Flutter Analysis of a Two-Dimensional Airfoil Containing Structural Hysteresis Noolinearities

199 Pages·1999·6.56 MB·English

Flutter Analysis of a Two-Dimensional Airfoil
Containing Structural Hysteresis Noolinearities
by
Brett M. Brooking, B.Eng. (Aerospace)
A thesis submitted to the
Faculty of Graduate Studies and Research
in partial fulfilment of
the requirements for the degree of
Master of Engineering (Aerospace)
Ottawa-Carleton Institute for
Mechanical & Aerospace Engineering
Carleton University-
Ottawa, Ontario
Canada
Q copyright 1998, B. Brooking
u*m
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Aeroelastic flutter is a dangerous phenomenon where aerodymnics interact uith
the structure of an aeronautical component to produce potenùally damaging oscillations.
Small concentrated structural nodineaities cmh ave signïficant effects on the flutter
behaviour and cm, in particul- cause large-amplitude oscillations at lower airspeeds
than for linear systerns-
This thesis documents an investigation of one particular type of noduiearïty,
hysteresis. The effects of introducing hyneresis into an otherwise linear aeroeiastic
system are detemiined. This work was a continuation of previous work that examined the
effects of bilinear and cubic type noiilinearities in a dynamic system consisting of a two-
dimensional airfoi1 having two degrees of fieedom. This thesis first outlines a method of
characterizhg the response of the system through the use of a fuite difference
formulation to produce a time marching simulation that traces the response of the system
through tirne. A second solution method oudined is a semi-anaiytical method using a
Descnbinp Function technique to approximate the amplitude of the response of the
system based on the nonlinearïv present.
Results are presented in the form of "maps" that indicate confi,wations where
Limit Cycle Oscillations (LCO) are induced, and in the form of amplitude versus airspeed
plots for the LCO cases. The simulation and describinp function methods were fomd to
compare with reasonably good accuracy. It was found that the Descnbing Function
solution tends to becone less accurate as the assumphons of sinusoida1 motion break
down.
I would Iike to thank Peter Ba.rrîngton, of Carleton Universis., my Master's thesis
supe~sorw, ho has provided me with guidance and assistance through this project. 1
would also like to thank my CO-supervisorB en Lee of the Institute for Aerospace
ResearchoN ational Research Council for his help and insight into this topic, and for the
use of the NRC facilities. I would also like to thank my third CO-supewisorD ick Kind, of
Carleton University. for going to great lengths ensure this project drew to cornpletion and
not forgetting me. Thanks aiso to the people at Davis Engineering for giving me tirne.
facilities and support to complete the project. The support of dl these people is greatly
appreciated-
This thesis is dedicated to Nicole. She is my light whenever it is dark.
TABLE OF CONTENTS
...
ABSTRACT .......................................................... i ~ i
1.0LNTRODUCTION ................................................... 1
1-1 Problem Definition .............................................. 1
1- 2 Scope of Present Work .......................................... - 3
2.0BACKGROUND .................................................... 5
2-1 Previous Work ................................................ - 5
3.0THEORY ......................................................... 12
~.lEquationsofr\noti~.n. ......................................... 13
3 .1.1 Quasi-Steady Aerodynamics .............................. 18
3.1.2 Unsteady Aerodynamics ................................. 3,3,-
3.3SolutionMethods .............................................. 23
3.2.1 Houbolt Finite Difference Method ........................ -36
3 2.1.1 Houbolt Method St arting Procedure ................2 8
3 -2-1 - 2 Houbolt's Recurrence F o d a ................... -29
3.2.2 Describing Function Approach ........................... -29
4OANGLYSIS ........................................................ 36
4.1 System Def-ition ............................................ -36
4.3 Time Marching Simulation Solutions .............................- 37
3.3 Describing Function Solutions ................................... -42
5.0 DISCUSSION AND COMPARISON OF RIESULTS ..................... -45
5.1 Finite Difference Simulations ................................... -46
5-1-1V alidation ........................................... -46
5-1- 2 Flutter Boundaries ......................................4 8
5.1.3 Ainpeed vs Limit Cycle Amplitude ....................... -50

LIST OF FIGURES
Figure Description
3.1 Schematic Diagram of Two-Dimensional Airfoi1 with Two Degrees
of Freedom
3 2 Diagram of Restoring Moment vs. Pitch Angle for Hysteresis
Spnng
3 -3 Block Diagram of Describing Function Method in Limit Cycle
S ystem
3 -4 Diagram of Restoring Moment vs Pitch Angle for Bilinear Spring
with Freeplay
4- 1 Sample Pitch/Heave Response for Hysteresis Nonlinearity
4.2 Damped Pitch Response for Hysteresis Nonlinearity
4.3 LCO Pitch Response for Hysteresis Nonlinearity
4.4 Damped Pitch Response for Hysteresis Nonlinearity
4.5 LCO Pitch Response for Hysteresis Nonlinearity
4.6 LCO Pitch Response for Hysteresis Nodineari:ty
4-7 Unstable Pitch Response for Hysteresis Nonlinearity
4.8 Hysteresis Spring Reversai of Pitch Direction on Slope
4.9 Hysteresis Spring Reversai of Pitch Direction on Flat
4-10 Root Locus Plot for Two Degree of Freedom Aufoii: o = 0.2
4.1 1 Pitch SprinJ Stifbess vs. Flutter Speed: o = 0.2
4.12 Describing Funcrion vs. LCO Amplitude
vii
5-1 Sinusoidal Type Pitch Response for Hysteresis Nodine-
Power Spectral Density of Pitch Response for Hysteresis
Nonlinearity
Pitch Response with Harmonic Structure for Hysteresis
Noniinearity
Power Spectral Densi- of Pitch Response for ETysteresis
Nonlinearïty
Results Plots for Hysteresis Nonlinearïty R u 0 6 fiom Table 1.1
Res1dts Plots for Hysteresis Nodinearity Run 10 fiom Table 4.1
Results Plots for Hysteresis Nonlinearity Run 14 fiom Table 4.1
Results Plots for Hysteresis Nonlinearity Run 15 fiom Table 4.1
Resdts PIots for Hysteresis Nonlinearity Run 16 fiom Table 4.1
Resdts Plots for Hysteresis Nonlinearity Run 100 fiom Table 4.1
Results Plots for Hysteresis Nonlinearity Run 2 0 1 fiom Table 4.1
Results Plots for Hysteresis NonIinearity Run 102 fiom Table 4.1
Results Plots for Hysteresis NodineanS. Run 103 fiom Table 4.1
Results Plots for Hysteresis Nonlinearity Run 1O 4 fiom Table 4.1
Results Plots for Hysteresis Nonlinearity Run 105 fiom Table 4-1
Results Plots for Hysteresis Nonlinearity Run 11 0 fiom TabIe 4- 1
Results Plots for Hysteresis Nonlinearity Run Il1 from Table 4-1
Rcsults PIots for Hysteresis Nonlinearity Run 25 fiom Table 4.1
Results Plots for Hysteresis Nonlineariq Run 27 fiom Table 4.1
Results Plots for Hysteresis Nonlinearity Run 28 fiom Table 4.1
Results Plots for Hysteresis Nonlinearïty Run 29 fiom Table 4.1
Results Plots for Hysteresis Nonlinearity Run 3 0 from Table 3.1
Results Plots for Hysteresis Nonlinearity Run 3 1 fiom Table 4.1
Results Plots for Hysteresis Nonlinearity Run 32 fiom Table 4.1
Results Plots for Hysteresis Nonlinearity Run 33 Eom Table 4.1
Resuits Plots for Hysteresis Nonlinearity Run 120 fiom Table 4.1
Results Plots for Hysteresis Nonlinearity Run 121 from Table 4.1
viii
5.28 azb,c Results Plots for Hysteresis Noniinearity Run 130 £iom Table 4.1
5.29 ab,c Results PLos for Hyaeresis Nonlinearity Run 13 1 from Table 4.1
5-30 a,blc Results Plots for Hysteresis Nonlinearity Run 132 from Tblz 4.1
5.3 1 ab$ Results Plots for Hysteresis Nodine* Run 133 from Table 3.1
amplitude of input sipal to Describing Function a(t) = B + Asin(wt+e)
position of elastic axis relative to midchord position
bias of input signal to Describing Function x(t) = B + Asin(ot+B)
semi-chord
coefficient of lifi
Linear heave smictural damping coefficient per unit span, see Equation 3.2
linear pitch structurai damping coefficient per unit spul. see Equation 3.2
nonlïnear function reruming the restoring moment in pitch
hction to show nonlinearity is srnall. F(a)= v fia)
heave position
mass moment of inertia about elastic axis per unit span
linear pitch sprinp stiaess
linear heave spring stfiess
lifi per unit span
airfoi1 mass per unit span
slope of hysteresis nonlinearity
moment per unit span