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NASA NTRS1989

A method for exponential propagation of large systems of stiff nonlinear differential equations

A new time integrator for large, stiff systems of linear and nonlinear coupled differential equations is described. For linear systems, the method consists of forming a small (5-15-term) Krylov space using the Jacobian of the system and carrying out exact exponential propagation within this space. Nonlinear corrections are incorporated via a convolution integral formalism; the integral is evaluated via approximate Krylov methods as well. Gains in efficiency ranging from factors of 2 to 30 are demonstrated for several test problems as compared to a forward Euler scheme and to the integration package LSODE.

Friesner, Richard A.↗

NASA NTRS1983

The response of cylindrical panels fabricated from symmetrically and unsymmetrically laminated composite materials

Equations are developed which govern the deflection response of long cylindrical panels subjected to a line load. The line load is directed toward the center of curvature of the panel, is located at an arbitrary point along the arc length of the panel, and is included at an arbitrary angle relative to the radial direction. Only the geometrically linear problem is considered and the spatial dependence in the problem is reduced to one independent variable, specifically, the arc length along the panel. The problem is thus solvable in closed form. Both symmetrically laminated and the less common unsymmetrically laminated simply supported panels are studied. The unsymmetrically laminated case was considered because the natural shape of an unsymmetric laminate is cylindrical. Results are presented which show the influence of the location and inclination of the line load on panel deflection. Shallow and deep panels are considered. Both the symmetric and unsymmetric panels exhibit similar behavior, the unsymmetric configurations being less stiff. Limited experimental results are presented.

Carper, D. M.↗

NASA NTRS1987

The effects of aeroelastic deformation on the unaugmented stopped-rotor dynamics of an X-Wing aircraft

A new design concept in the development of vertical takeoff and landing aircraft with high forward flight speed capability is that of the X-Wing. The X-Wing is a stiff, bearingless helicopter rotor system which can be stopped in flight and the blades used as two forward-swept wings and two aft-swept wings. Because of the unusual configuration in the fixed-wing mode, there is a high potential for aeroelastic divergence or flutter and coupling of blade vibration modes with rigid-body modes. An aeroelastic stability analysis of an X-Wing configuration aircraft was undertaken to determine if these problems could exist. This paper reports on the results of dynamic stability analyses in the lateral and longitudinal directions including the vehicle rigid-body and flexible modes. A static aeroelastic analysis using the normal vibration mode equations of motion was performed to determine the cause of a loss of longitudinal static margin with increasing airspeed. This loss of static margin was found to be due to aeroelastic 'washin' of the forward-swept blades and 'washout' of the aft-swept blades moving the aircraft aerodynamic center forward of the center of gravity. This phenomenon is likely to be generic to X-Wing aircraft.

Gilbert, Michael G.↗

NASA NTRS1993

Modeling of composite beams and plates for static and dynamic analysis

The main purpose of this research was to develop a rigorous theory and corresponding computational algorithms for through-the-thickness analysis of composite plates. This type of analysis is needed in order to find the elastic stiffness constants for a plate and to post-process the resulting plate solution in order to find approximate three-dimensional displacement, strain, and stress distributions throughout the plate. This also requires the development of finite deformation plate equations which are compatible with the through-the-thickness analyses. After about one year's work, we settled on the variational-asymptotical method (VAM) as a suitable framework in which to solve these types of problems. VAM was applied to laminated plates with constant thickness in the work of Atilgan and Hodges. The corresponding geometrically nonlinear global deformation analysis of plates was developed by Hodges, Atilgan, and Danielson. A different application of VAM, along with numerical results, was obtained by Hodges, Lee, and Atilgan. An expanded version of this last paper was submitted for publication in the AIAA Journal.

Hodges, Dewey H.↗

NASA NTRS1994

Interlaminar stress analysis of dropped-ply laminated plates and shells by a mixed method

A mixed method of approximation based on Reissner's variational principle is developed for the linear analysis of interlaminar stresses in laminated composites, with special interest in laminates that contain terminated internal plies (dropped-ply laminates). Two models are derived, one for problems of generalized plane deformation and the other for the axisymmetric response of shells of revolution. A layerwise approach is taken in which the stress field is assumed with an explicit dependence on the thickness coordinate in each layer. The dependence of the stress field on the thickness coordinate is determined such that the three-dimensional equilibrium equations are satisfied by the approximation. The solution domain is reduced to one dimension by integration through the thickness. Continuity of tractions and displacements between layers is imposed. The governing two-point boundary value problem is composed of a system of both differential and algebraic equations (DAE's) and their associated boundary conditions. Careful evaluation of the system of DAE's was required to arrive at a form that allowed application of a one-step finite difference approximation. A two-stage Gauss implicit Runge-Kutta finite difference scheme was used for the solution because of its relatively high degree of accuracy. Patch tests of the two models revealed problems with solution accuracy for the axisymmetric model of a cylindrical shell loaded by internal pressure. Parametric studies of dropped-ply laminate characteristics and their influence on the interlaminar stresses were performed using the generalized plane deformation model. Eccentricity of the middle surface of the laminate through the ply drop-off was found to have a minimal effect on the interlaminar stresses under longitudinal compression, transverse tension, and in-plane shear. A second study found the stiffness change across the ply termination to have a much greater influence on the interlaminar stresses.

Harrison, Peter N.↗

NASA NTRS1985

Thermomechanically induced pre- and postbuckling of general structure

An algorithmic solution strategy which enables handling the positive/indefinite stiffness characteristics associated with the pre and postbuckling of structures subject to complex thermomechanical loading fields was developed. The flexibility of the procedure is such that it can be applied to both finite difference and element type simulations. Due to the generality of the algorithmic approach developed, both kinematic and thermal/mechanical type material nonlinearity including inelastic effects can be treated. This includes the possibility of handling completely general thermomechanical boundary conditions. To demonstrate the scheme, the results of several benchmark problems are presented.

Padovan, J.↗

NASA NTRS1987

Structural concepts for large solar concentrators

The Sunflower large solar concentrator, developed in the early 1970's, is a salient example of a high-efficiency concentrator. The newly emphasized needs for solar dynamic power on the Space Station and for large, lightweight thermal sources are outlined. Existing concepts for high efficiency reflector surfaces are examined with attention to accuracy needs for concentration rates of 1000 to 3000. Concepts using stiff reflector panels are deemed most likely to exhibit the long-term consistent accuracy necessary for low-orbit operation, particularly for the higher concentration ratios. Quantitative results are shown of the effects of surface errors for various concentration and focal-length diameter ratios. Cost effectiveness is discussed. Principal sources of high cost include the need for various dished panels for paraboloidal reflectors and the expense of ground testing and adjustment. A new configuration is presented addressing both problems, i.e., a deployable Pactruss backup structure with identical panels installed on the structure after deployment in space. Analytical results show that with reasonable pointing errors, this new concept is capable of concentration ratios greater than 2000.

Hedgepeth, John M.↗

NASA NTRS1999

Understanding the Potential of Aeroelastic Couplings to Stabilize Ground and Air Resonance in a Soft-Inplane Tiltrotor

The tiltrotor offers the best mix of hovering and cruise flight of any of the current V/STOL configurations. One possible improvement on the tiltrotors of today designs would be using a soft-inplane hingeless hub. The advantages to a soft-inplane hingeless hub range from reduced weight and maintenance to reduced vibration and loads. However, soft-inplane rotor systems are inherently in danger of the aeromechanical instabilities of ground and air resonance. Furthermore tiltrotors can be subject to whirl flutter. At least in part because of the potential for air and ground resonance in a soft-inplane rotor, the Bell XV-15, the Bell-Boeing V-22 Osprey, and the new Bell Augusta 609 have stiff-inplane, gimballed rotors which do not experience these instabilities. In order to design soft-inplane V/STOL aircraft that do not experience ground or air resonance, it is important to be able to predict these instabilities accurately. Much of the research studying the stability of tiltrotors has been focused on the understanding and prediction of whirl flutter. As this instability is increasingly well understood, air and ground resonance for a tiltrotor need to be investigated. Once we understand the problems of air and ground resonance in a tiltrotor, we must look for solutions to these instabilities. Other researchers have found composite or kinematic couplings in the blades of a helicopter helpful for ground and air resonance stability. Tiltrotor research has shown composite couplings in the wing to be helpful for whirl flutter. Therefore, this project will undertake to model ground and air resonance of a soft-inplane hingeless tiltrotor to understand the mechanisms involved and to evaluate whether aeroelastic couplings in the wing or kinematic couplings in the blades would aid in stabilizing these instabilities in a tiltrotor.

Howard, Anna K. T.↗

NASA NTRS2006

Structure Detection of Nonlinear Aeroelastic Systems with Application to Aeroelastic Flight Test Data

This viewgraph presentation reviews the applicability of NARMAX structure detection to aeroelastic systems. In conclusion, the simulation results demonstrate bootstrap approach for structure computation of aircraft structural stiffness provided a high rate of true model selection: 1. T-test and stepwise regression methods had difficulty providing accurate results 2. Work contributes to understanding of the use of structure detection for modelling and identification of aerospace systems. 3. Limitation of model complexity that can be studied with these structure computation techniques 4. Result of the large number of candidate terms, for a given model order, and the data length required to guarantee convergence 5. Another approach to structure computation problem uses a least absolute shrinkage and selection operator (LASSO)

Kukreja, Sunil L.↗

NASA NTRS1988

Structural identification and damage assessment of large space structures

Two different methods are proposed for identifying the structural properties of large orbiting space structures under ordinary service loads, and for assessing potential damage due to impact or other extreme loadings. It is shown that the behavior of a structure in a weightless environment is nonlinear due to unloaded or lightly loaded connections, an effect which not only complicates structural control, but makes the problem of system identification more difficult than for ground based systems. Both proposed methods assume that the structure is subjected to loads imposed by prescribed self stressing systems sufficient to produce repeatable internal force systems in the structure. The first method is based on statical response and requires a survey of structural displacements produced by the self stressing systems. The displacements do not have to be determined completely (i.e., in three directions at each connection), but more displacement information produces more accurate structural stiffness information. It is anticipated that displacement measurements will be taken using on-board laser measurement devices. The second technique employs dynamic stress wave measurement techniques using on-board loading devices and strain gages to track stress wave propagation through the space structure. This approach, which is in its early stages of development, relies on an analysis of transit times of impulsive stress waves and changes in transit times and wave forms due to changes in structural parameters.

Nelson, Richard B.↗

NASA NTRS1986

First- and second-order sensitivity analysis of linear and nonlinear structures

This paper employs the principle of virtual work to derive sensitivity derivatives of structural response with respect to stiffness parameters using both direct and adjoint approaches. The computations required are based on additional load conditions characterized by imposed initial strains, body forces, or surface tractions. As such, they are equally applicable to numerical or analytical solution techniques. The relative efficiency of various approaches for calculating first and second derivatives is assessed. It is shown that for the evaluation of second derivatives the most efficient approach is one that makes use of both the first-order sensitivities and adjoint vectors. Two example problems are used for demonstrating the various approaches.

Haftka, R. T.↗

NASA NTRS1990

Finite element mesh refinement criteria for stress analysis

This paper discusses procedures for finite-element mesh selection and refinement. The objective is to improve accuracy. The procedures are based on (1) the minimization of the stiffness matrix race (optimizing node location); (2) the use of h-version refinement (rezoning, element size reduction, and increasing the number of elements); and (3) the use of p-version refinement (increasing the order of polynomial approximation of the elements). A step-by-step procedure of mesh selection, improvement, and refinement is presented. The criteria for 'goodness' of a mesh are based on strain energy, displacement, and stress values at selected critical points of a structure. An analysis of an aircraft lug problem is presented as an example.

Kittur, Madan G.↗

NASA NTRS1997

A Higher-Order Bending Theory for Laminated Composite and Sandwich Beams

A higher-order bending theory is derived for laminated composite and sandwich beams. This is accomplished by assuming a special form for the axial and transverse displacement expansions. An independent expansion is also assumed for the transverse normal stress. Appropriate shear correction factors based on energy considerations are used to adjust the shear stiffness. A set of transverse normal correction factors is introduced, leading to significant improvements in the transverse normal strain and stress for laminated composite and sandwich beams. A closed-form solution to the cylindrical elasticity solutions for a wide range of beam aspect ratios and commonly used material systems. Accurate shear stresses for a wide range of laminates, including the challenging unsymmetric composite and sandwich laminates, are obtained using an original corrected integration scheme. For application of the theory to a wider range of problems, guidelines for finite element approximations are presented.

Cook, Geoffrey M.↗

NASA NTRS1996

Dynamics and Control of Articulated Anisotropic Timoshenko Beams

The paper illustrates the use of continuum models in control design for stabilizing flexible structures. A 6-DOF anisotropic Timoshenko beam with discrete nodes where lumped masses or actuators are located provides a sufficiently rich model to be of interest for mathematical theory as well as practical application. We develop concepts and tools to help answer engineering questions without having to resort to ad hoc heuristic ("physical") arguments or faith. In this sense the paper is more mathematically oriented than engineering papers and vice versa at the same time. For instance we make precise time-domain solutions using the theory of semigroups of operators rather than formal "inverse Laplace transforms." We show that the modes arise as eigenvalues of the generator of the semigroup, which are then related to the eigenvalues of the stiffness operator. With the feedback control, the modes are no longer orthogonal and the question naturally arises as to whether there is still a modal expansion. Here we prove that the eigenfunctions yield a biorthogonal Riesz basis and indicate the corresponding expansion. We prove mathematically that the number of eigenvalues is nonfinite, based on the theory of zeros of entire functions. We make precise the notion of asymptotic modes and indicate how to calculate them. Although limited by space, we do consider the root locus problem and show for instance that the damping at first increases as the control gain increases but starts to decrease at a critical value, and goes to zero as the gain increases without bound. The undamped oscillatory modes remain oscillatory and the rigid-body modes go over into deadbeat modes. The Timoshenko model dynamics are translated into a canonical wave equation in a Hilbert space. The solution is shown to require the use of an "energy" norm which is no more than the total energy: potential plus kinetic. We show that, under an appropriate extension of the notion of controllability, rate feedback with a collocated sensor can stabilize the structure in the sense that all modes are damped and the energy decays to zero. An example, non-numeric, is worked out in some detail illustrating the concepts and theory developed.

Balakrishnan, A. V.↗

NASA NTRS1999

Turbomachinery Design Quality Checks to Avoid Friction Induced Structural Failure

A unique configuration of the P&W SSME Alternate Fuel Turbopump turbine disk/blade assembly, combined with a severe thermal environment, resulted in several structural anomalies that were driven by frictional contact forces. Understanding the mechanics of these problems provides new quality checks for future turbo machinery designs. During development testing in 1997 of the SSME alternate fuel turbopump at Stennis Space Center, several potentially serious problems surfaced with the turbine disk/blade assembly that had not been experienced in extensive earlier testing. Changes to the operational thermal environment were noted based on analytical prediction of modifications that affected performance and on stationary thermal measurements adjacent to the rotor assembly. A detailed structural investigation was required to reveal the mechanism of distress induced by the change. The turbine disk experienced cracking in several locations due to increased thermal gradient induced stress during start and shutdown transients. This was easily predictable using standard analysis procedures and expected once the thermal environment was characterized. What was not expected was the curling of a piston ring used for blade axial retention in the disk, indentation of the axial face of the blade attachment by a spacer separating the first and second stage blades, and most significantly, galling and cracking of the blade root attachment that could have resulted in blade release. Past experience, in gas turbine environments, set a precedent of never relying on friction for help and to evaluate it only in specific instances where it was obvious that it would degrade capability. In each of the three cases above, friction proved to be a determining factor that pushed the components into an unsatisfactory mode of operation. The higher than expected temperatures and rapid thermal transients combined with friction to move beyond past experience. The turbine disk/blade assembly configuration contributed to the potential for these problems to occur by limiting the radial deflection from thermals and centrifugal loading. The cooled solid bore configuration was chosen to improve rotordynamic stability by limiting the length of rotor overhang while still protecting the roller bearing by maintaining zero slope under the inner race. During a start transient, the rim area of the disk heats rapidly and expands axially and circumferentially and requires corresponding radial and axial growth of the disk to maintain relative positioning of the disk, blades, spacers and retainer rings. The stiffness, large thermal mass, and bore cooling flow combine to severely limit the disk rim radial growth which results in the potential for relative movement between these parts. Friction then becomes a player in the determination of component stress.

Moore, Jerry H.↗

NASA NTRS2013

Experimental and Analytical Characterization of the Macromechanical Response for Triaxial Braided Composite Materials

Increasingly, carbon composite structures are being used in aerospace applications. Their highstrength, high-stiffness, and low-weight properties make them good candidates for replacing many aerospace structures currently made of aluminum or steel. Recently, many of the aircraft engine manufacturers have developed new commercial jet engines that will use composite fan cases. Instead of using traditional composite layup techniques, these new fan cases will use a triaxially braided pattern, which improves case performance. The impact characteristics of composite materials for jet engine fan case applications have been an important research topic because Federal regulations require that an engine case be able to contain a blade and blade fragments during an engine blade-out event. Once the impact characteristics of these triaxial braided materials become known, computer models can be developed to simulate a jet engine blade-out event, thus reducing cost and time in the development of these composite jet engine cases. The two main problems that have arisen in this area of research are that the properties for these materials have not been fully determined and computationally efficient computer models, which incorporate much of the microscale deformation and failure mechanisms, are not available. The research reported herein addresses some of the deficiencies present in previous research regarding these triaxial braided composite materials. The current research develops new techniques to accurately quantify the material properties of the triaxial braided composite materials. New test methods are developed for the polymer resin composite constituent and representative composite coupons. These methods expand previous research by using novel specimen designs along with using a noncontact measuring system that is also capable of identifying and quantifying many of the microscale failure mechanisms present in the materials. Finally, using the data gathered, a new hybrid micromacromechanical computer model is created to simulate the behavior of these composite material systems under static and ballistic impact loading using the test data acquired. The model also quantifies the way in which the fiber/matrix interface affects material response under static and impact loading. The results show that the test methods are capable of accurately quantifying the polymer resin under a variety of strain rates and temperature for three loading conditions. The resin strength and stiffness data show a clear rate and temperature dependence. The data also show the hydrostatic stress effects and hysteresis, all of which can be used by researchers developing composite constitutive models for the resins. The results for the composite data reveal noticeable differences in strength, failure strain, and stiffness in the different material systems presented. The investigations into the microscale failure mechanisms provide information about the nature of the different material system behaviors. Finally, the developed computer model predicts composite static strength and stiffness to within 10 percent of the gathered test data and also agrees with composite impact data, where available.

Littell, Justin D.↗

NASA NTRS1966

Matrix methods and automation in structural engineering

The partial differential equations of motion of shell structures subject to arbitrary loads may be extremely difficult or even impossible to solve. Since the digital computer is now an available tool for the structural engineer, current research has been directed toward methods which involve matrix formulation of large systems of algebraic equations and matrix solutions for discrete elements rather than the solution of the partial differential equations. These methods require rapid and accurate computer solutions. Recognizing the accuracy problems inherent in working with large matrices, a comprehensive survey of available computer programs was performed for large matrix inversion and for eigenvalue and eigenvector solutions under a Research Grant from the National Aeronautics and Space Administration and the techniques are developed in this paper for using these programs most efficiently for structural applications. The contents of the paper include: automation of matrix compilation, methods of very large matrix inversion and solution of simultaneous equation, and techniques for finding eigenvalues and vectors. In addition, a finite element stiffness matrix approach developed at the Denver Research Institute for both plates and shells subject to arbitrary dynamic loads is described as it was instrumented with complete automation on the digital computer.

Anita S West↗

NASA NTRS1985

WRAP-RIB antenna technology development

The wrap-rib deployable antenna concept development is based on a combination of hardware development and testing along with extensive supporting analysis. The proof-of-concept hardware models are large in size so they will address the same basic problems associated with the design fabrication, assembly and test as the full-scale systems which were selected to be 100 meters at the beginning of the program. The hardware evaluation program consists of functional performance tests, design verification tests and analytical model verification tests. Functional testing consists of kinematic deployment, mesh management and verification of mechanical packaging efficiencies. Design verification consists of rib contour precision measurement, rib cross-section variation evaluation, rib materials characterizations and manufacturing imperfections assessment. Analytical model verification and refinement include mesh stiffness measurement, rib static and dynamic testing, mass measurement, and rib cross-section characterization. This concept was considered for a number of potential applications that include mobile communications, VLBI, and aircraft surveillance. In fact, baseline system configurations were developed by JPL, using the appropriate wrap-rib antenna, for all three classes of applications.

Freeland, R. E.↗