On primary resonances of weakly nonlinear delay systems with cubic nonlinearities

We implement the method of multiple scales to investigate primary resonances of a weakly nonlinear second-order delay system with cubic nonlinearities. In contrast to previous studies where the implementation is confined to the assumption of linear delay terms with small coefficients (Hu et al. in Nonlinear Dyn. 15:311, 1998 ; Ji and Leung in Nonlinear Dyn. 253:985, 2002 ), in this effort, we propose a modified approach which alleviates that assumption and permits treating a problem with arbitrarily large gains. The modified approach lumps the delay state into unknown linear damping and stiffness terms that are functions of the gain and delay. These unknown functions are determined by enforcing the linear part of the steady-state solution acquired via the method of multiple scales to match that obtained directly by solving the forced linear problem. We examine the validity of the modified procedure by comparing its results to solutions obtained via a harmonic balance approach. Several examples are discussed demonstrating the ability of the proposed methodology to predict the amplitude, softening-hardening characteristics, and stability of the resulting steady-state responses. Analytical results also reveal that the system can exhibit responses with different nonlinear characteristics near its multiple delay frequencies.