The work aims to propose an appropriate numerical simulation method for non-Gaussian random vibration processes generated according to the known information when considering non-Gaussian property of the actual random vibration. Based on such constraint conditions as mean, variance, skewness, kurtosis and PSD function (or autocorrelation function), the non-Gaussian random vibration was simulated. The autocorrelation matrix of the non-Gaussian process was obtained from the PSD. Through the formula derived from Hermite polynomial orthogonal property and polynomial chaos expansion, the covariance matrix of the standard normal distribution random process was constructed, and the spectral decomposition and principal component analysis were also performed. Finally, the simulated non-Gaussian vibration process was represented by Karhunen-Loeve expansion and polynomial chaos expansion. As the number of sampling points increased, the error between measured data and simulated data became smaller and smaller, besides, the proposed method had good simulation accuracy. Combined with polynomial chaos expansion, Karhunen-Loeve expansion and Monte Carlo method, the non-Gaussian random vibration process can be generated and accurate and effective simulation values of various statistical parameters can be obtained.