Detection of Damage in Beam from Measured Natural Frequencies Using Support Vector Machine Algorithm

Prashanth Shyamalaa, SubhajitMondalb, Sushanta Chakraborty*

aGraduate Student; bResearch Scholar, Communicating author; *Associate Professor Department of Civil Engineering, Indian Institute of Technology Kharagpur

bE-mail address:suman.subhajit@gmail.com

Abstract. Damage detection using vibration response is a vibrant area of research. Usually, measured natural frequencies are compared with the natural frequencies computed using a baseline finite element model and the objective function thus formed from the discrepancies of results are minimized using various algorithms. One such algorithm is the Support Vector Machine (SVM) algorithm which uses simulations of various damage scenarios. Damage is envisaged using stiffness loss in a member and the damaged responses are recorded and later on retrieved for comparison. The algorithm is found to be very robust for single damage cases of beam type of structures.

1 Introduction

Damage detection at early stages in structures has become substantial and also vital from serviceability and safety point of view. Many decades of research produced variety of methods to detect damage. Some methods need access to locations of damages in structures which are in most circumstances impractical. This limitation can be overcome if vibration responses of structure were used to identify damage which was extensively studied. The fundamental idea being that damage effects the stiffness, mass or energy dissipation, i.e. the damping properties of a system, which, in turn, alter the measured dynamic responses of that system. Changes in global dynamic response parameters, such as natural frequencies, mode shapes, modal damping factors or modal mass etc. are mostly used for damage detection. Doebling et al.[1] provided a comprehensive survey about various damage detection techniques employing vibration until 90s. Salawu and Williams [2] surveyed damage detection algorithms using natural frequencies alone. Among the various damage detection algorithms, gradient based or evolutionary or heuristic damage detection algorithms are very popular, SVM is one such heuristic based algorithm based on statistical learning mechanism. This algorithm gained popularity in last 4 decades which was developed and conceived by Vapnik [3], the acceptance and applications of which has increased recently [4,5]. The SVM formulation uses the Structural Risk Minimization (SRM) principle, which has been shown to be superior [6] to traditionally used Empirical Risk Minimization (ERM) principle employed by conventional Artificial Neural Network (ANN) algorithm. SRM basically search for an upper bound on the expected risk, whereas ERM minimizes the error on the training data itself. It is this

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difference which makes SVM more general as compared to ANN type of algorithms. In this paper regression form of SVM is used. Damage detection of a cantilever beam using modal displacements are used which showed promising results [7]. Several damage conditions are simulated and the natural frequencies are used to detect damages [8] using SVM. Regression analysis finds a best fit function by minimizing errors for linearly separable data, but it become difficult if the data are not linearly separable. The benefit of SVM is that a linearly non-separable data can be separated by projecting the data space to higher dimensions by adding the kernels function. The kernel functions convert the non-linearly separable data to linearly separable data in higher dimensions where we need not consider about the conversions of feature vectors( here in our case natural frequencies) to higher dimensions In this paper, SVM algorithm is used for damage identification (i.e. its location and severity) of a beam using natural frequencies only. The damage is simulated by reducing the Young’s modulus of the material locally within an elemental volume which indirectly reflects the localized stiffness loss. Numerically simulated data are generated from a converged numerical finite element model of an isotropic rectangular cantilever beam, simulating various damage scenarios and the corresponding natural frequencies are recorded. In this paper, the term measured data was referred to the data obtained from numerical simulations. The detection of damage and its severity of damage of a cantilever beam is demonstrated in the paper. 2 Mathematical Formulations

In the Support Vector Machine (SVM) algorithm natural frequency (x) and damage (y) can be correlated as (y,x) where x= {} is vector of natural frequencies and y being a scalar, depicting damage location or the material parameters [3]. Now this set of data can be regressed to determine or predict the values corresponding to damage location and severity. The approximation is represented by the equation

y= (w.x) + b. (1)

Here w describes the separating hyper-plane equation between the damage or undamaged class. The linear classifier w vector which divides the sample data points into their respective classes. The values of w can be found out as the optimum of y, which is a regression function and can be expressed as

minw,b,ξ,𝜉𝜉∗,𝜖𝜖12𝑤𝑤𝑇𝑇𝑤𝑤 + 𝐶𝐶( ∑ (𝜉𝜉𝑖𝑖𝑙𝑙

𝑖𝑖=1 + 𝜉𝜉𝑖𝑖∗) . (2)

Here, C is a variable parameter that encodes the cost of non-separation in the sample set, and 𝜉𝜉𝜉𝜉∗, 𝜉𝜉𝜉𝜉 are slack variables introduced so that the data could be made

308 Detection of Damage in Beam from Measured Natural Frequencies Using SVM Algorithm

separable. In order to account for the erroneous sample of data, a ε-insensitive loss function is introduced

𝑦𝑦 = �|𝑦𝑦𝑖𝑖 − ({𝑥𝑥}𝑖𝑖 , {𝑤𝑤}𝑖𝑖)|, 𝑖𝑖𝑖𝑖 |𝑦𝑦𝑖𝑖 − ({𝑥𝑥}𝑖𝑖 , {𝑤𝑤}𝑖𝑖)| > 𝜀𝜀𝜀𝜀 , 𝑒𝑒𝑙𝑙𝑒𝑒𝑒𝑒

� . (3)

The data is assumed to be linearly separable for the regression if the xi (natural frequencies) are taken just two. In our sets of problems tackled, it is initially decided that up to 8 natural frequencies be taken for each training sample to have a better representation of pattern in damage is using SVM’s regression model. Analyzing the data in higher dimensions may increase the volume of the sample space to a large extent and eventually the data will become sparse becomes sparse. This issue can be addressed by employing kernels functions in regression to model the sample space into higher dimensions. Usually, Kernels employing radial basis function are used in most cases and the same is employed in the present paper and can be mathematically expressed as

𝐾𝐾(𝑥𝑥,𝑦𝑦) = 𝑒𝑒−�

(𝑥𝑥−𝑥𝑥𝑖𝑖)2

2𝜎𝜎2 � . (4)

Here 𝜎𝜎 is the essential parameter in determination of the SVM regression model. The Cost of non-separation in samples is denoted by C. The ε is taken as 0.5 for the present problem. In this paper the code LIBSVM [9] of MATLAB[10] is used.

3 Results and discussion

3.1 Identifying location of damage and severity of damage in a cantilever beam

A cantilever beam having dimensions 1.0 m x 0.49m x 0.01 m. is considered first. The Young's modulus is assume to be 2E11 Pa and the Poisson’s ratio is considered as 0.28. The density of beam is assumed to be 7860 kg/m3 is to model the beam.

Fig. 1 Cantilever beam with partitions to represent damage locations

The damage location is introduced by creating a partition of size 10cm width and 5cm

depth. the damage was given by reducing the E value for that partition in intervals of

1% cumulatively till 50%. The results were simulated by imparting damage at one

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location for 50 levels of decrease in E value gives 50 samples of 8 natural frequencies.

So, a sum of 350 samples data is obtained for training data. For testing, the above

numerical model is used but damage introduced in other location. For the present

investigation only bending modes are considered as other modes show less influence

on this type of damage. The below figure shows natural frequencies along with their

mode shapes of first few modes.

Fig. 2 mode shapes and natural frequencies for cantilever beam

Along the cantilever beam few locations were chosen for the prediction of location

and Young's modulus (E) values, assuming no noise in the training data. Later random

noise was added to see the accuracy of prediction of values.

Figure 3 shows the pattern indicates natural frequencies is used as training data for the

regression model. Appropriate ‘C’ (cost parameter) and ‘σ’ (gamma) values are

chosen for the SVM regression model generations. There is no universal rule to select

the best parameters for SVM analysis and determined by trial and error. The values of

C and σ assume to be 512 and 0.5 respectively for the present case. The generated

model is tested on the values which were not in the training dataset. Table 1 shows the

prediction of the damage location and its percentage error. Table 2 shows the

prediction of Young's modulus obtained from the SVM algorithm for different level of

noise.

310 Detection of Damage in Beam from Measured Natural Frequencies Using SVM Algorithm

Fig 3. Percentage variation in natural frequency values (1 to 8) for 5 levels of reduction in E

value (5%,10%,20%,30%,40%) for damage location at (a) 5cm (b) 15cm (c) 25cm (d) 35cm

from fixed end of the cantilever beam.

Table 1. Prediction of damage location using SVM

Actual value (cm) Predicted value (cm) Error % 60 60.38 0.6 20 17.40 9.18

32.75 32.04 2.0 53.75 52.47 1.6

Table 2. Detection of severity of damage in the beam

Damage located at 56 cm from support

(actual Young's modulus 1.4E11) Damage located at 69 cm from support

(actual Young's modulus 1.4E11) noise (%) predicted value (1xE11) noise (%) predicted value

(1xE11) 1 1.3825 1 1.3969 2 1.3972 2 1.3964 4 1.3969 4 1.3894 8 1.3964 10 1.3885

10 1.3955 15 1.3871 15 1.3952 20 1.3824

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4 Conclusions

The SVM algorithm is found to be performing excellently to detect damages in various locations of an isotropic cantilever beam from measured natural frequencies only. The uniqueness of the detection with varying location and severity of damages are preserved, thus giving confidence about its future application. However, the algorithm has some difficulty in detecting multiple damages from measured natural frequencies alone and incorporation of more information in the form of frequency response functions, mode shapes etc. may be more appropriate. References 1. Doebling SW, Farrar CR and Prime MB. A summary review of vibration based damage

identification methods, Shock and Vibration Digest, 30(2):91-105.1998. 2. Salawu OS, Williams C. Damage location using vibration mode shapes, in: Proceedings of

the SPIE, vol.2251, Proceedings of the 12th International Modal Analysis Conference, pp.9 33– 941,1994.

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4. Charles R. Farrar, Keith Worden, Structural Health Monitoring: A Machine Learning Perspective by,ISBN: 978-1-119-99433-6,November 2012.

5. Charles RF, Hoon S, and Scott W. Doebling. Statistical Pattern Recognition, The 13thInternational Congress and Exhibition on Condition Monitoring and Diagnostic Engineering Management (COMADEM 2000), Houston, TX, USA, December 3-8, 2000.

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7. Satpal, SB, Guha A and Banerjee S. Damage identification in aluminum beams using support vector machine: Numerical and experimental studies. Structural. Control Health Monitoring., doi:10.1002/stc.1773.2015.

8. Burges C. A tutorial on support vector machines for pattern recognition”, In “Data Mining and Knowledge Discovery”. Kluwer Academic Publishers, Boston, (Vol. 2). 1998

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