多体动力学和非线性有限元联合仿真

A New Solution For Coupled Simulation Of Multi-Body Systems And Nonlinear Finite Element Models Giancarlo CONTI, Tanguy MERTENS, Tariq SINOKROT
(LMS, A Siemens Business)
Hiromichi AKAMATSU, Hitoshi KYOGOKU, Koji HATTORI
(NISSAN Motor Co., Ltd.)
1 Introduction
One of the most common challenges for flexible multi-body systems is the ability to properly take into account the nonlinear effects that are present in many applications. One particular case where these effects play an important role is the dynamic modeling of twist beam axles in car suspensions: these components, connecting left and right trailing arms and designed in a way that allows for large torsional deformations, cannot be modeled as rigid bodies and represent a critical factor for the correct prediction of the full-vehicle dynamic behavior.
The most common methods to represent the flexibility of any part in a multi-body mechanism are based on modal reduction techniques, usually referred to as Component Mode Synthesis (CMS) methods, which predict the deformation of a body starting from a preliminary modal analysis of the corresponding FE mesh. Several different methods have been developed and verified, but most of them can be considered as variations of the same approach based on a limited set of modes of the structure, calculated with the correct boundary conditions at each interface node with the rest of the mechanism, allowing to greatly reduce the size of system’s degrees of freedom from a large number of nodes to a small set of modal participation factors. By properly selecting the number and frequency range of the modes, as well as the boundary conditions at each interface node [1], it is possible to accurately predict the static and dynamic deformation of the flexible body with remarkable improvements in terms of CPU time: this makes these methods the standard approach to reproduce the flexibility of components in a multi-body environment. Still, an important limitation inherently lies in their own foundation: since displacements based on modal representation are by definition linear, any nonlinear phenomena cannot be correctly simulated. For example, large deformations like twist beam torsion during high lateral acceleration cornering maneuvers typically lead to geometric nonlinearities, preventing any linear solution from accurately predicting most of the suspension’s elasto-kinematic characteristics like toe angle variation, wheel center position, vertical stiffness.
One possible solution to overcome these limitations while still working with linear modal reduction methods is the sub-structuring technique [2]: the whole flexible body is divided into sub-structures, which are connected by compatibility constraints preventing the relative motion of the nodes that lie between two adjacent sub-structures. Standard component mode synthesis methods are used in formulating the equations of motion, which are written in terms of generalized coordinates and modal participation factors of each sub-structure. The idea behind it is that each sub-portion of the whole flexible structure will undergo smaller deformations, hence remaining in the linear flexibility range. By properly selecting the cutting sections it is usually possible to improve the accuracy of results (at least in terms of nodal displacements: less accuracy can be expected for stress and strain distribution). Another limitation of these methods is the preliminary work needed to re-arrange the FE mesh, although some CAE products already offer automatic processes enabling the user to skip most of the re-meshing tasks and hence reducing the modeling efforts.
An alternative approach to simulate the behavior of nonlinear flexible bodies is based on a co-simulation technique that uses a Multi-body System (MBS) solver and an external nonlinear Finite Element Analysis (FEA) solver. Using this technique one can model the flexible body in the external nonlinear FEA code and the rest of the car suspension system in the MBS environment. The loads due to the deformation of the body are calculated externally by the FEA solver and communicated to the MBS solver at designated points where the flexible body connects to the rest of the multi-body system. The MBS solver, on the other hand, calculates displacements and velocities of these points and communicates them to the nonlinear FEA solver to advance the simulation. This approach doesn’t suffer from the limitations that arise from the linear modeling of the flexibility of a body. This leads to more accurate results, albeit at the price of much larger CPU time. In fact, simulation results are strongly affected by the size of the communication time step between the two solvers: a better accuracy (and more stable solver convergence) can be generally obtained by using smaller time steps which require larger calculation times, as shown also in [3].
2 Overview of the activity
This paper presents the results of a benchmark activity performed in collaboration with Nissan Auto where a new FE-MBS variable-step co-simulation technique was used: a coupling at the iteration level currently implemented in commercial FEA package LMS SAMCEF Mecano [4] and general purpose multi-body system package LMS b Motion [5]. In this technique each solver uses its own integrator but only one Newton solver is used. In this case one solver is designated as the master and will be responsible for solving the Newton iterations. The coupled iterations continue until both solvers satisfy their own solution tolerances and convergence is achieved. The co-simulation process is organized by means of a supervisor code that manages the data exchange and determines the new time step of integration for both solvers. Further technical details on this “coupled simulation“ method, as well as a comparison with the variable-step co-simulation method, are available in [6].
A multi-body model of a rear twist beam suspension has been created, where the flexibility of the twist beam was simulated with three alternative modeling techniques to be compared:
- Component Mode Synthesis (Craig-Bampton method)
- Linear sub-structuring
- Nonlinear FE-MBS coupled simulation.
As a further step also the two bushings connecting the twist beam with the car body, originally modeled in b Motion as standard force elements with nonlinear stiffness and damping characteristics for all directions, have been replaced by two SAMCEF Mecano nonlinear flexible bodies.
Two different suspension events have been simulated in order to compare the results from the different modeling methods:
- Suspension roll (opposite wheel vertical travel applied at wheel centers)
- Braking in turn (dynamic loads applied at wheel centers).
Figure 1 shows the b Motion suspension model used for this activity, where the FE mesh models of twist beam and bushings are also displayed:
3 Modeling and simulations
3.1 Model validation
As a first step a multi-body model of rear suspension was created in b Motion with input data provided by Nissan Auto from a pre-existing model developed with another multi-body software package: hardpoints location, bodies mass and inertia data, kinematic and compliant connections characteristics, properties of coil springs, shock absorbers, end stop elements. Since the original model included a flexible twist beam based on a modal reduction method (Craig-Bampton) the same original mode set has been used to obtain a linear flexible representation of the twist beam in b Motion. Then a suspension roll has been simulated in both environments in order to validate Motion results with the data from the source model, obtained by applying a vertical displacement in opposite directions at the two wheel centers. The main elasto-kinematic suspension characteristics have been compared: toe and camber variation, wheel center longitudinal and lateral displacements, vertical stiffness. In fig.2 the vertical force at wheel center and the toe angle variation are plotted versus the wheel vertical displacement: the differences between the two models are negligible.
Fig. 1
b Motion multi-body model of rear suspension with flexible twist beam and bushings
3.2 Flexible twist beam modeling Once validated the b Motion model, the linear flexible twist beam was replaced by the two alternative modeling methods intended to take into account the geometric nonlinearities due to the large deformations of the beam element: sub-structuring and coupled simulation Motion – Mecano.
- Sub-structuring: the twist beam was cut in
3 sections along the central pipe, resulting
in 4 separate linear flexible bodies: the two
longitudinal arms + two symmetric halves of
the beam. Figure 3 shows the three cutting
sections used.
- Coupled simulation Motion – Mecano -
starting from the original Nastran FE mesh, the dynamic behavior of the full twist beam is calculated by the SAMCEF Mecano nonlinear solver through a specific Analysis Case added to the VL Motion model.
3.3 FE bushings modeling
As a further task of the activity, starting from the CAD representation of the geometry of the bushings connecting the twist beam with the car body a Mecano FE model of each bushing has been created and implemented into the b Motion mechanism to replace the original bushing force elements, modeled as nonlinear stiffness and damping curves in all six directions. Material properties for the rubber and metal parts of the bushings were not known in detail, so tentative values have been used for the rubber whereas the metal parts have been considered as rigid: although these assumptions were expected to have a major impact on results, the main purpose of this task was not to obtain accurate and correlated results, rather to prove the capability of the Motion-Mecano coupled simulation method to successfully solve multiple nonlinear flexible bodies in the same model.
3.4 Results comparison
Figure 4 shows the results of the suspension roll analysis for two of the most relevant outputs for the handling performance of a car: toe angle and wheel track variation, plotted vs. left wheel vertical displacement. The main outcome is that sub-structuring and coupled Motion-Mecano simulation (not including FE bushings) give very similar results, both different from the linear case: as expected, the linear approach gives reliable results only in a limited range of displacements, whereas for larger deformations of the twist beam a more accurate prediction of the behavior of the system can be obtained only by considering the nonlinear flexibility of the body.
In Fig.5 some of the results from the dynamic braking-in-turn maneuver are displayed, where during a cornering maneuver started at around 0.7s a braking force is applied after 1.5s. In this comparison the additional case with the two nonlinear FE bushings is also displayed: again, a remarkable difference can be detected between the linear case and the nonlinear FE-MBS coupled simulation; furthermore a clear effect from nonlinear FE bushings can be seen, although most likely affected by uncertainties on the material properties applied in the Mecano FE bushing models.
Fig. 3 Sub-structuring of the linear flexible twist beam Fig. 2
Comparison of results between b Motion model and source MBS model
4 Conclusions
In this paper the usage of a new FE-MBS co-simulation technique for an automotive application is compared with two alternative solutions to represent the nonlinear flexibility of a body in a multi-body mechanism. A b Motion rear suspension model with flexible twist beam has been created with the aim to simulate two typical handling events where the proper prediction of the large deformation of the twist beam strongly affects most of the elasto-kinematic characteristics of the suspension. The compared results show a clear difference between the linear approach, based on a modal representation of the flexibility of the body, and the alternative methods which allow a more correct prediction of the geometric nonlinearity.
This new b Motion – SAMCEF Mecano co-simulation technique allows also the simulation of multiple nonlinear flexible bodies in the same mechanisms as shown in this paper. Further studies are currently on-going to extend the usage of this solution to complex applications like flexible contact and friction forces, nonlinear material properties, thermal effects.
5 References
[1] Yoo W.S., Haug E.J.: “Dynamics of flexible mechanical systems using vibration and static correction
modes ”, Journal of Mechanisms, Transmissions and Automation in Design, 108, 315-322, 1985
[2] Sinokrot T.Z., Nembrini M., Toso A., Prescott W.C.: "A Comparison Of Sub-Structuring Synthesis And The
Cosimulation Approach In The Dynamic Simulation Of Flexible Multi-body Systems ", MULTIBODY
DYNAMICS 2011, ECCOMAS Thematic Conference, Brussels, Belgium, 4-7 July 2011
[3] Sinokrot T.Z., Nembrini M., Toso A., Prescott W.C.: "A Comparison Of Different Multi-body System
Approaches In The Modeling Of Flexible Twist Beam Axles ", Proceedings of the 8th International Conference on Multi-body Systems, Nonlinear Dynamics, and Control, August 28-31, 2011, Washington D.C., USA
[4] LMS International, b Online Help Manual , 2013.
[5] LMS Samtech, Samcef Online Help Manual – version 15.1, 2013.
[6] Sinokrot T., Jetteur P., Erdelyi H., Cugnon F., Prescott W.: "A New Technique for Stronger Coupling
between Multi-body System and Nonlinear Finite Element Solvers in Co-simulation Environments ",
MULTIBODY DYNAMICS 2013, ECCOMAS Thematic Conference, Zagreb, Croatia, 1-4 July 2013
Fig. 4
Suspension roll analysis: toe angle and wheel track variations
Fig. 5
Braking-in-turn analysis: wheel base and toe angle variations。