Phase plane analysis applied to non-explicit multibody vehicle models

Mantaras, Daniel A.; Luque, Pablo; Alonso, Marta · 2022 · Crossref

DOI: 10.1007/s11044-022-09846-9

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Summary

This paper addresses the limitation of traditional stability analysis methods for vehicle dynamics, which require explicit mathematical models and often rely on simplified representations like the bicycle model. Such simplifications neglect complex nonlinear components—such as bushings, flexible anti-roll bars, and detailed tire models—that significantly influence dynamic behavior, particularly near stability limits. The authors propose a new methodology for constructing stability maps (phase-plane analysis) for complex, non-explicit multibody vehicle models implemented in commercial Multibody Dynamics (MBD) software, specifically Adams®. This approach allows for the inclusion of highly nonlinear elements without requiring access to the underlying differential-algebraic equations. The methodology utilizes a "virtual force and torque" technique to generate phase-plane vector fields. Since explicit equations are unavailable in MBD software, the authors introduce external virtual lateral forces and yaw moments to the vehicle model. These inputs force the system into a dynamic equilibrium (steady state) at specific kinematic conditions defined by vehicle sideslip angle ($\beta_V$) and yaw rate ($\omega_{zV}$). By analyzing the magnitude of these virtual forces required to maintain equilibrium, the authors derive the corresponding time derivatives ($\dot{\beta}_V$ and $\dot{\omega}_{zV}$) that would exist in the unforced real system. This process is repeated across a grid of state space points to construct the vector field and identify equilibrium points. The method was validated using a detailed 3D multibody model of a passenger car with 144 degrees of freedom, including flexible bodies, nonlinear bushings, and Pacejka PAC2002 tire models. Simulations were conducted at a constant speed of 30 km/h for various steering angles. The resulting phase planes identified stable and saddle equilibrium points, which were verified against open-loop step-steer maneuvers. The study highlighted that at higher steering angles (0.3 rad), the vehicle exhibits nonlinear behavior due to tire saturation and suspension kinematics, specifically a reduction in slip angle caused by the saturation of the inner rear tire. This phenomenon, driven by the nonlinear interaction of the twistable rear cross-member and tire forces, was accurately captured by the proposed method. The significance of this work lies in its ability to provide accurate stability maps for complex vehicle models that traditional simplified methods cannot represent. The authors conclude that simplified models fail to predict vehicle behavior near stability limits, such as when tires are saturated, because they ignore critical nonlinearities. By enabling phase-plane analysis on full multibody models, this methodology supports more robust design and control system development for vehicles operating in nonlinear regimes, bridging the gap between high-fidelity simulation and stability analysis.

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