/ / Article

Research Article | Open Access

Volume 2018 |Article ID 2535409 | 14 pages | https://doi.org/10.1155/2018/2535409

# Construction of Three-Dimensional Road Surface and Application on Interaction between Vehicle and Road

Accepted04 Jan 2018
Published15 Feb 2018

#### 2. Time-Domain Model of Random Road Surface Roughness

##### 2.1. Power Spectra at Spatial Frequency and Time Frequency

The fitting expression of road surface power spectral density is as follows:where is spatial frequency; is the reciprocal of wavelength ; is reference spatial frequency, ; is frequency index, that is, the diagonal slope on log-log coordinate, which determines the power spectral density frequency structure of road surface; is the road surface power spectral density at reference spatial frequency called road surface roughness coefficient, .

When a car travels at through a road surface with a spatial frequency of , the equivalent temporal frequency (Hz) is

And the relation between power spectrum densities at temporal frequency and spatial frequency iswhere is the power spectral density at temporal frequency and is the power spectral density at spatial frequency.

##### 2.2. Description of Road Surface Spectrum Time-Domain Model

The PSD of road surface spectrum is statistics corresponding to a certain sort of road surface roughness. However, the reconstructed pavement elevation is not the only one for a given road surface PSD. The road function also corresponds with a sample function in the equivalent pavement elevation of the given road surface spectrum at a certain running velocity.

##### 2.3. Filtered White Noise Generation for Stochastic Road Surface

When a car travels at a constant speed , the time-domain road surface roughness power spectral density is expressed as follows, due to :When , . Therefore, if the lower cut-off angular frequency is taken into account, the actual power spectral density  could be expressed as follows: where is lower cut-off angular frequency

Equation (6) could be taken as the response of white noise excited first-order linear system. According to stochastic vibration theory,where is frequency response function; is white noise; is power spectral density that takes , soIt can be obtained as where is lower cut-off spatial frequency, ; is road surface roughness coefficient, ; is white noise with mean value equal to zero; is random elevation displacement of road surface, .

#### 3. Box-Counting Dimensions Method-Based Road Surface Parameter Extraction

The author calculates the fractal dimension of 5,000 m road surface random elevation and makes statistical and fractal analysis of road surface elevation data through box-counting dimensions method-based programming. Fractal dimension could be defined aswhere is measurement scale; is curve length measured with the th scale; is number of measurement scales; is fractal dimension of measurement curve.

Box-counting dimensions method calculates fractal dimension through covering fractal curve with small boxes with side length of . Some boxes are empty, while some cover part of the curve. The number of boxes is counted when the box is not empty. And then the number of nonempty boxes is regarded as . The followed thing is that the size of box gradually reduces, while the will immediately rise. Then the formula is obtained when :

The least square method is used to seek a series of and and to fit a straight line in log-log coordinate. The slope of straight line obtained should be the desired fractal dimension. Table 1 shows the fractal dimensions and standard deviations that correspond to road surface spectra at all levels attained from box-counting dimension method.

 Road surface grade A B C D E F G H Standard deviation (10−3 mm) 2.2 4.3 8.7 17.4 34.7 69.3 138.7 277.4 Fractal dimension 1.6003 1.6003 1.6003 1.6003 1.6003 1.6003 1.6003 1.6003

As shown in Table 1, there is no difference under different fractal dimensions with various levels, which indicates obvious self-similarity. The reason is that the inverse Fourier transform assures the coincidence of the straight line of road surface power spectral density obtained through simulation at log-log coordinate, which also reserves the similarity information of road surface roughness.

In the case of standard road surface, slight difference is observed between fractal dimensions of road surfaces at various levels obtained by box-counting dimensions method, and the mean value is approximately 1.6, which is taken as the fractal feature index for standard road surface. Since the standard deviation achieved with box-counting dimensions method falls in the international specified range of standard deviation, it could be taken as a statistical indicator for road surface grading. The above-noted two indicators constitute the grading criteria for reconstruction of three-dimensional standard road surface.

#### 4. Construction of Three-Dimensional Road Surface Spectrum with Road Surface Morphology Features

The road surface presents random performance and statistical self-similarity, which could be reconstructed through fractal Brownian motion (FBM). The midpoint displacement method is also known as the random midpoint displacement method, which is the simplest and classical method applied in FBM, especially for describing one-dimensional random process. In addition, the diamond-square algorithm is based on midpoint displacement method, which can produce two-dimensional or three-dimensional topography. It can not only simulate three-dimensional pavement, but also get a higher reconstruction accuracy certified in our article later.

##### 4.1. Diamond-Square Algorithm Principle

Diamond-square algorithm was originally proposed by Fourniew, Fussell, and Carpenter and is also known as diamond-quadrangle algorithm :(1)Initialization: the two-dimensional array is initialized and the same elevation value is assigned to the four angles. The size of each dimension is supposed to be 2’s nth power plus 1 (e.g., 33 × 33, 65 × 65, 129 × 129). Figure 1 shows the diamond-square algorithm process of a 5 × 5 array, where the elevation value of the 4 angles in Figure 1(a) is initialized and indicated with black spot. Actually, after five-iteration calculation, the road surfaces is separated by a distance of about 3 cm, which can meet our research requirement.(2)The “diamond” stage: as shown in Figure 1(b), a random value is generated using the four points forming a square at the midpoint of such square, that is, the intersection of two diagonals. The midpoint value is equal to the sum of the mean of four corner points’ values and the said random value. Diamond comes into being when a number of squares exist in grid, in which case the center point is indicated with black spot.(3)The “square” stage: use the four points forming diamond to generate a random value with the same value range as the previous step at the midpoint of that diamond. Also, this midpoint value is equal to the sum of the mean of four corner points’ values and the said random value as shown in Figure 1(c), thereby generating a square.(4)Iterate the process above for the specified number of times.

###### 4.1.1. Process of Determining One-Dimensional Stochastic Interpolation

The Brownian motion in two-dimensional plane generates three-dimensional morphology of landscape, which means that coordinates and in the plane bring about as the surface gradient of position . The change of gradient that occurs during constant speed travel along the straight line path in the plane is fractal Brownian motion. Assuming the travel distance in plane is (where: ), the curve gradient variation is given by [28, 29]The formula of stochastic interpolation algorithm is shown in where represents Gaussian random function and means the elevation value of related point. When , (12) leads to In (14), represents the Gaussian distribution when the mean value is equal to zero, and variance is equal to to .

Assuming , , the stochastic interpolation model iswhere .

Solving is the key to achieve the distribution of stochastic increment. The main process is described as follows: The first iteration, , according to (14)The variance of one-dimensional stochastic increment after iterations isThe one-dimensional stochastic increment submits to , .

###### 4.1.2. Process of Determining Two-Dimensional Stochastic Interpolation

First, define the matrix range of each point in two-dimensional space asDefineHence, each point in the two-dimensional space could be described asThese discrete points are subjected to the Gaussian distribution; the expected value is 0, while the variance is .

Then, the four vertexes are initialized aswhere a midpoint is needed; the initial iterative interpolation is expressed asThe first iteration leads toand the variance of two-dimensional stochastic increment after iterations isThen the stochastic increment of two-dimensional is subjected to .

##### 4.2. Fractal Brownian Motion Theory-Based Determination of Random Displacement

The random increment in diamond-quadrangle algorithm could be derived from fractal Brownian motion; that is,where is Hurst index, , . is the segment spacing after segmentations. The following formula is normally employed for numerical calculation of random displacement:orwhere “scale” means the scale factor and normally falls within : is a random function submitting to standard normal distribution ; represents the fractal parameter value of chosen regional terrain; means the number of iterations for stochastic midpoint displacement; stands for the segment spacing after segmentations.

The random increment of the first iteration isAccordingly, the random increment after iterations is

In this paper, five iterations are performed (i.e., ), and the sampling interval is 1. The number of iterations iswhere represents fractal parameter and means the segment spacing after iterations. Then the successive number of iterations is

The number of iterations is determined by researchers to satisfy research requirements. Figure 2 shows the flow chart of calculating the elevation of each point in three-dimensional road surface spectrum through fractal Brownian motion and diamond-square algorithm on the premise of five iterations.

#### 5. Reconstruction and Analysis of Three-Dimensional Rough Road Surface Spectrum

##### 5.1. Reconstruction of Three-Dimensional Rough Road Surface Spectrum

The foregoing theory is employed to simulate grade A~H standard highways; Figure 3 shows the simulation result after eight-time iterations. The road surface is 600 m in length and 33 m in width. The value is taken from random sample obtained through inverse Fourier transform of standard road surface power spectrum; to simplify the analysis, the direction and direction of road surface and the line segments in any plane share the same fractal characteristics, thus having the same Hurst index in simulation.

##### 5.2. Three-Dimensional Rough Road Surface Spectrum Reconstruction Accuracy Analysis

The three-dimensional road surface with 5000 m is reconstructed through fractal interpolation theory and the solution method for power spectral density. Compared with the power spectrum of two-dimensional road surface, the accuracy of road reconstruction proposed in this paper is verified. Due to space limitations, Figure 4 only list the comparison diagrams of standard power spectra for grades A~D between theory and simulation results.

As shown in Table 2, the power spectrum curve attained through the reconstruction of road surface roughness with ideal fractal curve is quite close to standard power spectrum, exhibiting extremely high reconstruction accuracy.

##### 5.3. Reconstruction of Three-Dimensional Road Surface Spectrum with Special Road Surface Features

Upon the establishment of three-dimensional stochastic road surface spectrum for each grade of road surface that reflects the road surface topography, some special road surface model can be also simulated to form the more authentic three-dimensional road model. According to transport industry standard JT/T713-2008 “Pavement Rubber Bump” , the profile of rubber bump should be approximately trapezoidal, and the bottom width and height of bump should be 300~400 mm and 30~60 mm, respectively. However, no complete standard has been issued for pavement contour curve in terms of specific cross section. The common bump cross-sectional profiles include trapezoid, circular arc, and parabola. Figure 5 shows the parabola-shaped profile.

The parabolic cross-sectional profile shown in Figure 5 is expressed as a mathematical equation:

#### 6. Model of Tire Contact with Three-Dimensional Rough Road Surface

The interaction between vehicle and road surface is an extremely complicated dynamic process that involves vehicle dynamics, pavement structural mechanics, and frictional mechanics. When a car travels on road surface, the roughness of road surface is transferred as displacement excitation via tires and suspension to car body and results in the random vibration of car body. The present section builds a 1/4 car body model and further introduces the process of tire contact with random road surface; the load characteristics on tire and tire footprint are integrated through test; the contact surface is considered to be composed of a finite number of points; the contact model is shown in Figure 7.

##### 6.1. Determination of Tire/Road Contact Area

Different pressure is applied to a heavy tire through self-developed actuator system. The test object is heavy-duty radial tire 10.00R20, the standard tire pressure is 830 kPa. The testing site is shown in Figure 8; the tire contact distribution is shown in Figure 9.

The contact area is 45,955 mm2 at a standard load of 30,000 N. Test results show that the width of footprint is 200 mm and the contact length is 230 mm. Since the distance between two adjacent points of three-dimensional road surface is 7.8125 mm, there should be 725 contact points under standard tire pressure and load, including 25 points in vertical direction and 29 points in horizontal direction.

##### 6.2. Two-DOF Vehicle Model

A Two-DOF 1/4 vehicle suspension model could be expressed by a spring and damper connected in parallel, while the tire could be expressed with a mass block and spring as shown in Figure 10, where is the unsprung mass, including rim, tire, and axle; is the sprung mass, including compartment and load; is tire stiffness coefficient; is suspension system stiffness coefficient; is tire damping constant; is damping constant of shock absorber in suspension system; is ground elevation (road surface roughness), a stochastic process; is vertical displacement of sprung part; is vertical displacement of unsprung part.

The system motion equation established based on Newton’s Second Law of Motion isOrwhere represents mass matrix; means damping matrix; is stiffness matrix; stands for displacement matrix; is an excitation matrix; , , and are mass, damping, and stiffness matrixes, respectively, as well as the real symmetric matrixes of , while is positive definite.

##### 6.3. Analysis of Interaction between Tire and Three-Dimensional Rough Road Surface

The subject studied is a heavy-duty automobile of a certain model with tire model 10.00R20; assuming the length and width of tire footprint area remain constant under a standard tire pressure of 830 kPa, the contact area could be determined through test. Select heavy-duty automobile parameters: , , , , ,  N·s/m; tire specifications 10.00R20 , .

Take the tire-road surface contact model as a model where there are a finite number of contact points, all of which have the same stiffness; calculate the vehicle-road coupling system under random excitation of road surface using the multipoint-and-plane contact of three-dimensional road surface spectrum and the two-dimensional curve of any profile section, including the vertical acceleration of car body, suspension distortion, and tire force as shown in Figures 11 and 12, where the dotted line represents multipoint-and-plane contact model, while the real line represents single point contact model.

###### 6.3.1. Response Analysis under Random Three-Dimensional Road Surface Spectrum

Assuming the road surface length in the case of grade-C random road surface, single point contact model and multipoint-and-plane contact model are used, respectively, to determine the dynamic response of vehicle-road coupling system as shown in Figure 11.

As shown in Figure 11, the peak values of car body response and tire force of multipoint-and-plane contact model, that is, plane contact model, are much smaller than that of point contact model; the peak value of car body acceleration is smaller by 47%, while the root mean square value is smaller by 57.6%; the peak value of suspension distortion is smaller by 60%, while the root mean square value is smaller by 56%; the peak value of tire force is smaller by 46.7%, while the root mean square value is smaller by 54%. This means the plane contact between tire and ground has buffering and inclusive effect on road surface.

###### 6.3.2. Analysis of Response under Bump Excitation

A parabola-shaped road bump model is built by assuming the road bump width and height to be 300 mm and 50 mm, respectively. As shown in Figure 12, single point contact model and multipoint-and-plane contact model are used to achieve the dynamic response of vehicle-road coupling system.

According to Figure 12, the overall values of car body response and tire force of multipoint-and-plane contact model are smaller than that of point contact model, but the peak value changes slightly when the vehicle is passing through the bump. Furthermore, there is little difference in action time between plane contact model and point contact model.

#### 7. Conclusion

There is no difference in fractal dimensions at various levels obtained with box-counting dimensions method, which show obvious fractal characteristics of road surface irregularity.

The multiscale characterization method is proposed to reconstruct standard-grade road surface morphology through FBM. The contrast analysis between reconstructed spectrum of three-dimensional road surface and theoretical three-dimensional spectrum of two-dimensional road surface in terms of power spectral density demonstrates the higher reconstruction accuracy.

Two-DOF vehicle model is taken as example to identify the difference between three-dimensional tire-rough road surface multipoint-and-plane contact model and traditional point contact model in terms of the response to car body acceleration, suspension distortion, and tire force. It is included that the multipoint-and-plane contact model reflects the real vertically inclusive characteristics of tire, especially under the excitation of bump.

This paper does not take into account such special conditions of road surface as turning, uphill, and downhill, which are to be subjected to more in-depth studies based on practice in future.

#### Conflicts of Interest

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

#### Acknowledgments

This work was supported by National Natural Science Foundation of China (Grants nos. 11572207, 11472180) and Natural Science Foundation of Hebei Province (Grant no. A2016210103).

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