Abstract
Analytical solutions describing free transverse vibrations with large amplitude of axially loaded Euler–Bernoulli beams for various end restrains resting on a Winkler one-parameter foundation are obtained using the Adomian modified decomposition method (AMDM). The AMDM allows the governing equation to become a recursive algebraic equation, and, after some additional simple mathematical operations, the equations can be cast as an eigenvector problem whose solution results in the calculation of natural frequencies and corresponding closed-form series solution of the mode shapes. Important to the use of the Adomian modified decomposition method is the treatment of the nonlinear Fredholm integral coefficient, which forms part of the governing equation. In addition to the calculation of natural frequencies and mode shapes, investigations are made of the effects on the free vibrations of the Winkler parameter and of increasing the axial loading.
Highlights
Uniform slender beams resting on an elastic foundation, while subjected to axial loading, are common in structural systems undergoing actual operating conditions
Clamped-Free Uniform Beam. e first case considered is the clamped-free uniform beam resting on an elastic foundation and experiencing axial compressive force e case was chosen to test the accuracy of the Adomian modified decomposition method (AMDM)
E case of the clamped-free uniform beam without an elastic foundation support or without any axial force and with small vibration amplitude was first chosen to test the accuracy of the AMDM as comparisons can be made with what is already given in the literature
Summary
Uniform slender beams resting on an elastic foundation, while subjected to axial loading, are common in structural systems undergoing actual operating conditions. The adomian modified decomposition method [21, 22] is utilized to calculate free transverse vibration characteristics of axially loaded Euler–Bernoulli beams with various end restrains, resting on a Winkler one-parameter foundation. E method is chosen as it has proved efficient and accurate [23, 24] for solving linear and nonlinear differential equations, and it has the advantage of computational simplicity It does not involve linearization, discretization, perturbation, or a priori assumptions, which may alter the physics of the problem considered [21]. The coefficients cannot be determined exactly, and the solutions can only be approximated by a truncated series nm− 10 Cm xm
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