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NASA NTRS1990

Dynamic substructuring by the boundary flexibility vector method of component mode synthesis

Component mode synthesis (CMS) is a method of dynamic analysis, for structures having a large number of degrees of freedom (DOF). These structures often required lengthy computer CPU time and large computer memory resources, if solved directly by the finite-element method (FEM). In CMS, the structure is divided into independent components in which the DOF are defined by a set of generalized coordinates defined by displacement shapes. The number of the generalized coordinates are much less than the original number of physical DOF, in the component. The displacement shapes are used to transform the component property matrices and any applied external loads, to a reduced system of coordinates. Reduced system property matrices are assembled, and any type of dynamic analysis is carried out in the reduced coordinate system. Any obtained results are back transformed to the original component coordinate systems. In all conventional methods of CMS, the mode shapes used for components are dynamic mode shapes, supplemented by static deflected shapes. Historically, all the dynamic mode shapes used in conventional CMS are the natural modes (eigenvectors) of components. A new method of CMS, namely the boundary flexibility vector method of CMS, is presented. The method provides for the incorporation of a set of static Ritz vectors, referred to as boundary flexibility vectors, as a replacement and/or supplement to conventional eigenvectors, as displacement shapes for components. The generation of these vectors does not require the solution of a costly eigenvalue problem, as in the case of natural modes in conventional CMS, and hence a substantial saving in CPU time can be achieved. The boundary flexibility vectors are generated from flexibility (or stiffness) properties of components. The formulation presented is for both free and fixed-interface components, and for both the free and forced vibration problems. Free and forced vibration numerical examples are presented to verify the accuracy of the method and the saving in CPU time. Compared to conventional methods of CMS, the results indicate that by using the new method, more accurate results can be obtained with a substantial saving in CPU time.

Abdallah, Ayman Ahmed↗

NASA NTRS1990

Magnetic forces in high-T(sub c) superconducting bearings

In September 1987 research at Cornell levitated a small rotor on superconducting bearing at 10,000 rpm. In April 1989 a speed of 120,000 rpm was achieved in a passive bearing with no active control. The bearing material used was YBa2Cu3O7. There is no evidence that the rotation speed has any significant effect on the lift force. Magnetic force measurements between a permanent rare-earth magnet and high T(sub c) superconducting material versus vertical and lateral displacements were made. A large hysteresis loop results for large displacements, while minor loops result for small displacements. These minor loops seem to give a slope proportional to the magnetic stiffness, and are probably indicative of flux pinning forces. Experiments of rotary speed versus time show a linear decay in a vacuum. Measurements of magnetic drag forces of a magnetic dipole over a high-T(sub c) superconducting disc of YBCO show that the drag force reaches a constant value, independent of the speed. Dampling of lateral vibrations of levitated rotors were measured which indicates that transverse flux motion in the superconductor will create dissipation. As a result of these force measurements, an optimum shape for the superconductor bearing pads which gives good lateral and axial stability was designed. Recent force measurements on melt-quench processed superconductors indicate a substantial increase in levitation force and magnetic stiffness over free sintered materials. As a result, application of high-T(sub c) superconducting bearings are beginning to show great promise at this time.

Moon, F. C.↗

NASA NTRS1990

Postbuckling response of long thick isotropic plates loaded in compression including higher order transverse shearing effects

Buckling and postbuckling results for aluminum plates loaded in compression are presented. The buckling results were plotted to show the effects of thickness on the stress coefficient. Buckling results are given for various length-to-width ratios. Postbuckling results for plates with transverse shearing flexibility are compared to results from classical theory for various width-to-thickness ratios. The plates are considered to be long with side edges simply supported, with edges free of stress and the plates are subjected to longitudinal compressive displacement. Characteristic curves indicating the average longitudinal direct stress resultant as a function of the applied displacements are calculated based on four different theories: Classical von Karman, first-order shear deformation, higher-order shear deformation, and three-dimensional flexibility. Present results indicate that the three-dimensional flexibility theory gives the lowest and therefore, most accurate results. The higher-order shear deformation theory has fewer unknowns than the three-dimensional flexibility but is not as accurate. The figures presented show that small differences occur in the maximum stress resultants and the transverse displacements calculated when the effects of transverse shear are included.

Stein, Manuel↗

NASA NTRS1990

A comparison of single-cycle versus multiple-cycle proof testing strategies

An evaluation of single-cycle and multiple-cycle proof testing (MCPT) strategies for SSME components is described. Data for initial sizes and shapes of actual SSME hardware defects are analyzed statistically. Closed-form estimates of the J-integral for surface flaws are derived with a modified reference stress method. The results of load- and displacement-controlled stable crack growth tests on thin IN-718 plates with deep surface flaws are summarized. A J-resistance curve for the surface-cracked configuration is developed and compared with data from thick compact tension specimens. The potential for further crack growth during large unload/reload cycles is discussed, highlighting conflicting data in the literature. A simple model for ductile crack growth during MCPT based on the J-resistance curve is used to study the potential effects of key variables. The projected changes in the crack size distribution during MCPT depend on the interactions between several key parameters, including the number of proof cycles, the nature of the resistance curve, the initial crack size distribution, the component boundary conditions (load vs. displacement control), and the magnitude of the applied load or displacement. The relative advantages of single-cycle and multiple-cycle proof testing appear to be specific, therefore, to individual component geometry, material, and loading.

Hudak, S. J., Jr.↗

NASA NTRS1989

Automated generation of influence functions for planar crack problems

A numerical procedure for the generation of influence functions for Mode I planar problems is described. The resulting influence functions are in a form for convenient evaluation of stress-intensity factors for complex stress distributions. Crack surface displacements are obtained by a least-squares solution of the Williams eigenfunction expansion for displacements in a cracked body. Discrete values of the influence function, evaluated using the crack surface displacements, are curve fit using an assumed functional form. The assumed functional form includes appropriate limit-behavior terms for very deep and very shallow cracks. Continuous representation of the influence function provides a convenient means for evaluating stress-intensity factors for arbitrary stress distributions by numerical integration. The procedure is demonstrated for an edge-cracked strip and a radially cracked disk. Comparisons with available published results demonstrate the accuracy of the procedure.

Sire, Robert A.↗

NASA NTRS1990

A variational justification of the assumed natural strain formulation of finite elements. I - Variational principles. II - The C(0) four-node plate element

The assumed natural strain formulation of finite elements is interpreted from a variational standpoint. The approach is based on hybrid extensions of the Reissner-type functional which uses the strains and displacements as independent fields. Consideration is restricted to linear elasticity. The four-node C(0) plate-bending quadrilateral is used as a specific example to illustrate the application of the present interpretation. A key finding is that any change in the strain-displacement interpolation from the variationally consistent interpolation must be associated in some way to the addition of incompatible displacement modes.

Militello, Carmelo↗

NASA NTRS1990

Lateral restoring force on a magnet levitated above a superconductor

The lateral restoring force on a magnet levitated above a superconductor is calculated as a function of displacement from its original position at rest using Bean's critical-state model to describe flux pinning. The force is linear for small displacements and saturates at large displacements. In the absence of edge effects the force always attracts the magnet to its original position. Thus it is a restoring force that contributes to the stability of the levitated magnet. In the case of a thick superconductor slab, the origin of the force is a magnetic dipole layer consisting of positive and negative supercurrents induced on the trailing side of the magnet. The qualitative behavior is consistent with experiments reported to date. Effects due to the finite thickness of the superconductor slab and the granular nature of high-Tc materials are also considered.

Davis, L. C.↗

NASA NTRS1990

Improved approximations for control augmented structural synthesis

A methodology for control-augmented structural synthesis is presented for structure-control systems which can be modeled as an assemblage of beam, truss, and nonstructural mass elements augmented by a noncollocated direct output feedback control system. Truss areas, beam cross sectional dimensions, nonstructural masses and rotary inertias, and controller position and velocity gains are treated simultaneously as design variables. The structural mass and a control-system performance index can be minimized simultaneously, with design constraints placed on static stresses and displacements, dynamic harmonic displacements and forces, structural frequencies, and closed-loop eigenvalues and damping ratios. Intermediate design-variable and response-quantity concepts are used to generate new approximations for displacements and actuator forces under harmonic dynamic loads and for system complex eigenvalues. This improves the overall efficiency of the procedure by reducing the number of complete analyses required for convergence. Numerical results which illustrate the effectiveness of the method are given.

Thomas, H. L.↗

NASA NTRS1990

Nonlinear analysis of compressively loaded linked-plate structures

A method is presented for the geometrically nonlinear analysis of the equilibrium behavior of a general class of compressively loaded, prismatic, linked-plate structures. The semi-analytical method accounts for geometric shape imperfections, and allows the use of laminated composite plate properties. Buckling eigenfunctions are used as displacement shape functions, and the associated second-order displacement fields are incorporated. The nonlinear plate equilibrium equations are expressed in terms of the assumed displacement form, and the Galerkin method is used to obtain a set of algebraic equations which are nonlinear in the modal amplitudes and the load parameter. Buckling eigensolutions for the class of structures under consideration can be obtained from the VIPASA computer code; thus, subject to some additional restrictions in loading and plate anisotropy, structural configurations which can be analyzed with regard to buckling response by VIPASA can be analyzed with regard to geometrically nonlinear response using the new method.

Stoll, Frederick↗

NASA NTRS1990

An optical technique to measure ion engine grid distortion due to differential thermal expansion

This paper describes an optical technique developed for measuring small differential grid displacements due to thermal expansion of an ion thruster accelerator system. The technique is based on confocal scanning optical microscope type II. For the measurements of small displacements where there are distances on the order of a meter or more between the lens plane and the sample, some of the optical components are moved while the sample is kept fixed. The feasibility of applying this technique to measure the thermally induced ion thruster grid displacements was demonstrated in a bench-top simulation. It is noted that this technique can also provide information on grid movement resulting from thermal transients such as the start-up.

Trava-Airoldi, V. J.↗

NASA NTRS1990

Status of structural analysis of 30 cm diameter ion optics

Three structural finite element programs are compared with theory, experimental data, and each other to evaluate their usefulness for modeling the thermomechanical deflection of ion engine electrodes. Two programs, NASTRAN and MARC, used a Cray XMP and the third, Algor, used an IBM compatible personal computer. The shape of the applied temperature gradient greatly affects off-axis displacement, implying that an accurate temperature distribution is required to analyze new designs. The use of bulk material constants to model the perforated electrodes was investigated. The stress and displacement predictions are shown to be sensitive to the temperature gradient and the Young's modulus, and insensitive to number of nodes, above some minimum value, and the Poisson ratio used. The models are shown to be useful tools for evaluating designs. Experimental measurement of temperatures and displacements was identified as the most critical area for further work.

Macrae, Gregory S.↗

NASA NTRS1990

Status of structural analysis of 30 cm diameter ion optics

Three structural finite element programs are compared with theory, experimental data, and each other to evaluate their usefulness for modeling the thermomechanical deflection of ion engine electrodes. Two programs, NASTRAN and MARC, used a Cray XMP and the third, Algor, used an IBM compatible personal computer. The shape of the applied temperature gradient greatly affects off-axis displacement, implying that an accurate temperature distribution is required to analyze new designs. The use of bulk material constants to model the perforated electrodes was investigated. The stress and displacement predictions are shown to be sensitive to the temperature gradient and the Young's modulus, and insensitive to number of nodes, above some minimum value, and the Poisson ratio used. The models are shown to be useful tools for evaluating designs. Experimental measurements of temperatures and displacements was identified as the most critical area.

Macrae, Gregory S.↗

NASA NTRS1990

Periodic trim solutions with hp-version finite elements in time

Finite elements in time as an alternative strategy for rotorcraft trim problems are studied. The research treats linear flap and linearized flap-lag response both for quasi-trim and trim cases. The connection between Fourier series analysis and hp-finite elements for periodic a problem is also examined. It is proved that Fourier series is a special case of space-time finite elements in which one element is used with a strong displacement formulation. Comparisons are made with respect to accuracy among Fourier analysis, displacement methods, and mixed methods over a variety parameters. The hp trade-off is studied for the periodic trim problem to provide an optimum step size and order of polynomial for a given error criteria. It is found that finite elements in time can outperform Fourier analysis for periodic problems, and for some given error criteria. The mixed method provides better results than does the displacement method.

Peters, David A.↗

NASA NTRS1989

GPS measurements of strain accumulation across the Imperial Valley, California: 1986-1989

The Global Positioning System (GPS) data collected in southern California from 1986 to 1989 indicate considerable strain accumulation across the Imperial Valley. Displacements are computed at 29 stations in and near the valley from 1986 to 1988, and at 11 sites from 1988 to 1989. The earlier measurements indicate 5.9 +/- 1.0 cm/yr right-lateral differential velocity across the valley, although the data are heavily influenced by the 1987 Superstition Hills earthquake sequence. Some measurements, especially the east-trending displacements, are suspects for large errors. The 1988 to 1989 GPS displacements are best modeled by 5.2 +/- 0.9 cm/yr of valley crossing deformation, but rates calculated from conventional geodetic measurements (3.4 to 4.3 cm/yr) fit the data nearly as well. There is evidence from GPS and Very Long Base Interferometry (VLBI) observations that the present slip rate along the southern San Andreas fault is smaller than the long-term geologic estimate, suggesting a lower earthquake potential than is currently assumed. Correspondingly, a higher earthquake potential is indicated for the San Jacinto fault. The Imperial Valley GPS sites form part of a 183 station network in southern California and northern Baja California, which spans a cross-section of the North American-Pacific plate boundary.

Larsen, Shawn↗

NASA NTRS1990

The method of lines in analyzing solids containing cracks

A semi-numerical method is reviewed for solving a set of coupled partial differential equations subject to mixed and possibly coupled boundary conditions. The line method of analysis is applied to the Navier-Cauchy equations of elastic and elastoplastic equilibrium to calculate the displacement distributions in various, simple geometry bodies containing cracks. The application of this method to the appropriate field equations leads to coupled sets of simultaneous ordinary differential equations whose solutions are obtained along sets of lines in a discretized region. When decoupling of the equations and their boundary conditions is not possible, the use of a successive approximation procedure permits the analytical solution of the resulting ordinary differential equations. The use of this method is illustrated by reviewing and presenting selected solutions of mixed boundary value problems in three dimensional fracture mechanics. These solutions are of great importance in fracture toughness testing, where accurate stress and displacement distributions are required for the calculation of certain fracture parameters. Computations obtained for typical flawed specimens include that for elastic as well as elastoplastic response. Problems in both Cartesian and cylindrical coordinate systems are included. Results are summarized for a finite geometry rectangular bar with a central through-the-thickness or rectangular surface crack under remote uniaxial tension. In addition, stress and displacement distributions are reviewed for finite circular bars with embedded penny-shaped cracks, and rods with external annular or ring cracks under opening mode tension. The results obtained show that the method of lines presents a systematic approach to the solution of some three-dimensional mechanics problems with arbitrary boundary conditions. The advantage of this method over other numerical solutions is that good results are obtained even from the use of a relatively coarse grid.

Gyekenyesi, John P.↗

NASA NTRS1991

A study of the response of nonlinear springs

The various phases to developing a methodology for studying the response of a spring-reinforced arch subjected to a point load are discussed. The arch is simply supported at its ends with both the spring and the point load assumed to be at midspan. The spring is present to off-set the typical snap through behavior normally associated with arches, and to provide a structure that responds with constant resistance over a finite displacement. The various phases discussed consist of the following: (1) development of the closed-form solution for the shallow arch case; (2) development of a finite difference analysis to study (shallow) arches; and (3) development of a finite element analysis for studying more general shallow and nonshallow arches. The two numerical analyses rely on a continuation scheme to move the solution past limit points, and to move onto bifurcated paths, both characteristics being common to the arch problem. An eigenvalue method is used for a continuation scheme. The finite difference analysis is based on a mixed formulation (force and displacement variables) of the governing equations. The governing equations for the mixed formulation are in first order form, making the finite difference implementation convenient. However, the mixed formulation is not well-suited for the eigenvalue continuation scheme. This provided the motivation for the displacement based finite element analysis. Both the finite difference and the finite element analyses are compared with the closed form shallow arch solution. Agreement is excellent, except for the potential problems with the finite difference analysis and the continuation scheme. Agreement between the finite element analysis and another investigator's numerical analysis for deep arches is also good.

Hyer, M. W.↗

NASA NTRS1991

Weak Hamiltonian finite element method for optimal control problems

A temporal finite element method based on a mixed form of the Hamiltonian weak principle is developed for dynamics and optimal control problems. The mixed form of Hamilton's weak principle contains both displacements and momenta as primary variables that are expanded in terms of nodal values and simple polynomial shape functions. Unlike other forms of Hamilton's principle, however, time derivatives of the momenta and displacements do not appear therein; instead, only the virtual momenta and virtual displacements are differentiated with respect to time. Based on the duality that is observed to exist between the mixed form of Hamilton's weak principle and variational principles governing classical optimal control problems, a temporal finite element formulation of the latter can be developed in a rather straightforward manner. Several well-known problems in dynamics and optimal control are illustrated. The example dynamics problem involves a time-marching problem. As optimal control examples, elementary trajectory optimization problems are treated.

Hodges, Dewey H.↗

NASA NTRS1990

A weak Hamiltonian finite element method for optimal control problems

Hodges, Dewey H.↗