Application of the Two-Sided Approximations Method to the Static Deflection Analysis of an Elastic Beam Under Various Boundary Conditions in a Microelectromechanical System Model

• Дата: 2025
• Автори: Савченко, Антон, Гвоздєв, Микита
• Формат: Стаття
• Мова: Українська
• Опубліковано: Кам'янець-Подільський національний університет імені Івана Огієнка 2025
• Онлайн доступ: http://mcm-math.kpnu.edu.ua/article/view/347483
• Назва журналу: Mathematical and computer modelling. Series: Physical and mathematical sciences

The article addresses a boundary value problem for a fourth-order semilinear differential equation that describes the static deflection of a beam in microelectromechanical systems (MEMS) under the action of electrostatic forces. Various types of beam end conditions are considered: fixed-fixed (clamped), which generates Dirichlet boundary conditions, and simply supported, which generates Navier boundary conditions. The corresponding boundary value problem is solved using the method of two-sided approximations based on the use of appropriate Green’s functions. This approach is chosen because it allows not only for the construction of an approximate solution but also for establishing theoretical conditions for the existence of a solution to the original problem, while providing a convenient a posteriori error estimation. The research is grounded in reducing the boundary value problem to a nonlinear Hammerstein integral equation, which is analyzed using the theory of nonlinear operators in semi-ordered Banach spaces. An iterative process for finding a positive solution has been constructed, and conditions guaranteeing its two-sided convergence have been established. To analyze the algorithm’s effectiveness, a series of computational experiments for various system parameter values were conducted. A comparative analysis of the obtained results was performed. The study investigates the variation in the maximum beam deflection and analyzes the impact of boundary conditions on the system’s stability. The novelty of this work lies in the development and application of the two-sided approximations method scheme to fourth-order equations modeling beam deflection in MEMS under different types of end conditions. The results of the study can be utilized in the design of micro-switches, gas sensors, micro-tweezers, and other components of modern microsystem technology to predict their static behavior and optimize operational parameters. Also, the results obtained in this work can be extended to two- and three-dimensional problems, as well as (in combination with the Rothe method) to unsteady-state processes.