Author(s)

Moussa Sali^1,2*, Fabien Kenmogne², Jean Bertin Nkibeu³, Abdou Njifenjou^4,5

¹Laboratory of Materials, Mechanics and Civil Engineering, National Higher Polytechnic School of Maroua, University of Maroua, Maroua, Cameroon.
²Department of Civil Engineering, Advanced Teachers Training College of the Technical Education, University of Douala, Douala, Cameroon.
³Laboratory Engineering Civil and Mechanics, National Advanced School of Engineering, University of Yaoundé 1, Yaoundé, Cameroon.
⁴Laboratory of Energy, Materials, Modeling and Method (E3M), National Advanced School of Engineering, University of Douala, Douala, Cameroon.
⁵Laboratory of Mathematical Engineering and Information Systems, National Advanced School of Engineering, University of Yaoundé 1, Yaoundé, Cameroon.

The behavior of beams with variable stiffness subjected to the action of variable loadings (impulse or harmonic) is analyzed in this paper using the successive approximation method. This successive approximation method is a technique for numerical integration of partial differential equations involving both the space and time, with well-known initial conditions on time and boundary conditions on the space. This technique, although having been applied to beams with constant stiffness, is new for the case of beams with variable stiffness, and it aims to use a quadratic parabola (in time) to approximate the solutions of the differential equations of dynamics. The spatial part is studied using the successive approximation method of the partial differential equations obtained, in order to transform them into a system of time-dependent ordinary differential equations. Thus, the integration algorithm using this technique is established and applied to examples of beams with variable stiffness, under variable loading, and with the different cases of supports chosen in the literature. We have thus calculated the cases of beams with constant or variable rigidity with articulated or embedded supports, subjected to the action of an instantaneous impulse and harmonic loads distributed over its entire length. In order to justify the robustness of the successive approximation method considered in this work, an example of an articulated beam with constant 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 $\overline{\tau }$ ) of calculus, and the difference between the values obtained by the two methods was small. For example for ($h=1/8$ , $\overline{\tau }=1/64$ ), the difference between these values is 17%.

Successive Approximations Method, Direct Integration, Differential Equations, Beams of Variable Stiffness, Quadratic Parabola, Impulse and Harmonic Loads

Sali, M. , Kenmogne, F. , Nkibeu, J. and Njifenjou, A. (2024) Numerical Study of the Vibrations of Beams with Variable Stiffness under Impulsive or Harmonic Loading. World Journal of Engineering and Technology, 12, 401-425. doi: 10.4236/wjet.2024.122026.