Application of the Method of Two-Sided Approximations to the Analysis of the Influence of Beam End Conditions on the Static Deflection of a Beam in Microelectromechanical Systems

• Datum: 2026
• 1. Verfasser: Савченко, Антон
• Format: Artikel
• Sprache: Ukrainisch
• Veröffentlicht: Кам'янець-Подільський національний університет імені Івана Огієнка 2026
• Online Zugang: https://mcm-math.kpnu.edu.ua/article/view/361173
• Назва журналу: Mathematical and computer modelling. Series: Physical and mathematical sciences

The article investigates the boundary value problem for a semilinear fourth-order differential equation that describes the static deflection of a beam in microelectromechanical systems (MEMS). Unlike previous studies, in which only the classical conditions of clamped ends and simple support were considered, six types of boundary conditions are investigated in this paper: clamped-clamped, simply supported (pinned-pinned), clamped-free (cantilever), clamped-sliding (clamped-guided), clamped-pinned, and pinned-sliding (pinned-guided) end conditions of the beam. The considered configurations cover the main design schemes of modern MEMS devices – from microswitches and resonators to atomic force microscopy probes, micromirrors, and piezoelectric energy harvesters. To investigate this problem, the method of two-sided approximations in a semi-ordered Banach space of continuous functions is applied. The original boundary value problem is reduced to a nonlinear Hammerstein integral equation with an isotone operator, the kernel of which is the corresponding Green’s function. For each of the six types of beam end conditions, the Green’s function is constructed, and the maximum dimensionless deflection of the beam under a unit uniformly distributed load is calculated. Based on the isotone property of the integral operator, an invariant cone segment is constructed, and a theorem on the existence and uniqueness of a positive solution to the boundary value problem is formulated, to which the iterative process converges in a two-sided manner. A computational experiment is conducted using the parameter values of a real MEMS actuator. For each type of end condition, the largest value of the applied voltage at which the sufficient conditions for the convergence of the method are satisfied is calculated, and approximate solutions are constructed. An ordering of the six types of beam end conditions with respect to this voltage value is established, which provides a quantitative basis for selecting the optimal type of beam end condition when designing MEMS devices with different operating characteristics. The novelty of the work lies in developing the method of two-sided approximations for application to boundary value problems with a set of six physically significant types of boundary conditions. This makes it possible to obtain, for each configuration, verified approximations of the deflection with a priori two-sided estimates within the constructively established safe operating range of the applied voltage. The obtained results can be directly applied in the design of microswitches, microresonators, sensors, micromirrors, and piezoelectric microactuators. The developed scheme of the method can be extended to generalizations of the model – variable dielectric permittivity, partial electrostatic loading, nonlinearities of other types, as well as (in combination with the Rothe method) to the non-stationary case.