ABSTRACT The derivation of linearized equations and subsequent eigenvalue analysis is the basis for tasks such as frequency‐domain response analysis, control design, and stability assessment for mechanical systems. However, for general multibody systems with redundant or nonholonomic constraints, practical challenges persist in achieving numerically stable linearization and reliable eigenvalue analysis. This study provides numerically reliable linearization and eigenvalue analysis algorithms of Lagrange's equations of the first kind for general multibody systems, and forms a systematic computational framework. First, a robust linearization methodology is developed to automatically generate linearized state‐space equations. Coordinate partitioning is employed not only to eliminate redundant constraints and dependent coordinates, but also to significantly simplify the calculations. Second, a direct eigenvalue analysis on Redundant Coordinate Set Jacobians is performed instead of decomposing the state matrix, for its better numerical conditioning and sparsity. The equivalence of the eigenvalue problems is strictly proven theoretically. The methodology is validated by various benchmark problems and practical cases. The proposed approach has been implemented and released with a general‐purpose rigid‐flexible coupled multibody dynamics software INTESIM‐FMBD (v7.0), and provides a reliable numerical tool for linearizing general mechanical systems.