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Computer solutions of linear programming problems are outlined. Information covers vector spaces, convex sets, and matrix algebra elements for solving simultaneous linear equations. Dual problems, reduced cost analysis, ranges, and error analysis are illustrated.
The effects of various gust alleviation methods on the random blade response in flapping were studied analytically, assuming a rigid rotor support. The analytical model assumes rigid flapping blades with elastic root restraints. Linearized equations which are approximately valid at low lift conditions were used. Because of the interblade coupling from the feedback devices, the method of multiblade generalized coordinates was most convenient and was extended to include coning, differential coning, and warping of the rotor. The numerical examples cover the dynamic stability characteristics as affected by feedback gains of three- to six-bladed rotors. The number of blades had large effects on stability limits and modal time functions at these limits. The random flapping response of the blades to atmospheric turbulence was determined at 1.6 rotor advance ratio using feedback gains below the stability limit. The most effective reduction of the flapping response per unit gain was achieved with a rotor coning angle feedback.
The dynamical stability of the contact component in a semi-detached binary system is investigated by a linear time dependent analysis. The boundary condition imposed on the photosphere by the Roche lobe is assumed to be one of constant pressure over the whole surface of the lobe, which is treated as being spherically symmetric. The eigenvalues of the linear equation of motion are evaluated for sequences of stellar envelope models under the constraint of this boundary condition. An extensive region in the Hertzprung-Russell diagram is found to be dynamically unstable. This region coincides almost precisely with that predicted by a previous quasi-static study. Time scales for the growth of the instability are also evaluated. Color observations of several novae and U Geminorum stars are discussed in an attempt to provide semi-empirical tests of the proposed theoretical model of the outbursts. In particular the track in the H-R diagram of the U Geminorum star, EM Cygni, is shown to be consistent with an outburst which is purely a temperature change at constant radius, as is demanded by the theory.
The application of the nonsteady Lax-Wendroff technique to problems with asymptotically periodic solution which offers a potentially powerful method for the investigation of the interaction of rotating and stationary blade rows in turbomachinery is reported. A technique for specifying boundary conditions with phase lag was developed to accomplish this. A complete nonlinear analysis is carried out numerically to determine the entire flow field without recourse to the assumption of small disturbances of linear equations which underlie the previous acoustic theories. The result, obtained for the case of equal number of rotor and stator blades shows that transonic flow can be handled without difficulty. In addition, the program is not limited with regard to blade thickness, camber or loading. Extension of this method to incorporate viscous wakes and to analysis of fully three dimensional configuration is feasible, and would greatly expand its utility in practical applications.
Propagation of waves in curved ducts and pipes belong to the class of motion which is characterized by wave patterns totally different from those known in straight ducts or in unlimited space. The curvilinear boundaries are responsible for the appearance of a continuous standing radial wave which in turn affects the transmitted tangential waves. The purpose of this paper is to solve the problem of propagation of long acoustic waves in slightly and sharply bent ducts. This problem has been only partially analyzed by various authors. In this study two acoustic systems are considered which allow determination of the basic modes of motion and describe the transition and distortion of plane waves as they propagate down the curved channel. A detailed study of the behavior of waves in junctions between straight and curved ducts is also given. Solutions and expressions for principal modes of the wave are obtained by using the linearized equation of motion solved for its characteristic values. This original approach required a novel use of Bessel functions to determine the characteristic values of the steady and the decaying fields of motion.
Study of the behavior of weak electrostatic waves in a collisionless magnetoplasma supporting a steady large amplitude whistler wave. All waves are assumed to propagate parallel to a uniform background magnetic field B sub zero. In the presence of the whistler wave fields each particle executes an oscillatory motion parallel to B sub zero, in addition to a translation along B sub zero and transverse motions. This oscillation causes the Landau resonance to be replaced by a series of new resonances between particles and the electrostatic modes. A distribution function for the perturbed plasma is constructed by solving the Vlasov equation, linearized in the electrostatic wave amplitudes. A dispersion relation is obtained and solved approximately for the growth/damping rate of the perturbations. Growing electrostatic modes are found to be approximately uncoupled. Trapped particles have a strong influence on the stability of the system.
A stochastic projected gradient algorithm is proposed which can be used for finding a constrained optimum point for a concave or convex objective function subject to nonlinear constraints which form a connected region even when only a noisy estimate of the objective function is available. For a constraint described by a single linear equation, convergence to the constrained optimum value is proved, and the rate of convergence of the algorithm to the constrained optimum value is determined. The algorithm is applied to the nonlinear problem of obtaining automatically an array of detectors which forms a beam in a desired direction in space in the presence of interfering noise so as to maximize the SNR subject to a constraint on the super-gain ratio.
Investigation of a stochastic projected gradient algorithm which can be used to find a constrained optimum point for a concave or convex objective function subject to nonlinear constraints. When the constraint consists of only one linear equation, convergence to the constrained optimum value is proven, and bounds are obtained for the rate of convergence of the algorithm to the constrained optimum value. This algorithm is then applied to the nonlinear problem of automatically making an array of detectors form a beam in a desired direction in space when unknown interfering noise is present so as to maximize the signal-to-noise ratio subject to a constraint on the supergain ratio.
An analytical study of the combined pilot-aircraft system to develop a procedure for computing pilot ratings for the lateral-directional response. In the analytical procedure the pilot is represented by a linear, second-order pilot model, and the aircraft by linear equations of motion. Two levels of pilot response were described by the pilot model. The first level contains a static gain and a second-order lag function with a time constant of 0.2 second. The second level added a 1-sec lead time constant. The second level is assumed to represent a greater effort on the part of the pilot. If the system response characteristics can be achieved with the first-level pilot model, it is concluded that the aircraft will be given a pilot rating of satisfactory. If the aircraft characteristics are such that the second-level pilot model is required to achieve the prescribed system response, it is concluded that the aircraft will be rated tolerable.
A pilot-vehicle-display model is used to study information and display requirements and the effects on system performance and reliability of pilot-induced randomness, wind gusts, configurational changes, etc. A brief description of a control theoretic systems model is given and its use and validity are demonstrated by applying it in a piloted approach to landing situation. The analysis procedure assumes that the vehicle dynamics are represented by linearized equations of motion.
Shape and inner flow of a tridimensional spark are studied. The spark is created by focusing a laser beam in a gas. A second order fully non-linear equation is derived for the radial velocity on the axis of symmetry in the neighborhood of the origin. Solutions to that equation display the existence of a forbidden region near the focus, thus indicating the limits of applicability of a small perturbation solution.
A new method has been developed for calculating the pressure distribution and aerodynamic characteristics of wing-body-tail combinations in subsonic and supersonic potential flow. A computer program has been developed to perform the numerical calculations. The configuration surface is subdivided into a large number of panels, each of which contains an aerodynamic singularity distribution. A constant source distribution is used on the body panels, and a vortex distribution having a linear variation in the streamwise direction is used on the wing and tail panels. The normal components of velocity induced at specified control points by each singularity distribution are calculated and make up the coefficients of a system of linear equations relating the strengths of the singularities to the magnitude of the normal velocities. The singularity strengths which satisfy the boundary condition of tangential flow at the control points for a given Mach number and angle of attack are determined by solving this system of equations using an interactive procedure. Once the singularity strengths are known, the pressure coefficients are calculated, and the forces and moments acting on the configuration determined by numerical integration.
A two-dimensional detailed study of the behavior of long waves in curved ducts and in junctions between straight and curved ducts will be given. The mathematical treatment of the problem utilizes the method of separation variables. Solutions and expressions for principal mode of the wave are obtained by using the linearized equation of motion solved for its characteristic values. The unavoidable approximations in the numerical solutions of the cylindrical functions are due to use of series expansion of Bessel functions and from restrictions necessary to solve infinite matrices.
Linearized equations of motion describing a compressible, unsteady, viscous gas under the influence of an externally applied fluctuating force field are examined. The response of the medium is split into a near-field viscous wake mode and a far-field acoustic wave mode. The resulting modal equations are then used to predict both the undulating viscous wake and the far-field acoustic wave emitted by a thin wing (modeled by a dipole force field) in a cylindrical slug jet. The directivity pattern of the acoustic wave which propagates into the quiescent region beyond the jet is compared with available experiments. The peak frequency of the broad-band noise generated by upstream turbulence is also calculated and compared to published data.
Many petrologic problems, which may be expressed as a set of linear equations, have been solved by least-squares analysis. In many cases insufficient attention has been paid to the physical conditions of the model resulting in incorrect application of the method. This paper presents a systematic treatment of the application of least-squares analysis to petrologic problems including the direct utilization of physical constraints and weighting factors in the problem, and the assessment of uncertainties in the solution. As an example, least-squares analysis is used to examine, in detail, the mass balance equations for lunar rock 12040 and to determine the consistency of the available analytical data.
We have computed populations of 16 levels of Si III-V and radiation fields in all connecting transitions; in particular the first six Si III triplet levels, including the 4553 line, and the first six Si IV levels including 4089. The computations were done for four non-LTE H-He model atmospheres, provided by Auer and Mihalas. Estimates of corresponding MK types are B1.5 V, B0.5 V, O9 V, and O6. Solutions were obtained by iterating the linearized equations of radiative transfer and statistical equilibrium, except that for less important lines an approximate equivalent two-level atom treatment was used. Continuous opacities of C, N, O, and Ne were included. All abundances were solar values.
A dynamic analysis is presented for a spin stabilized spacecraft employing four radial wire antennas with tip masses, a configuration first employed in the IMP-J spacecraft. The use of wires in place of the usual booms represents the ultimate in weight reduction at the expanse of flexibility. The satellite is modelled as a 14 degree of freedom system, and the linearized equations of motion are found. The lowest order vibrational modes and natural frequencies of the gyroscopically coupled system are then determined. Because the satellite spin rate is decreased by antenna deployment, a spin-up maneuver is needed. The response of the time varying mode equations during spin-up is found, for the planar modes, in terms of Bessel functions and a Struve function of order -1/4. Because tables of the latter are not readily available, the particular solution is expressed in various forms including an infinite series of Bessel functions and a particularly useful asymptotic expansion.
Two representative algorithms used on a serial computer and presently executed on the Control Data Corporation 6000 computer were adapted to execute efficiently on the Control Data STAR-100 computer. Gaussian elimination for the solution of simultaneous linear equations and the Gauss-Legendre quadrature formula for the approximation of an integral are the two algorithms discussed. A description is given of how the programs were adapted for STAR and why these adaptations were necessary to obtain an efficient STAR program. Some points to consider when adapting an algorithm for STAR are discussed. Program listings of the 6000 version coded in 6000 FORTRAN, the adapted STAR version coded in 6000 FORTRAN, and the STAR version coded in STAR FORTRAN are presented in the appendices.