Reproducing kernel particle method for free and forced vibration analysis

A reproducing kernel particle method (RKPM) is presented to analyze the natural frequencies of Euler–Bernoulli beams as well as Kirchhoff plates. In addition, RKPM is also used to predict the forced vibration responses of buried pipelines due to longitudinal travelling waves. Two different approaches, Lagrange multipliers as well as transformation method , are employed to enforce essential boundary conditions. Based on the reproducing kernel approximation, the domain of interest is discretized by a set of particles without the employment of a structured mesh, which constitutes an advantage over the finite element method. Meanwhile, RKPM also exhibits advantages over the classical Rayleigh–Ritz method and its counterparts. Numerical results presented here demonstrate the effectiveness of this novel approach for both free and forced vibration analysis.