Resonance responses in a two-degree-of-freedom viscoelastic oscillator under randomly disordered periodic excitations

•Resonance responses of TDOF viscoelastic system with nonlinear coupled terms are investigated.•(Random) jump, (random) saturation, and double-jumping phenomena are explored.•Viscoelastic parameters have great influence on the steady-state responses.•The increase of noise intensity causes the steady-state solutions changing from a limit cycle to a diffused one. An investigation is presented for primary resonance and internal resonance of two-degree-of-freedom (TDOF) viscoelastic system with some complex nonlinear coupled terms under randomly disordered periodic excitations. The assumed viscoelastic damping depending on the past history of motion is chosen as the form of convolution integrals over an exponentially decaying kernel function. Then, the steady-state responses of the TDOF system are discussed in detail through the method of multiple scales (MMS), which is employed to derive the determined and stochastic differential equations of amplitude and phase modulation for the internal and external resonance modes. Further, the numerical simulation method is absorbed to test the effectiveness and accuracy of the theoretical analysis solutions for steady-state moments with the changing excitation amplitude. The appearance of (random) jump, (random) saturation, and double-jumping can be explored by the development of steady-state moments with different excitation amplitudes and frequency, which also show the resonance bandwidth increases with the excitation amplitude. Besides, viscoelastic damping parameters have great influence on the steady-state responses under the different detuning conditions, which accelerate or delay the appearance of the above mentioned phenomena. Numerical simulation results of phase portraits tell that the increase of the intensity of the random excitation leads to the steady-state solutions changing from a limit cycle to a diffused limit cycle, thus, such multi-valued steady-state responses produce jump from one stable branch to another also response to the previous analysis conclusion.