System Identification of Pipe Cylinder Behavior Caused By Vortex Induced Vibration

MOHAMMED JAWAD MOHAMMED AND INTAN Z. MAT DARUS Department of Applied Mechanics and Design

Faculty of Mechanical Engineering UniversitiTeknologi Malaysia

81310, Johor Bahru MALAYSIA

msc.mohammed83@gmail.comintan@fkm.utm.my Abstract: - A dynamic modeling of pipe cylinder caused by vortex induced vibration (VIV) using system identification method is investigated in this paper. The input and the output data collected from experimental proceduresare used for modeling the system. Least Square (LS), Recursive Least Square (RLS) and Neural Network time series are used to predict the dynamic response model for pipe riser in offshore engineering. LS and RLS represented based on Auto-regressive external input (ARX) modelwhile, the Neural Network time series included three methods: Neural Network (NARX) based on the Nonlinear Auto-Regressive with External (Exogenous) Input, Neural Network (NAR) based on the Nonlinear Auto-Regressive and Nonlinear Input-Output Neural Network. The performance of all methods validated and compared through the mean squared error (MSE) of the one-step-ahead prediction. Finally, the results shown that the Neural Network based on Nonlinear Auto-Regressive and External Input (NARX) is more accurate to predict the dynamic behavior of the system from other methods which recorded the lowest MSE (1.2714×10-9 ) with8 neuron atthe hidden layer (NE) 8 and with 2 delays in the input output data. Key-Words: -System Identification, Recursive least Square, Neural Network, Vortex Induced Vibration. 1 Introduction Vortex induced vibration (VIV) of the bluff structure such as risers, spar platforms and pipelines are considered one of the significant challenges for a designer. In marine engineering structures, the VIV is very common and complex phenomenon lead to fatigue damage of pipe risers because the interaction between the created vortices and the bluff structure from one point of view and on the other hand when the vortex shedding frequency closed to the natural frequency of the cylinder (Resonance Phenomenon). The vortex induces vibration phenomenon happened, when an elastically mounted body placed in the perpendicular direction of flow; the vortices generated in the past of the cylinder and exerted oscillatory external force. As a result, the cylinder fluctuates occasionally to the flow direction. This phenomenon has been the focus point and gain attention from previous researchers [1-4]. Based on the semi-empirical methods, the behavior of vortex induced vibration can be divided into three portions: wake oscillator, single degree-of freedom models and force-decomposition model [5]. The first wake oscillator models depends on finding the equation between a fluid oscillator and the cylinder oscillator to predict the dynamic response[6]. The second portion is single degree-of-freedom models and depends on using single dynamic equation with

aero-elastic term. The third portion force-decomposition models depends on measuring the force components on the body[7]. System identification method used to identify the dynamic response characteristics during from creating the mathematical modelsto transfer function or equivalent mathematical description based on the experimental date of the vortex induced vibration behavior. Also, it commonly utilized to obtain the linear and nonlinear model structure. Nowdays, the nonlinear system identification method has become very important for researchers around the world, because the linear system identification at some times not give satisfactory outcomes despite the evolution in linear model[8]. There are three main parts presented in this paper, the data collected from theprevious paper, using Least Square (LS), Recursive Least Square (RLS) and Neural Network time series techniques to model the system identification and recently results and discussions. 2 Experimental Data 2.1 Data Extraction In this study, the input and output data used by previous researchers Shaharuddin and Mat Darus[9-

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10] has been utilized. The researchers used the rigid cylinder with flexible support in the crossflow vibration. Table 1 shows the specification and the parameter values from the experimental test carried out by Shaharuddin and Mat Darus.

Table 1. Obtained Parameters [9][10] Parameter Symbol Value Unit

Cylinder Diameter D 50 mm Cylinder Length L 1110 mm

Aspect Ratio L/D 22.2 Dimensionless Cylinder mass m 2.95 kg

Mass ratio m* 1.18 Dimensionless Natural Frequency

in water ƒw 1.11 Hz

Damping ratio in water ζ 0.1007 Dimensionless

System Stiffness k 265.34 N/m The input (from accelerometer A) and output (from accelerometer B) data which included 33000 data obtained from Shaharuddin and Mat Darusare as shown in Fig. 1 [9-10].

Fig.1 Accelerometer Positions for Detecting and

Observing Data of Experimental Setup Diagram [9-10]

The accelerometer A represents the detected input data while, the accelerometer B represents the observed output for system identification. Fig. 2 and 3 shown the input and output amplitudes with time.

Fig.2Input Amplitude with Simple Time Series for

Pipe Cylinder

Fig.3 Output Amplitude with Simple Time Series

For Pipe Cylinder 3 System Identification In this study, the system identification methods: Least Square (LS), Recursive Least Square (RLS) and Neural Network time series used to identify the dynamic response characteristics for vortex induced vibration behavior [11]. 3.1 Model Structure 3.1.1 Linear Auto-Regressive external input model (ARX) In this paper, the linear ARX model as shown below [12] is used to represent the input and the output data for the system. This ARX model is utilized for both Least Square (LS) and Recursive Least Square (RLS) algorithms.

where

Thus

3.1.2 Nonlinear Auto-Regressive model (NRA) NAR is one of the methods used to predict the output data for nonlinear model by using the input

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or the output data only. The equation of NAR model is [13]:

The represents the past value of the output. The

represents the nonlinear function which can be carried out by using intelligent methods such as Neural Network. 3.1.3 Nonlinear Auto-Regressive External Input model (NRAX) NARX model structure was employed in many researches for various purposes such as control and system identification. The NARX algorithm can be defined as follows [14]:

After neglecting the noise error and defining the and , the NARX became:

The and represent the past value of input and output respectively. While the represents the input delay. Finally the represents the nonlinear function which can be carried out by using intelligent methods such as Neural Network. 3.2 Least Square (LS) It is one of the simplest statistical methods for system identification used to find the transfer function for the model. The equations given by [15]:

where Y is the predicted output. φ is includes actual input and output parameters.β is includes parameters to be estimated. 3.3 Recursive Least Square (RLS) It is one of the successful predictionalgorithms to estimate frequently for unknown parameters with real time operation. The principle of Recursive Least Square depends on least square mathematical weighted. Also, Recursive Least Square can set the points to fit the curve. The equations can be written as follows [12]:

For one row, the matrix will became: …..,

Frequently, the value of new parameter β can be calculated from the equation below:

where:

3.4 Neural Network Time Series Neural Network is one of an artificial intelligent method which can be used for the nonlinear dynamic system identification. It consists of a number of neurons arranged in various layers. According to Fig. 4, Neural Network model includes at least three layers which are input layer, hidden layer and output layer. The equations can be formulated as follows: 3.4.1 Hidden layer

(14) (15)

Then

Where

are the actual input data of input network. are the actual output of input network.

the weights between input and hidden layers.

the bias weights for hidden layer. the summation values for hidden layer.

the final values for hidden layer. 3.4.2 Output layer

Where

the weights between the hidden and output layers.

the bias weights for the output layer. the summation values for the output layer. the final predicted value for the output layer or

neural network process.

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Fig.4 Neural Network Architecture

3.5 Method of validation The verification process of the results is considered one of the important ways to measure the performance of the mathematical algorithms which used in this paper. In this paper, the results are obtained and verified using Mean Square Error method (MSE). The equation of Mean Square Error is:

where: is the actual output. is the predicted output.

The data, which used is divided into two partitions: testing and partition for validating. 4 Results and Discussions 4.1 Least Square (LS) The input and output data which obtained from previous research [9-10] includes 33000 data. In LS, the input-output data are divided into two partitions. First 15000 data is used for training and the second partition is used for testing and validating. The LS composed based on linear ARX model. Tabulation of LS performance in MSE with different model order can be seen in Table 2. The results have shown that the lowest MSE value obtained at 2 mode order with MSE=1.7454×10-6. Using this order, the modelthat represent the cylindrical pipe performance caused by vortex induced vibration based on ARX model is obtained as:

Fig. 5shows the performance of actual and predicted output with time. While the Fig. 6, shows the predicted error with time for LS technique.

Table 2 Mean Square Error for LS Technique Mode order Mean Squared Error For LS Technique

2 1.7454×10-6 3 6.0390×10-6 4 1.8137×10-6 5 3.0205×10-4 6 0.0029 7 2.8905×10-4 8 7.8393×10-4 9 4.1927×10-5

10 0.0061 11 0.0164 12 3.7178×10-4

Fig.5 Amplitude of Actual and Predicted Output

Depends on LS

Fig6Error of the Predicted Output Depends on LS

4.2 Recursive Least Square Technique (RLS)

The data included 33000 data and divided into two partitions. First 15000 data used for training and the second partition used for testing and validating. It is composed based on the linear ARX model. Firstly, RLS is simulated with different model order ranging from 2 to11 with forgetting factor value 0.1 to find the best model order based on the lowest MSE. Then, RLS is simulated with different forgetting factor value ranging from 0.1 to 0.9 with the best model order which was found earlier.

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According to the results tabulated in Table 3, the lowest MSE is obtained using model order 2 with MSE= 1.0351×10-07. While in Table 4,it can be seen that the lowest MSE=1.0351×10-07is obtained at forgetting factor 0.1 and model order 2.

Table 3 Mean Square Error For RLS Technique at forgetting factor 0.1

Mode order Mean Squared Error For RLS Technique 2 1.0351×10-7 3 2.0759×10-7 4 2.6136×10-7 5 3.9896×10-7 6 3.1737×10-7 7 4.3386×10-7 8 5.6464×10-7 9 5.1704×10-7

10 6.0232×10-7 11 7.7963×10-7

Table 4 Mean Square Error For RLS Technique at

mode order 2 Forgetting

Factor Mean Squared Error For RLS

Technique at mode order 2 0.1 1.0351×10-7 0.2 2.0759×10-7 0.3 2.6136×10-7 0.4 3.9896×10-7 0.5 3.1737×10-7 0.6 4.3386×10-7 0.7 5.6464×10-7 0.8 5.1704×10-7 0.9 6.0232×10-7

From the above results, the best value of MSE is 1.0351×10-7 at mode order 2 and forgetting factor 0.1. Thus, the transfer function based on ARX model is:

Fig. 7 shows the performance of actual and predicted output with time. While the Fig. 8, shows the predicted error with time for RLS technique.

Fig.7Amplitude of Actual and Predicted Output Depends on RLS

Fig.8Error of the Predicted Output Depends on RLS 4.3Neural Network Time Series It is included for three methods to predict the dynamic response of the system which are: Neural Network based on Nonlinear Auto-Regressive external input model (NRAX), Neural Network based on Nonlinear Auto-Regressive (NAR) and Nonlinear Input-output Neural Network. All of these methods used the 33000 data which are divided into three parts. The first partition is 32100 data used for training, and the second part is 4950 data used for validating while the third partition is 4950 data used for testing. Also, all of these methods used firstly a different hidden neuron (NE) which ranged 1-11 with a number of delays equal 2 to find the Lowest MSE at which hidden layer. Then, used a different number of delays, which ranged 1-11 with the specific hidden neuron (NE) and found from the previous stage. According to Table 5, the results of NARX model shown that the lowest MSE in a number of hidden neurons (NE) 8 is 1.2714×10-9. According to Table 6, the results of NAR model shown that the lowest MSE in a number of hidden neurons (NE) 6 is 1.0532×10-8. According to Table 7, the results of Nonlinear input-output Neural Network model shown that the lowest MSE in a number of hidden neurons (NE) 9 is 5.3776×10-7. According to Table 8, the results of NARX model shown that the lowest MSE in the number of delay 2 is 1.2714×10-9. From the tabulated results, it was found that the lowest mean squares error, MSE = 1.2714×10-9was obtained using NARX model with number of hidden neurons (NE) = 8 and the number of delay 2 is better than from the other methods by using Neural Network. Also, Figs. 9 and 10 show the performance of NARX model and error prediction respectively. Figs. 11 and 12 show the performance of NAR model and error prediction respectively and Fig. 13 and 14 show the performance of nonlinear input-output Neural Network model and error prediction respectively.

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Table 5 Mean Square Error for NARX Model at Number of Delay 2

NE MSE for training MSE validated MSE tested MSE

Overall 1 7.06994×10-7 7.03911×10-7 7.30382×10-7 7.1048×10-7 2 1.51825×10-7 1.46421×10-7 1.63488×10-7 1.5353×10-7 3 2.51148×10-9 2.53456×10-9 2.49492×10-9 3.2278×10-9 4 2.25323×10-8 2.10880×10-8 2.24574×10-8 2.2795×10-8 5 1.90786×10-9 1.88947×10-9 1.91908×10-9 2.2781×10-9 6 2.33170×10-8 2.42179×10-8 2.39115×10-8 2.4056×10-8 7 8.42986×10-8 8.52992×10-8 8.58115×10-8 8.5672×10-8 8 4.73371×10-10 4.72190×10-10 4.76996×10-10 1.2714×10-9 9 3.27642×10-9 3.31040×10-9 3.41665×10-9 4.0278×10-9

10 6.28938×10-9 6.58235×10-9 6.65887×10-9 6.7772×10-9 11 1.60514×10-8 1.62700×10-8 1.75777×10-8 1.6682×10-8

Table 6 Mean Square Error for NAR Model at

Number of Delay 2 NE MSE for

training MSE validated MSE tested MSE Overall

1 1.21865 ×10-8 1.23348 ×10-8 1.47379 ×10-8 1.3618× 10-8 2 4.69192 ×10-7 4.44050 ×10-7 5.13385 ×10-7 4.7309× 10-7 3 6.160202 ×10-8 6.42999 ×10-8 6.11500 ×10-8 6.2567× 10-8 4 2.95893 ×10-7 2.89721 ×10-7 3.12913 ×10-7 2.9778× 10-7 5 1.01116 ×10-8 1.02313 ×10-8 1.00981 ×10-8 1.0532× 10-8 6 5.82502 ×10-9 5.97741 ×10-9 5.76880 ×10-9 6.6542× 10-8 7 9.52759 ×10-8 8.19346 ×10-8 1.00706 ×10-7 9.4854× 10-8 8 7.40069 ×10-9 6.37130 ×10-9 6.98316 ×10-9 7.7572× 10-8 9 2.54389 ×10-8 2.79919 ×10-8 2.74169 ×10-8 2.9366× 10-8

10 1.45673 ×10-8 1.48914 ×10-8 1.48435× 10-8 1.6209× 10-8 11 9.78359× 10-8 8.76832 ×10-8 9.03277× 10-8 9.5864× 10-8

Table 7 Mean Square Error For Nonlinear input-output Neural Network model at Number of Delay 2 NE MSE for

training MSE validated MSE tested MSE Overall

1 1.76778×10-6 1.72086× 10-6 1.86453×10-6 1.7753×10-6 2 6.25215×10-7 6.14274× 10-7 6.22246×10-7 6.2413×10-7 3 5.90255×10-7 6.085344×10-7 6.09983×10-7 5.9716×10-7 4 5.83823×10-7 5.74238× 10-7 5.79026×10-7 5.8231×10-7 5 6.44925×10-7 6.37583× 10-7 6.19089×10-7 6.3992×10-7 6 6.59038×10-7 6.79244× 10-7 6.60723×10-7 6.6249×10-7 7 5.52678×10-7 5.59595× 10-7 5.58973×10-7 5.5511×10-7 8 6.43524×10-7 6.49175× 10-7 6.47304×10-7 6.4492×10-7 9 5.32014×10-7 5.38511×10-7 5.48821×10-7 5.3776×10-7

10 5.37254×10-7 5.55222× 10-7 5.28715×10-7 5.4407×10-7 11 5.68340×10-7 5.68615× 10-7 5.68450×10-7 5.6848×10-7

Table 8 Mean Square Error For NARX Model at Number of Hidden Neuron 8

No. of

Delay

MSE for training

MSE validated MSE tested MSE

Overall

1 1.269×10-8 1.271×10-8 1.301×10-8 1.308×10-8 2 4.733×10-10 4.722×10-10 4.769×10-10 1.271×10-9 3 2.338×10-10 2.293×10-10 2.354×10-10 1.327×10-9 4 6.002×10-9 6.0448×10-7 5.968×10-7 6.452×10-9 5 1.855×10-8 1.850×10-8 1.893×10-8 1.954×10-8 6 3.096×10-8 3.125×10-8 1.095×10-8 3.218×10-8 7 3.378×10-8 3.613×10-8 3.430×10-8 3.496×10-8 8 1.364×10-9 1.397×10-9 1.369×10-9 2.823×10-9 9 2.085×10-9 2.010×10-9 1.989×10-9 3.747×10-9

10 2.786×10-9 2.711×10-9 2.861×10-9 2.879×10-8 11 7.702×10-9 7.706×10-9 8.067×10-9 8.918×10-9

Fig.9Amplitude of Actual and Predicted Output

Depends on NARX

Fig.10 Error of the Predicted Output Depends on

NARX

Fig.11Amplitude of Actual and Predicted Output

Depends on NAR

Fig.12 Error of the Predicted Output Depends on

NAR

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Fig.13 Amplitude of Actual and Predicted Output

Depends on Nonlinear input-output Neural Network

Fig. 14 Error of the Predicted Output Depends on

Nonlinear input-output Neural Network

5 Conclusion Dynamic model of the pipe cylinder caused by vortex induced vibration (VIV) using conventional and intelligent system identification methodsare investigated in this paper. Linear system identification has beenconductedutilizing LS and RLS methodsbased on Auto-regressive external input (ARX) model. The Neural Network time series have been conducted based on three methods: Neural Network (NARX) based on the Nonlinear Auto-Regressive with External (Exogenous) Input, Neural Network (NAR) based on the Nonlinear Auto-Regressive and Nonlinear Input-Output Neural Network. The results showed that the Neural Network based on Nonlinear Auto-Regressive External Input (NARX) is more accurate to predict the dynamic model as compared to other methods. Acknowledgment Ministry of Education (MOE) and UniversitiTeknologi Malaysia(UTM) for Research University Grant (Vote No.04H17).

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