## 1 Introduction  
Dynamic analysis of complex systems, such as vehicles, robots, and machining equipment, has been an area of extensive research, leading to the development of diverse methodologies. Predictive methods like Finite Element Modal Analysis (FEMA) and Dynamic Substructuring (DS) are widely used during the design phase to model and predict dynamic responses, providing insights into natural frequencies, mode shapes, and substructure interactions. On the other hand, diagnostic approaches such as Root Cause Analysis (RCA) and Transfer Path Analysis (TPA) focus on identifying and mitigating vibration issues in operational systems by analyzing energy transmission paths and the sources of vibrational behavior. [PERSON] et al. (2016) [1] discussed TPA as a diagnostic tool for vibration and noise issues, classifying it into classical, component-based, and transmissibility-based methods. Similarly, [PERSON] et al. (2021) [2] applied RCA combined with FEMA to diagnose vibration issues using structured cause-effect diagrams. [PERSON] et al. (2024) [3] advanced operational TPA by addressing the challenge of reliable motor controller unit analysis in electric vehicles under real-world conditions. However, their approach lacks a generalized framework for modeling system-wide dynamics. [PERSON], [PERSON], and [PERSON] (2008) [4] provided an extensive framework for DS, a methodology for analyzing complex systems by dividing them into smaller, manageable substructures for independent analysis. DS facilitates modular analysis while maintaining computational efficiency, utilizing techniques such as Component Mode Synthesis (CMS), Impulse-Based Substructuring (IBS), state-space coupling, and Frequency-Based Substructuring (FBS). CMS simplifies large finite element models by retaining key mode shapes (mode truncation), making it suitable for low-frequency applications. State-space coupling complements this by modeling substructures through first-order differential equations and enforcing compatibility at interfaces. The FBS and IBS approaches are effective for high-frequency applications due to their direct incorporation of receptance or Frequency Response Function (FRF) matrices, preserving damping and complex modes. While IBS leverages impulse response functions (IRFs) for transient or impact loading scenarios in the time domain, FBS synthesizes FRFs in the frequency domain, avoiding mode truncation and ensuring high-frequency dynamics are preserved. Among these, FBS has emerged as a modular approach for analyzing system dynamics by coupling subsystems using experimentally or numerically derived FRFs. Receptance coupling (RC), a subset of FBS, dynamically combines receptance matrices by enforcing compatibility (displacement/moment continuity) and equilibrium (force/moment balance) at subsystem interfaces. The concept of receptance coupling was first introduced by [PERSON] and [PERSON] [5], with subsequent advancements by [PERSON], [PERSON], and [PERSON] [6] in integrating structural coupling and modification within Finite Element Analysis (FEA). Further advancements were made by [PERSON] and [PERSON] [7], who explored translational and rotational parameters for structural coupling analysis using the finite-difference technique. Among the various FBS techniques, the generalized receptance coupling method introduced by [PERSON] in 1988 [8] and the Lagrange Multiplier Frequency-Based Substructuring (LM FBS) method developed by [PERSON] et al. in 2006 [9] are particularly notable and widely utilized. Since their pioneering contributions, the field of FBS and receptance coupling has evolved significantly. Recent innovations have mostly expanded upon these foundational works by incorporating advanced methodologies and cutting-edge technologies. [PERSON] and [PERSON] [10] quantified errors arising from neglecting rotational degrees of freedom (DOFs) in FRF coupling analysis, underscoring the importance of including rotational dynamics in structural analyses.