## 2 Modeling of Hybrid-Propellant Launch Vehicle
### Structural Model  
Mass, moment of inertia, and center of gravity are fundamental parameters in flight simulation within the structural model. The mass value of the launch vehicle has a decisive impact on acceleration, velocity, and maneuverability. The moment of inertia determines the response to angular motions, influencing maneuverability. The center of gravity represents the balance point of the launch vehicle and affects stability if not in an appropriate position. Accurate modeling and simulation of these parameters are crucial to understanding how the launch vehicle will behave in real-world conditions. Careful tracking of these parameters in flight simulation is critical for safe and effective flight operations. In the design phase, accurately determining these parameters plays a significant role in optimizing the launch vehicle's performance and achieving the desired flight characteristics. Therefore, the vehicle structure serves as the mechanical interface between launch vehicles and all subsystems, making it crucial to obtain the most optimal design of the structural configuration to meet mission requirements. For the structural models of the Minotaur-I launch vehicle, CATIA V5R19 product design software was utilized [24]. The geometry, stage characteristics, and dimensions utilized in both the structural and aerodynamic models are depicted in Figure 1. It is important to highlight that the dimensions of the final stage were updated and optimized using a hybrid rocket motor and included in the optimization process. The case material of the first stage, MM 55A1, is D6 AC steel, while the second stage, MM SR19, utilizes 6 Al-4V titanium. The third stage, Orion 50 XL, and the fourth stage, Orion 38, have monocque graphite epoxy as their case material. Structural modeling was performed using the CATIA V5R19 CAD program [38]. Throughout the optimization, structural characteristics were updated by assuming the hybrid motor is made of 6 Al-4V titanium also known as Grade 5 titanium, and mass and length calculations were performed in structural modeling. This material consists of 90% titanium, 6% aluminum, and 4% vanadium, offering an optimal combination of properties. With a density of 4430 kg/m\({}^{3}\) and a melting point around 1668 degrees Celsius, 6 Al-4V titanium exhibits a tensile strength ranging from 900 to 1100 MPa, a yield strength typically between 800 and 1000 MPa, and an elastic modulus of approximately 114,000 MPa. Its hardness falls within the range of 36 to 44 HRC, while its fatigue strength is commedable [39]. The embedded presence of material properties within the CATIA V5R19 program facilitates the structural modeling. Its versatility allows for further processing to tailor it to specific applications [40]. The objective of this study was to comprehensively incorporate subsystem models into the optimization process, assuming the hybrid motor is made of 6 Al-4V titanium, to iteratively update subsystems in each iteration cycle, aiming to obtain the most optimal design solution. In each iteration, due to changes in parameters such as geometry and mass values,  
Figure 1: The representation of the Minotaur-I launch vehicle geometry used in structural and aerodynamic subsystems, obtained from DATCOM. aerodynamic coefficients, inertia moments, and center of gravity values are recalculated and updated. However, in this study, since the focus was on evaluating the performance and efficiency of the hybrid propulsion system and trajectory of the launch vehicle, and no research was conducted on structural concerns, structural aspects such as structural resistance, vibration, strength, and composition have not been investigated or analyzed. Therefore, structural calculations and modeling were performed at a fundamental level and incorporated into the optimization cycle. For the first to third stages, the volumes of structural parts are determined based on known materials while the propellant volumes and heights are modeled using the propellant density (\(\rho_{p}\)) and structural part volume.