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Tapered beams are more efficient compared to uniform beams as they provide a better distribution of mass and strength and also meet special functional requirements in many engineering applications. In this paper, the linear and non-linear fundamental frequency parameter values of the tapered Timoshenko beams are evaluated by using the coupled displacement field (CDF) method and closed form expressions are derived in terms of frequency ratio as a function of slenderness ratio, taper ratio and maximum amplitude ratio for hinged-hinged and clamped-clamped beam boundary conditions. The effectiveness of the CDF method is brought out through the solution of the large amplitude free vibrations, in terms of fundamental frequency of tapered Timoshenko beams with axially immovable ends. The results obtained by the present CDF method are validated with the existing literature wherever possible.

Continuum solutions for solving the large amplitude free vibration problem of shear flexible beams using the energy method involves assuming suitable admissible functions for the lateral displacement and the total rotation. Use of even, single-term admissible functions leads to two coupled nonlinear temporal differential equations in terms of the lateral displacement and the total rotation, the solution of which is rather involved. This situation can be effectively tackled if one uses the concept of a coupled displacement field wherein the fields for lateral displacement and the total rotation are coupled through the static equilibrium equation of the shear flexible beam. This approach leads to only one undetermined coefficient, in the case of single-term admissible functions, which can easily be used in the principle of conservation of total energy, neglecting damping, to solve the problem. Finally, one gets a nonlinear ordinary differential equation of the Duffing type which can be solved using any available standard method. The effectiveness of the concept discussed above is brought out through the solution of the large amplitude free vibrations, in terms of the fundamental frequency, of uniform shear flexible beams, with axially immovable ends, using single-term admissible functions.

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