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The nonlinear equations of motion for a rigid rotor restrained by three flexible springs representing the flapping, lagging and feathering motions are derived using Lagrange's equations for arbitrary angular rotations. These are reduced to a consistent set of nonlinear equations using nonlinear terms up to third order.
An overview is presented of some of the promising washout schemes which have been devised. The four schemes presented fall into two basic configurations; crossfeed and crossproduct. Various nonlinear modifications further differentiate the four schemes. One nonlinear scheme is discussed in detail. This washout scheme takes advantage of subliminal motions to speed up simulator cab centering. It exploits so-called perceptual indifference thresholds to center the simulator cab at a faster rate whenever the input to the simulator is below the perceptual indifference level. The effect is to reduce the angular and translational simulation motion by comparison with that for the linear washout case. Finally, the conclusions and implications for further research in the area of nonlinear washout filters are presented.
The results and methodology used to derive linear models from a nonlinear simulation are presented. It is shown that averaged positive and negative perturbations in the state variables can reduce numerical errors in finite difference, partial derivative approximations and, in the control inputs, can better approximate the system response in both directions about the operating point. Both explicit and implicit formulations are addressed. Linear models are derived for the F 100 engine, and comparisons of transients are made with the nonlinear simulation. The problem of startup transients in the nonlinear simulation in making these comparisons is addressed. Also, reduction of the linear models is investigated using the modal and normal techniques. Reduced-order models of the F 100 are derived and compared with the full-state models.
First-order nonlinear interactions of Tollmien-Schlichting waves of different frequencies and initial amplitudes in boundary-layer flows are analyzed by using the method of multiple scales. For the case of two waves, a strong nonlinear interaction exists if one of the frequencies w2 is twice the other frequency w1. Numerical results for flow past a flat plate show that this interaction mechanism is strongly destabilizing even in regions where either the fundamental or its harmonic is damped in the absence of the interaction. For the case of three waves, a strong nonlinear interaction exists when w3 = w2- w1. This combination resonance causes the amplitude of the wave with the difference frequency w3 to multiply many times in magnitude in a short distance even if it is damped in the absence of the interaction. The initial amplitudes play a dominant role in determining the changes in the amplitudes of the waves in both of these mechanisms.
Future space vehicles will be relatively large and flexible, and active control will be necessary to maintain geometrical configuration. While the stresses and strains in these space vehicles are not expected to be excessively large, their cumulative effects will cause significant geometrical nonlinearities to appear in the equations of motion, in addition to the nonlinearities caused by material properties. Since the only effective tool for the analysis of such large complex structures is the digital computer, it will be necessary to gain a better understanding of the nonlinear ordinary difference equations which result from the time discretization of the semidiscrete equations of motion for such structures.
Presented in this paper are the results of three nonlinear computer programs, KRASH, ACTION and DYCAST used to analyze the dynamic response of a twin-engine, low-wing airplane section subjected to a 8.38 m/s (27.5 ft/s) vertical impact velocity crash condition. This impact condition simulates the vertical sink rate in a shallow aircraft landing or takeoff accident. The three distinct analysis techniques for nonlinear dynamic response of aircraft structures are briefly examined and compared versus each other and the experimental data. The report contains brief descriptions of the three computer programs, the respective aircraft section mathematical models, pertinent data from the experimental test performed at NASA Langley, and a comparison of the analyses versus test results. Cost and accuracy comparisons between the three analyses are made to illustrate the possible uses of the different nonlinear programs and their future potential.
Recent wind tunnel tests and analytical studies show that a store mounted on a pylon with soft pitch stiffness provides substantial increase in flutter speed of fighter aircraft and reduces dependency of flutter on mass and inertia of the store. This concept, termed the decoupler pylon, utilizes a low frequency control system to maintain pitch alignment of the store during maneuvers and changing flight conditions. Under rapidly changing transient loads, however, the alignment control system may allow the store to momentarily bottom against a relatively stiff backup structure in which case the pylon stiffness acts as a hardening nonlinear spring. Such structural nonlinearities are known to affect not only the flutter speed but also the basic behavior of the instability. The influence of pylon stiffness nonlinearities or the flutter characteristics of wing mounted external stores is examined.
The nonlinear propagation equations for sound generated by a constant speed blade tip are presented. Propagation from a subsonic tip is treated as well as the various cases that can occur at supersonic speeds. Some computed examples indicate that the nonlinear theory correlates with experimental results better than linear theory for large amplitude waves. For swept tips that generate a wave with large amplitude leading expansion, the nonlinear theory predicts a cancellation effect that results in a significant reduction of both amplitude and impulse.
The formulation basis for establishing the static or dynamic equilibrium configurations of finite element models of structures which may behave in the nonlinear range are provided. With both geometric and time independent material nonlinearities included, the development is restricted to simple one and two dimensional finite elements which are regarded as being the basic elements for modeling full aircraft-like structures under crash conditions. Representations of a rigid link and an impenetrable contact plane are added to the deformation model so that any number of nodes of the finite element model may be connected by a rigid link or may contact the plane. Equilibrium configurations are derived as the stationary conditions of a potential function of the generalized nodal variables of the model. Minimization of the nonlinear potential function is achieved by using the best current variable metric update formula for use in unconstrained minimization. Powell's conjugate gradient algorithm, which offers very low storage requirements at some slight increase in the total number of calculations, is the other alternative algorithm to be used for extremely large scale problems.
Newton's method for nonlinear mechanics problems replaces the governing nonlinear equations by an iterative sequence of linear equations. When the linear equations are linear differential equations, the equations are usually solved by numerical methods. The iterative sequence in Newton's method can exhibit poor convergence properties when the nonlinear problem has multiple solutions for a fixed set of parameters, unless the iterative sequences are aimed at solving for each solution separately. The theory of the linear differential operators is often a better guide for solution strategies in applying Newton's method than the theory of linear algebra associated with the numerical analogs of the differential operators. In fact, the theory for the differential operators can suggest the choice of numerical linear operators. In this paper the method of variation of parameters from the theory of linear ordinary differential equations is examined in detail in the context of Newton's method to demonstrate how it might be used as a guide for numerical solutions.
A linearly unstable first harmonic radial mode of oscillation (period 0.86 s) of a hot white dwarf with a hydrogen envelope is analyzed in the nonlinear regime by application of the Stellingwerf technique to a nonlinear hydrocode. The mass and equilibrium luminosity of the hot white dwarf are 1.3 solar masses, and 1.6 x 10 to the 37th erg/s, respectively. A stable nonlinear mode is found to exist, in which appreciable shock damping occurs at and near the surface - the shock damping being the dominant stabilizing effect. This enhances the possibility that such an object can produce a pulsating X-ray source.
Recent wind tunnel tests and analytical studies show that a store mounted on a pylon with 'soft' pitch stiffness provides substantial increase in flutter speed of fighter aircraft and reduces dependency of flutter on mass and inertia of the store. This concept, termed the decoupler pylon, utilizes a low-frequency control system to maintain pitch alignment of the store during maneuvers and changing flight conditions. Under rapidly changing transient loads, however, the alignment control system may allow the store to momentarily bottom against a relatively stiff backup structure in which case the pylon stiffness acts as a hardening nonlinear spring. Such structural nonlinearities are known to affect not only the flutter speed but also the basic behavior of the instability. This paper examines the influence of pylon stiffness nonlinearities on the flutter characteristics of wing-mounted external stores.
The effectiveness of using minimization techniques for the solution of nonlinear structural analysis problems is discussed and demonstrated by comparison with the conventional pseudo force technique. The comparison involves nonlinear problems with a relatively few degrees of freedom. A survey of the state-of-the-art of algorithms for unconstrained minimization reveals that extension of the technique to large scale nonlinear systems is possible.
An analytical method has been developed for studying the combined effects of rotational coupling and nonlinear aerodynamics on aircraft response for specified control inputs. The method involves the simultaneous solution of two nonlinear equations which are functions of angle of attack, roll rate, and the control inputs. The method is applied to a number of maneuvers for a fighter type aircraft. Time history responses verified the usefulness of the analysis for predicting a variety of response characteristics caused by interacting nonlinear aerodynamic and inertial effects, including spin conditions.
A numerical technique has been developed which generates linear perturbation models from nonlinear real-time aircraft vehicle simulations at Langley Research Center. The technique is very general and can be applied to simulations of any system described by nonlinear differential equations. Linear models about various trim conditions have been obtained from simulations of the Boeing 737, Northrop F5-E, Cessna 180, Sikorsky CH-54, and NASA/Army Rotor Systems Research Aircraft (RSRA). One use of the derived linear models is as a validation and verification tool for the nonlinear simulation.
An analytical method was developed for studying the combined effects of rotational coupling and nonlinear aerodynamics on aircraft response for specified control inputs. The method involves the simultaneous solution of two nonlinear equations which are functions of angle attack, roll rate, and control inputs. The method was applied to a number of maneuvers for a fighter-type aircraft. Time history responses verified the usefulness of the analysis for predicting a variety of response characteristics caused by interacting nonlinear aerodynamic and inertial effects, including spin conditions.
Finite difference solutions of nonlinear partial differential equations require discretizations and consequently grid errors are generated. These errors strongly affect stability and convergence properties of difference models. Previously such errors were analyzed by linearizing the difference equations for solutions. Properties of mappings of decadence were used to analyze nonlinear instabilities. Such an analysis is directly affected by initial/boundary conditions. An algorithm was developed, applied to nonlinear Burgers equations, and verified computationally. A preliminary test shows that Navier-Stokes equations may be treated similarly.
Perturbation procedures and associated computational codes for determining nonlinear flow solutions were developed to establish a method for minimizing computational requirements associated with parametric studies of transonic flows in turbomachines. The procedure that was developed and evaluated was found to be capable of determining highly accurate approximations to families of strongly nonlinear solutions which are either continuous or discontinuous, and which represent variations in some arbitrary parameter. Coordinate straining is employed to account for the movement of discontinuities and maxima of high gradient regions due to the perturbation. The development and results reported are for the single parameter perturbation problem. Flows past both isolated airfoils and compressor cascades involving a wide variety of flow and geometry parameter changes are reported. Attention is focused in particular on transonic flows which are strongly supercritical and exhibit large surface shock movement over the parametric range studied; and on subsonic flows which display large pressure variations in the stagnation and peak suction pressure regions. Comparisons with the corresponding 'exact' nonlinear solutions indicate a remarkable accuracy and range of validity of such a procedure.