## 1 Introduction
Building on foundational work by [PERSON] and [PERSON] [2], [PERSON] [3], and others, this study consolidates advancements in FBS techniques and extends their applicability to modern engineering challenges. Numerous studies have since explored the dynamic characteristics of systems composed of interconnected substructures, frequently utilizing dynamic substructuring techniques with FRF data. In 1985, [PERSON] and [PERSON] [4] proposed a method to identify the modal properties of structures using FRFs to derive system parameters. Among FBS approaches, the generalized receptance coupling method introduced by [PERSON] in 1988 [5] and the Lagrange multiplier Frequency-Based Substructuring (LM FBS) method introduced by [PERSON] et al. in 2006 [6] stand out as widely adopted techniques. These methods extend to the substructure decoupling problem, which is essential for determining the dynamic response of subsystems from the overall system. However, the novel contribution of this paper lies in applying these methods to non-linear dynamic systems through a reduced-order approach in the frequency domain, facilitating more accurate and computationally efficient predictions. Recent advancements have further refined FBS techniques. in 2017, [PERSON], [PERSON], and [PERSON] [7] employed FBS to investigate the dynamic behavior of subframes assembled with vehicle bodies. In 2019, [PERSON] et al. [8] introduced a method for deriving a road noise improvement factor within suspension systems using inverse substructuring, demonstrating the efficacy of FBS in practical automotive applications. In 2020, [PERSON], [PERSON] their team at BMW Group and VIBES.technology [9] demonstrated the application of dynamic substructuring in electric car drivetrain design, focusing on NVH issues. Further research by [PERSON] in 2019 [10] extended methods for measuring rotational receptance, a key factor in accurately estimating rotational dynamics. In 2017, [PERSON] and [PERSON] [11] focused on decoupling tire subsystems from suspension subsystems using FBS techniques, contributing to advancements in automotive NVH modeling. In 2022, [PERSON] and [PERSON] [12] investigated the impact of uncertainties in substructure measurements on FBS outcomes, emphasizing the need for precision in measurements. In 2021, [PERSON], [PERSON], and [PERSON] [13] introduced new algorithms to enhance the accuracy and reliability of FBS when coupling subsystems, contributing to the growing body of work aimed at refining FBS methodologies. [PERSON] et al. (2009) present a general framework for dynamic substructuring, classifying various methods developed over four decades, with an emphasis on dual and primal assembly approaches. This framework helps understand the mathematical foundations of substructuring techniques and addresses current challenges in experimental dynamic substructuring [14]. Their work builds on earlier research by [PERSON] and [PERSON] in 2003 [15], which introduced experimentally validated hybrid methods for vehicle axle noise simulation. In 2021, [PERSON] et al. [16] explored dynamic substructuring in railway vehicle dynamics, showcasing FBS's versatility across different industries. Similarly, in 2020, [PERSON] and [PERSON] [17] applied FBS to the aerospace sector, focusing on the dynamic analysis of satellite structures. Additionally, in 2022, [PERSON] et al. [18] examined the application of FBS in the context of robotic systems, particularly in modular robotic configurations. The evolution of FBS techniques is further evidenced by the work of [PERSON] et al. in 2020 [19], who utilized FBS for dynamic analysis in wind turbine blade structures, and that of [PERSON] et al. in 2023 [20], who applied FBS in the marine industry, focusing on ship hull vibration analysis. The recent work of [PERSON] et al. in 2021 [21] explored FBS for the dynamic analysis of civil engineering structures, such as bridges and high-rise buildings, highlighting its expanding relevance.