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The basic equations for the stability analysis of flow over three dimensional swept wings are developed and numerical methods for their solution are surveyed. The equations for nonlinear stability analysis of three dimensional disturbances in compressible, three dimensional, nonparallel flows are given. Efficient and accurate numerical methods for the solution of the equations of stability theory were surveyed and analyzed.
The probabilistic finite element computer program known as Numerical Evaluation of Stochastic Structures Under Stress (NESSUS) is being developed for the analysis of critical structural components for reusable space propulsion systems. First year efforts involve the formulation of the probabilistic analysis strategy and the development of a probabilistic linear analysis code. The ultimate goal of the 3-year program is the development of a finite element code capable of performing nonlinear dynamic analysis of structures having stochastic material properties, geometry, and boundary conditions and subjected to random loading. Three levels of sophistication are envisioned for the stochastic description of the structural problem, namely: (1) homogeneous random variable for stiffness, mass, damping, and external loading; (2) stochastic characterization of variables at the element level, with specified interelement correlations; and (3) stochastic interpolation of variables within a finite element. Two alternative probabilistic analysis methods will be developed, allowing for all three levels of modeling sophistication.
Galerkin finite-element approximations and Newton's method for solving free boundary problems are combined with computer-implemented techniques from nonlinear perturbation analysis to study solidification problems with natural convection in the melt. The Newton method gives rapid convergence to steady state velocity, temperature and pressure fields and melt-solid interface shapes, and forms the basis for algebraic methods for detecting multiple steady flows and assessing their stability. The power of this combination is demonstrated for a two-phase Rayleigh-Benard problem composed of melt and solid in a veritical cylinder with the thermal boundary conditions arranged so that a static melt with a flat melt-solid interface is always a solution. Multiple cellular flows bifurcating from the static state are detected and followed as Rayleigh number is varied. Changing the boundary conditions to approach those appropriate for the vertical Bridgman solidification system causes imperfections that eliminate the static state. The flow structure in the Bridgman system is related to those for the Rayleigh-Benard system by a continuous evolution of the boundary conditions.
The experimental results from a stitched VaRTM carbon-epoxy composite panel tested under uni-axial compression loading are presented along with nonlinear finite element analysis prediction of the response. The curved panel is divided by frames and stringers into six bays with a column of three bays along the compressive loading direction. The frames are supported at the frame ends to resist out-of-plane translation. Back-to-back strain gages are used to record the strain and displacement transducers were used to record the out-of-plane displacements. In addition a full-field-displacement measurement technique that utilizes a camera-based-stereo-vision system was used to record the displacements. The panel was loaded to 1.5 times the predicted initial buckling load (1st bay buckling load, P(sub er) from the nonlinear finite element analysis and then was removed from the test machine for impact testing. After impacting with 20 ft-lbs of energy using a spherical impactor to produce barely visible damage the panel was loaded in compression until failure. The buckling load of the first bay to buckle was 97% of the buckling load before impact. The stitching constrained the impact damage from growing during the loading to failure. Impact damage had very little overall effect on panel stiffness. Panel stiffness measured by the full-field-displacement technique indicated a 13% loss in stiffness after impact. The panel failed at 1.64 times the first panel buckling load. The barely visible impact damage did not grow noticeably as the panel failed by global instability due to stringer-web terminations at the frame locations. The predictions from the nonlinear analysis of the finite element modeling of the entire specimen were very effective in the capture of the initial buckling and global behavior of the panel. In addition, the prediction highlighted the weakness of the panel under compression due to stringer web terminations. Both the test results and the nonlinear predictions serve to reinforce the severe penalty in structural integrity caused by the low cost manufacturing technique to terminate the stringer webs, and demonstrates the importance of this type of sub-component testing and high fidelity failure analysis in the design of a composite fuselage.
A parallel adaptive dynamic relaxation (ADR) algorithm has been developed for nonlinear structural analysis. This algorithm has minimal memory requirements, is easily parallelizable and scalable to many processors, and is generally very reliable and efficient for highly nonlinear problems. Performance evaluations on single-processor computers have shown that the ADR algorithm is reliable and highly vectorizable, and that it is competitive with direct solution methods for the highly nonlinear problems considered. The present algorithm is implemented on the 512-processor Intel Touchstone DELTA system at Caltech, and it is designed to minimize the extent and frequency of interprocessor communication. The algorithm has been used to solve for the nonlinear static response of two and three dimensional hyperelastic systems involving contact. Impressive relative speedups have been achieved and demonstrate the high scalability of the ADR algorithm. For the class of problems addressed, the ADR algorithm represents a very promising approach for parallel-vector processing.
Nonlinear models of low gravity slosh, when coupled to spacecraft vibrations, predict intense nonlinear eigenfrequency shifts at zero gravity. These nonlinear frequency shifts are due to internal quadratic and cubic resonances between fluid slosh modes and spacecraft vibration modes. Their existence has been verified experimentally, and they cannot be correctly modeled by approximate, uncoupled nonlinear models, such as pendulum mechanical analogs. These predictions mean that linear slosh assumptions for spacecraft vibration models can be invalid, and may lead to degraded control system stability and performance. However, a complete nonlinear modal analysis will predict the correct dynamic behavior. This paper presents the analytical basis for these results, and discusses the effect of internal resonances on the nonlinear coupled response at zero gravity.
A next generation reusable launch vehicle (RLV) will require thermally efficient and light-weight cryogenic propellant tank structures. Since these tanks will be weight-critical, analytical tools must be developed to aid in sizing the thickness of insulation layers and structural geometry for optimal performance. Finite element method (FEM) models of the tank and insulation layers were created to analyze the thermal performance of the cryogenic insulation layer and thermal protection system (TPS) of the tanks. The thermal conditions of ground-hold and re-entry/soak-through for a typical RLV mission were used in the thermal sizing study. A general-purpose nonlinear FEM analysis code, capable of using temperature and pressure dependent material properties, was used as the thermal analysis code. Mechanical loads from ground handling and proof-pressure testing were used to size the structural geometry of an aluminum cryogenic tank wall. Nonlinear deterministic optimization and reliability optimization techniques were the analytical tools used to size the geometry of the isogrid stiffeners and thickness of the skin. The results from the sizing study indicate that a commercial FEM code can be used for thermal analyses to size the insulation thicknesses where the temperature and pressure were varied. The results from the structural sizing study show that using combined deterministic and reliability optimization techniques can obtain alternate and lighter designs than the designs obtained from deterministic optimization methods alone.
Wiener theory for analysis and synthesis of nonlinear physical systems applied to non- Gaussian and Gaussian inputs
A numerical and experimental investigation of the bending behavior of six eight-ply graphite-epoxy circular cylinders is presented. Bending is induced by applying a known end-rotation to each end of the cylinders, analogous to a beam in bending. The cylinders have a nominal radius of 6 inches, a length-to-radius ratio of 2 and 5, and a radius-to-thickness ratio of approximately 160. A (+/- 45/0/90)S quasi-isotropic layup and two orthotropic layups, (+/- 45/0 sub 2)S and (+/- 45/90 sub 2)S, are studied. A geometrically nonlinear special-purpose analysis, based on Donnell's nonlinear shell equations, is developed to study the prebuckling responses and gain insight into the effects of non-ideal boundary conditions and initial geometric imperfections. A geometrically nonlinear finite element analysis is utilized to compare with the prebuckling solutions of the special-purpose analysis and to study the buckling and post buckling responses of both geometrically perfect and imperfect cylinders. The imperfect cylinder geometries are represented by an analytical approximation of the measured shape imperfections. Extensive experimental data are obtained from quasi-static tests of the cylinders using a test fixture specifically designed for the present investigation. A description of the test fixture is included. The experimental data are compared to predictions for both perfect and imperfect cylinder geometries. Prebuckling results are presented in the form of displacement and strain profiles. Buckling end-rotations, moments, and strains are reported, and predicted mode shapes are presented. Observed and predicted moment vs. end-rotation relations, deflection patterns, and strain profiles are illustrated for the post buckling responses. It is found that a geometrically nonlinear boundary layer behavior characterizes the prebuckling responses. The boundary layer behavior is sensitive to laminate orthotropy, cylinder geometry, initial geometric imperfections, applied end-rotation, and non-ideal boundary conditions. Buckling end-rotations, strains, and moments are influenced by laminate orthotropy and initial geometric imperfections. Measured buckling results correlate well with predictions for the geometrically imperfect specimens. The postbuckling analyses predict equilibrium paths with a number of scallop-shaped branches that correspond to unique deflection patterns. The observed postbuckling deflection patterns and measured strain profiles show striking similarities to the predictions in some cases. Ultimate failure of the cylinders is attributed to an interlaminar shear failure mode along the nodal lines of the postbuckling deflection patterns.
Propellant tank slosh damping models for launch vehicle ascent analysis typically employ linear models for bare-wall damping, or assume a single, low-amplitude wave height for baffled configurations. A higher fidelity damping model would incorporate nonlinear effects to increase damping as slosh wave amplitude increases. This paper provides an overview of a technical assessment performed by the NASA Engineering and Safety Center (NESC) to evaluate lateral slosh damping as a function of lateral force (or wave) amplitude for multiple tank configurations. The increased fidelity of nonlinear slosh damping models, if leveraged, has the potential to reduce over-conservatism associated with the use of linear slosh damping models. This can provide design flexibility to enhance launch vehicle flight control performance, reduce baffle design requirements and/or increase robustness in targeted areas such as control-structure interaction during ascent.
A comprehensive computer program, designated BOSOR4, for analysis of the stress, stability, and vibration of segmented, ring-stiffened, branched shells of revolution and prismatic shells and panels is described. The program performs large-deflection axisymmetric stress analysis, small-deflection nonsymmetric stress analysis, modal vibration analysis with axisymmetric nonlinear prestress included, and buckling analysis with axisymmetric or nonsymmetric prestress. One of the main advantages of the code is the provision for realistic engineering details such as eccentric load paths, internal supports, arbitrary branching conditions, and a library of wall constructions. The program is based on the finite-difference energy method, which is very rapidly convergent with increasing numbers of mesh points.
A comprehensive computer program, designated BOSOR4, for analysis of the stress, stability, and vibration of segmented, ring-stiffened, branched shells of revolution and prismatic shells and panels is described. The program performs large-deflection axisymmetric stress analysis, small-deflection nonsymmetric stress analysis, modal vibration analysis with axisymmetric nonlinear prestress included, and buckling analysis with axisymmetric or nonsymmetric prestress. One of the main advantages of the code is the provision for realistic engineering details such as eccentric load paths, internal supports, arbitrary branching conditions, and a 'library' of wall constructions. The program is based on the finite-difference energy method, which is very rapidly convergent with increasing numbers of mesh points. The organization of the program is briefly described with the flow of calculations charted for each of the types of analysis. Overlay charts and core storage requirements are given for the CDC 6600, IBM 370/165, and UNIVAC 1108 versions of BOSOR4.
BOSOR4 computer program developed as comprehensive program for analysis of stress, stability, and vibration of complex, branched shells of revolution made of elastic materials. Used to analyze prismatic shells and panels. Performs large-deflection axisymmetric stress analysis, small-deflection nonsymmetric stress analysis, modal vibration analysis with axisymmetric nonlinear prestress included, and buckling analysis with axisymmetric or nonsymmetric prestress. One main advantage, provisions for such realistic engineering details as eccentric load paths, internal supports, arbitrary branching conditions, and "library" of wall constructions. Based on finite-difference energy method and offers very rapid convergence with increasing numbers of mesh points. Written in FORTRAN 77.
The analytical developments and experimental investigations performed in assessing the effect of internal friction on rotor systems dynamic performance are documented. Analytical component models for axial splines, Curvic splines, and interference fit joints commonly found in modern high speed turbomachinery were developed. Rotor systems operating above a bending critical speed were shown to exhibit unstable subsynchronous vibrations at the first natural frequency. The effect of speed, bearing stiffness, joint stiffness, external damping, torque, and coefficient of friction, was evaluated. Testing included material coefficient of friction evaluations, component joint quantity and form of damping determinations, and rotordynamic stability assessments. Under conditions similar to those in the SSME turbopumps, material interfaces experienced a coefficient of friction of approx. 0.2 for lubricated and 0.8 for unlubricated conditions. The damping observed in the component joints displayed nearly linear behavior with increasing amplitude. Thus, the measured damping, as a function of amplitude, is not represented by either linear or Coulomb friction damper models. Rotordynamic testing of an axial spline joint under 5000 in.-lb of static torque, demonstrated the presence of an extremely severe instability when the rotor was operated above its first flexible natural frequency. The presence of this instability was predicted by nonlinear rotordynamic time-transient analysis using the nonlinear component model developed under this program. Corresponding rotordynamic testing of a shaft with an interference fit joint demonstrated the presence of subsynchronous vibrations at the first natural frequency. While subsynchronous vibrations were observed, they were bounded and significantly lower in amplitude than the synchronous vibrations.
The variational-asymptotical method is a mathematical technique by which the three-dimensional analysis of laminated plate deformation can be split into a linear, one-dimensional, through-the-thickness analysis and a nonlinear, two-dimensional, plate analysis. The elastic constants used in the plate analysis are obtained from the through-the-thickness analysis, along with approximate, closed-form three-dimensional distributions of displacement, strain, and stress. In this paper, a theory based on this technique is developed which is capable of approximating three-dimensional elasticity to any accuracy desired. The asymptotical method allows for the approximation of the through-the-thickness behavior in terms of the eigenfunctions of a certain Sturm-Liouville problem associated with the thickness coordinate. These eigenfunctions contain all the necessary information about the nonhomogeneities along the thickness coordinate of the plate and thus possess the appropriate discontinuities in the derivatives of displacement. The theory is presented in this paper along with numerical results for the eigenfunctions of various laminated plates.
This paper describes the design, analysis, and nonlinear simulation results (batch and piloted) for a longitudinal controller which is scheduled to be flight-tested on the High-Alpha Research Vehicle (HARV). The HARV is an F-18 airplane modified for and equipped with multi-axis thrust vectoring. The paper includes a description of the facilities, a detailed review of the feedback controller design, linear analysis results of the feedback controller, a description of the feed-forward controller design, nonlinear batch simulation results, and piloted simulation results. Batch simulation results include maximum pitch stick agility responses, angle of attack alpha captures, and alpha regulation for full lateral stick rolls at several alpha's. Piloted simulation results include task descriptions for several types of maneuvers, task guidelines, the corresponding Cooper-Harper ratings from three test pilots, and some pilot comments. The ratings show that desirable criteria are achieved for almost all of the piloted simulation tasks.
This paper identifies the potential for unconstrained minimization algorithms of mathematical programming to be cost-effective with other conventional techniques of transient nonlinear structural analysis. With this in mind the authors have attempted to critically evaluate a few of the most commonly used algorithms for their effectiveness in solving structural problems involving geometric and material nonlinearities. The algorithms have been categorized as being zeroth order, first order and second order depending upon the order of the derivative of the function called for by the algorithm. The sensitivity of these algorithms to the accuracy of derivatives derived on the basis of finite difference operations clearly suggest using analytically derived derivatives to obtain better control on the number of minimizations required for convergence to the exact solution.
Runaway electrons exhibit kinetic instabilities with potentially beneficial consequences. The anomalous Doppler resonance between the electrons and whistler modes is a primary underlying mechanism. These instabilities require a first-principle nonlinear theoretical analysis, which would ultimately enable assessment of their impact on disruption mitigation and diagnostics, especially in ITER-relevant conditions. This paper presents recent progress in developing the required theoretical framework for isolated nonlinear resonances. By employing the generic bump-on-tail model, we predict the wave saturation levels in both near-threshold and strongly driven regimes. Remarkably, for the waves of interest, the parallel component of the wave vector dictates the parallel momentum of the resonant particles. This feature, together with the expected wave saturation level, provides a complete description of the resonance impact on the runaway electron distribution function. We show that the parametric dependence of the nonlinearly saturated mode provides a way to recover certain features of the runaway momentum distribution function.