Free vibration analysis of multiple-stepped beams by the differential quadrature element method

The differential quadrature element method (DQEM) is proposed for obtaining highly accurate natural frequencies of multiple-stepped beams with an aligned neutral axis. To increase the computational efficiency and accuracy, explicit formulas are provided for computing the weighting coefficients. Various examples with existing solutions are analyzed. To show the accuracy and efficiency of the DQEM, results are compared to analytical as well as numerical solutions by using finite element method (FEM) with very fine meshes. It is seen that the rate of convergence of the DQEM is very high and the DQEM can yield very accurate results with much less computational effort as compared to existing numerical methods. The results obtained by the DQEM can be exactly the same as the existing analytical solutions to five places of decimals with a small number of grid points. The proposed method is simple and efficient, and can be used to analyze beams with any step changes in cross-section conveniently.