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

This paper extends previously developed Bouc-Wen-based incremental collision models for convex viscoplastic bodies by incorporating time-dependent external forces, expanding the range of analytically favorable parameters to include corner cases, and validating the augmented models through additional parameter identification studies.

The Gist

Imagine two billiard balls crashing into each other. In the old days, physicists might have treated this like a perfect, bouncy rubber ball hitting a wall: it squishes, it pushes back, and it flies away. But real objects aren't perfect. They are a bit like sticky, squishy clay mixed with a spring. When they hit, they don't just bounce; they heat up, they deform, and they lose energy to friction. This is called "viscoplastic" behavior.

This paper is about building a better mathematical recipe to predict exactly how these squishy, sticky objects behave when they crash.

Here is the breakdown of what the authors did, using some everyday analogies:

1. The Old Recipe vs. The New Recipe

In a previous study (their "old recipe"), the authors created two mathematical models (let's call them Model A and Model B) to describe these crashes. These models were based on something called the Bouc-Wen model, which is like a fancy way of describing how a spring gets tired and changes shape after being stretched too many times (hysteresis).

• The Problem with the Old Recipe: It assumed the balls were floating in deep space, hit each other, and that was the only force acting on them.
• The Real World: In reality, things are rarely floating in space. A ball might be falling (gravity), or being pushed by a fan, or rolling down a ramp. These are external forces.

The Innovation: The authors updated their models to include these "pushes and pulls" from the outside world. Think of it like upgrading a video game physics engine. The old version only calculated what happened when two cars crashed. The new version calculates what happens when two cars crash while one of them is also being hit by a strong wind or rolling down a hill.

2. The Two Models: The "Parallel" and the "Series"

The authors use two different ways to mathematically describe the "squishiness" of the collision. Imagine you are trying to describe how a car's suspension works:

• Model A (The Parallel Connection): Imagine a shock absorber and a spring sitting side-by-side. When the car hits a bump, both the spring and the shock absorber work together at the same time to absorb the energy. This is the Bouc-Wen-Simon-Hunt-Crossley model.
• Model B (The Series Connection): Imagine a spring and a shock absorber connected end-to-end (like a chain). The force has to go through one, then the other. This is the Bouc-Wen-Maxwell model.

Both models are great at describing how energy is lost (dissipated) during a crash, but they do it in slightly different mathematical ways. The authors proved that both models are mathematically "safe" (they won't break or give crazy answers) even when you add those outside forces like gravity.

3. The "Corner Cases" (The Edge of the Map)

In their previous work, the authors only tested their models when the parameters were "nice and normal." It's like testing a car only on a smooth highway.

In this paper, they tested the models in the corner cases—the weird, extreme situations.

• What if the "stickiness" is zero?
• What if the "squishiness" is perfectly linear?
• What if the math gets really weird at the edges?

They proved that even in these weird, extreme scenarios, the models still work and don't explode mathematically. It's like proving your car can still drive even if you take it off-road, into the mud, or up a steep mountain, not just on the highway.

4. Testing the Models (The Lab Work)

To make sure their new "recipe" actually works, they didn't just do math on paper. They went to the lab (or used existing lab data) to test it.

• Test 1: The Steel vs. Aluminum Drop. They looked at data where balls were dropped onto plates. They showed their new models could predict how high the balls would bounce back (the "Coefficient of Restitution") just as well as, or better than, before.
• Test 2: The Baseball Bat. They looked at data of a baseball hitting a flat surface. This is a classic "squishy" collision. They showed both Model A and Model B could perfectly trace the shape of the force as the ball flattened and popped back out.
• Test 3: The Cart on a Ramp (The Big Win). This is the most important test. They used data from a cart rolling down a ramp and hitting a wall. Because the cart was rolling down a hill, gravity was constantly pushing it during the crash.
  • The old models (without the external force update) would have failed here.
  • The new models, which account for the "push" of gravity, matched the real-world data perfectly.

The Bottom Line

Think of this paper as a software update for physics.

The authors took two existing, powerful tools for simulating collisions and added a new feature: "External Force Support." They also made sure the tools are robust enough to handle weird, extreme settings.

Why does this matter?
If you are designing a robot that walks, a car safety system, or a video game, you need to know what happens when things crash while they are also being pulled by gravity or pushed by motors. This paper gives engineers the mathematical tools to simulate those complex, real-world crashes accurately.

In a nutshell: They took the math of "how things bounce," added the math of "how things are pushed," and proved that the combination works perfectly, even in the weirdest situations.

Technical Summary

1. Problem Statement

The paper addresses the modeling of binary direct collinear collisions between convex viscoplastic bodies. While the authors' previous work (Ref. [14]) established two incremental collision laws based on the Bouc-Wen differential model of hysteresis—the Bouc-Wen-Simon-Hunt-Crossley Collision Model (BWSHCCM) and the Bouc-Wen-Maxwell Collision Model (BWMCM)—several limitations remained:

• External Forces: The original models assumed that the contact force was the sole force acting on the bodies during collision. They could not account for external time-dependent forces (e.g., gravity, friction, or applied loads) acting on the bodies while in contact.
• Parameter Corner Cases: The analytical proofs regarding the existence, uniqueness, and boundedness of solutions for the BWMCM were restricted to specific parameter ranges, excluding physically plausible "corner cases" (e.g., $B=0$, $\Gamma \in \{-B, B\}$, or $p \in (1, 2)$).
• Validation Scope: Previous parameter identification studies were limited to specific datasets and did not fully validate the models' ability to handle external force effects.

2. Methodology

The authors extended the theoretical framework and analytical tools of the previous study to address the identified gaps.

A. Augmented Physical Model

The authors formulated a high-level physical model for two rigid bodies ($B_1$ and $B_2$) colliding in a 1D system.

• Equations of Motion: The system dynamics were described by an Initial Value Problem (IVP) incorporating both the contact force $F$ and external forces $u_1, u_2$.
• Effective Mass: The equations were reduced to a relative motion formulation using the effective mass $m = \frac{m_1 m_2}{m_1 + m_2}$ and a relative external force input $u$.
• Input Spaces: External forces were modeled as time-dependent inputs belonging to specific function spaces ($U_1$ for integrable functions or $U_\infty$ for bounded functions).

B. Model Augmentation

The two collision models were updated to include the external force input $u$:

1. Augmented BWSHCCM: The state equation for relative velocity $\dot{v}$ was modified to include the term $+\frac{1}{m}u$.
2. Augmented BWMCM: A new state-space formulation was derived for the BWMCM. Unlike the previous form, this new formulation ensures the state function is locally Lipschitz continuous for all admissible parameters ($p \ge 1$), guaranteeing solution uniqueness. The external force was similarly added to the velocity equation.

C. Nondimensionalization

The augmented models were nondimensionalized using characteristic scales ($T_c, X_c, Z_c$) derived from the system parameters. This resulted in the Nondimensionalized BWSHCCM (NDBWSHCCM) and Nondimensionalized BWMCM (NDBWMCM).

• A new mapping function $\mathbf{P}$ was defined to relate the "base parameters" (intrinsic to the material/geometry) to the dimensionless parameters, accounting for variations in initial velocity ($v_0$) and the external force input ($u$).

D. Analytical Proofs

Using Lyapunov stability theory, the authors provided rigorous proofs for the global existence, uniqueness, and uniform boundedness of solutions for both augmented models.

• Lyapunov Functions: New Lyapunov function candidates were constructed involving exponential terms of the integral of the input force magnitude.
• Corner Cases: The analysis was extended to cover the previously excluded parameter ranges (e.g., $B=0$, $\Gamma = \pm B$, $1 < p < 2$), proving that the models remain well-posed under these conditions.

E. Parameter Identification

The authors developed a methodology for identifying model parameters using experimental data.

• Objective: Minimize the mean squared error between experimental data (Coefficient of Restitution $e$ and collision duration $t_d$) and model predictions.
• Optimization: A multi-objective nonlinear constrained optimization problem was solved using the COBYQA algorithm.
• Datasets: Three distinct datasets were used:
  1. Kharaz and Gorham (2000): Normal impacts of spheres on plates (gravity-dominated, external forces ignored).
  2. Cross (2011): Hysteresis loops of a baseball impacting a flat surface.
  3. Villegas et al. (2021): Impacts of a spring-loaded cart on an incline (explicitly including gravity as an external force).

3. Key Contributions

1. Inclusion of External Forces: The primary contribution is the augmentation of the BWSHCCM and BWMCM to explicitly model time-dependent external forces ($u$) acting on colliding bodies. This allows the models to simulate scenarios like collisions on inclined planes or under varying load conditions.
2. Extended Analytical Validity: The paper extends the mathematical proof of well-posedness (existence and uniqueness of solutions) to corner cases that were previously unanalyzed, specifically for the BWMCM with parameters $B=0$, $\Gamma = \pm B$, and $p \in (1, 2)$.
3. Improved Model Formulation: A revised state-space representation for the BWMCM was introduced, which guarantees local Lipschitz continuity for all $p \ge 1$, resolving potential numerical issues present in the previous formulation.
4. Comprehensive Validation: The study validates the augmented models against three diverse experimental datasets, demonstrating that both models can accurately capture collision phenomena even when external forces are significant.

4. Results

• Analytical Results: The proofs confirm that for any bounded external input, the state variables (displacement, velocity, hysteretic displacement) and the contact force remain bounded for all time. This ensures the models do not exhibit unphysical blow-up behavior.
• Simulation Results (Application Example): A simulation of two balls colliding in a vertical pipe under gravity and an applied time-varying force showed that the models correctly predict the evolution of contact forces, velocities, and separation times.
• Parameter Identification:
  • Kharaz & Gorham: The models successfully fitted the CoR vs. velocity data. The study highlighted that CoR data alone is insufficient to uniquely determine all model parameters, as different parameter sets yielded similar CoR curves.
  • Cross (Baseball): Both models accurately reproduced the experimental hysteresis loops (Force vs. Displacement) for a baseball impact.
  • Villegas (Cart): The augmented models successfully fitted both CoR and collision duration data for a cart rolling on an incline, validating the capability to model gravity-driven collisions.
• Comparative Performance: The BWSHCCM and BWMCM showed comparable performance across all datasets. In some cases, the BWMCM provided slightly better fits to hysteresis data, while the BWSHCCM was robust for CoR prediction.

5. Significance

This work significantly advances the field of impact and contact modeling in multibody dynamics:

• Realism: By incorporating external forces, the models can now be applied to a wider range of real-world engineering problems, such as vehicle crashes on slopes, robotic manipulation under load, or granular flows in gravity fields.
• Robustness: The extension of analytical proofs to corner cases ensures that numerical simulations using these models will remain stable and physically meaningful across a broader spectrum of material properties and collision conditions.
• Validation Framework: The rigorous parameter identification framework provides a template for future researchers to calibrate Bouc-Wen-based collision models against diverse experimental data, moving beyond simple CoR matching to include hysteresis and duration analysis.
• Foundation for Future Work: The authors suggest that this framework can be extended to simultaneous multi-body collisions and other hysteresis models, paving the way for more complex dynamic simulations.

In conclusion, the paper transforms the Bouc-Wen-based collision models from idealized, isolated event simulators into robust, general-purpose tools capable of handling complex, force-driven collision environments.