Closed-form solutions for attenuation peaks and band boundaries of general monocoupled systems

Abstract

The unit cells connected periodically at a single node with only one degree of freedom is called a monocoupled system. Dispersion relations for such systems are studied widely; however, the analytical solution for salient features of the attenuation band, such as the number of peaks in a band and band merging, have been relatively unexplored. In this paper, a general theory for obtaining the attenuation characteristics of the general monocoupled system from the roots and poles of the rational polynomial of the dispersion relation is developed. The uniqueness of the developed rational polynomial method is that it can predict the attenuation peaks and the possibility of multiple peaks in an attenuation band due to coupling between the different band formation mechanisms in addition to standard band boundaries. The most general monocoupled system has been conceptualized by combining the three mechanisms, namely inertial amplifier, effective negative mass, and effective negative stiffness. This general system is named Inertial Amplifier Negative Mass Negative Stiffness (IANMNS). This designed monocoupled system degenerates into other seven subsystems as special cases, such as the Inertial Amplifier Negative Stiffness (IANS), the Inertial Amplifier Negative Mass (IANM), the Negative Mass Negative Stiffness (NMNS), the Inertial Amplifier (IA), the Negative Stiffness (NS), the Negative Mass (NM) and the Monoatomic system. The closed-form expressions for the peaks in attenuation level and bounds in terms of nondimensional frequency ratio and other governing parameters such as the inertial mass ratio, mass ratios of resonators one (embedded in main chain mass) and two (embedded in inertial mass), frequency ratios of resonators one and two, and angular parameter are derived for the IANMNS and all the other seven subsystems. The conditions for obtaining the double peaks and band merging are defined analytically.