Although the exact solutions for the free vibration problems regarding most of the non-uniform beams are not yet obtainable, this is not true for the special case when the equation of motion of a non-uniform beam can be transformed into that of an equivalent uniform beam. The nonlinearly tapered beam studied in this paper is a single-tapered beam with constant depth h0 and varying width b(x) along its length in the form b x( )  4 0 b xL [1 ( )]  , where b0 is the minimum width,  is the taper constant, x is the axial coordinate and L is the total beam length. For the case of no concentrated elements (CEs) attaching to it, the exact solution for its lowest several natural frequencies and the associated mode shapes has been appeared in the existing literature, however, the exact solution for the free vibrations of the last tapered beam carrying various CEs in various boundary conditions (BCs) is not found yet due to complexity of the problem. This is the reason why this paper aims at studying the title problem by using the continuous-mass transfer matrix method (CTMM). It is different from the general uniform (or multi-step) beam carrying various CEs in that the nonlinearly tapered beam itself as well as the attached translational and rotational CEs must all be transformed into the equivalent ones in the derivations. In addition to the solution accuracy, one of the salient merits of the proposed method is that the order of the characteristicequation matrix keeps constant (4  4) and does not increase with the total number of the CEs or the beam segments such as in the conventional finite element method (FEM), so that it needs Fig. 1. The vortex wind generator developed by Vortex Bladeless (2015). less than 0.2% of the CPU time required by the FEM to achieve the exact solutions. The CEs on the nonlinearly tapered beam include lumped masses (with eccentricities and rotary inertias), translational springs and rotational springs. The formulation of this paper is available for various classical or non-classical BCs. In addition to comparing with the existing available data, most of the numerical results obtained from the proposed method are also compared with those of the FEM and good agreement is achieved.

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