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A method of solving a class of linear matrix equations over various rings is proposed, using results from linear geometric control theory. An algorithm, successfully implemented, is presented, along with non-trivial numerical examples. Applications of the method to the algebraic control system design methodology are discussed.
The final equation for nitrogen was determined. In the work on the equation of state for nitrogen, coefficients were determined by constraining the critical point to selected critical point parameters. Comparisons of this equation with all the P-density-T data were made, as well as comparisons to all other thermodynamic data reported in the literature. The extrapolation of the equation of state was studied for vapor to higher temperatures and lower temperatures, and for the liquid surface to the saturated liquid and the fusion lines. A new vapor pressure equation was also determined which was constrained to the same critical temperature, pressure, and slope (dP/dT) as the equation of state. Work on the equation of state for oxygen included studies for improving the equation at the critical point. Comparisons of velocity of sound data for oxygen were also made between values calculated with a preliminary equation of state and experimental data. Functions for the calculation of the derived thermodynamic properties using the equation of state are given, together with the derivative and integral functions for the calculation of the thermodynamic properties using the equations of state. Summary tables of the thermodynamic properties of nitrogen and oxygen are also included to serve as a check for those preparing computer programs using the equations of state.
The region of validity of conventional rate equation formulations used to describe the excitation of flame gases for spectroscopic diagnostic purposes is examined by direct comparison with the density matrix formulation. The density matrix equations are formulated for an ensemble of atoms or molecules in the gas state. These equations are reduced to the steady-state limit for a two-level system under the effect of single-mode and multimode optical fields. The steady-state equations are shown to be valid only when the laser pulse rises slowly with respect to the collisional dephasing rate for single-mode excitation, and for certain special cases under multimode excitation.
While the changes in stresses caused by surfactant adsorption on non-deforming interfaces have been fairly well established, prior to this work, there were few studies addressing how surfactants alter stresses on strongly deforming interfaces. We chose the model problem of a drop in a uniaxial extensional flow to study these stress conditions To model surfactant effects at fluid interfaces, a proper description of the dependence of the surface tension on surface concentration, the surface equation of state, is required. We have adopted a surface equation of state that accounts for the maximum coverage limit; that is, because surfactants have a finite cross sectional area, there is an upper bound to the amount of surfactant that can adsorb in a monolayer. The surface tension reduces strongly only when this maximum coverage is approached. Since the Marangoni stresses go as the derivative of the surface equation of state times the surface concentration gradient, the non-linear equation of state determines both the effect of surfactants in the normal stress jump, (which is balanced by the product of the mean curvature of the interface times the surface tension), and the tangential stress jump, which is balanced by Marangoni stresses. First, the effects of surface coverage and intermolecular interactions among surfactants which drive aggregation of surfactants in the interface were studied. (see Pawar and Stebe, Physics of Fluids).
A case study of the application of an integrated rotorcraft identification method to the linear modeling of rotor system dynamics and rotor hub loads is presented. Applying the method to flight data obtained from the RSRA compound helicopter, the rotor derivatives in the rotor state dynamic equation are identified along with blade equivalent damping and spring periodic coefficients. A rigid blade flapping equation of motion can be derived from the identified rotor state equation. It is shown that the concept of rotor hub load derivatives is useful for small maneuvering loads. The hub load derivatives can be used to relate the hub dynamic load to rotor/fuselage motion and applied inputs.
The vibration-rotation interaction coupling in diatomic molecules is analyzed and first-order expressions for the energy levels are deduced. Corrections to the partition function to account for this coupling are derived. The effects of long-range intermolecular forces due to induced dipole-induced dipole interactions, dipole-induced dipole interactions, charge-dipole interactions, and charge-charge interactions are considered. London's quantized oscillator model of the induced dipole-induced dipole interaction is developed. The partition function including effects of such dispersion forces is derived and is shown to lead to the virial equation of state. For a model combining the hard sphere and long-range attractive type potentials, this is found to lead to the van der Waals equation of state. The equation of state near the critical point is presented in reduced form (thermodynamic variables expressed in units of critical point values). Finally, approximate corrections to the electronic partition functions are derived to account for perturbations of excited electronic states induced by neighboring neutral particles, heavy ions, and fast electron collisions.
Shell motion equations derived in terms of reference state from three-dimensional theory based on Cauchy motion laws for undeformed bodies
Three computer programs comprising six subroutines are used to calculate matrix stability and frequency response.
The novel idea presented is based on the observation that if a structure is controlled by distributed systems of sensors and actuators with limited authority, i.e., if the controller is allowed to modify only moderately the natural modes and frequencies of the structure, then it should be possible to apply root perturbation techniques to predict analytically the behavior of the total system. Attention is given to the root perturbation formula first derived by Jacobi for infinitesimal perturbations which neglect the induced eigenvector perturbation, a more general form of Jacobi's formula, first-order structural equations and modal state vectors, state-space equations for damper-augmented structures, and modal damping prediction formulas.
Simultaneous solution of the equation of motion, equation of state and energy equation including heating and cooling processes for interstellar medium gives for a collapsing cloud a lower mass limit which is significantly smaller than the Jeans mass for the same initial density. The clouds with higher mass than this limiting mass collapse whereas clouds with smaller than critical mass pass through a maximum central density giving apparently similar clouds (i.e., same Av, size and central density) at two different phases of its evolution (i.e., with different life time). Preliminary results of chemistry in such an evolving oscillatory cloud show significant difference in abundances of some of the molecules in two physically similar clouds with different life times. The problems of depletion and short life time of evolving clouds appear to be less severe in such an oscillatory cloud.
The airloads-inflow model presented is ideally suited for rotor aeroelasticity and control analysis, especially where eigenvalues are sought. The model, which is derived from thin-airfoil theory with an extension that allows for other airfoil effects, is of hierarchical-lift type and may be simplified to any degree desired; the inflow is also expanded in shape functions which may be truncated to any desired texture. Both models are written in closed-form, state-variable equations without hidden states and without integrals over the wake.
A theoretical comet nucleus model has been written which employs 3 volatiles: CO, CO 2 , and H 2 O. It allows variations for equations of state: the Clausius/Clapeyron equation as well as other equations based on the latest thermodynamic data: the Brown/Ziegler relationship for CO 2 and CO at low pressure and the Lowe expression for water vapor at low pressure. It allows for variations in the physics of the outflow, from an essentially throttling mechanism (pure sublimation to space), to the Knudsen regime, to the Dusty-Gas Dynamic model. Variations in several parameters are explored. Results are compared with data from actual comets. None of the variations in parameter space produce a result that compares well with the comet data. All models yield a total gas production between 10²7; and 10²8; per second (low). This suggests that secondary mechanisms peripheral to this modeling effort, such as the opening and closing of cracks, a conductivity expression which depends upon porosity as well as temperature, etc., may play a more important role in the cometary efflux process than previously thought.
Evolutionary solar models are computed using a new stellar evolution code, MOSEC (Modular Stellar Evolution Code). This code has been designed with carefully controlled truncation errors in order to achieve a precision which reflects the increasingly accurate determination of solar interior structure by helioseismology. A series of models is constructed to investigate the effects of the choice of equation of state (OPAL or MHD-E, the latter being a version of the MHD equation of state recalculated by the author), the inclusion of helium and heavy-element settling and diffusion, and the inclusion of a simple model of mixing associated with the solar tachocline. The neutrino flux predictions are discussed, while the sound speed of the computed models is compared to that of the sun via the latest inversion of SOI-NMI p-mode frequency data. The comparison between models calculated with the OPAL and MHD-E equations of state is particularly interesting because the MHD-E equation of state includes relativistic effects for the electrons, whereas neither MHD nor OPAL do. This has a significant effect on the sound speed of the computed model, worsening the agreement with the solar sound speed. Using the OPAL equation of state and including the settling and diffusion of helium and heavy elements produces agreement in sound speed with the helioseismic results to within about +.-0.2%; the inclusion of mixing slightly improves the agreement.
The steady-state Navier-Stokes equations are solved for hypersonic flow about blunt axisymmetric bodies. The equations of motion are solved by successive approximations using an implicit finite-difference scheme. The results are compared with viscous shock-layer theory, experimental data, and time-dependent solutions of the Navier-Stokes equations. It is demonstrated that viscous shock-layer theory is sufficiently accurate for the range of flight conditions normally encountered by entry vehicles.
The steady-state Navier-Stokes equations are solved for hypersonic flow about blunt axisymmetric bodies. The equations of motion are solved by successive approximations using an implicit finite-difference scheme. The results are compared with viscous shock-layer theory, experimental data, and time-dependent solutions of the Navier-Stokes equations. It is demonstrated that viscous shock-layer theory is sufficiently accurate for the range of flight conditions normally encountered by entry vehicles.
Tables of thermodynamic properties of nitrogen are presented for the liquid and vapor phases for temperatures from the freezing line to 2000K and pressures to 10,000 bar. The tables include values of density, internal energy, enthalpy, entropy, isochoric heat capacity, isobaric heat capacity velocity of sound, the isotherm derivative, and the isochor derivative. The thermodynamic property tables are based on an equation of state, P=P (p,T), which accurately represents liquid and gaseous nitrogen for the range of pressures and temperatures covered by the tables. Comparisons of property values calculated from the equation of state with measured values for P-p-T, heat capacity, enthalpy, latent heat, and velocity of sound are included to illustrate the agreement between the experimental data and the tables of properties presented here. The coefficients of the equation of state were determined by a weighted least squares fit to selected P-p-T data and, simultaneously, to isochoric heat capacity data determined by corresponding states analysis from oxygen data, and to data which define the phase equilibrium criteria for the saturated liquid and the saturated vapor. The vapor pressure equation, melting curve equation, and an equation to represent the ideal gas heat capacity are also presented. Estimates of the accuracy of the equation of state, the vapor pressure equation, and the ideal gas heat capacity equation are given. The equation of state, derivatives of the equation, and the integral functions for calculating derived thermodynamic properties are included.
An equation of state is presented for liquid and gaseous oxygen for temperatures from 100 R to 600 R and pressures to 5000 psia. The pressure-density-temperature data available from the published literature have been reviewed, and appropriate corrections have been applied to bring experimental temperatures into accord with the International Practical Temperature Scale of 1968. Representative comparisons of property values calculated from the equation of state to measured values are included to illustrate the accuracy of the equation of state. The coefficients of the equation of state were determined by a weighted least squares fit to selected published data, and simultaneously to isochoric heat capacity data, and to data which define the phase equilibrium for the saturated liquid and saturated vapor. The equation of state is estimated to be accurate for the liquid to within 0.1 percent in density, to within 0.2 percent for the vapor below the critical temperature and for states above the critical temperatures to 250 K, and within 0.1 percent for supercritical states at temperatures from 250 K to 300 K. The vapor pressure equation is accurate to within + or - 0.01 K between the triple point and the critical point.
FORTRAN computer programs were developed to calculate the thermodynamic properties of Freon 14 and air for isentropic expansion from given plenum conditions. Thermodynamic properties for air are calculated with equations derived from the Beattie-Bridgeman nonstandard equation of state and, for Freon 14, with equations derived from the Redlich-Quang nonstandard equation of state. These two gases are used in scale model testing of model rocket nozzle flow fields which requires simulation of the prototype plume shape with a cold flow test approach. Utility of the computer programs for use in analytical prediction of flow fields is enhanced by arranging card or tape output of the data in a format compatible with a method-of-characteristics computer program.