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Nonlinear rotorcraft analysis using symbolic manipulation

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Nonlinear rotorcraft analysis using symbolic manipulation*
George T. Flowers? and Benson H. Tongue~
Department of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA (Received March 1987; revised August 1987)
Analyzing a system of nonlinear equations for the presence of limit cycles is often a tedious and time-consuming proposition, even for relatively simple systems. The large amount of algebraic manipulation required is extremely difficult to perform accurately, and can quickly degenerate into an intractable problem. This paper presents the development of a math- ematical model for a helicopter with nonlinear damping in its landing gear. The use of symbolic manipulation in the derivation of the complex set of equations is stressed. The resulting equations represent an important ex- tension to the type of nonlinear equations that have previously been ana- lyzed, and allow a more accurate calculation of the system's response. The techniques used are described in detail and are applicable to the study of generic rotating systems.
Keywords:
symbolic manipulation, harmonic balance, rotorcraft oscilla- tions
Introduction
The analysis of nonlinear systems for the presence of limit cycles is one of the most important techniques for investigating the behavior of these systems, An attractive technique for accomplishing this is the har- monic balance procedure.' In this procedure one usu- ally assumes some periodic motion for the response, substitutes this assumed form into the equations of motion, expands the resulting expressions, and bal- ances the harmonics of the system to obtain a set of algebraic equations. These algebraic equations can then be solved for the unkhown parameters of the assumed motion, such as amplitude and frequency. Unfortu- nately, this technique can become very difficult to ap- ply to many systems because of the large number of terms that result from the expansion of the equations and the accompanying bookkeeping chore. What fol- lows is a discussion of how a limit cycle analysis was performed in a relatively quick and efficient manner using the techniques described above and employing * This work was supported by the Army Research Office, Contract No. 22557-EG. t Graduate research assistant. :[: Assistant professor. symbolic manipulation. 2.3 Previous nonlinear analyses ¢-8 have been confined to a more restricted nonlinear for- mulation than that considered in this paper.
Derivation of equations
The particular problem being investigated in this study is the effect of nonlinearities on the dynamic behavior of a helicopter experiencing a self-excited oscillation called ground resonance. 9 The primary feature of this phenomenon is a structural feedback between oscil- lations of the helicopter's blades and oscillations of the fuselage. A purely linear analysis predicts an expo- nentially increasing oscillation within a certain range of rotor speeds. If nonlinearities are considered, the previously divergent solutions are brought into finite amplitude limit cycles. Limit cycles are steady oscil- lations (not necessarily at a single frequency) that are often generated as the solution of nonlinear dynamical equations. The simplified model being studied is illus- trated in
Figure 1.
It consists of a mass, representing the helicopter fuselage, that is connected via springs and dampers to a stationary reference frame
(Figure l(a)).
The fuselage has two translational degrees of freedom, x and y, that correspond to an actual heli- copter's pitch and roll motions. A rotor, here with four
154 Appl. Math. Modelling, 1988, Vol. 12, April ©
1988 Butterworth Publishers
t_
Nonlinear rotorcraft analysis using symbolic manipulation: George T. Flowers
q
I I I
t
\ x ~, \\
Y tJ'
/ =
pdr
i /
Y ~.
Figure 2
Individual rotor blade model
i i
v
Figure I
Fuselage mass; (b) fuselage mass with rotor
rigid blades, rotates with respect to the fuselage mass at a constant speed w
(Figure l(b)).
The equations of motion of this system can be de- rived in the following way. First, obtain a position vector which describes the position of each point along a blade at any instant of time. The position vector for blade I, as shown in
Figure 2,
is P~
=
(x + /,-h COS(~I)+ rcos(¢l + ~:l))i + (y + /,.hsin($0+ rsin(¢l + sc0)j (1) The differentiation of this vector with respect to time produces the velocity vectors
vii = (.f - I~.t, wsin(Ot)
r(~o + ~:l)sin0kl + ~:l))i (2)
v2~ = (:~ + Ioho~cos(¢O
+ r(w + ~,)cos($, + ~,))j (3)
Vt =
Vii
+
vzt
(4)
A vector describing the kinetic energy of blade 1 can be then formed as the integral of the total kinetic energies of all the differential mass elements, p dr along the blade.
R
TI,
= ~ f (pVl * Vl) dr
(5)
0
where p is the mass per unit length of the rotor blades. Similar expressions for each of the other three blades can be obtained and the resulting expressions added to produce an expression for the total kinetic energy of all the blades. The kinetic energy of the fuselage is simply T = ½(m,.f 2 + m:..f 2) (6) where mx and my represent the fuselage inertias in pitch and roll motions. The total kinetic energy T is
T=Th+ T s
The potential energy of the fuselage is
Vy = ,,(k.,x- + ks.y z)
(7) and the potential energy of the blades is l=~k
vl, = ~.= ~,~:, 8)
where k.,, ks., and
ke,
are linear spring coefficients. The total potential energy V is
v=vs+ v,,
Appl. Math. Modelling, 1988, Vol. 12, April 155
Nonlinear rotorcraft analysis using symbolic manipulation. George T. Flowers
The Rayleigh dissipation functions associated with the system damping are Dx
= C.,2- + (9)
V .~ .
e,
= CyY z +
y-lyl
(10)
: C 9
Db, b,s¢~ k = 1,2,3,4 (11) where C, C:,, and
Cb,
are viscous damping coefficients and Vx and V:. are nonlinear hydraulic damping coef- ficients. The total Rayleigh dissipation D is The following definitions were used in the above equations for notational simplicity:
K.,. , K,.
, K,.i
P~ M.,. P7 pT, = '
•
C~.r Cy C b
P 2M,. P':' 2M,. /zh 21
I,,hS V ,. Vy
v~ = 1 ~ 2M,. v,. 2M,.
4
D=Dx+D:.+ ~]Db,
k=l
These expressions can be differentiated and substi- tuted into Lagrange's equations d(o~)
aT oV oD
(12)
to form the equations of motion. The technique de- scribed above is well suited to the capabilities of a symbolic manipulator 23 and can be applied to any ro- tating system. Symbolic manipulators are a relatively new option in the applied scientist's arsenal of com- putational tools. Put as simply as possible, symbolic manipulation programs will manipulate symbols in the same manner as a human analyst. Thus they can ex- pand terms such as (x + y)- and return the answer
x 2
+ 2xy + y2. They are also capable of symbolic inte- gration and differentiation, yielding 2y.9 when asked for
dy2/dt,
for instance. For this reason, such programs hold great appeal for use in system equation deriva- tions in which a great many terms are present. Since the entire procedure is computerized, the probability of errors is greatly reduced. Far from reducing the role of the human analyst, these programs take over the drudgery of massive differentiations, coordinate changes, expansions, etc., allowing the analyst to con- centrate on finding the best approach for a particular problem without undue regard to the time that an im- plementation of such an analysis would previously have taken. The particular program used in this study- was run on a CDC Vax 11/750 and proved relatively straightforward to use. The equations of motion that result from (12) are £ + 2tx.,.2 + ~,..fl.fl + p~x = ~ [~:~ sin(qJk + ~)
.k=l
+ (to + ~:~.)2cos(~Ok + ~:~)] (13) S N
~ [~ cos(4,~
+ ~k) - (w + ~k)2sin(qJ~ + ~)] (14) S ~k + 21Xb~k + P],~k + u],W sin(~k) = 7(£ sin(~ + ~k)-- j~cos(~k + ~k)) k = 1,2,3,4 (15)
Harmonic balance
Analyses of numerical integration results for the sys- tem of equations (13)-(15) indicate that the overall re- sponse of the rotorcraft fuselage is dominated by a single harmonic while the response of the blades in- cludes a constant offset as well as a primary harmonic (see
Figure 3
and 4). Such behavior indicates that each
X FUSELAGE MOTION
o
g.
uJ rE)
°:
, ,
Figure 3
.... i .... i .... i .... i .... i .... i , 0 4 8 12 16 20 24
FREQUENCY (rod./sec.)
Fuselage response versus frequency
o
Figure 4
Xi(1) BLADE LAG MOTION g:
i °
~o
u~
.... ~. .... ~ .... ~ .... ,;~ .... 2 o .... 2 ~
FREQUENCY (rad./sec.) Blade response amplitude versus frequency
156 Appl. Math. Modelling, 1988, Vol. 12, April
Nonlinear rotorcraft analysis using symbolic manipulation: George T. Flowers
of the dynamic variables associated with the fuselage can be modelled as a sinusoid at a given frequency and those associated with the blades can be approximated as a sinusoid at a given frequency with an additional bias term. Based on these analyses, the following forms were assumed for the limit cycle responses of the ro- torcraft: ~:l = H + A
COS(tobt)
~2 = H - A sin(tobt) ~3 = H - A cos(tot, ) ~4 = H + A sin(tobt)
X = X,.
COS(toet) + X~ sin(toet) y = Y,. cos(toet) +
Y,
sin(toet) (16) (17) (18) (19) (20) (21) Notice that in a standard harmonic balance procedure ~ the assumed forms for the dynamic variables are in terms of sinusoidal expansions at frequencies that are multiples of the principle harmonic of the motion. The type of motion that is assumed in this study differs from the standard procedure in that different frequen- cies are assumed for the fuselage and the blade motion and that a constant offset is introduced into the blade response. These variations increase the complexity of the analysis procedure, but are necessary if the be- havior of the system is to be adequately modelled. In order to perform a harmonic expansion of the equations of motion, several approximations are nec- essary. First, consider the term
klk],
which represents the force due to a hydraulic nonlinearity. This nonlin- earity will be approximated by the use of a describing function. The force due to such a nonlinearity when the x motion is expressed as -X cos(to, t) is ~ to~X- sin-'(toet) 0 < t -< 7r/to~
Fd
= [ _ to~X 2 sin2(to~,t )
7rlto e <- t <- 2¢rlto
(22) where
Fd
is the force produced by the damper. This can be expressed as a Fourier series
N
Fd = ~ b,,
sin(toent) (23)
;I = I
where
2 zr/¢o~,
b, = to--z~ J Fd sin(toxnt) dt
(24)
IT 0
Retaining only the first harmonic produces
Fa
= (8/37r)to~,X z sin(toet) (25) This approximation produces a force in which the am- plitude depends in a nonlinear manner on the fuselage oscillation amplitude and frequency. Another difficulty is posed by transcendental non- linear terms such as sin(A sin(toht)). For very small values of A (the amplitude of the blades), this is closely approximated by
A sin(tobt).
When A is not small, sub- stantial distortion of the amplitude of this harmonic occurs. In order to address this problem, a technique for expressing such relationships in terms of a har- monic expansion was developed. The procedure for this is as follows: A,~,, the component of sin(A sin(to~,t)) that is at the frequency rob (single-term harmonic balance ap- proach), is found from a Fourier decomposition to be
2 7r/tob
CO
/ sin(tobt) sin(A sin(tobt)) dt (26) ,o~ = 2-~
0
If a new variable, y~,, is defined as yh = sin(toht) (27) then expression (26) becomes
A,ob = I f yb(SinAyb)(l - y~,)-'adyo
(28) Integrating along the function Yt, and using integra- tion by parts finally allow an exact integral solution of the form A,ob = 2JI(A) where J~ is a Bessel function of the first order. The expressions for the higher harmonic terms can be obtained in a similar manner. One can then ap- proximate sin(A sin(toht)) in terms of as many harmon- ics as is desired. The procedure for determining the harmonic components of sin(A COS(tobt)), Cos(A sin(tobt)), and cos(A cos(tobt)) is similar. A complete expansion in terms of Bessel functions is m sin(A
sin(toht)) =2[~=oJ,_,,+,(A) in[(2n+,
1)toht]] (29)
sin(Ac°s(to t))=2[~=oJ2 +'(A),
× (- l)'cos[(2n +
i tobt]]
30)
,_1
c°s(Ac°s(toht))=J°(A)+2[~=o(-i)'l+l, × J2o,+l~(A)cos[2(n +
l)tobt] / (31)
..i
cos(A sin(toht)) =
Jo(A)
r
2 ~ Jz~,,+l~(A)cos[2(n + I)tobtl (32)
Ln=0
Thus a closed-form harmonic approximation for any desired number of retained harmonics can be imme- diately determined. For the work done in this paper, it was found that a two-term expansion provided a very good approximation of the full expressions. The above expressions were then substituted into the equations of motion, and the resulting expres- sions were expanded and simplified by symbolic ma- nipulation. 2 This results in a very large number of Appl. Math. Modelling, 1988, Vol. 12, April 157
Nonlinear rotorcraft analysis using symbolic manipulation: George T. Flowers
harmonic terms. For example, expansion of the term ~:l sin(~l + ~:~) results in the following terms:
~l sin ~bl + ~t)~
-½(Ac,.
cos(H)to~,(sin(tOgt) + sin((2tO- %)t))
-½Ac,.
sin(H)to~(cos(%t) + cos((2tO -
tO#)t))
at *
-4Acc2
cos(H)to~;(sm(tobt) + sin((3w,, - 2tO)t))
-**Ac,.,_
cos(H)tO~,(sin((2tO -
%)0 -
sin((4to - 3%)0)
-¼Acc~
sin(H)to~,(cos(to#t) + cos((2to - 3%)t))
-¼Ac,.z
sin(H)tO~,(cos((2to - %)t) + cos((4w - 3%)t))
~A
cos(H)n,.to2(cos(2tOt) +
½cos((2tOe - tO)t)) - {A cos(H)n,.to~cos((3to - 2%)0 -{A cos(H)n,.3to~(cos((to - 2to)t) + cos((3to - 4to#)t))
-I,A cos(H)n,.3to~(cos((3to -
2%)0 + cos((5to - 4%)t))
+ ½An,.
sin(H)to~(sin(tot) + ½sin((2% - to)t))
+ *4An,.
sin(H)tO~, sin((3tO - 2%)0
+ {An,.3
sin(H)to~(sin((4toe - 3tO)t) + sin((3tO - 2%)t))
+ ¼An,.3
sin(H)to~,(sin((3to -
2tO,,)t) +
sin((5tO - 4%)t))
Since the response of this system is dominated by Single harmonic (and constant terms in the case of the blades), only the first harmonic and constant terms are balanced. This procedure yields the following equa- tions:
al - (S/l)(bl
+ cl +dl + e0 + fl = 0 (33)
a2 - (S/l)(b2 + c2 + d2 +
e2) = 0 (34)
as - (S/l)(b3 +
c3) + d3 = 0 (35)
a4Xc -- b4xs --
(S/Mx)
X (C4 + d4 + e4 + f4)sin(H) = 0 (36)
a4Xs + b4xc -
(S/Mx)
x(c4+d4+e4+f4) cos(H)=0 (37)
asy,.- bsy., - (S/Mv)
x (cs+ds+e5 +fs) cos(H)=0 (38)
asy - s + bsy,. + (S/M:.)
x(cs+ds+es+f.0sin(H)=0 (39) where
al =
bl =
CI =
dl =
el =
fl =
a2 = b2 = c 2
(-tO~, + P~,)A - .,c,.tOr,(cos(H)x, -
sin(H)x,.)
½c,.tO2(cos(H)y,.-
sin(H)ys)
x4G.2tO2e(sin(H)x,. +
cos(H)x,)
¼c,.2tO -
g2(sin(H)y, + cos(H)y,.)
n .
cos H)v~tO ~
- 2p, htO~4 ½c,to2(- cos(H)x,. + sin(H)x3
½c,.to~(cos(H)y.,. +
sin(H)y,.) and
as
d2 = ~c,2to~(sin(H)x.,. +
cos(H)x,) e2 = 4c,2%(cos(H)y.~ + sin(H)y,.)
a3 =
P~,HA
b3 = n,.to2(cos(n)x, + sin(H)x.3
c3 = n,w~(cos(H)y.,. +
sin(H)y,.)
d 3 = - c,.
sin(H)vo2to 2 a4 = (P~ - to~)A b4 = -(2/zxto~. + (8/37r)vxAto~)
C 4 = --(tl t. --
ns)(to 2 1-
to~,/4)
d4
=
(Co
+ C.,.)(tO~ + 2OJtOb)
e4 = (c,2 -- G2)(tOtOb +
tO~,/2)
f4 = (r/,.3 --
ns3)AtO~,/4
as = (p} - tO~)A b5 = - (2/z:.to,, +
(8/3rr)£,.AtO~) c5 = (n,. + n,)(tO 2 + to~,A/4) d s = (c,. +
c,)(w~ -
2tOtO~,)
-- Cs2)(roJ2
5 = (G.2 ~ tOtOb)
f 5
= (n,.3 -- n,.3)AtO~/4
the transcendental nonlinearities are approximated sin(A sin(tobt)) = n, sin(tobt) +
ns3
sin(3tot) sin(A COS(tobt)) = nc COS(tO --
bt)
+
r/c3
cos(3tot) cos(A COS(tobt)) = C, + C,2
COS(2tobt)
Cos(A sin(to --
bt)) = c, + c,2
COS(2tobt)
158 Appl. Math. Modelling, 1988, Vol. 12, April

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