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The nonlinear flap-lag coupled oscillation of torsionally rigid rotor blades in forward flight is examined using a set of consistently derived equations by the asymptotic expansion procedure of multiple time scales. The regions of stability and limit cycle oscillation are presented. The roles of parametric excitation, nonlinear oscillation, and forced excitation played in the response of the blade are determined.
Installation practices are recommended that will increase rate of heat exchange between the thermometric sensing element and the cryogenic fluid, in addition to bringing about a reduction in the rate of undesired heat transfer to higher temperature objects. Formulas and numerical data are given that help to estimate the magnitude of heat conduction errors and of time lag in response.
A solution is presented to an optimization problem for time lag systems by the classical method of Lagrange multipliers in a Banach space. Following terminology and assumption definitions, the regularity and controllability of the Lagrange multipliers problem is discussed, and a set of necessary conditions for an optimal control is derived. In conclusion, the solution existence, uniqueness, and sufficiency are established.
The aeroelastic instability of the coupled nonlinear flap-lag motion of a torsionally rigid helicopter blade is treated by using the perturbation method in multiple time scales. The nonlinearities present in the equations are those arising from the inclusion of moderately large deflections in the inertia and aerodynamic loading terms. The stability boundaries, amplitudes of nonlinear response, and conditions for existence of limit cycles are obtained analytically. Thus the different roles played by the forcing, parametric excitation, and nonlinear coupling in affecting the solution can be easily identified. Numerical results illustrating the behavior of the blade are presented.
A bipropellant stability model is presented in which feed system inertance and capacitance are treated along with injection pressure drop and distinctly different propellant time lags. The model is essentially an extension of Crocco's and Cheng's monopropellant model to the bipropellant case assuming that the feed system inertance and capacitance along with the resistance are located at the injector. The neutral stability boundaries are computed in terms of these parameters to demonstrate the interaction among them.
The velocity lag of micrometer size spherical particles is theoretically determined for gas particle mixtures passing through a stationary normal shock wave and also for particles embedded in an oscillating gas flow. The particle sizes and densities chosen are those considered important for laser Doppler velocimeter applications. The governing equations for each flow system are formulated. The deviation from Stokes flow caused by inertial, compressibility, and rarefaction effects is accounted for in both flow systems by use of an empirical drag coefficient. Graphical results are presented which characterize particle tracking as a function of system parameters.
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.
Installation practices are recommended that will increase rate of heat exchange between the thermometric sensing element and the cryogenic fluid and that will reduce the rate of undesired heat transfer to higher-temperature objects. Formulas and numerical data are given that help to estimate the magnitude of heat-conduction errors and of time lag in response.
In this analysis of the phase-lag effect we utilize more extensive primary proton and helium spectral data from balloon studies of the University of New Hampshire group, from the Pioneer 8 and 9 spacecraft, and from ground-level monitors during 1965-72. If the rigidity dependence of the diffusion coefficient at earth did not change from 1968-72, then the data imply the nonseparability of the radial and rigidity parts of the diffusion coefficient.
Design of a discrete optimal regulator requires the a priori knowledge of a mathematical model for the system of interest. Because a second-order model with transport lag is very amenable to control computations and because this type of model has been used previously to represent certain high order single input-single output processes, an adaptive controller was designed based upon adjustment of controls computed for such a model. An extended Kalman filter was utilized for tracking the model parameters which were subsequently used to update a set of optimal control gains. Favorable results were obtained in applying this procedure to the control of several examples including a ninth order nonlinear process.
An experimental investigation was conducted of the dynamic response and inplane stability associated with a new soft-inplane helicopter rotor. The unique feature of this rotor was the use of an internal elastomeric damper to restrain the blade inplane motion about the lead-lag hinge. The properties of the elastomer were selected to provide both a nominal first inplane frequency ratio of 0.65 and sufficient damping to eliminate the need for additional external damping sources to prevent ground resonance on a typical fuselage structure. For this investigation a 1/5-scale aeroelastic model was used to represent the rotor. The four-blade model had a diameter of 3.05 m (10 ft) and a solidity of 0.103. The first out-of-plane frequency ratio was 1.06. The model was tested in hover and in forward flight up to an advance ratio of 0.45. At each forward speed the rotor lift was varied up to simulated maneuver conditions. The measured rotor loads and response were within acceptable limits, and no adverse response qualities were observed. Moderate out-of-plane hub moments were measured, even for zero lift, to indicate the beneficial control power available for this design. Blade inplane stability testing indicated that the rotor system damping remained at moderate levels throughout the operating envelope.
The differential equations describing the flap-lag-torsional motion of a flexible rotor blade including third-order nonlinearities were derived for hover and forward flight. Making use of the two boundary conditions, those equations were reduced to a set of three integro partial differential equations written in terms of the flexural deflections and the torsional variable.
Development of a method of analysis capable of predicting accurately the fracture behavior of unidirectional hybrid (buffer strip) composite laminates was studied. Three particular solutions are discussed in detail: broken fibers in a unidirectional half-plane; adjoined half planes of different fiber and matrix properties; and the solution of two half planes bounding a third distinct region of finite width. This finite width region represents a buffer strip and primary attention is given to the potential of this strip to arrest a crack that originates in one of the half planes. A materials modeling approach using the classical shear lag assumption to describe the stress transfer between fibers was analyzed. Explicit fiber and matrix properties of the three regions are retained, and changes in the laminate behavior as a function of the relative material properties, buffer strip width, and initial crack length are discussed.
The aeroelastic stability and response problem of the coupled flap-lag-torsional dynamics of an isolated hingeless rotor blade in forward flight is considered. Linear, quasi-steady aerodynamics below stall is included. The spatial dependence of the partial differential nonliner equations of motion is discretized using a multimodal Galerkin method. The nonlinear time dependent equilibrium position (steady state response) about which the equations are linearized is obtained by solving a sequence of linear periodic response problems (quasilinearization). Results illustrating blade behavior in forward flight for both soft-in-plane and stiff-in-plane designs are presented. Quasilinearization provides a clear indication of the cases when nonlinear terms due to moderate deflections are important. The results indicate that nonlinearities affect system stability much more than system response. This implies that for obtaining blade vibrations and loads, approximate analytical models based on linearized formulations could be employed, whereas for determining blade stability a more accurate nonlinear analysis would be required.
The Lagged Average Forecast (LAF) method differs from the Monte Carlo Forecast (MCF) method in the definition of the ensemble of initial states which are used to generate the ensemble of forecasts. The LAF initial states are the current analysis and the forecasts made from previous analyses verifying the current time. Thus the LAF ensemble is composed of forecasts which are made by a regular operational system of numerical weather prediction and the LAF method is therefore operationally attractive. The application of the authors' previous ideas and results to an operational model requires the resolution of what might be called the degrees of freedom problem, i.e., how to obtain a homogeneous sample large enough to calculate stable statistics. It is suggested that this problem may be solved by carefully modeling the required statistics in terms of a small set of parameters and then estimating only these few parameters from the data. It is noted that there may be considerable information in each initial ensemble relating to the predictability of each particular case, and that this information may be incorporated in the model of the statistics.
The fracture behavior of a debonded zone of finite width with no longitudinal damage in the unidirectional ply is predicted and the solution is then extended to include longitudinal matrix yielding and splitting in the unidirectional ply at the crack tip. The shear-lag assumption is used to describe the shear transfer between fibers. The fracture behavior of the laminate is studied as a function of initial crack length, constraint ratio, and width of the debonded zone. Results indicate that debonding can reduce the maximum fiber stress at the crack tip on the order of ten percent. This effect is maximum for a debond width of two or three fiber spacings and is independent of the initial crack length. As the debond width grows beyond this point, the maximum stress increases. For widths of about ten fiber spacings or more, the maximum fiber stress is larger than for the fully bonded case. In the presence of longitudinal matrix damage the same general behavior is found; however, the location of the maximum fiber stress is quite complex.
Estimating time and time-lag in time-of-flight velocimeters is investigated. Statistics of a filtered Poisson point process is given. A Maximum Likelihood Estimator is compared with suboptimum estimators in terms of robustness. For a dominating background combined spatial and temporal processing can improve the robustness compared with purely temporal processing. Schemes for the spatial filters are given.
The effects of pretwist, precone, setting angle, Coriolis forces and second degree geometric nonlinearities on the natural frequencies, steady state deflections and mode shapes of rotating, torsionally rigid, cantilevered beams were studied. The governing coupled equations of flap lag extensional motion are derived including the effects of large precone and retaining geometric nonlinearities up to second degree. The Galerkin method, with nonrotating normal modes, is used for the solution of both steady state nonlinear equations and linear perturbation equations. Parametric indicating the individual and collective effects of pretwist, precone, Coriolis forces and second degree geometric nonlinearities on the steady state deflection, natural frequencies and mode shapes of rotating blades are presented. It is indicated that the second degree geometric nonlinear terms, which vanish for zero precone, can produce frequency changes of engineering significance. Further confirmation of the validity of including those generated by MSC NASTRAN. It is indicated that the linear and nonlinear Coriolis effects must be included in analyzing thick blades. The Coriolis effects are significant on the first flatwise and the first edgewise modes.