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This paper presents an approach for solving accurate approximate analytical solutions for strong nonlinear oscillators called improved amplitude-frequency formulation. For illustrating the accuracy of the method, we also solved equations with He's energy balance method and compared results. New algorithms offer promising approaches, which are useful for nonlinear oscillations. We find that these attained solutions not only benefit from a high degree of accuracy, but are also uniformly valid in the whole solution domain which is so simple to do and effective. The studied equations are the general motion equation and the non-dimensional nonlinear differential equation of motion for the relativistic oscillator, which their solution can be useful for researchers to extend this ability into their other works.

In this article, large deflection and rotation of a nonlinear beam subjected to a coplanar follower static loading is studied. It is assumed that the angle of inclination of the force with respect to the deformed axis of the beam remains unchanged during deformation. The governing equation of this problem is solved analytically for the first time using a new kind of analytic technique for nonlinear problems, namely, the homotopy analysis method (HAM). The present solution can be used in wide range of load and length for beams under large deformations. The results obtained from HAM are compared with those results obtained by fourth order Range Kutta method. Finally, the load-displacement characteristics of a uniform cantilever under a follower force normal to the deformed beam axis are presented.