Application of the spectral geometry method to vibration analysis of pipes with complex boundary conditions

Abstract Vibration characteristic analysis of pipe structures is of great importance for ensuring the safe and stable operation of industrial systems. This paper develops a free vibration analysis method for pipes based on the Spectral Geometry Method (SGM). The method improves the conventional Fourier series by introducing auxiliary functions, constructing a space of displacement admissible functions that precisely satisfy arbitrary elastic boundary conditions and circumferential continuity requirements. Arbitrary boundary constraints are simulated using virtual spring techniques. Based on linear elastic theory, the expressions for the strain energy, kinetic energy, and spring potential energy of the pipe system are systematically derived, and the eigenvalue equation is established via the Rayleigh-Ritz method. Numerical examples demonstrate the favourable convergence of the proposed method. The computational results under different boundary conditions show excellent agreement with finite element solutions, confirming the method’s effectiveness and generality in handling arbitrary boundary conditions. Parametric analysis further reveals the influence patterns of wall thickness and length-to-diameter ratio on natural frequencies. This study provides a reliable theoretical tool for the dynamic design of pipe systems under complex boundary conditions.