Behavior of a flexible mesh plate placed in an electrostatic field

The object of study is a flexible mesh plate with clamped edges. A stationary electrode is arranged in parallel with the plate at some distance from it. The electric field between the plate and electrode with a given potential difference is created by an external source. The plate is attracted (deflected) towards the electrode and comes in equilibrium when a balance between the electric (Coulomb force) and elastic forces is reached. When the potential difference increases, the plate moves to a new equilibrium position. The state equations of a geometrically nonlinear plate and boundary conditions based on the Kirchhoff hypotheses are derived from the Ostrogradsky-Hamilton variational principle. An isotropic, homogeneous material is considered. The scale effects are taken into account by means of the couple stress theory. It is assumed that the fields of displacement and rotation are not independent. Geometric nonlinearity is taken into account according to Von Karman's theory. The mesh structure of the plate was modeled using the continuum theory developed by G. I. Pshenichny, which made it possible to replace the system of regular ribs by a continuous layer. Based on equilibrium conditions for a rectangular element, the relations between stresses arising in an equivalent smooth plate and stresses in the ribs were derived. The Lagrange multiplier method was used to determine the mesh plate physical ratios. The Bubnov-Galerkin method was applied to numerically solve a system of differential equations describing the nonlinear oscillations of the mesh plate under consideration. The mathematical model, solution algorithm and software package were verified by comparing the author's calculation results with the full-scale experiment data and with the results obtained by other authors. The paper investigates the influence of plate mesh structure geometry, constant voltage value, and geometric non-linearity on the natural frequency of the clamped plate. Numerical results are given for the graphene plate.