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 Title
 Dynamic stall and threedimensional wake effects on trim, stability and loads of hingeless rotors with fast Floquet theory.
 Creator
 Chunduru, Srinivas Jaya., Florida Atlantic University, Gaonkar, Gopal H., College of Engineering and Computer Science, Department of Ocean and Mechanical Engineering
 Abstract/Description

This dissertation investigates the effects of dynamic stall and threedimensional wake on isolatedrotor trim, stability and loads. Trim analysis of predicting the pilot's control inputs and the corresponding periodic responses is based on periodic shooting with the fast Floquet theory and damped Newton iteration. Stability analysis, also based on the fast Floquet theory, predicts damping levels and frequencies. Loads analysis uses a forceintegration approach to predict the rotatingblade...
Show moreThis dissertation investigates the effects of dynamic stall and threedimensional wake on isolatedrotor trim, stability and loads. Trim analysis of predicting the pilot's control inputs and the corresponding periodic responses is based on periodic shooting with the fast Floquet theory and damped Newton iteration. Stability analysis, also based on the fast Floquet theory, predicts damping levels and frequencies. Loads analysis uses a forceintegration approach to predict the rotatingblade root shears and moments as well as the hub forces and moments. The blades have flap bending, lag bending and torsion degrees of freedom. Dynamic stall is represented by the ONERA stall models of lift, drag and pitching moment, and the unsteady, nonuniform downwash is represented by a threedimensional, finitestate wake model. Throughout, full bladestallwake dynamics is used in that all states are included from trim to stability to loads predictions. Moreover, these predictions are based on four aerodynamic theoriesquasisteady linear theory, quasisteady stall theory, dynamic stall theory and dynamic stall and wake theoryand cover a broad range of system parameters such as thrust level, advance ratio, number of blades and blade torsional frequency. The investigation is conducted in three phases. In phase one, the elastic flaplagtorsion equations are coupled with a finitestate wake model and with linear quasisteady airfoil aerodynamics. The investigation presents convergence characteristics of trim and stability with respect to the number of spatial azimuthal harmonics and radial shape functions in the wake representation. It includes a comprehensive parametric study over a broad range of system parameters. The investigation also includes correlation with the measured lagdamping data of a threebladed isolated rotor operated untrimmed. In the correlation, three structural models of the rootflexureblade assembly are used to demonstrate the strengths and the weaknesses of lagdamping predictions. Phase two includes dynamic stall in addition to threedimensional wake to generate trim and stability results over a comprehensive range of system parameters. It addresses the degree of sophistication necessary in blade discretization and wake representation under dynamically stalled conditions. The convergence and parametric studies isolate the effects of wake, quasisteady stall and dynamic stall on trim and stability. Finally, phase three predicts the rotating blade loads and nonrotating hub loads; the predictions are based on the blade, wake and stall models used in the preceding trim and stability investigations. Although an accurate evaluation of loads requires a more refined blade description, the results isolate and demonstrate the principal dynamic stall and wake effects on the loads.
Show less  Date Issued
 1995
 PURL
 http://purl.flvc.org/fcla/dt/12426
 Subject Headings
 Floquet theory, Helicopters, Rotors (Helicopters), Vibration (Aeronautics)
 Format
 Document (PDF)
 Title
 FastFloquet method of hingelessrotor trim and stability including comparisons with experiments and approximate methods.
 Creator
 Ma, Guifa., Florida Atlantic University, Gaonkar, Gopal H., College of Engineering and Computer Science, Department of Ocean and Mechanical Engineering
 Abstract/Description

The trim and stability of an isolated hingeless rotor in forward flight are predicted for two coning angles with advance ratio, shaft angle and collective pitch variations. These predictions are correlated with measurements from a test model with four softinplane, softtorsion blades. The test was conducted by the U.S. Army Aeroflightdynamics Directorate at Ames. The collective pitch and shaft angle are set prior to each test point, and the rotor is trimmed as follows: the longitudinal and...
Show moreThe trim and stability of an isolated hingeless rotor in forward flight are predicted for two coning angles with advance ratio, shaft angle and collective pitch variations. These predictions are correlated with measurements from a test model with four softinplane, softtorsion blades. The test was conducted by the U.S. Army Aeroflightdynamics Directorate at Ames. The collective pitch and shaft angle are set prior to each test point, and the rotor is trimmed as follows: the longitudinal and lateral cyclic pitch controls are adjusted through a swashplate to minimize the 1/rev flapping moment at the 12% radial station. The database includes the cyclic pitch controls, steady rootflap moment and lag regressivemode damping. The predictions are based on a modal approach with both nonrotating and rotating modes, the ONERA dynamic stall models of lift, drag and pitching moment, and a threedimensional statespace wake model. The periodic shooting method, with damped Newton iteration and the fastFloquet theory, is used to predict the cyclic pitch controls and the corresponding periodic responses; the equivalent Floquet transition matrix (EFTM) comes out as a byproduct. The eigenvalues and eigenvectors of the EFTM lead to the frequencies and damping levels. The steady rootflap moment is calculated by both the force integration and modedeflection methods. Although exact, the fastFloquet theory requires a finitestate representation of all states and is not applicable to numerically and experimentally generated data of response histories. Therefore, the stability is also predicted by three related approximations: generalized Floquet (fastFloquet) theory and Sparse Time Domain (STD) technique. These approximations can be applied with a finitestate representation of an arbitrary number of states and to response histories; their convergence characteristics and accuracy are examined as well. Two major findings are: (1) The dynamic wake dramatically improves the correlation of the lateral cyclic pitch controls, and (2) all three approximations have excellent convergence characteristics and the converged values agree well with the exact values.
Show less  Date Issued
 2000
 PURL
 http://purl.flvc.org/fcla/dt/12648
 Subject Headings
 Floquet theory, Rotors (Helicopters), Stability of helicopters
 Format
 Document (PDF)
 Title
 Dynamic stall and threedimensional wake effects on isolated rotor trim and stability with experimental correlation and parallel fastFloquet analysis.
 Creator
 Subramanian, Shanmugasundaram., Florida Atlantic University, Gaonkar, Gopal H., College of Engineering and Computer Science, Department of Ocean and Mechanical Engineering
 Abstract/Description

Hingeless rotors are susceptible to instabilities of the leadlag or lag modes, which are at best weakly damped. The lag mode derives its damping primarily from the complex rotor flow field that is driven by interdependent dynamics of airfoil stall and rotor downwash or wake. Therefore, lagdamping prediction requires an aerodynamic representation that adequately accounts for quasisteady stall, dynamic stall and threedimensional dynamic wake. Accordingly, this dissertation investigates these...
Show moreHingeless rotors are susceptible to instabilities of the leadlag or lag modes, which are at best weakly damped. The lag mode derives its damping primarily from the complex rotor flow field that is driven by interdependent dynamics of airfoil stall and rotor downwash or wake. Therefore, lagdamping prediction requires an aerodynamic representation that adequately accounts for quasisteady stall, dynamic stall and threedimensional dynamic wake. Accordingly, this dissertation investigates these stall and wake effects on lag damping and demonstrates the strengths and weaknesses of the aerodynamic representation with a comprehensive experimental correlation. The database refers to a threebladed rotor operated untrimmed and to a fourbladed rotor operated trimmed; for both rotors, the blade collective pitch and shaft tilt angles are set prior to each test run. The untrimmed rotor is tested with advanceratios as high as 0.55 and shaft angles as high as 20°, and it has intentionally builtin structural simplicity: torsionally stiff blades and no swash plate. The trimmed rotor has torsionally soft blades; it is trimmed in the sense that the longitudinal and lateral cyclic pitch controls are adjusted through a swash plate to minimize l/rev root flap moment. Therefore, for the untrimmed rotor, the database refers to lagdamping levels, and for the trimmed rotor, it refers to lagdamping levels as well as to trim results of lateral and longitudinal cyclic pitch controls and steady root flap moments. The dynamic stall representation is based on the ONERA models of lift, drag and pitching moment, and the unsteady wake is described by a finitestate threedimensional wake model. The rootflexureblade assembly of the untrimmed rotor is represented by a rootrestrained rigid flaplag model as well as by an elastic flaplagtorsion model. Similarly, the trimmed rotor is represented by an elastic flaplag torsion model. The predictions are from three aerodynamic theories ranging from a quasisteady stall theory to a fairly comprehensive dynamic stall and wake theory. This dissertation also addresses the computational aspects of lagdamping predictions by parallel F!oquct analysis based on classical and fast Floquet theories. In a typical trimmed flight, the Floquet analysis comprises (i) trim or equilibrium analysis, (ii) generation of the Floquet transition matrix (FTM) about the trim position, and (iii) eigenanalysis of the FTM. The trim analysis involves the computations of the unknown control inputs that satisfy flight conditions of required thrust and forcemoment equilibrium as well as the initial conditions that guarantee periodic forced response. The shooting method is increasingly used for the trim analysis since it generates the FTM as a byproduct and is not sensitive to damping levels. The QR method is used almost exclusively for the FTM eigenanalysis. Presently, the Floquet analysis with shooting and QR methods is widely used for smallorder systems (number of states or order M < 100). However, it has been found to be practical neither for design nor for comprehensiveanalysis models that lead to large systems (A11 > 100); the run time on a conventional sequential computer is simply prohibitive. Nevertheless, all three parts of Floquet analysis can be algorithmically structured such that they lend themselves well to parallelism or concurrent computations. Furthermore, the conventional Floquet analysis requires integrations of equations of motion through one complete period T; and the bulk of the run time is for repeated integrations over one period. However, for rotors with Q identical blades, it is computationally advantageous to use the fast Floquet analysis, which requires integration through a period T/Q. Accordingly, this dissertation develops parallel algorithms for classical Floquet analysis with classical shooting and for the fast Floquet analysis with fast shooting; in each case the FTM eigenanalysis is baseJ on a parallel QR tibrary routine. The computational reliability· of the sequential anJ parallel Floquet analyses is quantified by (i) the condition number of the converged Jacobian matrix in Newton iteration of trim analysis, (ii) the condition numbers of the FTM eigenvalues of interest, and (iii) the corresponding residual errors of the eigenpairs (eigenvalue and the corresponding eigenYector). These algorithms are applied to study (i) linear flap stability with dynamic wake, (ii) nonlinear flaplag stability with dynamic wake under propulsive or flighttrim conditions. and (iii) noniinear fiapiag stabiiity with dynamic staii and wake under flight trim conditions. The parallel and sequential algorithms are compared with respect to computational reliability, saving in run time and growth of run time with increasing system order. Other parallel performance metrics such as speedup, efficiency, and sequential and parallel fractions are included as well. The computational reliability figures of the four algorithms  classical and fastFloquet analyses each in sequential and parallel modes  are comparable. The fastFloquet analysis brings in nearly Qfold reduction in run time in both the sequential and parallel modes; that is, its advantages apply equally to both the modes. 'While the run times for the classical and fastFloquet analyses in sequential mode grow in between quadratically and cubically with the system order, the corresponding run times in parallel mode are far shorter and more importantly remain nearly constant. These results offer considerable promise in making largescale Floquet analysis practical for rotorcrafts with identical as well as with dissimilar blades.
Show less  Date Issued
 1996
 PURL
 http://purl.flvc.org/fcla/dt/12462
 Subject Headings
 Rotors (Helicopters), Stalling (Aerodynamics), Drag (Aerodynamics), Wakes (Aerodynamics), Floquet theory
 Format
 Document (PDF)
 Title
 Parallelcomputing concepts and methods toward largescale floquet analysis of helicopter trim and stability.
 Creator
 Nakadi, Rajesh Mohan., Florida Atlantic University, Gaonkar, Gopal H., College of Engineering and Computer Science, Department of Ocean and Mechanical Engineering
 Abstract/Description

The rotorcraft trim solution involves a search for control inputs for required flight conditions as well as for corresponding initial conditions for periodic response or orbit. The control inputs are specified indirectly to satisfy flight conditions of prescribed thrust levels, rolling and pitching moments etc. In addition to the nonlinearity of the equations of motion and control inputs, the control inputs appear not only in damping and stiffness matrices but also in the forcingfunction or...
Show moreThe rotorcraft trim solution involves a search for control inputs for required flight conditions as well as for corresponding initial conditions for periodic response or orbit. The control inputs are specified indirectly to satisfy flight conditions of prescribed thrust levels, rolling and pitching moments etc. In addition to the nonlinearity of the equations of motion and control inputs, the control inputs appear not only in damping and stiffness matrices but also in the forcingfunction or input matrix; they must be found concomitantly with the periodic response from external constraints on the flight conditions. The Floquet Transition Matrix (FTM) is generated for perturbations about that periodic response; usually, a byproduct of the trim analysis. The damping levels or stability margins are computed from an eigenanalysis of the FTM. The Floquet analysis comprises the trim analysis and eigenanalysis and is routinely used for small order systems (order N < 100). However, it is practical for neither design applications nor comprehensive analysis models that lead to large systems (N > 100); the execution time on a sequential computer is prohibitive. The trim analysis takes the bulk of this execution time. Accordingly, this thesis develops concepts and methods of parallelism toward Floquet analysis of large systems with computational reliability comparable to that of sequential computations. A parallel shooting scheme with damped Newton iteration is developed for the trim analysis. The scheme uses parallel algorithms of RungeKutta integration and linear equations solution. A parallel QR algorithm is used for the eigenanalysis of the FTM. Additional parallelism in each iteration cycle is achieved by concurrent operations such as perturbations of initial conditions and control inputs, followup integrations and formations of the columns of the Jacobian matrix. These parallel shooting and eigenanalysis schemes are applied to the nonlinear flaplag stability with a threedimensional dynamic wake (N ~ 150). The stability also is investigated by widely used sequential schemes of shooting with damped Newton iteration and QR eigenanalysis. The computational reliability is quantified by the maximum condition number of the Jacobian matrices in the Newton iteration, the eigenvalue condition numbers and the residual errors of the eigenpairs. The saving in computer time is quantified by the speedup, which is the ratio of the execution times of Floquet analysis by sequential and parallel schemes. The work is carried out on massively parallel MasPar MP1, a distributedmemory, singleinstruction multipledata or SIMD computer. A major finding is that with increasing system order, while the parallel execution time remains nearly constant, the sequential execution time increases nearly cubically with N. Thus, parallelism promises to make largescale Floquet analysis practical.
Show less  Date Issued
 1994
 PURL
 http://purl.flvc.org/fcla/dt/15085
 Subject Headings
 Floquet theory, HelicoptersControl systems, Rotors (Helicopters), Parallel processing (Electronic computers)
 Format
 Document (PDF)