Free vibration analysis of multiple-stepped beams by the discrete singular convolution

In this paper, free vibration of beams with multiple step changes are successfully analyzed by using the modified discrete singular convolution (DSC). The jump conditions at the steps are used to overcome the difficulty in using ordinary DSC for dealing with ill-posed problems. An nth-order interpolation polynomial is established on each side of the step. Then two or three jump conditions together with two continuous conditions at steps are used to relate the two polynomials. Five examples are analyzed. Results obtained by the modified DSC and existing solutions are compared. It is seen that the DSC results agree well with existing analytical, numerical, and experimental data. It is also found that the DSC results are even more accurate than the data obtained by the differential quadrature element method for much higher mode frequencies. The present research extends the application range of the DSC algorithm.