Stodola-Vianello iteration method for the free flexural vibration frequencies of Shimpi’s single variable shear deformable beams
Résumé
The natural vibration frequency analysis of beams is vital for their design against resonance failures because such failures occur when the excitation load frequencies of vibration coincide with such natural frequencies. This work presents a single variable shear deformable beam equation formulated using Shimpi’s displacement field assumptions. This results in a quadratic shear stress profile over the depth and a satisfaction of the transverse shear stress-free boundary conditions. The governing equation is obtained using a first principles consideration and equilibrium method as a partial differential equation (PDE) which is non-homogenous for forced vibrations and homogeneous for free vibrations. The study then used the Stodola-Vianello iteration method to solve the resulting homogeneous PDE for simply supported boundary conditions and harmonic response. The problem reduced to an iterative problem of algebra involving the computation of an ( n + 1 )th vibratory modal shape function from an n th shape function that satisfies the boundary conditions. This work used a sinusoidal shape function which is exact for the simply supported boundary condition investigated. The use of boundary conditions solved the integration constants involved. Application of the convergence rule led to the eigenequation from which the eigenvalues were found. The eigenvalues were presented for the first four modes of vibration and for a rectangular beam. It was found that for l / h varying from 5 to 100, the natural vibration frequencies were identical with the ω n values obtained using Navier method for other thick beam vibration problems. It was also found that ω n was close to the exact values for all vibration modes and for all values of l / h between 5 and 100. For all vibration modes and all considered l / h values negligible differences, were observed between the ω n obtained using SVIM and the exact values obtained by previous researchers.
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