Detection of damages in simple elements.

Gillich, Gilbert-Rainer ; Birdeanu, Elena Daniela ; Gillich, Nicoleta 等

1. INTRODUCTION

Structures are often affected by damages, which reduce substantially their performances, leading to high risk of hazards. Classical tests on complex structures require a lot of time, involvement of important resources, being consequently expensive. Structural health monitoring systems offer the possibility to diagnose and predict structural failures through embedded sensing, actuation and data management. Thus, operating costs can be reduced along with increased safety. This technology has been quick developed in the last period, especially after the wireless sensors appearance.

The main research directions regarding structural health monitoring which are already in implementation estate use different methods of investigation, namely:

* Vibration based techniques (Fritzen et al., 1998)

* Piezo-electric sensors

* Fiber-optic sensors

* Sensors using electrical resistance

* Sensor using capacitive methods

* Sensors using low frequency electromagnetic sensors (Balageas et al., 2008)

The obtained results using the above mentioned techniques can be processed and used for damage detection and diagnostic in three ways:

* Signal-based in time domain, frequency domain and time-frequency domain;

* Model-based using quantitative mathematical models;

* Model-based using qualitative model rules; The authors present in this paper a statistical model for a

damaged beam developed after a vibration-based technique, this model being able to create the necessary link between the beam frequency and the depth of the damage.

2. EXPERIMENTS AND RESULTS

The experiment aims to provide trustful data, proper for use to build a statistical model which allows to know the depth of the damage in a simple element, fixed to one edge and free to the other one, as showed in figure 1.

In our case, the damage is artificially produced on one of the two horizontal faces of the beam, with a controlled depth d and width a, not allowing the frontal faces of the damage the come in contact one with the other during vibrations.

[FIGURE 1 OMITTED]

The last imposed condition refers to the system's symmetry. As consequence of this condition, the beam's behaviour will not depend on which face (top or bottom) is placed the damage.

If on the free end of the beam act a vertical force F, the beam suffer the deformation [Delta]. By force removal, the beam gets a pseudo-periodic damped motion, characterized by the (pseudo) frequency f until is established the equilibrium. The deformation of the free end S and the frequency f depend, among other physical and mechanical properties of the beam, on the moment of inertia I. The moment of inertia has one value for the undamaged beam, decreasing by appearance of the damage. Therefore, it can be established a relation between the frequency of the free motion of the beam and the depth of its damage d. This relation can be used as a statistical model which described the damage evolution. We have to mention that for elements with different mechanical properties and/or shapes there are different relations (Ghita et al., 2007).

Our concern was to establish a relation for an elastomeric beam with parallelepiped cross section, fixed on one end and free on the other end.

To be able to realise the measurements, we have developed an experimental stand. The system (figure 2), composed by a laptop, a NI cDAQ-9172 compact chassis with NI 9234 modules and a Kistler 8772 accelerometer has been used to acquire the signals. The damages are produces for seven levels of depth, starting with 0,3 mm and ending with 2,1 mm, by 0,3 mm steps. The damages are placed in the middle of the beam, on the upper horizontal face.

The power spectrums for the damaged and the undamaged beams are presented in figure 3.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

It can be observed that the processed damages lead to a frequency decrement, this mechanism being used to prove the existence and propagation of damages.

A series of measurement for five distinct beams have been performed, in order to obtain consistent data for analyze. For all five situations, the frequency decreased from around 42 Hz for the undamaged beam to 30 Hz for the beam with a 2,1 mm depth damage.

The results have been also validated by calculus, using normality tests (Pearson's chi-square [chi square] test). Only few results have been eliminated due to some errors provoked by mechanical causes. The summarized results of the measurements are presented in table 1, being the start point of the statistical model elaboration.

3. THE STATISTICAL MODEL

As method to define the statistical model was choose the regression analysis method. This technique is used for the modelling and analysis of numerical data consisting of values of a dependent (response) variable and of one independent variable. The dependent variable in the regression equation is defined as a function of independent variables, corresponding parameters (constants), and an error term. The error term is treated as a random variable, representing unexplained variation of the dependent variable. Parameters are estimated to give a "best fit" of the data. Most commonly "the best fit" is evaluated using the least squares method, but also have been used other criteria, too.

The obtained values presented in table 1 must be modelled by a function which is a nonlinear combination of parameters (Bethea et al., 1985). Therefore, some types of curves can be considered: exponential, logarithmic, polynomial, etc.

Using the eight values of frequency fn obtained during the measurements performed for the eight levels of damage, results a diagram like the one presented in figure 4. The best fitting regression curve is a forth degree polynomial one, as showed in the equation below.

d = [a.sub.1] x [f.sup.4] - [a.sub.2] x [f.sup.3] + [a.sub.3] - [f.sup.2] - [a.sub.4] - f + [a.sub.5] (1)

For the experiment performed by the authors, the coefficients [a.sub.n] have been determined as following:

[a.sub.1] = 0,000046

[a.sub.2] = 0,004926

[a.sub.3] = 0,179855

[a.sub.4] = 2,272029

[a.sub.5] = 0,029

[FIGURE 4 OMITTED]

Therefore, for already known values of the frequency, which are easy to be measured even on load structures, it is possible to find the depth of a damage situated in the centre of the beam using the relation (1).

It is also possible to find relations for other types of damages, which appear on other places or have other orientation on the beam, by correlating more parameters of the acquired signals (elongation, velocity, acceleration, etc.). This is one direction in which the authors intend to continue their future researches. Another direction for model generalisation is to study the influence of mechanical properties and the geometrical characteristics of the beam.

4. CONCLUSION

Structural health monitoring is a cheep technology and easy to be implemented, able to provide useful information about the structures safety, essential in various domains like aerospace, mechanical or civil engineering.

The model developed by the authors has been validated by numerous measurements, proving a high accuracy of the model.

In order to adapt the model for all types of elements of the structures, has to be introduce additional constants, leading to a compact model able to simulate damages having a large diversity of shapes.

5. REFERENCES

Balageas, D.; Fritzen, C.P. & Guemes, A. (2005). Structural Health Monitoring, ISTE Ltd., ISBN 978-1-905209-01-9, London

Bethea, R.M.; Duran B.S. & Boullion, T.L. (1985). Statistical Methods for Engineers and Scientists, Marcel Dekker Inc., ISBN 0-8247-7227-X, New York

Fritzen, C.P.; Jennewein D. & Kiefer T. (1998). Damage Detection Based on Model Updating Methods, Mechanical Systems & Signal Processing., 12 (1), 1998, pp 163-186, ISSN 0888-3270

Ghita, E.; Gillich, G.R.; Bordeasu, I.; Voda, M. & Troi C. (2007). Aspects concerning the behavior of polymers under stress, Mat. Plast., 44 (2), 2007, pp 158-162, ISSN 0025/5289

Pastrama, S.D.; Hadar, A.; Jiga, G & abara, I. (2008). Numerical study of appearance and development of damages in composite structures, Annals of DAAM for 2008 & Proceedings of 19th International DAAAM Symposium, Katalinic, B. (Ed.), pp 1035-1036, ISSN 1726-9679, Trnava, October 2008

Tab. 1. Correlation between damage depth and frequency

Depth [mm] 0 0,3 0,6 0,9 1,2 1,5 1,8 2,1

Frequency 42 41,3 40,1 39,2 37 34,5 32 30
[Hz]