Geometrically nonlinear forced vibrations of shells based on the domains with cut-outs are investigated. Classical nonlinear shallow-shell theories retaining in-plane inertia is used to calculate the strain energy; the shear deformation is neglected. A mesh-free technique based on classic approximate functions and the R-function theory is used to build the discrete model of the nonlinear vibrations. This allowed for constructing the sequences of admissible functions that satisfy given boundary conditions in domains with complex geometries. Shell displacements are expanded by using Chebyshev orthogonal polynomials. A two-step approach is implemented to solve the problem: first a linear analysis is conducted to identify natural frequencies and corresponding natural modes to be used in the second step as a basis for nonlinear displacements. The system of ordinary differential equations is obtained by using Lagrange approach on both steps. The convergence of the solution is studied by using different multimodal expansions. The pseudo-arclength continuation method and bifurcation analysis are used to study the nonlinear equations of motion. Numerical responses are obtained in the spectral neighbourhood of the lowest natural frequency. When possible, obtained results are compared to those available in the literature.

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