MB simulations for vehicle dynamics: reduction through parameters estimation Gubitosa Marco The aim of this activity is to propose a methodology applicable for parameters estimation in vehicle dynamics, aiming at generating reduced models to be adopted for functional analyses and real time simulations with the focus on enabling a model conversion scheme, allowing building a communication bridge between the 1D and the 3D simulation domains. The benchmark case, i.e. the reference high fidelity model, is defined in the 3D multibody environment of LMS Virtual.Lab Motion, while the simplified model is developed symbolically with the help of Maple and implemented in the block scheme oriented interface of Imagine.Lab AMESim. To estimate the physical and structural parameters of the detailed benchmark model for use in the functional model, it is important to make clear how and to what extent the response of the system depends on each parameter. Therefore sensitivity analysis and optimization loops are programmed to firstly define the most effective contribution to the behaviour of the simplified model and secondly asses a good correlation in the dynamic performance. Reference vehicle model The use of MBS software allows the modeling and simulation of a range of vehicle subsystem representing the chassis, engine, driveline and body areas of the vehicle as shown in Figure 1, where is intended that multibody system models for each of those areas are integrated to provide a detailed representation of the complete real vehicle. Figure 1: Integration of subsystems in a full vehicle model and detail of the vehicle dynamic areas of interest Here also the modeling of road and driver are included as elements of what is considered a full vehicle system model. Restricting the discussion of full vehicle system models to a level appropriate for the vehicle dynamics, a detailed modeling of the suspension systems, anti roll bars, steering system and tires is needed as evidenced with ovals in Figure 1. Of course to
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MB simulations for vehicle dynamics: reduction
through parameters estimation
Gubitosa Marco
The aim of this activity is to propose a methodology applicable for parameters estimation in
vehicle dynamics, aiming at generating reduced models to be adopted for functional analyses
and real time simulations with the focus on enabling a model conversion scheme, allowing
building a communication bridge between the 1D and the 3D simulation domains.
The benchmark case, i.e. the reference high fidelity model, is defined in the 3D multibody
environment of LMS Virtual.Lab Motion, while the simplified model is developed
symbolically with the help of Maple and implemented in the block scheme oriented interface
of Imagine.Lab AMESim. To estimate the physical and structural parameters of the detailed
benchmark model for use in the functional model, it is important to make clear how and to
what extent the response of the system depends on each parameter. Therefore sensitivity
analysis and optimization loops are programmed to firstly define the most effective
contribution to the behaviour of the simplified model and secondly asses a good correlation in
the dynamic performance.
Reference vehicle model The use of MBS software allows the modeling and simulation of a range of vehicle subsystem
representing the chassis, engine, driveline and body areas of the vehicle as shown in Figure 1,
where is intended that multibody system models for each of those areas are integrated to
provide a detailed representation of the complete real vehicle.
Figure 1: Integration of subsystems in a full vehicle model and detail of the vehicle dynamic areas of interest
Here also the modeling of road and driver are included as elements of what is considered a
full vehicle system model. Restricting the discussion of full vehicle system models to a level
appropriate for the vehicle dynamics, a detailed modeling of the suspension systems, anti roll
bars, steering system and tires is needed as evidenced with ovals in Figure 1. Of course to
complete the model and give the possibility to run simulations with an acceptable realistic
level of the vehicle’s response, models of driver inputs as well as engine and driveline final
effects should be comprised. Besides the inertia characteristics of the sprung (and un-sprung)
masses have to be included. A creation of a virtual environment is than an aspect that can be
considered.
Set up of the MB model
Figure 2: Front double wishbone and Rear 5-link suspension (example of the Acura RL and TSX 2008)
The double wishbone front suspension is constructed with short upper wishbones, lower
transverse control arms and longitudinal rods whose front mounts absorb the dynamic rolling
stiffness of the radial tires. The spring shock absorbers are supported via fork-shaped struts on
the transverse control arms in order to leave space to the crank shafts and are fixed within the
upper link mounts.
As its name indicates, the rear suspension employs five links. The hub-carrier/spring-shock-
absorber mount is located by five tubular links: a trailing link, lower link, lower control link,
upper link, and upper leading link. Car manufacturers claim that this system gives even better
road-holding properties, because all the various joints make the suspension almost infinitely
adjustable.
The linkages of the suspension parts are realized partially with bushings, representing the
compliance elements, and for some of them ideal joints have been included, realizing a
kinematically constrained mechanism as represented in Figures 3 and 4.
The bushing element available in Virtual Lab Motion defines a six degree-of-freedom element
between two bodies, producing forces along and torques about the three principal axes of the
element attachments. The bushing characteristics are defined as a combination of six values of
stiffness and six values of damping which are normally defined by non linear spline curves.
The equation below describes the formulation for forces in the bushing.
F1 = Kz + DŜ + FK(z) + FD(Ŝ)
F2=-F1
where
F2 and F1 are the force vectors applied to body 1 and 2
K and D are the stiffness and damping matrixes
z and Ŝ are the relative displacement and velocity vectors between the two bodies
FK(z) and FD(Ŝ) are the forces expressing stiffness and damping as functions of relative
displacement and velocity in a nonlinear sense.
A similar formulation is used for the calculation of torque reactions in function of relative
rotation and rotation velocity between the connected bodies.
Table 1: Connection types (Joints and Bushings) for the front suspension
Figure 3: Linkage of the Front Suspension system
Table 2: Connection types (Joints and Bushings) for the rear suspension
Figure 4: Linkage of the Rear Suspension system
Assumptions and force elements considered
Antiroll Bars
They're also known as sway-bars or anti-sway-bars. The function of the anti-roll bars is to
reduce the body roll inclination during cornering and to influence the cornering behavior in
terms of under- or over-steering. The anti-roll bar is usually connected to the front, lower
edge of the bottom suspension joint. It passes through two pivot points under the chassis,
usually on the subframe and is attached to the same point on the opposite suspension setup.
Hence, the two suspensions are not any more connected only due to the subframe and the
chassis, but effectively they are joined together through the anti-roll bar. This connection
clearly affects each one-sided bouncing.
Figure 5: Anti-roll bar loaded by vertical forces
In the model here implemented it has been considered a lumped torsional stiffness granted
to a bushing element located at the mid-point connection of the two bodies representing the
left and right portions of the antiroll-bar.
Damper, springs and end stops
Figure 6: Example of a spring dumper structure with notable elements listed
The force elements included in the strut here shown are re reproduced in the MB model as
non-linear splines for the damping characteristic and linear stiffnesses for the main
spring and end stops.
Figure 7: Setting the damping curves
Front Suspension
Stiffness 48000 N/m Coil Spring
Preload 5800 N
AntiRoll Bar Torsional Stiffness 2000 Nm/rad
Stiffness 350000 N/m Bump Stop
Clearance 40 mm
Stiffness 680000 N/m Rebound Stop
Clearance 50 mm
Rear Suspension
Stiffness 36000 N/m Coil Spring
Preload 3500 N
AntiRoll Bar Torsional Stiffness 500 Nm/rad
Stiffness 350000 N/m Bump Stop
Clearance 26 mm
Stiffness 700000 N/m Rebound Stop
Clearance 75 mm
Table 3: Characteristics of the Force Elements
Steering system
The simplest and also most common steering system to be created is the rack and pinion
steering system. Firstly the rotations of the steering wheel are transformed by the steering box
to the rack travel which is travels along a straight rail activated by the rotations of a pinion. At
the extremities of the rack two tie rods permit the transformation of this translational
movement to the rotation around the steering axis of the suspension. Hence the overall
steering ratio depends on the ratio of the steering box and the kinematics of the steering
linkage.
Table 4: Connection types for the steering mechanism
Figure 7: Representation of the joints and configuration of the mechanism of the steer
In Figure 7 the hierarchical organization of the joints is shown. Here is possible to see (in the
block scheme) two green arrows indicating the revolute joint of the steer on the chassis and
the translational joint of the rack. This means that there is a correlation between the two (set
by a relative driver) which is programmed by the steering ratio.
Tires modelling
An accurate modelling of the tire force elements is achieved by including the so called TNO-
MF tire (version 6.0), which is based on the renowned ‘Magic Formula’ tire model of
Pacejka. The model takes as input a series of parameters (i.e. a vector with more than 100
elements) for each calculation to be performed, which are empirically determined coefficients
that address the complexity of the model.
Equations of motion
Virtual.Lab Motion is based on a Cartesian coordinates approach for the assembly of the
equations of motion. The solver uses Euler parameters to represent the rotational degrees of
freedom (avoiding therefore the intrinsic singularity of the angular notation) and Lagrangian
formulation for the assembly and generation of equations of motion. The joints between
bodies are expressed in a set of algebraic equations, subsequently assembled in a second
derivative structure, obtaining finally a set of Differential Algebraic Equations (DAEs)
packable in the following form:
( ) ( )( )
( )
=
γ
qqQ
λ
q
0qΦ
qΦqM tT ,, &&&
Here
M is the mass matrix
q is the vector of the generalized coordinates
Q is the vector of the generalized forces applied to the rigid bodies
λλλλ is the vector of the Lagrange multipliers
ΦΦΦΦ is the Jacobian of the constraint forces
γ the right-hand-side of the second derivative the constraint equations
This model includes 52 bodies; therefore a total of 52 x 7 = 364 configuration parameters
are used by the pre-processor to build the set of equations of motion.
For the settling configuration, in which the vehicle is just let rest on the ground with null
initial conditions, joints and drivers are for a total of 234, therefore leaving the system with
130 degrees of freedom.
While setting up a manoeuvre, instead, additional constraint is added to the system in terms of
position driver on the steering wheel, commanded in open loop, and forces are acting on the
wheel’s revolute joints to represent the driving torque. Moreover, non-zero initial conditions
at velocity and position level are added to every body.
Between the different solvers solution proposed in the Virtual.Lab Motion (here below
reported), the BDF has been selected, granting a good stable behaviour for such a stiff system.