# A Nonlinear CFD/Multibody Incremental-Dynamic Model for A Constrained Mechanism
## 1 Introduction  
Analyzing the equations of motion for a multibody mechanism is an intricate issue in the field of dynamic mechanics, since several dynamic equations with high-order nonlinearities are essentially required to be solved [1, 2]. If a multibody mechanism enlarges with some active and passive joints moving in the air, the difficulties of analysis of the system will be amplified due to the initial and boundary conditions of the model [3]. Such mechanisms can drift away from their joint connections during their movement in the air owing to the set of complicated aerodynamic forces, such as drag, shear, and normal [4]. Constraint force equation (CFE) has been widely used for dynamic analysis of multibody mechanisms [5, 6, 7, 8, 9, 10, 11]. In CFE, the constraint forces and momentums are calculated using multibody dynamic equations, and then incorporated into the model as a separate-external force on each part/component of a multibody mechanism. Although CFE is in wide using for various types of multibody mechanisms under different initial and boundary conditions, the main disadvantage of that is the external forces applying on a mechanism should be determined during the procedure as a known function and cannot be dependent on the spatial location of each part/component of a multibody mechanism. The constraint forces in the CFE method are computed based on the external forces, so thevalues of external forces should be known independent of the spatial positions, which is in contrast with the usual performance of the ANSYS Fluent(r) software, where the external forces at each moment are estimated by the spatial position of the mechanism and complex numerical analysis corresponding to finite element method (FEM) and computational fluid dynamics (CFD) [12; 13; 14]. Therefore, using CFE seems costly as it needs to be performed in two different software, one for dynamic coupling, e.g., Matlab(r) (MathWorks, Natick, MA, USA) and one for solving aerodynamic equations and simulating the multibody system, e.g., ANSYS Fluent(r) (ANSYS Fluent Software: CFD Simulation, Canonsburg, PA, USA). This is why the application of user-defined functions (UDF) is significant, as it is faster and more accurate than those approaches using so far, especially CFE. UDF is flexible and powerful, and allow to modify the behavior of ANSYS Fluent(r) to satisfy the particular modeling needs, such as modifications in the initial and boundary conditions, material models, and various physical transformations [15]. However, it should be noted that due to the complexity of the UDF programming, its application in analyzing the dynamics of a multibody mechanism has not been extended. A problem corresponding to analyzing the dynamic separation of multiple bodies in the air is mostly related to store separation from aircraft [16; 17], for example, the separation of the X-15 research vehicle from the B-52 carrier aircraft [18; 19]. In such cases, the store and the X-15 vehicle are much smaller than that of the parent vehicle. However, there would be some other class of stage separation problem, which involves the separation of two vehicles of comparable sizes. In such cases, these would be defined as multi-stage reusable launch vehicles where the integrity of each stage is important after separation [20; 21]. NASA conducted extensive research on stage separation of multi-stage reusable launch vehicles since the 1960s. These studies mostly focused on the problem of separation of generic two-stage reusable launch vehicles. However, NASA is always seeking to develop a better and more comprehensive stage separation tool that included wind tunnel testing, as well as the development and validation of CFD and engineering level simulation tools [22; 23]. With these considerations in mind, a stage separation analysis and simulation tool was developed called Conceptual Separation, which is already implemented into a commercial multibody dynamic analysis software ADAMS(r).