# On Trade-Off Relationship between Static and Dynamic Lateral Stabilities of Articulated Heavy Vehicles

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- If the leading unit is understeer, that is, ${K}_{L,us}>0$, the tractor/semi-trailer is stable regardless of whether the trailing unit is understeer or oversteer;
- (2)
- If the extent of trailer oversteer is relatively larger than its counterpart of the tractor, the tractor/semi-trailer is unstable when the forward-speed approaches the respective critical-speed, leading to jackknifing in ‘trailer-swing’, as shown in Figure 1c;
- (3)
- If the tractor is oversteer and the extent of tractor oversteer is relatively larger than its counterpart of the trailer, the tractor/semi-trailer is unstable when the forward-speed approaches the corresponding critical-speed, leading to jackknifing in ‘tractor-swing’, as seen in Figure 1b.

## 2. Vehicle System Modelling

#### 2.1. Linear 3-DOF Yaw-Plane Model

**M**,

**D**, and

**E**are provided in the Appendix A, and the state variable vector (

**x**) and control variable vector (

**u**) are defined by the following:

#### 2.2. Nonlinear Three-Dimensional TruckSim Models

#### 2.3. Performance Indicators for Evaluating Static and Dynamic Stabilities of AHVs

#### 2.3.1. Understeer Gradient for Assessing Static Stability

#### 2.3.2. Eigenvalue-Analysis-Based Yaw-Damping Ratio for Evaluating Dynamic Stability

**A**, for which a pair of complex eigenvalues is represented in Equation (31), the respective damping ratio can be determined by the following:

#### 2.4. Tractor/Semi-Trailer Design Cases

^{3}. For simplicity, the positions of the trailer CGs with the two payload arrangements are assumed to be the same as that of the zero payload one. The mass moments of inertia of these payloads are calculated based on this assumption.

## 3. Static Stability Evaluation

#### 3.1. Static Stability Assessment Using ${K}_{L,us}$

#### 3.2. Static Stability Assessment Using Handling Diagram

## 4. Dynamic Stability Evaluation

#### 4.1. Dynamic Stability Assessment Using Eigenvalue Analysis

**A**, expressed in Equation (30) is utilized to calculate the eigenvalues considering the single and double tractor rear-axle arrangements with three different trailer payload schemes. Provided a pair of complex eigenvalues, the damping ratio is calculated using Equation (32).

#### 4.1.1. Eigenvalue Analysis for Single Tractor Rear-Axle Configuration

#### 4.1.2. Eigenvalue Analysis for Double Tractor Rear-Axle Configuration

#### 4.2. Dynamic Stability Assessment Using Simulated Open-Loop SLC Testing Maneuver

#### 4.2.1. Simulation Results for Single Tractor Rear-Axle Configuration

#### 4.2.2. Simulation Results for Double Tractor Rear-Axle Configuration

## 5. Trade-Off Relationship between Static and Dynamic Stabilities

- (1)
- Regardless of the tractor rear-axle configurations, increasing the trailer payload leads to the decrease in the understeer gradient of the leading vehicle unit, ${K}_{L,us}$, while increasing the trailer payload results in the increase in the damping ratio at the specified speed;
- (2)
- Given a trailer payload, compared with the single tractor rear-axle configuration, the double tractor rear-axle structure not only increases the understeer gradient of the leading vehicle unit, ${K}_{L,us}$, but also increases the damping ratio at the specified speed.

## 6. Conclusions

- (1)
- Regardless of the tractor rear-axle configurations, increasing the trailer payload leads to the degradation of the static stability of the tractor/semi-trailer, while increasing the trailer payload is beneficial for enhancing the dynamic stability of the AHV.
- (2)
- Compared with the single tractor rear-axle configuration, the double tractor rear-axle structure not only strengthens the static stability of the AHV, but also enhances the lateral dynamic stability of the vehicle.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**M**,

**D**, and

**E**are expressed as follows:

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**Figure 1.**Three unstable yaw-motion modes of a tractor/semi-trailer: (

**a**) trailer-sway/snaking; (

**b**) tractor swing; and (

**c**) trailer swing.

**Figure 3.**Two nonlinear 3-D TruckSim models for tractor/semi-trailer with the configurations as follows: (

**a**) S_S+SSS (tractor with single rear-axle); and (

**b**) S_SS+SSS (tractor with double rear-axles).

**Figure 4.**Tractor/semi-trailer design cases: (

**a**) tractor with single rear-axle; (

**b**) tractor with rear tandem-axle group; (

**c**) dimensions (in mm) of 9000 kg trailer payload; and (

**d**) dimensions (in mm) of 20,000 kg trailer payload.

**Figure 5.**Handling diagram of the tractor/semi-trailer featuring the single tractor rear-axle arrangement with different trailer payloads.

**Figure 6.**Handling diagram of the AHV featuring the double tractor rear-axle arrangement with different trailer payloads.

**Figure 7.**Damping ratio vs. forward-speed for single tractor rear-axle arrangement with: (

**a**) zero trailer payload (case 1); (

**b**) 9000 kg trailer payload (case 2); and (

**c**) 20,000 kg trailer payload (case 3).

**Figure 8.**Damping ratio vs. forward-speed for double tractor rear-axle arrangement with (

**a**) zero trailer payload (case 4); (

**b**) 9000 kg trailer payload (case 5); and (

**c**) 20,000 kg trailer payload (case 6).

**Figure 10.**Time-history of trailer yaw-rate for the single tractor rear-axle configuration with three trailer payload arrangements at forward-speed of (

**a**) 50.0 m/s and (

**b**) 70.0 m/s.

**Figure 11.**Time-history of trailer yaw-rate for the double tractor rear-axle configuration with three trailer payload arrangements at forward-speed of (

**a**) 65.0 m/s and (

**b**) 80.0 m/s.

**Table 1.**Values of vehicle system parameters for the two rear-axle group configurations with curb trailer weight.

Parameters | Single Rear-Axle Tractor | Double Rear-Axle Tractor |
---|---|---|

L_{ft}, distance from tractor front axle to its CG, m | 1.11 | 1.385 |

L_{rt}, distance from tractor rear-axle group to its CG, m | 2.39 | 4.25 |

L_{wt}, distance from tractor CG to fifth wheel, m | 2.64 | 4.35 |

L_{fs}, distance from fifth wheel to trailer CG, m | 5.5 | 5.5 |

L_{rs}, distance from trailer CG to its axle, m | 2.4 | 2.4 |

m_{t}, tractor mass, kg | 5496 | 7878 |

m_{s}, trailer mass, kg | 7807 | 7807 |

${I}_{zt}$, yaw moment of inertia of tractor, kg·m^{2} | 34,800 | 19,965 |

${I}_{zs}$, yaw moment of inertia of trailer, kg·m^{2} | 150,000 | 150,000 |

${C}_{f}$, front tire cornering stiffness of tractor (combined), N/rad | 200,000 × 2 | 200,000 × 2 |

${C}_{r}$, rear tire cornering stiffness of tractor (combined), N/rad | 50,000 × 4 | 50,000 × 8 |

${C}_{s}$, tire cornering stiffness of trailer (combined), N/rad | 40,000 × 12 | 40,000 × 12 |

Tractor/Semi-Trailer with Single Tractor Rear-Axle | Tractor/Semi-Trailer with Double Tractor Rear-Axles | |||||
---|---|---|---|---|---|---|

Zero Trailer Payload (Case 1) | 9000/kg Trailer Payload (Case 2) | 20,000/kg Trailer Payload (Case 3) | Zero Trailer Payload (Case 4) | 9000/kg Trailer Payload (Case 5) | 20,000/kg Trailer Payload (Case 6) | |

${K}_{L,us}$, rad/g | 0.0200 | −0.0546 | −0.1188 | 0.0408 | 0.0100 | 0.0090 |

**Table 3.**Damping ratios of the least-damped motion mode for different tractor/semi-trailer design cases at the forward-speed of 31 m/s.

Tractor/Semi-Trailer with Single Tractor Rear-Axle | Tractor/Semi-Trailer with Double Tractor Rear-Axles | |||||
---|---|---|---|---|---|---|

Zero Trailer Payload (Case 1) | 9000/kg Trailer Payload (Case 2) | 20,000/kg Trailer Payload (Case 3) | Zero Trailer Payload (Case 4) | 9000/kg Trailer Payload (Case 5) | 20,000/kg Trailer Payload (Case 6) | |

Damping ratio | 0.1924 | 0.4378 | 0.4781 | 0.3123 | 0.6300 | 0.6900 |

**Table 4.**${K}_{L,us}$ values and damping ratios of the least-damped motion mode at forward-speed of 31.0 m/s for different tractor/semi-trailer design cases.

Tractor/Semi-Trailer with Single Tractor Rear-Axle | Tractor/Semi-Trailer with Double Tractor Rear-Axles | |||||
---|---|---|---|---|---|---|

Zero Trailer Payload (Case 1) | 9000/kg Trailer Payload (Case 2) | 20,000/kg Trailer Payload (Case 3) | Zero Trailer Payload (Case 4) | 9000/kg Trailer Payload (Case 5) | 20,000/kg Trailer Payload (Case 6) | |

${K}_{L,us}$, rad/g | 0.0200 | −0.0546 | −0.1188 | 0.0408 | 0.0100 | 0.0090 |

Damping ratio | 0.1924 | 0.4378 | 0.4781 | 0.3123 | 0.6300 | 0.6900 |

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**MDPI and ACS Style**

Sharma, T.; He, Y.
On Trade-Off Relationship between Static and Dynamic Lateral Stabilities of Articulated Heavy Vehicles. *Designs* **2024**, *8*, 103.
https://doi.org/10.3390/designs8050103

**AMA Style**

Sharma T, He Y.
On Trade-Off Relationship between Static and Dynamic Lateral Stabilities of Articulated Heavy Vehicles. *Designs*. 2024; 8(5):103.
https://doi.org/10.3390/designs8050103

**Chicago/Turabian Style**

Sharma, Tarun, and Yuping He.
2024. "On Trade-Off Relationship between Static and Dynamic Lateral Stabilities of Articulated Heavy Vehicles" *Designs* 8, no. 5: 103.
https://doi.org/10.3390/designs8050103