A mathematical model describing nonlinear vibrations of size-dependent rectangular plates is proposed. The plates are treated as the Cosserat continuum with bounded rotations of their particles (pseudo-continuum). The governing partial differential equations (PDEs) and boundary/initial conditions are obtained using the von Kármán geometric relations, and they are yielded by the energetic Hamilton principle. The derived mixed-form PDEs are reduced to ordinary differential equations and algebraic equations (AEs) using (i) the Galerkin–Krylov–Bogoliubov method (GKBM) in higher approximations, and then they are solved with the help of a combination of the Runge–Kutta methods of the second and fourth order, (ii) the finite difference method (FDM), and (iii) the Newmark method. The convergence of FDM vs the interval of the space coordinate grids and of GKBM vs the number of employed terms of the approximating function is investigated. The latter approach allows for achieving reliable results by taking account of almost infinite-degree-of-freedom approximation to the regular and chaotic dynamics of the studied plates. The problem of stability loss of the size-dependent plates under harmonic load is also tackled.
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