Numerical Analysis of the Behavior of Beams with Variable Stiffness under Impulsive or Harmonic Loading Using Successive Approximation Method

In this study, the Successive Approximation Method (SAM), is employed for the numerical analysis of vibrations of beams with variable stiffness under impulse or harmonic loading. The main idea of the SAM consists of substituting the desired function and its derivatives with a polynomial (spline) of the same type, for example, the cubic spline. Using the SAM some integration algorithm is established and applied to examples of beams with variable stiffness, under variable loading, and the different cases of supports chosen in the literature. The cases of beams with constant or variable rigidity with articulated or embedded supports were calculated and subjected to the action of an instantaneous impulse and harmonic loads distributed over its entire length. To justify the robustness of the SAM considered in this work, an example of an articulated beam with variable stiffness subjected to a distributed harmonic load was calculated analytically, and the results obtained compared to those found numerically for various steps (spatial h and temporal ) of calculus, and the difference between the values obtained by the two methods was small. For example for (h = 1/8,
= 1/64), the difference between these values is 17%. Despite the fact that the proposed method is proven to be effective for beams, its application to the calculation of massive bodies (3D) is necessary and constitutes a perspective for future investigations.