Gaonkar, Gopal H.

This thesis presents a complete method of modeling the autospectra of turbulence
in closed form via an expansion series using the von Kármán model as a basis function. It
is capable of modeling turbulence in all three directions of fluid flow: longitudinal,
lateral, and vertical, separately, thus eliminating the assumption of homogeneous,
isotropic flow. A thorough investigation into the expansion series is presented, with the
strengths and weaknesses highlighted. Furthermore, numerical aspects and theoretical
derivations are provided. This method is then tested against three highly complex flow
fields: wake turbulence inside wind farms, helicopter downwash, and helicopter
downwash coupled with turbulence shed from a ship superstructure. These applications
demonstrate that this method is remarkably robust, that the developed autospectral
models are virtually tailored to the design of white noise driven shaping filters, and that these models in closed form facilitate a greater understanding of complex flow fields in
wind engineering.

Future helicopters will require all-weather capability for stabilized flight through severe atmospheric turbulence. This requirement has brought into focus the effect of turbulence on handling qualities. Accordingly, there is renewed interest in modeling and simulating turbulence and predicting turbulence-induced rotor oscillations. This thesis addresses three fundamental aspects of the problem: (1) modeling and simulation of turbulence including cross-correlation; (2) three-dimensional dynamic-wake effects on rotor response to turbulence and (3) prediction of turbulence and response statistics. The analysis is based on the theory of isotropic and homogeneous turbulence and Taylor's frozen-field approximation. Quasisteady airfoil aerodynamics and a three-dimensional wake are used. Both the isolated blades and isolated rotors are treated. The parallelization is carried out on a massively parallel MasPar SIMD computer. Major conclusions include: (i) The effects of cross-correlation are negligible when two stations lie on the same blade and appreciable when two stations lie on different blades. (ii) In modeling the three-dimensional wake, 3 harmonics are required and dynamic wake has dominant influence on response statistics. (iii) With increasing comprehensiveness of helicopter-turbulence modeling, the sequential execution times increase dramatically; by comparison, the parallel execution times are far lower and, more significantly, remain nearly constant.

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 forcing-function 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 Runge-Kutta 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,
follow-up integrations and formations of the columns of the Jacobian matrix.
These parallel shooting and eigenanalysis schemes are applied to the
nonlinear flap-lag stability with a three-dimensional 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 MP-1, a distributed-memory, single-instruction
multiple-data 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 large-scale Floquet analysis practical.

The atmospheric turbulence that a blade station experiences is called blade-fixed
turbulence. It can qualitatively differ from the conventional body-fixed turbulence
such as experienced by an element of the body or fuselage. This difference is due
to the rotational ,-elocity, which causes fore-and-aft motions of the blade station
through the turbulence waves. A closed-form solution of a frequency-time spectrum
for the dominant vertical turbulence velocity at an arbitrary blade station is
dc,·eloped. This solution makes it possible to explain qualitatively the turbulence
cllcrgy transfer due to rotational velocity from the low-frequency region (< 1P or
1/ rcv.) to the high-frequency(> 1P) region with the occurrence of spectral peaks
and split peaks at 1P /2, 1P, 3P /2, 2P etc. Comparison of blade responses to bladeand
body-fixed turbulence is also presented over a comprehensive range of turbulcuce
scale length and advance ratio; the comparison covers frequency-time spectra,
correlations including standard deviations, and average threshold-crossing rates of
a flapping blade. A major contribution is to formulate both the cyclostationary
turbulence and blade response by the frequency-time spectra, which predict simultaneously
the time- ancl frequency-dependent characteristics such as the energy
culltained in the frequency and time intervals. For low-altitude and low-advanceratio
flights, such as nap-of-the earth or NOE flights, rotational velocity effects on
turbulence modeling qualitatively affect the prediction of turbulence ancl response
statistics.

Rotating frame turbulence, or RFT, refers to the actual turbulence experienced by the helicopter blades and requires noneulerian description and rotational sampling of measurements. In the stationary case of axial flight, as investigated earlier, its spectra has peaks centered at integer multiples of rotational speed P, as in wind turbines. In forward flight, as investigated here, its instantaneous or frequency-time spectra has split peaks centered at P/2, P, 3P/2, 2P etc. Though nonstationary, it is wide sense cyclostationary in that its autocorrelation function R(t1,t2) = R(t1 + 2m pi, t2 + 2n pi) for integers m = n only. The major RFT characteristics--spectral peaks, the consequent transfer of energy essentially from the low-frequency region (<1P) to the high-frequency region (>1P) and cyclostationarity--cannot be predicted by conventional space-fixed description. However, these characteristics are simultaneously predicted by the instantaneous spectra, and for their qualitative and parametric investigation, a closed-form solution of an instantaneous spectrum is presented for a space-fixed turbulence model. The RFT effects on the blade response statistics of rms values and average threshold crossing rates are presented as well. The blade model includes flap bending degrees of freedom and dynamic stall effects. The blade response statistics demonstrate that RFT effects are appreciable for low-advance ratio and low-altitude flight conditions and that dynamic stall increases gust sensitivity.

Presently three schemes are used to generate the governing equations of motion. These schemes are: (1) general purpose processors such as REDUCE, MACSYMA and MAPLE, (2) a special purpose symbolic processor DEHIM--Dynamic Equations for Helicopter Interpretive Models and (3) completely numerical approaches such as AGEM--Automatic Generation of Equations of Motion. With REDUCE as a representative multipurpose processor in scheme 1, comparative aspects of these three schemes have been studied by applying them to the same set of problems. These problems range from a linear model of a single blade with one degree of freedom to a mildly nonlinear three-bladed rotor model with several degrees of freedom. The derivation process includes the nonlinear equations and the perturbed linear equations about a user-supplied equilibrium state in a rotating frame and then the multiblade equations, which represent transformation into a nonrotating frame using multiblade coordinates. (Abstract shortened with permission of author.)

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 soft-inplane, soft-torsion 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 root-flap moment and lag regressive-mode 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 three-dimensional state-space wake model. The periodic shooting method, with damped Newton iteration and the fast-Floquet 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 root-flap moment is calculated by both the force integration and mode-deflection methods. Although exact, the fast-Floquet theory requires a finite-state 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 (fast-Floquet) theory and Sparse Time Domain (STD) technique. These approximations can be applied with a finite-state 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.

Hingeless rotors are susceptible to instabilities of the lead-lag 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, lag-damping prediction requires an aerodynamic
representation that adequately accounts for quasisteady stall, dynamic stall and
three-dimensional 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 three-bladed 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 advance-ratios
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 lag-damping 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 finite-state three-dimensional wake model. The root-flexure-blade assembly of the untrimmed
rotor is represented by a root-restrained rigid flap-lag model as well as by an elastic
flap-lag-torsion 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 lag-damping 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 force-moment 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 small-order
systems (number of states or order M < 100). However, it has been found to be
practical neither for design nor for comprehensive-analysis 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 flight-trim conditions. and (iii)
noniinear fiap-iag 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 fast-Floquet analyses each in sequential and parallel
modes - are comparable. The fast-Floquet analysis brings in nearly Q-fold
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
fast-Floquet 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 large-scale Floquet analysis practical for rotorcrafts with
identical as well as with dissimilar blades.

This dissertation investigates the effects of dynamic stall and three-dimensional wake on isolated-rotor 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 force-integration approach to predict the rotating-blade 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 three-dimensional, finite-state wake model. Throughout, full blade-stall-wake dynamics is used in that all states are included from trim to stability to loads predictions. Moreover, these predictions are based on four aerodynamic theories--quasisteady linear theory, quasisteady stall theory, dynamic stall theory and dynamic stall and wake theory--and 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 flap-lag-torsion equations are coupled with a finite-state 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 lag-damping data of a three-bladed isolated rotor operated untrimmed. In the correlation, three structural models of the root-flexure-blade assembly are used to demonstrate the strengths and the weaknesses of lag-damping predictions. Phase two includes dynamic stall in addition to three-dimensional 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.

The trim analysis for the initial state and control inputs that satisfy response periodicity and flight conditions, and the Floquet eigenanalysis for a few largest eigenvalues of the Floquet transition matrix (FTM) are investigated. In the trim analysis, the convergence of Newton iteration is investigated in computing the periodic initial state and control inputs sequentially and in parallel. The trim analysis uses the shooting method and two h-versions of temporal finite element methods, one based on displacement formulation and the other on mixed formulation of displacements and momenta. In each method, both the sequential and in-parallel schemes are used, and the resulting nonlinear equations are solved by damped Newton iteration with an optimally selected damping parameter. The reliability of damped Newton iteration, including earlier-observed divergence problems, is quantified by the maximum condition number of the Jacobian matrices of the iterative scheme. For illustrative purposes, rigid flap-lag and flap-lag-torsion models based on quasisteady aerodynamics are selected. Demanding trim analysis conditions are included by considering advance ratios or dimensionless flight speeds twice as high as those of current helicopters. Concerning the Floquet eigenanalysis, the feasibility of using the Arnoldi-Saad method, one of the emerging subspace iteration methods, is explored as an alternative to the currently used QR method, which is not economical for partial eigenanalysis. The reliability of the Arnoldi-Saad method is quantified by the eigenvalue condition numbers and the residual errors of the eigenpairs. In the three trim analysis methods, while the optimally selected damping parameter provides almost global convergence, the in-parallel scheme requires much less machine time than the conventional sequential scheme; both schemes have comparable reliability of the Newton iteration without and with damping. The Arnoldi-Saad method takes much less machine time than the QR method with comparable reliability.