Nonlinear normal modes for vibratory systems under harmonic excitation

Nonlinear normal modes of a self-excited system driven by parametric and external excitations

“…Presented above formulation is used for resonant curves determination of the self-, parametrically and externally excited system near the main parametric resonances. The solutions are received from (13) by using analytical solutions (29) assuming that in the resonance the vibrations modes are close to the mentioned above NNMs found for the nonlinear conservative system. Solving together (29) and Eq.…”

“…The solutions are received from (13) by using analytical solutions (29) assuming that in the resonance the vibrations modes are close to the mentioned above NNMs found for the nonlinear conservative system. Solving together (29) and Eq. (7) we get the resonance curve presented in Fig.8(a).…”

“…To get vibration amplitudes in the resonance regions it is necessary to solve equation (29) together with nonlinear modal forms of the considered system formulated by Eq. (7).…”

“…where the amplitude a and the phase φ are determined from modulation equations (28) or for a steady state from algebraic equation (29).…”

“…Then time has to be introduced as an additional coordinate. To get the modal equation it is necessary to apply a special numerical Galerkin-based technique taking into account a time coordinate [ 29]. This method allows for finding NNMs for every instant of time.…”