design from being realized. One of these is the lack of availability of comprehensive total vehicle simulation models with high-fidelity subsystem representations. A second issue inhibiting virtual design of large artifacts is the tendency for model complexity to be determined in a nonsystematic fashion. Models with insufficient fidelity have limited or no predictive ability, while over-
International Journal of Heavy Vehicle Systems: L.S. Louca, D.G. Rideout, J.L. Stein, and G.M. Hulbert Generating Proper Dynamic Models for Truck Mobility and Handling
complex models present a needlessly onerous parameter estimation problem and a design optimization search space of excessive dimension. Efficient, integrated vehicle system models enable concurrent design, where changes in one component are reflected in the design of related components and subsystems. In addition, a predictive simulation can assist in quantifying parameters associated with subjective driver feel and overall driveability that are difficult to measure and yet very important for customer acceptance. 1.1 System model integration issues in virtual vehicle design Attempts to model the entire vehicle system or vehicle subsystems differ in the fidelity of individual models as well as in the methodologies used to integrate the various subsystems. Significant improvements in the integration methodology have emerged based on object-oriented programming environments with graphical interfaces, such as 20SIM, AMESim, EASY5, MATLAB/SIMULINK, etc. These computer environments provide flexibility in connecting and interchanging models of components and subsystems, which makes model assembly an easy and undemanding task. As a result, engineers can model many complex physical phenomena to generate system models capable of predicting every possible variable of interest. This is common practice due to the lack of rigorous metrics or systematic approaches to determine the required complexity (physical phenomena) for accurately predicting the variables of interest. In addition, computing resources are becoming faster and cheaper, alluring engineers into developing models that are even more complicated. The issue of model complexity can nonetheless not be treated cavalierly. In practice, subsystem models are often generated by domain experts who construct models for their own needs and may not necessarily know the context in which the subsystem model will be used within a system model. These subsystem models may contain unnecessary information and thus, when assembled, result in an overly large and complex system model. This occurs frequently in the automotive and aerospace industries. For example, automobile ride quality models do not typically need extensive representations of the suspension geometry and body modes, though these effects are often included because suspension and body experts generate comprehensive models that they need for their own analysis and design needs. More specifically, a finite element model is necessary for analyzing the body vibration, however, a few dynamic modes of the body might be sufficient for studying ride quality. Detailed system models require simulation times that are too long to be practical in design studies where thousands of evaluations of the model may be required. In addition, the interpretation of model results becomes more difficult when unnecessary items included in the model are not important to the response variables under study. It is the premise of this work, therefore, that the effective use of computer aided modeling and simulation necessitates the generation of the least complicated models possible. 1.2 Systematic treatment of model complexity Various algorithms have been developed and implemented to help automate the production of dynamic system models that are accurate yet simple (smaller in size). Wilson and Stein (1995) developed the concept of a “proper model” – the model described by the minimal set of physical parameters required to predict dominant system dynamics - along with an algorithm that deduces the “appropriate” system model complexity from subsystem models of variable complexity. Additional work on deduction algorithms for generating proper models has been reported by Ferris et al. (1994), Ferris and Stein (1995), and Walker et al. (2000). While these algorithms have been implemented and demonstrated to work well (Stein and Louca, 1996), they are limited to linear systems because the modeling metric used in the deduction process requires the system eigenvalues. Another limitation is that these algorithms address the issue of model complexity by adding/removing only Page 2 of 24
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International Journal of Heavy Vehicle Systems: L.S. Louca, D.G. Rideout, J.L. Stein, and G.M. Hulbert Generating Proper Dynamic Models for Truck Mobility and Handling
compliant elements (e.g., mechanical springs), which are only a subset of the physical phenomena comprising a dynamic system. In an attempt to overcome the limitations of eigenvalue-based model deduction techniques Louca et al. (1997), Louca (1998), and Louca and Stein (2002) introduced a model reduction technique that also generates proper models. This approach uses an energy-based metric that can be applied to nonlinear as well as to linear dynamic systems and considers the importance of all energetic elements (inertial, compliant and resistive) in a model. The contribution of each element in the system is ranked according to an energy metric named activity. Elements with low activity are eliminated from the model to generate a reduced model that is less complicated but still accurately predicts the system behavior. By varying the resolution of the models in an integrated simulation environment, while retaining physically meaningful variables, a simulation can be tailored to specific applications and scenarios. In addition, computational efficiency can be improved due to the smaller model size and the elimination of high frequency dynamics that allow larger integration time steps. It is the objective of this work to present and establish the efficacy of the model reduction technique in a vehicle modeling context. The reduction methodology is demonstrated in two case studies involving different types of models, for different types of analyses at different stages of the vehicle design process. In the first case study, a heavy-duty military tractor semi-trailer model is developed to simulate a handling maneuver. The vehicle dynamics are described by a three-dimensional multibody representation with a large number of degrees of freedom. The second case study models a medium-sized commercial truck for fuel economy analysis. The commercial truck model contains engine and drivetrain subsystems, and a pitch plane description of the vehicle dynamics. The two models use different formulations and have different levels of complexity; however, they can both be analyzed and reduced. The energy-based metric is used to generate less complicated and more efficient models, and to identify the principal parameters for a given design scenario. The paper is organized as follows. The background on the energy-based model reduction metric and algorithm is presented in Section 2. Next, the description of the military truck model and its reduction are given in Section 3. Then, the model of the medium size commercial truck is presented in Section 4 along with the model reduction analysis. Finally, discussion and conclusions are given in Sections 5 and 6, respectively. 2. BACKGROUND - MODEL REDUCTION The original work on energy-based metrics for model reduction (Louca et. al., 1997; Louca et al., 1998; Louca and Stein, 2002) is briefly presented in order to provide a sub-stratum for the contributions of this paper - the extended implementations and demonstration of the original idea. The reduction procedure starts with a highfidelity (“full”) model, and removes unimportant parts. To maintain “proper model” properties the reduction must proceed by removing physical phenomena or/and quantities from the model (e.g., remove a damper or a mass). By operating on the physics of the model, the process yields a physically meaningful description of the system after model reduction. This approach is in contrast with model reduction techniques used in the automatic control area (Moore, 1981), in which the states and parameters of the reduced models do not retain their physical significance. Such models have limited usefulness for design and redesign analyses since the design parameters are not explicit or not even present in the reduced model. The elimination of physical phenomena requires the identification of those elements that are not important, and therefore, do not contribute to the overall system behavior. The power of each element is used for identifying elements that can be eliminated. The power associated with each element in the system provides an indication of the element’s contribution to the total behavior of the system (at least as far as energy is Page 3 of 24
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International Journal of Heavy Vehicle Systems: L.S. Louca, D.G. Rideout, J.L. Stein, and G.M. Hulbert Generating Proper Dynamic Models for Truck Mobility and Handling
concerned). An element with high power associated with it stores (energy storage element) or absorbs (dissipative elements) a sizeable portion of the power that is supplied into the system, and therefore contributes significantly to the system behavior. However, because power is a function of time and can be oscillatory, its use as a modeling metric would lead to varying instantaneous estimates of elements’ importance. In order to eliminate time dependence, the absolute value of power is integrated to give a time invariant metric:
A=
P(t ) dt ,
(1)
0
where P (t ) is the element power and is the time window over which the metric is calculated. This scalar metric has units of energy (time integral of power) and represents the amount of energy that flows in and out of the element over the time window . Therefore, this new metric has a physical meaning. The energy that flows into and out of an element is a measure of how active this element is (how much energy passes through it), and consequently why the quantity in Eq. (1) is termed “activity”. It is assumed that the higher the activity, the higher the contribution of the element to the overall system behavior. The activity, as defined above, provides a measure of absolute, but not relative importance of the elements. To provide relative information the total activity (Atotal) of the model is defined as: k
k
Atotal = Ai = Pi (t ) dt , i=1
(2)
i=1 0
where Ai is the activity of the i-th element as given by Eq. (1). The total activity represents the total amount of energy that flows in and out of the system over the period . This quantity is used to calculate a normalized metric called “element activity index” or just “activity index”. The activity index AIi is calculated for each element in the model, and represents the portion of the total system energy that is flowing through that element.
A AI i = i = Atotal
P (t ) dt i
0
k
i = 1,K, k
(3)
P (t ) dt j
j=1 0
With the activity index defined as a relative metric for addressing element importance, the Model Order Reduction Algorithm (MORA) is constructed. The first step of MORA is to calculate the activity index for each element in the system for a given system excitation and initial conditions. Next, the activity indices are sorted to identify the elements with high activity (most important) and low activity (least important). With the activity indices sorted, the model reduction proceeds given some engineering specifications. These specifications are defined by the modeler who then converts them into a threshold of the total activity (e.g., 99.9%) that he/she wants to include in the reduced model. This threshold defines the borderline between the retained and eliminated model elements. The elimination process is shown in Figure 1 where the sorted activity indices are summed starting from the most important element until the specified threshold is reached. The element which, when included, increments the total activity above the threshold, is the last element included in the reduced model. The elements that are above this threshold are removed from the model. The elimination of the low activity elements is done by removing these elements from the model description.
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International Journal of Heavy Vehicle Systems: L.S. Louca, D.G. Rideout, J.L. Stein, and G.M. Hulbert Generating Proper Dynamic Models for Truck Mobility and Handling
100
Threshold 80
Removed Element s
[ %]
60
Cumulat ive Act ivit y Index 40
Sort ed Act ivit y Index 20
1
2
3
k-2
k-1
k
Element Ranking
Figure 1: Activity index sorting and element elimination In general, the solution of the time response calculation for highly nonlinear systems must be done numerically. Therefore, the calculations of all quantities needed for the activity index are obtained by means of numerical integration. First, a numerical integration of the model produces the time response of the system states along with the necessary outputs for calculating the power. Then the element power is calculated using the element constitutive laws. Finally, the activity and activity index are calculated using the expressions in Eq. (2) and (3), respectively. The above calculations provide the necessary information in order to apply the reduction algorithm (MORA). More information on the activity metric and the different approaches of calculating the activity are given in previous work by Louca (1998). 2.1 Multi-dimensional Activity The activity metric as defined above is applicable to physical elements that are only allowed to move in one dimension at a time. For example, systems with rotating shafts and translating masses connected together can be easily handled with the previously described methodology since the elements have a single degree of freedom (1DOF). However, the activity as it is defined in Eq. (1) cannot be applied to rigid bodies moving in the threedimensional space with up to six degrees of freedom. Such bodies are present in a vehicle handling or ride model. For bodies moving in multi-dimensional space, we encounter the apparent problem that the scalar activity must somehow be calculated from force and velocity vectors. One way to resolve this is to assume that the activity for each DOF emanates from a different element even though there is only one physical element. This means that the power for each DOF is calculated using the corresponding component of force and velocity. Therefore, for a rigid body or constraint force with p degrees of freedom the corresponding activity is defined as:
Aij =
0
Pj (t ) dt =
e (t ) f (t ) dt j
j
j = 1,K, p ,
(4)
0
where e, f p are the generalized force and velocity vectors. This approach to activity calculation can produce a higher degree of reduction than that suggested by a cumulative activity for all dimensions of motion. For example, the mass of an object can be allowed to move in six directions (translational and rotational) but in a given system and for a given scenario, only a subset of its DOFs might be important and have high activity. Page 5 of 24
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International Journal of Heavy Vehicle Systems: L.S. Louca, D.G. Rideout, J.L. Stein, and G.M. Hulbert Generating Proper Dynamic Models for Truck Mobility and Handling
This extension of activity, along with the methodology described earlier in this section, provides all the necessary tools for applying the activity metric to many practical automotive systems. 3. CASE STUDY I: VEHICLE HANDLING MODEL REDUCTION Vehicle handling and stability, which are critical for heavy trucks, are typically predicted by means of modeling and simulation. Recent models have been developed using modern general-purpose multibody simulation programs (Antoun et al., 1986) that automatically generate the equations of motions given the geometric description of the truck, constraints, and forces acting on the rigid bodies. On the other hand, earlier models (Gillespie et al., 1982) were based on general multibody formulations and were highly customized for the specific needs of vehicle handling prediction. In both cases the models are based on assumptions of the critical components that need to be included for obtaining accurate predictions. Depending on the modeler’s knowledge and understanding of the system, these assumptions may lead to an oversimplified or overcomplicated model. The model reduction methodology described in the previous section can be used to systematically verify the modeling assumptions and evaluate the complexity of such vehicle models. A tractor semi-trailer is selected as the first case study for the implementation of MORA. First the model description is given, followed by the model reduction and the generation of a series of reduced models. The system is an M916A1/870A2 military heavy-duty tractor semi-trailer for which a pitch and yaw plane schematic is shown in Figure 2. This truck is used for hauling heavy military equipment, e.g., M1 tanks; therefore, handling behavior and stability are very important considerations during the design. The tractor and trailer, which are connected through the hitch, have three axles each. The propulsion to accelerate the truck is provided by the first three axles (1-2-3) of the tractor. For studying the handling behavior of this vehicle, a model with just the vehicle dynamics is developed - no powertrain is included in the model. The model is used to predict the dynamic three-dimensional motion in response to steering and braking inputs. Generally, directional control, roll stability, and stopping distance are of particular concern. This model was originally developed by Sayers and Riley (1996) and will henceforth be referred to as the “full model.” The model consists of eight rigid bodies for the sprung masses of the tractor and trailer, and the six axles. The first axle (Axle 1) has single tires on each side while all other axles (Axle 2, 3, 4, 5, and 6) have dual tires on each side. Given this configuration the complete tractor semi-trailer has 22 tires. These rigid bodies are constrained by joints and forces/moments in order for the model to capture the actual vehicle behavior. The tractor is allowed to move in all six DOFs (3 Translational and 3 Rotational) while the trailer has three kinematic degrees of freedom since the hitch constraint allows three rotations and no translation with respect to the tractor. The axles are constrained with respect to the tractor or trailer, and may translate along the body-fixed z-axis (jounce) and rotate around the x-axis (roll). Finally, the wheels are allowed to rotate around the y-axis (pitch) of the axles. Given this configuration the model has the following 33 rigid body degrees of freedom: Tractor: 6 DOF (3 translational and 3 rotational) Trailer: 3 DOF (3 rotational) 6 Axles: 2 DOF (roll and jounce) 12 Wheels: 1 DOF (spin) In addition to the kinematic constraints between the rigid bodies, there are forces (and/or moments) acting between two bodies at specific points. For example, the suspension forces (F S), depicted in Figure 3, act between the axle and the sprung mass (tractor or trailer). On the other hand, the tire forces (FT) act between the tire and the inertial frame in order to provide the required forces to support the weight and move the vehicle on the desired path. The following 122 forces and moments are present in this tractor semi-trailer: Page 6 of 24
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International Journal of Heavy Vehicle Systems: L.S. Louca, D.G. Rideout, J.L. Stein, and G.M. Hulbert Generating Proper Dynamic Models for Truck Mobility and Handling
12 suspension spring forces between axle and tractor or trailer 12 suspension dampers forces between axle and tractor or trailer 66 tire forces (22 vertical, 22 longitudinal, and 22 lateral) between tire and inertial frame 1 aerodynamic drag force between tractor and inertial frame 22 tire aligning moments in yaw DOF between tire and inertial frame 6 axle moments in roll DOF between axle and tractor or trailer 3 hitch moment (1 yaw, 1 pitch, and 1 roll) between tractor and trailer Pit ch plane z
z
y
y
x
Hit ch
x
Tract or
Trailer
Axle 1 Axle 6
Axle 5
Axle 4
Axle 3
Axle 2
Yaw plane
y y
z x
z x
Figure 2: M916A1/870A2 tractor-semitrailer
FS FT
FS FT
Figure 3: Constraint forces (roll plane) The full model is generated by synthesizing the above rigid bodies and forces/moments. The full model has 91 states and approximately 120 parameters. The equations of motion are formulated using Kane’s method in the AUTOSIM multibody processing and code generation computer environment (Sayers, 1991). The resulting Page 7 of 24
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International Journal of Heavy Vehicle Systems: L.S. Louca, D.G. Rideout, J.L. Stein, and G.M. Hulbert Generating Proper Dynamic Models for Truck Mobility and Handling
model is highly nonlinear due to the nonlinear constraint forces and the three-dimensional rigid body kinematics. In addition to the multibody analysis, the AUTOSIM model was modified appropriately in order to calculate the element activities as defined in Eq. (1) and (4). More details on the model development, multibody formulation and vehicle parameters can be found in the work by Sayers and Riley (1996). This is not a large model by multibody dynamics standards, but is still sufficiently complicated and large enough to provide a meaningful model reduction problem and vigorously test the reduction algorithm. 3.1 Full Model Response and Activities A specific maneuver must be selected in order to calculate activity and reduce the model. For this paper, it is assumed that the vehicle is traveling with a constant speed of 60 mph and at time t = 1 s the driver steers to avoid an obstacle (lane change maneuver). A simple proportional-integral controller is used to provide the appropriate torques at the driven wheels in order to maintain a constant speed. This maneuver is assumed to be executed by turning the steering wheel first left to avoid the obstacle, and then right to resume the original heading in the adjacent left lane (see Figure 4). The full model is used to calculate the system response as it is performing this lane change maneuver. Using the steering input in Figure 4, a numerical simulation is run to predict the behavior of the system. Figure 5 shows the trajectory of the tractor, as it first avoids the obstacle and then resumes the original heading. This is a relatively severe maneuver producing a maximum of about 0.2 g of lateral acceleration.
Steering [deg]
40
20
0
-20
-40
0
1
2
3
4
5 Time [s]
6
7
8
9
10
Figure 4: Steering input for a lane change maneuver
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International Journal of Heavy Vehicle Systems: L.S. Louca, D.G. Rideout, J.L. Stein, and G.M. Hulbert Generating Proper Dynamic Models for Truck Mobility and Handling
30
Y position [ft]
25 20 15 10 5 0
0
100
200
300
400 500 X position [ft]
600
700
800
900
Figure 5: Tractor trajectory for the lane change maneuver (yaw plane) The simulation also produces the required outputs needed for calculating the power, the activity, and finally the activity index of each energy element in the model. There are 169 elements (121 forces and 8 rigid bodies with 6 directions each) for which activities are calculated in order to determine their relative importance for this maneuver. The inertial forces of rigid bodies are projected onto each degree of freedom to produce six activities for each body, i.e., three for translation and three for rotation as given by Eq. (4). In addition, the activity of all active forces is calculated. The sorted activity indices along with the cumulative activity indices for this steering maneuver are plotted in Figure 6 and the numerical values given in Appendix I. Notice in Figure 6 that the cumulative activity is at approximately 80% after including the first 13 most important (active) elements. The most important elements are the trailer translational inertia in the x and y-axis, the force-generating elements in the tire model in the x-axis and finally the tractor translational inertia in the x and y-axis. As an example of each rigid body having 6 inertial-based activities, rank numbers 29, 48, 134, 136, 163 and 131 in Appendix I correspond to the respective x-axis translation, y-axis translation, z-axis translation, x-axis rotation, yaxis rotation, and z-axis rotation for the first axle. Notice that the translating inertia in the x-axis is the most important inertia while the rotation of the axle about the y-axis (roll) is the least. The activity of this axle in the rotation about the y-axis is identically zero since the axles have no pitch moment of inertia (the same holds for all six axles). In addition, some other elements at the bottom of the list have zero activity since their parameter was set to zero for the specific scenario. 100 Cumulative Activity Index
80
[%]
60 40 20 0
Element Activity Index 20
40
60
80 100 Element Ranking
120
140
160
Figure 6: Sorted activity indices and cumulative index Page 9 of 24
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International Journal of Heavy Vehicle Systems: L.S. Louca, D.G. Rideout, J.L. Stein, and G.M. Hulbert Generating Proper Dynamic Models for Truck Mobility and Handling
3.2 Reduced Models A series of reduced models are produced based on the activity index results in Appendix I. The last elements on the list, the suspension dampers for the axles of the trailer, have zero activity because their parameters are identically zero in the full model. The next elements that have small activity indices are rotational inertial effects (yaw and roll) for all six axles of the system. These effects were eliminated from the model resulting in a reduced model with only 81% of its original parameters (the specific elements eliminated are shown in Appendix I). Figure 7 compares the tractor lateral acceleration and yaw rate as predicted by the reduced model (81%) versus the full model. The predictions from the two models are almost identical. There is only a small discrepancy in the prediction of the lateral acceleration and yaw rate of the tractor, two outputs that are known to be very sensitive to the obstacle avoidance maneuver being studied. Lateral Acceleration [g]
0.2 Full Reduced
0.1 0 -0.1 -0.2
0
1
2
3
4
5 Time [s]
6
7
8
9
10
0
1
2
3
4
5 Time [s]
6
7
8
9
10
Yaw Rate [deg/s]
4 2 0 -2 -4 -6
Figure 7: Comparison of the full and 81% model in predictng tractor lateral acceleration and yaw rate (no axle rotational inertias) The next elements to be eliminated, according to the activity index, are the axle inertial effects in the z-axis (jounce). With these items removed, a new model (the 77% model) is generated and compared to the full model. The results for lateral acceleration and yaw rate are given in Figure 8. The discrepancies are only marginally larger than the 81% reduction model. Such differences would likely be considered negligible by a vehicle designer even though 23% of the elements have been eliminated. Another selected variable of interest is the suspension force (leaf spring and damper) at the front axle, shown in Figure 9 as predicted by the 77% model vs. the 81% model. Notice that the curves are quite similar except for high frequency effects that are eliminated from the 77% model. Page 10 of 24
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International Journal of Heavy Vehicle Systems: L.S. Louca, D.G. Rideout, J.L. Stein, and G.M. Hulbert Generating Proper Dynamic Models for Truck Mobility and Handling
Lateral Acceleration [g]
0.2 Full Reduced
0.1 0 -0.1 -0.2
0
1
2
3
4
5 Time [s]
6
7
8
9
10
0
1
2
3
4
5 Time [s]
6
7
8
9
10
Yaw Rate [deg/s]
4 2 0 -2 -4 -6
Figure 8: Comparsion of the full and 77% model in predictng tractor lateral acceleration and yaw rate (no axle rotational inertias and no axle mass in the verticaldirection)
Damper Force [lbf]
30 81% model 77% model
20 10 0 -10 0
1
2
3
4
5 Time [s]
6
7
8
9
10
Figure 9: Comparsion of two reduced models (77% vs. 81%) in predictng front axle suspension force In the final level of reduction, the compliant elements producing the tire aligning moments are eliminated from the model. Twenty-two more elements are removed from the model resulting in a net 37% reduction in the number of elements as compared to the full model. This “63-% model” is used to predict the tractor lateral acceleration and yaw rate. The results depicted in Figure 10 show an accuracy that is still quite good. The reduced model does over estimate the lateral acceleration and yaw rate because the aligning moments are not there to counteract any increase in acceleration. Page 11 of 24
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International Journal of Heavy Vehicle Systems: L.S. Louca, D.G. Rideout, J.L. Stein, and G.M. Hulbert Generating Proper Dynamic Models for Truck Mobility and Handling
Lateral Acceleration [g]
0.2 Full Reduced
0.1 0 -0.1 -0.2
0
1
2
3
4
5 Time [s]
6
7
8
9
10
0
1
2
3
4
5 Time [s]
6
7
8
9
10
Yaw Rate [deg/s]
5
0
-5
Figure 10: Comparsion of the full and 63% model in predictng tractor lateral acceleration and yaw rate (no axle rotational inertias, no axle mass in the vertical direction, and no tire aligning moments) 4. CASE STUDY II: INTEGRATED VEHICLE MODEL REDUCTION To predict vehicle mobility and fuel economy, a proper model is needed that predicts vehicle acceleration and braking in the absence of steering. Analyzing this type of behavior requires a model that includes all vehicle components and not just the vehicle dynamics. A powertrain model is needed, in addition to the vehicle dynamics, to accurately predict engine speed and torque, and transmission gear ratios that determine the traction forces and therefore the vehicle acceleration. This application differs from the first case study in that the vehicle dynamics interact with the powertrain (engine and drivetrain) through the wheels. The presence of thermal, hydraulic, and mechanical components allows MORA to demonstrate its ability to handle multi-energy domain systems and not strictly mechanical systems. The vehicle system is composed of the engine, drivetrain, and vehicle dynamics, as shown schematically in Figure 11. The engine is connected to the torque converter (TC), whose output shaft is then coupled to the transmission (T), propeller shaft (PS), differential (D) and two driveshafts (DS) that connect the differential to the driven wheels. The vehicle is rear driven and has solid front and rear axles with leaf spring suspensions. This configuration models an International 4700 4x2 truck, powered by a turbocharged, intercooled engine, and equipped with a four speed automatic transmission. However, the same structure and component models can be used for modeling most two-axle vehicles after minimal modifications.
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International Journal of Heavy Vehicle Systems: L.S. Louca, D.G. Rideout, J.L. Stein, and G.M. Hulbert Generating Proper Dynamic Models for Truck Mobility and Handling yf
Vehicle Dynamics
xf
Body
y
θ
F x
y
CG
y sf
x rf
rf
xsf rf
SF y
θ
r
xr
Front Axle&Wheel
Y
R y rr y
sr
X x
sr
SR x rr
Rear Axle&Wheel θ rr
DS
DS
PS
T
Drivetrain
C
TC
D-R
IM ICM
T
EM
Engine
Figure 11: Schematic of the integrated vehicle system The model is developed using the 20SIM modeling environment (20SIM, 2000), which supports hierarchical modeling and allows the physical modeling of subsystems and components using the bond graph formulation (Rosenberg and Karnopp, 1983; Karnopp et al., 1990; Brown, 2001). The model is developed with the intention of using the model reduction approach described in the background section. Therefore, all components and subsystems are allowed to have the maximum possible complexity, i.e., all physical phenomena are included in the model without speculating about their significance to the system behavior. The system model complexity is later addressed by applying the energy-based metric. Given that an element’s activity is correlated with its effect on overall system dynamic response, MORA will be applied to the complete set of model elements and no distinction is made based on the subsystem to which each element belongs. 4.1 Full Model The complexity of this system necessitates the use of a hierarchical representation. At the top level of the model hierarchy are the engine, drivetrain and vehicle dynamics, excited by the environment (see Figure 12). One source of excitation is the driver who produces the required throttle and brake commands in order to follow Page 13 of 24
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International Journal of Heavy Vehicle Systems: L.S. Louca, D.G. Rideout, J.L. Stein, and G.M. Hulbert Generating Proper Dynamic Models for Truck Mobility and Handling
a time based predefined velocity profile (driving cycle). The steering input is neglected and the vehicle is constrained to move only on the pitch plane. The other excitation comes from the road, which is usually uneven and prescribes a velocity (or displacement) to the tires. The road excitation is applied to both front and rear tires as a function of their position. A detailed description of the system, subsystem, and component models is given by the previous work of Louca et al. (2001), however, a brief description of the models is given next.
file input
Driver
driving cycle
Engine
Vehicle Dynamics
Drivetrain
slope rear
slope front
Figure 12: Top-level vehicle representation The engine model includes the inertial effects of all rotating parts and energy losses due to friction, which are lumped into single inertia and dissipative elements, respectively. The engine brake torque is produced from a steady state lookup table as a function of the engine speed and fueling rate. The specifications for the engine used in this work correspond to the International 7.3 L V8 engine type T444E. The engine torque delivery is controlled by a fuel injection controller that provides a signal for the mass of fuel injected based on driver demand, environmental conditions, and current engine operating conditions. Special functions include corrections for insufficient boost pressure and the speed governing. The engine is connected to the drivetrain, which consists of the torque converter, transmission, propshafts, differential, and drive shafts. The torque converter input shaft, on one end, and the drive shaft, on the other end, are the connecting points for the engine and the vehicle dynamics models. The inputs are the engine speed and wheel rotational speed, and the outputs are the torque to the wheels and load torque on the engine. An additional input is the driver demand signal, which is used by the shift logic to determine if an upshift or downshift event is required. The specific elements used in the drivetrain component models are as follows. The torque converter is the fluid clutch by which the engine is coupled to the transmission. The pump is connected rigidly to the engine output shaft, and the turbine to the transmission input shaft. The stator is connected to the torque converter housing via a one-way clutch. The fluid coupling has characteristics of a gyrator since the pump and turbine torques are determined by the turbine and pump speeds. The model also includes the turbine inertia. This gyrator is a nonlinear and non-power conserving element that accounts for the energy losses due to the fluid circulation. The gyrator modulus is calculated from quasi-static experimental data. Two measured quantities define the nonlinear modulus: a) torque ratio and b) capacity factor. These quantities are measured experimentally as a function of the speed ratio (transmission over engine speed). Page 14 of 24
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International Journal of Heavy Vehicle Systems: L.S. Louca, D.G. Rideout, J.L. Stein, and G.M. Hulbert Generating Proper Dynamic Models for Truck Mobility and Handling
The central element of the transmission is a non-power conserving transformer that models gear efficiency and different gear ratios for the four gears. The speed reduction in each gear is assumed ideal, while the torque multiplication is reduced by a gear efficiency factor. The transmission model consists of a series spring-inertiaspring arrangement and the non-power conserving transformer in between. In parallel with each spring is a viscous damper that models structural damping. The transmission fluid churning losses are modeled as a variable nonlinear resistance that varies depending on the gear number. The charging pump is modeled as a constant torque loss throughout the entire operating range. The two propshafts, joined by a universal joint, are modeled as an inertia-spring-inertia-spring-inertia series arrangement, again with small viscous damping. Following the propshafts is the differential, which includes an input compliance, axle inertia, and output compliance. The fluid churning losses for the differential/axle cooler are modeled as a resistive element with constant and linear terms. The differential is modeled as a non-power conserving transformer with ideal speed reduction but non-ideal torque multiplication based on a given gear efficiency. The shift logic module is also included to capture the behavior of the automatic transmission. The inputs to the shift logic are the transmission output shaft speed and the driver demand, and the outputs are gear number, speed ratio, and torque ratio. A shift logic map determines the current gear number, and whether or not an upshift or downshift event is to be initiated. During a gearshift, the speed reduction and torque multiplication ratios vary from the initial to the final value according to a blending function. The blending function models the torque and speed ratio variations that occur during the shift event, as clutches and bands engage and disengage. The vehicle dynamics subsystem includes the wheels, tires, axles, suspensions, and body - a collection of rigid bodies that are allowed to move in 2-dimensional space subject to forces/moments and rigid constraints (see Figure 11). The forces/moments act at specific points of two bodies, e.g., the rear suspension acts between point SR of the axle and point R of the body. In addition, two bodies can be restrained to move only in a specific direction or trajectory by a rigid constraint, e.g., the front axle is constrained to move on the y f-axis of the body. This is a typical two-dimensional multibody system and the bond graph methodology is used to derive the equations of motion. The body is modeled as a rigid body that is allowed to move horizontally and vertically, and to rotate around an axis normal to the X-Y plane (pitch). Three inertias are used to represent the dynamics in the three degrees of freedom. The body also includes two points for connecting the front and rear axles. Each point is located at a constant position (xF, yF) and (xR, yR) relative to the center of gravity (CG). Each axle is modeled as a rigid body that is constrained to move on an axis that has its origin at the attachment point (F or R) and is perpendicular to the y-axis of the body (xf, yf) or (xr, yr). The axle is also constrained by the suspension that is modeled as a linear spring and damper connected in parallel. The wheel masses are lumped to the axle mass; however, the wheel moment of inertia and bearing viscous losses at the wheel hub are separately modeled. The drive torque from the drivetrain is applied to the hub to accelerate the wheel. A simple brake model with coulomb and viscous friction is included to generate the required torque for decelerating and/or stopping the vehicle. In addition, the tire rolling resistance is added to the model as an additional source of energy loss. The tire in the vertical direction is modeled as a linear spring and damper connected in parallel. The longitudinal traction force is calculated using the Pacejka model (Pacejka and Bakker, 1993), which is calculated as a nonlinear function of wheel slip and normal tire load. Finally, the aerodynamic drag is included as a quadratic function of vehicle speed.
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International Journal of Heavy Vehicle Systems: L.S. Louca, D.G. Rideout, J.L. Stein, and G.M. Hulbert Generating Proper Dynamic Models for Truck Mobility and Handling
4.2 Full Model Response and Activities The full model is used to simulate full-throttle acceleration from standstill on a flat road, braking to 30 mph, acceleration to 50 mph, and finally braking to bring the vehicle to a full stop. Initially, the engine is idling with the brakes applied, and at t = 5 s the brakes are released and full throttle is applied to start the cycle. The driver model automatically adjusts the throttle and brake signal in order for the vehicle to follow the prescribed speed profile. The response of the vehicle during this maneuver is shown in Figure 13. The vehicle first accelerates to reach maximum speed while upshifting through all four gears. Then the brakes are applied to decelerate the vehicle to 30 mph and downshift from 4th to 3rd gear. The vehicle accelerates again to reach 50 mph with an upshift to 4th gear. Finally, the brakes are applied to bring the vehicle to full stop and the gears downshift from 4th to 1st gear.
Vehicle speed [mph]
70 60 50 40 30 20 10 0
0
10
20
30
40
50 Time [s]
60
70
80
90
100
0
10
20
30
40
50 Time [s]
60
70
80
90
100
Gear number[-]
5 4 3 2 1 0
Figure 13: Vehicle response for the acceleration/braking maneuver There are 55 energy elements in the model. The time window used to calculate the activity is the time needed for the vehicle to complete both the acceleration and braking maneuver. The sorted element activity indices and cumulative activity indices for this maneuver are plotted in Figure 14 and the numerical values given in Appendix II. Notice that the cumulative activity reaches approximately 80% after including the first five most important elements. The most important element (44% of the total activity) is the vehicle mass in the forward direction. The least important elements are the damping of the flexible shafts in the drivetrain. The most active elements are related to the dynamics in the longitudinal direction, while the elements in the vertical and pitch directions are at the bottom of the list as the least important elements. Also, the powertrain elements associated with energy losses have much higher activities than the energy storage elements (inertias and compliances). Page 16 of 24
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International Journal of Heavy Vehicle Systems: L.S. Louca, D.G. Rideout, J.L. Stein, and G.M. Hulbert Generating Proper Dynamic Models for Truck Mobility and Handling
100 Cumulative Activity Index
80
[%]
60 40 20 0
Activity Index 5
10
15
20
25 30 35 Element Ranking
40
45
50
55
Figure 14: Sorted activity indices and cumulative index 4.3 Reduced Model A reduced model is generated based on the above activity index results. The element elimination starts with the least important element (Drivetrain\Transmission\B1) and continues with elements of higher activity. The last 36 elements are eliminated from the model, resulting in a reduced model that maintains 99.22% of the total activity. Most of the eliminated elements are the inertial, compliant and resistive effects of the shafts in the drivetrain. Other eliminated elements include the suspension and tire compliances, body moment of inertia and inertia of the axles in the vertical direction. The elimination of these low activity elements produces a reduced model that has no pitch or vertical motion, which might be expected since we have a heavy vehicle accelerating on a flat road. The reduced model only includes the body and axle masses in the longitudinal direction along with the aerodynamic drag. In addition, the wheel inertias, rolling resistances, brake losses, and tire slips are included. Finally, the churning losses of the differential, the charging and churning losses of the transmission, the transmission and differential gear efficiency and engine inertia have high activity, and therefore are included in the reduced model. The reduced model is recompiled and used to predict the vehicle response for the same acceleration/braking maneuver. Figure 15 compares the vehicle speed and acceleration as predicted by the reduced model versus the full model. The vehicle speed predictions from the two models are almost identical. There is only a small discrepancy in the prediction of the vehicle acceleration. Note that the responses are quite similar, except for the high frequency effects that are eliminated from the reduced model. The computational efficiency of the reduced model is improved by a factor of 2 when using the Euler fixed-step integration algorithm. A comparison of variables associated with the engine performance is also performed to assess the accuracy of the reduced model over all vehicle components and not just the vehicle dynamics. Figure 16 shows the comparison of engine speed and torque, which are required for accurate fuel economy predictions. The disagreement is minimal between the full and reduced models. Notice that the high frequency oscillations are not present in these variables, even in the full model, since they are filtered by the fluid coupling of the torque converter. The full model predicts 0.5238 kg fuel consumption for performing this maneuver, while the reduced model predicts 0.5231 kg of fuel. This amounts to 0.13% loss in accuracy, which is negligible compared to the improvements in the model complexity and computational efficiency. Page 17 of 24
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International Journal of Heavy Vehicle Systems: L.S. Louca, D.G. Rideout, J.L. Stein, and G.M. Hulbert Generating Proper Dynamic Models for Truck Mobility and Handling
Vehicle Speed [mph]
70
50 40 30 20 10 0
2
Vehicle Acceleration [m/s ]
Full Reduced
60
0
10
20
30
40
50 Time [s]
60
70
80
90
100
0
10
20
30
40
50 Time [s]
60
70
80
90
100
2 1 0 -1 -2 -3 -4
Figure 15: Comparison of the full and reduced model (vehicle dynamics)
Engine Speed [rpm]
3000 Full Reduced
2500 2000 1500 1000
Engine Torque [N*m]
500
0
10
20
30
40
50 Time [s]
60
70
80
90
100
0
10
20
30
40
50 Time [s]
60
70
80
90
100
600 400 200 0
Figure 16: Comparsion of the full and reduced model (engine) Page 18 of 24
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International Journal of Heavy Vehicle Systems: L.S. Louca, D.G. Rideout, J.L. Stein, and G.M. Hulbert Generating Proper Dynamic Models for Truck Mobility and Handling
5. DISCUSSION While some outcomes of the model reduction procedure may seem intuitive to an expert in vehicles, they may provide critical information to a modeler with less domain expertise. In addition, systematically identifying the elements that contribute the most to a specific maneuver is in general not a trivial task. Generating the proper model with MORA provides critical information to the engineer. Perhaps the most readily assessable benefit is that the activity identifies the important parameters (elements) relative to a particular scenario. Thus, even if the model is not reformulated into reduced form, having a rank ordered list of the parameter importance directs the designer towards the design features that can produce the greatest effect on the system. Another benefit of ranking parameter importance is that the time and effort (cost) of obtaining model parameters can be reduced. A full model can be created with low precision parameter values. Then, upon learning which parameters are important, resources can be directed into obtaining higher precision values of the important parameters. Design optimization would also be facilitated by identification of unimportant parameters that need not be perturbed to achieve the design objective. Beyond the benefits described above, another expected benefit from the elimination of energy storage elements from the model is computational efficiency. The computational efficiency of the reduced model should be improved by both the reduction of the number of states and the increase of the integration time step due to elimination of high frequency content from the model. However, the removal of inertial elements from a model, such as in the handling case study, produces models that cannot be formulated into explicit equations using classical methods. Thus, formulation of the reduced models remains an important research topic for not only computational efficiency but also for automating the generation of proper models as described previously. In addition, based on current knowledge, the designer would be required to eliminate elements systematically until the model complexity is just sufficient to produce the necessary accuracy. Presumably, this process could be automated for effortless element elimination, though this topic remains as future work. Using MORA, a significant reduction in the model size is shown to be possible without sacrificing useful accuracy. However, no attempt has been made in this work to quantify accuracy as a function of activity. The assumption is that energy elements with low activity cannot contribute significantly to the dynamics of an energetic based dynamic system. The relation between activity and accuracy has been systematically addressed by Sendur et al. (2003); Sendur et al. (2002); Sendur (2002), by means of a quantitative metric for assessing the accuracy of reduced models as compared to full models. Given this measure of accuracy, a model can be reduced by eliminating low activity elements until the accuracy of the variables of interest reaches a physically intuitive threshold. The specific modeling and simulation environment (e.g., 20SIM, AutoSim) is immaterial to the application of MORA as long as the time series of conjugate power variables such as velocity and force, pressure and flow rate can be extracted. In fact, a previous effort has demonstrated the utilization of MORA using the ADAMS simulation environment (Christensen et al., 2000). The application of MORA is also insensitive to the formulation used to generate the equations of motion. In this work the bond graph and Kane’s formulations were used; however, MORA can be used in many other formulations used in everyday engineering practice. 6. SUMMARY AND CONCLUSIONS MORA has been used to reduce multibody vehicle dynamics and integrated engine-drivetrain-vehicle models. The time-domain reduction technique is applicable to highly nonlinear models in which rigid bodies move in multiple dimensions, and in which multiple subsystems (with elements from different energy domains) are Page 19 of 24
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International Journal of Heavy Vehicle Systems: L.S. Louca, D.G. Rideout, J.L. Stein, and G.M. Hulbert Generating Proper Dynamic Models for Truck Mobility and Handling
connected. In the first case study, a tractor semi-trailer model with 33 rigid body degrees of freedom is subjected to a lane change steering input and reduced to the proper model that predicts lateral acceleration, yaw rate, and damper force. The algorithm is shown to generate proper models of highly nonlinear multibody vehicle dynamics models that correctly predict transient behavior. A series of three reduced models was produced by defining different thresholds of activity. Results show that by removing 19% to 37% of the total number of elements from the full model, reduced models are produced that generate predictions within acceptable limits. It was also shown that the frequency content of the models appears to be reduced as the model size is reduced according to the activity metric. In the second case study, engine, powertrain, and vehicle dynamics subsystem models are assembled into an integrated vehicle mobility model. The proper model is generated to predict engine and vehicle variables for a full throttle acceleration followed by braking to a stop. In generating the reduced model, the energy-based metric is successfully applied across subsystem boundaries and is shown to be effective for integrated vehicle simulations. MORA results show that by removing 36 out of the 55 elements of the full model, a reduced model is produced which generates almost identical predictions as the full model. For acceleration and braking of this vehicle, the pitch motion does not need to be included in the model. Moreover, the computational efficiency of the reduced model is significantly improved. The energy based modeling metric can be used for the methodical reduction of automotive system models. Such reduced models can decrease the simulation based design cycle time by focusing on the most important elements of the systems and by reducing the computation time. The reduction in model complexity and size is achieved with minimal sacrifices in model accuracy. It is evident that activity and MORA can have a significant impact on modeling and simulation. Automation of MORA is feasible, and it would make the tool even more powerful and valuable to engineers. ACKNOWLEDGMENT The authors gratefully acknowledge the support of this work by the U.S. Army Tank Automotive Command (ARC DAAE 07-94-Q-BAA3) through the Automotive Research Center (ARC), University of Michigan. Steve Riley of MSC software is recognized for his work on modeling the M916 military truck. The contributions of Dan Grohnke, Steve Gravante and Xinqun Gui of International Truck and Engine Corporation are also gratefully acknowledged. REFERENCES 20SIM, 2003. 20SIM Pro User’s Manual, Version 3.3. The University of Twente - Controllab Products B.V. Enschede, The Netherlands. AMESim, 2002. Reference Manual, Version 4.0. IMAGINE Software, Roanne, France. Antoun, R.J., P.B. Hackert, M.C. O'Leary, and A. Sitchin, 1986. Vehicle Dynamic Handling Computer Simulation: Model Development, Correlation, and Application Using Adams. SAE paper 860574. Brown, F.T., 2001. Engineering System Dynamics: A Unified Graph-Centered Approach. Marcel Dekker, ISBN 0-8247-0616-1, New York, NY.
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Christensen, B.G., J.B. Ferris, and J.L. Stein, 2000. An Energy-Enhanced Design of Experiments Method Applied to Multi-Body Models. Proceedings of the 2000 ASME IMECE Conference, Dynamic Systems and Control Division, DSC-VOL. 69-1, pp. 527-534. Published by ASME, ISBN 0-7918-1931-0, New York, NY. Easy5, 2000. User’s Manual. The Boeing Company, Seattle, WA. Ferris, J.B., J.L. Stein, and M.M. Bernitsas, 1994. Development of Proper Models of Hybrid Systems. Proceedings of the 1994 ASME International Mechanical Engineering Congress and Exposition - Dynamic Systems and Control Division, Symposium on Automated Modeling: Model Synthesis Algorithms, pp. 629-636, November, Chicago, IL. Published by ASME, Book No. G0909B, New York, NY. Ferris, J.B., and J.L. Stein, 1995. Development of Proper Models of Hybrid Systems: A Bond Graph Formulation. Proceedings of the 1995 International Conference on Bond Graph Modeling, pp. 43-48, January, Las Vegas, NV. Published by the Society for Computer Simulation, ISBN 1-56555-037-4, San Diego, CA. Gillespie, T.D. and C.C. MacAdam, 1982. Constant Velocity Yaw/Roll Program User’s Manual. The University of Michigan Transportation Research Institute, UMTRI-82-39, Ann Arbor, MI. Karnopp, D.C., D.L. Margolis, and R.C. Rosenberg, 1990. System Dynamics: A Unified Approach. WileyInterscience, New York, NY. Louca, L.S., 1998. An Energy-Based Model Reduction Methodology for Automated Modeling. Ph.D. Thesis. The University of Michigan. Ann Arbor, MI. Louca, L.S., J.L. Stein, G.M. Hulbert, and J.K. Sprague, 1997. Proper Model Generation: An Energy-Based Methodology. Proceedings of the 1997 International Conference on Bond Graph Modeling, pp. 44-49, January, Phoenix, AZ. Published by the Society for Computer Simulation, ISBN 1-56555-103-6, San Diego, CA. Louca, L.S. and J.L. Stein, 2002. Ideal Physical Element Representation from Reduced Bond Graphs. Journal of Systems and Control Engineering: Special Issue on Bond Graphs, Vol. 216, No. 1, pp. 73-83. Published by the Professional Engineering Publishing, ISSN 0959-6518, Suffolk, United Kingdom. Louca, L.S., J.L. Stein, and D.G. Rideout, 2001. Integrated Proper Vehicle Modeling and Simulation Using a Bond Graph Formulation. Proceedings of the 2001 International Conference on Bond Graph Modeling, Vol. 33, No. 1, pp. 339(345, January, Phoenix, AZ. Published by the Society for Computer Simulation, ISBN 1(56555(221(0, San Diego, CA. MATLAB/SIMULINK, 2002. User’s Manual, Version 6.5. The MathWorks Inc., Natick, MA. Moore, B.C., 1981. Principal Component Analysis in Linear Systems: Controllability, Observability, and Model Reduction. IEEE Transactions on Automatic Control, Vol. AC-26/1, pp. 17-32. Pacejka, H.B. and E. Bakker, 1993. Tyre Models for Vehicle Dynamics Analysis. Proceedings of the 1st International Colloquium on Tyre Models for Vehicle Dynamics Analysis. Journal of Vehicle System Dynamics, Vol. 21, No. Supplement, pp. 1-18. Published by Swets and Zeitlinger, Lisse, The Netherlands. Rosenberg, R.C., and D.C. Karnopp, 1983. Introduction to Physical System Dynamics. McGraw-Hill, New York, NY. Sayers, M.W., 1991. Symbolic Vector/Dyadic Multibody Formalism for the Tree-Topology Systems. Journal of Guidance, Control, and Dynamics, Vol. 14-6, pp. 1240-1250. Published by the American Institute of Aeronautics and Astronautics, ISSN 0731-5090, Reston, VA. Page 21 of 24
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International Journal of Heavy Vehicle Systems: L.S. Louca, D.G. Rideout, J.L. Stein, and G.M. Hulbert Generating Proper Dynamic Models for Truck Mobility and Handling
Sayers, M.W. and S.M. Riley, 1996. Modeling Assumptions for Realistic Multibody Simulation of the Yaw and Roll Behavior of Heavy Trucks. SAE Paper 960173. Sendur, P., J.L. Stein, L.S. Louca, and H. Peng, 2003. An Algorithm for the assessment of reduced dynamic system models for design. Proceedings of the International Conference on Simulation and Multimedia in Engineering Education, pp. 92-101, Orlando, FL. Published by the Society for Modeling and Simulation International Simulation, ISBN 1(56555(261(1, San Diego, CA. Sendur, P., J.L. Stein, L.S. Louca, and H. Peng, 2002. A Model Accuracy and Validation Algorithm. Proceedings of the 2002 ASME International Mechanical Engineering Congress and Exposition, New Orleans, Louisiana. Published by American Society of Mechanical Engineers, ISBN 0-7918-1692-3, New York, NY. Sendur, P., 2002. Proper System Models for Design: Model Accuracy and Validity. Ph.D. Thesis. The University of Michigan, Ann Arbor, MI. Stein, J.L. and L.S. Louca, 1996. A Template-Based Modeling Approach for System Design: Theory and Implementation. Transactions of the Society for Computer Simulation International. Published by the Society for Computer Simulation, ISSN 0740-6797/96, San Diego, CA. Walker, D.G., J.L. Stein, and A.G. Ulsoy, 2000. An input-Output Criterion for Linear Model Deduction. Transactions of the ASME: Journal of Dynamic Systems, Measurement, and Control, Vol. 122, No. 3, pp. 507-513. Published by ASME, ISSN 0022-0434, New York, NY. Wilson, B.H. and J.L. Stein, 1995. An Algorithm for Obtaining Proper Models of Distributed and Discrete Systems. Transactions of the ASME: Journal of Dynamic Systems, Measurement, and Control, Vol. 117, No. 4, pp. 534 - 540. Published by ASME, ISSN 0022-0434, New York, NY.
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Appendix I: Multibody M916 tractor semi-trailer model activities 19%
Element
Ran k
Cumulative [%]
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85
1.313E-01 2.321E-01 3.326E-01 3.862E-01 4.371E-01 4.879E-01 5.385E-01 5.890E-01 6.392E-01 6.894E-01 7.392E-01 7.890E-01 8.093E-01 8.176E-01 8.249E-01 8.320E-01 8.382E-01 8.444E-01 8.505E-01 8.566E-01 8.625E-01 8.685E-01 8.743E-01 8.798E-01 8.853E-01 8.907E-01 8.957E-01 9.007E-01 9.055E-01 9.104E-01 9.152E-01 9.200E-01 9.248E-01 9.295E-01 9.342E-01 9.379E-01 9.416E-01 9.451E-01 9.485E-01 9.516E-01 9.546E-01 9.574E-01 9.602E-01 9.627E-01 9.644E-01 9.661E-01 9.675E-01 9.689E-01 9.702E-01 9.714E-01 9.727E-01 9.739E-01 9.751E-01 9.762E-01 9.773E-01 9.784E-01 9.795E-01 9.805E-01 9.816E-01 9.824E-01 9.833E-01 9.840E-01 9.848E-01 9.856E-01 9.863E-01 9.871E-01 9.878E-01 9.883E-01 9.889E-01 9.895E-01 9.901E-01 9.906E-01 9.912E-01 9.917E-01 9.922E-01 9.927E-01 9.932E-01 9.936E-01 9.940E-01 9.944E-01 9.948E-01 9.952E-01 9.956E-01 9.959E-01 9.962E-01
Reduction
Page 23 of 24
Activity [%] 1.313E+01 1.007E+01 1.005E+01 5.363E+00 5.086E+00 5.082E+00 5.056E+00 5.051E+00 5.023E+00 5.019E+00 4.981E+00 4.979E+00 2.035E+00 8.277E-01 7.307E-01 7.080E-01 6.227E-01 6.190E-01 6.109E-01 6.055E-01 5.953E-01 5.930E-01 5.880E-01 5.482E-01 5.481E-01 5.389E-01 5.007E-01 4.955E-01 4.868E-01 4.849E-01 4.823E-01 4.803E-01 4.780E-01 4.739E-01 4.711E-01 3.711E-01 3.677E-01 3.437E-01 3.395E-01 3.171E-01 3.000E-01 2.765E-01 2.764E-01 2.504E-01 1.738E-01 1.676E-01 1.420E-01 1.376E-01 1.315E-01 1.245E-01 1.241E-01 1.234E-01 1.224E-01 1.099E-01 1.090E-01 1.076E-01 1.072E-01 1.066E-01 1.059E-01 8.351E-02 8.312E-02 7.909E-02 7.583E-02 7.538E-02 7.483E-02 7.441E-02 7.080E-02 5.859E-02 5.835E-02 5.759E-02 5.570E-02 5.517E-02 5.502E-02 5.233E-02 5.070E-02 5.060E-02 4.950E-02 4.333E-02 4.078E-02 4.020E-02 3.964E-02 3.606E-02 3.592E-02 3.472E-02 3.336E-02
23% Element
Element
Reduction
Name
Trailer Inertia X Right Tire 1 X Left Tire 1 X Trailer Inertia Y Left Tire 3i X Left Tire 2i X Right Tire 3i X Right Tire 2i X Right Tire 2o X Right Tire 3o X Left Tire 3o X Left Tire 3o X Tractor Inertia X Tractor Inertia Y Right Tire 1 Y Left Tire 1 Y Right Tire 3i Y Left Tire 3i Y Right Tire 3o Y Left Tire 3o Y Right Tire 6i Y Axle 3 Inertia X Left Tire 6i Y Right Tire 6o Y Axle 2 Inertia X Left Tire 6o Y Right Tire 5i Y Left Tire 5i Y Axle 1 Inertia X Right Tire 5o Y Right Tire 2i Y Left Tire 2i Y Left Tire 5o Y Right Tire 2o Y Left Tire 3o Y Right Tire 4i Y Left Tire 4i Y Right Tire 4o Y Left Tire 4o Y Axle 6 Inertia X Axle 3 Inertia Y Axle 4 Inertia X Axle 5 Inertia X Axle 2 Inertia Y Axle 6 Inertia Y Trailer Inertia ZZ Axle 5 Inertia Y Axle 1 Inertia Y Axle 4 Inertia Y Suspension Stiffness R3 Suspension Stiffness L3 Suspension Stiffness L2 Suspension Stiffness R2 Left Tire 1 Z Left Tire 4o Z Left Tire 6o Z Right Tire 4o Z Right Tire 1 Z Right Tire 6o Z Left Tire 5o Z Right Tire 5o Z Suspension Stiffness L5 Left Tire 4i Z Left Tire 6i Z Right Tire 4i Z Right Tire 6i Z Suspension Stiffness R5 Left Tire 5i Z Right Tire 5i Z Suspension Stiffness L4 Right Tire 3o Z Left Tire 3o Z Suspension Stiffness L6 Suspension Stiffness R4 Right Tire 2o Z Left Tire 3o Z Suspension Stiffness R6 Suspension Stiffness L1 Suspension Stiffness R1 Right Tire 3i Z Left Tire 3i Z Right Tire 2i Z Left Tire 2i Z Axle 4 Stiffness XX Axle 6 Stiffness XX
37%
Ran k
Cumulative [%]
86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169
9.965E-01 9.967E-01 9.969E-01 9.971E-01 9.973E-01 9.975E-01 9.977E-01 9.979E-01 9.981E-01 9.983E-01 9.985E-01 9.987E-01 9.988E-01 9.990E-01 9.991E-01 9.991E-01 9.992E-01 9.992E-01 9.993E-01 9.993E-01 9.994E-01 9.994E-01 9.994E-01 9.995E-01 9.995E-01 9.995E-01 9.996E-01 9.996E-01 9.996E-01 9.997E-01 9.997E-01 9.997E-01 9.997E-01 9.998E-01 9.998E-01 9.998E-01 9.998E-01 9.999E-01 9.999E-01 9.999E-01 9.999E-01 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00
Element
Activity [%] 2.494E-02 2.232E-02 2.226E-02 2.171E-02 1.992E-02 1.939E-02 1.933E-02 1.923E-02 1.919E-02 1.898E-02 1.810E-02 1.768E-02 1.763E-02 1.347E-02 7.628E-03 7.422E-03 5.826E-03 5.219E-03 4.665E-03 4.280E-03 4.174E-03 3.573E-03 3.550E-03 3.417E-03 3.381E-03 3.071E-03 3.050E-03 3.032E-03 2.939E-03 2.911E-03 2.890E-03 2.889E-03 2.790E-03 2.699E-03 2.665E-03 2.593E-03 2.590E-03 2.385E-03 2.296E-03 2.281E-03 2.225E-03 1.396E-03 6.241E-04 5.906E-04 4.681E-04 3.938E-04 3.938E-04 3.938E-04 3.071E-04 1.408E-04 1.323E-04 1.229E-04 1.160E-04 9.871E-05 9.264E-05 8.118E-05 6.734E-05 6.734E-05 5.918E-05 5.586E-05 5.328E-05 5.210E-05 3.889E-05 3.696E-05 3.236E-05 2.621E-05 7.799E-06 7.764E-06 3.681E-06 3.663E-06 3.379E-06 3.360E-06 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00
Reduction Element
Name
Trailer Inertia XX Left Tire 4i X Left Tire 6i X Left Tire 5i X Left Tire 5o X Right Tire 4i X Right Tire 6i X Left Tire 6o X Left Tire 4o X Right Tire 5i X Right Tire 5o X Right Tire 6o X Right Tire 4o X Tractor Inertia ZZ Axle 2 Stiffness XX Axle 3 Stiffness XX Trailer Inertia Z Axle 1 Stiffness XX Right Tire 1 ZZ Left Tire 1 ZZ Hitch Stiffness XX Right Tire 3o ZZ Right Tire 3i ZZ Left Tire 3i ZZ Left Tire 3o ZZ Right Tire 2o ZZ Right Tire 2i ZZ Right Tire 6o ZZ Left Tire 2i ZZ Left Tire 3o ZZ Right Tire 6i ZZ Left Tire 6o ZZ Left Tire 6i ZZ Right Tire 5o ZZ Right Tire 5i ZZ Left Tire 5o ZZ Left Tire 5i ZZ Right Tire 4o ZZ Right Tire 4i ZZ Left Tire 4o ZZ Left Tire 4i ZZ Tractor Inertia XX Axle 5 Stiffness XX Trailer Inertia YY Tractor Inertia Z Axle 1 Inertia ZZ Axle 3 Inertia ZZ Axle 2 Inertia ZZ Axle 1 Inertia Z Tractor Inertia YY Axle 1 Inertia XX Axle 3 Inertia Z Axle 6 Inertia Z Axle 5 Inertia Z Axle 2 Inertia Z Axle 4 Inertia Z Axle 4 Inertia ZZ Axle 6 Inertia ZZ Axle 5 Inertia XX Axle 5 Inertia ZZ Axle 6 Inertia XX Axle 4 Inertia XX Suspension Damping L1 Axle 3 Inertia XX Suspension Damping R1 Axle 2 Inertia XX Suspension Damping L5 Suspension Damping R5 Suspension Damping R6 Suspension Damping L6 Suspension Damping R4 Suspension Damping L4 Axle 6 Inertia YY Axle 5 Inertia YY Axle 4 Inertia YY Axle 3 Inertia YY Axle 2 Inertia YY Axle 1 Inertia YY Hitch Stiffness ZZ Hitch Stiffness YY Suspension Damping R3 Suspension Damping R2 Suspension Damping L3 Suspension Damping L2
Last printed December 29, 2003
International Journal of Heavy Vehicle Systems: L.S. Louca, D.G. Rideout, J.L. Stein, and G.M. Hulbert Generating Proper Dynamic Models for Truck Mobility and Handling
Note: There are two tires (left and right, e.g., element numbers 5 and 7) for the first axle and four tires (one inside (2i) and one outside (2o) at each side, e.g., element numbers 8 and 9) for all the other axles. X, Y, Z are the three translational degrees of freedom. XX, YY, ZZ are the three rotational degrees of freedom. L1 and R1 are the left and right side of axle 1, respectively. 2o and 2i are the outside and inside tires of axle 2, respectively. The last 12 (rank 158-169) elements have identically zero activity because they were already removed from the given full model.
Appendix II: Integrated class VI vehicle model activities Rank 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55
Page 24 of 24
Activity [%] 4.39180e+001 1.00088e+001 9.92285e+000 8.92438e+000 7.28187e+000 2.87526e+000 2.70224e+000 2.69429e+000 1.63414e+000 1.56451e+000 1.55910e+000 1.26180e+000 1.21185e+000 8.42608e-001 7.72682e-001 5.90806e-001 5.48879e-001 4.78390e-001 4.28669e-001 3.16462e-001 1.69558e-001 8.01511e-002 6.52237e-002 2.48577e-002 2.41970e-002 2.12537e-002 1.59390e-002 1.26725e-002 1.21678e-002 8.66206e-003 8.16314e-003 7.77640e-003 3.09554e-003 2.82317e-003 1.19879e-003 1.13653e-003 1.01974e-003 7.56227e-004 4.62251e-004 4.54751e-004 2.53804e-004 2.50870e-004 1.29187e-004 1.21779e-004 2.58300e-005 2.36366e-005 2.30720e-005 1.20926e-005 6.64077e-006 2.60259e-006 2.13250e-006 1.47409e-006 1.29486e-006 2.12440e-007 1.55985e-008
Cumulative [%] 43.91802 53.92680 63.84965 72.77403 80.05590 82.93116 85.63340 88.32769 89.96183 91.52633 93.08543 94.34723 95.55908 96.40169 97.17437 97.76518 98.31405 98.79244 99.22111 99.53758 99.70713 99.78729 99.85251 99.87737 99.90156 99.92282 99.93876 99.95143 99.96360 99.97226 99.98042 99.98820 99.99129 99.99412 99.99532 99.99645 99.99747 99.99823 99.99869 99.99915 99.99940 99.99965 99.99978 99.99990 99.99993 99.99995 99.99997 99.99999 99.99999 99.99999 100.00000 100.00000 100.00000 100.00000 100.00000
Element Name VehiclePitch\body\Mx VehiclePitch\tire_rear\B_brake VehiclePitch\tire_front\B_brake VehiclePitch\body\Raero Drivetrain\TC\FluidCoupling VehiclePitch\axle_rear\Mx VehiclePitch\tire_rear\B_slip VehiclePitch\tire_rear\B_rolling VehiclePitch\axle_front\Mx Drivetrain\Prop_Diff\differential VehiclePitch\tire_front\B_rolling VehiclePitch\tire_front\B_slip Drivetrain\Prop_Diff\B_churning VehiclePitch\tire_rear\Jwheel Engine\J Drivetrain\Transmission\T_charging Drivetrain\Transmission\gear Drivetrain\Transmission\B_churning VehiclePitch\tire_front\Jwheel Drivetrain\Transmission\I1 Drivetrain\Transmission\I2 Drivetrain\TC\I_turbine VehiclePitch\axle_front\C_susp Drivetrain\Prop_Diff\I_axle VehiclePitch\axle_rear\C_susp VehiclePitch\tire_front\C_tire Drivetrain\Prop_Diff\I2 VehiclePitch\tire_rear\C_tire VehiclePitch\body\My Drivetrain\Prop_Diff\C_axle_out Drivetrain\Prop_Diff\I1 Drivetrain\Prop_Diff\I3 Drivetrain\Transmission\C2 VehiclePitch\body\J Drivetrain\Prop_Diff\C1 Drivetrain\Prop_Diff\C2 Drivetrain\Prop_Diff\C_axle_in VehiclePitch\axle_rear\My Drivetrain\Transmission\C3 VehiclePitch\axle_front\My Engine\B VehiclePitch\axle_front\B_susp VehiclePitch\tire_rear\B_bearing VehiclePitch\tire_front\B_bearing Drivetrain\Transmission\B2 Drivetrain\Transmission\C1 VehiclePitch\axle_rear\B_susp Drivetrain\Prop_Diff\B_axle_out VehiclePitch\tire_front\B_tire Drivetrain\Prop_Diff\B_axle_in Drivetrain\Prop_Diff\B1 VehiclePitch\tire_rear\B_tire Drivetrain\Prop_Diff\B2 Drivetrain\Transmission\B3 Drivetrain\Transmission\B1
Last printed December 29, 2003
