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Various topics related to the control of large space structures are discussed. Equations of motion for distributed systems, eigenvalue problems, modal equations, control implementation, and the Langley beam experiment are discussed.
A steady, axisymmetric model of the general circulation is developed as a basis for climate stability studies. The model includes the effects of heating, rotation, and internal friction, but neglects topography. It is assumed that the axisymmetric flow may be modeled by making the Boussinesq and deep convection approximations. The hydrostatic assumption is not made, thus permitting the advective terms to be included in the vertical equation of motion. The initial set of five primitive equations is reduced to three equations in terms of the zonal velocity, meridional streamfunction, and the potential temperature perturbation. The application of the Boussinesq, deep convection, and quasi-geostrophic assumptions limits the ranges of the heating and rotation rates. For values not too far from typical atmospheric values, the model produces a stability boundary separating Hadley from Rossby flow. The boundary is characterized by a particular value of vertical wind shear, which suggests that baroclinic instability is the primary mechanism for the loss of stability. The initial growth rates are largest for longitudinal waves 4-7, also in agreement with studies of baroclinic instability.
Tools required to understand and predict in terms of its underlying causes the vibratory excitation arising from meshing spiral bevel gears are developed. A generalized three component transmission error of meshing spiral bevel gears is defined. Equations are derived that yield the three components of the generalized transmission error in terms of deviations of tooth running surfaces from equispaced perfect spherical involute surfaces and tooth/gearbody elastic deformations arising from the three components of the generalized force transmitted by the meshing gears. A method for incorporating these equations into the equations of motion of a gear system is described. Equations are derived for the three components of the generalized force transmitted by the gears which are valid whenever inertial effects of the meshing gears and their supports are negligible. Bearing offsets from the positions occupied by the shaft centerlines of perfect spherical involute bevel gears and bearing/bearing support flexibilities enter into the computation of these forces.
Methods of grid adaptation are reviewed and a method is developed with the capability of adaptation to several flow variables. This method is based on a variational approach and is an algebraic method which does not require the solution of partial differential equations. Also the method was formulated in such a way that there is no need for any matrix inversion. The method is used in conjunction with the calculation of hypersonic flow over a blunt nose. The equations of motion are the compressible Navier-Stokes equations where all viscous terms are retained. They are solved by the MacCormack time-splitting method and a movie was produced which shows simulataneously the transient behavior of the solution and the grid adaptation. The results are compared with the experimental and other numerical results.
The methods of grid adaption are reviewed and a method is developed with the capability of adaption to several flow variables. This method is based on a variational approach and is an algebraic method which does not require the solution of partial differential equations. Also the method has been formulated in such a way that there is no need for any matrix inversion. The method is used in conjunction with the calculation of hypersonic flow over a blunt nose body. The equations of motion are the compressible Navier-Stokes equations where all viscous terms are retained. They are solved by the MacCormack time-splitting method. A movie has been produced which shows simultaneously the transient behavior of the solution and the grid adaption.
Methods of grid adaption are reviewed and a method is developed with the capability of adaption to several flow variables. This method is based on a variational approach and is an algebraic method which does not require the solution of partial differential equations. The method has also been formulated in such a way that there is no need for any matrix inversion. The method is used in conjunction with the calculation of hypersonic flow over a blunt nose. The equations of motion are the compressible Navier-Stokes equations where all viscous terms are retained. They are solved by the MacCormack time-splitting method and a movie has been produced which shows simultaneously the transient behavior of the solution and the grid adaption. The results are compared with the experimental and other numerical results.
This paper describes the use of maximum likelihood parameter estimation unknown parameters appearing in a nonlinear vehicle motion filter. The filter uses the kinematic equations of motion of a rigid body in motion over a spherical earth. The nine states of the filter represent vehicle velocity, attitude, and position. The inputs to the filter are three components of translational acceleration and three components of angular rate. Measurements used to update states include air data, altitude, position, and attitude. Expressions are derived for the elements of filter matrices needed to use air data in a body-fixed frame with filter states expressed in a geographic frame. An expression for the likelihood functions of the data is given, along with accurate approximations for the function's gradient and Hessian with respect to unknown parameters. These are used by a numerical quasi-Newton algorithm for maximizing the likelihood function of the data in order to estimate the unknown parameters. The parameter estimation algorithm is useful for processing data from aircraft flight tests or for tuning inertial navigation systems.
Flow through pump components, such as an inducer and an impeller, is efficiently simulated by solving the incompressible Navier-Stokes equations. The solution method is based on the pseudocompressibility approach and uses an implicit-upwind differencing scheme together with the Gauss-Seidel line relaxation method. the equations are solved in steadily rotating reference frames and the centrifugal force and the Coriolis force are added to the equation of motion. Current computations use a one-equation Baldwin-Barth turbulence model which is derived from a simplified form of the standard k-epsilon model equations. The resulting computer code is applied to the flow analysis inside a generic rocket engine pump inducer, a fuel pump impeller, and SSME high pressure fuel turbopump impeller. Numerical results of inducer flow are compared with experimental measurements. In the fuel pump impeller, the effect of downstream boundary conditions is investigated. Flow analyses at 80 percent, 100 percent, and 120 percent of design conditions are presented.
Montgomery (1959) first studied how Alfven waves steepen to form collisionless shocks by use of a two-fluid model. He solved the coupled Lorentz equations of motion for electrons and ions and Maxwell equations by a perturbation technique and showed that linearized polarized Alfven waves in the course of time steepen to form shocks. We have extended this work and have obtained exact simple wave solutions and show that in the limit of slow fluid time scales, we recover the earlier result of Montgomery. For an arbitrary fluid time scale, the solutions include elliptically polarized Alfven waves which also steepen and evolve into shocks. These nonlinear Alfven waves are important in the study of the dynamic solar wind discontinuities.
This study presents a transient nonlinear finite element analysis within the realm of a multi-body dynamics formulation for determining the dynamic response of a moderately thick laminated shell undergoing a rapid and large rotational motion and nonlinear elastic deformations. Nonlinear strain measure and rotation, as well as 'the transverse shear deformation, are explicitly included in the formulation in order to capture the proper motion-induced stiffness of the laminate. The equations of motion are derived from the virtual work principle. The analysis utilizes a shear deformable shallow shell element along with the co-rotational form of the updated Lagrangian formulation. The shallow shell element formulation is based on the Reissner-Mindlin and Marguerre theory.
A recently developed method for solving optimal trajectory problems uses a piecewise-polynomial representation of the state and control variables, enforces the equations of motion via a collocation procedure, and thus approximates the original calculus-of-variations problem with a nonlinear-programming problem, which is solved numerically. This paper identifies this method as a direct transcription method and proceeds to investigate the relationship between the original optimal-control problem and the nonlinear-programming problem. The discretized adjoint equation of the collocation method is found to have deficient accuracy, and an alternate scheme which discretizes the equations of motion using an explicit Runge-Kutta parallel-shooting approach is developed. Both methods are applied to finite-thrust spacecraft trajectory problems, including a low-thrust escape spiral, a three-burn rendezvous, and a low-thrust transfer to the moon.
This paper introduces two new seven-parameter representations for spacecraft attitude dynamics modeling. The seven parameters are the three components of the total system angular momentum in the spacecraft body frame; the three components of the angular momentum in the inertial reference frame; and an angle variable. These obey a single constraint as do parameterizations that include a quaternion; in this case the constraint is the equality of the sum of the squares of the angular momentum components in the two frames. The two representations are nonsingular if the system angular momentum is non-zero and obeys certain orientation constraints. The new parameterizations of the attitude matrix, the equations of motion, and the relation of the solution of these equations to Euler angles for torque-free motion are developed and analyzed. The superiority of the new parameterizations for numerical integration is shown in a specific example.
The accurate modelling of the translational behavior of a drag-free satellite in an almost circular near-earth orbit is investigated. All short and long period position fluctuations in the satellite's coordinates down to 1 m are determined. The general zonal and tesseral harmonic effects are considered as well as lunar and solar effects. A noncanonical approach is followed. A mean orbital plane is chosen so that no short period out-of-plane fluctuations greater than second order occur. Short period radial and cross-track fluctuations and in-track fluctuations are computed. Long period and secular rates of the mean elements which define the slowly varying plane of reference are determined to third order in the small quantities. The equations of motion are integrated analytically away from critical inclination and tesseral resonance. The resonant situation of a sun-synchronous orbit is discussed as a special case. The long-period behavior of a lunar orbiter is also considered, including the effects due to the inclination of the earth's apparent orbit about the moon and those described by Cassini laws on the equations of motion. Both resonant and nonresonant low orbits, and high orbits are discussed.
In Hořava-Lifshitz gravity, a putative consistent theory of quantum gravity for which there is evidence for both black hole thermodynamics and a holographic construction, spacetime is endowed with a preferred dynamical spacelike foliation. The theory has a leaf reparameterization symmetry that is neither global nor local gauge, hyperbolic and elliptic equations of motion, a lack of splittability, and universal horizon black hole solutions. The reparameterization symmetry is “mixed”: it is a local symmetry in one coordinate yet global on each leaf. More broadly it is an example of both unfree and projectable gauge symmetries. The mixed symmetry and associated charge has not yet been accounted for in calculations of universal horizon thermodynamics in Hořava-Lifshitz gravity. This has led to problems, in particular the failure of the first law in a class of asymptotically AdS solutions where the normal to the leaves of the foliation is not aligned with the time translation Killing vector at infinity. We show how the dynamics of the charge corresponding to this symmetry coupled with the other features above resolves this issue. We then briefly comment how this mixed symmetry, the corresponding charge, and the elliptic equations of motion also conspire to evade recent holographic arguments for only local gauge fields in consistent theories of quantum gravity due to the lack of splittability of the elliptic equation and associated mode.
Here we develop a new method to describe electromagnetic observables of open-shell nuclei with two nucleons outside a closed shell. This approach combines the equation-of-motion coupled-cluster method for such systems and the Lorentz integral transform technique, expanding the applicability of coupled-cluster theory for electromagnetic observables beyond closed-(sub)shell nuclei. To validate this new approach, we compute the non-energy-weighted dipole sum rule and the dipole polarizability of 16,24 O in both the closed-(sub)shell and the new equation-of-motion coupled-cluster frameworks, finding agreement within error bars. We then analyze the evolution of the dipole polarizability along the oxygen and calcium isotopic chains. Our predictions agree well with available experimental data and other available theoretical calculations for the closed-(sub)shell 16,22 O and the open-shell 18 O. In the calcium isotopes, we observe that our dipole polarizability predictions for open-shell nuclei are lower than those of closed-(sub)shell nuclei. We expect that our predictions for 24 O and 54,56 Ca will motivate future experimental studies at the drip line.
MATLAB Stability and Control Toolbox (MASCOT) utilizes geometric, aerodynamic, and inertial inputs to calculate air vehicle stability in a variety of critical flight conditions. The code is based on fundamental, non-linear equations of motion and is able to translate results into a qualitative, graphical scale useful to the non-expert. MASCOT was created to provide the conceptual aircraft designer accurate predictions of air vehicle stability and control characteristics. The code takes as input mass property data in the form of an inertia tensor, aerodynamic loading data, and propulsion (i.e. thrust) loading data. Using fundamental nonlinear equations of motion, MASCOT then calculates vehicle trim and static stability data for the desired flight condition(s). Available flight conditions include six horizontal and six landing rotation conditions with varying options for engine out, crosswind, and sideslip, plus three take-off rotation conditions. Results are displayed through a unique graphical interface developed to provide the non-stability and control expert conceptual design engineer a qualitative scale indicating whether the vehicle has acceptable, marginal, or unacceptable static stability characteristics. If desired, the user can also examine the detailed, quantitative results.
Based on the integration of classical differential equations of motion of bearing elements, a real-time dynamic simulation of ball bearing performance is coupled with simultaneous modeling of thermal interactions between the bearing elements. The transient heat generations are time averaged over a significantly larger time step before they are used to compute the changes in bearing temperature fields. Such an averaging algorithm eliminates the numerical difficulties associated with simultaneous integration of mechanical thermal differential equations with vastly different time scales. Under stable operation, the step change in temperatures at each thermal time step converges to a steady-state solution. Following is a list of accomplishments in the present investigation: (1) A real-time dynamic performance of both 440C and hybrid ball bearings in a LOX environment is modeled in both transient and steady-state time domains, as the inner race accelerates to a prescribed operating speed. (2) The frictional interactions at the ball to race contacts are modeled by experimentally measured traction coefficients as a function of slide-to-roll ratios. (3) Bearing heat generation, as defined by the heat transferred to circulating LOX, is validated against experimental values determined by measured LOX temperatures under prescribed pressures and flow rates. The predicted heat generation is in good agreement with the experimental values. (4) The steady-state solutions are shown to be independent of initial conditions. This establishes convergence of the numerical integration of a differential equation of motion of the bearing elements. (5) Churning and drag losses in the circulating LOX constitutes the majority of bearing heat generation under the experimental test conditions. (6) The heat dissipated in frictional interactions in hybrid bearings is significantly lower in hybrid bearings in comparison to that simulated in all-steel bearings, while the churning and drag losses are relatively unchanged.
This report describes the development of an aeroelastic analysis capability for composite helicopter rotor blades with straight and swept tips, and its application to the simulation of helicopter vibration reduction through structural optimization. A new aeroelastic model is developed in this study which is suitable for composite rotor blades with swept tips in hover and in forward flight. The hingeless blade is modeled by beam type finite elements. A single finite element is used to model the swept tip. Arbitrary cross-sectional shape, generally anisotropic material behavior, transverse shears and out-of-plane warping are included in the blade model. The nonlinear equations of motion, derived using Hamilton's principle, are based on a moderate deflection theory. Composite blade cross-sectbnal properties are calculated by a separate linear, two-dimensional cross section analysis. The aerodynamic loads are obtained from quasi-steady, incompressible aerodynamics, based on an implicit formulation. The trim and steady state blade aeroelastic response are solved in a fully coupled manner. In forward flight, where the blade equations of motion are periodic, the coupled trim-aeroelastic response solution is obtained from the harmonic balance method. Subsequently, the periodic system is linearized about the steady state response, and its stability is determined from Floquet theory.