etd@IISc Collection:
http://hdl.handle.net/2005/27
Tue, 20 Feb 2018 06:12:48 GMT2018-02-20T06:12:48ZInverse Problems in Free Vibration Analysis of Rotating and Non-Rotating Beams and its Application to Random Eigenvalue Characterization
http://hdl.handle.net/2005/3139
Title: Inverse Problems in Free Vibration Analysis of Rotating and Non-Rotating Beams and its Application to Random Eigenvalue Characterization
Authors: Sarkar, Korak
Abstract: Rotating and non-rotating beams are widely used to model important engineering struc-tures. Hence, the vibration analyses of these beams are an important problem from a structural dynamics point of view. Depending on the beam dimensions, they are mod-eled using diﬀerent beam theories. In most cases, the governing diﬀerential equations of these types of beams do not yield any simple closed-form solutions; hence we look for the inverse problem approach in determining the beam property variations given certain solutions.
The long and slender beams are generally modeled using the Euler-Bernoulli beam theory. Under the premise of this theory, we study (i) the second mode tailoring of non-rotating beams having six diﬀerent boundary conditions, (ii) closed-form solutions for free vibration analysis of free-free beams, (iii) closed-form solutions for free vibration analysis for gravity-loaded cantilever beams, (iv) closed-form solutions for free vibration analysis of rotating cantilever and pinned-free beams and (v) beams with shared eigen-pair. Short and thick beams are generally modeled using the Timoshenko beam theory. Here, we provide analytical closed-form solutions for the free vibration analysis of ro-tating non-homogeneous Timoshenko beams. The Rayleigh beam provides a marginal improvement over the Euler-Bernoulli beam theory without venturing into the math-ematical complexities of the Timoshenko beam theory. Under this theory, we provide closed-form solutions for the free vibration analysis of cantilever Rayleigh beams under three diﬀerent axial loading conditions - uniform loading, gravity-loading and centrifu-gally loaded.
We assume simple polynomial mode shapes which satisfy the diﬀerent boundary conditions of a particular beam, and derive the corresponding beam property variations. In case of the shared eigenpair, we use the mode shape of a uniform beam which has a closed-form solution and use it to derive the stiﬀness distribution of a corresponding axially loaded beam having same length, mass variation and boundary condition. For the Timoshenko beam, we assume polynomial functions for the bending displacement and the rotation due to bending. The derived properties are demonstrated as benchmark analytical solutions for approximate and numerical methods used for the free vibration analysis of beams. They can also aid in designing actual beams for a pre-specified frequency or nodal locations in some cases. The eﬀect of diﬀerent parameters in the derived property variations and the bounds on the pre-specified frequencies and nodal locations are also studied for certain cases.
The derived analytical solutions can also serve as a benchmark solution for diﬀerent statistical simulation tools to find the probabilistic nature of the derived stiﬀness distri-bution for known probability distributions of the pre-specified frequencies. In presence of uncertainty, this flexural stiﬀness is treated as a spatial random field. For known probability distributions of the natural frequencies, the corresponding distribution of this field is determined analytically for the rotating cantilever Euler-Bernoulli beams. The derived analytical solutions are also used to derive the coeﬃcient of variation of the stiﬀness distribution, which is further used to optimize the beam profile to maximize the allowable tolerances during manufacturing.Sun, 18 Feb 2018 18:30:00 GMThttp://hdl.handle.net/2005/31392018-02-18T18:30:00ZA Study of the Characteristics of Gas-On-Liquid Impinging Injectors
http://hdl.handle.net/2005/3126
Title: A Study of the Characteristics of Gas-On-Liquid Impinging Injectors
Authors: Rakesh, P
Abstract: The work presented here pertains to investigations on gas-on-liquid type of impinging injectors with a generic approach with prospective applications in several areas, and at places with particular emphasis on cryogenic or semi-cryogenic liquid propellant rockets. In such
rockets, one of the components arrives at the injector in a gaseous phase after passing through the regenerative coolant passages or a preceding combustion stage. Most often, the injectors in such systems are of shear coaxial type. The shear coaxial injectors suffer from several disadvantages like complexity in design, manufacture and quality control. Adoption of impinging jet configuration can alleviate these problems in addition to providing further benefits in terms of cost, robustness in high temperature environment and manifolding.
However, there is very little literature on gas-on-liquid injectors either in this context or in any other Even for the simplest form of impinging injectors such as like-on-like doublets, literature provides no conclusive direction at describing a spray from the theoretical models of physical mechanisms. Empirical approach is still the prime mode of obtaining a proper understanding of the phenomena. Steady state spray characterization includes mainly of describing the spatial distribution of liquid mass and drop size distribution as a function of geometric and injection parameters. The parameters that are likely to have an impact on spray characteristics are orifice diameter, ratio of orifice length to diameter, pre-impingement length of individual jets, inter orifice distance, impingement angle, jet velocity and condition of the jet just before impingement. The gas-on- liquid configuration is likely to experience
some qualitative changes because of the expansion of the gas jet. The degree to
which each one of the above variables influences the drop size and mass distribution having implication to combustion performance forms the core theme of the thesis. A dedicated experimental facility has been built, calibrated and deployed exhaustively.
While spray drop size measurement is done largely by a laser diffraction instrument, some of the cases warranted an image processing technique. Two different image processing algorithms are developed in-house for this purpose. The granulometric image processing method developed earlier in the group for cryogenic sprays is modified and its applicability to gas-on-liquid impinging sprays are verified. Another technique based on the Hough transform which is feature extraction technique for extracting quantitative information has also been developed and used for gas-on-liquid impinging injectors. A comparative study of conventional liquid-on-liquid doublet with gas-on-liquid impinging injectors are first made to establish the importance of studying gas-on-liquid impinging injectors. The study identifies the similarities and differences between the two types and highlights the features that make such injectors attractive as replacements to coaxial configuration. Spray structure, drop-size mass distributions are quantified for the purpose
of comparison. This is followed by a parametric study of the gas-on-liquid impinging injectors carried out using identified control variables. Though momentum ratio appeared to be a suitable parameter to describe the spray at any given impingement angle, the variations due to impingement angle had to be factored in. It was found that normal gas momentum to liquid mass is an apt parameter to generalize the spray characteristics. It was also found that using identical nozzles for desired mass ratio could lead to rather large deflection of the spray which may not be acceptable in combustion chamber design. One way of overcoming this is to work with unequal orifice sizes for gas and liquid. It was found that using smaller gas orifice for a given liquid orifice resulted in lower SMD (Sauter Mean Diameter of the spray) for constant gas and liquid mass flow rates. This is attributable to the high dynamic
pressure of gas in the case of smaller gas orifices for the same mass flow rate. The impinging liquid jets with unequal momentum in the doublet configuration would
result in non-uniform mass and mixture ratio distribution within the combustion chamber
which may have to operate under varying conditions of mass flow rates and/or mixture
ratio. The symmetrical arrangement of triplet configuration can eliminate this problem at the same time generating finely atomized spray and a homogeneous mixture ratio. In view of the scanty literature available in this field, the atomization characteristics of the spray
generated by liquid centered triplet jets are examined in detail. It was found that as in the case of gas-on-liquid impinging doublets, normal gas momentum to liquid mass is an ideal parameter in describing the spray. Variants of this configuration are studied recently for many other applications too. As done in the case of doublets, efforts have also been made to compare gas centered triplet to liquid-liquid triplet. It was found that the trend of SMD of gas centered triplet is different from that of liquid-liquid triplets, thus pointing to a different mechanism in play. The SMD in the case of liquid-liquid triplets decreases monotonically with increasing specific normal momentum. It is to be noted that specific normal momentum is an ideal
parameter for describing the spray characteristics of liquid-liquid triplets and doublets. In the case of gas centered triplet the SMD first increases and then decreases with specific normal momentum, the inversion point depends on the gas mass flow rate for a constant specific normal momentum.
The thesis concludes with a summary of the major observations of spray structures for
all the above injector configurations and quantifies the parametric dependencies that would be of use to engineering designSat, 17 Feb 2018 18:30:00 GMThttp://hdl.handle.net/2005/31262018-02-17T18:30:00ZOn Three Dimensional High Lift Flow Computations
http://hdl.handle.net/2005/3127
Title: On Three Dimensional High Lift Flow Computations
Authors: Gopalakrishna, N
Abstract: Computing 3D high lift flows has been a challenge to the CFD community because of three important reasons: complex physics, complex geometries and large computational requirements. In the recent years, considerable progress has been made in understanding the suitability of various CFD solvers in computing 3D high lift flows, through the systematic studies carried out under High Lift Prediction workshops. The primary focus of these workshops is to assess the ability of the CFD solvers to predict CLmax and αmax associated with the high lift flows, apart from the predictability of lift and drag of such flows in the linear region. Now there is a reasonable consensus in the community about the ability of the CFD solvers to predict these quantities and fresh efforts to further understand the ability of the CFD solvers to predict more complex physics associated with these flows have already begun.
The goal of this thesis is to assess the capability of the computational methods in predicting such complex flow phenomena associated with the 3D High-Lift systems. For evaluation NASA three element Trapezoidal wing configuration which poses a challenging task in numerical modeling was selected. Unstructured data based 3D RANS solver HiFUN (HiFUN stands for High Resolution Flow Solver for UNstructured Meshes) is used in investigating the high lift flow. The computations were run fully turbulent, using the one equation Spalart-Allmaras turbulence model.
A summary of the results obtained using the flow solver HiFUN for the 3D High lift NASA Trapezoidal wing are presented. Hybrid unstructured grids have been used for the computations. Grid converged solution obtained for the clean wing and the wing with support brackets, are compared with experimental data. The ability of the solver to predict critical design parameters associated with the high lift flow, such as αmax and CLmax is demonstrated. The utility of the CFD tools, in predicting change in aerodynamic parameters in response to perturbational changes in the configuration is brought out. The solutions obtained for the high lift configuration from two variants of the Spalart-Allmaras turbulence model are compared. To check the unsteadiness in the flow, particularly near stall, unsteady simulations were performed on static grid. Lastly, hysteresis on lower leg of lift curve is discussed, the results obtained for quasi-steady and dynamic unsteady simulations are presented. Inferences from the study on useful design practices pertaining to the 3D high lift flow simulations are summarized.Sat, 17 Feb 2018 18:30:00 GMThttp://hdl.handle.net/2005/31272018-02-17T18:30:00ZFlutter Susceptibility Assessment of Airplanes in Sub-critical Regime using Ameliorated Flutter Margin and Neural Network Based Methods
http://hdl.handle.net/2005/3124
Title: Flutter Susceptibility Assessment of Airplanes in Sub-critical Regime using Ameliorated Flutter Margin and Neural Network Based Methods
Authors: Kumar, Brijesh
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.Fri, 16 Feb 2018 18:30:00 GMThttp://hdl.handle.net/2005/31242018-02-16T18:30:00Z