Two cubic spline based approximation schemes for the estimation of structural parameters associated with the transverse vibration of flexible beams with tip appendages are outlined. The identification problem is formulated as a least squares fit to data subject to the system dynamics which are given by a hybrid system of coupled ordinary and partial differential equations. The first approximation scheme is based upon an abstract semigroup formulation of the state equation while a weak/variational form is the basis for the second. Cubic spline based subspaces together with a Rayleigh-Ritz-Galerkin approach were used to construct sequences of easily solved finite dimensional approximating identification problems. Convergence results are briefly discussed and a numerical example demonstrating the feasibility of the schemes and exhibiting their relative performance for purposes of comparison is provided

Rosen, I. G.

English

NASA Technical Reports Server

NASA Contractor Report 172468
ICASE REPORT NO. 84-51 NASA-CR-172468
19850003071
ICASE
APPROXIMATION METHODS FOR INVERSE PROBLEMS
INVOLVING THE VIBRATION OF BEAMS WITH TIP BODIES
I. G. Rosen
Contract No. NASI-17070
September 1984
INSTITUTE FOR COMPUTER APPLICATIONS IN SCIENCE AND ENGINEERING
NASA Langley Research Center, Hampton, Virginia 23665
Operated by the Universities Space Research Association
National Aeronautics and
Space Administration ]LANGLEYRESEARCHCENTER
Langley Research Center •LIBRARY,NASA
Hampton.Virginia 23665 H,',_.".'.2TOrJ,VIRGINIA
https://ntrs.nasa.gov/search.jsp?R=19850003071 2020-03-20T21:27:08+00:00Z

APPROXIMATION METHODS FOR INVERSE PROBLEMS
INVOLVING THE VIBRATION OF BEAMS WITH TIP BODIES*
I. G. Rosen+
University of Southern California
Abstract
We outline two cubic spllne based approximation schemes for the
estimation of structural parameters associated with the transverse vibration
of flexible beams with tip appendages. The identification problem is
formulated as a least squares fit to data subject to the system dynamics which
are given by a hybrid system of coupled ordinary and partial differential
equations. The first approximation scheme is based upon an abstract semigroup
formulation of the state equation while a weak/varlatlonal form is the basis
for the second. Cubic spline based subspaces together with a Raylelgh-Ritz-
Galerkln approach was used to construct sequences of easily solved finite
dimensional approximating identification problems. Convergence results are
briefly discussed and a numerical example demonstrating the feasibility of the
schemes and exhibiting their relative performance for purposes of comparison
is provided.
*This research was supported in part by the Air Force Office of Scientific
Research under Control Number 84NM277.
+This research was carried out in part while the author was a member of the
technical staff of the C. S. Draper Laboratory in Cambridge, MA and in part by
the National Aeronautics and Space Adlmlnstratlon under NASA Contract No.
NASI-17070 while the author was a visiting scientist at the Institute for
Computer Applications in Science and Engineering, NASA Langley Research
Center, Hampton, VA 23665.
i

In this short paper we briefly outline two cubic spline based
approximation schemes for the solution of inverse problems involving the
vibration of flexible beams with attached tip bodies. The identification
problem is formulatedas the least squaresfit to data of a hybrid system of
coupled partialand ordinarydifferentialequationsdescribingthe dynamicsof
the beam and tip bodies. The resulting optimizationproblem is infinite
dimensionaland as such, necessitatesthe use of some form of approximation.
The schemeswe have developedare based upon the constructionof a sequenceof
approximating identificationproblems in which the underlying constraining
state equations are semi-discretefinite dimensional approximationsto the
infinite dimensional distributed system which governs the original
identification problem. Our study includes both theoretical convergence
resultsand numericaltesting.
Although our general approach applies to a broad class of problems, to
illustrate the two methods, we consider the problem of identifying the
spatially invariant flexural stiffness ql and linear mass density q2 for a
beam of length £ clamped at one end and cantilevered at the other with a tip
(point) mass of magnitude q3 which is also to be identified. Using the
Euler-Bernoulli theory and elementary Newtonian mechanics, the system of
-2-
q2utt(t,x) = -qlUxxxx(t,x) + f(t,x) x _ (0,£), t € (0,T) (I)
q3utt(t,£) = qlUxxx(t,£) + g(t) t € (0,T) (2)
describing the transverse deflection of the beam and tip mass is obtained
where f and g denote externally applied lateral loads (see [2]). The boundary
conditions are given by
u(t,0)= Ux(t,0)= Uxx(t,£)= 0 t € (0,T). (3)
The initial conditions are of the form
u(0,x)= ¢(x), ut(0,x)= ¢(x) x € [0,_]. (4)
The identification problem is stated formally as
(ID): Flnd q = (ql,q2,q3)T E Q, Q a compact subset of _R'3,
which minimizes
v _ _(ti'xj)J_q;u(',';q)) = _ lu(ti,xj;q) - 12t=1 j=l
subject to u(.,.;q) being a solution to (I) - (4) where
[_(tl,xj)}l=l,...,9 denote a set of given displacement
j=l,...,p
observations.
-3-
The first approximation scheme, which is described in detail in [6], is
based upon the recasting of (I) - (4) as an abstract evolution equation set in
an infinite dimensional Hilbert space. Let H = RxH0(0,£) with inner product
<(q'_),(_,_)>H = q_ + <_'$>0 where {Hk,<-,.>k} denote the usual Sobolev
spaces and Sobolev inner products. Define the operators M0(q):H.H and
A0(q):DCH+H by M0(q)(n,_) = (q3q,q2 _) and
A0(q)_(£),_ ) I-ql_'''(£),ql_ .... ) respectively where
D = {(q,_)cH:_H4(0,£), _(0) = $'(0) = 0, $''(£_ = 0, n = _(£)}. Assume
f € L21[0,T],H0(0,£)) , g C L2(0,T) , _ _ H2(0,£) and _ € H0(0,£) and let
A
F0(t) = _g(t),f(t,.)), _ = I_(£),_) and _ = _(£),_) where _(£) € R is
i
specified if it is not well defined. We then rewrite (I) - (4) as an abstract
second-order system in H;
M0(q) D_u(t) + A0(q)u(t ) = F0(t), t _ (0,T) (5)
A
u(0) Dtu(0) (6)
= d
where u(t) = _u(t,£),u(t,-))_ H and Dt _-_ . Let
V = {(n,_)cH:_H2(0,£),_(0) = _'(0) = 0,q = _(£)} and define
L:VCH+H by L_(£),_) = (0,_''). The operator A0(q) can then be written in
factored form as A0(q) = qlL*L where L*:Dom(L*)CH.H is given by
L*(_,$) = (-$'(£),$'') for (n,$)cDom(L*) = ((n,_)cH : ._H2(0,£), _(£) = 0}.
Let Z = H x H with inner product
<z,W>q = <qlzl,Wl> H + <M0(q)z2,w2> H. Choosing our state as
z(t) = _Lu(t),Dtu(t) ) we rewrite (5), (6) as the first-order system in Z given
by
-4-
_(t) = A(q)z(t) + F(t;q), z(0) = z0 (7)
where A(q):Dom(L*)×VCZ+Z is given in matrix form by
A(q) = [-qlM0(0q)-IL * 0L]
F(t;q) = (0,M0(q)-IF0(t)) and z0 = (L$,_) ($ is assumed to be an element in
V). The operator A(q) is densely defined and skew self adjoint. It
therefore generates (see [8]) a CO group of unitary operators
{S(t;q): -_ < t < =} on Z. The mild solution (strong or classical if F and
z0 are sufficiently regular) t0 (7) is then given by (see [5])
t
z(t) = S(t;q)z 0 + f S(t-T;q)F(T;q)dT. (8)0
Problem (ID) can then be written as
(IDA): Find qcQ which minimizes J(q;Cl(-)z(';q) ) subject to
z(-;q) being given by (8) where the operators
CI(X):Z.R are defined by
x T
Cl(X)I(q,¢),(_,,) ) = f f O(o)d_dT for x € [0,£].O0
For each N = 1,2..., let S3(AN) denote the space of cubic spline
functions on the interval [0,g] corresponding to the uniform partition
AN={0,_,g --N''2£..,£} (see [7]). Let WN = {(q,_)gH:_S3(AN),_(£) = 0},
VN = {(q,0)cH:0ESB(AN),¢(0) = 0"(0) = 0,_ = ¢(g)} and ZN = WN x VN. Then
-5-
ZN is a 2N+4 dimensional subspace of Z which satisfies
zNCDomIA(q)) = Dom(L*)xV for all q_Q. Let pN denote the orthogonal
q
projection of Z onto ZN wlth respect to the <-,-> inner product.
q
Define AN(q):Z N + ZN by AN(q) = pNA(q) and noting that AN(q) is a linear
q
operator defined on a finite dimensional space (and is therefore bounded),
consider the initial value problem
z (t) = AN(q)zN(t) + P (t)F(t;q), t € (0,T)
zN(0) = P_z0,
and its solution
zN(t) = exp(AN(q)t)P_z0
t
+ f exp(AN(q)(t-T))pNF(T;q)d_ t € [0,T]. (9)0
The approximating identification problems for the first method take the form
(iDNI): Find qeQ which minimizes JIq;Cl(.)zN(-;q)) subject
to zN(-;q) being given by (9).
Under rather mild assumptions it can be shown that each of the problems
--N {q-N} contains a(IDN 1) admits a solution q and that the sequence
_k . -- k . Using standardconvergent subsequence k}, q e Q as =.
approximation results from linear semigroup theory (see [5]) and the
-6-
properties of spllne functions (see [7]) to establish convergence of the state
aPproximations, it can then be argued that _ is a solution to problem (ID).
The second approximation scheme involves the rewriting of (I) - (4) in an
equivalent weak/variational form (see [4]). Let H and V be as they have
been defined previously. Define the inner product on V, <'''>V by
<_'$>V = <L$'L_>H" We have the usual dense embeddlngs V C H C V" where V"
is the dual of V. We consider the weak form of (I) - (4)
<M0(q)D (t),8> H + alu(t),8;q ) = <F0(t),B>H, B € V, t € (0,T) (I0)
_( ^ Dt_( ^O) = @, O) = _ (11)
where the bilinear form a(.,-;q) on VxV is defined by
a(_,_;q) = ql<L$,L$>H = <_(q)_,$>H" The derivatives in the definition of
the operator AO(q) are interpreted in the distributional sense and AO(q)
is considered to be an element in-_-_(V,V'), the space of continuous linear
operators from V into V'.
We define a Galerkin approximation GN to u as the solution to
<M0(q)D2uN(t),_N>H + a(uN(t),_N q) = <F0(t),_N>H ' _N g VN, t E (0,T)(12)
uN(0) = pN_, DtuN(0 ) = pN_ (13)
where VN is as it was defined above and pN is now the orthogonal
projection of H onto VN with respect to the <','>H inner product.
-7-
The approximating identification problems are then given by
(IDN2): Find q_Q which minimizes J[q;C2(-)uN(.;q) ) subject to
_N(.;q) being the solution to (12), (13) where the operators
C2(x):V.R are defined by C2(x)I_(£),_) = _(x),x£[0,£].
Using standard variational arguments of the type found in [3] it can be
argued that under sufficient regularity assumptions, for {qN},cQ with
qN.q,
we have _N(qN)._(q,) in V and DtuN(qN).Dtu(q*) in H as
N+_ where _N(qN) is the solution to (12), (13) corresponding to qN and
u(q*) is the solutlon to (I0), (II) corresponding to q*. This in turn
yields a convergence result analogous to the one stated above for the first
scheme. A more detailed discussion of these results appears in [I].
We demonstrate the feasibility of our methods with a simple example. We
consider a beam of length 1 and use the two schemes described above to
estimate its stiffness ql, its linear mass density q2 and the magnitude of
a tip mass q3" We assumed that the system was initially at rest (#=@=0)
and then excited via the distributed lateral load f(t,x) = eXsln 2_t and the
point force g(t) = 2e-2t acting on the tip mass. Displacement observations
at positions xj = .5, .75, 1.0, j=1,2,3, at times ti = .5i, i=1,2,...,I0,
were generated using the "true" parameter values ql = 1.0, q2 = 3.0 and
q3 = 1.5, the first two natural modes of the system and a standard Galerkin
method. The approximating identification problems (IDN I) and (IDN2) were
solved using an iterative Levenberg-Marquardt scheme with the approximating
state equations solved at each iteration using a variable order Adams i
Predictor Corrector method. The "start-up" values for the unknown parameters
-8-
0 .7, q_ = 2.7 and q_ = 1.7. Our results for methods Iwere taken to be ql =
and 2 are given in Tables 1 and 2 respectively below.
Table 1
--N ---N --N
N ql q2 q3 CPU (m:s)
2 1.00083 2.99817 1.50090 0:08.70
3 1.00072 3.00317 1.49868 0:29.38
4 1.00061 2.99227 1.50141 1:01.19
5 1.00009 2.98711 1.50195 1:35.26
6 1.00061 2.99039 1.50144 2:45.20
Table 2
--N --N ---N
N ql q2 q3 CPU (m:s)
2 1.00057 3.04455 1.48957 0:09.19
3 1.00067 3.01256 1.49707 0:22.10
4 1.00027 3.00922 1.49721 0:57.57
5 1.00016 2.98936 1.50262 1:22.52
6 .99912 2.99720 1.49952 2:52.76
-9-
REFERENCES
[I] Banks, H. T. and I. G. Rosen, "A Galerkin Method for the Estimation of
Parameters in Hybrid Systems Governing the Vibration of Flexible Beams
with Tip Bodies," CSDL Report No. R1724, C. S. Draper Laboratory,
Cambridge, MA, June 1984.
[2] Clough, R. W. and J. Penzlen, Dynamics of Structures, McGraw Hill, New
York, 1975.
[3] Dupont, T., "L2-Estlmates for Galerkln Methods for Second-order
Hyperbolic Equations," SlAM J. Numer. Anal., Vol. I0, 1967, pp. 880-889.
[4] Lions, J. L., Optimal Control of Systems Governed by Partial Differential
Equations, Springer-Verlag, New York, 1971.
[5] Pazy, A., Semlgroups of Linear Operators and Applications to Partial
Differential Equations, Springer-Verlag, New York, 1971.
[6] Rosen, I. G., "A Numerical Scheme for the Identification of Hybrid
Systems Describing the Vibration of Flexible Beams with Tip Bodies,"
ICASE Report No. 84-23, NASA CR-172388, NASA Langley Research Center,
Hampton, VA, June 1984.
[7] Schultz, M. H., Spllne Analysis, Prentice-Hall, Englewood Cliffs, NJ,
1973.
[8] Yoslda, K., Functional Analysis, Sprlnger-Verlag, New York, 1966.
1. Report No. NASA CR-172468 2. GovernmentAccessionNo. 3. Recipient'sCa,Jlog No.
ICASE Report No. 84-51
5. Report Date
4. _n_lO N METHODS FOR INVERSE PROBLEMS INVOLVING THE September 1984
VIBRATION OF BEAMS WITH TIP BODIES 6. PerformingOrganizationCode
7. Author(s) 8. PerformingOrganizationReportNo.
I. G. Rosen 84-51
., 10. Work Unit No.
9. P.erforminaOroanizationlgameandAddress
instiCut_ _or Computer Applications in Science
and Engineering 11. Contract or Grant No.
Mail Stop 132C, NASA Langley Research Center NASI-17070
Hampton, VA 23665 13. Typeof ReportandPeriodCovered
12. Sponsoring Agency Nameand Address Contractor Report
National Aeronautics and Space Administration 14. SponsoringAgencyCode
Washington, D.C. 20546
505-31-83-01
15. Supplementary Notes
Langley Technical Monitor: J. C. South, Jr.
Final Report
16. Abstract
We outline two cubic spline based approximation schemes for the estimation of
structural parameters associated with the transverse vibration of flexible beams with
tip appendages. The identification problem is formulated as a least squares fit to
data subject to the system dynamics which are given by a hybrid system of coupled
ordinary and partial differential equations. The first approximation scheme is based
upon an abstract semigroup formulation of the state equation while a weak/variational
form is the basis for the second. Cubic spline based subspaces together with a
Rayleigh-Ritz-Galerkin approach was used to construct sequences of easily solved
finite dimensional approximating identification problems. Convergence results are
briefly discussed and a numerical example demonstrating the feasibility of the
schemes and exhibiting their relative performance for purposes of comparison is
provided.
i
17. Key Words (Suggested by Author(s)) 18. Distribution Statement
distributor parameter systems, 18 - Spacecraft Design, Testing & Perf.
identification approximation, 39 - Structural Mechanics
variational methods 64 - Numerical Analysis
66 - Systems Analysis
Unclassified - Unlimited
19, SecuriwQa_if.(ofthisreport) 20. SecurityCla_if.(ofthis_) 21, No. of Pa_s 22. Dice
Unclassified Unclassified Ii A02
ForsalebytheNationalTechnicalInformationService,Springfield,Vir£inia22161 NASA-Langley, 1984




Approximation methods for inverse problems involving the vibration of beams with tip bodies

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