Flutter' 'Analysis of an Airplane With Multiple Structural Nonlinearities in the Control. System

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NASA Technical Paper 1620
Flutter'' Analysis of anA irplane
With Multiple Structural
' Nonlinearities in the Control. System
> , -
Elmar J. Breitbach
MARCH 1980
II
TECH LIBRARY KAFB. NY
NASA Technical Paper 1620
Flutter Analysis of anA irplane
With Multiple Structural
Nonlinearitiesi n the Control System
Elmar J. Breitbach
Latlgley ResearchC euter
Hunzptorr, Virgiuia
National Aeronautics
and Space Administration
Scientific and Technical
Information Office
1980
SUMMARY
Experience has shown that the flutter prediction process for airplanes can
be greatly affected by strong concentrated nonlinearities which may be local-
ized in the linking elements of the control mechanism, in the pivoto f joints
variable-sweep-wing systems, and in the connecting points between wing- and
pylon-mounted external stores. The principle of equivalent linearization
offers an efficient possibility for solving the related nonlinear flutter equa-
tions in the frequency domaaisn a complement to the well-known time domain
procedures. Taking as an example an airplane with nonlinear control character-
istics, it is demonstrated how the equivalent linearization approaccahn be
extended to rather complicated systems with multiple sets of strongly inter-
acting, concentrated nonlinearities.
INTRODUCTION
Routine flutter analyses generally imply linearized representation of both
the structural and the aerodynamic properties.T his approximation has proved
to be a useful basis for the flutter clearance of a large number of aircraft
prototypes. There remains, nevertheless, a significant number of flutter cases
suffering from ratherp oor agreement between analysis and test results. Many
of these disagreements can be traced to structural nonlinearitiAe ss.u rvey of
the various types of structural nonlinearities, their physical sources, and
their effects on aircraft vibration and flutter is given in refere1n, cew hich
indicates that strong concentrated nonlinearities are a common feature of the
control systems of mechanically controlled airplanes. From referen2c, e which
presents a new experimental-numerical approach to determining the dynamic char-
acteristics of hydraulic aircraft control actuators, it becomes obvious that
flutter of aircraft with hydraulic controls may also be greatly affected by
strong concentrated nonlinearities. References 3 and 4 focus on the special
case of a modern variable-sweep-wing fighter airplane with concentrated non-
linearities in the wing pivot mechanism and in the corresponding single-point
external store suspension system. Reference5 and, in particular, reference 6
describe several concepts of how the governing equations of airplanes with con-
trol system nonlinearities can conveniently be formulated in terms of consis-
tent sets of both measured modal data and nonlinear force-deflection diagrams.
The nonlinear flutter equations can be solved in the time domain by using ana-
log computer techniques (see refs.5 and 7) or by numerical integration. In
addition to this time domain approach, promising attempts have been made to
solve nonlinear flutter problems in the frequency domain by employing the prin-
ciple of equivalent linearization (see ref8. ) . The effectiveness and accuracy
of this equivalent linearization approach were impressively demonstrated for a
semispan wing-aileron model wiat hs ingle nonlinearity in the aileron hinge;
the calculated and the wind-tunnel test results agreed very well (seeI ) .r ef.
Application of the equivalent linearization approach to systems with more
than one nonlinearity creates some additional, though still solvable, difficul-
ties. These difficulties are associated with an incompatibility between the
input data representing the equivalent stiffness and damping properties of the
nonlinearities involved and the corresponding output deflections. In coping
with this problem, a recent investigation (re9f). describes the application
of a method called the describing function method’ (re1f0.) to the special
case of a flexible missile control surface with simple undamped free-play non-
linearities in both the roll and pitch degree of freedom of the root support
stiffness.
The particular concern of the present sitsu dtyh e extension of the equiv-
alent linearization conceptto the flutter analysis of complete airplanes with
strong hysteresis-type nonlinearities in the control system. Antisymmetrical
flutter of a sailplane involving strongly interacting rudder and aileron non-
linearities is useda s a realistic example to demonstrate the applicability of
the method proposed.
SYMBOLS
A,B,C mass, damping, and stiffness matrices, respectively, defined in terms
of physical deflections
bv iscousd ampingc oefficiento fc ontrols urfaceh inge
C stiffnessc oefficiento fc ontrols urfaceh inge
,
fr efq uency w/Zrr
F forceo rm omenta ctingo nc ontrols urface
9 absolutea mplitudev alue,s eee quation( 31)
hb endingd eflectiono fq uarter-chordl ineo fl iftings urface
0
j imagiunnai rty,
II ha If -chor d leng th
fMl fi gMhan tcu hm ber
M,D,K generalized mass, damping, and stiffness matrices, respectively
AM,AD,AK generalized matrices of mass, damping, and stiffness changes,
respectively
lA slightly modified form of the equivalent linearization approach.
2
N number of controls involved in flutter case
P column matrix of external forces
colum matrix of generalized coordinates
9
Q column matrix of generalized forces
R matrix of unsteady aerodynamic forces Rr relatedt o normal
modes Qr
t time
U column matrix of physical deflections
V flight speed
01 rotation about quarter-chord ,line of lifting surface
B control surface rotation about hinge line
Y damping lossa ngle, 25
r
diagonalm atrix ofd amping loss angles
8 matching function,s eee quation (30)
5 damping expressed as ratio to critical damping
11 absolute amplitude value,s eee quation (29)
A diagonalm atrix of squarev alues of circular normal frequencies
W r = 2Tfr
P aird ensity
T controls urface chord length ratio (see fig. 11 )
@ integrationv ariable, (fit
@ modal matrix of normaml odes 6,
w circular frequency
Subscripts :
A pariolep reornti es
L prloipn eeratri es
NL nonlinear properties
3
. . .,
r normal mde index, r = 1, 2, n
R rudder properties
VrFI,(J indices of concentratend o nlinearitieisn volved
Superscripts:
F flutter speed
. T tramnsaptro isxe d
0 vstaalrut eisn g
NONLINEAR EQUATIONS OF MOTION
Reference 6 offers a choice of several modal synthesis concepts which can
conveniently be used to establish the aeroelastic equationso fm otionf ort he
flutter analysis of airplanesw iths trongc oncentratedc ontrols ystemn onlin-
earities. Ina ccordancew itho ne of thesec oncepts( concept I1 of ref. 6) , the
originallyn onlineara irplane structure is physicallyc onverted to an artifi-
cially linearized test configuration by replacingt hen onlineare lements by
linears tiffnessesw ith olw damping. The normal mode characteristics of the
linearized test configurations erve as a consistent basis for the calculation
of botht heu nsteadya erodynamicr eactionsa nd a set of nonlinearc oupling
terms retransformingt he test configuration to thea ctuals ystem. The nonlin-
earities can be determined statically in the formo ff orce-deflectiond iagrams
or dynamically by direct measurement of equivalent stiffness andd ampingv alues
versusv ibrationa mplitude. The equationso fm otion of them odifiedl inearized
test configuration,f ormulatedi n terms of physicald eflections,c an be written
in matrix notation as follows:
..
AU + BLU + CLU = P (1 1
where
A mass matrix
viscous damping ma tr ix
BL
stiffness matrix
CL
P column matrix of externalf orces,f ori nstance,u nsteadya erodynamic
forces
U column matrixo fp hysicadl eflections; u and u are first- and
second-orderd ifferentialsw ithr espect to time t
Thed ynamicb ehavior of the unchanged nonlinears ystem may be described by
..
Au + B; + Cu = P (2)
4
I
where
B = BL - ABL + AB&
(3)
- +
C = CL ACL ACm
and where ACL and ABL denote the stiffness and damping properties of the
artificial linear elements and ACNL and AB& denote the amplitude-dependent
stiffnpss and damping of the replaced nonlinearities.
Development of the arbitrary deflection vector u in a series expansion
of the normal mdes 0, of the linearized test configuration yields
u = @q (4)
where
@ modal matrix containing normal modes Qr as columns
9 column vectoor f generalized coordinates
Substituting the modal transformation (eq. (4)) into equation (2), premultiply-
ing by QT, and taking into account equation (3) lead to the generalized equa-
tions of motion of the unmodified nonlinear system
where
7
M = (PTA@
with Bur denoting the control rotation in the section where the control force
- -
is applied. Accordingly, the matrices A%L ABL and &NL LkL degenerate
to the 1 x 1 matrices
5
1
- -
%L,V &L,V = hL,V(BV) bL,V
- -
&m,v &L,V = cNL,v(Bv) CL,V
where cL,v and bL,V definet hea rtificialh inges tiffnessa nd damping of
the Vth control surface and %L, v(B v) and h,( B y) definet hea mplitude-
dependent stiffness and damping oft he replaced nonlinearity of the vth control
surf ace. Hence,
N xN
- E - avTrm,v(Bv) -
AKNL k L = (&m,v M L , ~ ) = c L , v ~ ~ v
V=l V=l
The normal modes Qr of the linearized test configurations atisfyt he or t hogo-
nality condition,
QTA@ = M
aTCL@ = ALM = RL
where
M diagonma algt ore ifn x e ralized masses M,
2
KL diagonma la trogix fe neralizesdt i ffnesses K L , ~= W L , ~ M ~
AL diagonam l atriosx f q uarve a lues of circulanr o rmaf rl equencieus L lr
The generalized damping .m atrix DL, which is notn ecessarilyd iagonal, was
definedi ne quation (6) Without damping coupling,m atrix DL also becomes
diagonalw itht heg eneralized damping elements DL,~.
The unsteadya erodynamic forces P generally dependo n time t, flight
Mach number Mf, flighst peed V, and air density P. Developing P in a
series expansiono fu nsteadya erodynamicf orces Rr related to then ormal
or
modes leads to
Q = QT R(Mf,V,P,t) q (11)
where
AS mentionedp reviously,a pplicationo ft hee quivalentl inearizationa pproach
to nonlinear flutter problems requires a transformation of the differential
equation (5) intot hef requency domain. Accordingly, bya ssuming simple har-
monic motions,
6
jwt ( j = 4 7 ) (1 3)
q(t) = qe
where W is thec ircularf requency,e quation (5) reduces to
+ - + + - + -
[ a 2 M jU(DL ADL AhL) KL AKL AKNL QT R(Mf,V,P,u)]q = 0 (14 )
Solutions of this equation can be obtained mcch more easily by expressing the
viscous damping forces in terms of complex stiffnesses or damping loss angles.
By 90 doing,e quation (14) becomes
r
1
N N
L
V=l
Damping can also be expressed by 5 as a ratio to the critical damping. The
,
relation between 5 and Y is 5 = Y/2. In equation (15) denotetsh e
diagonalm atrixo f the damping loss angles yr associated with the general-
izeds tiffnesses Kr. The matrices AK~,v and AKNL v are definedi n equa-
( 6 ~ )
tion (9). The damping loss angles YL,v and YNL,~ represent the
structural damping coefficients associated with the hinges tiffnesses CL,V
and ~ N Lv ( Bv), respectively, which are definedi ne quation (8). The matrices
r,
M, AL, and @w,h ich describe the dynamic behavioro ft he modified line.a r-
izeds ystem,c an be measuredi n a fairly simple groundv ibration test (GVT)
These modal data and some related geometrical data are giveni n detail in
appendix A for the sailplane taken as ane xample of a nonlinears ystem.
Because of the high aspect ratio and thec omparatively low maximum speed of
thiss ailplane,i ncompressibles tript heory is used to calculate the unsteady
aerodynamicf orces based on the measured mode shapes Qr. Them ethod of deter-
mining the nonlinear terms YNL,~(BV) and AKNL,~ is described int he follow-
ings ection.
EQUIVALENT LINEARIZATION APPROACH
As is known from reference 8, ane lastodynamics ystemw ithn onlinear
stiffness andd amping elementsc an be approximately described as a linear sys-
tem for constant-amplitudev ibrations at anya rbitrarya mplitudel evel. The
fundamental idea of thise quivalentl inearizationa pproach is based ont he
assumptiont hat a nonlineare lastomechanicale lementc an be approximately
replaced by a linear substitute element with equivalent stiffness andd amping
energies when activated at equivalenta mplitudel evels. The accuracy of the
approach,d ependingo nt he special problem to be investigated,c an be assessed
by proceduresd escribed, for instance,i nr eferences 11 and 12 for application
to systems subjected to simple harmonice xcitation.I na ddition,r eference 12
shows a simple way to solve problems with preloaded nonsymnetricn onlinearities,
such as those arisingi ns ystemss ubjected to maneuver loads.
7
In accordance with reference 8 the equivalent linear coefficients of a
nonlinear force-deflection diagramc an be calculated from
where cm(B) and YNL(B) definteh e complex stiffness,
f3.
anwd here thef orce F is a nonlinearf unction of thed eflection Inte-
gration is carriedo ut over a full period of oscillation using 4 = w t as
integration variable.
In viewo f the particular flutter case to be dealt with subsequently, two
special types of bilinearf orce-deflection diag.r ams,s ketched in figures 1
and 2, aree valuated by meanso f equation (16) The diagram in figure 1 is
characterized by a olw stiffness cl for
-B1 6 f3 6 B1 (18 )
where B1 denotes the amplitudec orresponding to the maximum stroke of the
controls urf ace. For
thes tiffness assumes the much higherv alue c2 because of kinematic limita-
tion beyond theb locking point (see figs. 1 and 2). Hence it follows that
where
B1
41 = arcsin -
B
Figure 2 illustrates a bilinearh ysteresis-typef orce-deflection diagram. For
amplitudes belaw theb lockingp oint according toe quation (la), thee quivalent
stiffness andd amping valuesa re
8