Abstract This paper presents a theoretical derivation and reports on a numerical verification of a model-free method for identification of added masses in truss structures. No parametric numerical model of the monitored structure is required, so that there is no need for initial model updating and fine tuning. This is a continuation and an improvement of a previous research that resulted in a time-domain identification method, which was tested to be accurate but very time-consuming. A general methodology is briefly introduced, including the inverse problem, and a numerical verification is reported. The aim of the numerical study is to test the accuracy of the proposed method and its sensitivity to various parameters (such as simulated measurement noise and decay rate of the exponential FFT window) in a numerically controlled environment. The verification uses a finite element model of the same real structure that was tested with the time-domain version of the approach. A natural further step is a lab verification based on experimental data

Grzegorz Suwała

Jan Biczyk

Łukasz Jankowski

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IPPT Reports 4f/2013                                                                                                   1 
 
Nonparametric identification of added masses 
in frequency domain: a numerical study 
 
Grzegorz Suwała1, Jan Biczyk2, Łukasz Jankowski1,* 
 
1
Institute of Fundamental Technological Research (IPPT PAN) 
Department of Intelligent Technologies 
ul. Pawińskiego 5b 
02-106, Warsaw, Poland 
2
Adaptronica sp.z o.o. 
ul. Szpitalna 32 
05-092 Łomianki k. Warszawy 
*
ljank@ippt.pan.pl 
 
received DD MMM 20xx, in final form DD MMM 20xx, published online DD MMM 20xx 
 
 
Abstract 
This paper presents a theoretical derivation and reports on a numerical 
verification of a model-free method for identification of added masses in truss 
structures. No parametric numerical model of the monitored structure is 
required, so that there is no need for initial model updating and fine tuning. 
This is a continuation and an improvement of a previous research that resulted 
in a time-domain identification method, which was tested to be accurate but 
very time-consuming. A general methodology is briefly introduced, including 
the inverse problem, and a numerical verification is reported. The aim of the 
numerical study is to test the accuracy of the proposed method and its 
sensitivity to various parameters (such as simulated measurement noise and 
decay rate of the exponential FFT window) in a numerically controlled 
environment. The verification uses a finite element model of the same real 
structure that was tested with the time-domain version of the approach. 
A natural further step is a lab verification based on experimental data. 
 
IPPT Reports 4f/2013                                                                                                   2 
 
Introduction 
This paper presents a derivation and reports on a numerical verification of 
a frequency-domain version of a nonparametric approach to identification of added 
masses in truss structures. The general approach has been recently developed in 
IPPT PAN [1–3], and it is based on the essentially nonparametric methodology of 
the virtual distortion method (VDM) [4]. The monitored structure is characterized in 
a purely experimental way, by means of its impulse response functions. As a result, 
no parametric numerical modelling is required, which obviates the need for model 
updating and fine-tuning that is typical for other model-based methods.  
Most of the low-frequency identification methods used in global structural 
health monitoring (SHM) can be classified into two general groups: (1) Model-based 
methods that rely on a parametric numerical model of the monitored structure [5,6]. 
Their appealing feature is the physicality of the model and its identified 
modifications. However, an accurate parametric model is not easy to obtain and 
update. (2) Pattern recognition methods, which rely on a database of numerical 
fingerprints that are extracted from the experimentally measured responses [7]. No 
parametric modeling is required. The identification rarely goes beyond detection or 
approximate localization of the modification.  
The developed approach exploits the advantages of both groups: it makes use of 
a nonparametric model of the monitored structure based on experimentally measured 
data, but it allows parametrically expressed modifications to be identified. In [1], 
a time-domain version of the approach has been proposed and experimentally 
verified. It proved to be accurate, and thanks to the iterative CGLS solution scheme, 
provided a good control over numerical regularization of the computed time-domain 
response [8]. However, the fundamental equation is a system of linear integral 
equations of the Volterra type, whose solution is significantly time-consuming. This 
paper develops and verifies a frequency-domain approach, which uses the fast 
Fourier transform (FFT) to solve the equations. A significant reduction in 
computation time is attained (up to four orders of magnitude). However, this is at the 
cost of losing the control of the process of numerical regularization: regularization in 
frequency domain seems to be relatively weakly researched and understood. The 
aim of this study is to test the accuracy of the proposed method, as well as its 
sensitivity to various parameters, such as the decay rate of the exponential FFT 
window (which seems to play the role of the regularization parameter) and the 
simulated measurement noise. The verification uses a finite element model of the 
same real structure that was tested with the time-domain version of the approach. 
 3 
Nonparametric modeling and identification 
The direct problem 
Mass modifications are modeled with the equivalent pseudo-loads that act in the 
involved degrees of freedom (DOFs) of the original unmodified structure. The 
influence of the pseudo-loads on the response is computed using a convolution with 
the experimentally obtained local impulse-responses. The original, time-domain 
solution presented in [1] is transferred here to frequency domain, which converts the 
original Volterra integral equation into a series of simple decoupled linear equations. 
The problem is formulated in terms of the finite element method. The 
unmodified structure is assumed to be linear and to satisfy the equation of motion: 
                              (1) 
where M, C and K denote respectively the structural matrices of mass, damping and 
stiffness, f(t) is the external testing excitation and x
L
(t) denotes the corresponding 
response of the unmodified structure (reference response). In frequency domain, 
(1) takes the following quasi-static form: 
                 (2) 
where   is the angular frequency,       and      denote respectively the complex 
amplitudes of the reference response and the excitation, and D( ) is the complex 
dynamic stiffness matrix, 
                 , (3) 
whose inverse             is called the dynamic compliance matrix and 
allows (2) to be represented in the direct form: 
                          . (4) 
The added masses,               , are represented in terms of the 
modification   (m) to the original mass matrix  . The mass matrix   of the 
modified structure is thus given by 
          , (5) 
while its dynamic stiffness matrix can be expressed as 
                      (6) 
As a result, the response      of the modified structure (the original structure with 
added masses) satisfies the following counterpart of (2): 
                     , (7) 
where the vector        denotes the pseudo-loads that act in the involved DOFs of 
the unmodified structure to model the inertial effects of the added masses, 
                   . (8) 
Equations (7), (4) and (8) yield: 
                      . (9) 
 4 
Equation (9) can be substituted into (8) to yield the following linear equation: 
                                   , (10) 
where I is the identity matrix of the appropriate dimensions. Notice that       is 
a diagonal matrix with non-vanishing entries only in the DOFs that correspond to 
the added masses. As a result, the pseudo-loads        vanish in all other DOFs 
and (10) is reduced to a small system with dimensions 3N x 3N, where N is the 
number of the added masses. Similarly, only the corresponding small submatrix of 
the full dynamic compliance matrix     . Based on (9) and (10), the direct problem 
is easily solved: for each angular frequency   and vector m of the added masses, 
(10) is solved and the resulting pseudo-loads are substituted into (9) to compute the 
corresponding response of the modified structure. 
The inverse problem 
The inverse problem is formulated in the form of an optimization problem of 
minimization of a certain objective function with respect to the vector m. In general, 
any objective function that expresses a discrepancy between the modeled response 
     of the modified structure and its actually measured response       is 
plausible. Here, the following form is used: 
      
                   
              
, (10) 
where            is an indicator function of the considered frequency range, 
which should coincide with the frequency range of the testing excitation f(t). 
Numerical verification 
The verification uses a model of the setup tested experimentally in [1], see 
Fig. 1. Each element cross-sectional area is 66 mm
2
, the density is 7800 kg/m
3
 and 
the Young modulus is 210 GPa. The sampling frequency is 65.5 kHz, all the sensors 
are accelerometers and the time interval of 230 ms is sampled. Four different masses 
(1.36 kg, 2.86 kg, 3.86 kg, 5.36 kg) are added in turn to one of the nodes M1, M2 
or M3. The location of the modification is assumed to be known. The identification 
is performed separately for each of the 12 considered modification cases, and the 
identification range is 0 kg to 10 kg with the step of 0.01 kg. As the testing 
excitation an impact by a modal hammer is used. Its placement is selected so that it 
excites at least two bending vibration modes. The objective function is based on the 
frequency range up to 485 Hz, which has been determined by including the spectral 
fringes (of the impact) with the energy above 10% of the maximum fringe energy. 
 5 
 
Figure 1. Truss structure used in the numerical example. 
 
Figure 2(a) shows the dependence of the root mean square relative identification 
error on the decay rate of the FFT exponential window (defined by the value of the 
window at the end of the time interval) and on the simulated measurement noise (per 
cent of the signal rms, applied to the accelerations and excitation). Figure 2(b) plots 
typical objective functions (decay rate 10
-5
, simulated measurement error 10% rms). 
 
 
 
(a) (b) 
Figure 2. (a) Root mean square relative identification errors in dependence on the decay 
rate of the FFT exponential window and the simulated measurement noise. (b) Typical 
objective functions (FFT window decay rate 10
-5
, measurement noise 10% rms). 
Conclusions 
The numerical study confirms that the developed frequency-domain version of 
the identification method is accurate and relatively insensitive to (simulated) 
measurement noise. The decay rate of the exponential window used with FFT seems 
to play the role of the regularization parameter. The method performs up to four 
 6 
orders of magnitude faster than its previously developed time-domain counterpart. 
A lab verification based on experimental data is the natural further step. 
Acknowledgments 
Financial support of Structural Funds in the Operational Programme–Innovative 
Economy (IE OP) financed from the European Regional Development Fund, 
Projects “Innowacyjne technologie dla poprawy bezpieczeństwa małego lotnictwa 
SWING (Safe-Wing)” (POIG.01.04.00-14-100/09, POIG.04.01.00-14-100/09) and 
“Modern material technologies in aerospace industry” (PKAERO, POIG.0101.02-
00-015/08), and of the Polish National Science Centre Project “AIA” (DEC– 
2012/05/B/ST8/02971) is gratefully acknowledged. 
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Nonparametric identification of added masses in frequency domain: a numerical study

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Nonparametric identification of added masses in frequency domain: a numerical study

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