Free vibration analysis of tapered columns under self-weight using pseudospectral method

Abstract

Abstract. This paper deals with the vibration of tapered column which is affected by gravity using a pseudospectral formulation. The formulation is simple and easy-to-implement and is capable of dealing with different end conditions. Numerical examples of the effects of taper, cross section shapes and gravity on the vibration of columns are illustrated. The effectiveness of the pseudospectral method for vibration analysis of tapered heavy columns is validated by comparing the results with numerical techniques such as the numerical initial value method and differential quadrature method.

Vibrations of flexible pylons with time‐dependent mass attachments under ground induced motions

“…We now employ the eigenfunctions and wave numbers corresponding to the stand-alone cantilever of Equation ( 12), along with normalized unit values for the pylon's generalized masses 𝑚 𝑛 . Pre-multiplying Equation (19) with Φ 𝑗 (𝑥), 𝑗 = 1, 2, … , ∞ and integrating along the pylon's length 𝐿 so to take advantage of the orthogonality property of the eigenfunctions, yields the following system of differential equations:…”

“…The governing equation was derived by Hamilton's principle for both Bernoulli‐Euler and Timoshenko beam models, and analytical solutions were obtained by Frobenius’ method. Finally, Sudheer et al 19 . studied the vibrations of a tapered column under gravity loads using a pseudo‐spectral formulation capable of accounting for various types of end conditions.…”

“…The governing equation was derived by Hamilton's principle for both Bernoulli-Euler and Timoshenko beam models, and analytical solutions were obtained by Frobenius' method. Finally, Sudheer et al 19 studied the vibrations of a tapered column under gravity loads using a pseudo-spectral formulation capable of accounting for various types of end conditions. The effectiveness of the Chebyshev polynomials used as trial functions for representing the transient column displacement was validated by comparing results with other numerical techniques, and examples were presented for examining the influence of tapering, gravity, and cross section shape on the vibrations of heavy columns.…”