Combining the weight function approach with the perturbation method and analysis of the euler–bernoulli equations of motion

This study proposes a new solution technique based on the combination of the weight function approach and the perturbation method to investigate the dynamic behavior of Euler–Bernoulli beams both analytically and numerically. In this thesis, the advantages of using both methods in the solution of nonlinear and position-dependent dynamic systems are evaluated theoretically and practically. The study consists of four main sections. The first section discusses the fundamental concepts of vibrational motion, including self-adjointness and solvability conditions, and summarizes the theoretical foundations of numerical solution approaches such as the Galerkin, Petrov–Galerkin, collocation, and least squares methods. In the second section, the numerical solution method based on weight functions is detailed and the general solution procedure applicable to position-dependent dynamic problems is presented. In the third section, the developed general vibration model was analyzed using perturbation techniques. Various cases were evaluated separately for different values of the system parameters, and physically consistent solutions were obtained by eliminating secular terms. In the fourth section, the dynamic behavior of multiple spring-supported Euler–Bernoulli beams subjected to axial parametric loading was examined as a case study. The generated differential equations were nondimensionalized, and results were obtained free of material and geometry effects. The solution processes were carried out separately using the Galerkin, Petrov–Galerkin, collocation, and least squares methods, and the obtained natural frequency values were compared with the CPU times.