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NASA NTRS1986

Plates and shells containing a surface crack under general loading conditions

The severity of the underlying assumptions of the line-spring model (LSM) are such that verification with three-dimensional solutions is necessary. Such comparisons show that the model is quite accurate, and therefore, its use in extensive parameter studies is justified. Investigations into the endpoint behavior of the line-spring model have led to important conclusions about the ability of the model to predict stresses in front of the crack tip. An important application of the LSM was to solve the contact plate bending problem. Here the flexibility of the model to allow for any crack shape is exploited. The use of displacement quantities as unknowns in the formulation of the problem leads to strongly singular integral equations, rather than singular integral equations which result from using displacement derivatives. The collocation method of solving the integral equations was found to be better and more convenient than the quadrature technique. Orthogonal polynomials should be used as fitting functions when using the LSM as opposed to simpler functions such as power series.

Joseph, Paul F.↗

NASA NTRS1987

A hybrid perturbation Galerkin technique with applications to slender body theory

A two step hybrid perturbation-Galerkin method to solve a variety of applied mathematics problems which involve a small parameter is presented. The method consists of: (1) the use of a regular or singular perturbation method to determine the asymptotic expansion of the solution in terms of the small parameter; (2) construction of an approximate solution in the form of a sum of the perturbation coefficient functions multiplied by (unknown) amplitudes (gauge functions); and (3) the use of the classical Bubnov-Galerkin method to determine these amplitudes. This hybrid method has the potential of overcoming some of the drawbacks of the perturbation method and the Bubnov-Galerkin method when they are applied by themselves, while combining some of the good features of both. The proposed method is applied to some singular perturbation problems in slender body theory. The results obtained from the hybrid method are compared with approximate solutions obtained by other methods, and the degree of applicability of the hybrid method to broader problem areas is discussed.

Geer, James F.↗

NASA NTRS1987

An eigensystem realization algorithm using data correlations (ERA/DC) for modal parameter identification

A modification to the Eigensystem Realization Algorithm (ERA) for modal parameter identification is presented in this paper. The ERA minimum order realization approach using singular value decomposition is combined with the philosophy of the Correlation Fit method in state space form such that response data correlations rather than actual response values are used for modal parameter identification. This new method, the ERA using data correlations (ERA/DC), reduces bias errors due to noise corruption significantly without the need for model overspecification. This method is tested using simulated five-degree-of-freedom system responses corrupted by measurement noise. It is found for this case that, when model overspecification is permitted and a minimum order solution obtained via singular value truncation, the results from the two methods are of similar quality.

Juang, Jer-Nan↗

NASA NTRS1987

Viscous damping of Alfven normal modes in non-uniform plasmas

The Alfven normal mode in a nonuniform, viscous plasma is investigated. Because the ideal hydromagnetic equation is singular in a nonuniform magnetic field, viscosity is included in order to regularize the equation, analogous to the removal of the singular point by resistivity (Mok and Einaudi, 1985). The eigenvalue equation is then solved numerically for a particular magnetic configuration. The real part of the frequency of this viscous normal mode is found to be similar to the one in the resistive case, while the damping is shown to be comparable, and sometimes to exceed, the resistive effect under certain conditions in which the Lundquist number is sufficiently large. The damping rate is evaluated for various plasma conditions corresponding to different parts of the solar atmosphere. The effects of viscosity are found to dominate resistivity in the quiet sun corona and solar wind.

Mok, Y.↗

NASA NTRS1987

Perturbation analysis of internal balancing for lightly damped mechanical systems with gyroscopic and circulatory forces

Approximate expressions are developed for internally balanced singular values corresponding to the modes of mechanical systems with gyroscopic forces, light damping, and small circulatory forces. A brief overview is first given of the balanced realization model reduction method, including a discussion of recent work. The models considered are defined, and a perturbation analysis is used to show that the modal representation becomes asymptotically balanced as damping reduces to zero. The approximate balanced singular values are calculated, and a simple example of a flexible, dual-spin spacecraft is given as an illustration of the results.

Blelloch, P. A.↗

NASA NTRS1987

A new approach to the effect of sound on vortex dynamics

Analytical results are presented on the effect of acoustic radiation on three-dimensional vortex motions in a homogeneous, slightly compressible, inviscid fluid. The flow is considered as linear and irrotational everywhere except inside a very thin cylindrical core region around the vortex filament. In the outside region, a velocity potential is introduced that must be multivalued, and it is shown how to compute this scalar potential if the motion of the vortex filament is prescribed. To find the motion of this singularity in an external potential flow, a variational principle involving a volume integral that must exclude the singular region is considered. A functional of the external potential and vortex filament position is obtained whose extrema give equations to determine the sought-after evolution. Thus, a generalization of the Biot-Savart law to flows with constant sound speed at low Mach number is obtained.

Lund, Fernando↗

NASA NTRS1986

Finite-volume scheme for transonic potential flow about airfoils and bodies in an arbitrarily shaped channel

A conservative finite-volume difference scheme is developed for the potential equation to solve transonic flow about airfoils and bodies in an arbitrarily shaped channel. The scheme employs a mesh which is a nearly conformal O mesh about the airfoil and nearly orthogonal at the channel walls. The mesh extends to infinity upstream and downstream, where the mapping is singular. Special procedures are required to treat the singularities at infinity, including computation of the metrics near those points. Channels with exit areas different from inlet areas are solved; a body with a sting mount is an example of such a case.

South, Jerry C., Jr.↗

NASA NTRS1988

Numerical solution methods for viscoelastic orthotropic materials

Numerical solution methods for viscoelastic orthotropic materials, specifically fiber reinforced composite materials, are examined. The methods include classical lamination theory using time increments, direction solution of the Volterra Integral, Zienkiewicz's linear Prony series method, and a new method called Nonlinear Differential Equation Method (NDEM) which uses a nonlinear Prony series. The criteria used for comparison of the various methods include the stability of the solution technique, time step size stability, computer solution time length, and computer memory storage. The Volterra Integral allowed the implementation of higher order solution techniques but had difficulties solving singular and weakly singular compliance function. The Zienkiewicz solution technique, which requires the viscoelastic response to be modeled by a Prony series, works well for linear viscoelastic isotropic materials and small time steps. The new method, NDEM, uses a modified Prony series which allows nonlinear stress effects to be included and can be used with orthotropic nonlinear viscoelastic materials. The NDEM technique is shown to be accurate and stable for both linear and nonlinear conditions with minimal computer time.

Gramoll, K. C.↗

NASA NTRS1988

Advances in dual algorithms and convex approximation methods

A new algorithm for solving the duals of separable convex optimization problems is presented. The algorithm is based on an active set strategy in conjunction with a variable metric method. This first order algorithm is more reliable than Newton's method used in DUAL-2 because it does not break down when the Hessian matrix becomes singular or nearly singular. A perturbation technique is introduced in order to remove the nondifferentiability of the dual function which arises when linear constraints are present in the approximate problem.

Smaoui, H.↗

NASA NTRS1987

Damage growth in composite laminates with interleaves

The influence of interleaves on the fracture behavior of laminated composites is investigated. The geometry of the composite consists of a cracked layer, bonded between two half-planes, which is separated by thin, low modulus, interleaves. The interleaves are modeled as uncoupled tension and shear springs. A stress analysis is carried out for a crack which may grow up to and spread along the interface. Integral transform techniques are used to develop the solution in terms of singular integral equations. An asymptotic analysis reveals logarithmic singularities in the critical stresses for the case of a broken layer. The equations are solved numerically and the effect of damage on critical stresses is studied. Initial results indicate that the interleaf gives a significant reduction in stresses in the adjacent regions and, if properly selected, should result in a more damage tolerant laminate.

Goree, James G.↗

NASA NTRS1988

Convergence of strain energy release rate components for edge-delaminated composite laminates

Strain energy release rates for edge delaminated composite laminates were obtained using quasi 3 dimensional finite element analysis. The problem of edge delamination at the -35/90 interfaces of an 8-ply composite laminate subjected to uniform axial strain was studied. The individual components of the strain energy release rates did not show convergence as the delamination tip elements were made smaller. In contrast, the total strain energy release rate converged and remained unchanged as the delamination tip elements were made smaller and agreed with that calculated using a classical laminated plate theory. The studies of the near field solutions for a delamination at an interface between two dissimilar isotropic or orthotropic plates showed that the imaginary part of the singularity is the cause of the nonconvergent behavior of the individual components. To evaluate the accuracy of the results, an 8-ply laminate with the delamination modeled in a thin resin layer, that exists between the -35 and 90 plies, was analyzed. Because the delamination exists in a homogeneous isotropic material, the oscillatory component of the singularity vanishes.

Raju, I. S.↗

NASA NTRS1988

A near optimal guidance algorithm for aero-assisted orbit transfer

The paper presents a near optimal guidance algorithm for aero-assited orbit plane change, based on minimizing the energy loss during the atmospheric portion of the maneuver. The guidance algorithm makes use of recent results obtained from energy state approximations and singular perturbation analysis of optimal heading change for a hypersonic gliding vehicle. This earlier work ignored the terminal constraint on altitude needed to insure that the vehicle exits that atmosphere. Thus, the resulting guidance algorithm was only appropriate for maneuvering reentry vehicle guidance. In the context of singular perturbation theory, a constraint on final altitude gives rise to a difficult terminal boundary layer problem, which cannot be solved in closed form. This paper will demonstrate the near optimality of a predictive/corrective guidance algorithm for the terminal maneuver. Comparisons are made to numerically optimized trajectories for a range or orbit plane angles.

Calise, Anthony J.↗

NASA NTRS1988

Maneuvering strategies using CMGs

This paper considers control strategies for maneuvering spacecraft using Single-Gimbal Control Momentum Gyros (CMGs). A pyramid configuration using four gyros is utilized. Preferred initial gimbal angles for maximum utilization of CMG momentum are obtained for some known torque commands. Feedback control laws are derived from the stability point of view by using the Liapunov's Second Theorem. The gyro rates are obtained by the pseudo-inverse technique. The effect of gimbal rate bounds on controllability are studied for an example maneuver. Singularity avoidance is based on limiting the gyro rates depending on a singularity index.

Oh, H. S.↗

NASA NTRS1989

Supercritical flow past a symmetrical bicircular arc airfoil

A numerical scheme is developed for computing steady supercritical flow about symmetrical airfoils, applying it to an ellipse for zero angle of attack. An algorithmic description of this new scheme is presented. Application to a symmetrical bicircular arc airfoil is also proposed. The flow field before the shock is region 1. For transonic flow, singularity can be avoided by integrating the resulting ordinary differential equations away from the body. Region 2 contains the shock which will be located by shock fitting techniques. The shock divides region 2 into supersonic and subsonic regions and there is no singularity problem in this case. The Method of Lines is used in this region and it is advantageous to integrate the resulting ordinary differential equation along the body for shock fitting. Coaxial coordinates have to be used for the bicircular arc airfoil so that boundary values on the airfoil body can be taken with one direction of the coaxial coordinates fixed. To avoid taking boundary values at + or - infinity in the coaxial co-ordinary system, approximate analytical representation of the flow field near the tips of the airfoil is proposed.

Holt, Maurice↗

NASA NTRS1989

On the freestream matching condition for stagnation point turbulent flows

The problem of plane stagnation point flow with freestream turbulence is examined from a basic theoretical standpoint. It is argued that the singularity which arises from the standard kappa-epsilon model is not due to a defect in the model but results from the use of an inconsistent freestream boundary condition. The inconsistency lies in the implementation of a production equals dissipation equilibrium hypothesis in conjunction with a freestream mean velocity field that corresponds to homogeneous plane strain - a turbulent flow which does not reach such a simple equilibrium. Consequently, the adjustment that has been made in the constants of the epsilon-transport equation to eliminate this singularity is not self-consistent since it is tantamount to artificially imposing an equilibrium structure on a turbulent flow which is known not to have one.

Speziale, Charles G.↗

NASA NTRS1989

Integrated tools for control-system analysis

The basic functions embedded within a user friendly software package (MATRIXx) are used to provide a high level systems approach to the analysis of linear control systems. Various control system analysis configurations are assembled automatically to minimize the amount of work by the user. Interactive decision making is incorporated via menu options and at selected points, such as in the plotting section, by inputting data. There are five evaluations such as the singular value robustness test, singular value loop transfer frequency response, Bode frequency response, steady-state covariance analysis, and closed-loop eigenvalues. Another section describes time response simulations. A time response for random white noise disturbance is available. The configurations and key equations used for each type of analysis, the restrictions that apply, the type of data required, and an example problem are described. One approach for integrating the design and analysis tools is also presented.

Ostroff, Aaron J.↗

NASA NTRS1989

Implementation of equivalent domain integral method in the two-dimensional analysis of mixed mode problems

An equivalent domain integral (EDI) method for calculating J-intergrals for two-dimensional cracked elastic bodies is presented. The details of the method and its implementation are presented for isoparametric elements. The total and product integrals consist of the sum of an area of domain integral and line integrals on the crack faces. The line integrals vanish only when the crack faces are traction free and the loading is either pure mode 1 or pure mode 2 or a combination of both with only the square-root singular term in the stress field. The EDI method gave accurate values of the J-integrals for two mode I and two mixed mode problems. Numerical studies showed that domains consisting of one layer of elements are sufficient to obtain accurate J-integral values. Two procedures for separating the individual modes from the domain integrals are presented. The procedure that uses the symmetric and antisymmetric components of the stress and displacement fields to calculate the individual modes gave accurate values of the integrals for all problems analyzed. The EDI method when applied to a problem of an interface crack in two different materials showed that the mode 1 and mode 2 components are domain dependent while the total integral is not. This behavior is caused by the presence of the oscillatory part of the singularity in bimaterial crack problems. The EDI method, thus, shows behavior similar to the virtual crack closure method for bimaterial problems.

Raju, I. S.↗

NASA NTRS1989

Robust computer-aided synthesis and optimization of linear multivariable control systems with varying plant dynamics via AUTOCON

AUTOCON is an automated computer-aided design tool for the synthesis and optimization of linear multivariable control systems based upon user-defined control parameter optimization. Violations in stability and performance requirements are computed from constraints on Single Input/Single Output (SISO) open- and closed-loop transfer function frequency responses, and from constraints on the singular-value frequency responses of Multiple Input/Multiple Output (MIMO) transfer functions, for all critical plant variations. Optimum nonlinear programming algorithms are used in the search for local constrained solutions in which violations in stability and performance are caused either to vanish or be minimized for a proper selection of the control parameters. Classical control system stability and performance design can, in this way, be combined with modern multivariable robustness methods to offer general frequency response loop-shaping via a computer-aided design tool. Complete Nichols, Nyquist, Bode, singular-value Bode magnitude and transient response plots are produced, including user-defined boundary responses. AUTOCON is used to synthesize and optimize the lateral/directional flight control system for a typical high-performance aircraft.

Lefkowitz, C. P.↗