Fault Identification in Structures in the Presence of Missing Data
Tshilidzi Marwala and Lungile Mdlazi School of Electrical and Information Engineering, University of the Witwatersrand, P/Bag 3, Wits, 2050, South Africa, E-mail:
[email protected]
Abstract This paper introduces a method for classification of faults in mechanical systems in the presence of missing input data entries. The method is based on auto-associative neural networks, where the input and the output of the network are the same. From the trained network, an error equation with missing inputs as design variables is constructed. A genetic algorithm is then used to solve for the missing input data entries. The proposed method is tested on a fault classification problem in a population of cylindrical shells. It was found that the proposed method is able to estimate missing data entries to an average accuracy of 92.5%. Furthermore, the proposed method results in a classification accuracy of 94% when used with a database that has missing input data entries, while the full database set results in a classification accuracy of 96%. 1. Introduction Vibration data has been used with varying degrees of success to identify faults in structures. This work includes work conducted by Worden et al. [1], who used neural networks to identify damage in a twelve-member structure. The input to the neural network was the strain measurements from members of the structure. Leath and Zimmerman [2] successfully applied neural networks to identify damage on a four-element cantilevered beam. Marwala [3] used probabilistic neural networks to classify faults in structures and more examples on the use of neural networks for damage detection may be found in [4]. When neural networks are applied in real life, one of the main problems encountered is the failure of sensors. If one of the sensors fails, the neural network is unable to make a decision because of the incomplete input data set. This problem is normally addressed by using an average value calculated from the unavailable sensor’s history data over some defined period. In this paper a new method of estimating missing data entries in the input database is proposed. This method is based on auto-associative models combined with genetic algorithms. The proposed method is tested on fault classification in a population of cylindrical shells. 2. Mathematical Background 2.1. Neural Networks In this study an auto-associative neural network is constructed. Auto-associative neural networks are neural networks that have the same input and output. The multi-layer perceptron (MLP) neural network is used to construct the auto-associative neural network. The MLP architecture contains a hyperbolic tangent basis function in the hidden units and linear basis functions in the output units [5]. A schematic illustration of the MLP is shown in Figure 1. The relationship between the output y and input x can be written as follows [5]:
yk = Here,
M j =0
w
(2) kj
tanh
d i =0
w(ji1 ) xi
(1)
w(ji1 ) and w(ji2 ) indicate weights in the first and second layer, respectively, going from input i to hidden unit j,
M is the number of hidden units, d is the number of output units.
Output Units
yc
y1
z0
z1
zM
Hidden Units
bias
xd
x1
x0
Input Units Figure 1. Feed-forward network having two layers of adaptive weights The model in Figure 1 is able to take into account the intrinsic dimensionality of the data. Models of this form can approximate any continuous function to arbitrary accuracy if the number of hidden units M is sufficiently large. Training the neural network identifies the weights in equations 1 and in this paper scaled conjugate gradient method [6] is used in the training. 2.2. Auto-associative Networks and Missing Data Auto-associative neural networks are models in which the neural network is trained to recall its inputs. This means that whenever an input is presented to the network the output is equal to the input. Equation 1 may thus be rewritten in simplified form as:
{ y } = f ({ w },{ x })
(2)
Here {y} is the output vector, {x} is the input vector, f is a function and {w} is the mapping weight vector. For an auto-associative networks {x}={y} and therefore Equation 2 may be re-written as follows:
{ x } = f ({ w },{ x })
(3)
For a perfectly mapped system, Equation 3 holds, however, for a realistic mapping there will be some error. Equation 3 is therefore be re-written as:
{ e } = { x } − f ({ w },{ x })
(4)
The sum of squares of both the left hand side and the right hand side of Equation 4 will give:
{E}=
M i =1
({ x } − f ({ w },{ x }))2
(5)
Here M is the size of the input vector. For a situation where some of the input values are unknown, the input data may be divided into known xk and unknown components xu and thus Equation 5 may be written as follows:
{E}=
M
xu
i =1
xk
−f
{w},
xu
2
xk
(6)
From Equation 6, the unknown component xu data is estimated from the known component xk by minimizing the error in equation 6. It is absolutely important that a global minimum error be achieved because a local minimum error results with the incorrect estimation of the unknown component xu. In this study global optimum method genetic algorithm is used to find the global optimum solution. In the next section genetic algorithm is explained. 2.3. Genetic Algorithms (GA) The GA was inspired by Darwin’s theory of natural evolution [7, 8]. Genetic algorithm is a simulation of natural evolution where the law of the survival of the fittest is applied to a population of individuals. This natural optimization method is used for optimization, and in this paper, to minimize the error in Equation 6. The GA is implemented by generating a population and creating a new population by performing the following procedures: (1) crossover; (2) mutation; (3) and reproduction [7]. The crossover operator mixes genetic information in the population by cutting pairs of chromosomes at random points along their length and exchanging over the cut sections. This has a potential of joining successful operators together. Arithmetic crossover technique [7, 8] is used in this paper. The mutation operator picks the chromosomes at random and inverts it. This has a potential of introducing to the population new information. In this paper, non-uniform mutation [7, 8] is used. Reproduction takes successful chromosomes and reproduces them in accordance to their fitness functions. In this paper normalized geometric selection method is used [7, 8]. 2.4. Dynamics In this study, modal properties i.e. natural frequencies and mode shapes are used for fault classification. For this reason, these parameters are described in this section. Modal properties are related to the physical properties of the structure. In the time domain, all elastic structures may be described in terms of their distributed mass, damping and stiffness matrices through the following expression [9]:
[ M ]{ X ' ' } + [ C ]{ X ' } + [ K ]}{ X } = { F }
(7)
Here [M], [C] and [K] are the mass, damping and stiffness matrices respectively, and {X}, {X′} and {X′′} are the displacement, velocity and acceleration vectors, respectively, while {F} is the applied force vector. When th Equation 7 is transformed into the modal domain to form an eigenvalue equation for the i mode, then [9]: 2
( −ω i [ M ] + jω i [ C ] + [ K ]){ φ }i = { 0 } where
(8)
j = − 1 , ω i is the ith complex eigenvalue, with its imaginary part corresponding to the natural frequency
ωi, {0} is the null vector, and { φ }i is the ith complex mode shape vector with the real part corresponding to the normalized mode shape {φ}i. From Equation 8, it may be deduced that the changes in the mass and stiffness matrices cause changes in the modal properties of the structure. Therefore, the modal properties can be identified through the identification of the correct mass and stiffness matrices. 3. Experimental setup: Cylindrical structure In this section the experimental setup used to validate the proposed procedure is presented. The experiment was performed on a population of cylinders. The cylinders were supported by inserting a sponge rested on a bubblewrap, to simulate a ‘free-free’ environment [see Figure 2] and the details of this may be found in [10].
Accelerometer
Fault Substructure 1
Substructure 3 Substructure 2
Excitation position
Sponge
Bubble wrap
Figure 2. Illustration of a cylindrical shell showing the positions of the impulse, accelerometer, substructures, fault position and supporting sponge Conventionally, a ‘free-free’ environment is achieved by suspending a structure with light elastic bands. A ‘freefree’ environment was implemented so that rigid body modes, which do not exhibit bending or flexing, could be identified. These modes occur at a frequency of 0Hz and can be used to calculate the mass and inertia properties. In this study, the interest was not on the rigid body modes. Testing the cylinders suspended is approximately the same as testing it while resting on a bubble-wrap, because the frequency of cylinder-on-wrap was below 100Hz. The first natural frequency of cylinders was over 300Hz, which was several orders of magnitude above the natural frequency of a cylinder on a bubble-wrap. This means that the cylinder on the wrap was effectively de-coupled from the ground. It should be noted that the use a bubble-wrap introduced damping to the structure but the damping was small and therefore did not influence the identification of modes. The impulse hammer test was performed on each of the 20 steel seam-welded cylindrical shells (1.75 ± 0.02mm thickness, 101.86 ± 0.29mm diameter and of height 101.50 ± 0.20mm). The impulse was applied at 19 different locations as indicated in Figure 2: 9 on the upper half of the cylinder and 10 on the lower half of the cylinder. A sponge was inserted inside the cylinder to control boundary conditions. Control of the boundary conditions was achieved by rotating the sponge every time a measurement was taken. The top impulse positions were located 25mm from the top edge and the bottom impulse positions were located 25mm from the bottom edge of the cylinder. The angle between two adjacent impulse positions was 36°. Each cylinder was divided into three equal substructures and holes of 10-15 mm in diameter were introduced at the centers of the substructures to simulate faults. For one cylinder, the first type of fault was a zero-fault scenario. This type of fault was given the identity [0 0 0], indicating that there were no faults in any of the three substructures. The second type of fault was a one-fault-scenario, where a hole was located in any of the three substructures. The other possible one-fault-scenarios were [1 0 0], [0 1 0], and [0 0 1] indicating a hole in substructures 1, 2 or 3 respectively. The third type of fault was a two-fault scenario, where one hole was located in two of the three substructures. The three possible two-fault-scenarios were [1 1 0], [1 0 1], and [0 1 1]. The final type of fault was a three-fault-scenario, where a hole was located in all three substructures, and the identity of this fault was [1 1 1]. This resulted in a total of 8 different types of fault-scenarios.
Because the zero-fault scenarios and the three-fault scenarios were over-represented, twelve cylinders were picked at random and additional one- and two-fault cases are measured after increasing the magnitude of the holes. This was done before the next fault case was introduced to the cylinders. Only a few fault-cases were selected because of the limited computational storage space available. For each fault-case, acceleration and impulse measurements were taken. The types of faults that were introduced (i.e. drilled holes) did not influence the damping. Each cylinder was measured three times under different boundary. The number of measurements sets taken for undamaged population was 60 (20 cylinders × 3 for different boundary conditions). All the possible fault types and their respective number of occurrences are listed in Table 1. Table 1. The number of different types of fault-cases generated Fault [0 0 0] [1 0 0] [0 1 0] [0 0 1] [1 1 0] Number 60 24 24 24 24
[1 0 1] 24
[0 1 1] 24
[1 1 1] 60
The impulse and response data were processed using the Fast Fourier Transform (FFT) to convert the time domain impulse history and response data into the frequency domain. The data in the frequency domain were used to calculate the FRFs. The sample FRF results from an ensemble of 20 undamaged cylinders are shown in Figure 3. This figure indicates that the measurements were generally repeatable at low frequencies and not repeatable at high frequencies. Axisymmetric structures such as cylinders have repeated modes due to their symmetry [11]. The presence of an accelerometer and the imperfection of cylinders destroyed the axisymmetry of the structures. The incidence of repeated natural frequencies was destroyed making the process of modal analysis easier to perform [12]. 5
10
4
10
3
10
2
Inertance (m/s 2/N)
10
1
10
0
10
−1
10
−2
10
−3
10
−4
10
0
500
1000
1500
2000 2500 3000 Frequency (Hz)
3500
4000
4500
5000
Figure 3. The measured frequency response functions from a population of cylinders
4. Testing the Proposed Procedure From the data measured in Section 3, 10 parameters were selected. The auto-associative network with 10 inputs and 10 outputs was constructed and it was found that 10 hidden units provided the optimal network. The autoassociative network was trained using scaled conjugate method. After network training, one of the inputs was assumed to be unknown and then estimated using the genetic algorithm. The genetic algorithm used arithmetic cross-over, non-uniform mutation and normalized geometric selection as described before. The genetic algorithm had a population of 20 and was run for 25 generations. The missing data were estimated to 92.5% accuracy. The estimated values were used in the classification of faults and a fault classification accuracy of 94% was observed. When the complete database was used, a fault classification accuracy of 96% was achieved. When average values were used to replace of the missing entries, the fault classification accuracy was 75%. 5. Conclusion In this study, a method based on auto-associative neural networks and genetic algorithms is proposed to estimate missing input data entries. This procedure was tested on a population of cylindrical shells. Missing entries were estimated at an average accuracy of 92.5%. When the estimated entries were used to classify faults in structures it is was found that the classification accuracy is 94% while the original data gave the accuracy of 96%. References
[1] Worden, K.A., Ball, A., and Tomilinson, G., “Neural networks for fault location” In Proceedings of the 11th International Modal Analysis Conference, pp. 47-54, 1993.
[2] Leath, W.J. and Zimmerman, D.C., (“Analysis of neural network supervised training with application to
structural damage detection” Damage and Control of Large Structures, Proceeding of the 9th Virginia Polytechnic Institute and State University Symposium, pp. 583-594, 1993. [3] Marwala, T., “Probabilistic fault identification using a committee of neural networks and vibration data” American Institute of Aeronautics and Astronautics, Journal of Aircraft, 38(1), pp. 138-146, 2001. [4] S.W. Doebling, C.R. Farrar, M.B. Prime, D.W. Shevitz, Damage identification and health monitoring of structural and mechanical systems from changes in their vibration characteristics: a literature review Los Alamos National Laboratory Report LA-13070-MS, 1996. [5] Bishop, C.M., Neural Networks for Pattern Recognition. Oxford University Press, Oxford, UK, 1995. [6] Møller, M., “A scaled conjugate gradient algorithm for fast supervised learning” Neural Networks, Vol. 6, pp. 525-533, 1993. [7] Holland J., Adaptation in Natural and Artificial Systems, Ann Arbor: University of Michigan Press, 1975. [8] D.E. Goldberg, Genetic algorithms in search, optimization and machine learning, Addison-Wesley, Reading, MA, 1989. [9] D.J. Ewins, Modal Testing: Theory and Practice, Research Studies Press, Letchworth, U.K, 1995. [10] T. Marwala. Fault identification using neural networks and vibration data. Ph.D. Thesis, Department of Engineering, University of Cambridge, UK, 2000. [11] Royton, T.J., Spohnholtz, T., and Ellington, W.A., “Use of non-degeneracy in nominally axisymmetric structures for fault detection with application to cylindrical geometries” Journal of Sound and Vibration, Vol. 230, pp. 791-808, 2000. [12] Maia, N.M.M., Silva, J.M.M., and Sampaio, R.P.C., “Localization of damage using curvature of the frequencyresponse-functions” In Proceedings of the 15th International Modal Analysis Conference, pp. 942-946, 1997.
