Applications of representation transformation and secular perturbation theory to coupled Duffing equations

In physics, the non-linear mode coupling is an important strategy to manipulate the mechanical properties of a vibrational system. Compared with the single-mode nonlinear system, the complex systems with two- or multi-mode nonlinear coupling have garnered considerable attention, among which the analytical solutions to the coupled Duffing equations are widely studied to solve nonlinear coupling. The fact is that the solving of the Duffing coupling equations generally starts with the eigenmodes solution of the linear equations. The trial solution of the coupled equations is the linear superposition of the eigenmodes. Under the secular perturbation theory and similar conditions, the Duffing coupling equation degenerates into two decoupled equations. However, thus far most of the solution methodologies are too complicated to unravel the underlying physical essence clearly. In this paper, first, by applying the representational transformation to the linear terms of the first-order coupled Duffing equations and the secular perturbation theory for the nonlinear terms, a decoupled expression of the first-order Duffing equations is derived, which can be solved more straightforwardly. Subsequently, in order to verify the correctness of the method, we design a coupled tuning fork mechanical vibration system, which consists of two experimental instruments to provide driving force and receive signals, two tuning forks and springs. The amplitude spectra are measured by an experimental instrument of forced vibration and resonance (HZDH4615), which provides a periodic driving signal for the tuning fork. The numerical fitting by software is employed to clarify the mechanism of the spectrum. Theoretically, the obtained fitting parameters can also evaluate some important attributes of the system. Most strikingly, due to the nonlinear coupling the splitting of the resonant peak and the phenomenon of “hysteresis loop” are clearly observed in the experiment. The research shows that the experimental results perfectly match the theoretical results obtained before. The method of solving coupled nonlinear equations in this article provides a solution and improvement of flexible adoption of nonlinear theory. On the other hand, it can be extended to coupled light and electricity systems, offer certain guidance for understanding the dynamic behavior of coupled systems, and will be conductive to the quantitative examination of numerous nonlinear coupling devices.