18 J. Sills et al. jointed struts providing the aft connections for each booster to the CS), vehicle stabilizer system (VSS) coupling/preloads (see Sect. 3.4), mobile launcher (ML) extensible column/preloads (see Sect. 3.5), and the transient behavior and decay characteristics of the preloads with changing boundary conditions as the vehicle separates from the ML. DGS seamlessly integrates into the SLS multibody modal synthesis framework and accurately simulates all stacking without the use of artificial external loads. The preloads are automatically reflected in the load indicator recoveries without any corrections or post-processing steps. All stages of stacking can be simulated including booster “toe-in” (for CS installation) and CS fueling/cryogenic shrinkage. DGS starts with an initial deformed geometry state and solves for the final system deformed geometry state, inclusive of all preloads and geometric nonlinear effects. This process can proceed iteratively until desired convergence is achieved. With this, executing linearized parametrics to, for instance, exercise different potential aft strut orientations is avoided since geometric nonlinear effects are captured in the final deformed geometry. To reduce the number of iterations, the initial deformed geometry state can include 1G and thermal effects in order to quickly converge to the final state geometry. In practice, this is done and fully converged solutions are achieved in only two iteration steps. The transient contribution (twang) and decay characteristics of the stacking and cryo-induced preloads is of considerable interest to the SLS program. To accurately capture this, DGS algorithms work together with Henkel-Mar nonlinear pad separation algorithms [2] which operate on the longitudinal and lateral separating DoFs at the ML to booster interface. As these separating DoFs release in the manner dictated by the interface geometries, interface flexibilities, interface loads, and external loads, the interaction of DGS and Henkel-Mar automatically generates the transient time-history, inclusive of the twang and decay characteristics, of the internal strain energy release. A series of DGS verification problems involving a system of ball-jointed struts are provided in the next section to help make concepts crystal clear. The first set of these problems has a closed form solution provided by Timeshenko and Gere [3]. To verify DGS capability for capturing larger strut rotations, the Timshenko problem is extended to larger preloads. Here, the linear analytical solution no longer holds and DGS is compared with a Newton-Raphson nonlinear solution of the Timoshenko problem. 3.2 DGS Verification Problems 3.2.1 Verification Problem #1: Ball Jointed Strut Problem – Preloads (Linear Case) Figure 3.1 depicts the ball-joint strut problem Timoshenko and Gere [3] utilized as the verification example. The vertical strut has an unstressed length of initial length (L) +change in length ( L). It is coupled into the diagonal struts resulting in compression of the vertical strut and associated extension/rotation of the inclined struts. The objective is to predict the strut loads, and the vertical displacement at the vertex. The parameters used were: E=10e6 psi, A=1 in2, L=10 in, and L=0.125 in. Timoshenko and Gere [3] provided the following closed form solution for the strut loads as applicable for small displacements: F2 = 2EA( L)cos 3 β L 1+2cos3β Fig. 3.1 Ball Jointed Strut Verification Problem (Timoshenko)
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