## 1 Introduction  
Many engineering designs incorporate fastening multiple components together using some form of bolted joint. It is common to utilize this class of design because of the benefits they provide. Using a component-based design allows for modularity and a reduction in complexity. However, these components can dynamically interact with one another when they encounter forces in their service environment. Often, it is essential to the success of these missions that these components' dynamics are understood before implementing them into service. To achieve this understanding, we must simulate model dynamic environments in the laboratory. Researchers often utilize the traditional techniques of mounting components of interest onto rigid fixtures, meaning that the fixtures are designed in a way that they will not influence the dynamics measured during the testing. This is achieved by ensuring they exhibit no elastic motion in the frequency range to be tested on the component of interest [1]. The connections to these \"rigid fixtures\" are not characteristic of the components' service environment, and oftentimes, important aspects, such as coupling, may be ignored during the analysis. To prompt an investigation into this, researchers from Sandia National Laboratories (SNL) and Kansas City National Security Campus introduced the Box Assembly with Removable Component (BARC) to serve as a benchmark structure for investigating boundary condition problems. By introducing a non-rigid fixture to mount a component of interest on, researchers can begin to account for the effects that other bodies might introduce during a component's service environment. In fields such as aerospace, reducing weight can be critical to an aircraft's ability to perform. However, certain removal of material may result in a reduction in stiffness and ultimately affect the system's structural dynamics. The BARC has been tested with many variations in its assembly and dimensions. Researchers have investigated topics including modal analysis, impedance matching, parametric modeling of an optimized fixture, and many more [1; 2; 3]. These studies have incorporated a critical design alteration, a central cut on the top of the box assembly (BA). The original BARC design, which can be seen in Sandia's documentation, was labeled as the \"uncut\" version and includes a BA made from a box beam [4]. This cut introduces a decrease in the rigidity of the system and can be characteristic of engineering decisions that have to be made in aerospace, automotive, and even medical applications where varying widths of the central cut may be necessary to meet certain design requirements. Although some researchers may utilize adhesives to avoid some possible nonlinearities present in the bolted-joint connections of the BARC, our focus is on investigating these nonlinearities and how altering the width of the central cut on the BA affects the number and type of nonlinearities observed, as well as the dynamical properties of the BARC system. Because altering cut widths may be necessary as a continuation to improve and steward designs of the past, it is essential for scientists and engineers alike to understand the effects that altering central-cut widths has on the system's dynamics and, ultimately, the mission's success. Though the focus of this investigation is the nonlinear characteristics of the BARC system with various cut widths, linear behaviors, including the natural properties of the system, will also be characterized. The system can demonstrate linear and nonlinear behaviors/responses based on its output in relation to the input it receives. If there is no correlation between the input and the output, then the behavior of the system can be deemed nonlinear [5; 6]. Examples of nonlinearities that can be present in the system are nonlinear softening, nonlinear hardening, and nonlinear damping. Nonlinear softening and hardening differ in relation to the bending curve of the system's frequency response.