Equilibrium feedback control in linear games with quadratic costs
Equilibrium strategies for linear games with quadratic costs in Hilbert space, deriving nonlinear equation in allowed operator feedback spaces
Lukes, D. L.↗
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Results for “Nonlinear equations”
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Nonlinear response of gas-lubricated shrouded Rayleigh-step thrust pad
Quasi-static pressure characteristics of gas lubricated thrust bearing with shrouded Rayleigh step pads, solving nonlinear equation of motion
Cheng, C.↗
Finite stretching of a circular plate of neo-Hookean material.
The analytical solution presented is based on the assumption that the deformed thickness of the plate is approximately constant. The nonlinear equations governing finite axisymmetric deformations of a circular plate made of neo-Hookean material are used in the analysis. The variation of circumferential extension ratio and the variation of deformed thickness are shown in graphs.
Biricikoglu, V.↗
A variable-step-size robust delta modulator.
Description of an analytically obtained optimum adaptive delta modulator-demodulator configuration. The device utilizes two past samples to obtain a step size which minimizes the mean square error for a Markov-Gaussian source. The optimum system is compared, using computer simulations, with a linear delta modulator and an enhanced Abate delta modulator. In addition, the performance is compared to the rate distortion bound for a Markov source. It is shown that the optimum delta modulator is neither quantization nor slope-overload limited. The highly nonlinear equations obtained for the optimum transmitter and receiver are approximated by piecewise-linear equations in order to obtain system equations which can be transformed into hardware. The derivation of the experimental system is presented.
Song, C. L.↗
Nonlinear instability of a helicopter blade.
The nonlinear equations of flap-lag coupled with oscillation of a torsionally rigid blade are derived. The solution is established by a simplified asymptotic expansion procedure of multiple time scales. The region of stability and limit cycle oscillation and its comparison with numerical results are presented.
Tong, P.↗
Analytic prediction of airplane equilibrium spin characteristics
The nonlinear equations of motion are solved algebraically for conditions for which an airplane is in an equilibrium spin. Constrained minimization techniques are employed in obtaining the solution. Linear characteristics of the airplane about the equilibrium points are also presented and their significance in identifying the stability characteristics of the equilibrium points is discussed. Computer time requirements are small making the method appear potentially applicable in airplane design. Results are obtained for several configurations and are compared with other analytic-numerical methods employed in spin prediction. Correlation with experimental results is discussed for one configuration for which a rather extensive data base was available. A need is indicated for higher Reynolds number data taken under conditions which more accurately simulate a spin.
Adams, W. M., Jr.↗
Optimal trajectory generation for mechanical arms
A general method of generating optimal trajectories between an initial and a final position of an n degree of freedom manipulator arm with nonlinear equations of motion is proposed. The method is based on the assumption that the time history of each of the coordinates can be expanded in a series of simple time functions. By searching over the coefficients of the terms in the expansion, trajectories which minimize the value of a given cost function can be obtained. The method has been applied to a planar three degree of freedom arm.
Iemenschot, J. A.↗
Pole and zero placement in multivariable control systems
A method is proposed for designing multivariable systems based on an alternate derivation of Davison's theorem on pole placement and the solution of the nonlinear equations for the feedback gains by the least square error method. Output feedback is used to control a complex dynamical system. The freedom in design, after allocating poles, is used to place zeros and/or satisfy other design objectives. This method results in algorithms which are computationally attractive. However, this is done at a considerable sacrifice in terms of the design freedom available. For a system with m inputs and p outputs only m + p variables are available instead of mp variables.
Sridhar, B.↗
Application of pole-placement theory to helicopter stabilization systems.
This paper is concerned with the problem of designing a controller for a complex dynamical system using output feedback. The system selected for the study is the Boeing-Vertol CH-46 tandem rotor helicopter. Feedback gains are obtained by a least square solution of the nonlinear equations derived from pole-placement theory.
Sridhar, B.↗
Toward understanding the conical arc-chamber driver
The mechanical and electrical characteristics of the conical arc-chamber driver and the 1450-microfarad capacitor bank are described. The system dynamic performance can be approximated by a second-order nonlinear equation. The discharge of the capacitor bank into the conical arc-chamber was investigated at a variety of initial conditions. Details of the discharge are explained and compared for typical performance with both hydrogen and helium arc-chamber gases. At peak current the hydrogen plasma resistivity is significantly higher than that of the helium plasma and this, in part, accounts for the better shock speed performance with hydrogen. The lower melting and boiling point exploding wires of aluminum, copper, and stainless steel produced higher shock speed performance than the apparently nonexploding tungsten wire.
Livingston, F. R.↗
A variable step incremental procedure
Description of a variable step incremental procedure for the solution of nonlinear equations in finite element structural analysis. The proposed procedure is effective in improving the accuracy of the basic incremental technique and in providing, in addition, an accurate estimate of the discretization error. The proposed approach is highly appropriate for solving problems for which the user has no a priori estimate of the step size to use.
Thomas, G. R.↗
A model of the effect of the satellite photosheath on a double floating probe system
Description of a model developed to predict the interactions between multiple electrostatic probes separated by distances that are shorter than or comparable to the characteristic plasma shielding length. The computation involves crude approximations accounting for space charge effects, currents due to the emission and exchange of photoelectrons, and currents from the ambient plasma. Floating potentials are calculated by requiring current balance for each probe and solving by iterating the resulting coupled nonlinear equations. Although only qualitative accuracy is claimed, predictions using the model are found to be in good agreement with observations from the Explorer 45 satellite. The Explorer 45 satellite measures the potential difference between a pair of symmetrical spherical floating probes. The Explorer 45 data show that with increasing altitude, first, the sunward sphere will typically assume a more negative potential than the antisunward sphere; second, the polarities of the probes in the region L = 2 to 4 may switch, sometimes several times; and, finally, near the plasmapause the detector output may saturate, the sunward sphere being more negative.
Cauffman, D. P.↗
Importance of helicopter dynamics to the mathematical model of the helicopter
A mathematical model of the helicopter requires appropriate representation of the constituent elements of rotor dynamics. General-purpose programs that model a variety of configurations for a broad range of operating conditions result in varying and incompatible levels of sophistication. Analysis of specific dynamic problems facilitates the identification of configuration parameters which determine system behavior. For the present analysis, the nonlinear equations of a torsionally rigid hingeless rotor are linearized about an equilibrium condition to determine flap-lag stability characteristics in hover. A collocation method was used to obtain the coupled natural frequencies and modes. These modes allow exact treatment of the effect of elastic coupling which more than compensates for the destabilizing inertial coupling. The sensitivity of damping to the number of modes was found to be small, and reasonable accuracy was obtained the first flapwise and edgewise coupled modes. The range of destabilizing precone was found to be small.
White, W. F., Jr.↗
A multilevel control system for the large space telescope
A multilevel scheme was proposed for control of Large Space Telescope (LST) modeled by a three-axis-six-order nonlinear equation. Local controllers were used on the subsystem level to stabilize motions corresponding to the three axes. Global controllers were applied to reduce (and sometimes nullify) the interactions among the subsystems. A multilevel optimization method was developed whereby local quadratic optimizations were performed on the subsystem level, and global control was again used to reduce (nullify) the effect of interactions. The multilevel stabilization and optimization methods are presented as general tools for design and then used in the design of the LST Control System. The methods are entirely computerized, so that they can accommodate higher order LST models with both conceptual and numerical advantages over standard straightforward design techniques.
D D Siljak↗
A globally converging secant method with applications to boundary value problems
This paper presents a new algorithm for solving systems of nonlinear equations, which combines elements of the method of local variations with a sequential secant method. The new algorithm converges superlinearly and its convergence does not depend upon a good initial guess at a solution.
Polak, E.↗
Upwind second-order difference schemes and applications in unsteady aerodynamic flows
Explicit second-order upwind difference schemes in combination with spatially symmetric schemes can produce larger stability bounds and better numerical resolution than symmetric schemes alone. However, if conservation form is essential, a special operator is required for transition between schemes. An operational approach has been devised for deriving transition operators so that strict conservation and local consistency are maintained. Various aspects of hybrid schemes are studied numerically for model linear and nonlinear equations. To demonstrate the utility of combining two different algorithms, MacCormack's explicit, noncentered, second-order method is combined with a completely upwind version, and numerical solutions of the Euler equations are obtained for two-dimensional, transonic flows with embedded supersonic regions and shock waves. The general utility of the operational approach for combining schemes is emphasized by deriving a second-order conservative scheme for the steady transonic small-disturbance potential equation.
Warming, R. F.↗
An analysis of the strong zonal circulation within the stratosphere of Venus
A dynamic model is presented for the strong zonal circulation within the stratosphere of Venus which was observed by Mariner 10 and Venera. Rotational effects are neglected, a compressible and radiating atmosphere is considered, and diurnal radiative heating is shown to be negligible in the lower stratosphere (below 85 km). Analysis of the model indicates that propagating internal gravity waves generated by solar heating of the upper stratosphere may induce mean zonal velocities within the upper and lower stratosphere which would have a retrograde direction. The nonlinear equations of motion and energy are solved by an approximate analytical method to determine the magnitude of the zonal velocity, which is found to increase from zero at the tropopause to about 200 m/sec at the 85-km level. The calculated velocity near the UV cloud level compares favorably with the observed value of 100 m/sec.
Ramanathan, V.↗
Estimation of transonic aircraft aerodynamics to high angles of attack
A recently developed method for estimating transonic aircraft aerodynamics to high angles of attack which is especially suited to computerized aircraft design and optimization studies is described. Following theoretical and empirical guidelines, explicit nonlinear equations are formulated for subsonic, transonic, and supersonic speeds; new algorithms are used for assessing compressibility effects and for analyzing transonic, shock-dominated flows which adhere to Laitone's limit Mach-number criterion. Viscous dominated flows are not analyzed. The transonic influence of viscosity is indirectly accommodated by assigning designated inputs rather than extracted solutions for the chordwise locations of shock and separation. The method is extended to complete aircraft configurations by including accounts for nose lift, the wing downwash field, and tail lift. Several comparisons of experiment and estimate are included.
Axelson, J. A.↗