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NASA NTRS1992

A robust direct-integration method for rotorcraft maneuver and periodic response

The Newmark-Beta method and the Newton-Raphson iteration scheme are combined to develop a direct-integration method for evaluating the maneuver and periodic-response expressions for rotorcraft. The method requires the generation of Jacobians and includes higher derivatives in the formulation of the geometric stiffness matrix to enhance the convergence of the system. The method leads to effective convergence with nonlinear structural dynamics and aerodynamic terms. Singularities in the matrices can be addressed with the method as they arise from a Lagrange multiplier approach for coupling equations with nonlinear constraints. The method is also shown to be general enough to handle singularities from quasisteady control-system models. The method is shown to be more general and robust than the similar 2GCHAS method for analyzing rotorcraft dynamics.

Panda, Brahmananda↗

NASA NTRS1992

Dynamic analysis of flexible gear trains/transmissions - An automated approach

In this paper an automated algorithmic method is presented for the dynamic analysis of geared trains/transmissions. These are treated as a system of interconnected flexible bodies. The procedure developed explains the switching of constraints with time as a result of the change in the contacting areas at the gear teeth. The elastic behavior of the system is studied through the employment of three-dimensional isoparametric elements having six degrees-of-freedom at each node. The contact between the bodies is assumed at the various nodes, which could be either a line or a plane. The kinematical expressions, together with the equations of motion using Kane's method, strain energy concepts, are presented in a matrix form suitable for computer implementation. The constraint Jacobian matrices are generated automatically based on the contact information between the bodies. The concepts of the relative velocity at the contacting points at the tooth pairs and the subsequent use of the transmission ratios in the analysis is presented.

Amirouche, F. M. L.↗

NASA NTRS1993

Time-variant analysis of rotorcraft systems dynamics - An exploitation of vector processors

In this paper a generalized algorithmic procedure is presented for handling constraints in mechanical transmissions. The latter are treated as multibody systems of interconnected rigid/flexible bodies. The constraint Jacobian matrices are generated automatically and suitably updated in time, depending on the geometrical and kinematical constraint conditions describing the interconnection between shafts or gears. The type of constraints are classified based on the interconnection of the bodies by assuming that one or more points of contact exist between them. The effects due to elastic deformation of the flexible bodies are included by allowing each body element to undergo small deformations. The procedure is based on recursively formulated Kane's dynamical equations of motion and the finite element method, including the concept of geometrical stiffening effects. The method is implemented on an IBM-3090-600j vector processor with pipe-lining capabilities. A significant increase in the speed of execution is achieved by vectorizing the developed code in computationally intensive areas. An example consisting of two meshing disks rotating at high angular velocity is presented. Applications are intended for the study of the dynamic behavior of helicopter transmissions.

Amirouche, F. M. L.↗

NASA NTRS1993

Navier-Stokes calculations for the unsteady flowfield of turbomachinery

The Multistage Unsteady Turbomachinery Code is presently used to numerically simulate the flowfield generated by rotating machinery. The code, which simplifies the Reynolds-averaged Navier-Stokes equations via the thin-layer approximation, is an implicit finite-volume scheme with flux Jacobians that are evaluated by flux-vector splitting, while residual fluxes are evaluated by flux-difference splitting. Attention is given to the computational results this obtained for three engineering problems, involving propfan, rotating blade, and both rotating and static blade flows.

Chen, Jen P.↗

NASA NTRS1993

Finite element implementation of state variable-based viscoplasticity models

The implementation of state variable-based viscoplasticity models is made in a general purpose finite element code for structural applications of metals deformed at elevated temperatures. Two constitutive models, Walker's and Robinson's models, are studied in conjunction with two implicit integration methods: the trapezoidal rule with Newton-Raphson iterations and an asymptotic integration algorithm. A comparison is made between the two integration methods, and the latter method appears to be computationally more appealing in terms of numerical accuracy and CPU time. However, in order to make the asymptotic algorithm robust, it is necessary to include a self adaptive scheme with subincremental step control and error checking of the Jacobian matrix at the integration points. Three examples are given to illustrate the numerical aspects of the integration methods tested.

Chang, T. Y. P.↗

NASA NTRS1992

Robot path planning for space-truss assembly

Construction, repair, and maintenance of space-based structures will require extensive planning of operations in order to effectively carry out these tasks. The path planning algorithm described here is a general approach to generating paths that guarantee collision avoidance for a single chain nonredundant or redundant robot. The algorithm uses a graph search of feasible points in position space, followed by a local potential field method that guarantees collision avoidance among objects, structures, and the robot arm as well as conformance to joint limit constraints. This algorithm is novel in its computation of goal attractive potential fields in Cartesian space, and computation of obstacle repulsive fields in robot joint space. These effects are combined to generate robot motion. Computation is efficiently implemented through the computation of the robot arm Jacobian and not the full inverse arm kinematics. These planning algorithms have been implemented and evaluated using existing space-truss designs, and are being integrated into the RPI-CIRSSE Testbed environment.

Muenger, Rolf↗

NASA NTRS1992

Human low vision image warping - Channel matching considerations

We are investigating the possibility that a video image may productively be warped prior to presentation to a low vision patient. This could form part of a prosthesis for certain field defects. We have done preliminary quantitative studies on some notions that may be valid in calculating the image warpings. We hope the results will help make best use of time to be spent with human subjects, by guiding the selection of parameters and their range to be investigated. We liken a warping optimization to opening the largest number of spatial channels between the pixels of an input imager and resolution cells in the visual system. Some important effects are not quantified that will require human evaluation, such as local 'squashing' of the image, taken as the ratio of eigenvalues of the Jacobian of the transformation. The results indicate that the method shows quantitative promise. These results have identified some geometric transformations to evaluate further with human subjects.

Juday, Richard D.↗

NASA NTRS1993

Helicopter trim analysis by shooting and finite element methods with optimally damped Newton iterations

Helicopter trim settings of periodic initial state and control inputs are investigated for convergence of Newton iteration in computing the settings sequentially and in parallel. The trim analysis uses a shooting method and a weak version of two temporal finite element methods with displacement formulation and with mixed formulation of displacements and momenta. These three methods broadly represent two main approaches of trim analysis: adaptation of initial-value and finite element boundary-value codes to periodic boundary conditions, particularly for unstable and marginally stable systems. In each method, both the sequential and in-parallel schemes are used, and the resulting nonlinear algebraic equations are solved by damped Newton iteration with an optimally selected damping parameter. The impact of damped Newton iteration, including earlier-observed divergence problems in trim analysis, is demonstrated by the maximum condition number of the Jacobian matrices of the iterative scheme and by virtual elimination of divergence. The advantages of the in-parallel scheme over the conventional sequential scheme are also demonstrated.

Achar, N. S.↗

NASA NTRS1993

A natural partitioning scheme for parallel simulation of multibody systems

A parallel partitioning scheme based on physical-co-ordinate variables is presented to systematically eliminate system constraint forces and yield the equations of motion of multibody dynamics systems in terms of their independent coordinates. Key features of the present scheme include an explicit determination of the independent coordinates, a parallel construction of the null space matrix of the constraint Jacobian matrix, an easy incorporation of the previously developed two-stage staggered solution procedure and a Schur complement based parallel preconditioned conjugate gradient numerical algorithm.

Chiou, J. C.↗

NASA NTRS1992

A quadratic-tensor model algorithm for nonlinear least-squares problems with linear constraints

A new algorithm for solving nonlinear least-squares and nonlinear equation problems is proposed which is based on approximating the nonlinear functions using the quadratic-tensor model by Schnabel and Frank. The algorithm uses a trust region defined by a box containing the current values of the unknowns. The algorithm is found to be effective for problems with linear constraints and dense Jacobian matrices.

Hanson, R. J.↗

NASA NTRS1993

Three-dimensional unstructured grid Euler computations using a fully-implicit, upwind method

A method has been developed to solve the Euler equations on a three-dimensional unstructured grid composed of tetrahedra. The method uses an upwind flow solver with a linearized, backward-Euler time integration scheme. Each time step results in a sparse linear system of equations which is solved by an iterative, sparse matrix solver. Local-time stepping, switched evolution relaxation (SER), preconditioning and reuse of the Jacobian are employed to accelerate the convergence rate. Implicit boundary conditions were found to be extremely important for fast convergence. Numerical experiments have shown that convergence rates comparable to that of a multigrid, central-difference scheme are achievable on the same mesh. Results are presented for several grids about an ONERA M6 wing.

Whitaker, David L.↗

NASA NTRS1993

Analysis of implicit second-order upwind-biased stencils

Truncation error and stability properties of several implicit upwind schemes for the two-dimensional Euler equations are examined. The schemes use linear data reconstruction methods to achieve second-order flux integrations where the implicit Jacobian operators are first order. The stability properties of the schemes are examined by a Von Neumann analysis of the linearized, constant-coefficient Euler equations. The choice of the data reconstruction method used to evaluate the flux integral has a dramatic effect on the convergence properties of the implicit solution method. In particular, the typical one-dimensional data reconstruction methods used with structured grids exhibit poor convergence properties compared to the unstructured grid method considered. Of the schemes examined, the one with the superior convergence properties is well-suited for both unstructured and structured grids, which has important implications for the design of implicit methods.

Roberts, Thomas W.↗

NASA NTRS1993

Strongly coupled radiative transfer and Joule heating in the cathode of an arc heater

Radiation and Joule heating in the electrode region of an arc heater are discussed. Radiative transport equations for a true axisymmetric geometry are used. A subsonic code is developed to numerically solve the fluid equations with strongly coupled radiation and Joule heating representative of a high pressure and high current arc heater. Analytic expression for the divergence of radiative heat flux derived previously is used. Jacobians of the radiation term are derived. The Joule heating term is computed using a previously developed code. The equilibrium gas model consists of seven species. The fluxes are differenced using Van Leer flux splitting. Using this code, the effects of radiative cooling on the thermodynamic parameters of the arc core are discussed.

Durgapal, P.↗

NASA NTRS1991

Dynamic analysis and control of lightweight manipulators with flexible parallel link mechanisms

The flexible parallel link mechanism is designed for increased rigidity to sustain the buckling when it carries a heavy payload. Compared to a one link flexible manipulator, a two link flexible manipulator, especially the flexible parallel mechanism, has more complicated characteristics in dynamics and control. The objective of this research is the theoretical analysis and the experimental verification of dynamics and control of a two link flexible manipulator with a flexible parallel link mechanism. Nonlinear equations of motion of the lightweight manipulator are derived by the Lagrangian method in symbolic form to better understand the structure of the dynamic model. A manipulator with a flexible parallel link mechanism is a constrained dynamic system whose equations are sensitive to numerical integration error. This constrained system is solved using singular value decomposition of the constraint Jacobian matrix. The discrepancies between the analytical model and the experiment are explained using a simplified and a detailed finite element model. The step response of the analytical model and the TREETOPS model match each other well. The nonlinear dynamics is studied using a sinusoidal excitation. The actuator dynamic effect on a flexible robot was investigated. The effects are explained by the root loci and the Bode plot theoretically and experimentally. For the base performance for the advanced control scheme, a simple decoupled feedback scheme is applied.

Lee, Jeh Won↗

NASA NTRS1993

Path planning for robotic truss assembly

A new Potential Fields approach to the robotic path planning problem is proposed and implemented. Our approach, which is based on one originally proposed by Munger, computes an incremental joint vector based upon attraction to a goal and repulsion from obstacles. By repetitively adding and computing these 'steps', it is hoped (but not guaranteed) that the robot will reach its goal. An attractive force exerted by the goal is found by solving for the the minimum norm solution to the linear Jacobian equation. A repulsive force between obstacles and the robot's links is used to avoid collisions. Its magnitude is inversely proportional to the distance. Together, these forces make the goal the global minimum potential point, but local minima can stop the robot from ever reaching that point. Our approach improves on a basic, potential field paradigm developed by Munger by using an active, adaptive field - what we will call a 'flexible' potential field. Active fields are stronger when objects move towards one another and weaker when they move apart. An adaptive field's strength is individually tailored to be just strong enough to avoid any collision. In addition to the local planner, a global planning algorithm helps the planner to avoid local field minima by providing subgoals. These subgoals are based on the obstacles which caused the local planner to fail. A best-first search algorithm A* is used for graph search.

Sanderson, Arthur C.↗

NASA NTRS1993

Multigrid properties of upwind-biased data reconstructions

The multigrid properties of two data reconstruction methods used for achieving second-order spatial accuracy when solving the two-dimensional Euler equations are examined. The data reconstruction methods are used with an implicit upwind algorithm which uses linearized backward-Euler time-differencing. The solution of the resulting linear system is performed by an iterative procedure. In the present study only regular quadrilateral grids are considered, so a red-black Gauss-Seidel iteration is used. Although the Jacobian is approximated by first-order upwind extrapolation, two alternative data reconstruction techniques for the flux integral that yield higher-order spatial accuracy at steady state are examined. The first method, probably most popular for structured quadrilateral grids, is based on estimating the cell gradients using one-dimensional reconstruction along curvilinear coordinates. The second method is based on Green's theorem. Analysis and numerical results for the two dimensional Euler equations show that data reconstruction based on Green's theorem has superior multigrid properties as compared to the one-dimensional data reconstruction method.

Warren, Gary P.↗

NASA NTRS1994

Robust inverse kinematics using damped least squares with dynamic weighting

This paper presents a general method for calculating the inverse kinematics with singularity and joint limit robustness for both redundant and non-redundant serial-link manipulators. Damped least squares inverse of the Jacobian is used with dynamic weighting matrices in approximating the solution. This reduces specific joint differential vectors. The algorithm gives an exact solution away from the singularities and joint limits, and an approximate solution at or near the singularities and/or joint limits. The procedure is here implemented for a six d.o.f. teleoperator and a well behaved slave manipulator resulted under teleoperational control.

Schinstock, D. E.↗

NASA NTRS1994

Benchmark problems in computational aeroacoustics

A recent directive at NASA Langley is aimed at numerically predicting principal noise sources. During my summer stay, I worked with high-order ENO code, developed by Dr. Harold Atkins, for solving the unsteady compressible Navier-Stokes equations, as it applies to computational aeroacoustics (CAA). A CAA workshop, composed of six categories of benchmark problems, has been organized to test various numerical properties of code. My task was to determine the robustness of Atkins' code for these test problems. In one category, we tested the nonlinear wave propagation of the code for the one-dimensional Euler equations, with initial pressure, density, and velocity conditions. Using freestream boundary conditions, our results were plausible. In another category, we solved the linearized two-dimensional Euler equations to test the effectiveness of radiation boundary conditions. Here we utilized MAPLE to compute eigenvalues and eigenvectors of the Jacobian given variable and flux vectors. We experienced a minor problem with inflow and outflow boundary conditions. Next, we solved the quasi one dimensional unsteady flow equations with an incoming acoustic wave of amplitude 10(exp -6). The small amplitude sound wave was incident on a convergent-divergent nozzle. After finding a steady-state solution and then marching forward, our solution indicated that after 30 periods the acoustic wave had dissipated (a period is time required for sound wave to traverse one end of nozzle to other end).

Porter-Locklear, Freda↗