Geometric Theory of Functional Differential Equations
Functional differential equations for determining n-dimemensional vector spaces - Liapunov function asymptotic theory of linear systems, and Floquet theory
Hale, J. K.↗
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Motion and stability of a spinning spring-mass system in orbit.
Equations of motion of elastic dumbbell satellite and two-mass spring-connected satellite in orbit, using energy and Floquet theory to investigate spinning motion stability
Crist, S. A.↗
Spin stability of dual-spin spacecrafts.
The spin-axis attitude stability in torque-free environment of dual-spin spacecrafts with energy losses in both bodies is studied by analytical means. A new mathematical model, which assumes the high speed rotor to be symmetrical, is introduced and analyzed. The linearized variational system of equations for the model with periodic coefficients is transformed to a kinematically similar autonomous system (according to Floquet and Liapunov) which can be analyzed routinely using established procedure. As an example, this technique is used on a recent mathematical model of the SAS-A spacecraft to establish steady state stability over a wide range of speeds.
Guha, A. K.↗
A new model for dual-spin satellites.
A new mathematical model for the analysis of dissipative dual-spin satellites has been developed. This model makes explicit use of the fact that the high speed rotor is symmetric by modeling the rotor energy dissipation with symmetrically placed dampers. This permits the use of a quasi-holonomic transformation for the reduction of the periodic system of variational equations to an autonomous system in accordance with the Floquet Theory and the Liapunov Reducibility Theorem.-
Guha, A. K.↗
The dynamic behavior of articulated pipes conveying fluid with periodic flow rate.
The plane motion of two rigid, straight articulated pipes conveying fluid is examined. In contrast to previous work, the flow rate is not taken as constant, but is allowed to have small periodic oscillations about a mean value, as would be expected in a pump-driven system. It is shown that in the presence of such disturbances, both parametric and combination resonances are possible. When the system can also admit loss of stability by static buckling or by flutter, it is found that the presence of small periodic disturbances constitutes a destabilizing effect. Floquet theory and converging infinite determinant expansions are used to illustrate a basic difference between systems which lose stability by divergence and those that lose stability by flutter. An algebraic criterion is obtained for the minimum amplitude of flow-rate oscillation required for the system to be affected by the presence of small disturbances.
Bohn, M. P.↗
Aeroelastic stability of coupled flap-lag motion of hingeless helicopter blades at arbitrary advance ratios
Equations for large amplitude coupled flap-lag motion of a hingeless elastic helicopter blade in forward flight are derived. Only a torsionally rigid blade excited by quasi-steady aerodynamic loads is considered. The effects of reversed flow together with some new terms due to radial flow are included. Using Galerkin's method the spatial dependence is eliminated and the equations are linearized about a suitable equilibrium position. The resulting system of homogeneous periodic equations is solved using multivariable Floquet-Liapunov theory, and the transition matrix at the end of the period is evaluated by two separate methods. Computational efficiency of the two numerical methods is compared. Results illustrating the effects of forward flight and various important blade parameters on the stability boundaries are presented.
Friedmann, P.↗
Flap-lag dynamics of hingeless helicopter blades at moderate and high advance ratios
Equations for large amplitude coupled flaplag motion of a hingeless elastic helicopter blade in forward flight are derived. Only a torsionally rigid blade exicted by quasi-steady aerodynamic loads is considered. The effects of reversed flow together with some new terms due to forward flight are included. Using Galerkin's method the spatial dependence is eliminated and the equations are linearized about a suitable equilibrium position. The resulting system of equations is solved using multivariable Floquet-Liapunov theory, and the transition matrix at the end of the period is evaluated by two separate methods. Results illustrating the effects of forward flight and various important blade parameters on the stability boundaries are presented.
Friedman, P.↗
Aeroelastic stability of periodic systems with application to rotor blade flutter
The dynamics of a helicopter blade in forward flight are described by a system of linear differential equations with periodic coefficients. The stability of this periodic aeroelastic system is determined, using multivariable Floquet-Liapunov theory. The transition matrix at the end of the period is evaluated by: (1) direct numerical integration, and (2) a new, approximate method, which consists in approximating a periodic function by a series of step functions. The numerical accuracy and efficiency of the methods is compared, and the second method is shown to be superior by far. Results illustrating the effect of the periodic coefficients and various blade parameters are presented.
Friedmann, P.↗
Specialists Meeting on Rotorcraft Dynamics, Moffett Field, Calif., February 13-15, 1974, Proceedings
Analysis of specific problems in rotorcraft dynamics. Topics include hingeless rotor theory, dynamic stall modelling, periodic systems identification, analysis of complex systems with phasing matrices, flapping stability, flap-lag dynamics at high advance ratios, finite element analysis and fuselage free-vibration characteristics, coupled rotor/frame vibration methods, gust response characteristics with unsteady stall effects, antiresonance theory, cyclic feathering motions and dynamic loads, control load envelope shaping, rotor aeroelasticity, use of Floquet theory, theory of proprotors and tilt-rotors, two-bladed teetering rotors, stability of air and ground resonance, vertical-plane pendulum absorbers, multicyclic jet-flap control, engine/frame interface dynamics, and others. The minutes of the question and answer periods following the presentations are presented in the supplement. Individual items are announced in this issue.
Source record↗
An elementary magnetic attitude control system
A three-axis closed-loop attitude-control system for earth-observatory momentum-bias spacecraft is proposed. A horizon scanner and magnetometer are employed to measure attitude errors; no yaw sensor is required. Appropriate control signals are generated and used to command variable-strength electromagnets, which interact with the geomagnetic field to reduce observed errors. This system provides for initial acquisition, precession control, nutation damping, and pitch-axis momentum control. All functions are performed autonomously - no ground-station interaction is required. Analytical expressions predicting system response are compared with numerical solutions of the governing equations, and with the results of the application of Floquet Theory.-
Stickler, A. C.↗
Bragg interactions in periodic media
The interaction of electromagnetic waves of wavelength lambda with periodic structures of spatial period lambda are studied. The emphasis of the work is on Bragg interactions where lambda approximately equal to 2 lambda/N and the Bragg order N takes on the values 1, 2,.... An extended coupled waves (ECW) theory is developed for the case N greater or equal to 2 and the results of the theory are found to compare favorably with the exact results of Floquet theory. Numerous numerical results are displayed as Brillouin diagrams for the first few Bragg orders. Moreover, explicit expressions for coupling coefficients, bandgap shifts and bandgap widths are derived for singly periodic media. Particular note is taken of phase speeding effects.
Jaggard, D. L.↗
Waves in active and passive periodic structures - A review
The theory and recent applications of waves in periodic structures are reviewed. Both the Floquet and coupled waves approach are analyzed in some detail. The theoretical part of the paper includes wave propagation in unbounded and bounded active or passive periodic media, wave scattering from periodic boundaries, source radiation (dipole, Cerenkov, transition, and Smith-Purcell) in periodic media, and pulse transmission through a periodic slab. The applications part covers the recent development in a variety of fields: distributed feedback oscillators, filters, mode converters, couplers, second-harmonic generators, deflectors, modulators, and transducers in the fields of integrated optics and integrated surface acoustics. Work on insect compound eyes, mechanical structures, ocean waves, pulse compressions, temperature waves, and cholesteric liquid crystals, and particles interaction with crystals is briefly reviewed, especially in the case of zeolite crystals and superlattices. Recent advances in fabrication techniques for very fine gratings are also covered.
Elachi, C.↗
Higher-order Bragg coupling in periodic media with gain or loss
Wave propagation in longitudinally periodic media with gain or loss is considered. An extended coupled-waves approach and a Floquet approach are used to study index and gain (or loss) coupling in periodic media at the first few Bragg resonances by the use of Brillouin diagrams. Phase speeding or slowing that is dependent upon the type of coupling occurs in periodic media. Even and odd resonances display different dispersion characteristics. Several examples of coupling due to multiharmonic periodicities are also covered. Coupling or feedback strength can be continuously varied by changing the relative phase of multiharmonic periodicities.
Jaggard, D. L.↗
Efficient numerical treatment of periodic systems with application to stability problems
Two efficient numerical methods for dealing with the stability of linear periodic systems are presented. Both methods combine the use of multivariable Floquet-Liapunov theory with an efficient numerical scheme for computing the transition matrix at the end of one period. The numerical properties of these methods are illustrated by applying them to the simple parametric excitation problem of a fixed end column. The practical value of these methods is shown by applying them to some helicopter rotor blade aeroelastic and structural dynamics problems. It is concluded that these methods are numerically efficient, general and practical for dealing with the stability of large periodic systems.
Friedmann, P.↗
Evaluation of MOSTAS computer code for predicting dynamic loads in two bladed wind turbines
Calculated dynamic blade loads were compared with measured loads over a range of yaw stiffnesses of the DOE/NASA Mod-O wind turbine to evaluate the performance of two versions of the MOSTAS computer code. The first version uses a time-averaged coefficient approximation in conjunction with a multi-blade coordinate transformation for two bladed rotors to solve the equations of motion by standard eigenanalysis. The second version accounts for periodic coefficients while solving the equations by a time history integration. A hypothetical three-degree of freedom dynamic model was investigated. The exact equations of motion of this model were solved using the Floquet-Lipunov method. The equations with time-averaged coefficients were solved by standard eigenanalysis.
Kaza, K. R. V.↗
Review of analysis methods for rotating systems with periodic coefficients
Two of the more common procedures for analyzing the stability and forced response of equations with periodic coefficients are reviewed: the use of Floquet methods, and the use of multiblade coordinate and harmonic balance methods. The analysis procedures of these periodic coefficient systems are compared with those of the more familiar constant coefficient systems.
Dugundji, J.↗
Modal control of an unstable periodic orbit
Floquet theory is applied to the problem of designing a control system for a satellite in an unstable periodic orbit. Expansion about a periodic orbit produces a time-periodic linear system, which is augmented by a time-periodic control term. It is shown that this can be done such that (1) the application of control produces only inertial accelerations, (2) positive real Poincareexponents are shifted into the left half-plane, and (3) the shift of the exponent is linear with control gain. These developments are applied to an unstable orbit near the earth-moon L(3) point pertubed by the sun. Finally, it is shown that the control theory can be extended to include first order perturbations about the periodic orbit without increase in control cost.
Wiesel, W.↗
Some analysis methods for rotating systems with periodic coefficients
Two of the more common procedures for analyzing the stability and forced response of equations with periodic coefficients are reviewed: the use of Floquet methods, and the use of multiblade coordinate and harmonic balance methods. The analysis procedures of these periodic coefficient systems are compared with those of the more familiar constant coefficient systems. Previously announced in STAR as N82-23702
Dugundji, J.↗