# Road Surface Profile Synthesis: Assessment of Suitability for Simulation

^{*}

## Abstract

**:**

## 1. Introduction

_{d}(n

_{0}), identified as power spectral density (PSD) of the vertical road profile displacement, and the waviness $w$) to describe the road profile. The road profiles are classified as belonging to one of the classes (from A to H) provided by ISO 8608 on the basis of the G

_{d}(n

_{0}) parameter values, while keeping the constant waviness value of $w=2$.

## 2. Road Profile Simulation

#### 2.1. White Noise Filtration

#### 2.2. Sinusoidal Approximation

^{th}harmonic, $N=\left({\Omega}_{\mathrm{U}}-{\Omega}_{\mathrm{L}}\right)/\Delta \Omega $ is the number of frequency bands into which the total PSD spectrum is divided, ${\Omega}_{\mathrm{U}},{\Omega}_{\mathrm{L}}$ are the upper and lower spatial frequencies in the PSD spectrum (rad/m), $\Delta \Omega $ is the width of each frequency band, ${\Omega}_{i}={\Omega}_{\mathrm{L}}+\left(i-1\right)\Delta \Omega $ is the spatial frequency of i

^{th}harmonic or i

^{th}frequency band. The phase angle of i

^{th}harmonic ${\phi}_{i}$ was uniformly distributed in the interval (0, 2π).

#### 2.3. Moving Average of White Noise

## 3. Implementation of Simulations

#### 3.1. White Noise Filtration

#### 3.2. Sinusoidal Approximation

#### 3.3. Moving Average of White Noise

_{j}as a uniform random number in the range (0.25, 4) and normalized it to match Equation (8). For the moving average of white noise, the discretization step set $\mathrm{d}x=0.05\mathrm{m}$, is equal to the reciprocal of the sampling frequency. Note that in the code, the same sample of a Gaussian white noise has been used to generate the Gaussian and non-stationary Laplace models of the road profile. The same function kernel g(x) implemented with different parameters allows facilitating a visual comparison of the simulated profiles. Moreover, due to this reason, all random generations take the same seed value, and the time-domain data are converted to the spatial domain. The total length of the profile was chosen as L

_{p}= 1000 m.

## 4. Simulation Results and Discussion

#### 4.1. White Noise Filtration

#### 4.2. Sinusoidal Approximation

_{i}is used as a full road profile (Figure 3c). However, the influence of waves with the long period disappears, and the abrupt shift can appear at any place (Figure 3d).

#### 4.3. Moving Average of White Noise

## 5. Conclusions

^{2}and low-frequency boundary α. The method is simple, but in the case of unadjusted parameter values, the profile floats away and therefore, the roughness amplitude and period draughts away. This method is sensitive to the virtual driving velocity, as well. The positive side in this method is the feature, which is the case when the variation σ

^{2}fits the road profile with ω = 2, according to the corresponding ISO standard. This method does not generate an identical profile picture for every 100 m of road profile, as it happens using the method of harmonic functions.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Simulated road profile using the white noise filtration method for three velocity values v = 10, 20, and 30 m/s: (

**a**) With the required minimal sample time value (spectra are shifted vertically for clarity), (

**b**) with the fixed sample time value 0.05/30 s.

**Figure 3.**Simulated road profiles in the spatial domain using sinusoidal approximation: (

**a**) Without modification, (

**b**) with modification, (

**c**) without windowing and with random φ

_{I}; and (

**d**) small range of (

**a**,

**c**) cases.

**Figure 4.**Normalized probability density of simulated (

**a**) separate blocks and (

**b**) whole profiles using a method of sinusoidal approximation.

**Figure 5.**Comparison of simulated road profile z(x) using the moving average of white noise: (

**a**) Stationary Gaussian model, (

**b**) non-stationary Gaussian model, (

**c**) non-stationary Laplace model, (

**d**) differences of them.

**Figure 6.**Comparison of non-parametric estimates of probability densities (dashed line with symbol) with (

**a**) standardized normal distribution, (

**b**) fitted Laplace probability density function (solid lines).

**Figure 7.**Power spectral density (PSD) spectra of the ISO 8608 road profile (dashed line) and generated road profiles using: 1—White noise filtration, 2—sinusoidal approximation and moving average of white noise, 3—stationary Gaussian model, 4—non-stationary Gaussian model, 5—non-stationary Laplace model.

Method | Characteristics | Validation | Ref. |
---|---|---|---|

White noise filtration; sinusoidal approximation; moving average of white noise | White noise; PSD; Kurtosis | Comparison to the ISO 8608 standard, evaluation of Gaussianity | This work |

White noise filtration | White noise; coordinates | Evaluation of PSD | [22] |

Sinusoidal approximation; first order filter | PSD | - | [23] |

Gaussian white noise; superposition of harmonics | White noise; PSD; spatial frequency | Comparison to the ISO 8608 standard | [24] |

Sinusoidal approximation | Spatial frequency | - | [25] |

Hybrid (Gaussian and white noise) | Power spectrum; vehicle speed; standard deviation | - | [31] |

Laplace processes | Kurtosis | Comparison to the real road | [32] |

Gaussian random field | PSD roughness | Steady state analysis | [34] |

Gaussian random field | PSD | - | [35] |

Method | Advantages | Disadvantages |
---|---|---|

White noise filtration | Direct use in Simulink. | Only three parameters: $\mathrm{Low}\mathrm{frequency}\mathrm{cut}\mathrm{off}\mathsf{\alpha};\mathrm{velocity}v,\mathrm{and}\mathrm{variance}{\sigma}^{2}=4{G}_{\mathrm{d}}\left({\Omega}_{0}\right)$. One must use the appropriate simulation parameters to get correct results (velocity, sample time). |

Sinusoidal approximation | Differentiations without going through numerical differentiation. Maximum possible parameters.Possible various PSD approximations. | Repeatability of profile segments or abrupt shift. Greater computation time. Only stationary Gaussian processes. |

Moving average of white noise | Possible various PSD approximations. No repeatability of profile segments. Possible stationary or non-stationary Gaussian/Laplace processes. |

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**MDPI and ACS Style**

Lenkutis, T.; Čerškus, A.; Šešok, N.; Dzedzickis, A.; Bučinskas, V.
Road Surface Profile Synthesis: Assessment of Suitability for Simulation. *Symmetry* **2021**, *13*, 68.
https://doi.org/10.3390/sym13010068

**AMA Style**

Lenkutis T, Čerškus A, Šešok N, Dzedzickis A, Bučinskas V.
Road Surface Profile Synthesis: Assessment of Suitability for Simulation. *Symmetry*. 2021; 13(1):68.
https://doi.org/10.3390/sym13010068

**Chicago/Turabian Style**

Lenkutis, Tadas, Aurimas Čerškus, Nikolaj Šešok, Andrius Dzedzickis, and Vytautas Bučinskas.
2021. "Road Surface Profile Synthesis: Assessment of Suitability for Simulation" *Symmetry* 13, no. 1: 68.
https://doi.org/10.3390/sym13010068