# Contact Analysis and Friction Prediction of Non-Gaussian Random Surfaces
## 1 Introduction  
The friction between contact surfaces is demonstrated as an important source of nonlinear vibrations in many mechanical and structural systems. Moreover, the vibration could help reflect the health status of the machine [1, 2], which has been studied widely [3, 4, 5, 6, 7, 8]. Therefore, it is vital to investigate the friction characteristics. The friction behavior at the macroscopic length scale is highly correlated with the microscopic topography of the contacting surfaces, which determines the contact and lubrication status of machinery elements [9]. Over the past few decades, many analytical and numerical contact models for rough surfaces which assumed a Gaussian distribution of surface heights have been developed. In fact, some certain types of surface finishing and the running-in processes of machine operation often results in the asymmetric distribution of roughness heights. Compared with Gaussian surface, non-Gaussian surface reflects different tribological properties due to the impacts of skewness and kurtosis. Therefore, a thorough awareness of contact performance and tribological property of non-Gaussian contact surfaces is of great concern for enhancing effectiveness and reducing friction losses. The solution of contact models demands the input of an either two- or three-dimensional rough profile which can be acquired either from digital output of the profilometer or from numerical simulations. Compared with the measured rough surfaces, the numerical method has many advantages in high efficiency and low cost. Additionally, it can efficiently generate arbitrarily and sufficiently large samples of profiles [10, 11, 12]. The engineering surface can be mathematically approximated as a description of stochastic process. In addition, most of the statistical parameters of an engineering surface can be sufficiently described with the use of the height distribution function and the autocorrelation function (ACF) [13]. Thus, it is convenient to generate surfaces with predetermined ACFs and the height distribution functions to take the place of engineering surfaces [14]. [PERSON] [15] used linear transformation of a random matrix and moving average (MA) model to generate Gaussian and non-Gaussian rough surfaces with specified ACFs. However, [PERSON]'s method requiredsolving a nonlinear system of equations to obtain an autocorrelation coefficient matrix, which is both storage space and time limiting. In order to deal with those disadvantages, [PERSON] [10] transformed the nonlinear system of equations into a nonlinear optimization problem and used the nonlinear conjugate gradient method (NCGM) with secant line search to solve. In addition, based on a 2D digital filter and Fast Fourier Transform (FFT), a fast and convenient procedure firstly proposed by [PERSON] and [PERSON] [16] also were applied to generate rough surfaces. [PERSON] [17; 18] found that there were large errors between the intended ACF and the actual ACF for surfaces generated by FFT methods, especially when autocorrelation lengths became long. In addition, he modified the FFT method to obtain Gaussian and non-Gaussian surface with large wavelengths. The FFT algorithm also was adopted to the computation process of the optimization problem first presented by [PERSON], and many advances were made in terms of computing power and efficiency [12; 19; 20]. These have made [PERSON]'s method to generate surfaces with large autocorrelation accurately. Hence, the improved [PERSON]'s method is adopted to generate non-Gaussian surfaces in this paper. During the past decades, many analytical and numerical models have been proposed for contact simulation of rough surfaces. In general, existing contact models can be divided into two broad categories: stochastic methods and deterministic methods. The original stochastic model was constructed by [PERSON] and [PERSON] [21].