Exact solutions of Euler–Bernoulli beams

In numerous real-world applications, transverse vibrations of beams are nonlinear in nature. It is a task to solve nonlinear beam systems due to their substantial dependence on the 4 variables of the system and the boundary conditions. To comprehend the nonlinear vibration characteristics, it is essential to do a precise parametric analysis. This research demonstrates an approximation solution for odd and even nonlinear transverse vibrating beams using the Laplace-based variation iteration method, and the formulation of the beams depends on the Galerkin approximation. For the solution of the nonlinear differential equation, this method is efficient as compared to the existing methods in the literature because the solutions exactly match with the numerical solutions. The Laplace-based variation iteration method has been used for the first time to obtain the solution to this important problem. To demonstrate the applicability and precision of the Laplace-based iteration method, several initial conditions are applied to the governing equation for nonlinearly vibrating transverse beams. The natural frequencies and periodic response curves are computed using Laplace-based VIM and compared with the Runge–Kutta RK4 method. In contrast to the RK4, the results demonstrate that the proposed method yields excellent consensus. The Lagrange multiplier is widely regarded as one of the most essential concepts in variational theory. The result obtained are displayed in the table form. Highlights The highlights of the solution of the Euler–Bernoulli beam equation with quintic nonlinearity using Lagrange multiplier are: 1. Introducing the constraint of the boundary conditions into the equation using Lagrange multipliers. 2. Formulating the equations for the Lagrange multipliers and the deflection of the beam. 3. Solving the resulting system of algebraic equations using numerical methods. 4. Obtaining the deflection of the beam as a function of its length and the applied load. 5. Analyzing the behavior of the beam under different loads and boundary conditions.