A class of nonlinear damping models is introduced with application to flexible flight structures characterized by low damping. Approximate solutions of engineering interest are obtained for the model using the classical averaging technique of Krylov and Bogoliubov. The results should be considered preliminary pending further investigation

Balakrishnan, A. V.

English

NASA Technical Reports Server

5/89- 1346 3
Revised November 1987
SOME NONLINEAR DAMPING MODELS IN FLEXIBLE STRUCTURES
A.V. Balakrishnan t
Electrical Engineering Department
UCLA
September 1987
Abstract
We introduce a class of nonlinear damping models with application to flexible flight
structures characterized by low damping. We are able to obtain approximate solutions of
engineering interest for our model using the classical "averaging" technique of Krylov and
Bogoliubov. The results should be considered preliminary pending further investigation.
t Paper presented at NASA SCOLE Workshop, November 1987, Colorado Springs, Colorado.
https://ntrs.nasa.gov/search.jsp?R=19890004092 2020-03-20T04:41:51+00:00Z
1. Introduction
The problem of characterizing the damping mechanism in flexible structures has
received renewed attention in recent years in connection with the need to stabilize flexible
flight structures such as antennas deployed in space. The damping models even when
simplified to be linear appear to lead to rather complex mathematics if the structure is
described by partial differential equations and much progress has been made (the analyticity
of the generated semigroup has been shown to be essential). But experimental evidence as
in SCOLE [ 1 ] seems to support the need for nonlinear models -- the decrement is much
smaller than predicted by linear models. Some of the difficulty inherent in handling
nonlinear models is offset by the fact that damping, whatever its nature, is still small. This
opens up in particular the feasibility of obtaining approximate solutions using the classical
averaging method of Krylov-Bogoliubov [ 2].
In this paper we study a class of nonlinear models and approximate the response by
the Krylov-Bogoliubov technique. We use a modal expansion and neglect off-diagonal
terms. The emphasis is on useful engineering solutions rather than abstract mathematics.
We begin in Section 2 with the primary nonlinear damping model for the simplest
system -- the one-dimenisonal or single-mode case. We emphasize in particular one
feature that emerges, viz., the potential lack of identifiability from response data. In Section
3 we generalize to the multi- (non-finite-) dimensional case. In Section 4 we show the
relevance of the Krylov-Bogoliubov technique for approximating solutions to nonlinear
boundary feedback. We may mention that there is much work -- even classical in nature
-- on nonlinear oscillations such as the nonlinear pendulum where the spring constant is no
longer linear; however, relatively little attention appears to have been paid to the small
nonlinear damping term case.
-2-
2. Single-mode Example
To illustrate ideas, let us begin with a one-dimensional (single-mode)
d((O + F_.D(x, 50 + O_2x(t) = 0
where the dots indicate time-derivatives, as usual. We assume that:
the energyso that for E(t),
we have
example:
(2.1)
D(x, Jc)dc > 0 (2.2)
(2.3)E(O = _-(,i:(t) 2 + 0_2x(t) 2)
d E(o = -eD(x, <_ 0
satisfyingthe energy non/ncreaserequirement.The particularchoice for
shall make is:
D(x, 2) =
are normegative integers,
(2.4)
where
1 12' 
Ko(a) - 2_ D(a sin 0, ao_ cos 0) cos 0 dO (2.8)
0
0 < o_,13 and 0 < ot+13 < 1; 0<_<1, 0<7<1.
For small enough E we may apply the averaging method of Krylov-Bogoliubov [2, 5].
Thus, we write for the approximate solution:
x(t) = a(t) sin (cot + 0(t)) (2.6)
where the amplitude function a(t) and the phase function 0(t) are slowly varying over
the period T = 2rdco. According to the K-B approximation [2]:
da E
go(a) (2.7)
where m, n
2_i + _/x2"lxl _ _c<2"+1)_II3 (2.5)
D(x, de) we
-3-
and
-_ Po(a)dt = (2.9)
2_
1 jPo(a) = _-_
0
D(asin¢, ao_ cos ¢) sin ¢ d_}. (2.10)
Now we can readily calculate that for our choice, because of (2.2),
Po(a) = 0
JKo(a) -- 2o_a co cos2_ de
0
l+13y 2m+2n+l+ct+_ [_ f 2_+ co a
o
sin2"O c0s2n+2_ Isin ¢1a Icos _11_de I
= _2_a + a2m+2n*l+a*f_yO)2n+l+f_t
where
2_1
2re f sin2"_ cos2"_ cos2_ Isin _1a Icos _1g d0
0
and is a constant less than 1/2. Hence letting p -- 2rn + 2n, we have
da
-E(O3_a + aP+l+c_+f_O32n+fJ'y[t)dt -
We may set _ = 1 without loss of generality since we may absorb it into
Then
t
yielding
a(t) = a(O)e -t_; rl
I.
a(t)
- 0)_a + aP+l+a+130)2n+13yId
a(0)
1
+ a(O)P+a+13_ 2"-1+13 _-_:(1-e-'°_;(P+a+f_))] -p+a+f_-
We can readily verify that for _ = 0, we have
1
a(t) = a(O)[1 + a(O)P+a+f3o_2n+f3yB(p + ot + p)t]-P+_-T-ff-_
(2.11)
(2.12)
and y.
(2.13)
(2.14)
(2.15)
-4-
The case ), = 0 is evenmoreobvious.Onesalientfact that emergesimmediatelyfrom
(2.14)and(2.15)is that it wouldbedifficult to resolve p + ct + 13 into its components
from response data, unless we can change to.
Note also from (2.15) that the rate of decay is not exponential in t and further the
decrement over any integral multiple of the period depends on the initial amplitude as well
as the frequency of oscillation. Finally for inegral k and
2rtk
t -
to
we have, taking logarithms and setting c = p + cc + I_
log
a(0) - -2nk;
_ lc log [1 -2_k_c 1a(O)cto2n-l+f-_TP" (1 - e ) (2.16)+ _
For small _ this is well approximated by
1 log (1 + 2rck)_a(0) c) ,
where
_, = (.02n+13-1 71.t "
The slope (as a function of k)
2n_'a (0) c (2.17)
= -2n_ c(1 + 2nk)_a(0) c)
and hence the linear damping term is yielded by the asymptotic slope as k -_ _, while
for small k there is a marked curvature which depends also on the initial amplitude
a(0). The inital (at k = 0) slope
= -2=_- 2rtXa(0)c (2.18)
c
is larger (in absolute value). The second derivative being positive, the curve is convex
-- CUP. This is in excellent qualitative agreement with SCOLE damping data: see
Figure 1 where amplitude is plotted on logarithmic scale (period = 5 seconds).
ORICtI_IAL PAGE IS
OF P_)OR QUALITY
r_
' i
apn]!ldmV
0
t i .
0
0
0
0
0
0
0
m
0
N
m
W
0
!
Z
-5-
OF QUALIfy
To get another version of (2.16) we may replace (2.7) by the more exact formula
a(twT) a(t)
- _.. Ko(a(t)) (2.19)T =
I.O
and hence using
a k = a(kT) (2.20)
we would have
eT
ak+ 1 = a k - 7._ Ko(ak)
so that
ra.+ll : (1_ __earKo(ak)I_log k ak J co ak Y
which under our "small damping" assumption, may be replaced by
(2.21)
ak+l e.T Ko(ak)
log ak tO a k (2.22)
c 2n-l+_
= -¢27r(_ + ak)'l.tto ). (2.23)
3. Multidimensional Generalization
Analogous entirely to the one-dimensional case, we may write the general nonlinear
dynamic equation for flexible structures [2] as
M_(t) + _(x(t), 5c(t)) + Ax(t) = 0 (3.1)
where the state x(t) ranges in a separable (real) Hilbert space _t; M is a self-
adjoint positive definite (with bounded inverse) operator on _t onto _t; A is a self-
adjoint nonnegative definite closed linear operator with domain dense in _t and with
compact resolvent; we shall (for simplicity) assume that zero is in the resolvent set of
r
A. In the linear case
_(x(t), 5c(0) : D±(t) (3.2)
where D is also a self-adjoint nonnegative definite closed linear operator whose
domain includes that of _-. In the most important case we further specify that
6O
-6-
D% = 2{kc0kM % (3.3)
where (0k } are the M-orthonormalized eigenfunctions of A with eigenvalues (o2 such
that
A0k -- 0)2 M0k . (3.4)
Here _k is the damping ratio. If _k -- _ and we have strict proportional damping --
see [3] for more -- D is then essentially the positive square root of A (except for M).
More generally we require that lim _k > _ > 0. In the nonlinear analogue of (2.5) we
k
set
(3.5)
_(_)),Ok) -- 0 j;e k
and more generally for x, y such that
2 4m +20t 2n+2+213 2Y- (o2kbk + Y_ak bk _k < *"
where
(3.6)
we define:
ak = [x, 0kl ; bk = [Y, Ok] (3.7)
•2n+ 1[_(x, y), _k] = _k a_ '_ [ak[a Ok Ibk113 +
where, as before, m and n are nonnegative integers and that
0 < a, [3 < 1; ct+[_ < 1 ;
Note that
Hence if
[_(x,y), y] > 0
E(t) = _ {lAx(t), x(t)l + [M±(t), k(t)]}
for every x and y.
we have that
2 _k(ok bk (3.8)
tE(t) = -[_(x(t), k(t)), 5c(t))l <
Or, the energy is nonincreasing. Using the modal expansion
6/
0 < _'. (3.9)
o. (3.1o)
-7-
we see that for each k
iik(t ) + o2_ ak(t)
x(t) = Y_ak(t ) #k (3.11)
+ ak(t)21ak(t)laak(t)lak(t)l I_ + 2_k0lkak(t ) = 0. (3.12)
We can therefore invoke the K-B averaging procedure obtaining the approximate solution
ak( 0 = Ak(t ) sin (cokt + 0k)
1
[ 2n+13-1 _/kg (1- e-t;kokc)l-'i-Ak(t ) = Ak(O ) e-t;k_k 1 + Ak(O) c O_k
And for _k--0,
1
l 1Ak(t ) = Ak(O ) 1 + Ak(O) c O)k Ykgct
where, as before,
(3.13)
(3.14)
2_1 f sin2mO cos 2"# Isin #l a Icos _l I_ cos2# d0 . (3.15)
0
For ot + 13= 0, we can give a kernel representation. Thus
z = _(x, y) = Y. Yi _i[#i, Xl2m [_i' y]2n+ 1
where
Y.y_ < 0_; yi_>O;
and for the concrete realization R = L2(0, L), the corresponding "kernel" would be
W(s, G 1 ..... (Y2m, $1 ..... S2n+ 1) = Z _i ¢i(S) I_i(G1) "'" I_i(G2m) (Pi(Sl) "'" ¢_i(S2n+ 1) (3.16)
i
and
L L
Z(S) = f "'" ; W(s, O I ..... (J2m, s1 ..... S2n+l) x(GI) x(G2) "'" x((J2m)
o o (3.17)
× y(s 1) ""y(s2n+l) dG 1 do 2 .-. dG2m ds I ... ds2n+l •
A plausible model in this case would be to rewrite (3.1) as
Mk'(t) + _2(x(t), Dk(t)) + 2_DSc(t) + Ax(t) = 0. (3.18)
-8-
which will satisfy (3.10), since the _'i in (3.16) are nonnegative.In the notationof
(3),the"roll" equationsfor examplewill havetheform:
pAi_,(t, s) +
L
_::
0 0
El, u_'"(t, s) - 2_/pAEI, i)3u_(t' s)
at
L
W(s, a I ..... a2,,,, s 1, .... s2,+l ) x u,(t, al) "" ut(t, 02,,, )
_3u_( t, Sl) _3u_( t, S2n+l)
X 3t _2S _t _ZS dal "'" dc2m dsl "'" ds2"+l
ffi 0.
It is clear that we may generalize (3.17) without recourse to modes. The "nonlocal"
nature of the operator should hardly be surprising, since this is already so in the
linear case if we want strict proportionality (_k = 4) for example.
4. Application to Nonlinear Boundary Feedback
In this section we shall apply the K-B averaging technique to obtain approximate
solution to the response of a flexible structure to nonlinear boundary feedback control. The
control effort is small so that the K-B approximation is reasonable. We follow [4] for the
model where the "boundary" is finite-dimensional. Thus we have in the same setting as
Section 3, but omitting the natural damping term:
M_'(t) + Bf(B*±(t)) + Ax(t) = 0
onto _t and f(.) maps R m into R"
[f(u),u] > 0 for u# 0. (4.2)
where B means R"
Using the modal expansion as in Section 3:
x(t) = _. ak(t ) _lc
ak t) + + [f(gaj(t)bj), bk] = 0
G3
we obtain
and is such that
(4.1)
(4.3)
-9-
where
B*0k = bk-
Taking the approximation
we see that setting
[f(Z itj(t)b)),bk] = [f(_k(t)bk),bk]
ak(t ) = Ak(t ) sin (cokt + 0k(t))
that
d#k(t) = 0
Ak(t = K°(Ak(t))fDk
where
2n1
Ko(a) - 21_ f [f(aco k cos Obk), bk] cos O dO.
o
To simplify matters further let us take
m = 1.
Then
[f(aoJkb k cos 0), bit] = bit f(abkok COS 0) •
We shall take:
f(u) = _,tan-1 u
which is consistent with (4.2). Then (4.3) becomes
iik(t) + o_2ak(t) + _l_b2tan-1 _k(t) = 0 (4.4)
Hence
Ko(a)
2_1
- 2re f
o
_,bk tan -1 (abkco k cos 0) cos 0 dO
"2/_2t"_2 -- 1 ]X [,11+ ..k.ok
2o)k
Ak(t) dAk(t) _,
_2t.,2t,_2 = - _ dt .[41 + .vk,. k - 1
To solve this, let
z(t) (4.5)
-10-
so that
dz(t) - b 2 _, dr. (4.6)
2('¢r_ - 1)
Let
F(z) = e"17 (_z - 1), z_> 1. (4.7)
Then
F'(z) > 0 for z> 1
and hence we may define the inverse function
F(z) = y ; z = F-l(y).
Thus (4.6) has the solution:
z(t) = F -1 [F(z(O))e -b_'t] , (4.8)
where
unless Ak(0 ) = 0.
= :bko_kAk(t) "_F-I(F(1 + Ak(O)2bko_k)e - I (4.9)
Note that
z(0) = 1 + Ak(0)2b2¢o 2 > 1
F-l(y) _ 1 as y_0
and hence z(t) decreases monotonically to 1 and hence the amplitude Ak(t ) decays to
zero asymptotically.
Note that the decay rate depends on the control effort 2_/_2 as well as the initial
amplitude. Of course we have in (4.4) yet another nonlinear damping model. Following
(2.22) we have:
222 12r_, Ix/1 + a]bko3 k - 1
- 2_ [. aj
where
2_
aj = A_(jT) ; T -
COk
-11-
References
, [1]
/
[2]
[31
[4]
[5]
Proceedings of the 3rd Annual NASA SCOLE Workshop, November 1986. NASA
Technical Memorandum 89075. Edited by L.W. Taylor.
N. Krylov and N. Bogoliubov. Introduction to Non-Linear Mechanics, Princeton: Prince-
ton University Press, 1949.
A.V. Balakrishnan. Control of Flexible Flight Structures. In: Volume dedicated to
J.L. Lions. To be published by Gauthier-Villars, Paris, 1988.
A.V. Balakrishnan. Stability Enhancement of Flexible Structures by Nonlinear Bound-
ary Feedback. In: Proceedings of lFIP Working Conference on Boundary Control and
Boundary Variations, Nice, June 1986.
S.C. Hayashi. Nonlinear Oscillations in Physical Systems. New York: McGraw-Hill,
1964.
66


Some nonlinear damping models in flexible structures

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