Wheel shimmy suppression through the piecewise nonlinear energy sink: elimination of detrimental isolas

Dear colleagues and researchers,
I am Yan Kai Ying from the University of Technology. It is an honor to present our latest research on the piecewise nonlinear energy sink for wheel shimmy suppression. This work was conducted in collaboration with Dr. Han Yu from Shanghai Jiao Tong University and Professor Habib from Budapest University of Technology and Economics.

Wheel shimmy is a self-excited vibration phenomenon commonly observed in various wheeled vehicles. As illustrated in the accompanying video, electric bicycles experience shimmy during specific events, particularly when traversing rough terrain. This issue is especially pronounced in advanced vehicles featuring independent wheel systems, such as those with four-wheel steering, in-wheel motors, and steer-by-wire technology, due to their inherent reduced damping and stiffness.

Our study concentrated on a single wheel system shimmy model, as illustrated in the figure on the left. We developed a dynamic model that integrates the governing equation of the single wheel system, the magic formula tire model, and the stretched-string tire model. These three equations are utilized to describe the rotation relative to the kingpin, the lateral force exerted by the tire, and the relationship between the slip angle and the rotational angle. Furthermore, the magic formula tire model can be simplified into a polynomial form.

Our study examines the application region of the single wheel system shimmy model, as illustrated in the accompanying figure. This model effectively characterizes the behavior of the system within the critical velocity range. Bifurcations occur at both boundaries of this range, which is influenced significantly by system parameters including damping, sharpness, and chair darkness.

Several researchers have investigated shimmy suppression using Tuned Mass Dampers (TMD) and cubic Nonlinear Energy Sink (NES) as common solutions. The configurations are mounted on the wheel frame. The system incorporating the vibration absorber is derived by integrating the single wheel system model with the governing equation of the vibration absorber. As a result, we have determined the linear stability characteristics of both the TMD and the CNS, with the initial figure illustrating the TMD.

Several researchers have investigated shimmy suppression using Tuned Mass Dampers (TMD) and cubic Nonlinear Energy Sink (C-NES) in common solutions. These devices are typically mounted on the wheel frame. The wheel system equipped with the vibration absorber is derived from the integration of the single wheel system model and the governing equations of the vibration absorber. As a result, we have assessed the linear stabilities of both the TMD and C-NES, as illustrated in the first figure.

While the TMD clearly enhances the linear stability of the system, the C-NES has a comparatively lesser impact on linear stability. In the accompanying figure, I also present both the Generalized Mass Damper (GMD) and the Cubic Energy Sink (CES). For a specific system represented by the red dot in the figure, the mechanisms of stabilization differ between TMD and C-NES. The TMD achieves stabilization by modifying the zero attractor stability, while the C-NES operates through a different mechanism.

The analysis of the feasible parameter domain for semi-suppression using TMD is illustrated in the first figure. The red contours represent various initial values. Although the feasible domain for TMD is narrow, it demonstrates consistent performance even with large initial perturbations.

We analyzed the feasible parameter domain for the C-NES, which differs from that of the TMD. The C-NES exhibits a broader feasible domain; however, this domain is notably constrained by large initial perturbations.

The performance analysis of the Tuned Mass Damper (TMD) and the Controlled Nonlinear Energy Sink (C-NES) demonstrates distinct characteristics. The tuned TMD exhibits superior performance; however, its sensitivity to system parameter changes can lead to detuning. In contrast, the performance of the C-NES is compromised by the presence of undesired isolation.

In summary, the weaknesses of both systems are evident. While the tuned TMD effectively suppresses shimmy under large perturbations, it lacks robustness, as even minor parameter variations can result in detuning. Conversely, the C-NES offers greater robustness compared to the TMD, but its performance deteriorates under large perturbations due to the influence of undesired isolation.

The application of the Tuned Mass Damper (TMD) and the Controlled Nonlinear Energy Sink (C-NES) illustrates their performance characteristics. The tuned TMD demonstrates superior performance; however, variations in system parameters can lead to detuning. The performance of the C-NES is compromised by the presence of undesired isolation effects.

In summary, the limitations of both the TMD and the C-NES are evident. While the tuned TMD effectively suppresses shimmy at large perturbations, it lacks robustness, as minor parameter changes can lead to detuning. Conversely, the C-NES exhibits greater robustness but performs poorly under large perturbations due to the adverse effects of undesired isolation.

To address these challenges, we propose a piecewise characteristic damping model. We will analyze the attractors of the specific Passive Nonlinear Energy Sink (PNES) and the C-NES. The PNES has fewer attractors, while the C-NES maintains higher attractors, with low and high attractors corresponding to strongly modulated responses and single-period responses, respectively.

The verification diagram for the Piecewise Nonlinear Energy Sink (P-NES) and the Continuous Nonlinear Energy Sink (C-NES) is presented. The P-NES does not exhibit a high-amplitude branch, which is characteristic of systems utilizing the C-NES.

A critical value significantly influences the performance of the P-NES system, with its stiffness being a fundamental determinant of performance. In situations where shimmy is present, the system demonstrates a strong veridical response.

The P-NES effectively limits the high-amplitude branch found in C-NES systems. Furthermore, the P-NES shows superior performance in low damping regimes, whereas the C-NES is less effective in these conditions.

The verification diagram for the P-NES and C-NES systems is presented. The P-NES system lacks a high-amplitude branch, which is characteristic of systems utilizing the C-NES. The critical value is significant in determining the performance of the P-NES system, as its stiffness fundamentally influences its effectiveness. Similar to the C-NES, when the shimmy is sustained, the system exhibits a robust veridical response. Consequently, the P-NES effectively mitigates the high-amplitude branch found in C-NES systems. Furthermore, the P-NES outperforms the C-NES in low damping scenarios, where the latter proves to be ineffective.

We also examine the feasible parameter domain of the TNS. It is preferable for the upper lambda 2 to be a large value. Within the feasible domain, the wheel rotational angle shows a strong correlation with the response. However, outside this region, the response diminishes. In low damping conditions, lambda 1 can vary from 0 to the maximum damping value of the C-NES. This indicates a significantly larger range. Notably, the feasible stiffness domain of the P-NES closely resembles that of the C-NES.

We analyze the feasible parameter domain of the Tuning Nonlinear System (TNS) and identify key characteristics. The upper limit of lambda 2 should be set to a large value. Within the feasible domain, the wheel rotational angle shows a strong correlation with system response; outside this domain, the response becomes negligible. In low damping scenarios, lambda 1 can vary from 0 to the maximum damping value of the CNES, indicating a significantly larger range. Notably, the feasible stiffness domain of the P-NES closely resembles that of the CNES.

This figure illustrates the performance comparison between the P-NES and C-NES. The P-NES not only eliminates unwanted isolation but also exhibits superior vibration suppression capabilities compared to the C-NES. However, both the P-NES and C-NES are less effective than tuned Tuned Mass Dampers (TMD) in vibration suppression. Nevertheless, they possess larger feasible parameter domains, indicating greater robustness to variations in system parameters compared to TMDs. In conclusion, the proposed P-NES effectively overcomes the limitations of both TMD and C-NES, providing enhanced robustness and performance for shimmy suppression.

This figure illustrates the performance comparison of the opals. The larger P-NES not only eliminates the undesirable isola but also exhibits superior vibration suppression performance compared to the C-NES. However, both the P-NES and C-NES are less effective than the tuned TMD in vibration suppression. They possess extensive feasible parameter domains, indicating greater robustness to variations in system parameters compared to the TMD. In summary, our proposed P-NES effectively overcomes the limitations associated with both the TMD and C-NES, providing enhanced robustness and performance for shimmy suppression.