-NASA Contractor Report 178259 

r 

ICASE REPORT NO. 87-10 I 

I 

I. 

ICASE 

INVERSE PROBLEMS IN THE MODELING OF 

VIBRATIONS OF FLEX1 BLE BEAMS 

H. T. Banks 

R. K. Powers 

I. G. Rosen 

(LBSA-CR-178259) INVEESE EbCELEtS IN THE 1483-20717 

HCDELING OF VIEHAIICNS OF ElEXJEIE BEALJS Final Resort (NASA) 24 p CSCL 12A I Unclas 

~3164 45172 

INSTITUTE FOR COMPUTER APPLICATIONS IN SCIENCE AND ENGINEERING 

NASA Langley Research Center, Hamptoa, Virginia 23665 

Operated by the Universities Space Research Association It, 

NationalAermuticsand 

Space --- Administration 

Hampton,Wginia 23665 
i 

INVERSE PROBLEMS IN THE MODELING OF 

VIBRATIONS OF FLEXIBLE BEAMS+ 

* 

H. T. Banks 

Center for Control Sciences 

Division of Applied Mathematics 

Brown University 

Providence, RI 0291 2 

R. K. Powers 

Department of Mathematics and Statistics 

University of Arkansas 

Fayetteville, AK 72701 

** 

I. G. Rosen 

Department of Mathematics 

University of Southern California 

Los Angeles, 'CA 90089 

ABSTRACT 

The formulation and solution of inverse problems for the estimation of 

parameters which describe damping and other dynamic properties in distributed 

models for the vibration of flexible structures is considered. Motivated by a 

slewing beam experiment, the identification of a nonlinear velocity dependent 

term which models air drag damping in the Euler-Bernoulll equation is investigated. Galerkin techniques are used to generate finite dimensional approximations. Convergence estimates and numerical results are given. The modeling 

of, and related inverse problems for the dynamics of a high pressure hose line 

feeding a gas thruster actuator at the tip of a cantilevered beam are then 

considered. Approximation and convergence are discussed and numerical results 

involving experimental data are presented. 

'Invited lecture, Conference on Control Theory for Distributed Parameter 

Systems and Applications, Vorau, Austria, July 7-11, 1986. 

Part of this research was supported under the National Aeronautics and Space 

Administration under NASA Contracts No. NASl-17070 and NAS1-18107 while the 

authors were visiting scientists at the Institute for Computer Applications in 

Science and Engineering (EASE), NASA Langley Research Center, Hampton, VA. 

* 

This research was supported in part by the National Science Foundation under 

NSF Grant MCS-8504316, the Air Force Office of Scientific Research under 

Contract AFOSR-84-0398, and the National Aeronautics and Space Administration 

under NASA Grant NAG-1-517. 

** This research was supported in part by the Air Force Office of Scientific 

Research under Contract AFOSR-84-0393. 

i 
The purpose of this note is to illustrate and explain some of the ideas underlying the use of 

parameter estimation techinques in investigating damping and other dynamic phenomena in several 

classes of distributed models for flexible structures. We do this in the context of two specific 

examples drawn from experimental structures. In the first example we present techniques and results 

that can be used to study nonlinear aspects of viscous damping (nonlinear air drag) in slewing 

maneuvers with flexible beam like structures. We shall describe some fundamental questions, 

present a model for which an estimation problem is of importance, and then show how this inverse 

problem can be approximated for computational purposes. We do this in a weak or variational setting 

and give convergence arguments to provide a theoretical foundation for the numerical schemes we 

have used. These convergence results are then followed by presentation of a numerical test example. 

Our methods have proved useful in current studies with experimental data, the results for which will 

be presented in detail elsewhere. 

In a second example, we outline techniques that have been useful in developing accurate models 

for the dynamic effects of a flexible gas hose/tip mass/thruster apparatus when it is attached to a 

flexible beam to provide an active control system (the so-called "RPL experiment"). Since detailed 

mathematical arguments for this project are given elsewhere [BGRW], we shall only outline the 

model, the approximation ideas, and present a summary of numerical results obtained when using our 

general approach with experimental data. 

2, Nonlinear DamDing in Slewing Maneuvers 

In a series of papers [a, [JHJ, [JHR], Juang and his co-workers describe experiments carried 

out to demonstrate the feasibility of actively controlling (stabilizing) vibrations of a beam during 

slewing maneuvers. Experiments were conducted with a 1 meter steel beam and with a 3.9 meter 

aluminum honeycomb solar panel cantilevered in a vertical plane and rotated in the horizontal plane. 

Each was attached to a torquing motor at the hub of rotation or "root" of the beam. In typical 

experiments, the beams were slewed 30" to 45' in 1.5 to 4.5 seconds. Strain gauges were located at 

the root and at .22 and .5 of the length 8 of the beam. An angle potentiometer (at the root) measured 

angular displacements during the slew. These measurements (strain, yxx, and angular displacement, 

e) were used as feedback to the motor which was then used to suppress the vibrations via an LQR 

theoretical formulation for the feedback control laws. 

A series of slewing maneuver experiments were carried out in a laboratory at NASA Langley 

Research Center with possible effects due to air damping present. These were then followed by 

repetition of the experiments in a vacuum chamber. Experiment and theory were in good agreement 

in the vacuum chamber experiments, but there were significant discrepancies between the theoretical 

model based simulations and the experimental data obtained in the laboratory setting. It is important 

1 
to understand the variation in responses of controlled flexible structures in such differing 

environments since many future large spacecraft studies will, by necessity, involve model 

extrapolation and design based on laboratory performance only without the benefit of vacuum 

chamber comparisons. 

It has been suggested that the inaccuracies in the Juang, et. al. simulations most likely resulted 

from two types of model error: (1) the absense of a viscous damping term to represent the nonlinear 

air damping and (2) dissipation at the fixed end of the cantilevered beams was not included in the 

model even though there was some loose "play" in the clamped or "built-in" end of the beams. This 

latter mechanism for energy absorption presumably should be modeled by some type of nonlinear 

boundary conditim in place of the usual no displacement, no slope boundary conditions: y = yx = 0. 

Figure 2.1 

Here we focus on a nonlinear air damping component of the model as suggested by Juang, et. 

al., and show that one might effectively compute this using the experimental data in a least squares 

setting. We use a formulation proposed by Juang and his co-workers; this model can be derived by 

combining first principle energy considerations with laboratory findings on nonlinear drag forces. If 

e(t) and y(t,x) denote respectively angular displacement (from some reference angle) at time t and 

bending deflection along the beam at time t and position x as depicted in Figure 2.1 above, the model 

including actuator dynamics has the form (see [JHR] for a more detailed discussion) 

de ay d4Y + c3f(x- + $ + CJX- + 

d28 + pxa2Y 

a? d? dt ax4 (2.1) p- dt x) + E1 - = 0 de ay 

d20 4 a2y e de dy e de 

(2.2) 1, - + I, PX 7 dx + Joc3f (x- dt + xdx + jo c4{xx + $1 xdx = T(t) 

d? at 

de d'8 T(t) = koe,(t) - k dt - k, - 

d? 

(2.3) 

JY a2Y a3Y y(t,O) = -(t,O) = 0, 

ax ax2 ax3 (2.4) - (t,-e) = - (tJ) = 0. 

2 
r 

Here e, represents the applied armature voltage (this term involves strain feedback in the control 

problem), the terms involving c3 and c4 represent viscous damping and the coefficients p and E1 are 

the usual beam material parameters, linear mass density and flexural stiffness, and IB is the moment 

ji px2dx about the axis of rotation. The damping function f is assumed to have the form f(v) = vlvl 

for v in some bounded region. Of course, one can assume without loss of generality based upon 

physical considerations that f becomes bounded as v becomes large. 

A number of state space formulations of this model can be investigated in the context of 

identification and control problems including the following: 

de 

dt (i) 

(ii) The time history for 8 is assumed known, but 6 is treated as a state (if 8(t) is known 

from experimental data, it does follow that 0(t) can be effectively obtained by 

differentiation of the data), 

(iii) Both 8 and 6 are measured reliably so that each can be treated as a known quantity. 

Both 8 and 6 = - are treated as states, 

For our discussions here of estimation of the damping coefficients, we assume that (iii) holds so 

that one can combine (2.1) and (2.2) to also eliminate 0 as an unknown in the model. We do this 

before formulating a least squares problem for estimation of the damping. We also include a 

Kelvin-Voigt material damping term in the beam dynamics equation. 

We consider then the coupled system (actuator dynamics plus transverse vibrations of the beam) 

forO<x<&,t>O 

(2.5) 4 4 

py, + pi6 + c3f(x0 + yt> + c4{x8 + y,} + E1 D y + cDI D y, = 0, 

with appropriate boundary conditions for t 2 0 

and appropriate initial conditions. Here and below we shall use the notation D = a/ax. Assuming 

that 8(t) and 6(t) are known for 0 I t 5 T, where T is the duration of the experiment, we may solve 

for 8 in equation (2.6), subsitute this into equation (2.5), and obtain a single nonlinear partial 

differential equation for the bending deflection y. Upon doing this we obtain 

.. 

where 

3 
(2.10) 

(2.1 1) 

(2.12) 

and 

(2.13) 

PX F,(t,x,y,) - ----I" f(s6 + yt> s ds, 

I,+%! 

The boundary conditions (2.7), (2.8) involving no displacement, no slope at x = 0 and no moment, 

no shear at x = 8, can be equivalently written as 

Using this system, we may formulate an estimation problem for the parameter q= (ql,q2,q3,~) 

= (EI, cDI, c3,c4), given observations ei, i = 1, ..., M, for the root strain D2y(ti,0). The problem 

becomes one of choosing from some admissible set of parameters Q c R4 (we treat here only the 

constant parameter case - variable coefficients can be readily treated with appropriate modifications of 

the arguments - see [BCl], [BC2], [BCR], [BRl] for details) a parameter q* which minimizes over 

Q the least squares criterion 

M 

i= 1 

J(q) = I D2y(ti,0;q) - ei 

where y(., ; q) is the solution of (2.9), (2.14) for a given set of initial conditions y(0,x) = @(x), 

y,(O,x) = Y(x). Least squares problems of this type possess a number of interesting aspects (infinite 

dimensional states, theoretical and computational ill-posedness, etc.) which have been discussed by 

us and others in a number of previous publications. Here we focus on one particular question: state 

approximation techniques and the convergence arguments for approximating parameter estimates 

(these convergence arguments are also an important part of theoretical results which guarantee a type 

of inverse method stability - i.e. continuous dependence of estimates on the observations - see [B]). 

Before turning to these convergence arguments, we rewrite the system in a variational form 

(conservative form) with states v = y,, w = D2y. 

We first rewrite (2.9) as 

(2.15) PV, + av,) + q3 F,(t,x,v) + Q F22(t,x,v) + g 

+q3f(xb + v) + ~(x6 +VI + q, D2w + q2 D4v = 0 

(2.16) W, = D% 

(2.17) v(0) = Y, w(0) = D2@, 

4 
I 

where we seek solutions v E H4 n Hi ,w E Hi with 

HE 6 (9~ H2(0,4?) I~p(0) = Dq(0) = 0) 

Hi E (TE H2(0,4?)I~(4?) = Dq(4?) = 0). 

Then an equivalent weak or variational form is: Find (v,w) E Hi x Ho satisfying the initial condtions 

(2.17) and 

We next make the following observations: for each of the terms involving 44 in equation (2.18), 

there is an analogue involving q3, the difference being that the q3 terms are nonlinear, the sl, terms are 

linear. Since our primary emphasis here is in presenting convergence arguments for estimation of the 

nonlinear damping coefficients and since these arguments readily extend to the case involving the 

linear Q terms, we, for the sake of exposition, shall assume throughout the remainder of our 

discussions that % = 0. Thus the system of interest to us is 

We approximate the system via the usual Galerkin procedures by choosing finite dimensional 

subspaces Hfj c HE and HN c 

specified below. The approximate system for vN E HE, wN E HN is given by 

satisfying certain approximation properties (as N + =) to be 

(2.23) < pv:+ Gv: ) + q3F1(t,-,vN) +q3f(xi) + VN> + g(t,.>, CpN > 

+ < qlwN +q2DhN, D2gN) = 0 for @E HE, 

(2.24) < wr- DhN, vN > = 0 for wN E HN, 

(2.25) VN(0) = PEV(O), wN(0) = PNw(0) 

where PE, PN are the orthogonal projections (in the Ho norm) of H(' onto Hfj, HN, respectively. We 

then may define a sequence of approximating parameter estimation problems consisting of minimizing 

over Q the criterion 

(2.26) MN 2 JN(q) = XI w (ti,O;q)-EiI 

i= 1 

where wN is the solution of (2.23) - (2.25) corresponding to q. 

5 
The procedure for arguing convergence of the approximate parameters GN (a minimizer of JN in 

(2.26)) is now well-documented in a number of our previous papers (see [B] for a summary). A 

crucial step entails the following: If [ qN} is an arbitrary sequence in Q with qN + q in Q, one must 

establish the convergence zN + 0, uN + 0 where zN(t) = vN - PE v, uN(t) = wN - Pw with vN, wN the 

solutions of (2.23)-(2.25) corresponding to qN and v,w the solutions of (2.20) - (2.22) 

corresponding to q. We outline the arguments to establish this convergence. 

N 

I 

Using 4 (2.20) - (2.22) and (2.23) - (2.25) with q = qN, we find for all cp E HE 

Thus we find (suppressing some of the obvious function arguments) 

for all cp E HF. Also, for yf E HN we find 

<uf!yf> = <wfJ-PNwt,yf> = <wy-wt+(I-PN)Wt,yf> 

(2.28) = < D%~ - D~V, w > + <(I - P~)w,, w > 

= < D2zN, yf > + < D2( PE - I)v, yf > + < (I - PN)w,, yf >. 

To obtain convergence estimates we make particular choices for cp and yf in (2.27) and (2.28). Let 

cp = zN E HE in (2.27) and yf = uN E HN in (2.28) and define = (I - PN)w,. 

Then from (2.28) we obtain, using the inequality ab I - a + Eb where E > 0 can be arbitrarily 

= D2( PE - I)v, 

12 2 

4E 

chosen, 

6 
r 

- 1 d ~2 2N 1 N2 N2 1 N2 N2 1 N2 -iU I SEID~ I+-I~ I +Elel I +-IU I +Eie2i +-lu I 

4E 4E 4E 2 dt 

(2.29) 

=&ID 2N z I+-IU 3 N2 I +Elel N2 I +&le2 N2 I. 

4E 

Before deriving an analogous estimate from (2.27), we require some hypotheses on the nonlinear 

damping function f which will in turn yield estimates on .L and Fl. Recall that we can expect f(v) to 

behave like vlvl= (sgn v) v2 for v in some neighborhood (not necessarily small) of the origin, while it 

becomes bounded for v large. Thus, for our theoretical considerations here we make the reasonable 

assumptions that f satisfies a Lipschitz condition 

as well as satisfying a boundedness condition 

for el, c2 real 

(2.3 1) I f(6) I I M. 

From (2.30) we readily obtain the estimates (we shall need these to use in estimates from (2.27) 

with cp = zN) 

< f(x6 +v) - f(x6 + PLv), z > I K I (I- PL) N v(t,x) I I z N (t,x) I dx NN 

(2.32) 

N N K N 2KN2 I KI(I-PL)v(t)Ilz (t)I 5- l(I-PLv(t)l +- IZ (t)I, 2 2 

where we have used 1.1 to denote either the absolute value or the H"(0,e) norm (the interpretation 

being clear from the usage). Similarly, we find 

(2.33) < f(x6 + PFv) - f(x0 + vN), zN> I K I vN - PFv I I zN I = K I zN(t) 12. 

Furthemore, using the definition (2.1 1) of Fl, we easily find 

NN N N < F,(t,*,v> - F1(t,*,P,v), z > I K I (I - PL)v(t,s) I s d s I z (t,x) I 

I (I - P;)v 12 + I zN(t) 121, (2.34) 

where the constant p depends in an obvious way only on 8 ,K,p, IB, k2. With similar calculations 

we find 

(2.35) < F1(t;, Pi v ) - F1(t, VN), ZN > 5 2p I zN(t) 12. 

Using the definition (2.10) of L we next find 

7 
(2.36) 

Also 

- -p2 1 d N 

-- - - [ J: xz (t,x) dx] I, + k2 2 dt 

- -p2 1 d N2 

-- - - < x,z (t)>. I,+k2 2 dt 

c3 N N 

< q (I - P;) Vt), zN > I D - I (I - PL) Vt I I z I I, + k2 3 

(2.37) 

I fi [ I (I - P;) Vt 12 + I zN(t) I 'I 

where the constant depends only on p, I,, b, e. 

Finally, we can now obtain a desired estimate from (2.27) with cp = zN. Using (2.32) - (2.37) 

we find 

+F[ I(I-P;)vt12+IzNIz] + lq3-q3 NI G lzNl 

N2 

N N2 [ l(I-pL)vl N2 +lZ I + Iq, I K Iz I 

"1 

'1 

(2.38) +Iqylp[ I(I-pL)VI N2 +lZ I +lq3 12p lz I +lq3-qy1Mfi lzNl 

+lq3 17 

+Iq~Il~~lID~z~l-<q~D N 2N z ,D 2N z > + Iet;'IID 2N z I 

where €ly = (qlw - qyPNw + q2D% - qFD2PF v). 

Noting that x, zN > I dx I zN I, we find that 

3 

N2 lz I -P N2 -P2 c N2 -P 

2 

(2.39) - <X,Z > 2 - - I z I = 3 

1 + 3l54p.e 'B +% 'B+% 

'2 3 since I, = px dx = p e /3. We assume that Q is a compact set and that the admissible parameter 

I set Q entails the constraints q2 2 v for some positive constant v. We then have 

I 8 
v I D2zN l2 I < q2D2zN, D2zN >. 

Using these estimates in (2.38) we obtain 

2 

TTt 

2N2 

+[ 

+ '[ G2+M28] Iq,-q, ~2 I + -lu k2 ~2 I +EIDz I 

4E 2 

2N2 1 N2 2N2 -VID~ I +- le, I +EID~ I 

4E 

or, assuming that I q3 - qyl + 0, 

[ PIZNt - <x,z > 5 ~lz~? + ?(t)+(2~-v)ID 2N2 z I 

'B + k2 'I (2.40) Z 

k2 N 2 

4E 

+-lu I 

where @ (t) is bounded and + 0 as N + - under the usual assumptions (see [B], [BCR],[BRl]) on 

the approximation properties of HL and HN (Le. that PE + I, PN + I in the desired topologies). FOI 

example, approximations based upon cubic splines in Ht and linear splines in H! would suffice. 

Finally combining (2.29) and (2.40) we obtain 

N2 N2 N + yllz I + y21u I +G (t) 

where GN is bounded with GN(t) + 0. Integrating this inequality and using the fact that zN(0) = 0, 

uN(0) = 0, we find 

Finally, using (2.39) and choosing E such that v - 3~ = 6 > 0 in this last inequality, we obtain 

9 
1 

2 

- 

(2.43) 

+ 

+ I zN(t) 12 + I uN(t) 12 ,[ '- 1 + 3k2/p8 ] 1 

where A(N) + 0. Thus, by the usual Gronwall arguments we have I zN(t) l2 + 0, I uN(t) l2 + 0, 

and j', I D2zN(s) l2 ds + 0, where zN(t) = vN(t) - PEv(t) and uN(t) = wN(t) - PNw(t). Under 

appropriate convergence properties for PE and p, we may then use the triangle inequality to obtain 

N N I v (t) - v(t) I + 0, I w (t) - w(t) I + 0, I D2vN(s) - D2v(s) ? ds + 0. 

We note that the above arguments can be used to establish convergence at each t of the strain in 

the Ho(0,8) norm. If we use the root strain D2y(t,0) in the least squares criterion, a stronger strain 

convergence (pointwise in the spatial coordinate) is needed to complete the theoretical development. 

Arguments in the spirit of those given above can be made to give strain convergence in the H1 norm 

which, of course, yields root strain convergence. Arguments of this nature have been given for 

similar problems elsewhere [BCK], [BR2]; they involve some technical detail and we shall not 

pursue the development here. Instead we turn to a brief discussion of some computational aspects of 

these schemes. 

The approximation and estimation schemes discussed above can be readily used to develop 

computational algorithms for estimation of damping coefficients (including those for the nonlinear 

viscous damping terms). We have developed and tested numerically some software packages based 

on these ideas. We are currently using the packages with experimental data provided by J. Juang; 

these results will be reported elsewhere. We close this section with a brief summary of findings for 

one of the numerical test examples we have investigated. 

A test example was considered for the system (2.9) with 8 = p = E1 = 1.0 assumed known and 

c4 = c, = 0. We sought to estimate the damping coefficient c3 from simulated data for the strain. 

That is, we chose a particular function y(t,x) = .5t2(x4 - 4x3 + 6x2) as the true solution of the system 

equation (2.9) with an appropriate forcing function added to the equation. The true parameter value 

c; = 2.0 was used in (2.9) along with choices of e(t) and e,(t) which were qualitatively similar to the 

corresponding experimental time histories available to us. For example, we used 

and c, on chosen so that 8 has a maximum amplitude of 4.5 at t = 1.75. The values k, = 52.136, kl 

10 
= 1672.96 and IB + k, = 32.95 were used in (2.9) along with &t) obtained by differentiating 0. 

For the approximating system we used modified cubic splines for both HE and HN; the basis for 

HE was the usual cubic B-spline basis modified to satisfy the boundary conditions of Hi while the 

usual B-splines modified to satisfy the boundary conditions of Hi were used to generate HN. In each 

case the subspaces HE, HN had dimension N + 1. In the fit criterion we used the root strain y,,(t,,O) 

at 29 equally spaced observations in the time interval 0 S t I 7. For the initial guess cs = 2.5, a 

residual of 1789 is obtained. For N = 4, the scheme produced a converged estimate c := 1.996 with 

a corresponding residual of .0088. A number of other test examples were studied with equally 

satisfactory findings. 

2L Pose Ef fects o n the Dv namics of t he RPJ, St ructure 

The RPL structure is an experimental apparatus which was designed and constructed at the 

Charles Stark Draper Laboratory in Cambridge, Massachusetts with funding supplied by the United 

States Air Force Rocket Propulsion Laboratory (RPL). Its primary function is to serve as a test bed 

for the purpose of investigating control algorithms and instrumentation (sensors, actuators, 

processors, etc.) for the large angle slewing of spacecraft with flexible appendages. It was designed 

to specifically incorporate those features which make control design for large flexible spacecraft an 

especially difficult and challenging problem. In particular, this includes light damping, high 

flexibility, a large number of, and closely spaced natural modes of vibration, difficult to model and 

coupled structural and actuator dynamics and dissipation mechanisms, etc. 

Figure 3.1 

11 
The structure itself consists of four aluminum beams, each of length = 4 feet, width b=6 

inches and thickness h = .125 inches, cantilevered in a symmetrical fashion to a central hub which is 

mounted on an air bearing table. The air bearing table allows for the near frictionless rotation of the 

entire structure about the vertical axis. Control actuation is achieved via nitrogen cold gas thrusters 

mounted at the tips of two opposing appendages. The other two beams are passive with masses at 

their tips serving only to maintain the over all symmetry of the structure. Nitrogen gas is supplied to 

the thrusters from storage tanks mounted to the central hub through stainless steel, wire mesh 

wrapped, flexible, high pressure hoses. Electro-mechanical valves control the expulsion of the gas 

from the thruster nozzles. Each appendage is instrumented with a linear accelerometer at the tip. Data 

from the sensors is recorded and control input signals are generated using a MINC 11/23 

microcomputer. 

Effective control design depends heavily upon the availability of a high fidelity model for the 

plant. In the case of the RPL structure, it is immediately clear that a model involving partial 

differential equations would be of some use. For the transverse vibration of the passive beams, a 

distributed parameter model based upon the Euler-Bernoulli equation together with appropriate 

boundary conditions describing the coupled motion of the tip mass and the rigid body rotation of the 

central hub would be adequate. For the active appendages (i.e., those with the tip thrusters) on the 

other hand, a more sophisticated model which also captures the coupled dynamic effects (i.e. 

additional mass, stiffness and dissipation, torsional motion, etc.) due to the motion of the flexible 

thruster hoses is needed. 

Since the transverse vibration of each of the individual appendages is decoupled, for our 

investigation here, we consider the problem of modeling the hose effects on the transverse vibration 

of a single cantilevered (i.e. clamped - free) beam (see Figure 3.2). 

i 

Figure 3.2 

We describe a model which was suggested by S. Gates of the Contwl and Flight Dynamics division 

of the Draper Laboratory wherein the hose is treated as a damped linear harmonic oscillator that is 

rigidly attached to the thruster assembly at the free end of the beam. More precisely, the hose is 

modeled as a proof mass which reacts against the tip or thruster mass via an elastic spring and a 

linear, viscous damper (see Figure 3.3) 

12 
I 

and 

I 3 L I 

Figure 3.3 

Letting u(t,x) denote the vertical displacement of the beam at time t and position x, 0 I x I 8, 

aU 

ax and assuming only small deformations ( Le. lu(t,x) I < < 8, I - (t,x) I << 1) 

we use the Euler-Bernoulli equation together with Kelvin-Voigt viscoelastic material damping to 

describe its transverse vibration . That is 

a2u a4 aU a4u 

at ax4 at ax4 (3.1) Py(t,x) + cD1 - -(t,x) + EI-(t,x) = 0 O<X< 8, t>O 

where p is the linear mass density, cD and E are respectively the material coefficient of viscosity and 

modulus of elasticity and I = bh3/12 is the second moment or moment of inertia of the uniform, 

rectangular cross section of the beam. 

At the free end of the beam, corresponding to x = 8, the motion of the tip mass mT and the 

hose mass m, are given by 

d2Y dy au 

d? 

(3.3) m, - (t) + c (- (t) - - (t,8)) -t k, (y(t) - u(t, 8)) = 0, H dt at 

t>O 

t >o 

respectively where y(t) is the vertical displacement of the hose mass at time t measured from the 

equilibrium position, f(t) is the thruster force at time t and C, and k, are respectively the hose 

damping and stiffness coefficients. Assuming that the rotational inertia due to the hose - thruster 

13 
assembly is negligible, we also have the zero moment condition 

at 

ax2 s(t,e) + E1 -(t, 8) = 0, t > 0 (3.4) CD' ax2 at 

at the free end. At the clamped end, we have the usual geometric boundary conditions of zero 

displacement, 

(3.5) u(t,O) =o, t>O 

and zero slope, 

aU 

ax (3.6) -(t,O) = 0, t > 0. 

Assuming that the system is initially at rest, we have the temporal boundary conditions or initial 

conditions given by 

aU 

(3.7) u(0,x) = 0, ?J+O,X) = 0, 0 2 x I e 

and 

dY -(O) = 0. dt (3.8) y(0) = 0, 

Our primary concern here is the inverse problem which is naturally associated with the 

mathematical model given by equations (3.1) - (3.8) above. The physical dimensions and mass 

properties of the beam and thruster assembly and the elastic properties of the material from which the 

beam is made are known. Also, the thruster output can be experimentally calibrated. Consequently 

the parameters e, p, E, mT and b and h (and therefore I) are known. The input function f is given as 

well. However, the coefficient of viscosity cD and the hose parameters mH, cH and kH must be 

determined via an identification procedure. Recalling that the structure is instrumented with a linear 

accelerometer at the tip of the beam, we formulate the following inverse or parameter identification 

problem. 

Given a known input f(t) and corresponding measured output z(t) for t E [tO,tl], determine the 

parameters q = ( cD, mH, cH, k,)T in a closed and bounded subset Q of R4, which minimize the least 

squares performance index 

- -- -- 

14 
a2u 

6 at 

(3.9) J(q) = I I 7 (t,e ; q) - z(t) 12 dt 

where u(-, - ; q) denotes the solution to the initial boundary value problem (3.1) - (3.8) 

corresponding to the choice of parameters q = (CD,mH, cH, kH) E Q. 

Implicit in the statement of the problem above is the well posedness of the initial boundary value 

problem, Le. the existence, uniqueness and regularity of solutions to the system (3.1) - (3.8). The 

infinite dimensionality of the distributed state constraints necessitates the development and use of 

some form of finite dimensional approximation. Both of these issues are most efficiently addressed 

via an abstract, functional analytic formulation of the system (3.1) - (3.8). 

Define the Hilbert space H = R2 x L,(O,&) endowed with the usual inner product and let V = 

{((,q,cp) E H: cp E H2(0,8), cp(0) = Dcp(0) = 0, q = cp( e)). Define the coercive bilinear forms c 

and k from V x V into R by c( ?,$) = cH(C -cp( 

kH(C-(P(e))(h-w(e))+EI<D2cp,D2W>for ? =(C, cp(e),cp)E Vand$=(h,v(e),yl)E V, 

the operator M E L(H,H) by M((,q,cp) = (m&, qq, pcp) and set F(t) = (O,f(t), 0 ) E H. Then the 

initial boundary value problem (3.1) - (3.8) can be rewritten in weak form as 

-w( e)) + cDI <D2q, D2ur> and k( &$) = 

(3.10) < M;;t,(t), 3 >H + c( ^u,(t), $ ) + k( GO), 3 ) = < F(t), 6 >H, t > 0,G E V 

h (3.11) G(0) = 0, uJ0) = 0, 

for $t> = (y(t), u(t,t), u(t, -1) E V. 

Depending upon the degree of smoothness imposed upon the input f as a function oft (i.e. b, 

Holder continuity, H', etc.), standard results from the theory of abstract parabolic equations (see m, 

[SI) can be used to demonstrate the existence and uniqueness of solutions to (3.10) - (3.11) with 

varying degrees of regularity. 

based Galerkin scheme. For each N = 1,2, . . . , let ( PN} ?' denote the usual cubic polynomial Bsplines defined on the interval [O,e] with respect to the uniform mesh (O,e/N, 2e/N, ..., 81 and 

which have been modified to satisfy p,"<O) = DPY(0) = O,j=1,2, ..., N+l. Set @= (l,O,O), Py= 

(0, pj (e), P? , j = 1,2,. . ., N + 1 and VN = span ( pj }j=o . The Galerkin equations corresponding to 

(3.10), (3.11) for P(t) E VN are given by 

We define finite dimensional approximations to the system (3.10), (3.1 1) using a cubic spline 

J J=1 

h 

"N N+l 

(3.13) P(0) = 0 *(O) = 0. 

15 
For = (CD, mH, CH, kH) E Q, let p(t; q) = (yN(t;q),uN(t,8;q), uN(t,.; 9)) denote the solution to the 

finite dimensional system (3.12), (3.13) corresponding to the choice of parameters q E Q and let 

h a2u 

at2 

u (t;q) be defined analogously with respect to the system (3.10), (3.1 1). If - (t,C;q) in the least 

a2UN 

at2 squares performance index J given by (3.9) is replaced with - (t,k; q) and the resulting finite 

dimensionally constrained estimation problem (henceforth referred to as the Nth approximating 

estimation problem) is solved, an approximation to the optimal parameters i is obtained. Indeed, it 

can be argued (using estimates in the spirit of those given in the previous section; see [BGRW] for 

the details) that iff is smooth and if (qN}i=l is a sequence in Q with qN + qo , 

AN N A 0 a2uN N a2u 0 then utt(- ; q )+ utt(- ; q ) in L,(O,T; H) (or equivalently, that - (.,C;q ) + (., C ; q ) at2 at 

in L2(0,T)) as N + 00. This in turn can then be used to show that if ;;" is a solution (which can be 

shown to exist) to the Nth approximating estimation problem, the sequence ( iN } i=l c Q admits a 

convergent subsequence (q }j=l with q + q as j + and is a solution to the original infiiite 

dimensionally constrained identification problem. 

-Nj co -Nj - 

m 

,I- -I 

> 

z 

Z' 

0 

I- 

< 

0 0.2 

H 

H 

5 0.0 -l 

W 

U 

U 

4 

0 2 Lf 

TIME IN SECONDS 

Figure 3.4 

16 
!. 

We tested our model and scheme using data produced by the following experiment carried out 

for us on the RPL structure by Dr. Michel A. Floyd, formerly of the Draper Laboratory. With the 

central hub held stationary and the structure initially at rest, a thruster line was pressurized to 300 psi 

and the corresponding jet fired for 50 milliseconds. Linear acceleration at the tip of the corresponding 

appendage was recorded over the period 0 to 5 seconds with a sampling interval of 5 milliseconds 

yielding 1000 measurements. With the accelerometer calibrated to 5 volts/g (g = 32 ft/sec2), the 

observations produced are plotted in Figure 3.4 above. 

From an FFT of the data we found the first three frequencies to be approximately .75 Hz, 7.5Hz 

and 14Hz. A visual inspection of the data immediately reveals the modes at .75Hz and 14Hz. The 

mode at 14Hz is a torsional mode, or, more pecisely, a twisting of the beam about its longitudinal 

axis. The excitation of this mode results from a combination of factors including the presence of the 

hose and the fact that it is connected to the thruster assembly at the top rather than the center of the 

beam and the opening and closing of the thruster valves. Since the torsional vibration is at a much 

higher frequency than are either of the first two transverse vibrational modes, and since it is relatively 

rapidly damped, in our study here we simply treated it as noise and left it unmodeled. A more 

detailed discussion of the torsional effects and its coupling into the accelerometer measurements of the 

transverse motion of the beam can be found in [BGRW]. 

The beam is made of a grade of aluminum having linear mass density p = .027 slug/ft and 

modulus of elasticity E = 15.84 x lo8 lb/(ft)2. We have 8 = 4ft and I = bh3/12 = 4.71 x 10-8(ft)4. 

The mass of the thruster assembly mT was determined to be .149 slug. A hose pressure of 300 psi 

was determined to be equivalent to a force of .2971b. We have therefore 

0.2971b 01 tl .05 

.05 < t 5 5.0. 

The index of approximation N was taken to be 4 throughout. A detailed study of the convergence 

properties of our scheme using test examples and simulation data was carried out with the results 

having been reported in BGRW]. 

We neglected the hose effects and internal damping (i.e., we took cD = mH = cH = k, = 0) and 

used the standard Euler-Bernoulli model for a cantilevered beam with tip mass to generate values for 

the linear acceleration at the tip. The resulting acceleration profile along with the experimentally 

observed data is plotted in Figure 3.5. If one is willing to accept the Euler-Bernoulli theory as a 

reasonable description for the transverse vibration of a long, slender, flexible beam, then it is clear 

from the figure that the hose effects are indeed significant and should be modeled. The residual was 

found to be 3.03. 

17 
x Data 

0 Fit 

u ’- 

84- 

B 

4 x 

X 

R I 

CD (lb SeC/ft2) 

127.40 

0 

I1 I I I I 

0.00 i .OG 2.60 3.00 4.00 

Time (Semnds) 

mH C, (lb.sec/ft) kH(lb/ft) 

.0801 .0078 .413 

Figure 3.5 

In applying our scheme we used the data over the interval 3.0 to 5.0 seconds (where the 

contribution from the torsional mode has been significantly damped) sampled at the rate of 10 

observations per second. By matching the first two modal frequencies of the data with the first two 

modal frequencies of the model we obtained crude estimates for mH and k, that could be used as 

start-up values for an iterative optimization procedure. Then, using our spline-based scheme along 

with an iterative Levenberg-Marquardt nonlinear least-squares routine to solve the approximating 

estimation problems to search over mH and k, optimal values for the hose mass and hose stiffness 

were obtained. Taking these values along with C, = 0 as start-up values, we used our scheme again 

to obtain optimal values for m,, C, and k,. Repeating this general procedure, we obtained the 

optimal values for the parameters cD, m,, C, and k, given in Table 3.1. The corresponding tip 

acceleration profile is plotted in Figure 3.6. The residual was computed to be .70 - a clear 

improvement. 

Table 3.1 

18 
0 4 x Data 

0 Fit 

N Ii 

+----l 1 -r-F-- 

$.GO i .oo 2.00 3.00 4.00 

Time (Seconds) 

Figure 3.6 

In summary, our results demonstrate the feasibility and effectiveness of using distributed 

parameter systems to model structural dynamics. Our finite dimensional approximation methods 

provide a viable means for estimating unknown parameters appearing in the models which can not be 

explicitly measured experimentally. Finally, our general approach appears to compare favorably (see 

[BGRW]) with other identification procedures for the estimation of parameters in models for 

structural dynamics which are commonly used in engineering practice. 

Acknowled- The authors would like to gratefully acknowledge Ms. Y. Wang of the 

Division of Applied Mathematics at Brown University for her assistance in carrying out the 

computations reported on in this note. In addition, they would also like to express their sincere 

appreciation to Dr. Jer-Nan Juang of the Structural Dynamics Branch of the NASA Langley Research 

Center for the numerous stimulating conversations and collaboration which motivated their efforts on 

the estimation of nonlinear damping in slewing beam maneuvers discussed in the first part of this 

paper. 

19 
Peference 

H.T. Banks, On a variational approach to some parameter estimation problems, 

Distributed Parameter Svsrems, Proceedings of the 2nd International Conference, Vorau, 

Austria, 1984, Volume 75 in Lecture Notes in Control and Information Sciences, 

Springer-Verlag, Berlin, 1985, 1 - 23. 

[BCl] H.T. Banks and J.M. Crowley, Estimation of material parameters in elastic systems, 

LCDS Rep No 84 - 20, Brown Univ., June 1984. 

[BC2] H.T. Banks and J.M. Crowley, Parameter identification in continuum models, ,IAstronautical Sci *& 33 (1985), 85 - 94. 

[BCK] H.T. Banks, J.M. Crowley and K.Kunisch, Cubic spline approximation techniques for 

parameter estimation in distributed systems, IEEE Trans. Automatic Cont rol AC-28 

(1983), 773-786. 

[BCR] H.T. Banks, J.M. Crowley, and I.G. Rosen, Methods for the identification of material 

parameters in distributed models for flexible structures, ICASE Rep No. 84-66, NASA 

Langley Research Center, Hampton, VA, Mat. Applicada e ComDut. 5 ( I986 1, to appear. 

[BGRW] H.T. Banks, S.S. Gates, I.G. Rosen and Y. Wang, The identification of a distributed 

parameter model for a flexible structure, ICASE Rep. No 86-72, NASA Langley 

Research Center, Hampton, VA; SIAM J. Cont rol and Om., submitted. 

[BRl] H.T. Banks and I.G. Rosen, Computational methods for the identification of spatially 

varying stiffness and damping in beams, ICASE Rep No. 86-70, NASA Langley 

Research Center, Hampton, VA; Control-Theoy and Advanced Technolom, to appear. 

[BR2] H.T. Banks and I.G. Rosen, Numerical schemes for the estimation of functional 

parameters in distributed models for mixing mechansims in lake and sea sediment cores, 

LCDS Rep No. 85-27, Brown University; Inverse Problems, to appear. 

A. Friedman, Partial Differential Eauations of Parabolic Tvue, Prentice Hall, Englewood 

Cliffs, New Jersey, 1964. 

[HJI L.G. Horta and J.N. Juang, Identifying approximate linear models for simple nonlinear 

systems, J. Gu idance, Cont rol and Dvnamics , to appear. 

20 
[JHI J.N. Juang and L.G. Horta, Effects of atmosphere on slewing control of a flexible 

structure, 27th Structures, Structural Dynamics, and Materials Conference, Paper No. 

86-1001- CP, May, 1986, San Antonio. 

[JHR] J.N. Juang, L.G. Horta, and H.H. Robertshaw, A slewing control experiment for 

flexible structures, J. Gu idance. Co ntrol and Dvnamics , to appear. 

[SI R.E. Showalter, Hilbert Svace Methods fo r Partial Differential Eauations, Pitman, 

London, 1977. 

21 
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H. T. Banks, R. K. Powers, and I. G. Rosen 

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Langley Technical Monitor: 

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Final Report 

Submitted to Proceedings of the 

Conference on Control Theory for 

Distributed Parameter Systems and 

Applications, Vorau, Austria 

16. Abstract 

The formulation and solution of inverse problems for the estimation of 

parameters which describe damping and other dynamic properties in distributed 

models for the vibration of flexible structures is considered. Motivated by a 

slewing beam experiment, the identification of a nonlinear velocity dependent 

term which models air drag damping in the Euler-Bernoulli equation is investigated. Galerkin techniques are used to generate finite dimensional approximations. Convergence estimates and numerical results are given. The modeling of, 

and related inverse problems for the dynamics of a high pressure hose line 

feeding a gas thruster actuator at the tip of a cantilevered beam are then 

considered. Approximation and convergence are discussed and numerical results 

involving experimental data are presented. 

inverse problems, approximation, 

elastic structures 

64 - Numerical Analysis 

66 - Systems Analysis 

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NASA-Langley, 1987