## 1 Introduction
This advancement has the potential to revolutionize metastructural design for industrial applications, enabling the creation of structures with multiple bandgaps and diverse functionalities. This framework not only enhances our understanding of metastructures but also provides novel methods for tuning bandgaps, thereby improving vibration isolation and facilitating energy harvesting. With implications spanning industrial machinery and noise cancellation, these advancements promise to revolutionize engineering practices by enabling more efficient and effective control mechanisms in various industrial applications. This work addresses the knowledge gap in linear internal coupling in metastructures, and aims to improve wave manipulation and dynamic control through a new mathematical framework, expanding the applications and functionalities of metastructures. It claims that transfer function methodology can model and control metastructure dynamics, including internally coupled resonators. It highlights a gap in understanding linear internal coupling effects on bandgap manipulation, and demonstrates the maintenance of primary bandgaps and the emergence of secondary bandgaps through internal coupling, suggesting adjustable resonator mass distribution for further tuning. This leads to the following research questions:  
* How does internal coupling affect the bandgap characteristics of a metastructure? * What is the role of internal coupling in enhancing or merging bandgaps for vibration isolation in continuous (distributed) metastructures? * Can the integration of control system theory and transfer function methodology lead to real-time adaptive tuning of metastructure bandgaps? * Can an alternative method, such as modifying the mass distribution on resonators, offer a practical way to alter bandgap characteristics without restructuring, while also being suitable for heavy-duty applications where piezoelectric solutions are less viable?The key contributions of this paper are as follows:  
* We enable the transfer function approach as an analysis method for metastructures, enhancing dynamic bandgap characteristics through the use of different functionalities and precise control engineering techniques. * We develop a mathematical method for formulating closed-form equations describing the behaviors of internally coupled resonators, providing a deeper understanding of their impact on metastructure dynamics. * We address the challenge of merging bandgaps from internally coupled and conventional resonators, offering insights into continuous vibration control in distributed systems. The structure of the remaining sections of this paper is as follows: Section 2 delves into the methodology, detailing the theoretical foundations and optimization strategies employed for bandgap generation. Results and discussion are presented in Section 3, where the implications of the applied methodologies are interpreted in the context of mechanical system design and enhancement. A \"Finite Element Study\" is detailed in Section 4, showcasing the vibrational behaviors and bandgap characteristics of the metastructures under study. This section includes a focused examination of the effects of spatial variations on bandgap properties, emphasizing the utility and implications of these findings for practical applications. Finally, Section 5 concludes the paper, summarizing the key findings and proposing directions for future research.