Modeling of tractor dynamic behavior during maneuvering on compound slopes for identifying the onset of rollover

Document Type : Research Paper

Authors

1 Department of Agricultural Machinery Engineering, College of Agriculture and Natural Resources, University of Tehran, Tehran, Iran

2 Department of Agricultural Machinery Engineering, College of Agriculture and Natural Resources, University of Tehran, Tehran, Iran.

10.22059/ijbse.2025.401891.665613

Abstract

Tractors play a vital role in the agricultural industry, and due to their inherent characteristics and operational conditions, rollover accidents are a probable occurrence. Despite numerous studies aimed at preventing and reducing the consequences of rollover, statistics show that this accident leads to thousands of fatalities and injuries each year—a trend that continues to rise. This study modelled the tractor stability using both numerical and analytical methods, and the performance of these approaches in predicting lateral and longitudinal stability was compared. The numerical models were developed using the Response Surface Methodology, and both models were validated by comparing them to the experimental data obtained from a scaled-down tractor prototype. In addition, sensitivity analysis was conducted to determine the relative contribution of each variable. This study mathematically modelled the influence of lateral and longitudinal slope angles, center-of-gravity position, speed, acceleration, and turning radius on tractor stability. Results indicated that the squared sine of the lateral slope angle (coefficient of 0.24) most strongly governed lateral stability, while the squared sine of the longitudinal slope angle (0.21) was the principal factor affecting longitudinal stability. Despite the analytical model considering the interaction between lateral and longitudinal slopes, the numerical results indicated that the interaction was not statistically significant. The coefficients of determination (R²) for lateral and longitudinal stability were 96.93% and 91.59% for the analytical model, 93.75% and 98.81% for the numerical model, and 65.31% and 81.16% for the advanced numerical model, respectively. The optimization framework integrated all relationships to evaluate overall stability.

Keywords

Main Subjects

 

EXTENDED ABSTRACT

Introduction

Tractors play a crucial role in modern agriculture, yet their inherent design poses significant rollover risks, causing thousands of fatalities annually. While Roll-Over Protective Structures (ROPS) remain a key safety measure, emerging technologies—such as active steering and momentum control—offer effective complements. Unlike most previous works emphasizing practical solutions, this study provides a theoretical and comparative analysis of prevention mechanisms. To reduce the high cost of physical testing, scaled-down prototypes and computer simulations were employed.

Methodology

This study enhances tractor safety through mathematical modelling of rollover dynamics using analytical and numerical approaches.

The numerical model applied Response Surface Methodology (RSM) in Design-Expert software to develop an optimized quadratic model, while the analytical model employed Newton’s laws and D’Alembert’s principle to describe forces under static and dynamic conditions.

A coordinate system consistent with the tractor geometry allowed direct sensor integration for parameter measurement. Key parameters reflected the goal of retrofitting older tractors with preventive systems.

Experimental Validation

A scaled-down tractor prototype was designed and fabricated with:

Dual-axis tilt capability

Adjustable centre-of-gravity

High-precision sensors, including:

Four wheel load cells (0.1 N resolution)

A dual-axis inclinometer (0.1° resolution)

Model Evaluation & Development

Initial Validation

Model outputs were compared with prototype test data using R² coefficients.

Combined longitudinal and lateral stability metrics were analysed using the desirability function.

Advanced Model Synthesis

A comprehensive numerical model was developed to overcome prototype limitations by:

Incorporating analytical model data

Adding key dynamic parameters (acceleration, velocity, and turning radius) absent in the earliest numerical model

Sensitivity Analysis

Parameter significance was quantified to guide the design of preventive safety systems.

Theoretical Contribution

The finalized model provides a theoretical framework for developing next-generation rollover prevention systems.

Results

Tractor stability on compound slopes was analysed using analytical (Newton’s law, D’Alembert’s principle) and numerical RSM models based on 35 runs. Both showed strong concordance, with R² values of 93.75/98.81% (numerical) and 91.59/96.93% (analytical) for lateral and longitudinal stability.

The sensitivity analysis based on normalized RSM coefficients revealed that lateral stability was most affected by the squared lateral slope (0.206), followed by the CoG height (0.020). This indicates that slope steepness has nearly tenfold the effect of CoG height (0.206 vs 0.020). For longitudinal stability, the squared longitudinal slope (0.185) and slope–position coupling term (0.177) showed dominant contributions, both considerably exceeding the effect of the squared longitudinal CoG position (0.087) and CoG height (0.023).

Notably, the Advanced numerical model uncovered additional complexity in lateral stability behavior, with quadratic lateral CoG position emerging as a critical factor and slope-angle terms appearing in multiple interaction components. These findings fundamentally reorient rollover prevention strategies, emphasizing that while longitudinal stability depends overwhelmingly on CoG positioning, lateral stability requires balanced consideration of both geometric and operational parameters. The models' strong agreement (all R²> 90%) despite differing methodologies underscores their reliability for safety system design.

Conclusions

This study developed analytical and numerical models to assess tractor rollover stability, achieving R² values of 96.93%/91.59% (analytical) and 93.75%/98.81% (numerical). Sensitivity analysis revealed that stability was most affected by sin² (slope). An advanced model showed excellent theoretical fit (R² 99.93%/99.17%) but reduced accuracy (R² 65.31%/81.17%) when validated with prototype data, likely due to limited experimental parameter ranges. These findings provide a framework for stability system design, though real-world calibration is proposed to address observed deviations.

Author Contributions

M.R. Shabani: Conceptualization, Methodology, Investigation, Formal Analysis, Data Curation, Writing - Original Draft

  1. Jafari: Supervision, Project Administration, Validation, Writing-Review & Editing
  2. Hajiahmad: Resources (experimental apparatus), Methodology, Validation, Writing-Review & Editing

S.S. Mohtasebi: Writing-Review & Editing, General Advisory

All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

Data available on request from the authors.

Acknowledgements

This research was conducted as part of the approved project No. 8929743 by the Vice Presidency for Research and Technology, Faculty of Agriculture, College of Agriculture and Natural Resources, University of Tehran. We sincerely appreciate their financial and scholarly support. Additionally, we would like to express our appreciation to the Department of Agricultural Machinery Engineering, College of Agriculture and Natural Resources, University of Tehran, for providing us with essential technical and workshop facilities.

Ethical considerations

This study did not involve human or animal subjects, experimental procedures, or sensitive data; consequently, ethical approval was deemed unnecessary. The authors upheld the standards of academic integrity throughout the conduct and reporting of this research

Conflict of interest

The author declares no conflict of interest.

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Iranian Journal of Biosystem Engineering
Volume 56, Issue 4
January 2026
Pages 47-69

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History

  • Receive Date: 14 September 2025
  • Revise Date: 12 December 2025
  • Accept Date: 22 December 2025

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