This book provides an in-depth exploration of random vibration and spectral analysis theory, while emphasizing the completeness of knowledge. It incorporates fundamental theories related to the book's content, drawing from various fields such as theoretical mechanics, material mechanics, mechanical vibration, and engineering mathematics. The focus is on studying and solving the problem of random vibration response analysis in mechanical systems, locomotives, vehicles, and other engineering disciplines.The content includes fundamentals of vibration, probability theory, random variables and random processes, spectral analysis, transfer characteristics of single-degree-of-freedom and multi-degree-of-freedom random vibrations, and random vibration of rail transit vehicles. The book also presents essential theories and methods such as integral transforms, correlation analysis of random vibrations, spectral analysis, random excitation-response analysis of linear systems, statistical characterization of random responses, and random vibration analysis.