Formulation of a complete structural uncertainty model for robust flutter prediction Brian Danowsky...

Formulation of a complete structural uncertainty model for robust flutter predictionBrian DanowskyStaff Engineer, ResearchSystems Technology, Inc., Hawthorne, [email protected](310) 679-2281 ex. 28

SAE Aerospace Control and Guidance Systems Committee Meeting #99

Copyright © Brian Danowsky, 2004. All rights reserved

Acknowledgement

Iowa State University

Dr. Frank R. Chavez NASA Dryden Flight Research Center

Marty Brenner

NASA GSRP Program


Outline Introduction to the Flutter Problem Purpose of Research Wing Structural Model Application of Unsteady Aerodynamics Complete Aeroelastic Wing Model Review of Robust Stability Theory Application of the Allowable Variation in the

Freestream Velocity Application of Parametric Uncertainty in the Wing

Structural Properties Conclusions and Discussion


Introduction to The Flutter Problem

Coupling between Aerodynamic Forces and Structural Dynamic Inertial Forces

Can lead to instability and possible structural failure.

Flight testing is still an integral part in estimating the onset of flutter.

Current flutter prediction methods only account for variation in flutter frequency alone, and do not account for variation in structural mode shape.

VIDEO


Flutter Points: Mach Number vs. Altitude

18000

19000

20000

21000

22000

23000

24000

25000

26000

27000

28000

29000

30000

31000

32000

33000

34000

35000

36000

37000

0.3 0.4 0.5 0.6 0.7 0.8 0.9

Mach Number

Alt

itud

e, ft

.

Purpose of Research

Flutter problem can be very sensitive to structural parameter uncertainty.


Wing Structural Model

Governing Equation of Unforced Motion for Wing

Modal Analysis: mode shapes and frequencies


Wing Structural Model


Application of the Unsteady Aerodynamics Aerodynamic Forces

Vector of panel forces

Vector of non-dimensional pressure coefficients

*Aerodynamic forces calculated in different coordinates than structure


Application of the Unsteady Aerodynamics Aerodynamic force: Pressure Coefficient

cP = vector of panel pressure coefficients

w = vector of panel local downwash velocities

AIC(k,Mach) = Aerodynamic Influence Coefficient matrix (complex)

Determined from the unsteady doublet lattice method


Complete Aeroelastic Wing Model

Since the structural model and the aerodynamic model have been established the complete model can be constructed

Representation of the Aeroelastic Wing Dynamics as a First Order State Equation Needed to Apply Robust Stability ( analysis) The dynamic state matrix will be a function of one

variable (U) Tailored for subsequent control law design, if desired



Coordinate TransformationAerodynamic force calculations in a different

domain than structural Modal Domain Approximation

Significantly reduce the dimension of the mass and stiffness matrices

h = HMatrix of retained mode shapes


Forced Aeroelastic Equation of Motion:

Flutter prediction can now be done: v-g method Not suitable to be cast as a 1st order state

equation AIC is not real rational in reduced frequency (k)




Unsteady Aerodynamic Rational Function Approximation (RFA)

With constant Mach number,approximate as:

If s = j, then p = jk



Atmospheric Density Approximation

Direct relationship between atmospheric density and freestream velocity

Coefficients are a function of Mach numberBased on the 1976 standard atmosphere

model



State Space RepresentationState Vector

First Order SystemOnly a function of velocity for a fixed constant Mach number


Nominal Flutter Point Results

V-g Flutter Point(no AIC or density approximation)

Flutter Point calculated using stability of ANOM


Nominal Flutter Point


Model with Uncertainty

The flutter problem can be sensitive to uncertainties in structural properties

A model accounting for uncertainty in structural properties is desired

An allowable variation to velocity must be accounted for to determine robust flutter boundaries due to uncertainty in structural properties

Robust flutter margins are found using Robust Stability Theory ( analysis)


Robust Stability

The Small Gain Theorem- a closed-loop feedback system of stable operators is internally stable if the loop gain of those operators is stable and bounded by unity


Robust Stability

The Small Gain Theorem


Robust Stability

: The Structured Singular Value- With a known uncertainty structure a less

conservative measure of robust stability can be implemented

stable if and only if


Application of the Allowable Variation in the Freestream Velocity Allowable variation to velocity must be accounted for to

determine robust flutter boundaries due to uncertainty in structural properties.

System can be formulated with a stable nominal operator, M, and a variation operator, .

M - constant nominal operator representing the wing dynamics at a stable velocity

– variation operator representing the allowable variation to the nominal velocity

Nominal flutter point can be determined using this M- framework which will match that found previously.


Application of the Allowable Variation in the Freestream Velocity

Velocity representation

Applied to Aeroelastic Equation of motion


Application of the Allowable Variation in the Freestream Velocity

Formulate M- model with polynomial dependant uncertainty definedStandard method to separate polynomial

dependant uncertainty (Lind, Boukarim) Introduce new feedback signals


Nominal Flutter Margin

Only V variation is considered


Application of Parametric Uncertainty in the Wing Structural Properties

Must expand M- model to account for uncertainty in structural parameters

Account for uncertainty in structural mode shape and frequency

Uncertain elements are plate structural properties:



Define uncertainty in any modulus (elasticity or density)

Structural mode shapes and frequencies are dependant on this:

derivatives calculated analytically (Friswell)



Apply J to Aeroelastic Equation of motion:

Note: 2nd order J2 terms are neglected



Formulate M- model V = VI

J = JI


Robust Flutter Margin Determination Uncertainty operator, , a function of 2

parameters (V, J) Calculation of is necessary


Robust Flutter Margin Determination Formulate frequency dependant model

1/s

s = j


Robust Flutter Margin Results



30% uncertainty in

*



30% uncertainty in

E*


Conclusions and Discussion

Complete Model Direct mode shape and frequency dependence on structural

parameters Analytical derivatives avoiding computational inaccuracies

State Space Model Aerodynamic RFA Flutter point instability matches V-g method Well-Suited for Subsequent Control Law Design if Desired

Method can be easily applied to a much more complex problem (i.e. entire aircraft)


Major Contributions of this Work

Inclusion of Mode Shape Uncertainty Traditionally only frequency uncertainty is considered

Dependence of Mode Shape and Frequency The uncertainty in both the structural mode shape and mode

frequency are dependant on a real parameter (E*,*) The individual mode shapes and frequencies are not

independent of one another

Complete M- model with Uncertainty Well suited for subsequent control law design taking structural

parameter uncertainty into account (Robust Control)


Areas of Future Investigation

Abnormal flutter point Instability reached with a decrease in velocity Abnormality due to Mach number dependence Wing created that would flutter at reasonable altitude

Limited range of valid velocities Due to Mach number dependence and standard

atmosphere


Questions?

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