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In this paper, we will discuss an internal state transition between synchronization and anti-synchronization in 2-dof system of 2-equivalent-mass oscillators, termed physical-coordinate model, which has a movable mass at the edge. The state transition is caused by target energy transfer between internal modal oscillators in a quasi-modal model of the system. Here, the internal modes of the quasi-modal correspond to synchronization and anti-synchronization in the physical-coordinate model. For realization of the idea, we will propose a method to control the internal modal coupling of the two synchronizations in the physical-coordinate model by using resonance condition in quasi-modal model. The method use a new boundary coordinate and new pseudo modes in the quasi-modal model. In addition, one of the angular frequencies of the new pseudo modes is internally resonated to the natural frequency of the internal quasi-mode. We derived equations to decide the mass and stiffness parameters of the 2-equivalent-mass oscillators by using the resonance condition. Moreover, numerical simulations were carried out to confirm the internal state transition in the physical coordinate model. In addition, an equation of the natural transfer period for the internal modal coupling was derived from a complex state transition matrix, which contains operators in Grover algorithm in quantum algorithm. At last, we discussed uni-directional phenomena of the internal state transition in the physical-coordinate model in case the model has proportional viscous damping.

Transition between synchronization and anti-synchronization states of 2-equivalent-mass oscillators which has a movable mass at the edge