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At least 667 records · Page 37
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NASA NTRS1982

Stability of compressible three-dimensional boundary-layer flows

For compressible three-dimensional flow, the method of multiple scales to formulate the three-dimensional stability problem and determine the partial-differential equations governing variations of the amplitude and complex wavenumbers is used. A method for following one specific wave along its trajectory to ascertain the characteristics of the most unstable disturbance is proposed. Numerical results using the flow over the X-21 wing as calculated from the Kaups-Cebeci code will be presented.

Reed, H. L.↗
NASA NTRS1982

Flow induced force of labyrinth seal

Flow induced instability force due to a labyrinth seal is analyzed. An approximate solution is given for the partial differential equation representing the flow in labyrinth seal and it is compared with the finite difference method in order to verify the accuracy of both methods. The effects of difference of inlet and outlet pressures of the seal, deflection of pressure and mass flow from the steady state, rotor diameter, seal clearance, seal interval and seal number on the flow induced force of the seal are investigated and it is known that some of these factors are very influential on the flow induced force.

Iwatsubo, T.↗
NASA NTRS1983

Canonical forms for nonlinear systems

Necessary and sufficient conditions for transforming a nonlinear system to a controllable linear system have been established, and this theory has been applied to the automatic flight control of aircraft. These transformations show that the nonlinearities in a system are often not intrinsic, but are the result of unfortunate choices of coordinates in both state and control variables. Given a nonlinear system (that may not be transformable to a linear system), we construct a canonical form in which much of the nonlinearity is removed from the system. If a system is not transformable to a linear one, then the obstructions to the transformation are obvious in canonical form. If the system can be transformed (it is called a linear equivalent), then the canonical form is a usual one for a controllable linear system. Thus our theory of canonical forms generalizes the earlier transformation (to linear systems) results. Our canonical form is not unique, except up to solutions of certain partial differential equations we discuss. In fact, the important aspect of this paper is the constructive procedure we introduce to reach the canonical form. As is the case in many areas of mathematics, it is often easier to work with the canonical form than in arbitrary coordinate variables.

Su, R.↗
NASA NTRS1983

Linear approximations of nonlinear systems

The development of a method for designing an automatic flight controller for short and vertical take off aircraft is discussed. This technique involves transformations of nonlinear systems to controllable linear systems and takes into account the nonlinearities of the aircraft. In general, the transformations cannot always be given in closed form. Using partial differential equations, an approximate linear system called the modified tangent model was introduced. A linear transformation of this tangent model to Brunovsky canonical form can be constructed, and from this the linear part (about a state space point x sub 0) of an exact transformation for the nonlinear system can be found. It is shown that a canonical expansion in Lie brackets about the point x sub 0 yields the same modified tangent model.

Hunt, L. R.↗
NASA NTRS1983

Singular Perturbations in Systems and Control

The application of singular perturbation theory to problems in systems and control is addressed. The types of systems considered include linear and nonlinear, discrete and continuous, and stochastic and deterministic. Both ordinary and partial differential equations are covered.

Ardema, M. D.↗
NASA NTRS1983

Development of a distributed-parameter mathematical model for simulation of cryogenic wind tunnels

A one-dimensional distributed-parameter dynamic model of a cryogenic wind tunnel was developed which accounts for internal and external heat transfer, viscous momentum losses, and slotted-test-section dynamics. Boundary conditions imposed by liquid-nitrogen injection, gas venting, and the tunnel fan were included. A time-dependent numerical solution to the resultant set of partial differential equations was obtained on a CDC CYBER 203 vector-processing digital computer at a usable computational rate. Preliminary computational studies were performed by using parameters of the Langley 0.3-Meter Transonic Cryogenic Tunnel. Studies were performed by using parameters from the National Transonic Facility (NTF). The NTF wind-tunnel model was used in the design of control loops for Mach number, total temperature, and total pressure and for determining interactions between the control loops. It was employed in the application of optimal linear-regulator theory and eigenvalue-placement techniques to develop Mach number control laws.

Tripp, J. S.↗
NASA NTRS1983

Modeling and control of distributed structures

An inherent incongruity in the modeling of controls for structures is discussed. Structures are basically distributed-parameter systems, described by partial differential equations, and control theory is concerned almost exclusively with discrete (in space) systems, described by ordinary differential equations. The standard approach to solving this dilemma is to discretize the system in space, which precludes the use of distributed controls. A different approach, known as the independent modal-space control method, is designed to eliminate the incongruity by bringing about a closer correspondence between modeling and control theory. The independent modal-space control method can treat distributed structures as well as discretized models and it permits design of both distributed and discrete-point controls.

Meirovitch, L.↗
NASA NTRS1983

Distributed system modeling of a large space antenna

A general approach for distributed parameter modeling of complex dynamical systems is described. The method consists of dividing the system in parts which can be modeled by simple partial differential equations and coupling the equations thus obtained by applying Hamilton's variational formalism to the entire system. The modeling of a large, offset-fed, wrap-rib antenna is presented to illustrate the approach. Although such models are perhaps not as precise as finite element models, they can be useful for initial physical insight and parametric design.

Hamidi, M.↗
NASA NTRS1983

Orbiting chains and rings

Two very different, highly flexible, space structures proposed during the last decade which involve analytical solution of certain partial differential equations are discussed. The first structure is an array, or hanging chain of aluminum beads which would serve as a convenient communicator if, under the influence of the Earth's gravity gradient, it assumes a local vertical orientation as it circles the Earth. Two passive schemes have been proposed for damping the rigid modes: twist the wire at the ends to provide non-zero moment of inertia about the vertical, thereby inducing relative motion of the two tips during rigid pitch or roll, and thus exercising a damper; and introduce weak lossy springs between the end sections and the main section; thereby providing linear coupling between the springs and all the in-plane (pitch) modes. The second structure is a complete ring of satellites cabled together at synchronous altitude. By a slight increase in altitude, the cable is in tension. This configuration is, however, unstable, and an active feedback control scheme is required to stabilize it. A possible scheme involves local cable length adjustment based on measurement of local altitude and shape variations and their rates.

Breakwell, J. V.↗
NASA NTRS1981

Multigrid solvers on parallel computers

Massively parallel computers, as considered in this investigation, are not yet available. However, a large-scale parallel computer cannot usefully be designed before the hypothetical algorithms which will employ it are studied. Most of the studies of parallel partial differential equations (PDE) solvers are based on solution techniques much slower (on sequential machines) than multigrid methods. Multigrid methods are highly parallelizable. Each of their processes can simultaneously be performed at all grid points. The present investigation is concerned with a preliminary exploration of the potential of multigrid, or, more generally, Multi-Level Adaptive Techniques (MLAT) on computers with many processors. Basic processes are considered, taking into account coarse-grid approximation, relaxation, coarse-grid corrections, full multigrid algorithms, nonlinear problems and eigenvalue problems, fine-to-coarse correction, and chains of problems. Details of parallel multigrid processing are also examined.

Brandt, A.↗
NASA NTRS1982

On current aspects of finite element computational fluid mechanics for turbulent flows

A set of nonlinear partial differential equations suitable for the description of a class of turbulent three-dimensional flow fields in select geometries is identified. On the basis of the concept of enforcing a penalty constraint to ensure accurate accounting of ordering effects, a finite element numerical solution algorithm is established for the equation set and the theoretical aspects of accuracy, convergence and stability are identified and quantized. Hypermatrix constructions are used to formulate the reduction of the computational aspects of the theory to practice. The robustness of the algorithm, and the computer program embodiment, have been verified for pertinent flow configurations.

Baker, A. J.↗
NASA NTRS1983

Computation of shock wave/target interaction

Computational results of shock waves impinging on targets and the ensuing diffraction flowfield are presented. A number of two-dimensional cases are computed with finite difference techniques. The classical case of a shock wave/cylinder interaction is compared with shock tube data and shows the quality of the computations on a pressure-time plot. Similar results are obtained for a shock wave/rectangular body interaction. Here resolution becomes important and the use of grid clustering techniques tend to show good agreement with experimental data. Computational results are also compared with pressure data resulting from shock impingement experiments for a complicated truck-like geometry. Here of significance are the grid generation and clustering techniques used. For these very complicated bodies, grids are generated by numerically solving a set of elliptic partial differential equations.

Mark, A.↗
NASA NTRS1983

Adaptive grids generated by elliptic systems

It is pointed out that a finite difference grid which moves with the solution of the partial differential equation being solved can improve the accuracy and efficiency of a numerical algorithm. This technique is particularly advantageous in the solution of problems involving boundary layers or shocks where a poorly chosen grid may give a numerical solution which is useless because of poor resolution or extreme oscillations. The present investigation is concerned with the development of a scheme which does not excessively distort the grid. The grid generation algorithm is based on the numerical solution of a system of elliptic differential equations. Holst and Brown (1981) have used a preliminary solution to move points on the boundary of the physical region and then resolved the problem on a new grid generated by an elliptic system. In the current investigation, the grid movement and the solution will develop simultaneously. The solution is used to modify the generating equations, thereby controlling the grid point distributions.

Mastin, C. W.↗
NASA NTRS1981

Optimal control and controller location for distributed parameter elastic systems

A class of systems governed by second order partial differential equations and driven by controllers located at points r sub i, i = 1, ..., k, is considered. A cost functional quadratic in the time derivative of the state and the control is associated with the system. The optimal controller locations are defined as the ones which minimize the maximum of the cost over all possible initial states. An analytical solution to the associated Riccati equation is presented providing a convenient expression for determining the optimal locations.

Hamidi, M.↗
NASA NTRS1983

Postbuckling of long orthotropic plates in combined shear and compression

The nonlinear large-deflection partial differential equations of von karman for orthotropic plates loaded in combined shear and compression are converted into a set of first-order nonlinear ordinary differential equations by assuming trigonometric functions in one direction. These equations are solved numerically using a two point boundary problem solver which makes use of Newton's method. Results are obtained which determine the postbuckling behavior of rectangular plates with loading up to about three times the buckling load. Both isotropic and orthotropic composite plates are considered. Results show that orthotropic plates may behave quite differently than isotropic plates and that in-plane boundary conditions are important for plates loaded in shear.

Stein, M.↗
NASA NTRS1983

Symmetric linear systems - An application of algebraic systems theory

Dynamical systems which contain several identical subsystems occur in a variety of applications ranging from command and control systems and discretization of partial differential equations, to the stability augmentation of pairs of helicopters lifting a large mass. Linear models for such systems display certain obvious symmetries. In this paper, we discuss how these symmetries can be incorporated into a mathematical model that utilizes the modern theory of algebraic systems. Such systems are inherently related to the representation theory of algebras over fields. We will show that any control scheme which respects the dynamical structure either implicitly or explicitly uses the underlying algebra.

Hazewinkel, M.↗
NASA NTRS1982

General curvilinear coordinate systems

The basic ideas of the construction and use of numerically-generated boundary-fitted coordinate systems for the numerical solution of partial differential equations are discussed. With such coordinate systems, all computation can be done on a fixed square grid in the rectangular transformed region regardless of the shape or movement of the physical boundaries. A number of different types of configurations for the transformed region and the basic transformation relations from a cartesian system to a general curvilinear system are given. The material of this paper is applicable to all types of coordinate system generation.

Thompson, J. P.↗
NASA NTRS1982

Elliptic grid generation

Various types of generating systems for boundary-conforming coordinate systems based on the numerical solution of systems of elliptic partial differential equations are discussed. Particular emphasis is given to the determination of functions in these equations which control the distribution of the curvilinear coordinate lines in the field.

Thompson, J. F.↗