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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_