In this paper, a dynamic model is adopted to investigate the stability and response characteristics of a vibrating system driven by four vibrators placed on two different rigid frames (RFs). Using the equations of motion of the system derived, the conditions for synchronization and stable operation of the system are studied by the average method and Hamilton’s rules, respectively. Based on the theoretical results obtained, some factors are further studied concerning the stable phase differences (SPDs), the coefficients for ensuring stability, and the vibration amplitudes of the two RFs in different resonant regions. These serve to reveal the stability and response characteristics of the system that determine the ultimate function of the vibrating machine. Finally, numerical simulations are carried out to examine the validity of the theoretical methods and numerical qualitative results. Based on the results from the theory and simulation analyses, it is suggested that the working region of the system should be selected in the sub-resonant region corresponding to the natural frequency (NF) of the main vibrating system in the x- and y-directions. In this case, the ideal relative circular motion for two RFs with a well isolation effect can be achieved, and the energy is saved.

中文翻译：

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四个同向耦合振动器驱动的双刚架振动系统的稳定性特性

在本文中，采用动态模型来研究由放置在两个不同刚性框架 (RF) 上的四个振动器驱动的振动系统的稳定性和响应特性。利用导出的系统运动方程，分别采用平均法和Hamilton规则研究了系统同步和稳定运行的条件。在得到的理论结果的基础上，进一步研究了两个RF在不同谐振区域的稳定相位差（SPD）、确保稳定的系数和振幅的一些因素。这些有助于揭示系统的稳定性和响应特性，这些特性决定了振动机的最终功能。最后，进行数值模拟以检验理论方法和数值定性结果的有效性。根据理论和仿真分析的结果，建议系统的工作区域应选择在与主振动系统的固有频率（NF）相对应的次谐振区域内。X- 和是的-方向。在这种情况下，可以实现两个射频的理想相对圆周运动，具有良好的隔离效果，节省了能量。