Nonlinear internal resonance of double-walled nanobeams under parametric excitation by nonlocal continuum theory

•The nonlinear internal resonance of double-walled nanobeams under parametric excitation is analyzed.•The stability analysis is presented based on the Routh–Huriwitz criterion.•The relation between the frequency and response is affected by the van der Walls interaction and displacement amplitude obviously. In the present work, the nonlinear internal resonance of double-walled nanobeams under the external parametric load is studied. The nonlocal continuum theory is applied to describe the nano scale effects and the nonlinear governing equations are derived by the multiple scale method. The parametric internal resonance is considered and the relation between the frequency and amplitude is discussed. From the numerical simulation, it can be observed that small scale effects are more obvious for short structures. Three different nonlinear cases can be found. The gap between the stable and instable regions is reduced by the van der Walls (vdW) interaction but enhanced by the excitation amplitude. Moreover, the dynamical motions of double-walled nanobeams are sensitive to the initial condition and excitation frequency.