In this paper, an analytical method is proposed for calculation of natural frequencies of a delaminated composite beam from both free and constrained mode frequencies. In previous studies, the frequencies of a delaminated composite beam were computed with assumption of occurring open or close delamination during the vibration. According to this assumption, two separated modes, i.e., “free mode” and “constrained mode”, are occurred in vibration of the delaminated beam. In fact, a delamination may breathe (open and close) during the vibration and the assumptions of the free or constrained mode models are not completely correct in the whole of the vibration period. For this reason, a new formulation is proposed for calculation of natural frequencies based on the breathing of delamination. The obtained results are compared with various theoretical and experimental results available in the literature. Thus, the effects of location and size of delamination can be investigated on the natural frequencies of delaminated beams.

In this paper, a differential quadrature element method (DQEM) is developed for free transverse vibration analysis of a non-uniform cantilever Timoshenko beam with multiple concentrated masses. Governing equations, compatibility and boundary conditions are formulated according to the differential quadrature rules. The compatibility conditions at the position of each concentrated mass are assumed as the continuity in the vertical displacement, rotation and bending moment and discontinuity in the transverse force due to acceleration of the concentrated mass. The effects of number, magnitude and position of the masses on the value of the natural frequencies are investigated. The accuracy, convergence and efficiency of the proposed method are confirmed by comparing the obtained numerical results with the analytical solutions of other researchers. The two main advantages of the proposed method in comparison with the exact solutions available in the literature are: 1) it is less time-consuming and subsequently moreefficient; 2) it is able to analyze the free vibration of the beams whose section varies as an arbitrary function which is difficult or sometimes impossible to solve with analytical methods.