# A Theoretical Model with Which to Safely Optimize the Configuration of Hydraulic Suspension of Modular Trailers in Special Road Transport

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## Abstract

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## 1. Introduction

_{D}as the drag force, ρ as the fluid density, v

_{w}as the velocity of the air relative to the truck, A as the area that is vertical to the relative motion and C

_{d}as the drag coefficient. The drag coefficient depends on the relative position and geometry of the payload. However, the equation is accurate only for isolated objects. It is normally employed for basic load shapes. The actual truck–trailer configuration can increase the aerodynamic forces. The greatest contributor to the longitudinal aerodynamic force acting in commercial tractor-trailer configurations is the space between tractor and semitrailer [22,23]. The space is expressed by $b/\sqrt{A}$, a non-dimensional form, in which b is the width of the space (gap) and A is the area. The drag force that the gap produces is zero for small gaps. It increases suddenly when $b/\sqrt{A}\approx 0.5$, but becomes stable at 0.7 [22,23]. The other parameter that raises the drag force is the yaw angle, especially in trucks that have sharp-edged cabs. Crosswinds on exposed highways and bridges can cause difficulties in driving in tall and long vehicles and the risk of a serious accident. The force of the crosswind on a conventional truck–trailer in North America as it crosses a bridge was examined by using Computational Fluid Dynamics (CFD) [24]. The authors validated the model with experimental data that previous studies had provided [25,26]. The conclusion was that wind pressure can be sufficient to overturn the truck–semitrailer at the wind speeds that were studied.

## 2. Materials and Methods

#### 2.1. Modular Trailers

#### 2.2. Stability Area

_{0}, y

_{0}, z

_{0}), (x

_{1}, y

_{1}, z

_{1}), and (x

_{2}, y

_{2}, z

_{2}) as the coordinates of the centers of the groups 0, 1, and 2, respectively. One can calculate these coordinates’ geometrically by using the location of the junction of the axles and cylinders or analytically is as follows:

_{i}as the number of group i pendulum axles, and (X

_{ij}, Y

_{ij}, and Z

_{ij}) are the pendulum axle fulcrum j’s coordinates in group i. Camber and inclination of the road change the spatial position of the cylinders and, therefore, the coordinates of the groups. These inclinations can increase or decrease the safety, depending on the stability area’s definition and the forces that are acting on the transport.

#### 2.3. Forces Acting upon the Transportation Model

_{load}, the mass of the trailer’s component i as m

_{i}, the acceleration of gravity as g and the weight as W. This force is exerted on the CoG. The latter is located at a point that corresponds to the weighted average of the CoGs of all items or parts that are being transported:

_{CoG}, y

_{CoG}, and z

_{CoG}) as coordinates of the final CoG and (x

_{i}, y

_{i}, and z

_{i}) as the coordinates of the component i’s CoG.

_{A}for the acceleration force, M for the total weight, and a for the acceleration. F

_{A}is applied at the CoG.

_{T}is the centrifugal force, M is the total weight, R is the turn’s radius, and v is the trailer’s velocity. F

_{T}is applied to the center of gravity. Depending on the turning direction, it can be positive or negative. The radius in Equation (8) is positive when it appears on the left. It is negative if it appears on the right (See Figure 4).

_{Dx}) and transversal (F

_{Dy}) drag forces that movement through a fluid creates:

_{Dy}for the longitudinal and transverse drag forces, ρ for the fluid’s density, v

_{w}for the velocity of the air, α

_{w}for the angle between the air velocity and the trailer’s longitudinal axle, and A

_{x}and Ay denote the reference areas. Finally, C

_{dy}and C

_{dx}are drag coefficients, and are dimensionless (they depend on the geometry, surface, and relative position). One should remember that three aerodynamic forces generally act on the trailer. They are a longitudinal force (axis X), a transversal force (axis Y), and a lifting force (axis Z). However, the equations for these forces are valid for certain objects. The drag force on the truck, trailer, and load together cannot be determined easily. Obtaining realistic values for the forces that are acting on the three axles requires CFD and/or wind tunnel experiments. Only transversal and longitudinal forces that act on the load have been considered in order to simplify the calculations, as they are not great and their effect is minor. The model assumes that all of the forces are applied at the CoG. Therefore, the total force on the trailer and its load is the vector total of the dynamic, aerodynamic, and static forces as expressed by:

_{x}, F

_{y}and F

_{z}. Thus, the vector of the force can be represented by Equation (12)

#### 2.4. Stability Calculation

_{i}, y

_{i}, z

_{i}) and (x

_{i+1}, y

_{i+1}, z

_{i+1}) as coordinates for two consecutive corners. The equation for the plane Ω is obtained by developing this expression:

_{CD}and v

_{ID}as the vectors between points (I,O) and (C,O), v

_{IO}× v

_{CO}as the cross product, and |v

_{IO}| and |v

_{CO}| as the modulus of these vectors.

#### 2.5. Reactions of the Suspension Groups

- Load (W) and reactions (F
_{1}, F_{2}, and F_{3}) are perpendicular to the stability plane. - There are no forces in X or Y and/or moment in Z. Thus, the equilibrium equations are three in number.
- The system affects quasi-static loading. That is, it is assumed that the time and mass do not influence the load.
- The ground is assumed to be a completely rigid plane.

_{cyl,j}), divide the force in group i (F

_{i}) by the number of cylinders in the group (n

_{i}):

_{i}as the force in group I and n

_{i}as the number of cylinders in the group. Calculate the pressure in each cylinder by the following equation:

_{cyl j}as the cylinder’s oil pressure j and A

_{j}as this cylinder’s area.

## 3. Results and Discussion

#### 3.1. Experimental Validation of the Proposed Model

- Trailer: SPMT 6-axle: weight 23.5 and maximum capacity 216.3 Tn.
- The oil pressure is supplied to the hydraulic cylinders at the axles by a Power Unit (PPU).
- Even surface: camber and slope equal zero.

- Weight: 42,500 kg.
- Dimensions: 5133 × 2650 × 2975 mm.
- CoG position: 97.5 × 0 × −180 mm.

_{i}is a result of the experiment, μ is the mean of this variable, and n is the number of cases (just repeated).

_{i}is the model’s result, E

_{i}is the experimental result, and n is the number of cases.

#### 3.2. Optimization Process Proposed

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Modular trailer axle compensation [28].

**Figure 6.**Forces in a 3-point suspension system: (

**a**) 3D coordinate systems with X, Y and Z axes (

**b**) 2D coordinate systems with X and Y axes.

**Figure 7.**Configuration of the experiments to validate the proposed regression model: Crane, trailer and load.

**Figure 11.**Optimized transport: 108.5 Tn cylindrical tank on 10 axle trailer (Intercombi model of Scheuerle company [38]).

Inputs | Symbol | Unit | Levels | ||
---|---|---|---|---|---|

−1 | 0 | +1 | |||

Longitudinal location of the load | X | mm | 0 | 675 | 1350 |

Transversal location of the load | Y | mm | 0 | 175 | 350 |

X | Y | Pressures [bar] | ||||
---|---|---|---|---|---|---|

A | B | C | D | |||

1 | 0 | 0 | 59 | 61 | 77 | 75 |

2 | 0 | 0 | 55 | 62 | 80 | 79 |

3 | 0 | 0 | 60 | 61 | 75 | 72 |

4 | 0 | 0 | 64 | 56 | 90 | 92 |

5 | 0 | 0 | 58 | 63 | 78 | 74 |

6 | 0 | 335 | 24 | 97 | 80 | 80 |

7 | 0 | 435 | 13 | 108 | 81 | 85 |

8 | 675 | 0 | 66 | 75 | 58 | 58 |

9 | 675 | 335 | 38 | 103 | 58 | 59 |

10 | 675 | 435 | 27 | 117 | 58 | 58 |

11 | 1350 | 0 | 79 | 85 | 39 | 40 |

12 | 1350 | 335 | 47 | 120 | 39 | 38 |

13 | 1350 | 435 | 38 | 127 | 38 | 38 |

Case | Range | Mean [μ] | SD [s] |
---|---|---|---|

Pressure A | 9 bar | 59.2 bar | 2.9 bar |

Pressure B | 7 bar | 60.6 bar | 2.4 bar |

Pressure C | 15 bar | 80.0 bar | 5.3 bar |

Pressure D | 20 bar | 78.4 bar | 7.2 bar |

Average values | 69.55 bar | 4.45 bar |

Case | Pressures | ||||
---|---|---|---|---|---|

A [bar] | B [bar] | C [bar] | D [bar] | MAPE [%] | |

1–5 | 59.0 | 58.0 | 84.4 | 84.4 | 4.4 |

6 | 28.0 | 89.3 | 84.4 | 84.4 | 8.8 |

7 | 18.3 | 98.7 | 84.4 | 84.4 | 13.6 |

8 | 69.8 | 68.9 | 62.6 | 62.6 | 7.4 |

9 | 38.6 | 100.2 | 62.6 | 62.6 | 4.6 |

10 | 29.2 | 109.6 | 62.6 | 62.6 | 7.6 |

11 | 80.7 | 79.8 | 40.9 | 40.9 | 3.8 |

12 | 49.4 | 111.1 | 40.9 | 40.9 | 6.2 |

13 | 40.1 | 120.4 | 40.9 | 40.9 | 6.4 |

Average: | 7.0% |

Characteristics | Value |
---|---|

Load | Cylindrical tank |

Dimensions | 19.5 × 3.2 × 2.5 m |

Load weight | 108,500 kg |

Trailer weight | 33,000 kg |

Initial CoG coordinates | 0.0 × 0.2 × 2.1 m |

Optimum CoG coordinates | 0.3 × 0.0 × 2.1 m |

Number of trailers | 2 |

Number of axles | 10 (4 + 6) |

Tipping angle (on even road) | 7.2° |

Maximum axle load | 21,600 kg |

Oil pressures | 8.1 × 8.1 × 5.9 bar |

Hydraulic configuration | 6, 7, 7 |

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**MDPI and ACS Style**

Escribano-García, R.; Corral-Bobadilla, M.; Somovilla-Gómez, F.; Lostado-Lorza, R.; Ahmed, A.
A Theoretical Model with Which to Safely Optimize the Configuration of Hydraulic Suspension of Modular Trailers in Special Road Transport. *Appl. Sci.* **2021**, *11*, 305.
https://doi.org/10.3390/app11010305

**AMA Style**

Escribano-García R, Corral-Bobadilla M, Somovilla-Gómez F, Lostado-Lorza R, Ahmed A.
A Theoretical Model with Which to Safely Optimize the Configuration of Hydraulic Suspension of Modular Trailers in Special Road Transport. *Applied Sciences*. 2021; 11(1):305.
https://doi.org/10.3390/app11010305

**Chicago/Turabian Style**

Escribano-García, Rubén, Marina Corral-Bobadilla, Fátima Somovilla-Gómez, Rubén Lostado-Lorza, and Ash Ahmed.
2021. "A Theoretical Model with Which to Safely Optimize the Configuration of Hydraulic Suspension of Modular Trailers in Special Road Transport" *Applied Sciences* 11, no. 1: 305.
https://doi.org/10.3390/app11010305