Incremental Collision Laws Based on the Bouc-Wen Model: Improved Collision Models and Further Results

In the article titled “The Bouc-Wen Model for Binary Direct Collinear Collisions of Convex Viscoplastic Bodies” and published in the Journal of Computational and Nonlinear Dynamics (Volume 20, Issue 6, June 2025), the authors studied mathematical models of binary direct collinear collisions of convex viscoplastic bodies that employed two incremental collision laws based on the Bouc-Wen differential model of hysteresis. It was shown that the models possess favorable analytical properties, and several model parameter identification studies were conducted, demonstrating that the models can accurately capture the nature of a variety of collision phenomena. In this article, the aforementioned models are augmented by modeling the effects of external forces as time-dependent inputs. Furthermore, the range of the parameters under which the models possess favorable analytical properties is extended to several corner cases that were not considered in the prior publication. Finally, the previously conducted model parameter identification studies are extended, and an additional model parameter identification study is provided in an attempt to validate the ability of the augmented models to represent the effects of external forces.

💡 Research Summary

The paper builds on the authors’ earlier work on two incremental collision laws derived from the Bouc‑Wen hysteresis model – the Bouc‑Wen‑Simon‑Hunt‑Crossley Collision Law (BWSHCCL) and the Bouc‑Wen‑Maxwell Collision Law (BWMCL). In the original study the models were limited to situations where only the contact force acted during a binary direct collinear impact of convex viscoplastic bodies. The present work augments both laws by introducing external forces as time‑dependent inputs, thereby allowing the models to represent realistic scenarios where bodies are subject to gravity, actuation, or other environmental loads while in contact.

The authors first formulate a high‑level one‑dimensional mechanical system: two convex rigid bodies collide along a common line, with relative displacement x, relative velocity v, and an effective mass m. The dynamics are expressed as an initial‑value problem (IVP) that includes the contact force F and the external force u(t). The contact force is modeled by the two incremental laws, each consisting of a nonlinear viscous dissipation element in parallel (BWSHCCL) or series (BWMCL) with a Bouc‑Wen hysteretic element. The state variables (x, z for BWSHCCL; r, y, z, w for BWMCL) obey coupled differential equations that are now driven by u(t).

A major theoretical contribution is the proof that, for external inputs belonging to either the space U₁ (continuous functions with finite L¹ norm) or U∞ (bounded continuous functions), the augmented differential systems admit unique, globally bounded solutions on any interval where the contact force remains non‑negative. The authors extend the existence‑uniqueness and boundedness results to several “corner cases” that were omitted previously: the stiffness parameter B may be zero, the hysteresis asymmetry γ may attain its extreme values ±B, and the non‑linear stiffness exponent p may lie in the interval (1, 2). These extensions broaden the applicability of the models to nearly elastic contacts, highly asymmetric hysteresis, and softer non‑linear stiffness regimes.

Both nondimensionalized (NDB‑WSHCCM, NDB‑WMCM) and dimensional forms are presented. Nondimensionalization introduces scaled variables X, Z, V (or R, Y, W) and dimensionless parameters κ, B, Γ, σ, etc., facilitating parameter studies and numerical implementation. The output functions are defined so that the relative displacement, velocity, and contact force can be recovered directly from the state vector.

The paper then details a comprehensive parameter‑identification campaign. Existing experimental data (e.g., the dataset from Fig. 9.5 in Ref.