Analytical solutions describing free transverse vibrations with large amplitude of axially loaded Eulerâ??Bernoulli beams for various\nend restrains resting on a Winkler one-parameter foundation are obtained using the Adomian modified decomposition method\n(AMDM). The AMDM allows the governing equation to become a recursive algebraic equation, and, after some additional simple\nmathematical operations, the equations can be cast as an eigenvector problem whose solution results in the calculation of natural\nfrequencies and corresponding closed-form series solution of the mode shapes. Important to the use of the Adomian modified\ndecomposition method is the treatment of the nonlinear Fredholm integral coefficient, which forms part of the governing\nequation. In addition to the calculation of natural frequencies and mode shapes, investigations are made of the effects on the free\nvibrations of the Winkler parameter and of increasing the axial loading.
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