Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

49, 3, pp. 665-683, Warsaw 2011

MODEL UPDATING IN STRUCTURAL DYNAMICS

THROUGH A CONFLUENCE ALGORITHM

Maria Chierichetti

Massimo Ruzzene

Georgia Institute of Technology, School of Aerospace Engineering, Atlanta, USA

e-mail: mchierichetti@gatech.edu; massimo.ruzzene@aerospace.gatech.edu

The identification of the dynamic response of a structure in the pre-
sence of structural degradation has potential practical use on health
monitoring systems and can contribute to improve the safety of ro-
torcraft flight and wind-turbine operations and to decrease their han-
dling costs. A combined numerical and experimental procedure, called
Property Identification Algorithm, updates the numerical model based
on a limited set of experimental measurements in order to accurately
predict the dynamic response of a system in the presence of structural
degradations. The algorithm is designed based onmodal decomposition
and discrete experimental measurements, and is formulated in the case
of periodic excitations. It is demonstrated that the updated dynamic
response represents an accurate map of the experimental response in
the domain. The paper describes the proposed algorithm and presents
validation cases.

Key word: displacement mapping, inverse problem, FEM, experimental
update, modal expansion

1. Introduction

The ability to assess the effects of changes in the physical properties of a
structure from experimental measurements is crucial for understanding its re-
al dynamic behavior because of the possible improvements in themaintenance
of critical dynamic components connected to it.Modifications inmass and stif-
fness affect the performance of the system and can result in an increase of the
vibrations of the structure and in higher fatigue loads. Therefore, the accurate
prediction of the vibration levels of a system could result in improvements of
Condition Based Maintenance processes.



666 M. Chierichetti, M. Ruzzene

Wind turbines and helicopter blades are particularly affected by this pro-
blem. Icing (Fig.1a) and accumulation of dirt and bugs (Fig.1b) significantly
affect the maintenance of wind turbines because of the additional mass intro-
duced on the structure. Ice build-up is dangerous because it can reduce the
generated power up to 25-50% of its design value, and the additionalmass (up
to 7-10% of the total mass) affects the dynamic behavior, causing excessive
vibrations, higher bendingmoments, imbalance of the rotor, andnoise (Corten
andVeldkamp, 2001; Khalfallaha andKoliub, 2007; Bolton, 2007). Accumula-
tion of dirt and bugs on the leading edge of the airfoil affects the aerodynamic
efficiency of the airfoil by up to 20%of its design value, and is particularly cri-
tical for offshore towers (and towers operating in sandy environments) somuch
that the cleaning of the blades is recommended every few weeks (Ibsen and
Liingaard, 2006; Andersen et al., 2008) and their operational life is reduced
to half of their design length with respect to ground-based towers. The initial
clean model of the wind turbine, therefore, does not satisfactorily represent
the actual behavior of the system.

Fig. 1. Examples of the effect of ice and bugs on wind-turbine and rotorcraft blades;
(a) ice on wind turbine, (b) bugs on wind turbine, (c) ice on helicopter blade

Rotorcraft operations are also greatly affected by icing accretion (Fig.1c).
Unlike aircraft propellers, helicopter blades collect ice along the entire leading



Model updating in structural dynamics... 667

edge of the airfoil at temperatures below 10◦C causing an increase in drag
(Palacios et al., 2008;Coffman, 1987), flowseparation andhighvibration levels
due both to uneven ice adhesion and progressive ice shedding from the blade
due to centrifugal and aerodynamic forces (Gent et al., 2000). One of the
consequences of this behavior is that the maximum available power of the
engine can be reached before the generation of the required torque to sustain
the required operation.

Moreover, damage detection systems of the helicopter main rotor are still
not generally available because of their high cost and complexity. Limited
rotor system fault detection is provided as part of usage monitoring as well
as automatic rotor track and balance systems. A comprehensive Health and
Usage Monitoring System (HUMS) should provide post-flight diagnostic ca-
pabilities through the processing of flight data at ground stations in order to
improve the detection of structural degradations. The focus of this paper is on
vibrationmonitoring during operations as ameans to achieve condition based
maintenance (Stevens, 2001). In-flight or post-flight data processing requires
simple and fast procedures for model updating so that changes in the system
can be rapidly detected and analyzed.

The goal of the proposed approach, called Property Identification Algo-
rithm, is to update the properties of a numerical model to achieve accurate
predictions of the dynamic response of the system. This result is achieved by
“tuning” a numerical model so that it accurately predicts the actual behavior
of the system in every part of its life based on a limited amount of isolated
measurement points so that the response at non-measured locations can be
accurately predicted. The use of experimental measurements to identify and
update the dynamical properties of a finite element model is known as model
updating, and it has been extensively applied to structural dynamic inFriswell
and Mottershead (1995) and Mottershead and Friswell (1993). The Property
Identification Algorithm, however, does not focus on the accurate extrapo-
lation of the changes in physical properties but is based on the Confluence
Approach presented in McColl et al. (2010), Chierichetti et al. (2010, 2011).
The approach presented in these papers computes corrections to the external
loads in order to improve the prediction of the dynamic response, and it as-
sumes accurate knowledge of the physical properties of the system, based on
very fewmeasurements of the response. Similar to the Confluence Algorithm,
the proposed approach improves the dynamic response of a periodic system
througha fast and simpleupdating technique. In contrast, theProperty Identi-
fication Algorithm achieves this objective by updating the physical properties
of the system and assuming accurate knowledge of the external loads. This



668 M. Chierichetti, M. Ruzzene

assumption requires either having an a priori accurate model of the loads or
achieving an accurate prediction of the loads through the Load Confluence
Algorithm.

The proposed approach is ideally suited to themonitoring of the response
of periodic systems such as wind-turbines and helicopters. It requires only
the measurement of the dynamic response of the system and therefore allows
modal identification under operations and in situations where the structure
is difficult to excite by externally applied forces, Brincker et al. (2003). The
predicted response of an initial model of the system is combined to reference
data for the definition of simple algebraic relations between the corrections of
mass and stiffness distributions and the difference between the measured and
numerical responses, as discussed in Section 2. In Section 3, the potentialities
of the algorithm are demonstrated through numerical validation in the case of
a simple lumped-parameters system for variations in mass, and/or stiffness.

2. Property Identification Algorithm

The goal of the Property Identification Algorithm is the mapping of the re-
sponse of a modified system through the identification of changes in dynamic
properties (natural frequencies andmodes). The application to rotating envi-
ronments admits the assumption that the applied loads, and as a consequence
the dynamic response, vary periodically with time and can be expanded thro-
ugh a Fourier series.

2.1. Concept

TheProperty Identification Algorithm consists of an initial numerical mo-
del of the structure, a set of experimental measurements at a limited number
of locations, and a procedure that estimates the difference in dynamic pro-
perties between the numerical model and the experimental measurements. A
block-diagram representation of the component of the procedure is depicted
in Fig.2. The applied loads are assumed to be known.

The dynamic model of the structure is defined by a mass matrix M and
a stiffness matrix K, the set of available experimental measurements is deno-
ted as eE(x), where x is a vector defining the location of the sensors, and
array eN(x) stores the numerical response. The following notation is used
throughout the paper: bold, lower case defines vectors, bold, upper case defi-
nes matrices.



Model updating in structural dynamics... 669

Fig. 2. Schematic of the Property Identification Algorithm

The procedure can be summarized as the following sequence of steps:
• A numerical model of the system is built from an initial guess of the
physical properties of the system.

• The solution of the initial model estimates the dynamic response eN(x)
at the sensors location x.

• The numerical andmeasured responses are compared and the error vec-
tor ∆e= eE −eN is calculated.

• Corrections for the mass and/or stiffness matrices are calculated based
on ∆e through a formulation of the problem in the modal domain, as
described in Section 2.2.

• A modified set of dynamic properties is found from the solution to the
new eigenvalue problemwith the updatedmass and stiffness matrices.

2.2. Modal procedure for property estimation

Structural degradation can occur in a system as a change in stiffness, or
mass, ormass and stiffness distributions.Threedifferent procedures have been
developed to analyze each of these problems separately.

2.2.1. Change in stiffness

Consider a general undamped linear system

Mü(t)+Ku(t)=F(t) (2.1)

where array u(t) stores the N degrees of freedom of the solution, and F(t)
is the array of the generalized applied loads. This system (defined as “initial”
system) is characterized by natural frequencies ω andmodes P.



670 M. Chierichetti, M. Ruzzene

The real model (defined as “reference”) of the system is

M̂¨̂u+ K̂û=F(t) (2.2)

It is assumed that the applied loads are accurately modeled and that the
reference system undergoes the same loading condition as the initial system.
The reference model is characterized by natural frequencies ω̂ andmodes P̂.
Supposean inaccuracy exists in theestimation of the stiffnessmatrix, therefore

M̂=M K̂=K−∆K (2.3)

with ∆K unknown. The eigenvectors of the modified system P̂ are assumed
to be a linear combination of the initial modes P through a matrix α of
coefficients

P̂=Pα (2.4)

The difference between Eq.(2.1) and (2.2) leads to

M(ü− ¨̂u)+K(u− û)−∆Kû=0 (2.5)

with u = Pq and û = P̂q̂ = Pαq̂. The difference in generalized modal
coordinates is defined as

∆z= q−αq̂ (2.6)

Therefore the difference in the dynamic response is defined as u− û=P∆z.
Pre-multipling Eq. (2.5) by P⊤, and defining the projection of the change in
the stiffness matrix in the modal domain as K=P⊤∆KP, Eq. (2.5) results

I∆z̈+ diag(ω2i )∆z+Kαq̂=0 (2.7)

The response of the system and the generalized coordinates can be related
through amatrix B and its pseudo-inverse as:

— numerical response

eN(x,t)=B(x)q(t) (2.8)

— experimental measurement

eE(x,t)= B̂(x)q̂(t) (2.9)

The definition of B depends on the reference quantities. In the case of
experimentalmeasurements, it depends on the type of sensors. For example, if
ei represents a displacement measurement at the location (xi,yi), Bij repre-
sents the contribute of the modal displacement j at the location i. If strain



Model updating in structural dynamics... 671

gages are considered, Bij represents the contribute of modal strain j at loca-
tion i. A reduced number ofmodes m can be used in the analysis, that has to
be lower than the number of sensors used to identify the response. Moreover,
only modes that can be identified by the choice of sensors can be included.
Assume that the relation between matrices B and B̂ is the same as the

relation between P and P̂, so that B̂ = Bα. This assumption is valid in a
linear or linearized framework. Thedifference between the initial response and
the reference signal is

eN −eE =B(q−αq̂)=B∆z (2.10)

and
∆z=B+∆e (2.11)

Equation (2.7) thus becomes

B
+
∆ë+ diag(ω2i )B

+
∆e+KB+ê=0 (2.12)

Theexternally applied loadsare supposedtobeperiodicaswell as the response
of the system, and can be expanded in a Fourier series of frequency Ω

∆e=∆e0+
M∑

j=1

[ecj cos(jΩt)+esj sin(jΩt)] (2.13)

The harmonic balance of Eq. (2.12) leads to

diag(ω2i )B
+
∆e0+KB

+
ê0 =0

diag(−j2Ω2+ω2i )B
+
∆ecj +KB

+
êcj =0 (2.14)

diag(−j2Ω2+ω2i )B
+
∆esj +KB

+
êsj =0

The unknown of the system is the change in the stiffnessmatrix in themodal
domain K. Assume that it is possible to approximate ∆K with a rank-1
approximation (see Appendix A for details) such that

K̂=K−∆K=K−hh⊤ (2.15)

and
K=P⊤∆KP=P⊤hh⊤P=ββ⊤ (2.16)

Eq. (2.14) becomes

diag(ω2i )B
+
∆e0+ββ

⊤
B
+
ê0 =0

diag(−j2Ω2+ω2i )B
+
∆ecj +ββ

⊤
B
+
êcj =0 (2.17)

diag(−j2Ω2+ω2i )B
+∆esj +ββ

⊤
B
+êsj =0

These non-linear system of equations is solved with a Newton-Raphson itera-
tive method.



672 M. Chierichetti, M. Ruzzene

2.2.2. Change in mass

The change in dynamical properties due to a modification in the mass
distribution of the system is analogous to the previously described procedure.
The differences between the two cases are underlined in the following. The
reference system is in this case characterized by an unknown change in the
mass matrix such that

M̂=M−∆M K̂=K (2.18)

The aim is in this case to reconstruct the difference in mass ∆M from the
observation of the response of the reference system at a limited number of
points, and to improve the prediction of the initial system. The difference
between Eq. (2.1) and (2.2), considering Eq. (2.18), leads to

M(ü+ ¨̂u)+K(u− û)−∆M¨̂u=0 (2.19)

The change in the mass matrix projected in the modal domain is defined as
M=P⊤∆MP and it is approximated as a rank-1matrix (AppendixA) such
that

M̂=M−∆M=M−gg⊤ (2.20)

and
M=P⊤∆MP=P⊤gg⊤P=ηη⊤ (2.21)

Amodal expansion of Eq. (2.19) leads to

I∆z̈+ diag(ω2i )∆z+Mα
¨̂q=0 (2.22)

that becomes
IB
+∆ë+ diag(ω2i )B

+∆e+MB+¨̂e=0 (2.23)

with the introductionof thepreviouslydefinedmatrix B.The samehypothesis
on the periodic nature of the external load is assumed. A Fourier expansion
of all quantities and harmonic balance of Eq. (2.23) lead to

diag(ω2i )B
+∆e0 =0

diag
(
1−

ω2i
j2Ω2

)
B
+
∆ecj +MB

+
êcj =0 (2.24)

diag
(
1−

ω2i
j2Ω2

)
B
+
∆esj +MB

+
êsj =0

The first equation on the zeroth coefficient is generally verified in itself and
only the other two equations will be brought forward. Introducing the rank-1
approximation of the modal mass matrix M, the solution to the non-linear
problem can be found through the Newton-Raphson procedure.



Model updating in structural dynamics... 673

2.2.3. Change in mass and stiffness

The previous procedures can be combined and generalized in a unique
algorithm in the case of a contemporaneous change in the mass and stiffness
matrices. It is therefore assumed that the reference system is characterized by
unknownmodifications in the mass and stiffness distribution so that

M̂=M−∆M K̂=K−∆K (2.25)

Following the same steps as previously described, the non-linear system of
equations in the frequency domain is

diag(ω2i )B
+
∆e0−ββ

⊤
B
+
ê0 =0

diag(−j2Ω2+ω2i )B
+
∆ecj −ββ

⊤
B
+
êcj + j

2
Ω
2
ηη
⊤
B
+
êcj =0 (2.26)

diag(−j2Ω2+ω2i )B
+∆esj −ββ

⊤
B
+êsj + j

2Ω2ββ⊤B+êsj =0

and can be solved with the Newton-Raphson iterative method.

3. Numerical validation

3.1. Concept

The approach used for the numerical validation of the algorithm is descri-
bed in this section. The initial model of the system is known, and its dynamic
response is calculated in a particular loading, and labeled as “initial”. Known
structuralmodifications either in themass and stiffness distribution (or both)
are then added to the initial model, and its dynamic response is computed in
the same loading condition and stored as “reference”. Theuse of a numerically
generated reference response is due to the lack of experimentalmeasurements.
The corrections to the initialmass and stiffnessmatrices are then foundby

comparison of the initial and reference response. The response of the updated
system is called “final”. The characteristics of the algorithm are assessed by
comparing thefinal and reference responses, and the identified changes inmass
and stiffness with the real, known modifications introduced in the reference
model.

3.2. Results

The algorithm is validated by the analysis of a 14 degrees of freedom
lumped-parameter system (Fig.3). A concentrated periodic load is applied



674 M. Chierichetti, M. Ruzzene

at degree of freedom 10, with excitation frequencies of 4.5rad/s and 9rad/s.
The natural frequencies of the initial system are enumerated in Table 1.

Fig. 3. Schematic of a lumped-parameters mass-spring system.

Table 1.Natural frequencies of the initial model

mode ωi [rad/s]

1 3.56

2 10.01

3 16.45

4 23.39

5 28.82

6 32.68

The ability of the algorithm to capture structural modifications both in
the mass and stiffness distribution is investigated, as specified in Table 2.

Table 2.Description of the analyzed cases

ID modes control points modification

1 1:14 1:14 k3 =108.5N/m

2 1:3 1,4,7,10,13 k3 =108.5N/m

3 1:6 1,3,5,7,9,11,13 k3 =108.5N/m

4 1:6 1,3,5,7,9,11,13 k3 =90N/m, k5 =15N/m, k9 =75N/m

5 1:3 1,4,7,10,13 m3 =0.15kg

6 1:6 1,3,5,7,9,11,13 m3 =0.05kg, k3 =90N/m

7 1:6 1,3,5,7,9,11,13 m3 =0.05kg, k3 =90N/m with noise

Themain factors that influence the ability of the approach to converge to
the exact solution are the number of modes in the modal expansion and the
choice of control points (both total number and location). Their influence is
investigated in this section as well as the influence of noise in the reference
signals.

Initially, a change in stiffness at degree of freedom 3 is introduced. A com-
plete modal expansion and a complete set of control points are considered
in Case 1 to ensure that the algorithm exactly captures the modifications of



Model updating in structural dynamics... 675

the systemwhen a complete modal expansion is used. However, when a lower
number of modes and control points is considered, as for example in Case 2
(threemodes and five control points), themodifications in natural frequencies
(Table 3) andeigenvectors (Figs.4) are capturedwithanaccuracy of 1%onthe
modes included in the modal expansion, while the change in stiffness cannot
be captured. This behavior is due to the use of a truncated modal expansion
that causes the stiffness matrix to lose its original structure: in fact, both
the initial and reference systems are characterized by a tridiagonal, sparse
stiffness matrix, while the final stiffness matrix loses its sparse characteristic.
The deformed shape of the system at a specified time instant, Fig.5, reveals
that even if the change in stiffness is not accurately captured, the dynamic
response is greatly improved with only three modes to represent the dynamic
response. Before the application of the Property Identification Algorithm, the
mean error with respect to the reference system is about 30%, while after the
application of the algorithm is reduced to 3%.

Table 3.Case 2. Initial and final error between the reference (ωr), the initial
and the identified (ωf) natural frequencies

ID ωr [rad/s] ωf [rad/s] Ein [%] Efin [%]

1 3.86 3.86 −7.7 0.02

2 10.55 10.63 −5.1 0.8

3 16.46 16.49 −0.1 0.2

4 23.84 23.39 −1.9 −1.9

5 30.12 28.82 −4.3 −4.3

6 39.46 32.68 −17.2 −17.2

Fig. 4. Case 2. Comparison of the eigenvectors of the system



676 M. Chierichetti, M. Ruzzene

Fig. 5. Case 2. Deformed shape of the system

The inclusion of three higher modes and two control points (Case 3) hi-
ghly improves the prediction of the modal properties, both of the natural
frequencies and of the eigenvectors. The errors on higher frequencies, such as
the 5th and 6th natural frequency, are reduced to about 1% as the lower fre-
quencies. However, the algorithm still encounters difficulties in the physical
representation of the change in stiffness. The dynamic response of the system
is accurately represented with a mean error of about 4%, as shown in Fig.6.

Fig. 6. Case 3. Deformed shape of the system

The presence of multiple modifications in the system is analyzed in Ca-
se 4 and represents a critical condition for the algorithm. As described in
Section 2.2, in fact, the change in the characteristic matrices is modeled as
a rank-1 matrix approximation, which corresponds to an elementary modifi-
cation of the system. Also in this case, the algorithm is able to improve the
natural frequencies of the system, but cannot accurately update the stiffness



Model updating in structural dynamics... 677

distribution. However, the time history of the system at different degrees of
freedom is identifiedwith an accuracy of about 5%(initial error of about50%),
as illustrated by Figs.7 and 8. The dynamic response is effectively mapped,
both at degrees of freedom used as control points (Fig.7a) and not used by
the identification approach (Fig.7b). A broader view of the behavior of the
whole system is given by the analysis of the deformed shape at a particular
time instant (Fig.8), and it confirms the exceptional improvements achieved
in the prediction of the final response of the whole system.

Fig. 7. Case 4. Time history at different degrees of freedom; (a) DOF 2 – not control
point, (b) DOF 11 – control point

Fig. 8. Case 4. Deformed shape of the system

A change in the mass distribution at degree of freedom 3 is introduced in
Case5.This analysis represents aparallel problem forCase2, and thebehavior
of the identification procedure is very similar. The natural frequencies and the
lowest eigenvectors are identifiedwith an accuracy of about 3%.As previously



678 M. Chierichetti, M. Ruzzene

underlined, the change in physical mass is not properly captured due to the
incompleteness of the modal expansion. The dynamic response is accurately
identified in the whole system, both at control points, Fig.9b, and at points
not considered as the reference, Fig.9a. The analysis of the deformed shape of
the system confirms this observation (Fig.10).

Fig. 9. Case 5. Time history at different degrees of freedom; (a) DOF 2 – not control
point, (b) DOF 11 – control point

Fig. 10. Case 5. Deformed shape of the system

Finally, simultaneous changes in the mass and stiffness distributions are
considered. In Case 6, six modes and seven reference points are considered in
the identification of the dynamic response.

As noticed in the previous sections, the algorithm can capture major mo-
difications inmodal properties with an incompletemodal expansionwithin an
accuracy of 10% (Table 4), while themass and stiffness distributionsmaintain
large inaccuracies: the algorithm cannot distinguish between changes in mass



Model updating in structural dynamics... 679

and stiffness. Figures 11 and12 reveal that thedynamic responseof the system
is accurately identified within an error of 5% at critical locations.

Table 4.Case 6. Initial and final error between the reference (ωr), the initial
and the identified (ωf) natural frequencies

ID ωr [rad/s] ωf [rad/s] Ein [%] Efin [%]

1 3.77 3.77 −5.5 0.1

2 8.51 8.51 17.6 0.05

3 14.93 14.92 10.2 −0.1

4 22.84 22.44 2.4 −1.8

5 29.81 29.58 −3.3 −0.8

6 39.37 35.37 −17.0 −10.2

Fig. 11. Case 6. Time history at different degrees of freedom; (a) DOF 2 – not
control point, (b) DOF 11 – control point

Fig. 12. Case 6. Deformed shape of the system



680 M. Chierichetti, M. Ruzzene

The noise contained in the reference signals is an important factor for the
convergence of the algorithm. Random noise has been added to the reference
response characterized by lower signal levels (DOF: 1,2,3,4). The noise level is
about 5% the amplitude of the considered signal. The results of the identifica-
tion procedure are shown in Figs.13 and 14. Themean error on the deformed
shape is reduced from about 25% before the application of the Property Iden-
tification Algorithm to less than 10% after the identification, and the error on
the natural frequencies is reduced from amaximum of 17% to 1.5%.

Fig. 13. Case 7. Time history at different degrees of freedom; (a) DOF 2 – not
control point, (b) DOF 11 – control point

Fig. 14. Case 7. Deformed shape of the system

It is therefore demonstrated that the algorithm is effectively capable of
extracting information on the complete map of the response from a minimal
set of reference/experimental data, and to improve the predictions of both the



Model updating in structural dynamics... 681

responseandmodal properties of the initialmodel in thepresence of structural
degradations, even in the presence of noise in the signals. These results are
promising for applications in more complex environments.

4. Conclusion

An innovative approach for the mapping of the response in the presence of
structural modifications has been formulated, which integrates minimal expe-
rimental data into the initial and inaccuratemodel of a system. Simple valida-
tion cases demonstrate that the algorithm is capable of mapping the response
of the system in the complete domain, based on discrete reference data of
the dynamic response. The procedure identifies the presence of changes in the
mass and stiffness distribution andupdates themodal properties of the system
to best represent the reference dynamic properties.

A. Rank-1 tensor approximation

A rank-1 tensor is the simplest possible tensor, and is defined as

A1 =uw
⊤ (A.1)

where u and w are columnvectors. In general, a tensor can always be expres-
sed as a linear combination of r rank-1 tensors, where r is the rank of the
tensor.
In the case of a dynamic system, a rank-1 approximation of the upda-

ted mass and stiffness matrices corresponds to an elementary modification of
the system. For example, in the case of a three-degrees-of-freedom, lumped-
parameter system

K=





k1+k2 −k2 0
−k2 k2+k3 −k3
0 −k3 k3






(A.2)
M= diag([m1,m2,m3])

If m2 and k2 are reduced by ∆m and ∆k respectively, the changes in the
mass and stiffness matrices are in fact



682 M. Chierichetti, M. Ruzzene

∆K=





∆k −∆k 0
−∆k ∆k 0
0 0 0




=∆k





1
−1
0





[
1 −1 0

]

(A.3)

∆M= diag([0,∆m,0]) =∆m





0
1
0





[
0 1 0

]

References

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Udoskonalanie modeli w dynamice konstrukcji za pomocą

algorytmu konfluencji

streszczenie

Identyfikacja dynamicznej odpowiedzi konstrukcji wykazującej cechy degradacji
strukturalnej ma praktyczne zastosowanie w układach monitorowania stanu i może
przyczynić się do poprawy bezpieczeństwa lotu śmigłowców oraz zwiększenia efek-
tywności pracy turbinwiatrowychprzy jednoczesnymobniżeniu kosztóweksploatacji.
Wpracyprzedstawiono tzw.Algorytm IdentyfikacjiWłaściwości (Property Identifica-
tionAlgorithm) będącykombinacjąnumerycznej i eksperymentalnej procedury, której
celem jest udoskonaleniemodelu przy ograniczonymzbiorze danych doświadczalnych
do precyzyjnego przewidywania dynamicznej odpowiedzi układu z degradacją struk-
turalną.Algorytmopartonadekompozycjimodalnej i dyskretnychpomiarówekspery-
mentalnych oraz sformułowano dla przypadkuwymuszeń harmonicznych.Wykazano,
że odpowiedzi dynamiczne modelu udoskonalonego pozwala na uzyskanie dokładnej
mapyrzeczywistychodpowiedziwzadanymobszarzeparametrów.Wartykuleopisano
zaproponowany algorytm i przedstawiono kilka przykładów jego weryfikacji.

Manuscript received March 1, 2011; accepted for print April 29, 2011