3rd Int. Conference on Inverse Problems in Engineering. Daniel Lesnic. University of Leeds. Leeds, West Yorkshire LS2 9JT

Transcription

1 Inverse Problems in Engineering: Theory and Practice 3rd Int. Conference on Inverse Problems in Engineering June 13-18, 1999, Port Ludlow, WA, USA APP RETRIEVING THE FLEXURAL RIGIDITY OF A BEAM FROM DEFLECTION MEASUREMENTS Daniel Lesnic Department of Applied Mathematics University of Leeds Leeds, West Yorkshire LS 9JT amt5ld@amsta.leeds.ac.uk ABSTRACT A rigorous investigation into the identication of the heterogeneous exural rigidity coecient from deection measurements recorded along a beam in the presence of a prescribed load is presented. Mathematically, the problem reduces to the need to solve the Euler-Bernoulli steady-state beam equation subject to appropriate boundary conditions. Conditions for the uniqueness and continuous dependence on the input data of the solution of the inverse problem for simply supported beams are established and, in particular, it is shown that the operator which maps an input deection into an output exural rigidity is Holder continuous. Since the inverse problem can be recast in the form of a Fredholm integral equation of the rst kind, numerical results obtained using various methods, such as Tikhonov's regularization, singular value decomposition and mollication are discussed. NOMENCLATURE E Modulus of elasticity I Moment of inertia L Length of the beam M Bending moment a Flexural rigidity f Transversely distributed load p Amount of noise u Transverse deection Inverse of the exural rigidity Regularization parameter Standard deviation INTRODUCTION In the Euler-Bernoulli beam theory, it is assumed that the plane cross-sections perpendicular to the axis of the beam remain plane and perpendicular to the axis after deformation, resulting in the transverse deection u of the beam being governed by the fourth-order ordinary dierential equation d a(x) d u dx dx (x) = f(x) <x<l (1) where L is the length of the beam, f is the transversely distributed load and the spacewise dependent conductivity a = EI (often called exural rigidity) is the product of the modulus of elasticity E and the moment of inertia I of the cross section of the beam about an axis through its centroid at right angles to the cross section. It should be noted that eqn.(1) can be split into a system of two secondorder ordinary dierential equations, namely, M(x) =a(x) d u (x) <x<l () dx d M = f(x) <x<l (3) dx where M is the bending moment. Using a simple nite element methodology, it can be seen that the primary variables 1 Copyright c 1999 by ASME

2 associated with eqn.(1) are the deection u and the slope du=dx, whilst the bending moment M = a(d u=dx ) and the shear force (d=dx)(ad u=dx ) are the secondary variables, any four values of which are usually specied on the boundary, namely, x = orx = L. The direct problem of the Euler-Bernoulli beam theory requires the determination of the deection u which satises eqn.(1) (or M and u satisfying eqns () and (3)), when a> and f are given and four boundary conditions (essential or natural) are prescribed. If at least one of these boundary conditions is on the deection then this direct problem is well-posed. However, there are other problems (termed inverse) that may be associated with the Euler-Bernoulli eqns (1)-(3). For simply supported beams an inverse load identication problem, which requires the determination of the load f(x) which satises eqn.(3) when a and M are given, has been investigated by Collins et al. (1994) in the practical context of recovering engineering loads from strain gauge data. This problem is ill-posed since it violates the stability of the solution, which relates to twice dierentiating numerically M, which is a noisy function. In this study, weinvestigate an inverse coecient identication problem which requires the identication of the positive, heterogeneous exural rigidity coecient a(x) ofa beam which satises eqn.(1), (or eqns () and (3)), when u and f are given. This problem is an extension to higherorder dierential equations of the inverse problem analysed by Marcellini (198) for the one-dimensional Poisson equation. Prior to this study, anumerical algorithm has been recently proposed by Ismailov and Muravey (1996) for supported plates, when f > and u. However, the theoretical investigation of the uniqueness of the solution is much more dicult since eqn.(1) does not, in general, have a unique solution, e.g. if a is a solution of eqn.(1) then for any linear function h, the function a 1 = a + h(d u=dx ) ;1 may still be a solution of eqn.(1). Therefore, a necesary condition for the uniqueness of the solution of eqn.(1) is that the set S = x [ L] j d u dx (x) = is not empty. However, at the other extreme, if S is a subset of [ L] of non-zero measure then a(x) is not identiable on S. Further, based on the physical argument that the properties of the beam, namely E and I, should be positive, and considering only applications in which there is always a positive nite load acting on the beam, we can assume that there are <f 1 f < 1 such thatf 1 f(x) f and (4) that there are < 1 < 1 and Q>such that a(x) belongs to the domain denition set A = fa L 1 ( L) j 1 a ka kqg (5) where k:k denotes the L ( L)-norm. For the present analysis we restrict ourselves to homogeneous boundary conditions to accompany the non-zero load f and the linear eqns (1)-(3). Also, since a> then the set S given by eqn.(4) is S = fx [ L] j M(x) =g. Thus boundary conditions which ensures that S 6= include u () = u (L) or M()M(L) =. If only Dirichlet and/or Neumann type boundary conditions P for u and M are considered then there 4 are in total k= Ck 4 C 4;k 4 = 7 of these possibilities. However, half of these possibilities are equivalent, based on the invariance of eqns (1)-(3) under the translation x 7! (L;x). The aim of this study is not to investigate all these possibilities but rather select realistic physical boundary conditions associated with beams that naturally occur in elasticity, such as beams supported at both ends, namely, u() = u(l) =M() = M(L) = (6) Other types of boundary conditions have been investigated elsewhere, see Lesnic et al. (1999). The plan of the paper is as follows. In section we prove that the operator u 7! a is Holder continuous and thus injective, i.e. the uniqueness and continuous dependence on the input data of the inverse problem. Further, when random noisy discrete input data is included, a regularization algorithm is developed in section 3 and the numerically obtained results are illustrated and discussed. MATHEMATICAL ANALYSIS We formally dene the operator U : A! L ( L) U(a) :=u a = ZZ M a M = ZZ f 8a A (7) as a formal general solution of eqns (1)-(3), where the integral sign R is understood in the sense that the constants of integration are to be determined by imposing the boundary conditions (6). For convenience, and in order to simplify the algebraic manipulations, a uniform load f 1 was assumed which in turn gives Copyright c 1999 by ASME

3 x(x ; L) M(x) = (8) Lemma 3. If a, b A then there is a positive constant C such that ka ; bk L Cku a ; u b k =3 L (14) u a (x) = Z x Z t dt ( ; L) d ; x a() L Lemma 1. The following inequality holds Z t dt ( ; L) d a() (9) ku a ; u bk 3= ku a ; u b k 1= ku a ; u b k (1) Proof: To evaluate the L ;norm ku a ; u b k we useintegration by parts, impose the boundary conditions (6) and apply the Holder inequality toobtain ku a;u bk = ; ku a ; u b k = ; (u a ; u b)(u a ; u b )dx ku a ; u bkku a ; u b k (11) (u a ;u b )(u a;u b )dx ku a ;u b kku a;u b k (1) Finally, combining the inequalities (11) and (1) yields the inequality (1). Lemma. If a A then there is a positive constant C such thatku a kc. Proof: If we denote = a ;1, = ;1 1 then wehave < < 1 and u a = M. Thus u a = M + M and using eqn.(8) we obtain the following estimates for ku a k, namely, ku a kk k L L = (13) 1 Proof: If we denote = a ;1, = b ;1, 1 = ;1 and = ;1,thenwehave < 1 1 < 1 and u a = M, u b = M. The key idea of the proof is based on removing (similar toacauchy nite part evaluation of singular integrals) the singularities of the function = u a=m (and similarly = u b =M) which, based on eqn.(8), is of the type x;1 (L ; x) ;1. Therefore we extend the functions u a, u b, and to the whole real axis by dening them to be zero outside of the interval [ L]. Then, for any >, using the Holder inequality, wehave the following identities and inequalities Z k ; k = j ; j dx (; )[(L; L+) Z + j ; j j u a ; u b j dx (;1 ;)[( L;)[(L+ 1) j M j Z (;1 ;)[( L;)[(L+ 1) ; 4 + ku a ; u b k! 1= 4 x (L ; x) dx =4 + 3= L ku a ; u L ; + j L ; j L ln j L + j b k 1= 4 ( + ku a ; u b k ;1= =L) (15) The right-hand side of expression (15), as a function of, attains its minimum at min = ku a ; u L b k =3 (16) and introducing this value into expression (15) we obtain Finally, since j j= ja j a j a j and thus k k ka k Q, from eqn.(13) it follows that there is a positive constant C>such that ku a kc, which concludes lemma. k ; k ( 4=3 + 1=3 )L ;=3 4=3 ku a ; u b k =3 (17) Finally, the estimate (14) is obtained using the inequality j a ; b j ; 1 j ; j and eqn.(17). 3 Copyright c 1999 by ASME

4 At this stage, using lemmas 1-3, we can conclude the following theorem: Theorem 1. (The Holder continuity of the inverse operator U ;1 (u) =a) If a, b A then there is a positive constant C such that ka ; bk Cku a ; u b k 1=9 (18) As an immediate consequence of this theorem we conclude the uniqueness and continuous dependence of the solution a A on the input data u. REGULARIZATION ALGORITHM In this section the solution of the inverse problem (1), which consists of nding a A when the additional data u a z (19) is given, is based on the rst-order regularization method of Tikhonov and Arsenin (1977), which requires the nding of the minimum with respect to = a ;1 A of the functional () =ku a ; zk + k k () where > is a regularization parameter to be prescribed. For simply supported beams subjected to a unit load we can obtain the deection u(x) as the solution of the second-order dierential equation However, the transformation of the dierential eqn.(1) into the integral eqn.() does not eliminate the ill-posedness of the problem, as such an equation is a Fredholm integral equation of the rst kind and constitutes a well-known example of an ill-posed problem due to its instability of the solution with respect to noise in the input data z as given by eqn.(19), see e.g. Tikhonov and Arsenin (1977). We now divide the interval [ L]into N equal parts by putting s j = jl=n for j = N, and on each subinterval [s j;1 s j ]forj = 1 N,we assume that is constant and takes its value at the midpoint ~s j =(s j + s j;1 )=, i.e. j [sj;1 s j]= (~s j )= j j = 1 N (4) Then eqn.() can be approximated by u(x) = NX j=1 j Z sj s j;1 G(x s)ds x L (5) If the deection u(x) is recorded only at K discrete locations x i =(i ; 1)L=(K) for i = 1 K, then eqn.(5) gives u i = u(x i )= where the integrals A ij = Z sj NX j=1 A ij j i = 1 K (6) s j;1 G(x i s)ds i = 1 K j = 1 N (7) u (x) = x(x ; L) (x) u() = u(l) = (1) can be evaluated analytically and their expressions are given by Integrating eqn.(1) twice we obtain u(x) = G(x s)(s)ds x L () where the Greens function G(x s) isgiven by A ij =(i ; 1)(1j 3 ; 18j ; 4Nj +1j +1jN +4Nj ; 6N ; 8N ; 3)L 4 )=(48KN 4 ) if i<j A ij =(i ; 1 ; K)(1j 3 ; 18j ; 1Nj + +1j +1Nj ; 4N ; 3)L 4 =(48KN 4 ) if i>j G(x s) = ( s (s;l)(x;l) L s x sx(s;l) L x s (3) A ij =[(i ; 1)j (3j ; 8Nj +6N ) +(;i +1+K)(j ; 1) 3 (3j ; 3 ; 4N)]L 4 =(48KN 4 ) +(i ; 1) 3 (i ; 4K ; 1)L 4 =(384K 4 ) if i = j(8) 4 Copyright c 1999 by ASME

5 In order to accommodate for the physical reality that in practice the measurements of u will never be exact or smooth, the deection u is contaminated with some errors, say = (x i ) for i = 1 K,given by a(x) z = u + (9).4 Thus, we have transformed the innite-dimensional illposed problem () into the nite-dimensional problem of nding the vector from the ill-conditioned system of linear equations z u = A (3) In order to deal with the ill-conditioned system of linear eqns (3), the minimization of the discretised version of eqn.() results in a stable numerical solution given by =(A tr A + R) ;1 A tr z (31) where R is the rst-order regularization matrix which has the components R ii = 1 for i = 1 and i = K, R ii = for i = (K ; 1), R i(i+1) = ;1 fori = 1 (K ; 1), R i(i;1) = ;1 for i = K and R ij = otherwise. For a simple typical benchmark example, namely that of a simply supported beam of length L = 1having a deection u(x) = x5 ; x3 + 7x, with the exural rigidity coecient to 6 6 be retrieved given by a(x) = 1, the solution given by (x+1) eqn.(31) has been employed. Also, in order to investigate the stability of the numerical solution, p% = 1% noise has been included in the measured deection z, asgiven by eqn.(9), where the components of the noise vector were generated using the NAG routine G5DDF, as Gaussian random variables with mean zero and standard deviation = p p max j u(x) j = ; (3) The numerically obtained results for a(x) = ;1 (x) given by eqn.(31), when K = N =andp =and =,and when p =1and =5 1 ;5,1 ;5,5 1 ;6 and 1 ;6, are shown in Fig.1 and also for comparison the exact solution is presented. From this gure it can be seen that when no noise is included in the data, i.e. p =, then a simple inversion = A ;1 z retrieves very accurately the exact solution Figure 1. THE NUMERICAL RESULTS ( ) FOR VARIOUS VALUES OF WHEN p% =1%,INCOMPARISON WITH THE EXACT SOLUTION a(x) = 1 (;;;). THE CIRCLES () REPRESENT THE NUMER- (1+x) ICAL RESULTS OBTAINED WHEN p =AND =. However, when noise is included in the data then as becomes very small, say 1 ;6, oscillations start to develop in the numerical solution which becomes unstable. Also as becomes large, say > 1 ;4, then the numerical solution tends to a constant value. However, there is a wide range of values of, say5 1 ;5 <<5 1 ;6 for which the estimate of the solution is stable and reasonably accurate. Nevertheless there is quite a signicant disagreement between the exact and the numerical solutions near the ends of the beam, and in fact the numerical solution may be considered a good estimate of the exact solution only in the range : <x<:8. Finally, it is reported that alternative numerical methods employed by the author, which are based on the singular value decomposition or the mollication method, also failed to produce accurate estimates of the solution near the ends of the xed beam. In a future study it is proposed that more powerful numerical methods which are based on parameterisation and/or constrained minimization procedures should be employed. Also, future work will be concerned with the inverse coecient identication problem associated with the Euler-Bernoulli unsteady-state beam theory. REFERENCES Collins, W.D., Quegan, S., Smith, D. and Taylor, D.A.W., Using Bernstein polynomials to recover engineer- x 5 Copyright c 1999 by ASME

6 ing loads from strain gauge data, Q. Jl Mech. appl. Math. 47, , Ismailov, I. and Muravey, L., Some problems of coecient control for elliptic equations of high order, Inverse and Ill-Posed Problems Conf. (IIPP-96), Moscow, 81 (abstract), Lesnic, D., Elliott, L. and Ingham, D.B., Analysis of coecient identication problems associated to the inverse Euler-Bernoulli beam theory, IMA J. Appl. Math. 6, , Marcellini, P., Identicazione di un coeciente in una equazione dierenziale ordinaria del secondo ordine, Ricerche Mat. 31, 3-43, 198. Tikhonov, A.N. and Arsenin, V.Y., Solutions of Ill- Posed Problems, Winston-Wiley, NewYork, Copyright c 1999 by ASME