equations of motion, define weak and strong solutions, and formulate the identification .... Let wcDom(A*) and y = A*w. ...... 1.00117 3.03186 1.49436 0.35 x 10-4.
A/A_ UA-i7_ _-J7
NASA ContractorReport 172537 ICASE REPORT NO.
IqASA.CR-172537
85-7
19B50011424
ICASE A GALERKIN METHOD FOR THE ESTIMATION OF PARAMETERS IN HYBRID SYSTEMS GOVERNING THE VIBRATION OF FLEXIBLE BEAMS WITH TIP BODIES
#
H. Thomas Banks I. Gary Rosen
Contract
No.
February
1985
NASI-17070
INSTITUTE FOR COMPUTER APPLICATIONS IN SCIENCE AND ENGINEERING NASA Langley Research Center, Hampton, Virginia 23665 Operated
by the Universities
Space Research
Association
[l fl fl7 National Aeronautics and Space Administration I.lll_iloy R_rch
Centl,
Hampton,Virginia 23665
r
I"!.'L7 ! o i,_OO LANGLEY RESEARCH CENTER LIBRARY,NASA H.'_,V,?]ON,. VIRGINIA
A GALERKINMETHOD FORT HE ESTIMATIONOF PARAMETERS IN HYBRID SYSTEMSGOVERNINGTHE VIBRATION OF FLEXIBLE BEAMS WITH TIP BODIES
H. Thomas Brown
Banks*
University
I. Gary Rosen** University of Southern California
ABSTRACT In this report we develop an approximation scheme of hybrid systems describing the transverse vibrations
for the identification of flexible beams with
attached tip bodies. In particular, problems involving the estimation of functional parameters (spatially varying stiffness and/or linear mass density, temporally and/or spatially varying loads, etc.) are considered. The identification problem is formulated as a least squares fit to data subject to the coupled system of partial and ordinary differential equations describing the transverse displacement of the beam and the motion of the tip bodies respectively. A cubic spline-based Galerkin method applied to the state equations in weak form and the discretization of the admissible parameter space yield a sequence of approximting finite dimensional identification problems. We demonstrate that each of the approximating problems admits a solution and that from the resulting sequence of optimal solutions a convergent subsequence can be extracted, the limit of which is a solution to the original identification problem. The approximating identification problems can be solved using standard techniques and readily available software. Numerical results for a variety of examples are provided. To besubmittedforpublicationintheJournalofMathanaticalAnalysisandApplications.
*This research was supported in part by the National Science Foundation under NSF Grant MCS-8205355, the Air Force Office of Scientific Research under Contract No. 81-0198 and the Army Research Office under Contract No. ARO-DAAG29-83-K-0029. **This research was supported in part Research under Grant No. AFOSR-84-0393.
by
the Air
Force
Office
of
Scientific
Part of this research was carried out while the authors were visiting scientists at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23665 which is operated under NASA Contract No. NASI-17070.
TABLE OF CONTENTS
Section
Page
1
INTRODUCTION ........................................... 1
2
THE IDENTIFICATIONPROBLEM.............................4
3
AN APPROXIMATIONSCHEME................................ 17
4
NUMERICALRESULTS...................................... 31
REFERENCES......................................................... 42
iii
SECTION1 INTRODUCTION
In this cation with
report
we develop an approximation
of systems describing attached
lem generally erning
tip
bodies.
bodies)
and initial
data.
varying),
the admissible
such is infinite constrained
of motion for this
dimensional
optimization
form of finite
with
appropriate
dimensional
geometric bound-
identification (spatially
problems, Moreover, if
and/or tempor-
parameter space is a function as well.
The solution
(gov-
the motion of
constraints.
are functional
problem, therefore,
dimensional
beams
type of prob-
(describing
The resulting
tend to have infinite
of flexible
system of coupled partial
equations
the parameters to be identified arily
vibration
of the beam) and ordinary
differential
ary conditions therefore,
The equations
take the form of a hybrid
the vibration
the tip
the transverse
scheme for the identifi-
space and as
of the resulting
necessitates
the use of some
approximation.
The scheme we develop here is based upon the reformulation equations Galerkin the state
of motion in weak form. method is used tO define equations.
the admissible approximating
a finite
identification
problems.
a convergence result
based Rayleigh-Ritz-
dimensional
dimensional
parameter space, we obtain
ments, we derive show next that
Using finite
A cubic spline
approximation
to
subspaces to discretize
a doubly indexed sequence of Using standard
for
each of the approximating
of the
the state identification
variational
approximations.
arguWe
problems admits a
solution
and that
from the resulting
sequence of optimal
a convergent subsequence can be extracted the original
infinite
dimensional
convergence results, feasibility
we present
whose limit
identification numerical
problem.
results
in this
report
to
In addition
to
which demonstrate the
represents
improvement over the method developed in [22]. a scheme which is computationally hypotheses on the admissible class of problems. and in a forthcoming standard
cantilevered,
simpler
Indeed, we have developed
and, by relaxing
are similar
of parabolic
in spirit
the necessary
systems, in [8]
boundary conditions
(e.g.,
for
hyperbolic
for
systems
beam equations
clamped, simply supported,
Other work regarding
problems in elasticity
to a wider
to those presented
paper by Banks and Crowley [5]
etc.).
a significant
parameter space, is applicable
Our results
in [7] in the context
inverse
is a solution
of our method.
The approach described
with
parameter values
approximation
can be found in [2],
methods for
[3],
[4],
[i0]
and
[14]. We simplify beam with
our presentation
an attached tip
however, our general body vibration equations
identification
We employ standard
on the interval
are denoted by Hk(a,b), products
and their
respectively.
the approximation
f : [O,T] * Z, we say that
The corresponding
space with
f_L2([O,T],Z)
2
if
scheme 4.
The Sobolev spaces of real
induced norms are denoted by O, El,pcL (0,_)with El,p > O, ocL2([O,T],HI(o,_)),gcL2(O,T),fcL2([O,T],HO(o,T) ), @cH2(O,_),and @cHO(o,_)with @(_) specifiedin R. Define the Hilbert space H = R x HO(o,_) with inner product
< (n,@),
(C,_) >H = n_ + < _,_ >0"
We then rewrite
Equations
(2.1)
MoD_u(t) + Aou(t)
YoU(t)l x=O = 0
=
as
Bo(t)O(t)
yiG(t)I x=O = 0
G(O)
where u(t)
- (2.5)
:
= (u(t,_),u(t,.))_H,
$
+ Fo(t)
tc(O,T)
y2G(t)l x=_ = 0
Dtu(O)
:
tc[O,T]
(2.8)
MO, AO, Bo(t ) and Yi'
$ = (_(_),@), i=0'1'2
Mo(n,@)
=
(mn,p@),
Ao(n,@)
=
(- DEI(_)D2@(_), D2EID2ap),
Bo(t)(n,
@) =
yi(q,@) =
(- a(t,_)D_(_), DI_ ,
(2.7)
_
Fo(t ) = (g(t),f(t,.)],
= (@(_),@) and the operators defined formally by
(2.6)
on H are
DoD_),
i = 0,1,2
respectively. There exist system (2.6)
several ways in which the notion
- (2.8)
can be made precise.
here are the ideas of a weak or variational solution.
6
of a solution
Of particular solution
interest
and a strong
to the to us
Define the Hilbert
V =
space {V, < .,.
{(n,@)cH : @cH2(0,£), @(0) : D@(O)= O, n : @(_)} , < (@(_)'@)'
It
>V} by
is not difficult
(_(_)'_)
to show that
Choosing H as our pivot
< D2@' D2¢ >0"
V can be densely embedded in H.
space, we have therefore
denotes the space of continuous second order initial
>V =
linear
that
functionals
VCHcV'
on V.
where V'
Consider the
value problem
< MoD_u(t)'
0 >H + aC_(t),
9)
= (2.9)
b(t)(_(t),_)
+ < Fo(t),
_(0)
where _ = (e(_),o)
=
$
and the bilinear
_ >H
Dt_(O )
tc(O,T),
:
_
forms a: VxV . R and b(t):
are given by
a($,_)
=
_cV
< EID2@, D2¢ >0
and
: 0
(2.10)
VxV . R
respectively.
A solution
weak or variational
G to (2.9)
solution
and (2.10) with
to (2.6)
- (2.8).
_(t)cV
Indeed,
if
is known as the deriva-
tives in (2.6) and (2.7) are taken in the distributional sense, A0 and Bo(t) become bounded linear operators from V into V' with < AO$'_ >H = a($,_) and
< Bo(t)$'_
where the H inner duality
pairing
Fo(t)cHcV' and (2.10)
product
>H :
is interpreted
between V and V' (see [1],
we have therefore that are two representations
b(t)($,_)
as its
natural
[19],
extension
[24]).
to the
Since
the systems (2.6) - (2.8) and (2.9) for the same initial value problem in
Vi .
Under the assumptions which we have made above, standard arguments (see [16], solution
[17])
can be used to demonstrate the existence
_ to (2.9)
D_u_L2([O,T],V'
and (2.10) with O_C([O,T],V),
DtOEC([O,T],H ) and
).
In order to characterize as an equivalent
of a unique
abstract
first
strong
solutions
we rewrite
order system and then rely
theory of semigroups and evolution
operators.
(2.6)
- (2.8)
upon the
Let Z = V x H with
product
< (Vl'hl)' (v2'h2)>Z = a(vl'v2) + < Mohl'h2 >H "
inner
We assume that EIcH2(O,_)and occl([o,T],HI(o,_)]-and define the operaA: Dom(A)cZ . Z by
Dom(A) = Dom(Ao) x V
where I is the identity
on V, M0 and AO are as they were defined
above, and
Dom(Ao) =
Similarly,
define
{$
=
(@(_), @) cV: @cH4(O,_), D2@(_) =
the operators
B(t):
Z . Z by
0 B(t)
0
= (t) I(O 1Bo
and let
A(t):
Dom(A)cZ
defined initial
by F(t) : (0, value problem
. Z be given MoZFo(t))
0}.
0 ]
by A(t)
= A + B(t).
, zoEZ by Zo= ($,_)
Dtz(t) = A(t)z(t) + F(t)
z(O)
=
zO.
Let
and consider
tc(O,T)
F(t)cZ
be
the
(2.11)
(2.12)
It is not difficult to argue that conservative. That is
the operator
< Az,z >Z :
0
A is densely defined
z_Dom(A).
and
(2.13)
Moreover, we have Theorem 2.1:
The operator
A:Dom(A)cZ
. Z is skew self
adjoint.
Proof We first
argue that
zcDom(A), A*z = -Az.
-AcA*.
Let Zl,Z 2 _Dom(A) with
< AZl'Z2 >Z =
a($I'$2)
course,
implies
+ < - A051'$2 >H
:
- < $I'-A052
-
0 - < EID2@1'D2@2>0
of V in performing that
zi = ($i'$i)"
and for
=
where we have used integration the definition
That is Dom(A)cDom(A*)
Dom(A*)cDom(A).
Let wcDom(A*) and y = A*w.
z_Dom(A)
< z,y >Z =
< z,A*w >Z =
i0
< Az,w >Z"
Recallingthat z,w,y_Z, let z = (_i,_2),w = "(Wl,W ^ ^ 2) and y = £Yl,Y2_Then 0
=
z " Z
=
a(_l,yI) + H- a(_2,_1) + H ^2
=
O
+ mz2(g)Y _ + o ^2 + o
o - DEI(g)D2Zl(g)w_
(2.14)
^ ,^I .2, ,^2 ^2,sH Let o_H2(O,g)be defined by where Y2 = £Y2'Y2;_H and w2 = £w2'w2) " ^1 D20
= p_, o(2) =
0
and DO(2) =
- mY2 •
Then substitutinginto (2.14)and integratingby parts, we obtain .
0
=
- DEI(2)D2zI(2)_Yl (_) + w_] ^2 D2 < D2EID2Zl,Yl+ w2 >0 + < z2, 0 - EID2Wl >0
which implies
(i)
-DEI(_)D2zI(_)(Yl (_) + w_) + g
tc(O,T),
Dt_N(o ) N.
We define
problems.
17
=
6NovN pN_
the following
(3,2) sequence of
(IDN) Find q = (m,EI,p,ao)cQwhich minimizesj(q;GN(q))where J is given by (2.17) and AN u (t;q) = ( UN(t,_;q),uN(t, (3.1), (3.2) corresponding to qcQ.
• •, q))
is the solution
to
Let {B-_., _j=-I N+I denote the standard cubic B-splines on the interval [0,_] correspondingto the uniform partitionAN = {0'N'N'"" _-- 2_____4} (see N N+I [21]). Let {Bj}j=1 denote the modified cubic B-splineswhich satisfy B_(O) = DB_(O) = O, j = 1,2,..N+1. That is
= BN. = _ J J
AN BN N Let Bj : (j(_),Bj)
N V cV
and let
j = 2,3...N+1.
be defined
N V
by
^N N+I =
SPaN _ .IBj_j=I"
The Galerkin equations (3.1), (3.1)take the form
wN(0)=
(WN)-IsN
_N(o)
18
= (wN)-I_ N
(3.4)
where
[A_]ij = I_EIO2_O2B ". 0 J [B (t)]i
=
j
o(t,. )DB".DB". I J
- i 0
[F_(t)]i = g(tlB_(_) +I °f(t,-)B_
[_N]i = _(_)B_(_)+ I_O_B_
[.N]_j = B_(_)B_(_)+ I_BNB N 0
N+I
^N i,j
: 1,2..N+1
and u (t)
=
jZ__ 1
N
1J
^N
wj (t)Bj.
Our convergence arguments are based upon the approximation properties
of spline functions. Let _3(A N) = SPAN{B-_N+I }j=-1 and let N N+I S3(AN) = SPAN{Bj}j= I. For @a function defined on the interval [0,_] I-N@denote that constraints
element in _3(AN) which satisfies
ITN_)(_-_) = _(_--_), j = 0,1,2...N,
19
let
the interpolatory
DII4_]I_-)=
D_I_- ) j = 0
and N and let
IN@denote that
interpolatory
constraints
D@(_). well
The interpolatory
defined
3.1:
be well
the
DcIN@)(_) :
defined whenever @is
and D@at the end points.
A similar
IN@.
the following properties
: @(_-_-_) j : 1,2...N,
I_@ will
at the node points
We require
Proposition
(IN@)(-_] spline
statement can be made for
approximation
element in S3(AN) which satisfies
two standard
of interpolatory
results splines
concerning
the
(see [23]).
For @cH2(0,£)
IIo
< C_N'2+klD2@I0
k = 0,I
where CIk is independent of @and N.
Proposition3.2: For @_H4(O,_)
2 where Ck is independent
of @ and N.
Lemma3.1
(1)
^ Let @N:
@N + (2)
[@(_),@)cV and let
@ in
^N @ :
N^ P@ =
N N I@ (_),@)"
H2( 0,_) and consequently;N. @ ^ in V.
pN . I strongly
in H.
2O
Then
Proof
(1)
I_"_lo< I_"_l.< l_ _l, l_"_ _lo I_ _lo < c_N21D2_Io_O asN_o where (INS) =
c(IN@)(4),IN¢].
< _"i_" _lo._"_lo_lo < _,i_" _io +_.i#_ _lo+c_,_l_lo H
N^ ^N ^N ^ ^N + aN(u - P u, Otv ) + a(u, mtv ) - aN(u, Dtv ) + bN(t)(v
^N
+ bN(t)(u,
^N bN N^ ^N , mtv ) + (t)(P u - u, Otv ) ^N Dtv ) - b(t)(u,
24
^N Dtv )
and (2.10)
or
1 Dt(H + aN(v^N, v^N)) =
,, ^ n + < (Mo - Mo)Dtu' N 2^ D'vN>" + DtamI(l
- pN)u, vN) - aN((l
+ Dt(a(u,
vN) -aN(u,
- pN)Dtu,
vN)) -(a(Dtu,
vN)
vN) -aN(Dt u, vN))
+ bN(t)(vN ' DtvN) + bN(t)(pN u-u,Dtv) ^ ^N N ^ ^N ^ ^N + b (t)(u,Dtv ) - b(t)(u,Dtv ). Integratingboth sides of the above expressionfrom 0 to t, invoking hypotheses(H1) - (H3) on Q and using standardestimateswe obtain min (mI, m2, m3)(IDtvNl_