Lindau, Jules Washington
13 February 2009
A more robust method for solving the governing equations of a one-dimensional stage-by-stage dynamic compression system model was developed and validated. The improved method was then applied to two-dimensional post-stall models. The improvement in robustness was achieved by modeling the governing equations with upwind differencing and use of implicit time integration. A special form of upwind flux, flux difference splitting with source term treatment, FDSS, was developed for the model. A two-dimensional axisymmetric model was developed to allow post-stall modeling of split flowpath systems such as turbofans. This model was an entirely new concept. Additionally, a two-dimensional axial-circumferential model of rotating stall cell development and propagation was developed based on previous work. All of the models developed applied upwind differencing techniques to improve upon central-difference methods. / Ph. D.
This thesis deals with high angle of attack behaviour of a generic delta wing model aircraft. A high angle of attack wind tunnel database has been generated for this aircraft and based upon the bifurcation analysis of the data and the results of extensive simulations, it has been shown in the thesis that the post stall behaviour of this aircraft is both unstable and unpredictable. Unpredictability of aircraft behaviour arises from the fact that the aircraft response is oscillatory and divergent; the aircraft state trajectories do not settle down to any stable limit set and very often exceed valid aerodynamic database limits. This unpredictability of behaviour raises a major difficulty in the design of a procedure to recover the aircraft to normal flight regime in case the aircraft stalls and departs accidentally. A new methodology has been presented in this thesis to recover such unstable aircraft. In this methodology, a nonlinear controller is first designed at high angles of attack. This controller is connected by the pilot after the departure of the aircraft and the controller drives the aircraft to a well-defined spin condition. Thus, the controller makes the post stall aircraft behaviour predictable. Then a set of automatic recovery inputs is designed to reduce aircraft rotations and to lower the angle of attack. The present aircraft model is unstable at low angle of attack flight conditions as well and therefore to stabilize the aircraft to a low angle of attack level flight, another controller is designed. The high angle of attack controller is disconnected and the low angle of attack controller is connected automatically during the recovery process. The entire methodology is tested using extensive non-linear six degree-of-freedom simulations and the efficacy of the technique is established. The nonlinear controller that stabilizes the aircraft to a spin condition is designed using feedback linearization. The stability of a closed loop system obtained using feedback linearization is determined by the stability of the zero dynamics of the open loop plant. It has been shown in literature that the eigenvalues of the linearized zero dynamics are the same as the transmission zeros of the linearized plant at the equilibrium point. It is also well known that the location of transmission zeros of a linear system can be changed by the choice of outputs. In this thesis it is shown that if it is possible to reassign the outputs, then the feedback linearization based design for a linear system becomes very similar to a controller design for eigenvalue assignment. This thesis presents a new two-step procedure to obtain a locally stable and optimally robust closed loop system using feedback linearization. In the first step of this procedure optimal locations of the transmission zeros are found and in the second step, optimal outputs are constructed to place the system transmission zeros at these locations. The same outputs can then be used to construct nonlinear feedback for the nonlinear system and the resultant closed loop system is guaranteed to be locally robustly stable. The high angle of attack controller is designed using this procedure and its performance is presented in the thesis. The stabilized spin equilibrium point of the closed loop system is also shown to have a large domain of attraction. Having designed a locally robust stabilizing controller, the thesis addresses the problem of the evaluation of robustness of the stability of the equilibrium point in a nonlinear framework. The thesis presents a general method to construct bounds on the additive perturbations of the system vector field over a large region in the domain of attraction of a stable equilibrium point using Lyapunov functions. If the system perturbations lie within these bounds, the system is guaranteed to be stable. The thesis first proposes a method to numerically construct a Lyapunov function over a large region in the domain of attraction. In this method a sequence of Lyapunov functions are constructed such that each function in the sequence gives a larger estimate of the domain of attraction than the previous one. The seminal idea for this method is obtained from the existing literature and this idea is considerably generalized. Using this method, it is possible to numerically obtain a Lyapunov function value at each point in the domain of attraction, but the Lyapunov function does not have an analytical form. Hence, it is proposed to represent this function using neural networks. The thesis then discusses a new method to construct perturbation bounds. It is shown that the perturbation bounds obtained over a large region in the domain of attraction using a single Lyapunov function is too conservative. Using the concept of sequence of Lyapunov functions, the thesis proposes three methods to obtain the least conservative bounds for an initial local Lyapunov function. These general ideas are then applied to the aircraft example and the bounds on the perturbation of the aerodynamic database are presented.
Murch, Austin Matthew
This work addressed aerodynamic modeling methods for prediction of post-stall flight dynamics of large transport aircraft. This was accomplished by applying historically successful modeling methods used on high-performance military aircraft to a transport configuration. The overall research approach involved integrating forced oscillation and rotary balance wind tunnel data into an aerodynamic model using several methods of blending these data. The complete aerodynamic model was integrated into a six degree-of-freedom simulation. Experimental data from free-spin wind tunnel testing was used to validate the aerodynamic modeling methods by comparing aerodynamic force and moment coefficients and also to validate the simulation performance by comparing spin mode characteristics and time histories. The aerodynamic model prediction of spin dynamics was generally very good using all of the blending methods studied. In addition, key spin mode characteristics were predicted with a high degree of accuracy. Overall, using the Hybrid Kalviste method of blending forced oscillation and rotary balance data produced the closest match to the free-spin data when comparing aerodynamic coefficients and spin mode characteristics. Several issues were encountered with the blending methods that were exacerbated by nonlinearities and asymmetries in the dynamic aerodynamic data. A new method of looking up dynamic aerodynamic data was proposed to address shortcomings in the blending methods and recommendations were provided on addressing issues with the dynamic aerodynamic data.
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Purpose - The paper presents a theoretical framework that describes the aerodynamics of a falling maple (Acer pseudoplatanus) seed. --- Methodology - A semi-empirical method is developed that provides a ratio stating how much longer a seed falls in air compared to freefall. The generated lift is calculated by evaluating the integral of two-dimensional airfoil elements using a preliminary falling speed. This allows for the calculation of the definitive falling speed using Blade Element Momentum Theory (BEMT); hereafter, the fall duration in air and in freefall are obtained. Furthermore, the input-variables of the calculation of lift are transformed to require only the length and width of the maple seed. Lastly, the method is applied to two calculation examples as a means of validation. --- Findings - The two example calculations gave percentual errors of 5.5% and 3.7% for the falling speed when compared to measured values. The averaged result is that a maple seed falls 9.9 times longer in air when released from 20 m; however, this result is highly dependent on geometrical parameters which can be accounted for using the constructed method. --- Research limitations - Firstly, the coefficient of lift is unknown for the shape of a maple seed. Secondly, the approximated transient state is yet to be verified by measurement. --- Originality / Value - The added value of this report lies in the reduction of simplifications compared to BEMT approaches. In this way a large amount of accuracy is achieved due to the inclusion of many geometrical parameters, even though simplicity is maintained. This has been accomplished through constructing a simple three-step method that is fundamental and essentially non-iterative.
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