An artificial neural network application to fault ...

An artificial neural network application to fault detection of a rotor bearing system Hamdi Taplak Vocational School of Kayseri, Erciyes University, Kayseri, Turkey, and

I˙brahim Uzmay and S¸ahin Yıldırım Department of Mechanical Engineering, Engineering Faculty, Erciyes University, Kayseri, Turkey Abstract Purpose – To improve the application neural networks predictors on bearing systems and to investigate the exact neural model of the ball-bearing system. Design/methodology/approach – A feed forward neural network is designed to model-bearing system. Two results are compared for finding the exact model of the system. Findings – The results of the proposed neural network predictor gives superior performance for analysing the behaviour of ball bearing undergoing loading deformation. Research limitations/implications – The results of the proposed neural network exactly follows desired results of the system. Neural network predictor can be employed in practical applications. Practical implications – As theoretical and practical study is evaluated together, it is hoped that ball-bearing designers and researchers will obtain significant results in this area. Originality/value – This paper fulfils an identified research results need and offers practical investigation for an academic career and research. Also, It should be very helpful for industrial application of ball-bearing systems. Keywords Bearings, Neural nets Paper type Research paper W Ijk

Nomenclature V e rC rGC rG m R Fs FU k z yst vn n h m N RMSE Zm i g(z) yi xI

¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼

angular velocity (1/s) eccentricity (mm) instant position of C instant position of G as relative instant position of G mass (gr) combine force static force vector external exciting force stiffness coefficient (N/m) displacement at Z axis (mm) static deflection (mm) natural frequency of the system rotating speed of the shaft learning rate of the NN momentum term of the NN iteration number of the NN, during training root mean square error activation function for the ith neuron in the mth the activation function in the hidden layer the ith output of a network the input vector to the network

bH j J dm i

¼ the weight of the connection between the kth neuron of the hidden layer ¼ the bias of the jth neuron in the hidden layer ¼ The error function ¼ The error signal of the ith neuron in the mth layer

1. Introduction The use of machine condition monitoring can provide considerable cost savings in many industrial applications especially where large rotating machines are involved, for example, generators in power stations. The monitoring of vibrations of these machines has been reported as being a useful technique for the analysis of their condition although many other machine parameters such as motor current, temperature or acoustic emission can be measured (Schoen et al., 1995). By using machine condition monitoring, faults in the machinery can be detected without having to shut down for maintenance. Maintenance can then be planned for the most convenient time to minimize the loss of revenue due to the machine being off-line. The use of scheduled maintenance with no indication of machine condition also can introduce faults in the machine which did not exist before the servicing: the replacing of good components with faulty ones would be detected using machine condition monitoring. In order to analyse the transient response of the system many condition monitoring techniques require extensive analyses of large amounts of data often collected when the machine is switched on or off. Sometimes the steady state conditions are estimated by measuring the machine vibrations for a short time. These data can be used to produce estimates

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Industrial Lubrication and Tribology 58/1 (2006) 32– 44 q Emerald Group Publishing Limited [ISSN 0036-8792] [DOI 10.1108/00368790610640082]

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An artificial neural network application Hamdi Taplak, I˙brahim Uzmay and S¸ahin Yıldırım

Industrial Lubrication and Tribology Volume 58 · Number 1 · 2006 · 32 –44

2. Vibration-based fault diagnosis of ball-bearing system

of the vibration spectra. Analysis of this information by a human expert can give indications of many faults. There are several advantages in automating this process, however these processes are still time consuming and consequently analysis in real-time is not possible (Schoen et al., 1995). With an online monitoring system, the machine is continuously monitored. This has two significant advantages in fault detection: firstly, short non-stationary effects, which periodical analysis could miss, will be detected; secondly, a fault which could have catastrophic effects in a short time could be detected and the machine can be shut down before serious damage occurs. If the machine can be run in a variety of no-fault conditions, the vibration monitoring system could also be used to validate the control system. The disadvantage of real-time monitoring algorithms for determining the machine condition is that they are limited by time constraints; often just calculating a time average of a parameter and comparing to a threshold. However, a realtime system could be used to detect anomalies and this data could then be passed on for more detailed analysis using more powerful, but slower analysis techniques (Mayes, 1994). Machine condition monitoring requires the recognition of patterns in noisy data. Feed-forward artificial neural networks (ANNs) have been used in a wide variety of pattern recognition applications including vibration monitoring. Their ability as a universal approximation allows the transformation of a set of input features for which the condition classes may be separated by a highly non-linear boundary to a set of outputs which can be easily separated with very little computational cost. This transformation can be trained using input features for known conditions and by minimizing the mean squared error for the output. This yields an optimal Bayes classifier (Haykin, 1994). Recently, fault detection and diagnosis has become an important area of investigations. An ANN design for fault identification in a rotor-bearing system has been developed (Vyas and Satishkumar, 2001). In that article, a neural network algorithm which was knowledge-based system has been used. Also, adaptability of various neural network architecture has been utilized to predict the vibration parameters of the system. ANNs have been used as non-linear autoregressive models of vibrations of rolling element bearings allowing fault diagnosis by comparing the mean square value of the prediction error for models of vibration under known machine conditions (Baillie and Mathew, 1996). Consequently, non-linear autoregressive modelling was applied to the vibrations of a rotating shaft, and an analysis of prediction error was extended to cover its higher-order uncorrelated statistical properties. In this paper, an experimental analysis and NN predictor investigation for vibration analysis of rotating shaft is presented. Vibration data from a rotating shaft under two different load conditions were employed for training and testing NN predictor. The paper is organized as follows; fault diagnosis of rotating-bearing system is described in Section 2. Section 3 presents the model of the system in detail. The dynamic structure of the system is represented in Section 4. The fundamentals of the experimental investigation are given in Section 5. Simulation with neural predictor is pointed out in Section 6. The simulation results for both approach is given in Section 7. The paper is concluded with Section 8.

During the course of the last half-century, rotating machineries have been studied in more and more detail. A through understanding of the principles of rotor dynamics is essential for engineers and scientists involved in the design and manufacturing of the transmission rotor or shaft for such system as turbine-generator system, vehicle motor driving system, etc. as well as many other fields, on which we find ourselves relying on to an increasing extent. Several modern dynamics texts also contain sections devoted solely to rotordynamics, a good example being that of Genta (1993). Vibration condition monitoring as helpful way to fault diagnosis has been examined (Taylor, 1995). Taylor also included information on the actual data analysis process – how the measured data should be processed for a diagnosis to be performed. Smalley et al. (1996) have presented a method for assessing the severity of vibration in terms of the probability of damage by analysis of vibration signals and its related cost, using the net present value method. The question of whether or not to shut down a machine for the purpose of maintenance was considered, and some guidelines were formulated comparing maintenance and down-time costs against the possible costs that would be incurred by damage. As condition monitoring systems become increasingly elaborate and complicated, the analysis of data provided by these systems also becomes more in-depth and involved. By using the Pareto distribution method for machinery diagnostic tests, Cempel (1991) showed that the method for the condition monitoring of tribovibroacoustical processes could be generalised for vibration processes and so used in vibration condition monitoring. The reliability graph drawn from vibration measurements can be easily transformed to a life-curve for a given machine, so that the Pareto distribution lends itself to assessing both the condition of a machine and its residual time to breakdown. For the diagnosis of anisotropy and asymmetry in rotating machinery, Lee et al. (1990) developed a method incorporating directional frequency response functions (dFRFs). Anisotropy and asymmetry may cause whirl, fatigue and instability, as well as influencing system characteristics such as unbalance and critical speeds. Complex modal testing was used to estimate the dFRFs. An example was presented, showing the proposed method to be very efficient in identifying anisotropy and asymmetry.

3. System modelling Let us consider rotor shown in Figure 1, which rotates with a constant angular speed V. The shaft of the rotor S is Figure 1 Schematic representation shaft-bearing system

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An artificial neural network application

Industrial Lubrication and Tribology

Hamdi Taplak, I˙brahim Uzmay and S¸ahin Yıldırım

Volume 58 · Number 1 · 2006 · 32 –44

supported rigidly at its ends. Assume that the shaft can be considered massless and flexible whereas the element E can be approximated by a particle of mass m. This particle is attached to the shaft at the centre of gravity G of the element E. The centre of gravity G is displaced by e from the geometrical centre of the shaft cross-section C. The distance e represents imbalance of the element E and can be considered as a small magnitude. To analyse motion of this system, let us introduce the inertial system of coordinates XY Z as it is shown in Figure 2. The instantaneous position of the centre C is determined by the position vector rC. The centre of gravity G rotates with respect to this centre with the angular velocity V. Since the angular velocity is constant, the relative instantaneous position of the centre of gravity G is determined by the angle Vt and the imbalance e (vector rGC). The absolute position of the centre of gravity G, in Figure 2 is denoted by rG. The vector Fs represents the static resultant force acting on the element E. R stands for the interaction force between the element being considered and the shaft (Krodkiewski, 2000).

€ þ kX ¼ F X s mX mY€ þ kY ¼ F Y s

yields the equilibrium position (Xs; Ys). Upon assuming the particular position in form: X ¼ Xs Y ¼ Ys

The total deflection of the shaft X, Y are sum of the static deflection Xs; Ys and the dynamic deflection x, y (Figure 3). X ¼ Xs þ x Y ¼ Ys þ y Introduction of equation (8) into the mathematical model 2.4 in equations which govern the:

Motion of the centre of gravity G is governed by the Newton law:

m€y þ ky ¼ meV2 sin Vt

ð9Þ

x€ þ v2n x ¼ q cos Vt

ð10Þ

y€ þ v2n y ¼ q sin Vt

ð11Þ

or:

ð1Þ

where according to Figure 2 where:

r G ¼ IðX þ e cos VtÞ þ JðY þ e sin VtÞ R ¼ 2IkX 2 JkY

ð6Þ

one may obtain the following formulae for coordinates of the equilibrium position which are usually referred to as the static deflection of the shaft: FXs FYs Ys ¼ ð7Þ Xs ¼ k k

4. System dynamics

mrr€ G ¼ R þ F s

ð5Þ

ð2Þ

rffiffiffiffi k vn ¼ ; m

q ¼ eV2

Upon multiplying the equation (2) by the imaginary unit i and adding the equations (10) and (11), one may obtain the equations of motion of the rotor in the following form:

F s ¼ IF X s þ JF Y s In the above formula k stands for stiffness of the shaft at the point C and XY are its coordinates. Introduction of equation’s (2) into equation (1), results in the following set of differential equations: mðY€ 2 eV2 sin VtÞ ¼ 2kY þ F Y s ð3Þ

z€ þ v2 z ¼ qeiVt

ð13Þ

where: z ¼ x þ iy ð14Þ The above equation governs motion of the rotor in the stationary system of coordinates xyz.

or after reorganization: € þ kX ¼ F X s þ meV2 cos Vt mX mY€ þ kY ¼ F Y s þ meV2 sin Vt

5. Experimental work

ð4Þ

As can be seen from the Figures 4 and 5, the system consists of a 27 mm diameter shaft with 1,000 mm length carrying a

The particular solution of the non-homogeneous equation (5): Figure 2 Representation of absolute position of centre of gravity G

34



Figure 3 Sum of the static deflection Xs; Ys and the dynamic deflection x, y

have been presented. Besides, these experiments have been repeated by replacing the used bearings with the new ones. The possible faults for the shaft have been investigated. In order to determine the effects of such various disturbances as mechanical faults, fatigue and unbalance, etc. on the system dynamics, several experimental analyses have been achieved under different working conditions. The results obtained for the system with one disc are shown in Figures 6-11. Here, such vibration parameters as amplitude, velocity and acceleration values measured for the left and right bearings are given as a spectrum. From the figures, the amplitude for the right side bearing is getting to increase. This is why the right bearing is farther from the driving source than the other one. The vibration parameters of the system for different running speeds, such as 450, 750, 1,050 and 1,350 rpm, are given in Figures 12-17, respectively. As can be seen from these figures, these parameters reach their upper values at the frequencies correspond to the critical speed.

centrally located steel disc weighing 1.64 kg. It is supported by identical rolling element bearings with type of UC 204 SKF, and driven by a 0.75 kW, 230 v AC electric motor, controlled via Siemens Junior Velocity Control Unit. CSI 2110 Machine analyser is used for data acquisition from the system. Firstly, experimental work is accomplished for analysing of the system behaviour without the load. Secondly, the case of loaded system is implemented as shaft with only one disc. In order to analyse dynamic behaviour of rotating machineries, a direct coupled rotor system is designed and manufactured, and the system for various working conditions in regard to vibration parameters have been tested. To observe the dynamic behaviour of the system, the planned experiments for De Laval Rotor System have been accomplished and the results dealing with fault diagnosis

6. Artificial neural networks Artificial NNs are made up of simple highly interconnected processing units called neurons each of which performs two functions: aggregation of its inputs from other neurons or the external environment and generation of an output from the aggregated inputs. The output from a neuron is fed to other neurons to which it is connected via weighted links. Through this simple structure NNs have been shown to be able to approximate most continuous functions to any degree of accuracy by the choice of an appropriate number of neurons and output activation.

Figure 4 Representation of the experimental apparatus (schematic)

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Figure 5 Representation of the experimental apparatus (photograph)

Figure 6 Amplitude for right bearing (450 rpm)

6.1 Feed forward neural networks Feed forward neural networks (FNNs) are made up of one or more hidden layers between the input and output layers as illustrated in Figures 18 and 19. The functionality of the network is determined by specifying the strengths of the connection paths called weights and the neuron activation function. The input layer distributes inputs to the first hidden layer. The inputs then propagate forward through the network and each neuron computes its output according to:

xm i

¼g

nX m21

! m21 Wm ij xj

þ

bm i

ð15Þ

j¼1

m where xm i is the output of the ith neuron in the mth layer. W ij is the weight of the connection between the jth neuron of the (m 2 1)th layer and the ith neuron of the mth layer and bm i is the bias of the ith neuron in the mth layer. bm i can be regarded

36



Figure 7 Amplitude for left bearing (450 rpm)

Figure 8 Velocity for right bearing (450 rpm)

For neurons in the hidden layers the activation function is often chosen to be: 1 ð17Þ gðzÞ ¼ 1 þ e2z

as the weight of the connection between a fixed input of unit value and neuron i in layer m. The function g(.) is called the neuron activation function. The argument: zm i ¼

nX m21

m21 Wm þ bm ij xj i

Because the activation is non-linear the neurons are said to be non-linear neurons. Since in system modelling applications the dynamic range of the output data may be greater than 1. The activation function of the output nodes is chosen to be linear and the output nodes are said to be linear neurons. Thus the ith output node performs a weighted sum of its inputs as follows:

ð16Þ

j¼1

is the activation for the ith neuron in the mth layer. The function g(.) is assumed to be differentiable and to have a strictly positive first derivative. 37




Volume 58 · Number 1 · 2006 · 32 –44

Figure 9 Velocity for left bearing (450 rpm)

Figure 10 Acceleration for right bearing (450 rpm)

Figure 11 Acceleration for left bearing (450 rpm)

38



Figure 12 Amplitudes for right bearing at 450, 750, 1,050 and 1,350 rpm running speeds

Figure 13 Amplitudes for left bearing at 450, 750, 1,050 and 1,350 rpm running speeds

yi ¼

nX m21

m21 WO ij xj

denotes the dimension of the parameter vector Q and is defined as n ¼ nH(nI þ 1) þ nHnO where nInH and nO refer to the number of neurons in the input layer in the hidden layer and in the output layer, respectively. The ith output of a network with a single layer of hidden units can then be defined by:

ð18Þ

j¼1

where m represents the output layer and W O ij is the weight of the connection between the jth neuron of the last hidden layer and the ith neuron of the output layer. Only networks with one hidden layer are considered in the present study because the results of Cybenko (1989) and Funahashi (1989) show that this is sufficient to approximate all continuous functions. Let Q ¼ ½u1 ; . . . ; un T represent all the unknown weights and the biases of the network where n

yi ¼

nH X j¼1

H WO ij xj

¼

nH X j¼1

WO ij g

nI X

! W Ijk xIk

þ

bH j

1 # i # nO

j¼1

ð19Þ 39




Volume 58 · Number 1 · 2006 · 32 –44

Figure 14 Velocities for right bearing at 450, 750, 1,050 and 1,350 rpm running speeds

Figure 15 Velocities for left bearing at 450, 750, 1,050 and 1,350 rpm running speeds

h iT where x I ¼ xII . . .xInI is the input vector to the network. W Ijk is the weight of the connection between the kth neuron of the input layer and the jth neuron of the hidden layer and bH j is the bias of the jth neuron in the hidden layer. A backpropagation algorithm is employed to update weights of the FNN. The Backpropagation learning algorithm topology which was employed for the neural network updating the weight can be described as follows; define the error function as:

J¼

nO 1X ð ydi ðtÞ 2 yi ðtÞÞ2 2 i¼1

ð20Þ

where ydi(t) are the ith desired outputs and yi(t) are the ith outputs of the network. This error function is to be minimised with respect to all the unknown parameters Q. In the steepest descent approach the parameter vector Q ¼ ½u1 ; u2 ; . . . ; un T is 40



Figure 16 Accelerations for right bearing at 450, 750, 1,050 and 1,350 rpm running speeds

Figure 17 Acceleration for left bearing at 450, 750, 1,050 and 1,350 rpm running speeds

adjusted using the increment vector ½Du1 ; Du2 ; . . . ; Dun T defined along the negative gradient direction of J: Dui ¼ 2h

›J ›ui

layer m of the network and setting ui ¼ W m ij the application of the chain rule gives rise to:

›J ›J ›yi m ¼ ›W ij ›yi ›W m ij

ð22Þ

›J ¼ 2ðydi 2 yi Þ ¼ 2dm i ›yi

ð23Þ

ð21Þ From equation (20):

Although the one-hidden layer model is used in the present application it is useful to derive the gradient of J for the general case and the result for the one-hidden-layer model can readily be obtained as a special case. Starting from the output 41




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Figure 18 A layered feed-forward ANN

Then:

›xim21 ¼ g 0 ðzm21 Þ i ›zim21

ð27Þ

›zim21 ¼ xm22 j ›W ijm21

ð28Þ

and:

g 0 ðzÞ ¼

›gðzÞ ›z

ð29Þ

and g(zi) is the activation of neuron i. By defining the error signal for the ith neuron of the (m 2 1)th layer as:

dm21 ¼ g 0 ðzim21 Þ i

nO X

m dm k W ki

ð30Þ

k¼1

Equation (20) can be rewritten as:

›J ¼ 2dim21 xm22 j ›W ijm21

Figure 19 The neural predictor of the system

ð31Þ

Similarly it can be shown that:

›J ¼ 2dm21 i ›bm21 i

ð32Þ

where bim21 is the bias input to neuron i in layer m 2 1. By carrying on this procedure equations (30)-(32) can be used as a general algorithm for updating weights in other layers. Equations (30)-(32) indicate how the error signals propagate backwards from the output layer of the network through the hidden layer to the input layer. Hence the name “backpropagation”. The steepest-descent minimisation of the error function defined in equation (20) produces the following increments for updating Q: m m21 ðtÞ DW m ij ðtÞ ¼ hw di ðtÞxj

Dbm i ðtÞ

¼

hb dm i ðtÞ

ð33Þ ð34Þ

where in the output layer:

dm i ðtÞ ¼ ydi ðtÞ 2 yi ðtÞ dm i

where is called the error signal of the ith neuron in the mth layer. From equation (22):

›yi ¼ xjm21 ›W m ij

and in other layers: 0 m dm i ðtÞ ¼ g ðzi ðtÞÞ

ð36Þ

The constants hw (0 , h w , 1) and h b(0 , h b , 1) represent the learning rates for the weights and biases, respectively. In practice a large value of the learning rate would be preferable because this would result in rapid learning. Unfortunately a large value of the learning rate can also lead to oscillation or even divergence. To help speed up learning but avoid undue oscillations a momentum term is usually included so that equations (33) and (34) become:

ð25Þ

Next consider the (m 2 1)th layer. Using the chain rule yields: nO X ›J ›J ›yk ›xm21 ›zm21 i £ m21 £ im21 £ m21 ¼ ›yk ›xi ›zi ›W ij ›W m21 ij k¼1

X dmþ1 ðtÞW mþ1 ðt 2 1Þ j ji j

ð24Þ

Thus:

›J m21 ¼ 2dm i xj ›W m ij

ð35Þ

ð26Þ

42

m m21 ðtÞ þ mw DW m DW m ij ðtÞ ¼ hw di ðtÞxj ij ðt 2 1Þ

ð37Þ

m m Dbm i ðtÞ ¼ hb di ðtÞ þ mb Dbi ðt 2 1Þ

ð38Þ



where mw and mb are momentum constants which determine m the effect of past changes of DW m ij ðtÞand Dbi ðtÞ on the current updating direction in the weight and the bias space, respectively. This effectively filters out high frequency variations in the error surface. To summarise the backpropagation algorithm updates the weights and thresholds of the networks according to: m m Wm ij ðtÞ ¼ W ij ðt 2 1Þ þ DW ij ðtÞ

ð39Þ

m m bm i ðtÞ ¼ bi ðt 2 1Þ þ Dbi ðtÞ

ð40Þ

and:

where the increments equations (37) and (38).

DW m ij ðtÞand

Dbm i ðtÞ

Table II Structural, training parameters and simulation error results of the system (with a centred load on the shaft)

are given in

Network topology

N

Correct percent

RMSE

450 750 1,050 1,350

0.1 0.1 0.1 0.1

0.5 0.5 0.5 0.5

1-10-3 1-10-3 1-10-3 1-10-3

500,000 500,000 500,000 500,000

78 77 83 76

0.02 0.023 0.017 0.026

FNN FNN FNN FNN

Network topology

N

Correct percent

RMSE

450 750 1,050 1,350

0.1 0.1 0.1 0.1

0.5 0.5 0.5 0.5

1-10-3 1-10-3 1-10-3 1-10-3

500,000 500,000 500,000 500,000

67 68 74 73

0.033 0.031 0.030 0.029

FNN FNN FNN FNN

References Baillie, D.C. and Mathew, J. (1996), “A comparison of autoregressive modelling techniques for fault diagnosis of rolling elements bearings”, Mechanical Systems and Signal Processing, Vol. 10 No. 1, pp. 1-17. Cempel, C. (1991), “Condition evolution of machinery and its assessment from passive diagnostic experiment”, Mechanical Systems and Signal Processing, Vol. 5 No. 4, pp. 317-26. Cybenko, G. (1989), “Approximations by superposition of a sigmoid function”, Mathematics of Control, Signals and Systems, Vol. 2, pp. 303-14. Funahashi, K. (1989), “On the approximate realisation of continuous mappings by neural networks”, Vol. 2, pp. 1983-92. Genta, G. (1993), Vibration of Structures and Machines: Practical Aspects, Springer-Verlag, New York, NY. Mayes, I.W. (1994), “Use of neural networks for on-line vibration monitoring”, Proceedings of the Institution of Mechanical Engineers. Part A, Vol. 208, pp. 267-74. Krodkiewski, J.M. (2000), Dynamics of Rotors, The University of Melbourne, Department of Mechanical and Manufacturing Engineering, Melbourne. Lee, C.W., Joh, Y.D. and Kim, Y.D. (1990), “Automatic modal balancing of flexible rotors during operation – computer-controlled balancing head”, Proceedings of the Institution of Mechanical Engineers. Part C – Mechanical Engineering Science, Vol. 204 No. 1, pp. 19-28. Schoen, R.R. et al., (1995), “An unsupervised on-line system for induction motor fault detection using stator current monitoring”, IEEE Transactions on Industry Applications, Vol. 31 No. 6, pp. 1274-9. Haykin, S. (1994), Neural Networks: A Comprehensive Foundation, Macmillan, New York, NY. Smalley, A.J., Baldwin, R.M., Mauney, D.A. and Millwater, H.R. (1996), “Towards risk based criteria for rotor

Table I Structural, training parameters and simulation error results of the system (without load on the shaft) m

m

In this paper, a neural network predictor approach has been presented for analyzing vibration parameters of the ball bearing-shaft system. The system has been employed with two different working conditions. The simulation results obtained have supported the theory that the FNN was able to represent different types of ball-bearing systems. The neural network was also trainable using the simple back propagation algorithm to analyse the ball-bearing system with two conditions. In future, this type of the NN could be used as a predictor of bearing system in experimental applications.

An experimental work and simulation study were carried out for vibration analysis of the rotor-bearing system. The experimental work was employed for two different loading conditions. Firstly, the system was driven without load. Secondly, the system was driven with a centrally load on the shaft. As a result, a certain increase in values of the vibration parameters of the system has been observed for the case of loaded condition compare to unloaded situation. Besides, for the defined cases, as the running speed increases, an increase in values of amplitude, velocity and acceleration parameters at various measurement points of the system has been found. In the case of the shaft-bearing system with centrally disc, if the frequencies range less than 6 Hz which is outside the stationary working condition is not considered, the amplitude values for both the ball-bearings are almost the same. It should be noted that the running speed of 450 rpm corresponds to 7.5 Hz. The study has illustrated the effectiveness of the ANN predictors for vibration, velocity and acceleration analysis in a shaft-bearing system. The NN is a feed forward network with one hidden layer including 10 neurons. The predictor has been developed to perform fault diagnosis to a good degree of accuracy. From the network results, it is concluded that the success of the testing process depends on two-training parameter, namely the learning rate and the momentum term. It has been proven that a FNN with non-linear hidden units, such as described and applied to vibration analysis, can be modelled as a function of arbitrary complexity. However, acceptable accuracy can be achieved in most applications using the Backpropagation training algorithm. Training, structural parameters and the results of the neural network for shaft-bearing system are given in Table I. The relevant results for the model system with centrally disc are also outlined in Table II. From the Table I, the good results of neural network for 1,050 rpm.

h

h

8. Conclusion

7. Simulation results

Neural n (rpm) network

Neural n (rpm) network

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system”, Mechanism and Machine Theory, Vol. 36, pp. 157-75.

vibration”, Proceedings of the Institution of Mechanical Engineers – Vibrations in Rotating Machinery, pp. 517-27. Taylor, J.I. (1995), “Back to the basics of rotating machinery vibration analysis”, Journal of Sound and Vibration, Vol. 29 No. 2, pp. 12-16. Vyas, N.S. and Satishkumar, D. (2001), “Artificial neural network design for fault identification in a rotor-bearing

Corresponding author S¸ahin Yıldırım can be contacted at: [email protected]

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