Flutter Susceptibility Assessment of Airplanes in Sub-critical Regime using Ameliorated Flutter Margin and Neural Network Based Methods

Abstract

As flight flutter testing on an airplane progresses to high dynamic pressures and high Mach number region, it becomes very difficult for engineers to predict the level of the remaining stability in a flutter-prone mode and flutter-prone mechanism when response data is infested with uncertainty. Uncertainty and ensuing scatter in modal data trends always leads to diminished confidence amidst the possibility of sudden decrease in modal damping of a flutter-prone mode. Since the safety of the instrumented prototype and the crew cannot be compromised, a large number of test-points are planned, which eventually results in increased development time and associated costs. There has been a constant demand from the flight test community to improve understanding of the con-ventional methods and develop new methods that could enable ground-station engineers to make better decision with regard to flutter susceptibility of structural components on the airframe. An extensive literature survey has been done for many years to take due cognizance of the ground realities, historical developments, and the state of the art. Besides, discussion on the results of a survey carried on occurrences of flutter among general aviation airplanes has been provided at the very outset. Data for research comprises results of Computational Aero elasticity Analysis (CAA) and limited Flight Flutter Tests (FFTs) on two slightly different structural designs of the airframe of a supersonic fixed-wing airplane. Detail discussion has been provided with regard to the nature of the data, the certification requirements for an airplane to be flutter-free in the flight-envelope, and the adopted process of flight flutter testing. Four flutter-prone modes - with two modes forming a symmetric bending-pitching flutter mechanism and the other two forming an anti-symmetric bending-pitching mechanism have been identified based on the analysis of computational data. CAA and FFT raw data of these low frequency flutter modes have been provided followed by discussion on its quality and flutter susceptibility of the critical mechanisms. Certain flight-conditions, at constant altitude line and constant Mach number lines, have been chosen on the basis of availability of FFT data near the same flight conditions. Modal damping is often a highly non-linear function of airspeed and scatter in such trends of modal damping can be very misleading. Flutter margin (FM) parameter, a measure of the remaining stability in a binary flutter mechanism, exhibits smooth and gradual variation with dynamic pressure. First, this thesis brings out the established knowledge of the flutter margin method and marks the continuing knowledge-gaps, especially about the applicable form of the flutter margin prediction equation in transonic region. Further theoretical developments revealed that the coefficients of this equation are flight condition depended to a large extent and the equation should be only used in small ‘windows’ of the flight-envelope by making the real-time flutter susceptibility assessment ‘progressive’ in nature. Firstly, it is brought out that lift curve slope should not be treated as a constant while using the prediction equation at constant altitudes on an airplane capable of transonic flight. Secondly, it was realized that the effect of shift in aerodynamic canter must be considered as it causes a ‘transonic-hump’. Since the quadratic form of flutter margin prediction equation developed 47 years ago, does not provide a valid explanation in that region, a general equation has been derived. Furthermore, flight test data from only supersonic region must be used for making acceptable predictions in supersonic region. The ‘ameliorated’ flutter margin prediction equation too provides bad predictions in transonic region. This has been attributed to the non-validity of quasi-steady approximation of aerodynamic loads and other additional non-linear effects. Although the equation with effect of changing lift curve slope provides inconsistent predictions inside and near the region of transonic-hump, the errors have been acceptable in most cases. No consistent congruency was discovered to some earlier reports that FM trend is mostly parabolic in subsonic region and linear in supersonic region. It was also found that the large scatter in modal frequencies of the constituent modes can lead to scatter in flutter margin values which can render flutter margin method as ineffective as the polynomial fitting of modal damping ratios. If the modal parameters at a repeated test-point exhibit Gaussian spread, the distribution in FM is non-Gaussian but close to gamma-type. Fifteen uncertainty factors that cause scatter in modal data during FFT and factor that cause modelling error in a computational model have been enumerated. Since scatter in modal data is ineluctable, it was realized that a new predictive tool is needed in which the probable uncertainty can be incorporated proactively. Given the recent shortcomings of NASA’s flutter meter, the neural network based approach was recognized as the most suitable one. MLP neural network have been used successfully in such scenarios for function approximation through input-output mapping provided the domains of the two are remain finite. A neural network requires ample data for good learning and some relevant testing data for the evaluation of its performance. It was established that additional data can be generated by perturbing modal mass matrix in the computational model within a symmetric bound. Since FFT is essentially an experimental process, it was realized that such bound should be obtained from experimental data only, as the full effects of uncertainty factors manifest only during flight tests. The ‘validation FFT program’, a flight test procedure for establishing such bound from repeated tests at five diverse test-points in safe region has been devised after careful evaluation of guide-lines and international practice. A simple statistical methodology has been devised to calculate the bound-of-uncertainty when modal parameters from repeated tests show Gaussian distribution. Since no repeated tests were conducted on the applicable airframe, a hypothetical example with compatible data was considered to explain the procedure. Some key assumptions have been made and discussion regarding their plausibility has been provided. Since no updated computational model was made available, the next best option of causing random variation in nominal values of CAA data was exercised to generate additional data for arriving at the final form of neural network architecture and making predictions of damping ratios and FM values. The problem of progressive flutter susceptibility assessment was formulated such that the CAA data from four previous test-points were considered as input vectors and CAA data from the next test-point was the corresponding output. General heuristics for an optimal learning performance has been developed. Although, obtaining an optimal set of network parameters has been relatively easy, there was no single set of network parameters that would lead to consistently good predictions. Therefore some fine-tuning, of network parameters about the optimal set was often needed to achieve good generalization. It was found that data from the four already flown test-points tend to dominate network prediction and the availability of flight-test data from these previous test-points within the bound about nominal is absolutely important for good predictions. The performance improves when all the five test-points are closer. If above requirements were met, the predictive performance of neural network has been much more consistent in flutter margin values than in modal damping ratios. A new algorithm for training MLP network, called Particle Swarm Optimization (PSO) has also been tested. It was found that the gradient descent based algorithm is much more suitable than PSO in terms of training time, predictive performance, and real-time applicability. In summary, the main intellectual contributions of this thesis are as follows: • Realization of that the fact that secondary causes lead incidences of flutter on airplanes than primary causes. • Completion of theoretical understanding of data-based flutter margin method and flutter margin prediction equation for all ranges of flight Mach number, including the transonic region. • Vindication of the fact that including lift-curve slope in the flutter margin pre-diction equation leads to improved predictions of flutter margins in subsonic and supersonic regions and progressive flutter susceptibility assessment is the best way of reaping benefits of data-based methods. • Explanation of a plausible recommended process for evaluation of uncertainty in modal damping and flutter margin parameter. • Realization of the fact that a MLP neural network, which treats a flutter mechanism as a stochastic non-linear system, is a indeed a promising approach for real-time flutter susceptibility assessment.