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. Detection of Damage in Beam from Measured Natural
Frequencies Using Support Vector Machine Algorithm
Prashanth Shyamalaa
, SubhajitMondalb
, Sushanta Chakraborty*
a
Graduate Student; b
Research Scholar, Communicating author; *
Associate Professor
Department of Civil Engineering, Indian Institute of Technology Kharagpur
b
E-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∗,𝜖𝜖
1
2
𝑤𝑤 𝑇𝑇
𝑤𝑤 + 𝐶𝐶( ∑ (𝜉𝜉𝑖𝑖
𝑙𝑙
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

3. 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)| > 𝜀𝜀
𝜀𝜀 , 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒
. (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−𝑥𝑥 𝑖𝑖)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.

5. 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.
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