A Method for Determining Model-Structure Errors ... - Michael I Friswell

Meccanica 34: 155–168, 1999. c 1999 Kluwer Academic Publishers. Printed in the Netherlands.

A Method for Determining Model-Structure Errors and for Locating Damage in Vibrating Systems JOHN E. MOTTERSHEAD1, MICHAEL I. FRISWELL2 and CRISTINEL MARES1 1 The University of Liverpool, Department of Engineering, Mechanical Engineering Division, Brownlow Hill;

Liverpool, L693GH U.K. 2 The University of Wales Swansea, Department of Mechanical Engineering, Swansea, U.K.

(Received: 6 May 1999; accepted in revised form: 1 June 1999) Abstract. A method is proposed for the determination of natural frequencies and mode shapes of a system which is constrained so that unknown stiffnesses are replaced by rigid connections. The constraint is not imposed physically but only in mathematics so that the behaviour of the constrained system is inferred from the unconstrained measurements. Since stiffnesses which are made rigid cannot experience any elastic strain they can have no effect on the inferred measurements. A procedure for comparing the inferred measurements with similarly constrained finite element predictions can be used to determine model-structure errors. Damage, such as a crack in a beam, can be located by comparing the inferred measurements from the structure in its undamaged and current states. It is demonstrated how unmeasured rotations may be constrained by using rigid-body modes and a reduction/expansion transformation from a finite element model. Sommario. Viene proposto un metodo per la determinazione delle frequenze proprie e dei modi di vibrazione di un sistema vincolato in modo tale che alcuni elementi elastici siano sostituiti da collegamenti rigidi. Il vincolo non viene imposto fisicamente, ma solo matematicamente, e pertanto il comportamento del sistema vincolato viene dedotto dalle misure sul sistema non vincolato. Poich´e gli elementi che sono resi rigidi non possono subire alcuna deformazione elastica, essi non hanno certamente alcun effetto sulle misure dedotte per il sistema vincolato. Una procedura che mette a confronto le misure dedotte per il sistema vincolato con le previsioni fornite da un modello ad elementi finiti con analoghi vincoli, pu`o essere utilizzata per determinare errori nella struttura del modello. Danni del tipo di una cricca su una trave possono essere localizzati confrontando le misure dedotte – per sistemi analogamente vincolati – da quelle effettuate sulla struttura non danneggiata e sulla struttura danneggiata. Si dimostra come si possono imporre vincoli sulle rotazioni (non misurate) utilizzando i modi di corpo rigido dell’elemento e una tecnica di riduzione/espansione dei gradi di libert`a di un modello ad elementi finiti. Key words: Model structure, Damage location, Vibrations

1. Introduction The purpose of finite element model updating [1–3] is to converge the predictions of a model upon physical vibration measurements. This can be achieved by adjusting the parameters of the model in an infinite number of different combinations [4]. So, a set of parameters is sought that can be adjusted consistently with engineering understanding of the test structure and the measurements taken from it. Regularisation [5] can be applied to reinforce the physical meaning of the parameters in the resulting system of ill-conditioned equations. But, in damage detection [6, 7] a major obstacle is presented to the application of model-based methods by the difficulty in achieving a physically meaningful updated model. The area known as model-structure determination [8] is about the capability of the finite element model to represent the physical system and should be applied with the purpose of

156 J.E. Mottershead et al. exposing (i) erroneous connectives, (ii) nonlinearities modelled as linear, (iii) unrepresentative joints and boundary conditions and (iv) errors in the constitutive equations such as a plate modelled as a beam, the omission of coupling between bending and torsion when curved beams are modelled as straight, and warping deformations neglected in torsion. Model-structure determination should be carried out before the model is parameterised for updating and requires the application of techniques that are different from the conventional system identification and parameter estimation methods now regularly applied in model updating. Freund and Ben-Haim [9, 10] and Mottershead [11] considered the problem of determining finite element connectivities from vibration measurements, and Gordis [12] and Gladwell [13] studied the related problem of disassembling a finite element model. In this article the method presented in [11] is extended to allow the detection of an unmodelled nonlinearity and to locate a crack in a damaged beam. The method works by applying rigid constraints across a chosen part of the structure which might for instance coincide with a single element in a finite element model. In the case of a simple spring the purpose of the rigid constraint is to prevent extension and compression so that the output of the complete system is entirely insensitive to the spring constant. The same principle can be applied to a beam element, in which case the deflection of that portion of the beam must be given by a linear combination of its rigid-body modes. It will be demonstrated that the constraint can be applied to the finite element model and also to receptance measurements. In the case of the latter it is possible to infer the natural frequencies and mode shapes of the constrained structure from the measurements, but the constraint does not have to be applied physically – the inferred data is obtained only by the application of algebra to the unconstrained measurements. So, for model-structure determination the predictions from a constrained finite element model may be compared to the inferred natural frequencies and mode shapes from measurements on the unconstrained system. Since stiffnesses which are replaced by rigid connections cannot experience any elastic strain they can have no effect on the inferred measurements or on the constrained predictions from the model. Therefore, when the rigid ‘element’ correctly locates an erroneous stiffness the constrained predictions and inferred measurements will be in agreement. Otherwise the ill-modelled connections will result in deviations between the measured and modelled results. Doebling et al. [6] carried out an extensive literature review on damage identification and Rytter [7] provided a classification of techniques to detect, locate and assess the extent of damage in structures. The methods described in [11] may be applied to locate damage. In that case two sets of inferred measurements are required from the system in its undamaged and current conditions. Damage will produce a disagreement between the two unless the damage site is located by the rigid constraint. In this paper the specific problem of locating damage in a beam is considered. The constraints involve rotations as well as translations at the ends of the beam ‘element’ to be made rigid. This introduces a practical problem because rotations are generally very difficult to measure and are often not measured at all. A reduction/expansion transformation from a finite element model of the complete structure is used to overcome the problem of unmeasured rotations. The rigid-body modes from a single finite element are used to form the constraints which must be applied to the measurements. In what follows we review the theory of rigid-body constraints as applied to an extension/compression spring in [11]. Then the application of such constraints to a beam is considered. Finally, a generalised theory is presented. Numerical examples are used to illustrate the application of the method to the problems of detecting an unmodelled nonlinearity in a spring and the location of a crack in a damaged beam.

Model Structure Determination 157 2. Rigid Constraints for a Simple Spring The case of an unknown stiffness,   1 −1 k=k , −1 1 between the ith and j th coordinates of a multi-degree-of-freedom system was considered in [11]. The essential details are repeated here for completeness. The undamped finite element dynamic stiffness equation, B(ω2)x = f,

(1)

B(ω2) = K − ω2 M,

(2)

K, M ∈ Rn×n ,

(3)

x, f ∈ Rn×1 ,

(4)

can be written as  b11 b12  b21 b22     bii bij   b j i bjj   bnn

    x1    x2          x   i   x  j         xn

         

          0           = fi ,       fj                      0

(5)

when external loads are applied only at the ith and j th coordinates. When xi = xj and the system is at a natural frequency then it is found that,       b11 b12    x1        b21 b22       0     x 2               (6)    xi  =  0  , bii + bij + bj i + bjj                        0  x n bnn fi = −fj , and the stiffness   1 −1 k −1 1 between i and j is annihilated in equation (6). The dynamic flexibility equation formed from the ith and j th columns of measured frequency responses, H(ω2)f = x,

(7)

158 J.E. Mottershead et al. can be written as     h1i h1j    x1   h2i h2j  ( )    x2            fi  hii hij  x . = i      hj i hjj  fj   x   j           hni hnj xn

(8)

When xi = xj , fj = −

fi (hii − hj i ) , hij − hjj

(9)

and at the natural frequency of the system (when fj = −fi ), hii − hij − hj i + hjj = 0, so that the mode-shape eigenvector can be determined from     x1  h1i − h1j              x2     h2i − h2j   = fi . xi  hii − hij                    xn hni − hnj

(10)

(11)

3. Rigid Constraints for a Beam The stiffness matrix of a beam element can be written as   6 3l −6 3l 2 2  2EI   3l 2l −3l l  k= 3  , l  −6 −3l 6 −3l  3l l 2 −3l 2l 2

(12)

where E is the elastic modulus, I the second moment of inertia and l is the length of the element. The rigid body modes have the form   1 − 12 l 0 1    8R =  (13) . 1  1 2l  0

1

If the element deflections, xe , are constrained to the rigid-body modes, then xe = 8 R α , or



xj xk



 =

(14) 8j 8k

 α, R

(15)

Model Structure Determination 159

Figure 1. Displacements of the constrained element.

where the superscripts j and k denote the two ends of the beam element and xj = (w j , θ j )T , xk = (w k , θ k )T as shown in Figure 1. Re-arrangement of equation (15) leads to 8k )−1 xk = Rxk , xj = 8 j α = 8 j (8 so that 

xj xk



 =

 R k x, I

where R is given by   1 −l R= . 0 1

(16)

(17)

(18)

Equation (17) specifies a constraint that transforms the full set of coordinates x to a constrained set x0 , x = Cx0 , by the connection matrix,   I 0 0 0 R 0   C= . 0 I 0

(19)

(20)

0 0 I Then for the constrained system one can write, B0 = CT BC,

(21)

f 0 = CT f,

(22)

where B is defined in equation (2) and C ∈ Rn×(n−2) , B0 ∈ R(n−2)×(n−2) and f 0 , x0 ∈ R(n−2)×1 . If the forces are applied only at the element degrees-of-freedom then equation (22) becomes,    T  fj R I = f 0, (23) fk

160 J.E. Mottershead et al. and at a natural frequency of the constrained system, the modified response x0 is in the null space of the dynamic stiffness matrix B0 . Thus, CT f = 0,

(24)

or in terms of the constraints on the applied force,    T  fj R I = 0, fk so that RT f j = −f k , and



fj fk



 =

I −RT

(25)  fj .

(26)

We are therefore able to construct   0  I    N= ,  −RT 

(27)

0 spanning the null-space of C, NT C = 0.

(28)

3.1. F REQUENCY R ESPONSE F UNCTIONS When the measurements are taken only at the nodes defining a single element, the dynamic flexibility equation, H(ω2) f = x, becomes " # Hjj (ω2 ) Hj k (ω2 ) Hkj (ω2 ) Hkk (ω2 )

(29)

fj fk

! =

xj xk

!

" =

R I

# xk .

(30)

By combining equations (26) and (30) and premultiplying by [I − R] one obtains an expression which is satisfied when ω is equal to a natural frequency of the measured system with an imposed rigid constraint, (Hjj (ω2 ) − RHkj (ω2 ) − Hj k (ω2 )RT + RHkk (ω2 )RT ) fj = 0,

(31)

and since fj 6 = 0, det(Hjj (ω2 ) − RHkj (ω2) − Hj k (ω2 )RT + RHkk (ω2 )RT ) = 0,

(32)

Model Structure Determination 161 or det(NT H(ω2 )N) = 0.

(33)

Equation (32) is the test function from which can be determined the natural frequencies of constrained system. 3.2. M ODEL R EDUCTION

AND THE

A PPLICATION

OF

R IGID B ODY C ONSTRAINTS

Equation (32) implies the measurement of point and cross receptances at rotational degreesof-freedom. Point receptances for rotations are especially difficult to measure. We will assume that no rotational measurements are available so that either, (i) the rotations must be recovered by expanding the measured translational frequency responses or (ii) the constraints defined previously in equation (15) must be applied entirely by means of the translations. 3.2.1. Expansion of measured frequency responses Consider the reduction/expansion transformation,     x1 I x= = Tx1 = x1 , T2 x2

(34)

where x1 are the retained (translational) coordinates and x2 are the omitted (rotational) coordinates. The measured frequency responses can be arranged in matrix form and expressed as H1 f1 = x1 ,

(35)

where f1 = TT f.

(36)

Thus, by combining equations (35) and (36) and by premultiplying by T it is found that, TH1 TT f = x,

(37)

where TH1 TT is the expanded n × n frequency response function matrix which contains the rotational coordinates as well as the measured translations. If T were exact, then each term in TH1 TT at a rotational coordinate would be identical to a measurement, were it available. We assume this to be the case and having determined the null-space matrix N from equation (28) we obtain a test equation equivalent to equation (32) having the form, det[NT (TH1 (ω2 )TT )N] = 0, or

" det NT

H1 (ω2 )

H1 (ω2 )TT2

T2 H1 (ω2 ) T2 H1 (ω2 )TT2

!

# N = 0.

(38)

The natural frequencies of the constrained system are determined when equation (38) is satisfied.

162 J.E. Mottershead et al. 3.2.2. Rotational constraints applied by means of the translations We observe that the rotational coordinates θ j and θ k in Figure 1 can be expressed in terms of j the translations by using the rows t2 , tk2 of the condensation matrix, j

j

x2 = [t2 ]x1 ,

(39)

x2k = [tk2 ]x1 .

(40)

Using equations (17), (18), (39) and (40) it is possible to combine the reduction and the rigidbody constraint to obtain " j # " #" # x1 1 −l x1k = . (41) j 0 1 tk2 x1 t2 x1 Now we can consider a reduced set of degrees-of-freedom x01 in which both x1 , x1k displacements are preserved and we eliminate, by using equation (41), two different displacements x1a , x1b from the active set x1 in order to introduce the rigid constraint expressed in equation (18). Thus, j

x01 = [x11 · · · x1a−1 , x1a+1 · · · x1b−1 , x1b+1 · · · x1 , x1k · · · x1m ]T ∈ R(m−2)×1 , j

and by rearranging equation (41) an expression can be written having the form ! x1a A = Bx01 , x1b or x1a x1b

(42)

(43)

! = A−1 Bx01 .

Then the final relation between the x1 and x01 is   I|0   x1 =  A−1 B  x01 , 0|I and the new connection matrix has the form,   I|0   C1 =  A−1 B  .

(44)

(45)

(46)

0|I The test equation can be written as det(NT1 H1 (ω2 )N1 ) = 0,

(47)

where, NT1 C1 = 0,

(48)

Model Structure Determination 163 and the constraint is enforced by the translational forces f1 spread across the complete structure. If the transformation T is exact then the constraint matrix C1 will also be exact. 4. A Generalised Theory for Rigid Constraints We begin with the constraint matrix C which may be applied in the finite element equations to enforce rigid-body behaviour in a part (or parts) of the structure. At a natural frequency the modified response x0 must be in the null-space of B0 , or in terms of the applied force, CT f = 0.

(49)

Because C is rectangular there exists a matrix N, with the same number of columns as constraints, such that CT N = 0.

(50)

It is clear that the columns of N span the null-space of C, so that from equation (49) f must be in the range of N, β, f = Nβ

(51)

β 6 = 0.

(52)

Putting this expression into equation (7) and using equations (19) and (50), it is found that at a natural frequency ω = ωn the general test function expressed in equation (33) is obtained. The columns of N are not unique, so the test equation (33) is not unique either. However its zeros will always define the natural frequencies of the constrained system. The mode shapes in the unconstrained system of coordinates can then by expressed as β. x = H(ωn2 )Nβ

(53)

The above theory can easily be applied to the cases of nonlinear and damped connections, as will be demonstrated in the following section. 5. Simulated Examples 5.1. A N ONLINEAR S PRING

WITH

DAMPING

The example problem illustrated in Figure 2 is of two aluminium cantilever beams connected by a nonlinear spring at an unknown location. Each beam has 10 elements and in the simulated experiment the spring is connected between nodes 6 on each beam. Table 1 gives

Figure 2. Arrangement of the two beams and the connecting spring.

164 J.E. Mottershead et al. Table 1. Dimensions of the two beams

Length Depth Breadth

Beam 1

Beam 2

1m 15 mm 50 mm

1m 20 mm 50 mm

the dimensions of the beams. The spring has linear and cubic coefficients of 100 kN/m and 2 GN/m3 , respectively. A viscous damper is included in parallel with the spring having a damping coefficient of 100 Ns/m. We aim to identify the two coordinates, one on each of the two beams, which are connected by the spring having a stiffness coefficient that we consider to be unknown. For any choice of two coordinates, the natural frequencies of the constrained system with a rigid link are obtained from the unconstrained frequency response measurements when equation (10) is satisfied, and the mode-shapes are determined from equation (11). The ‘measurements’ were obtained from a reduced 20 degree-of-freedom model of the system representing the transverse (translational) motions at each node, and the frequency response functions were generated by using a simulated stepped sine test [14] where the excitation frequency was increased in 1 rad/s steps. Figure 3 shows the transfer frequency response between the two ends of the spring, that is the translational degrees-of-freedom at node 6 of both beams. Reciprocity does not hold for all the frequency range, breaking down near the jumps in the response. Although the first resonance is predominantly linear, the second and third are clearly nonlinear. Table 2 shows the identified frequencies of the system with a rigid connection between the translational degrees-of-freedom of different pairs of nodes (the equal-numbered node on each beam) obtained from equation (10). Figure 4 shows the plot of the absolute value of the left-hand-side of equation (10) when the nonlinear spring is present between the two sixth nodes. Three natural frequencies are indicated where the plot goes to zero and equation (10) is satisfied. Linear interpolation cannot be used to locate the natural frequencies from the measurements because the frequency responses are complex, so the frequencies quoted in Table 2 are correct only to 1 rad/s. The constrained natural frequencies, inferred from the ‘measurements’, were compared to the predicted natural frequencies from the constrained model in order to locate the unmodelled connectivity. The rigid connection was placed at candidate positions and the natural frequencies were determined. The results, given in Table 2 clearly show that the frequencies correspond most closely when the rigid connection is placed between the two sixth nodes, which is the correct location of the nonlinear spring. 5.2. A C RACKED B EAM The example problem illustrated in Figure 5 is a steel beam having 1 m length and the section of 0.020 m × 0.020 m. The beam is divided in 10 elements and it is considered that the third element is damaged due to a crack of 4 mm length. The cracked beam-element is modelled by the method of Gounaris and Dimarogonas [15]. Table 3 gives the natural frequencies of the beam in its initial and damaged conditions. We aim to identify the damage location by comparing the zeros of the left-hand-side of

Model Structure Determination 165

Figure 3. Transfer frequency response functions showing lack of reciprocity.

Table 2. Table of inferred and analytical natural frequencies

Natural frequencies from measurements (rad/s) Natural frequencies from analytical model (rad/s)

Mode number

Rigid connection assumed between nodes 2 4 6 8 10

1 2 3 1 2 3

93 – 576 89.1 125.5 562.5

93 – 586 92.2 185.8 583.5

93 327 593 93.0 326.6 592.8

93 312.5 – 93.2 484.5 646.4

93 – 577.5 93.0 367.9 597.6

equation (47) obtained from the unconstrained frequency response measurements on the beam in its cracked and undamaged conditions. First a condensation of all the rotational degrees-of-freedom is carried out and for each rigid element the connection matrix C1 is obtained using equation (46) in which two displacements are constrained. With the corresponding N1 one can determine the value of the left-hand-side of equation (47). A plot of the test equation (47) when the element 3 was constrained, is shown in Figure 6.

166 J.E. Mottershead et al.

Figure 4. Left-hand side of equation (10) for the nonlinear example.

Figure 5. Clamped steel beam with a crack in the third element.

Different methods of reduction were used for numerical simulations and the most accurate location was obtained by using exact dynamic condensation. In the case of the simulation the selected frequency used in the condensation was the exact frequency of the measured H1 (ω2 ). Thus equation (47) might be more correctly written as, det(NT1 (ω2 )H1 (ω2 )N1 (ω2 )) = 0.

(54)

Efficient schemes for determining the zeros of the left-hand-side of equation (54) can be devised. The two sets of natural frequencies are compared in Table 4. The correct location of the crack is given for the case when the error in the natural frequencies is the least and it can be seen that when using the first four natural frequencies the crack is correctly located. It is well known that the crack produces a very localised stiffness reduction and in such cases the mode shapes of the initial and damaged system show no significant differences and give little information about the position of the crack. Table 3. Natural frequencies (rad/s) for the two states of the beam Modes

1

2

3

4

Undamaged Damaged

103.99 103.30

651.69 651.50

1825.14 1816.72

3579.09 3561.31

Model Structure Determination 167

Figure 6. Test equation when the third element is constrained. Table 4. Table of test equation zeros for the beam in its undamaged and current conditions Mode number

1 2 3 4

1

2

0.64 −0.68 −0.31 0.20

1.07 −1.34 −0.73 −0.38

3 0.00 0.00 0.00 0.00

Frequency error/(%) assumed rigid element 4 5 6 7 8 1.96 −0.93 −0.13 −0.02

−5.32 1.30 −0.18 −0.16

−8.60 −0.10 −0.97 −0.14

0.69 −0.20 −0.37 −0.06

−0.86 −0.53 −0.33 0.04

9

10

−2.80 0.29 −0.80 −1.32

−2.57 −1.21 −0.76 −0.23

6. Conclusions A theory has been presented to implement rigid-body constraints in measured frequency responses so that the constrained behaviour of the system can be inferred from the unconstrained measurements. The same constraints may be introduced into finite element equations. It has been demonstrated how the rigid constraints can be applied to a portion of a beam and the theory has been generalised to include other structure types. Methods for the application of the constraints which avoid the use of rotational measurements have been described. Numerical results have been presented to illustrate the use of the method in determining an unmodelled nonlinearity and locating a crack in a cantilever beam.

168 J.E. Mottershead et al. Acknowledgements The research reported in this article is supported by EPSRC grant GR/M08622. Dr Friswell gratefully acknowledges the support of the EPSRC through the award of an Advanced Fellowship.

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