An analytic solution of transversal oscillation of quintic non-linear beam with homotopy analysis method

non-linear vibration analysis of beam used in steel structures is of particular importance in mechanical and industrial applications. To achieve a proper design of the beam structures, it is essential to realize how the beam vibrates in its transverse mode which in turn yields the natural frequency of the system. Equation of transversal vibration of hinged–hinged flexible beam subjected to constant excitation at its free end is identified as a non-linear differential equation. The quintic non-linear equation of motion is derived based on Hamilton’s principle and solved by means of an analytical technique, namely the Homotopy analysis method. To verify the soundness of the results, a comparison between analytical and numerical solutions is developed. Finally, to express the impact of the quintic nonlinearity, the non-linear responses obtained by HAM are compared with the results from usual beam theory.

► This work derives the quintic non-linear governing equation of beam vibrations.
► non-linear transversal vibration equation is derived based on Hamilton’s principle.
► The governing equation is solved via an analytical approach, namely HAM.