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Based on the analysis of the geometric characteristics, distribution characteristics, and constraints of the cobblestone road, a three-dimensional (3D) parametric road modeling method based on geometric characteristics is proposed. The cobblestone model is established by the exponential oval equation, and the cobblestones distribution is determined by Monte Carlo stochastic search method. Then the 3D theoretical model of cobblestone road is obtained. To veritably simulate the cobblestone road, a generated 3D random road is fusing with the 3D cobblestone road model by alternating values, and the comparison between the real and modeling cobblestone roads is done in space domain and PSD curves. Then based on the establishment of the standard vehicle vibration model, the influences of the key geometric parameters of the cobblestone road on the vehicle vibration are analyzed by using the evaluation indices of the IRI and the RMS of the vehicle body acceleration. The proposed method can be extended to the 3D modeling of almost all strengthened test roads such as Belgian road, fish-scale pits road, ripple tracks, and washboard road and provides a 3D road modeling method with adjustable parameters, authenticity, and accuracy for the comprehensive construction of vehicle virtual test field.

With the rapid development and update of the vehicle, compared with the real vehicle driving test [

Road profile [

Currently, the methods of road profile acquisition [

The simulation methods can reconstruct the required road profiles according to the statistical characteristics of road roughness, which contain harmonic superposition method [

In terms of modeling of road geometry, 3D models can more accurately reflect the effect of roads on vehicles, especially when vehicles are subjected to lateral loading; thus more accurate test results are obtained in the virtual test based on 3D model [

In all kinds of strengthened test roads, such as long/short ripple tracks, washboard road, Belgian road, cobblestone road, and fish-scale pits road, they all rely on the specific “obstacles” that appear regularly on the roads to achieve the purpose of accelerated test. The geometric characteristics of the road obstacles determine the form and strength of the impact on vehicles. Due to the sensitivity and importance of the geometric characteristics of road obstacles on vehicle, even for the same type of strengthened test road, the roads with different geometric characteristics will have different effects on vehicles. Therefore, for the reconstruction of these roads, the geometric characteristics of the roads can be reconstructed accurately, and they can be better applied to road simulation or virtual test.

Therefore, taking the cobblestone road as the research object, the paper mathematically describes its geometric characteristics and establishes the 3D road model of cobblestone road based on the determination of its key geometry or distribution parameters. To achieve the effect of real road simulation, a more real 3D road model is obtained by integrating the theoretical model with the generated 3D random road. Then the comparison between modeling road and real road is done in space domain and frequency domain. Finally, the influence of cobblestone road geometric parameters on vehicle response is analyzed by establishing standardized vehicle model and evaluation indicators based on the existing 3D road model. The proposed method realizes fast 3D modeling of roads with different parameters and provides the 3D road model with adjustable parameters and real accuracy for building vehicle virtual test ground and carrying out virtual driving test

The cobblestone road is a kind of strengthened test road by placing the cobblestones with a certain grain size into the cement concrete sparsely and irregularly. The cobblestone road is the most used reliability/durability test road, which can provide a wide band of strong load for the whole vehicle (especially the steering system).

Compared with other strengthened test roads, the 3D modeling of cobblestone road is more complicated because of the random variation in the main parameters such as cobblestone size, cobblestone height, cobblestone distribution, and the shape of cobblestone itself. In this section, based on the exponential 2D oval equation, the 3D model of cobblestone road is gradually established according to the order of Oval curve, cobblestone mode, cobblestone coordinates, and cobblestone road.

The exponential oval equation evolves from the standard form of elliptic equation, which can be expressed as

Obviously, when

Oval curves under different shape parameters.

As shown in Figure

Obviously, (

Taking the derivative of (

When

According to the solution of (

Because

The solution of (

The relation between the ordinate and the abscissa of any point in the positive half of the longitudinal section can be expressed by plugging (

Equations (

As shown in Figure

The coordinate system of 3D cobblestone model.

Assume that the XOY plane is the ground. As the boundary of the vertical section, curve 1 determines the actual height and protruding characteristic of the upper part of the cobblestone. And as the boundary of the horizontal section contour, curves 2 and 3 determine the basic shape of the cobblestone.

The oval curves 2 and 3 have independent “height” parameters

The basic shape of the pebble’s horizontal surface profile under different parameters.

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When the boundaries expressed by three oval curves are determined, the cobblestone surface consists of the elliptic curves by parallel sliding along the

When

3D cobblestone model.

After determining the method of generating 3D cobblestone, they should be scattered on the road randomly according to a certain distribution law. Usually, the intervals between cobblestones are required on the cobblestone road, and it is hoped that the cobblestones can be evenly distributed on the road. Besides, the cobblestones cannot overlap and accumulate obviously.

To simplify the analysis, the horizontal area of all the cobblestones is considered to be a circle with a diameter that varies from the minimum particle size

When the length

However, even if it is simplified as circular, essentially, filling a number of circles with variable radius and without overlapping in a rectangle with a fixed boundary is still a classic Packing problem [

Flowchart of Monte Carlo searching for cobblestone center point.

As shown in Figure

When the maximum particle size of 300 mm and the minimum particle size of 150 mm are determined, Figures

Monte Carlo search results of the cobblestones distribution with different minimum intervals: (a)

From (

The center point coordinate of the cobblestone

At this time, because

The change of the coordinate system (grid) after the cobblestone rotation is shown in Figure

The coordinate system of cobblestones before and after rotation.

For any point on the cobblestone road, its height can be expressed as

As there are many road parameters of the cobblestone road, the road with different emphasis can be produced by the combination of different parameters. Here 3D models of three types of cobblestone roads are built by copying them from a domestic vehicle test ground. The shape and distribution parameters of the three kinds of cobblestone roads in the test ground are shown in Table

The parameters of the cobblestone roads in a vehicle test field.

Particle size | Spacing | Height | Density ^{2}) | |
---|---|---|---|---|

Cobblestone road A | 250~350 | | 80~120 | 3.5 |

Cobblestone road B | 150~300 | 200~400 | <120 | / |

Cobblestone road C | 70~80 | 100~200 | 20~35 | 30~35 |

As shown in Table

Besides, among the parameters mentioned in Table

Road A has the largest diameter, height, and moderate difference in height, and the road is mainly used for the assessment of heavy vehicles. So the shape parameter of the required cobblestone is large, resulting in maximum difference in shape to give the highest strength assessment of vehicle.

Road B has maximum difference in height and larger particle size, so larger vertical shape parameters

The cobblestone of road C is nearly spherical or hemispherical with small particle size and high density, so the shape parameter of cobblestone has almost no impact on vehicle excitation. The road provides rich high frequency excitation for medium or light vehicles at high speed, so smaller shape parameter is selected.

The 3D cobblestone road models with the length of 10 m and the width of 4 m according to the above parameters by the proposed method are shown in Figure

Three kinds of 3D cobblestone road models in a test field. (a) Cobblestone road A. (b) Cobblestone road B. (c) Cobblestone road C.

Although the 3D road model of cobblestone road has been established based on the mathematical description of the geometric characteristics, the 3D model is only a theoretical model of the cobblestone road, and there is still a big gap with the actual road. Therefore, based on the theoretical model of 3D cobblestone road, the 3D road model similar to the actual road is obtained by fusing a generated 3D random road with certain road roughness.

Among the existing methods of generating 3D random road model [

According to the standard ISO/DIS 8608 and GB/T 7031-2005, the road roughness is divided into eight levels by the road PSD, that is, Grade A to H. Considering that the actual test road itself is generally good pavement in addition to obstacles, the road of Grade A (at this time, there is

3D random road model of Grade A.

For the theoretical road height

Because the cobblestone surface of the cobblestone road itself is more smooth, the fusion is carried out by alternating values between the cobblestone theoretical height

The 3D cobblestone road model after the fusion of 3D random road is shown in Figure

3D cobblestone road after random road surface fusion.

The road model generated by the proposed method gives the X, Y, and Z coordinate values of all space points on the road; then the model is usually saved as a text file (or Excel file) in the form of 3D coordinate dot array. In order to apply the 3D road model to virtual simulation test, the 3D road model can be imported into the virtual simulation environment such as Virtual.Lab and Adams and used as a “road model” that can be identified by the simulation environment. Due to the cobblestone road being rigid road, the deformation of the road is negligible. Then the friction coefficient and other parameters are given on the surface of the road model in the virtual environment, and the simulation test research can be carried out.

When the vehicle runs through the cobblestone road, the contact between the tire and the road is a surface contact, which includes not only the tire deformation along the road but also the deformation along the width of the tire. At this time, the excitation input of the road on the vehicle can be regarded as the average of the road roughness on the whole contact surface. However, when the road is long enough, any road trajectory of cobblestone road has the same statistical characteristics in time and frequency domains, which can be compared to verify the accuracy of the cobblestone road model established in the paper. Therefore, a real cobblestone road with the similar geometric characteristics in a proving ground is used for comparison with the cobblestone road model, as shown in Figure

The real cobblestone road in picture.

Then, using a road profile acquisition system, the road profile of the real cobblestone road is obtained, and the local road profile is shown in Figure

Extracted road profile of cobblestone road. (a) The real road; (b) the modeling road.

Because the road profile is usually described in the form of PSD, the PSD of the road profiles in real and modeling cobblestone roads both need to be calculated through Fast Fourier Transform (FFT) for comparison, as shown in Figure

Space PSD of the cobblestone road profile.

Based on the establishment of the 3D cobblestone road model, the influence mode of the cobblestone road geometric parameters on the vehicle response is discussed preliminarily by establishing the standardized vehicle vibration model and evaluation indices.

In the field of road engineering, the International Roughness Index (IRI) is the most widely used quality evaluation index for road [

IRI standard vehicle model and its frequency response curves. (a) 1/4 vehicle vibration model. (b) The frequency response curves of the standard vehicle model.

According to Figure

The longitudinal height on arbitrary abscissa of the established strengthened test road (cobblestone road) is used as the road profile input of the standard vehicle model; then the vehicle responses are obtained.

In the paper, the IRI and the root mean square (RMS) of vehicle body vertical vibration are used as the indices to evaluate the road and its impact on the vehicle, respectively.

The IRI is the cumulative value of the relative displacement between the sprung mass and the unsprung mass per kilometer when driving on the corresponding road with the excitation of road roughness, expressed by m/km. It is an important index to evaluate road roughness. According to the definition of IRI, when the total length of the road is L (unit km),

Corresponding to the road profile represented by IRI, the RMS of vehicle body (here is represented by sprung mass) acceleration is the important index of vehicle vibration intensity, and its value is directly related to vehicle reliability and ride comfort. For the vehicle body acceleration

As shown in Figure

Considering that the vehicle/road vibration model represented by (

Geometric parameters of Belgian road and their range of variation.

Road Type | Geometric parameters | Change range/mm |
---|---|---|

Cobblestone road | Maximum height | 0~(10-300) |

Particle size | 20~(50-300) | |

Shape parameter | -0.4~0.4 | |

Interval | 200~1000 |

Note: “~” means parameters change randomly in this range; “-” means parameters increase equally in this range.

Effect of maximum height and longitudinal length of cobblestones on

As shown in Figure

The above studies proved that the established road model based on road geometric characteristics can quickly evaluate the effects of various parameters on vehicles and provides useful references for road types and parameters selection when test roads are built.

Taking cobblestone road as the research object, a 3D road modeling method based on geometric characteristics is proposed. The oval equation of cobblestone is established and discussed by determining the key geometric characteristics, and the cobblestones distribution on the cobblestone road is determined by Monte Carlo search method. Then the 3D theoretical reconstruction of the cobblestone road is completed, and a generated 3D random road is fused to get a more realistic 3D model of the cobblestone road. The comparison between the modeling road and real road is done in space domain and PSD curves.

Based on the standard vehicle vibration model, the IRI and the RMS of vehicle body acceleration are established as the evaluation indices of vehicle vibration, and the influence of key geometric parameters of the cobblestone road on the vehicle vibration was evaluated. Analysis results show that the sizes of cobblestone have a decisive influence on the vehicle vibration.

Although the method proposed in the paper is only verified under the cobblestone road situation, the method is applicable for 3D modeling of almost all strengthened test roads, such as the Belgian road, washboards, and ripple tracks and provides an effective way of 3D road modeling for building vehicle virtual test field.

Three-dimensional

Regular Grid Road

Curved Regular Grid

Road Definition File

Nondeterministic Polynomial

Power spectral density

Fast Fourier Transform

International Roughness Index

Root mean square.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work is supported by Weapon and Equipment Exploration Research Project (no. 7131255).