Nonlinear normal modes of a shallow arch with elastic constraints for two-to-one internal resonances

The nonlinear normal modes (NNMs) of an elastically constrained (EC) shallow arch in the case of two-to-one internal resonance are constructed, and the effects of the vertical and rotational elastic boundary constraints are studied. The multiple scales method is directly applied to obtain the second-order uniform-expansion solution and the modulation equations from the dimensionless integral-partial-differential equation of motion. The elastic constraints have a corresponding relationship with the coefficients of modulation equations and influence the natural frequencies and modes, as demonstrated by solving the algebraic eigenvalue equation. The stability of the uncoupled single-mode and coupled-mode motions for the nonlinear system is investigated. Then the shape functions, activation conditions and space-time evolutions accounting for the two-to-one internally resonant NNMs for vertical and rotational elastic constraints are examined. The results show that the vertical and rotational elastic constraints play a fundamental role in the nonlinear dynamic phenomena of the EC shallow arch.