Rotors (Helicopters)

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.

A numerical model for the simulation of three-dimensional normal blade-vortex interaction has been developed to study the bending and variation of core radius of the vortex due to the influence of the blade and the subsequent unsteady force on the blade. For thin blades, a procedure to enable instantaneous cutting of the vortex has been employed to study the vortex response to cutting. The vortex is represented by a filament model which includes axial flow within the core and non-uniform core area. The vortex is convected with self-induced velocities given by the Biot-Savart line integral, and the effect of the cylinder is obtained using a vortex sheet panel method. The governing equations for the vortex axial velocity have a form similar to that of the one-dimensional gas dynamics equations and admit "shock-like" discontinuities. The results indicate that the amount of vortex bending due to interaction with the blade is primarily dependent on the ratio of blade thickness T to ambient vortex core radius sigma o, although for a given amount of bending of the vortex axis, increase in cylinder forward speed results in a decrease in vortex core radius. For blades with T/sigma o < 0(1), very little bending is observed for attack angles under the stall limit. In the case of vortex cutting by a blade, vortex shocks and expansion waves are observed to propagate on the vortex axis away from the blade.

An experimental study of the vortex response to interaction with and cutting by a thin flat
plate or circular cylinders of various diameters has been performed. The direction of
motion of the flat plate (or circular cylinder) is normal to the vortex axis in the experiments.
The vortex is generated by withdraw of fluid at an orifice at the bottom of an "inner
cylinder" immersed in a rectangular tank, and the flow field is visualized with both water
soluble and immiscible dyes. In the experiments with circular cylinders, the bending of
the vortex is compared to computational predictions from [15], and the mechanism of
subsequent breakup of the vortex as it gets closer to the cylinder is studied. The vortex is
observed to bend farther without breakup for larger forward speeds of the circular cylinder.
Very little bending is observed when the vortex interacts with the flat plate, except for
angles of attack exceeding the stall limit Following cutting of the vortex by the flat plate or
circular cylinder, a vortex shock is observed to form and propagate up the vortex axis. No
vortex shock is observed on the opposite side of the blade. The various forms of these
vortex shocks have been photographed, and they appear very similar to travelling vortex
breakdowns. The propagation speed of the shocks is compared to an analytical solution for
instantaneous vortex cutting by a flat plate of zero thickness.

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.

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 dissertation investigates helicopter trim and stability during level bank-angle and diving bank-angle turns. The level turn is moderate in that sufficient power is available to maintain level maneuver, and the diving turn is severe where the power deficit is overcome by the kinetic energy of descent. The investigation basically represents design conditions where the peak loading goes well beyond the steady thrust limit and the rotor experiences appreciable stall. The major objectives are: 1) to assess the sensitivity of the trim and stability predictions to the approximations in modeling stall, 2) to correlate the trim predictions with the UH-60A flight test data, and 3) to demonstrate the feasibility of routinely using the exact fast-Floquet periodic eigenvector method for mode identification in the stability analysis. The UH-60A modeling and analysis are performed using the comprehensive code RCAS (Army's Rotorcraft Comprehensive Analysis System). The trim and damping predictions are based on quasisteady stall, ONERA-Edlin vi (Equations Differentielles Lineaires) and Leishman-Beddoes dynamic stall models. From the correlation with the test data, the strengths and weaknesses of the trim predictions are presented.