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NASA NTRS1957

Wall Interference in a Perforated Wind Tunnel

The theory of cascades, made up of a series of flat plates placed one behind the other, is extended to the case where the impinging stream is not uniform, and the deduced properties of this cascade-flow are then applied to the study of the wall interference between such as cascade-like boundary and a vortex-source type of singularity. It is shown that the induced velocities, produced by the presence of such a wall, are equal to what is obtained by action of a suitably chosen 'reflected' singularity situated on one side of the wall, together with the action of another suitably chosen 'transmitted' singularity placed on the other side. The concepts of a reflection factor and a transmission factor are introduced to characterize various types of tunnel boundary, whether this is composed of solid wall, open and closed sequences, or just a free fluid surface. These ideas are then extended to cover the situation in which a pair of such walls are allowed to coalesce, especially in the event that one of the walls consists of open and closed portions and the other is a fluid surface. This latter particular combination of boundaries is called a perforated wall. Finally, the interference arising from an arbitrary general singularity placed symmetrically in between two such perforated walls is analyzed, and these results are then applied to the determination of the particular kinds of wall geometry that will produce no interference effects in the case of a slender lifting wing and also in the case of a symmetric profile having a finite thickness.

Brescia, Riccardo↗
NASA NTRS2000

3D Higher Order Modeling in the BEM/FEM Hybrid Formulation

Higher order divergence- and curl-conforming bases have been shown to provide significant benefits, in both convergence rate and accuracy, in the 2D hybrid finite element/boundary element formulation (P. Fink and D. Wilton, National Radio Science Meeting, Boulder, CO, Jan. 2000). A critical issue in achieving the potential for accuracy of the approach is the accurate evaluation of all matrix elements. These involve products of high order polynomials and, in some instances, singular Green's functions. In the 2D formulation, the use of a generalized Gaussian quadrature method was found to greatly facilitate the computation and to improve the accuracy of the boundary integral equation self-terms. In this paper, a 3D, hybrid electric field formulation employing higher order bases and higher order elements is presented. The improvements in convergence rate and accuracy, compared to those resulting from lower order modeling, are established. Techniques developed to facilitate the computation of the boundary integral self-terms are also shown to improve the accuracy of these terms. Finally, simple preconditioning techniques are used in conjunction with iterative solution procedures to solve the resulting linear system efficiently. In order to handle the boundary integral singularities in the 3D formulation, the parent element- either a triangle or rectangle-is subdivided into a set of sub-triangles with a common vertex at the singularity. The contribution to the integral from each of the sub-triangles is computed using the Duffy transformation to remove the singularity. This method is shown to greatly facilitate t'pe self-term computation when the bases are of higher order. In addition, the sub-triangles can be further divided to achieve near arbitrary accuracy in the self-term computation. An efficient method for subdividing the parent element is presented. The accuracy obtained using higher order bases is compared to that obtained using lower order bases when the number of unknowns is approximately equal. Also, convergence rates obtained using higher order bases are compared to those obtained with lower order bases for selected sample

Fink, P. W.↗
NASA NTRS2011

Normalization of Gravitational Acceleration Models

Unlike the uniform density spherical shell approximations of Newton, the con- sequence of spaceflight in the real universe is that gravitational fields are sensitive to the nonsphericity of their generating central bodies. The gravitational potential of a nonspherical central body is typically resolved using spherical harmonic approximations. However, attempting to directly calculate the spherical harmonic approximations results in at least two singularities which must be removed in order to generalize the method and solve for any possible orbit, including polar orbits. Three unique algorithms have been developed to eliminate these singularities by Samuel Pines [1], Bill Lear [2], and Robert Gottlieb [3]. This paper documents the methodical normalization of two1 of the three known formulations for singularity-free gravitational acceleration (namely, the Lear [2] and Gottlieb [3] algorithms) and formulates a general method for defining normalization parameters used to generate normalized Legendre Polynomials and ALFs for any algorithm. A treatment of the conventional formulation of the gravitational potential and acceleration is also provided, in addition to a brief overview of the philosophical differences between the three known singularity-free algorithms.

Eckman, Randy A.↗
NASA NTRS2016

Normalization and Implementation of Three Gravitational Acceleration Models

Unlike the uniform density spherical shell approximations of Newton, the consequence of spaceflight in the real universe is that gravitational fields are sensitive to the asphericity of their generating central bodies. The gravitational potential of an aspherical central body is typically resolved using spherical harmonic approximations. However, attempting to directly calculate the spherical harmonic approximations results in at least two singularities that must be removed to generalize the method and solve for any possible orbit, including polar orbits. Samuel Pines, Bill Lear, and Robert Gottlieb developed three unique algorithms to eliminate these singularities. This paper documents the methodical normalization of two of the three known formulations for singularity-free gravitational acceleration (namely, the Lear and Gottlieb algorithms) and formulates a general method for defining normalization parameters used to generate normalized Legendre polynomials and Associated Legendre Functions (ALFs) for any algorithm. A treatment of the conventional formulation of the gravitational potential and acceleration is also provided, in addition to a brief overview of the philosophical differences between the three known singularity-free algorithms.

Eckman, Randy A.↗
NASA NTRS1972

A layered composite with a broken laminate

The problem of a laminate composite in the presence of a crack located normal to the bond lines is considered. Stress analysis of the limiting case when the crack extends to the bond lines is carried out. An integral transform technique is used to formulate the problem in terms of a singular integral equation. The power of stress singularity around the crack tip terminating at the interface is obtained using the above technique. The singular integral equation is solved numerically and the effect of material properties on the stress intensity factor is calculated.

Gupta, G. D.↗
NASA NTRS1972

An integral equation approach to the strip problem

A semi-infinite strip held rigidly on its short end is considered. Loads in the strip at infinity (far away from the fixed end) are prescribed. Integral transform technique is used to provide an exact formulation of the problem in terms of a singular integral equation. Stress singularity at the strip corner is obtained from the singular integral equation which is then solved numerically. Stresses along the rigid end are determined and the effect of the material properties on the stress intensity factor is presented. The method can also be applied to the problem of a laminate composite with a flat inclusion normal to the interfaces.

Gupta, G. D.↗
NASA NTRS1973

The problem of a finite strip compressed between two rough rigid stamps

A finite strip compressed between two rough rigid stamps is considered. The elastostatic problem is formulated in terms of a singular integral equation from which the proper stress singularities at the corners are determined. The singular integral equation is solved numerically to determine the stresses along the fixed ends of the strip. The effect of material properties and strip geometry on the stress intensity factor is presented graphically.

Gupta, G. D.↗
NASA NTRS1973

Interaction between a circular inclusion and an arbitrarily oriented crack

The plane interaction problem for a circular elastic inclusion imbedded into an elastic matrix which contains an arbitrarily oriented crack is considered. Using the existing solutions for the edge dislocations as Green's functions, first the general problem of a through crack in the form of an arbitrary smooth arc located in the matrix in the vicinity of the inclusion is formulated. The integral equations for the line crack are then obtained as a system of singular integral equation with simple Cauchy kernels. The singular behavior of the stresses around the crack tips is examined and the expressions for the stress intensity factors representing the strength of the stress singularities are obtained in terms of the asymptotic values of the density functions of the integral equations. The problem is solved for various typical crack orientations and the corresponding stress intensity factors are given.

Erdogan, F.↗
NASA NTRS1973

2-SPEED, a single-gimbal control moment gyro attitude control system.

In 2-SPEED (Two Scissored Pair Ensemble, Explicit Distribution) four single-gimbal control moment gyros (SGCMGs) configured into two scissored pairs are combined with an explicit distribution of angular momentum between pairs to produce a system relatively insensitive to the singularity problems which have plagued other SGCMG concepts. In this system, the singularity surfaces in momentum space degenerate to discrete curves. Further, the system permits a smooth passage through these remaining singularities with, at worst, a temporary delay while momentum redistribution takes place. Finally, CMG-out operation is possible within the full volume of the reduced momentum envelope.

Crenshaw, J. W.↗
NASA NTRS1973

An integral equation approach to the semi-infinite strip problem.

A semi-infinite strip held rigidly on its short end is considered. Loads in the strip at infinity (far away from the fixed end) are prescribed. The integral transform technique is used to provide an exact formulation of the problem in terms of a singular integral equation. The stress singularity at the strip corner is obtained from the singular integral equation, which is then solved numerically. Stresses along the rigid end are determined, and the effect of the material properties on the stress-intensity factor is presented. The method can also be applied to the problem of a laminate composite with a flat inclusion normal to the interfaces.

Gupta, G. D.↗
NASA NTRS1973

A layered composite with a broken laminate.

The problem of a laminate composite in presence of a crack located normal to the bond lines is considered. Stress analysis of the limiting case when the crack extends to the bond lines is carried out. Integral transform technique is used to formulate the problem in terms of a singular integral equation from which the power of stress singularity around the crack tip terminating at the interface is obtained. The singular integral equation is solved numerically and the effect of material properties on the stress intensity factor is calculated.

Gupta, G. D.↗
NASA NTRS1974

Limitations to the small scale yielding approximation for crack tip plasticity

Recent finite-element results by S. G. Larsson and A. J. Carlsson suggest a limited range of validity to the 'small scale yielding approximation,' whereby small crack tip plastic zones are correlated in terms of the elastic stress intensity factor. It is shown with the help of a model for plane strain yielding that their results may be explained by considering the non-singular stress, acting parallel to the crack at its tip, which accompanies the inverse square-root elastic singularity. Further implications of the non-singular stress term for crack tip deformations and fracturing are examined. It is suggested that its effect on crack tip parameters, such as the opening displacement and J-integral, is less pronounced than its effect on the yield zone size.

Rice, J. R.↗
NASA NTRS1973

Aircraft maneuver optimization of reduced-order approximation

Review of recent work in recasting the energy type of approximation to aircraft flight in terms of singular perturbation theory and in extending this theory to three-dimensional maneuvers. Singular perturbations for differential equations arising in optimal control are first examined for a system of fairly general form but low order. The attitude dynamics for optimal flight of a rocket in vacuum is then studied as an introductory, and fairly transparent, example. The question of the choice of variables is then discussed. Finally, optimal aircraft flight in various reduced-order approximations is investigated. In particular, the problem of three-dimensional aircraft flight is formulated for singular perturbation treatment, and possibilities for decoupling into several lower-order problems are illustrated. The use of reduced-order approximation facilitates numerical computations by reducing the number of multiplier initial values that must be determined simultaneously and by improving the conditioning of the differential equations.

Kelley, H. J.↗
NASA NTRS1975

A steering law for three double-gimbal control moment gyro systems

A steering law for three double-gimbal control moment gyro (CMG) systems is proposed. This steering law is applicable to systems with almost any configuration of CMGs and the CMG-out operation needs no special modification. Examples of three double gimbal CMG systems in an orthogonal configuration and in a parallel configuration are shown along with the results of digital simulations. Simulation results show that any command torque can always be met except when the system is in a singular state and that, whenever the system is in, or close to, a singularity, the steering law drives the system state out of the vicinity of the singularity.

Yoshikawa, T.↗
NASA NTRS1974

A note on a broken layer in an orthotropic laminate composite

An orthotropic laminate composite containing a completely broken layer is considered. The problem is formulated in terms of integral transforms and then reduced to a singular integral equation which is solved numerically. The strength of stress singularity at the crack tip is determined from a characteristic equation which is obtained by studying the dominant part of the singular integral equation near the end points. The stress intensity factors are given for various material properties.

Arin, K.↗
NASA NTRS1975

The scatter of obliquely incident plane waves from a corrugated conducting surface

A physical optics solution is presented for the scattering of place waves from a perfectly conducting corrugated surface in the case of waves incident from an arbitrary direction and for an observer far from the surface. This solution is used to compute the radar cross section of the surface in the case of backscatter from irregular (i. e. , stochastic) corrugations and is used to point out a correction to the literature on this problem. An interesting feature of the solution is the occurrence of singularities in the scattered fields. These singularities appear to be a manifestation of focussing by the surface at its "stationary" points. Whether or not the singularities occur in the solution depends on the manner in which one restricts the analysis to the far field.

David M Le Vine↗
NASA NTRS1974

Mixed mode stress field effect in adhesive fracture

Numerical or analytical analyses were performed on seven different test specimens including blister test, 90-degree peel test, torsion test, and various cone tests. These specimens are in general subjected to complex stress fields having various amounts of Mode I, Mode II, and Mode III loads. The specimens were then constructed using polymethyl methacrylate for the adherends and a transparent polyurethane elastomer (Solithane 113) for the adhesive. This combination permitted direct observation of the bondline as load was applied. Although initial debonds as well as bond end termination singularities were present in all specimens, in some cases the debond did not initiate at the singularity points as would normally have been expected. An explanation for this behavior is presented, as well as a comparison of loading mode effect on those specimens for which the debond did propagate from a bond terminus singular point.

Anderson, G. P.↗
NASA NTRS1975

Interaction between a circular inclusion and an arbitrarily oriented crack

The plane interaction problem for a circular elastic inclusion embedded in an elastic matrix which contains an arbitrarily oriented crack is considered. Using the existing solutions for the edge dislocations as Green's functions, first the general problem of a through crack in the form of an arbitrary smooth arc located in the matrix in the vicinity of the inclusion is formulated. The integral equations for the line crack are then obtained as a system of singular integral equations with simple Cauchy kernels. The singular behavior of the stresses around the crack tips is examined and the expressions for the stress-intensity factors representing the strength of the stress singularities are obtained in terms of the asymptotic values of the density functions of the integral equations. The problem is solved for various typical crack orientations and the corresponding stress-intensity factors are given.

Erdogan, F.↗