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This paper presents a modified continuous-mass (model) transfer matrix method (CTMM) to determine the natural frequencies and associated mode shapes of a uniform or non-uniform beam with various classical (or non-classical) boundary conditions (BCs) and carrying multiple sets of concentrated elements with each set consisting of a point mass (with eccentricity and rotary inertia), a translational spring and a rotational spring. To this end, a continuous non-uniform free–free beam is subdivided into several uniform beam segments (each having distributed mass) and any two adjacent beam segments are connected by a node at which various concentrated elements being attached. Next, the transfer matrix for the integration constants of arbitrary two adjacent beam segments joined at an intermediate node is derived, and then the characteristic equation of the entire vibrating system is derived by combining all transfer matrices for all intermediate nodes and considering the BCs of the entire free–free beam. It has been found that, based on the foregoing formulation for a non-uniform free–free beam, one may easily obtain the mathematical model for a uniform or non-uniform beam with various BCs and carrying various concentrated elements by only adjusting the magnitudes of cross-sectional area and length of each beam segment and those of the concentrated elements (such as the lumped mass m i with eccentricity e i and rotary inertia J i , the translational spring with stiffness k t,i and/or the rotational spring with stiffness k r,i ) attached to each node. The reliability of the presented results has been confirmed by comparing them with those of the existing literature or the conventional finite element method (FEM) and good agreement is achieved.

In the conventional lumped-mass (model) transfer matrix method (LTMM), one must use different frequency equation and the associated initial parameters to determine the natural frequencies and the corresponding mode shapes of a beam with different boundary (supporting) conditions as one may see from the Appendix at the end of this paper. Besides, the eccentricity of each attached lumped mass is also neglected in the existing LTMM. The purpose of this paper is to present a modified LTMM so that one may easily determine the natural frequencies and the corresponding mode shapes of a multi-step Timoshenko beam with various boundary (supporting) conditions and carrying various concentrated elements with eccentricity of each lumped mass considered, by using the same formulation developed from a beam with “free–free” boundary conditions.To this end, by considering the effect of rotary inertia of each beam segment and the eccentricity of each attached lumped mass, the transfer matrix for a typical station carrying three kinds of concentrated elements is derived. Meanwhile, by considering the shear deformation of each beam segment, the transfer matrix for a typical field is also deduced. It has been found that, by changing the magnitude (from zero to infinity) of each concentrated element, one may easily model many problems studied in the exiting literature concerned. The reliability of the presented approach has been confirmed by the good agreement between the numerical results of this paper and those of the existing literature or the conventional finite element method (FEM).

In general, the exact solutions for natural frequencies and mode shapes of non-uniform beams are obtainable only for a few types such as wedge beams. However, the exact solution for the natural frequencies and mode shapes of an immersed wedge beam is not obtained yet. This is because, due to the “added mass” of water, the mass density of the immersed part of the beam is different from its emerged part. The objective of this paper is to present some information for this problem. First, the displacement functions for the immersed part and emerged part of the wedge beam are derived. Next, the force (and moment) equilibrium conditions and the deflection compatibility conditions for the two parts are imposed to establish a set of simultaneous equations with eight integration constants as the unknowns. Equating to zero the coefficient determinant one obtains the frequency equation, and solving the last equation one obtains the natural frequencies of the immersed wedge beam. From the last natural frequencies and the above-mentioned simultaneous equations, one may determine all the eight integration constants and, in turn, the corresponding mode shapes. All the analytical solutions are compared with the numerical ones obtained from the finite element method and good agreement is achieved. The formulation of this paper is available for the fully or partially immersed doubly tapered beams with square, rectangular or circular cross-sections. The taper ratio for width and that for depth may also be equal or unequal.

The purpose of this paper is to present a simple approach for eliminating the “dependent” torsional angles existing in the reduction gears of a gear-branched system so that this system may be modelled as an equivalent straight-geared (or direct-transmitted) system. Then the overall mass matrix, damping matrix, stiffness matrix, and load vector of the direct-transmitted system are obtained with the conventional finite element method (FEM) by assembling the elemental property matrices of all the shaft elements contained in the torsional system. Based on the overall property matrices determined, the equations of motion of the whole vibrating system are defined. Solution of the equations of motion gives the dynamic responses and solution of the associated eigenvalue equation provides the natural frequencies and the mode shapes of the system. A simple technique is also presented to study the influence of the shaft mass on the natural frequencies of a torsional system.