Model
Digital Document
Publisher
Florida Atlantic University
Description
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.
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.
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