Beams are very important element in structures because of its widespread usage in steel construction such as buildings, bridges, aircraft arm, aerospace vehicles and many other industrial usages. To attain 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. The main objective of present study is to obtain highly accurate analytical solutions for vibrations of a beam with quintic nonlinearity. Two new analytical methods are the improved energy balance method, and the global residue harmonic balance method. Subsequently, the analytical results are compared with the numerical solution of the exact equation in order to evaluate the correctness of the applied approaches.

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