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The state of the art is assessed in substructure coupling for dynamic analysis. A general formulation, which permits all previously described methods to be characterized by a few constituent matrices, is developed. Limited results comparing the accuracy of various methods are presented.
A method is described to derive a generalized coordinate model consisting of flexible cantilever modes and rigid body modes from a physical coordinate model using a direct matrix abstraction procedure alter to rigid format 3. This model can readily be coupled to other substructures using modal synthesis techniques. It allows the use of a reduced size model for structural analyses while maintaining the capability of recovering the accelerations, forces, stresses, etc., from the original, large, complex model. This output recovery is accomplished with the use of a loads transformation matrix which relates the output parameters to the modal coordinate accelerations. In addition, a method is described to synthesize structural models consisting of hybrid coordinates for use in dynamic response analyses where one structure is described using physical coordinates and the other using generalized modal coordinates.
A technique for condensing dynamic loading points is described. The method was applied to substructure transient solutions and was found to be very effective.
Development and analysis of the Learjet 54/55 fuselage NASTRAN model is presented. This structure was broken down into several substructures to make the modeling and analysis effort more manageable. Since the geometry was fairly complex in some areas and in order to provide flexibility for future configuration studies, a series of local coordinate systems were used to describe the model. This work was accomplished using several different computer systems, the more recent of which were connected by a high speed data communications lines to perform the tasks of model generation and NASTRAN analysis.
Surface number density contour maps based on Dressler's (1976, 1980) studies of 65 rich clusters are presented to demonstrate the frequent occurrence of substructure in these dense systems. A prescription is given for examining the significance of such structure and it is shown that structure on scales equal to or larger than 0.24 Mpc (H0 = 100 km/s Mpc) is present in approximately 40% of the clusters in this sample. Many of the remaining clusters are elongated. The contour plots are useful for comparison with X-ray surface brightness maps. Together with available velocity data, they also serve warning that many rich clusters may not be simply characterized by a single core radius and velocity dispersion.
Automation reduces input length and potential data-entry errors. Matrix Automated Reduction and Coupling (MARC) program used for combining NASTRAN substructural models with primary structural model MARC also constructs job-control language (JCL) stream for NASTRAN batch job that utilizes previously-written user library of dynamic models. Minimizes lengthy input and reduces potential data-entry errors. MARC procedure used in assembling Space Shuttle orbiter dynamic models since 1983 and reduced NASTRAN modeling input time by as much as 50 percent. MARC program written in FORTRAN IV for interactive execution.
Substructuring methods are in common use in mechanics problems where typically the associated linear systems of algebraic equations are positive definite. Here these methods are extended to problems which lead to nonpositive definite, nonsymmetric matrices. The extension is based on an algorithm which carries out the block Gauss elimination procedure without the need for interchanges even when a pivot matrix is singular. Examples are provided wherein the method is used in connection with finite element solutions of the stationary Stokes equations and the Helmholtz equation, and dual methods for second-order elliptic equations.
The exact FORTRAN computer program BUNVIS-RG which uses an exact stiffness matrix method to find eigenvalues and modes of three-dimensional planes, is presented in detail. It is shown that BUNVIS-RG solution times can be predicted fairly accurately from two computer dependent constants, the number of nodes and members in the frame, and the maximum node number difference of any pair of connected nodes. The program's use of repetitive geometry, stayed column substructures and nonuniform member options was found to enhance efficiency and range of application.
The application of FEM modeling techniques to the analysis of torsional vibrations in complex rotating systems is described and demonstrated, summarizing results reported by Soares (1985). A substructuring approach is used for determination of torsional natural frequencies and resonant-mode shapes, steady-state frequency-sweep analysis, identification of dynamically unstable speed ranges, and characterization of transient linear and nonlinear systems. Results for several sample problems are presented in diagrams, graphs, and tables. STORV, a computer code based on this approach, is in use as a preliminary design tool for drive-train torsional analysis in the High Altitude Wind Tunnel at NASA Lewis.
Component mode synthesis was used to analyze different types of structures with MSC NASTRAN. The theory and technique of using Multipoint Constraint Equations (MPCs) to connect substructures to each other or to a common foundation is presented. Computation of the dynamic response of the system from shack spectrum inputs was automated using the DMAP programming language of the MSC NASTRAN finite element code.
Microstructural observations were conducted on NaCl single crystals after creep. The microstructure after a stress increase followed by a stress decrease consisted primarily of cells; no significant number of subgrains were observed although they were present in the microstructures produced during uninterrupted tests. Prestraining in the exponential creep regime produced an uniform distribution of dislocations and a few subboundaries which transformed to equiaxed subgrains when tested in the power law creep region. This substructure was similar to that observed in an as-received specimen deformed to an equivalent strain. Prior creep in the power law creep region produced equiaxed subgrains whose boundaries were found to be mechanically stable when the specimen was retested in the exponential creep region. The role of subboundary migration in the formation of new subgrains is discussed.
It is demonstrated here that complete checkout of a basic substructure can be done under the special circumstance of a sliding connection with offsets. Stiff bar connectors make this possible so long as the bar coordinates are aligned with the displacement coordinates at the sliding surface.
A generic unit cell model which includes a unique fiber substructuring concept is proposed for the development of micromechanics equations for continuous fiber reinforcement ceramic composites. The unit cell consists of three constituents: fiber, matrix, and an interphase. In the present approach, the unit cell is further subdivided into several slices and the equations of micromechanics are derived for each slice. These are subsequently integrated to obtain ply level properties. A stand alone computer code containing the micromechanics model as a module is currently being developed specifically for the analysis of ceramic matrix composites. Towards this development, equivalent ply property results for a SiC/Ti-15-3 composite with 0.5 fiber volume ratio are presented and compared with those obtained from customary micromechanics models to illustrate the concept. Also, comparisons with limited experimental data for the ceramic matrix composite, SiC/RBSN (Reaction Bonded Silicon Nitride) with a 0.3 fiber volume ratio are given to validate the concepts.
The NASTRAN Manuals in the substructuring area are all geared toward instant success, but the solution paths are fraught with many traps for human error. Thus, the probability of suffering a fatal abort is high. In such circumstances, the necessity for diagnostics that are user friendly is paramount. This paper is written in the spirit of improving the diagnostics as well as the documentation in one area where the author felt he was backed into a blind corner as a result of his having committed a data oversight. This topic is aired by referring to an analysis of a particular structure. The structure, under discussion, used a number of local coordinate systems that simplified the preparation of input data. The principle features of this problem are introduced by reference to a series of figures.
This paper develops a parallelizable multilevel constrained nonlinear equation solver. The substructuring process is automated to yield appropriately balanced partitioning of each succeeding level. Due to the generality of the procedure, both sequential, partially and fully parallel environments can be handled. This includes both single and multiprocessor assignment per individual partition. Several benchmark examples are presented. These illustrate the robustness of the procedure as well as its capacity to yield significant reductions in memory utilization and calculational effort due both to updating and inversion.
Using a method for constraining the dynamical state of a galaxy cluster by examining the moments of its x-ray surface brightness distribution, we determine the statistics of cluster substructure for a sample of 40 Abell clusters. Using x-ray observations from the Einstein Observatory Imaging Proportional Counter (IPC), we measure the first moment M1(r), the ellipsoidal orientation angle theta2(r), and the axial ratio eta(r) at several different radii in the cluster. We determine the effects of systematics such as x-ray point source emission, telescope vignetting, Poisson noise, and characteristics of the IPC by measuring the same parameters on an ensemble of simulated cluster images. Due to the small band-pass of the IPC, the ICM emissivity is nearly independent of temperature so the intensity at each point in the IPC images is simply proportional to the emission measure calculated along the line of sight through the cluster (e.g. Fabricant et al. 1980). Therefore, barring a change superposition of two x-ray emitting clusters, a significant variation in the image centroid M1(r) as a function of radius indicates that the center of mass of the intra-cluster medium (ICM) varies with radius. We argue that such a configuration (essentially an m = 1 component in the ICM density distribution) is a non-equilibrium component; it results from an off-center subclump or a recent merger in the ICM.
The stability and robustness of the controlled system obtained via the substructure control synthesis (SCS) method of Su et al. (1990) were examined using a six-bay truss model, and employing an LQG control design method to obtain controllers for two separate structures. It is found that the assembled controller provides a stability in this instance. A qualitative assessment of the stability robustness of the system with controller designed with the SCS method is provided by obtaining a controller using the complete truss model and comparing the robustness of the corresponding closed-loop systems.
We use the moments of the X-ray surface brightness distribution to constrain the dynamical state of a galaxy cluster. Using X-ray observations from the Einstein Observatory IPC, we measure the first moment FM, the ellipsoidal orientation angle, and the axial ratio at a sequence of radii in the cluster. We argue that a significant variation in the image centroid FM as a function of radius is evidence for a nonequilibrium feature in the intracluster medium (ICM) density distribution. In simple terms, centroid shifts indicate that the center of mass of the ICM varies with radius. This variation is a tracer of continuing dynamical evolution. For each cluster, we evaluate the significance of variations in the centroid of the IPC image by computing the same statistics on an ensemble of simulated cluster images. In producing these simulated images we include X-ray point source emission, telescope vignetting, Poisson noise, and characteristics of the IPC. Application of this new method to five Abell clusters reveals that the core of each one has significant substructure. In addition, we find significant variations in the orientation angle and the axial ratio for several of the clusters.