Thrusting trajectory minimization program for orbital transfer maneuvers
Computer program for transfer orbit trajectory optimization based on Lagrange multipliers, Newton-Raphson method, and power series
Green, R. N.↗
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Analysis of an arched outer-race ball bearing considering centrifugal forces
A Newton-Raphson method of iteration was used in evaluating the radial and axial projection of the distance between the ball center and the outer raceway groove curvature center (V and W). Fatigue life evaluations were made. The similar analysis of a conventional bearing can be directly obtained from the arched bearing analysis by simply letting the amount of arching be zero (g = 0) and not considering equations related to the unloaded half of the outer race. The analysis was applied to a 150-mm angular contact ball bearing. Results for life, contact loads, and angles are shown for a conventional bearing (g = 0) and two arched bearings (g = 0.127 mm (0.005 in.), and 0.254 mm (0.010 in.)). The results indicate that an arched bearing is highly desirable for high speed applications. In particular, for a DN value of 3 million (20,000 rpm) and an applied axial load of 4448 N (1000 lb), an arched bearing shows an improvement in life of 306 percent over that of a conventional bearing. At 4.2 million DN (28,000 rpm), the corresponding improvement is 340 percent. It was also found for low speeds, the arched bearing does not offer the advantages that it does for high speed applications.
Hamrock, B. J.↗
Analysis of an arched outer-race ball bearing considering centrifugal forces.
Thrust-load analysis of a 150-mm angular contact ball bearing, taking into account centrifugal forces but omitting gyroscopics, elastohydrodynamics, and thermal effects. A Newton-Raphson method of iteration is used to evaluate the radial and axial projection of the distance between the ball center and the outer raceway groove curvature center. Fatigue life of the bearing is evaluated. Results for life, contact loads, and angles are given for a conventional bearing and two arched bearings.
Hamrock, B. J.↗
Thermochemical nonequilibrium in atomic hydrogen at elevated temperatures
A numerical study of the nonequilibrium flow of atomic hydrogen in a cascade arc was performed to obtain insight into the physics of the hydrogen cascade arc. A rigorous mathematical model of the flow problem was formulated, incorporating the important nonequilibrium transport phenomena and atomic processes which occur in atomic hydrogen. Realistic boundary conditions, including consideration of the wall electrostatic sheath phenomenon, were included in the model. The governing equations of the asymptotic region of the cascade arc were obtained by writing conservation of mass and energy equations for the electron subgas, an energy conservation equation for heavy particles and an equation of state. Finite-difference operators for variable grid spacing were applied to the governing equations and the resulting system of strongly coupled, stiff equations were solved numerically by the Newton-Raphson method.
Scott, R. K.↗
Extraction from flight data of longitudinal aerodynamic coefficients for F-8 aircraft with supercritical wing
The longitudinal aerodynamic derivatives of the F-8 aircraft with supercritical wing were obtained from flight data by a parameter-extraction algorithm at Mach numbers of 0.8, 0.9, and 0.98. A set of derivatives were obtained from which calculated aircraft responses were correlated almost identically with actual flight responses. In general, the trends of the extracted derivatives obtained by the algorithm agreed with those obtained by a Newton-Raphson method and with preliminary data from the Langley 8-foot transonic pressure tunnel. The wind-tunnel damping derivatives were, however, substantially higher than the converged damping derivatives possibly because of Reynolds number differences between flight and model tests.
Williams, J. L.↗
A modified Newton-Raphson analysis of flight measurements of the trailing vortices of a heavy jet transport
A modified Newton-Raphson method has been used to determine the parameters in equations describing the vortex flow to obtain a best match with flight measurements of the flow behind the C-5A airplane. The flight measurements were made using a specially instrumented T-33 airplane which passed as closely as possible to the centers of the trailing vortices at several distances behind the C-5A airplane. The flight measurements were transformed to flow velocity relative to an inertial frame of reference. The assumed form of the flow consisted of the superposition of two counterrotating, finite core vortices. The positions of the vortex centers, their total circulation, the effective eddy viscosity and measurement bias were the parameters adjusted. The assumed form of vortex flow fit well the measured velocities for the numerous sets of data, both flaps up and down for the C-5A airplane. The resulting values of total circulation, however, were about two-thirds that expected of a wing with an elliptical loading. A partial explanation of the less than expected circulation is a dip in the spanwise lift distribution at the airplane's center line. The distance between the trailing vortices at the smallest times encountered is somewhat less than that expected for an elliptical wing loading.
Taylor, L. W., Jr.↗
Evaluation of importance of lateral acceleration derivatives in extraction of lateral-directional derivatives at high angles of attack
A theoretical investigation was conducted to determine the importance of the lateral acceleration (beta) derivatives in the extraction of lateral-directional stability derivatives for swept wing airplanes at high angles of attack. Representative values of lateral acceleration derivatives in yaw and roll (Cn beta and Cl beta) were used in a computer program to generate representative flight motions at several angles of attack and altitudes. The computer-generated motions were then subjected to a parameter identification process based on a modified Newton-Raphson method. Two identification techniques were evaluated, one which included the beta derivatives and one which neglected them. The results of the study indicate that omission of the beta derivatives from mathematical models used in the derivative-extraction techniques can produce erroneous values for the lateral-directional stability derivatives particularly at high angles of attack, where the beta derivatives are large. The largest errors occur in the dynamic derivatives, but large errors may also occur in the static derivatives for cases in which the beta derivatives have large effects on the flight motions of the airplane. In addition, the resulting identified mathematical models provide poor motion prediction as well as erroneous predictions of dynamic modal characteristics. These results strongly indicate that the effects of beta derivatives should be considered in any attempt to extract lateral-directional aerodynamic parameters at high angles of attack.
Nguyen, L. T.↗
An iterative procedure for obtaining maximum-likelihood estimates of the parameters for a mixture of normal distributions
A general iterative procedure is given for determining the consistent maximum likelihood estimates of normal distributions. In addition, a local maximum of the log-likelihood function, Newtons's method, a method of scoring, and modifications of these procedures are discussed.
Peters, B. C., Jr.↗
An effective solution to the nonlinear, nonstationary Navier-Stokes equations for two dimensions
A sequence of approximate solutions for the nonlinear, nonstationary Navier-Stokes equations for a two-dimensional domain, from which explicit error estimates and rates of convergence are obtained, is described. This sequence of approximate solutions is based primarily on the Newton-Kantorovich method.
Gabrielsen, R. E.↗
Preliminary design of composite wings for buckling, strength and displacement constraints
An unstiffened panel buckling constraint for balanced, symmetric laminated composites is included on the global design level in a mathematical programming structural optimization procedure for designing wing structures. Constraints are introduced by penalty functions, and Newton's method based on approximate second derivatives of the penalty terms is used as the search algorithm to obtain minimum-mass designs. Constraint approximations used during the optimization process contribute to the computational efficiency of the procedure. A criterion is developed that identifies the appropriate conservative form of the constraint approximations that are used with the optimization procedure. Minimum-mass design results are obtained for a multispar high-aspect-ratio wing subjected to material strength, minimum-gage, displacement, panel buckling and twist constraints. The material systems considered for the examples are all graphite-epoxy, graphite-epoxy with boron-epoxy spar caps, and all aluminum. The composite material designs are shown to have an advantage over the aluminum designs since they can often satisfy additional constraints with only small mass increases.
Starnes, J. H., Jr.↗
NEWSUMT: A FORTRAN program for inequality constrained function minimization, users guide
A computer program written in FORTRAN subroutine form for the solution of linear and nonlinear constrained and unconstrained function minimization problems is presented. The algorithm is the sequence of unconstrained minimizations using the Newton's method for unconstrained function minimizations. The use of NEWSUMT and the definition of all parameters are described.
Miura, H.↗
Programs for analysis and resizing of complex structures
The paper describes the PARS (Programs for Analysis and Resizing of Structures) system. PARS is a user oriented system of programs for the minimum weight design of structures modeled by finite elements and subject to stress, displacement, flutter and thermal constraints. The system is built around SPAR - an efficient and modular general purpose finite element program, and consists of a series of processors that communicate through the use of a data base. An efficient optimizer based on the Sequence of Unconstrained Minimization Technique (SUMT) with an extended interior penalty function and Newton's method is used. Several problems are presented for demonstration of the system capabilities.
Haftka, R. T.↗
A cubic extended interior penalty function for structural optimization
This paper describes an optimization procedure for the minimum weight design of complex structures. The procedure is based on a new cubic extended interior penalty function (CEIPF) used with the sequence of unconstrained minimization technique (SUMT) and Newton's method. The Hessian matrix of the penalty function is approximated using only constraints and their derivatives. The CEIPF is designed to minimize the error in the approximation of the Hessian matrix, and as a result the number of structural analyses required is small and independent of the number of design variables. Three example problems are reported. The number of structural analyses is reduced by as much as 50 per cent below previously reported results.
Prasad, B.↗
Research in nonlinear structural and solid mechanics
Nonlinear analysis of building structures and numerical solution of nonlinear algebraic equations and Newton's method are discussed. Other topics include: nonlinear interaction problems; solution procedures for nonlinear problems; crash dynamics and advanced nonlinear applications; material characterization, contact problems, and inelastic response; and formulation aspects and special software for nonlinear analysis.
Mccomb, H. G., Jr.↗
A general algorithm for the solution of Kepler's equation for elliptic orbits
An efficient algorithm is presented for the solution of Kepler's equation f(E)=E-M-e sin E=0, where e is the eccentricity, M the mean anomaly and E the eccentric anomaly. This algorithm is based on simple initial approximations that are cubics in M, and an iterative scheme that is a slight generalization of the Newton-Raphson method. Extensive testing of this algorithm has been performed on the UNIVAC 1108 computer. Solutions for 20,000 pairs of values of e and M show that for single precision, 42.0% of the cases require one iteration, 57.8% two and 0.2% three. For double precision one additional iteration is required.
Ng, E. W.↗
Preliminary design of composite wing-box structures for global damage tolerance
A procedure is presented that incorporates the influence of potential global damage conditions into the design process for minimum-mass wing-box structures. The procedure is based on mathematical-programming optimization techniques. Material-strength, minimum-gage, and panel-buckling constraints are introduced by penalty functions, and Newton's method with approximate second derivatives of the penalty terms is used as the search algorithm to obtain minimum-mass designs. A potential global damage condition is represented by a structural model with the damaged components removed. Example minimum-mass designs are obtained that simultaneously satisfy the constraints of the damaged and undamaged configurations of both graphite-epoxy and aluminum wing-box structural models. These examples are designed with and without the influence of potential damage conditions, and results indicate that for equal mass cases the residual strength of a damaged structure is higher when the influence of potential damage is properly included in the design from the outset. Results of these examples also identify the minimum structural mass increase required to increase residual strength levels.
Starnes, J. H., Jr.↗
Postbuckling of orthotropic composite plates loaded in compression
The nonlinear large deflection equations of von Karman are written for 'specially' orthotropic plates. The equations are then manipulated to determine the parameters required to establish postbuckling behavior. It is found that only two new parameters are needed beyond those required for buckling. By assuming trigonometric functions in one direction, the plate equations are converted into ordinary nonlinear differential equations which are solved numerically using a two point boundary problem solver that makes use of Newton's method. The postbuckling behavior is obtained for simply supported and clamped, long, rectangular, orthotropic plates covering the complete range of dimensions and material properties.
Stein, M.↗
A family of solution algorithms for nonlinear structural analysis based on relaxation equations
A family of hierarchical algorithms for nonlinear structural equations are presented. The algorithms are based on the Davidenko-Branin type homotopy and shown to yield consistent hierarchical perturbation equations. The algorithms appear to be particularly suitable to problems involving bifurcation and limit point calculations. An important by-product of the algorithms is that it provides a systematic and economical means for computing the stepsize at each iteration stage when a Newton-like method is employed to solve the systems of equations. Some sample problems are provided to illustrate the characteristics of the algorithms.
Park, K. C.↗