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Identification of the nonlinear ship rolling motion equation using the measured response at sea
Ocean Engineering 31 (2004) 2139–2156 www.elsevier.com/locate/oceaneng
Identification of the nonlinear ship rolling motion equation using the measured response at sea Ayman B. Mahfouz InCoreTec Incorporated, Baine Johnston Center, Suite 804, 10 Fort William Place, St. John’s, Newfoundland, Canada A1C 1K4 Received 26 October 2003; accepted 28 June 2004
Abstract This paper describes a new robust method for the identification of the parameters in the equation describing the rolling motion of a ship using only its measured response at sea. Those parameters are the linear and nonlinear damping and restoring parameters. The random decrement equations as well as the auto- and cross-correlation equations are derived for a ship performing rolling motion in random beam waves. The linear and nonlinear parameters in the equation of motion are identified using a combination of the random decrement technique, auto- and cross-correlation functions, a linear regression algorithm, and a neural networks technique. The combination of the classical parametric identification techniques and a neural networks technique provides robust results and does not require a large amount of computer time. The proposed method would be particularly useful in identifying the nonlinear damping and restoring parameters for a ship rolling under the action of unknown excitations effected by a realistic sea. Numerically generated data and experimental data for the ship rolling motion are used to test the accuracy and the validity of the method. It is shown that the method is reliable in the identification of the parameters of the equation of the rolling motion using only the measured response at sea. # 2004 Elsevier Ltd. All rights reserved. Keywords: Ship hydrodynamics; Dynamical stability; Neural networks; Random decrement signature; Ship motion; Parametric identification techniques
Corresponding author. Tel.: +709-739-1760; fax: +709-739-7780. E-mail address: [email protected] (A.B. Mahfouz).
0029-8018/$ - see front matter # 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2004.06.001
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1. Introduction The stability of a ship in realistic sea is mainly dependent on its rolling motion, which is the most critical factor in defining ship survivability. In fact, most available approaches for the assessment of ships’ survivability in a seaway are based on the study of rolling motion. Two approaches are used in this assessment: a static approach (quasi-static) and a dynamic one. The former is based on the minimum value that the metacenteric height, GM, should have and the shape of the static stability curve (GZ–h). This approach is still being applied in the assessment of ship’s stability criterion. The latter approach is based on the analysis of the stability of the equation of the rolling motion. This involves constructing a model for a ship rolling in a realistic sea. Using this model, the occurrences of capsizing modes can be predicted (see Haddara, 1992; Roberts et al., 1991, 1994). The linear restoring parameters can be easily obtained from ship hydrostatics; however, the damping parameters are not. Therefore, the main objective of this paper is to develop a reliable and robust method that can be used in the identification of the parameters involved in the equation describing the roll motion of a ship in a random sea using only its measured random response. Since the wave excitation is usually not measurable, the long-term objective of this work is to provide a reliable methodology to assess the ship’s survivability at sea. It has been shown that the roll auto-correlation function satisfies the same equation of motion as the free roll response (see Haddara and Xu, 1997; Haddara, 1992). This is true for lightly damped motions. When the damping increases, the auto-correlation function becomes more influenced by the wave excitations (see Haddara and Xu, 1997). In this work, a new robust method is presented to identify the linear and nonlinear damping and restoring parameters involved in the rolling equation without prior knowledge of the wave excitation. This has been achieved using a combination of the random decrement technique, auto- and cross-correlation functions, a multiple linear regression algorithm, and a neural networks technique. 2. Identification method The identification method is based on the derivation of the random decrement equations and the auto-correlation functions for the rolling motion. Linear regression is used to identify the linear parameters and neural networks are used to identify the nonlinear rolling function. Details of the identification method are presented in the following sections. 2.1. Random decrement equations The random decrement technique is an averaging technique that has been used successfully in the on-line failure detection and damping identification of linear structures (see Cole, 1973). In general, the response of a single degree of freedom system has two main components: the transient and the steady state. The random
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decrement technique is based on the fact that when the general response of a single degree of freedom system is averaged, the contribution of the steady state component to the expected value of the response vanishes when the excitation function is a Gaussian white noise random process. Then, the expected value of the random response is represented only by the contribution of its transient component caused by the initial conditions (see Haddara, 1992). This ensemble average is the random decrement signature, which can be calculated from the random roll response using methods described in the literature (see Haddara, 1992; Cole, 1973). A forced second order differential equation is a reasonable model for a ship performing rolling motion in random seas. The main parameters of this model are the virtual moment of inertia, the damping moment, the restoring moment, and the exciting moment. This equation can be normalized with respect to the virtual moment of inertia. In this case, the equation has three unknowns only (see Haddara, 1992). The equation describing the rolling motion of a ship excited by random beam waves can be written as (see Haddara, 1992, 1995): € þ NðU_ Þ þ DðUÞ ¼ KðtÞ U
ð1Þ
where U is the roll angle, NðU_ Þ and D(U) are the nonlinear damping and restoring moments per unit virtual moment of inertia of the ship, and K(t) is the wave exciting moment per unit virtual moment of inertia of the ship. A dot over the variable denotes the differentiation with respect to time. The wave excitation K(t) is assumed to be a zero mean Gaussian, white random process (see Haddara, 1992; Roberts et al., 1991, 1994). Using the following change of variables: y1 ¼ U
ð2Þ
y2 ¼ U_
ð3Þ
Eq. (1) can be easily replaced by two first order differential equations as shown in Eq. (4) (see Haddara, 1992): Y_ ¼ F ðY Þ þ EðtÞ
ð4Þ
where Y_ ¼ ½y1 y2 T F ðY Þ ¼ ½y2 Nðy2 Þ Dðy1 ÞT EðtÞ ¼ ½0 KðtÞT Assuming that the process Y(t) is a Markov process, the Fokker–Planck equation can be used to describe the conditional probability density function for the random process, Y(t) as shown (see Haddara, 1992): @P @ @ 1 @2 ¼ ðy2 PÞ þ ð½Nðy2 Þ þ Dðy1 ÞPÞ þ ðW2 PÞ @t @y1 @y2 2 @y22
ð5Þ
where W2 is the variance of the exciting function K(t) and can be calculated if K(t)
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is measurable: W2 ¼ E ½KðtÞ KðtÞ where E[ ] denotes the expected value. The solution of Eq. (5) subject to the initial condition PðY ; tjYo Þ ¼ dðy1 y1o Þdðy2 y2o Þ yields the conditional probability density function which describes the Markov process Y(t) completely. Where yio, i ¼ 1; 2 are the initial conditions for the roll displacement and the roll rate, respectively. Multiplying Eq. (5) by the variables y1(t) and y2(t), respectively, and integrating the whole equations over the range of the two variables, the propagation of the expected values of y1(t) and y2(t) can be obtained as: l_ 1 ¼ l2
ð6Þ
l_ 2 ¼ E½Nðy2 Þ þ Dðy1 Þ
ð7Þ
where l1 ¼ l2 ¼
ðð y1 PðY ; tÞ dy1 dy2 ðð y2 PðY ; tÞ dy1 dy2
The evaluation of the averages on the right-hand side of Eq. (7) poses a problem since the probability density function P(Y,t), for the Markov process Y(t), is yet to be determined. Replacing these averages by their Taylor series expansion about l1 and l2 solves this problem (see Haddara, 1992). Retaining the first order terms from this expansion, Eqs. (6) and (7) can be combined in a one second order differential equation as given in Eq. (14). The damping moment, N(y2), can be expressed as the sum of two terms: a linear term in the roll rate and a nonlinear term. Two forms have been suggested to represent the nonlinear term: the quadratic form suggested by Froude and the cubic form suggested by Haddara (1995). In this work, the following expressions for the damping and restoring moments are used (see Haddara and Hinchey, 1995; Haddara, 1992): Nðy2 Þ ¼ 2fx/ ½1 þ e1 y22 y2 Dðy1 Þ ¼
x2/ ½1
þ
e2 y21 y1
ð8Þ ð9Þ
where f, e1 and e2 are the nondimensional linear damping coefficient, the nonlinear damping coefficient, and the nonlinear restoring moment coefficient, respectively, x/ is the linear natural frequency. Using Eqs. (8) and (9), the nonlinear damping and restoring moments in Eq. (4) can be written in the following form: Nðy2 Þ þ Dðy1 Þ ¼ 2fx/ y2 þ x2/ y1 þ F ðy1 ; y2 Þ
ð10Þ
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where F ðy1 ; y2 Þ ¼ 2fx/ e1 y32 þ x2/ e2 y31
ð11Þ
The expected value of Eq. (10) is obtained as ðy1 ; y2 Þ E½Nðy2 Þ þ Dðy1 Þ ¼ 2fx/ l2 þ x2/ l1 þ F
ð12Þ
where ðy1 ; y2 Þ ¼ 2fx/ e1 E½y3 þ x2 e2 E½y3 F 2 1 /
ð13Þ
Eq. (13) represents the lamped nonlinear terms of the damping and restoring moments. Eqs. (6) and (7) are combined as: ðy1 ; y2 Þ ¼ 0 €1 þ 2fx/ l_ 1 þ x2/ l1 þ F l
ð14Þ
It is obvious that the approximate equation describing the expected value of the roll response, l1 satisfies the differential equation, which describes the free rolling ðy1 ; y2 Þ, in Eq. (14) motion of a ship. The linear terms and the nonlinear function, F are identified in the following two sections, respectively. 2.2. Auto- and cross-correlation functions The derivation of the auto- and cross-correlation equations corresponding to Eq. (1) is based on the Fokker–Planck equation (5). The latter equation is to be multiplied by yi ðtÞ yj ðt þ sÞ Ps ðYo Þ; i ¼ 1; 2 and j ¼ 1; 2 each in time and integrating the whole equations over the range of the two variables. The following equations are obtained: R_ 11 ¼ R21 ðð _R21 ¼ ½Nðy2 Þ þ Dðy1 Þy1 ðt þ sÞPðY ; tÞPs ðY Þ dy1 dy2
ð15Þ
R_ 12 ¼ R22 ðð R_ 22 ¼ ½Nðy2 Þ þ Dðy1 Þy2 ðt þ sÞPðY ; tÞPs ðY Þ dy1 dy2
ð17Þ
ð16Þ
ð18Þ
Substituting Eqs. (8) and (9) into Eqs. (16) and (18) one can get the following equations as: ðð R_ 21 ¼ x2/ R11 2fx/ R21 F ðy1 ; y2 Þy1 ðt þ sÞPðY ; tÞPs ðY Þ dy1 dy2 ð19Þ R_ 22 ¼ x2/ R12 2fx/ R22
ðð
F ðy1 ; y2 Þy2 ðt þ sÞPðY ; tÞPs ðY Þ dy1 dy2
ð20Þ
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Eqs. (19) and (20) can be written as R_ 21 ¼ x2/ R11 2fx/ R21 E½F ðy1 ; y2 Þy1 ðt þ sÞ
ð21Þ
R_ 22 ¼ x2/ R12 2fx/ R22 EðF ðy1 ; y2 Þy2 ðt þ sÞ
ð22Þ
Eqs. (21) and (22) have been identified using a multiple linear regression algorithm. In the multiple linear regression cases, certain tests of hypotheses about the model coefficients are helpful in measuring the usefulness of the model (see Montgomery, 1997). Two main statistic indexes are important in measuring the usefulness of the model: P-value and R-sq. In addition, there is another criterion for model acceptance known as the multicollinearity of the predictors. This is related to the variance inflation factor (VIF) (see Dingman and Sharma, 1997; Montgomery, 1997). The computed values of the VIF exceeding 10 indicate potentially detrimental multicollinearity effects. From the regression process derived for Eqs. (21) and (22) using MINITAB software, it is found that when the last term in each equation is included in the models, the calculated VIF is more than 10 and the P-value is more than 0.05. It is concluded that the last term in these equations is insignificant in the models (Eqs. (21) and (22)). Therefore, these terms are removed from the equations. As a result, the final strong relationship in these equations can be written as: R_ 21 ¼ x2/ R11 2fx/ R21
ð23Þ
R_ 22 ¼ x2/ R12 2fx/ R22
ð24Þ
The linear parameters in Eqs. (23) and (24) are identified only from the measured roll response using a multiple linear regression algorithm as described above (see Montgomery, 1997). The auto- and cross-correlation functions are calculated from the roll response using (see Haddara and Xu, 1997): Rik ðsÞ ¼
N p s X 1 yi ðjÞyk ðj þ sÞ ðNp sÞ j¼1
ð25Þ
where Np and s are total number of points in the roll response, and the time difference, respectively. 2.3. Neural networks In general, artificial neural networks try to mimic the biological network (see Haddara and Hinchey, 1995). The present artificial neural networks are considered much simpler compared to the biological networks especially in the number of neurons, size, and construction complexity (see Flood and Kartam, 1994). The combination of the neural networks technique with the standard parametric techniques provides a means for the identification of the nonlinearity in the roll equation without a prior knowledge of the form of the nonlinearity (see Haddara and Hinchey, 1995).
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Fig. 1. Feedforward neural network.
The nonlinearity in the equation describing the rolling motion of a ship sailing in ðy1 ; y2 Þas shown in Eq. (14). A continuous record of random seas is the function F the measured roll response is used with Eq. (14) to identify the nonlinear function using a neural networks technique. A single hidden layer feedforward neural network, as shown in Fig. 1, is used to identify the nonlinear function in Eq. (14). The input to the network is a vector having the expected value of the measured roll response, the expected value of the roll rate, and a bias. The hidden layer may consist of several neurons. In this paper, a value of 11 neurons gives a sufficient accuracy in the identification process. The output layer consists of one neuron whose output is the identified nonlinear function (see Haddara, 1995; Flood and Kartam, 1994). The input to the hidden layer neuron ith is the weighted sum of the inputs expressed as X Aj ¼ wij li ; i ¼ 1; . . . ; 3 and j ¼ 1; . . . ; 12 ð26Þ i;j
where wij is the synaptic weight of the ith neuron in the input layer to the jth neuron in the hidden layer. l3 is the bias and is equal to 1.0. The weighted sum of the inputs into hidden layer neurons are processed by a sigmoidal function f, expressed as f ðAj Þ ¼
1 1 þ eðAj Þ
ð27Þ
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Then, the nonlinear function F ðy1 ; y2 Þ is equal to the weighted sum of the outputs of the hidden layer neurons expressed as
ðy1 ; y2 Þ ¼ F
12 X bj f ðAj Þ
ð28Þ
i¼1
where bj is the synaptic weight of neuron j in the hidden layer. Using an arbitrary starting set of weights, an initial value of the nonlinear function is obtained. The obtained value of that function is substituted back in Eq. (14), which is integrated numerically using a Runge–Kutta algorithm. The corresponding roll response obtained from the integration is compared to the input roll response to the network. The difference between these two responses is the error. The synaptic weights wij and bj are then updated and the process is repeated in an iterative fashion until the error in the response is minimized. A simple flowchart for the neural networks technique is shown in Fig. 2. The nonlinear damping and restoring parameters e1 and e2, respectively, are then obtained using a multiple linear regression algorithm for Eq. (13) (see Dingman and Sharma, 1997; Montgomery, 1997).
Fig. 2. Neural networks technique.
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3. Validation Numerically generated data and experimental data for the ship rolling motion are used to test the accuracy and the validity of the method. 3.1. Numerically generated data A mathematical model that has been used in the generation of the random roll response is given as (see Wu, 1992): € þ 2fx/ ðU_ þ e1 U_ 3 Þ þ x2 ðU þ e2 U3 Þ ¼ U /
70 X
0:07sinðxi t þ /i Þ
ð29Þ
i¼1
The values of the linear and nonlinear damping and restoring coefficients used are given in Table 1. The wave exciting moment is composed of 70 sinusoidal components of constant amplitude of 0.07 1/s2. The phase angle, /i, between these components is taken as a random variable uniformly distributed between 0 and 2p. The frequency band for the excitation is taken between 2.0 and 5.0 rad/s, with components’ frequencies to ensure that the excitation is a broadband process (see Haddara and Hinchey, 1995; Haddara, 1992). Four case studies are used in this work to test the accuracy of the proposed method in the identification of the linear and nonlinear parameters in the rolling equation. The values of the linear and nonlinear damping and restoring coefficients corresponding to each case are given in Table 1. Close attention has been given to the selection of these values in order to cover the range between the light damping motion and the heavy damping motion cases as shown in the table for case 1 and case 4, respectively. The proposed method in this work has been applied to the data generated using Eq. (29). The results are shown in Table 1. It is obvious in this table that the estimated linear parameters are close to the actual ones while the nonlinear parameters are not. This does not mean that the identification process using the proposed method has failed. This is obvious when the estimated parameters are substituted back in Eq. (29) and the predicted free roll response is obtained and it is found that the predicted response is close to the simulated ones as shown in Figs. 3–6. Table 1 Comparison between the actual and the estimated parameters from the numerical simulation Case
1 2 3 4
Actual
Predicted
f
x/
e1
e2
f
x/
0.0240 0.1500 0.2000 0.2500
3.39 5.00 5.00 5.00
3.204 3.204 3.204 3.204
0.148 0.148 0.148 0.148
0.0270 0.1735 0.2237 0.2754
3.38 5.02 4.92 5.00
e1 0.8336 0.1039 0.0574 3.4521
e2 0.6541 1.9274 0.3190 51.4314
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Fig. 3. Comparison between the simulated and the predicted free response [Case # 1].
The results for the lightly damped motion case are shown in Fig. 3. This figure shows a comparison between the generated free roll response and the predicted one. Figs. 4–6 show similar results for the heavy damping motions. The agreement between the generated and the predicted free roll response for the light damping case is excellent; however, and for the heavy damping motions, it is good. Even when a common excitation is applied for both the actual and the estimated parameters in Eq. (29), it produces an excellent agreement between the generated and predicted roll responses as shown in Figs. 7–10. In these figures the
Fig. 4. Comparison between the simulated and the predicted free response [Case # 2].
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Fig. 5. Comparison between the simulated and the predicted free response [Case # 3].
Fig. 6. Comparison between the simulated and the predicted free response [Case # 4].
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Fig. 7. Comparison between the simulated and the predicted regular response [Case # 1].
Fig. 8. Comparison between the simulated and the predicted regular response [Case # 2].
wave excitation is taken as: F ðtÞ ¼ 0:2sinð2:5t þ pÞ
ð30Þ
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Fig. 9. Comparison between the simulated and the predicted regular response [Case # 3].
In addition, the proposed method highlights that the estimated nonlinear parameters of damping and restoring moments in the rolling equation are no longer to be close to the actual ones.
Fig. 10. Comparison between the simulated and the predicted regular response [Case # 4].
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The predicted free response using the proposed method is excellent as shown in the previous figures. In conclusion, the numerical simulation proves that the accuracy of the proposed method in the identification of the linear and nonlinear parameters in the rolling equation is high even in the case of the heavily damped motions. After testing the accuracy of the proposed method, there is an essential need to validate it. The only way for this to happen is to use an experimental data as shown in the next section. 3.2. Experimental data Experimental data have been obtained by testing a Series 60 Model in the towing tank of Memorial University. Table 2 shows the main particulars of the model. JONSWAP wave spectrum with different significant wave heights and different modal frequencies was used as an exciting moment in the tank. Three main runs were used to validate the proposed method with each being carried out twice. For the first two runs (A5401 and A5601), JONSWAP spectra were taken with constant wave modal frequency of 0.5 Hz and with two significant wave heights of 4 and 6 cm, respectively. For the third run (A7401), the wave modal frequency and the significant wave height were taken to be 0.7 Hz and 4 cm, respectively. In addition, calm water experiments have been carried out to measure the free roll motion response (A0005) for the model. This has been conducted by giving the model an initial angle of heel. After the model is initially inclined, it is then released and the free rolling response is measured. The random and free roll motion responses are measured using the available data acquisition system at the wave tank. Table 2 Hydrostatic particulars for Series 60 Model (without appendages) block coefficient 0.60, and scale = 1:40 Length between perpendiculars Length on water line Waterline beam at midships Draught at midships Draught at maximum section Maximum draught Draught above the keel Maximum section forward of midships Area of midships section Longitudinal center of buoyancy Vertical center of buoyancy above datum (KB) Wetted surface area Volume of displacement Displacement in fresh water Longitudinal center of floatation (LCF) Area of waterline plane Transverse metacentric radius (BM)
3.0480 m 3.0920 m 0.4065 m 0.1625 m 0.1625 m 0.1625 m 0.1625 m 0.0380 m 0.1295 m2 +0.0455 m 0.0870 m 1.5924 m2 0.1206 m3 120.45 kg +0.1155 m 0.8768 m2 0.0767 m
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Table 3 Estimated linear and nonlinear parameters from experiments Run
A5401 A5601 A7401
Actual f
x/
0.0161 0.0196 0.0189
3.38 3.41 3.41
e1 5.8316 0.2646 0.6297
e2 2.2908 0.1828 0.3355
Using only the measured random roll response, the linear and nonlinear parameters in the equation describing the rolling motion of a ship are identified using the proposed method. The results are shown in Table 3. A better way to validate the proposed method is to use the predicted parameters to generate the predicted random and free responses of the system, which can be compared with the measured ones. Since the wave excitation is nonmeasurable, the nonhomogenous equation of the predicted system cannot be integrated to predict the corresponding random response. In addition, the comparison between the measured random decrement, auto-correlation signatures, and the predicted ones cannot be conducted. Therefore, the validation of the proposed method is limited only to the comparison of the measured free response (A0005) and the predicted one. The predicted free motion response is obtained by the numerical integration of Eq. (29) corresponding to the identified parameters for each run. The comparison of the predicted and measured free responses is excellent as shown in Figs. 11–13.
Fig. 11. Comparison between the measured and the predicted free response [A5401].
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Fig. 12. Comparison between the measured and the predicted free response [A5601].
Fig. 13. Comparison between the measured and the predicted free response [A7401].
4. Conclusions It has been shown from the experimental results that the effect of varying a significant wave height on the agreement between the random decrement, auto-corre-
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lation, and free response signatures is not significant. However, varying the wave modal frequency may cause a considerable effect on that agreement. In addition, the proposed method highlights that the estimated nonlinear parameters of damping and restoring moments in the rolling equation are no longer to be closed to the actual ones. The accuracy of the proposed method has been tested using numerical generated data and has been validated using experimental data. The proposed method is shown as an accurate method in the identification of the parameters involved in the equation describing the rolling motion of a ship in a realistic sea using only the measured response at sea. Approximate theoretical basis for estimating the free nonlinear roll response of a ship from its steady state response in random waves has been investigated. It has been proven that the free response for the nonlinear rolling motion can be obtained using the proposed method as well as from either the propagation of the expected value of the rolling motion response in a random sea, or from the calculation of the auto-correlation function of the steady state random rolling response. In all cases the measurement of the wave excitation is not required. The linear and nonlinear damping and restoring moments parameters predicted using the proposed method seem to be more accurate since a good agreement between the predicted free response and the measured one is excellent. The error in the damped natural frequency of rolling motion predicted using the proposed method is less than 1%. The method developed in this paper will serve as a very practical tool for estimating the linear and nonlinear damping and restoring moments parameters and the roll natural frequency for a ship sailing in a realistic sea. This will provide a means for an immediate estimation of the margin against capsizing for a ship sailing in a certain conditions.
Acknowledgements The author is grateful for the financial support from NSERC (the Natural Sciences and Engineering Research Council of Canada) and InCoreTec Incorporated. The author would like to thank Professor M.R. Haddara for his support and guidance, and for providing constructive comments. Also, the author would like to thank Mrs. Mona El-Tahan, President & COO of InCoreTec Inc., and Dr. Hussein El-Tahan, Vice President of InCoreTec Inc. for their support. In addition, the author acknowledges the assistance of Miss Shana Murphy and Mr. David Chafe of InCoreTec for reviewing the paper.
References Cole Jr., H.A., 1973. On-line failure detection and damping measurement of aerospace structures by random decrement signatures. NASA CR-2205. Dingman, S.L., Sharma, K.P., 1997. Statistical development and validation of discharge equations for natural channels. Journal of Hydrology 199, 13–35.
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Flood, I., Kartam, N., 1994. Neural network in civil engineering I: principles and understanding. Journal of Computing in Civil Engineering 8 (2), 131–148. Haddara, M.R., 1992. On the random decrement for nonlinear rolling motion. In: Proceedings of the 12th International Conference on Offshore Mechanics and Arctic Engineering, Calgary, Canada, pp. 283–288. Haddara, M.R., 1995. On the use of neural network techniques for the identification of ship stability parameters at sea. In: Proceedings of the 14th International Conference on Offshore Mechanics and Arctic Engineering, Copenhagen, Denmark, pp. 127–135. Haddara, M.R., Hinchey, M., 1995. On the use of neural network techniques in the analysis of free roll decay curves. International Shipbuilding Progress 42 (430), 166–178. Haddara, M.R., Xu, J., 1997. On the identification of ship coupled heave and pitch motions using neural networks. Ocean Engineering 26 (5), 381–400. Montgomery, D., 1997. Design and Analysis of Experiments. John Wiley & Sons, New York, NY. Roberts, J.B., Dunne, J.F., Debonos, A., 1991. Estimation of ship roll parameters in random waves. Proceedings of the 10th International Conference on Offshore Mechanics and Arctic Engineering, Safety and Reliability, vol. II. Stavanger, Norway, pp. 97–105. Roberts, J.B., Dunne, J.F., Debonos, A., 1994. Stochastic estimation methods for non-linear ship roll motion. Probabilistic Engineering Mechanics 9, 83–93. Wu, X., 1992. Roll motion parameter identification using response in random waves. Master’s Thesis, Faculty of Engineering and Applied Science, Memorial University of Newfoundland.
Identification of the nonlinear ship rolling motion equation using the measured response at sea Ayman B. Mahfouz InCoreTec Incorporated, Baine Johnston Center, Suite 804, 10 Fort William Place, St. John’s, Newfoundland, Canada A1C 1K4 Received 26 October 2003; accepted 28 June 2004
Abstract This paper describes a new robust method for the identification of the parameters in the equation describing the rolling motion of a ship using only its measured response at sea. Those parameters are the linear and nonlinear damping and restoring parameters. The random decrement equations as well as the auto- and cross-correlation equations are derived for a ship performing rolling motion in random beam waves. The linear and nonlinear parameters in the equation of motion are identified using a combination of the random decrement technique, auto- and cross-correlation functions, a linear regression algorithm, and a neural networks technique. The combination of the classical parametric identification techniques and a neural networks technique provides robust results and does not require a large amount of computer time. The proposed method would be particularly useful in identifying the nonlinear damping and restoring parameters for a ship rolling under the action of unknown excitations effected by a realistic sea. Numerically generated data and experimental data for the ship rolling motion are used to test the accuracy and the validity of the method. It is shown that the method is reliable in the identification of the parameters of the equation of the rolling motion using only the measured response at sea. # 2004 Elsevier Ltd. All rights reserved. Keywords: Ship hydrodynamics; Dynamical stability; Neural networks; Random decrement signature; Ship motion; Parametric identification techniques
Corresponding author. Tel.: +709-739-1760; fax: +709-739-7780. E-mail address: [email protected] (A.B. Mahfouz).
0029-8018/$ - see front matter # 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2004.06.001
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1. Introduction The stability of a ship in realistic sea is mainly dependent on its rolling motion, which is the most critical factor in defining ship survivability. In fact, most available approaches for the assessment of ships’ survivability in a seaway are based on the study of rolling motion. Two approaches are used in this assessment: a static approach (quasi-static) and a dynamic one. The former is based on the minimum value that the metacenteric height, GM, should have and the shape of the static stability curve (GZ–h). This approach is still being applied in the assessment of ship’s stability criterion. The latter approach is based on the analysis of the stability of the equation of the rolling motion. This involves constructing a model for a ship rolling in a realistic sea. Using this model, the occurrences of capsizing modes can be predicted (see Haddara, 1992; Roberts et al., 1991, 1994). The linear restoring parameters can be easily obtained from ship hydrostatics; however, the damping parameters are not. Therefore, the main objective of this paper is to develop a reliable and robust method that can be used in the identification of the parameters involved in the equation describing the roll motion of a ship in a random sea using only its measured random response. Since the wave excitation is usually not measurable, the long-term objective of this work is to provide a reliable methodology to assess the ship’s survivability at sea. It has been shown that the roll auto-correlation function satisfies the same equation of motion as the free roll response (see Haddara and Xu, 1997; Haddara, 1992). This is true for lightly damped motions. When the damping increases, the auto-correlation function becomes more influenced by the wave excitations (see Haddara and Xu, 1997). In this work, a new robust method is presented to identify the linear and nonlinear damping and restoring parameters involved in the rolling equation without prior knowledge of the wave excitation. This has been achieved using a combination of the random decrement technique, auto- and cross-correlation functions, a multiple linear regression algorithm, and a neural networks technique. 2. Identification method The identification method is based on the derivation of the random decrement equations and the auto-correlation functions for the rolling motion. Linear regression is used to identify the linear parameters and neural networks are used to identify the nonlinear rolling function. Details of the identification method are presented in the following sections. 2.1. Random decrement equations The random decrement technique is an averaging technique that has been used successfully in the on-line failure detection and damping identification of linear structures (see Cole, 1973). In general, the response of a single degree of freedom system has two main components: the transient and the steady state. The random
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decrement technique is based on the fact that when the general response of a single degree of freedom system is averaged, the contribution of the steady state component to the expected value of the response vanishes when the excitation function is a Gaussian white noise random process. Then, the expected value of the random response is represented only by the contribution of its transient component caused by the initial conditions (see Haddara, 1992). This ensemble average is the random decrement signature, which can be calculated from the random roll response using methods described in the literature (see Haddara, 1992; Cole, 1973). A forced second order differential equation is a reasonable model for a ship performing rolling motion in random seas. The main parameters of this model are the virtual moment of inertia, the damping moment, the restoring moment, and the exciting moment. This equation can be normalized with respect to the virtual moment of inertia. In this case, the equation has three unknowns only (see Haddara, 1992). The equation describing the rolling motion of a ship excited by random beam waves can be written as (see Haddara, 1992, 1995): € þ NðU_ Þ þ DðUÞ ¼ KðtÞ U
ð1Þ
where U is the roll angle, NðU_ Þ and D(U) are the nonlinear damping and restoring moments per unit virtual moment of inertia of the ship, and K(t) is the wave exciting moment per unit virtual moment of inertia of the ship. A dot over the variable denotes the differentiation with respect to time. The wave excitation K(t) is assumed to be a zero mean Gaussian, white random process (see Haddara, 1992; Roberts et al., 1991, 1994). Using the following change of variables: y1 ¼ U
ð2Þ
y2 ¼ U_
ð3Þ
Eq. (1) can be easily replaced by two first order differential equations as shown in Eq. (4) (see Haddara, 1992): Y_ ¼ F ðY Þ þ EðtÞ
ð4Þ
where Y_ ¼ ½y1 y2 T F ðY Þ ¼ ½y2 Nðy2 Þ Dðy1 ÞT EðtÞ ¼ ½0 KðtÞT Assuming that the process Y(t) is a Markov process, the Fokker–Planck equation can be used to describe the conditional probability density function for the random process, Y(t) as shown (see Haddara, 1992): @P @ @ 1 @2 ¼ ðy2 PÞ þ ð½Nðy2 Þ þ Dðy1 ÞPÞ þ ðW2 PÞ @t @y1 @y2 2 @y22
ð5Þ
where W2 is the variance of the exciting function K(t) and can be calculated if K(t)
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is measurable: W2 ¼ E ½KðtÞ KðtÞ where E[ ] denotes the expected value. The solution of Eq. (5) subject to the initial condition PðY ; tjYo Þ ¼ dðy1 y1o Þdðy2 y2o Þ yields the conditional probability density function which describes the Markov process Y(t) completely. Where yio, i ¼ 1; 2 are the initial conditions for the roll displacement and the roll rate, respectively. Multiplying Eq. (5) by the variables y1(t) and y2(t), respectively, and integrating the whole equations over the range of the two variables, the propagation of the expected values of y1(t) and y2(t) can be obtained as: l_ 1 ¼ l2
ð6Þ
l_ 2 ¼ E½Nðy2 Þ þ Dðy1 Þ
ð7Þ
where l1 ¼ l2 ¼
ðð y1 PðY ; tÞ dy1 dy2 ðð y2 PðY ; tÞ dy1 dy2
The evaluation of the averages on the right-hand side of Eq. (7) poses a problem since the probability density function P(Y,t), for the Markov process Y(t), is yet to be determined. Replacing these averages by their Taylor series expansion about l1 and l2 solves this problem (see Haddara, 1992). Retaining the first order terms from this expansion, Eqs. (6) and (7) can be combined in a one second order differential equation as given in Eq. (14). The damping moment, N(y2), can be expressed as the sum of two terms: a linear term in the roll rate and a nonlinear term. Two forms have been suggested to represent the nonlinear term: the quadratic form suggested by Froude and the cubic form suggested by Haddara (1995). In this work, the following expressions for the damping and restoring moments are used (see Haddara and Hinchey, 1995; Haddara, 1992): Nðy2 Þ ¼ 2fx/ ½1 þ e1 y22 y2 Dðy1 Þ ¼
x2/ ½1
þ
e2 y21 y1
ð8Þ ð9Þ
where f, e1 and e2 are the nondimensional linear damping coefficient, the nonlinear damping coefficient, and the nonlinear restoring moment coefficient, respectively, x/ is the linear natural frequency. Using Eqs. (8) and (9), the nonlinear damping and restoring moments in Eq. (4) can be written in the following form: Nðy2 Þ þ Dðy1 Þ ¼ 2fx/ y2 þ x2/ y1 þ F ðy1 ; y2 Þ
ð10Þ
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where F ðy1 ; y2 Þ ¼ 2fx/ e1 y32 þ x2/ e2 y31
ð11Þ
The expected value of Eq. (10) is obtained as ðy1 ; y2 Þ E½Nðy2 Þ þ Dðy1 Þ ¼ 2fx/ l2 þ x2/ l1 þ F
ð12Þ
where ðy1 ; y2 Þ ¼ 2fx/ e1 E½y3 þ x2 e2 E½y3 F 2 1 /
ð13Þ
Eq. (13) represents the lamped nonlinear terms of the damping and restoring moments. Eqs. (6) and (7) are combined as: ðy1 ; y2 Þ ¼ 0 €1 þ 2fx/ l_ 1 þ x2/ l1 þ F l
ð14Þ
It is obvious that the approximate equation describing the expected value of the roll response, l1 satisfies the differential equation, which describes the free rolling ðy1 ; y2 Þ, in Eq. (14) motion of a ship. The linear terms and the nonlinear function, F are identified in the following two sections, respectively. 2.2. Auto- and cross-correlation functions The derivation of the auto- and cross-correlation equations corresponding to Eq. (1) is based on the Fokker–Planck equation (5). The latter equation is to be multiplied by yi ðtÞ yj ðt þ sÞ Ps ðYo Þ; i ¼ 1; 2 and j ¼ 1; 2 each in time and integrating the whole equations over the range of the two variables. The following equations are obtained: R_ 11 ¼ R21 ðð _R21 ¼ ½Nðy2 Þ þ Dðy1 Þy1 ðt þ sÞPðY ; tÞPs ðY Þ dy1 dy2
ð15Þ
R_ 12 ¼ R22 ðð R_ 22 ¼ ½Nðy2 Þ þ Dðy1 Þy2 ðt þ sÞPðY ; tÞPs ðY Þ dy1 dy2
ð17Þ
ð16Þ
ð18Þ
Substituting Eqs. (8) and (9) into Eqs. (16) and (18) one can get the following equations as: ðð R_ 21 ¼ x2/ R11 2fx/ R21 F ðy1 ; y2 Þy1 ðt þ sÞPðY ; tÞPs ðY Þ dy1 dy2 ð19Þ R_ 22 ¼ x2/ R12 2fx/ R22
ðð
F ðy1 ; y2 Þy2 ðt þ sÞPðY ; tÞPs ðY Þ dy1 dy2
ð20Þ
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Eqs. (19) and (20) can be written as R_ 21 ¼ x2/ R11 2fx/ R21 E½F ðy1 ; y2 Þy1 ðt þ sÞ
ð21Þ
R_ 22 ¼ x2/ R12 2fx/ R22 EðF ðy1 ; y2 Þy2 ðt þ sÞ
ð22Þ
Eqs. (21) and (22) have been identified using a multiple linear regression algorithm. In the multiple linear regression cases, certain tests of hypotheses about the model coefficients are helpful in measuring the usefulness of the model (see Montgomery, 1997). Two main statistic indexes are important in measuring the usefulness of the model: P-value and R-sq. In addition, there is another criterion for model acceptance known as the multicollinearity of the predictors. This is related to the variance inflation factor (VIF) (see Dingman and Sharma, 1997; Montgomery, 1997). The computed values of the VIF exceeding 10 indicate potentially detrimental multicollinearity effects. From the regression process derived for Eqs. (21) and (22) using MINITAB software, it is found that when the last term in each equation is included in the models, the calculated VIF is more than 10 and the P-value is more than 0.05. It is concluded that the last term in these equations is insignificant in the models (Eqs. (21) and (22)). Therefore, these terms are removed from the equations. As a result, the final strong relationship in these equations can be written as: R_ 21 ¼ x2/ R11 2fx/ R21
ð23Þ
R_ 22 ¼ x2/ R12 2fx/ R22
ð24Þ
The linear parameters in Eqs. (23) and (24) are identified only from the measured roll response using a multiple linear regression algorithm as described above (see Montgomery, 1997). The auto- and cross-correlation functions are calculated from the roll response using (see Haddara and Xu, 1997): Rik ðsÞ ¼
N p s X 1 yi ðjÞyk ðj þ sÞ ðNp sÞ j¼1
ð25Þ
where Np and s are total number of points in the roll response, and the time difference, respectively. 2.3. Neural networks In general, artificial neural networks try to mimic the biological network (see Haddara and Hinchey, 1995). The present artificial neural networks are considered much simpler compared to the biological networks especially in the number of neurons, size, and construction complexity (see Flood and Kartam, 1994). The combination of the neural networks technique with the standard parametric techniques provides a means for the identification of the nonlinearity in the roll equation without a prior knowledge of the form of the nonlinearity (see Haddara and Hinchey, 1995).
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Fig. 1. Feedforward neural network.
The nonlinearity in the equation describing the rolling motion of a ship sailing in ðy1 ; y2 Þas shown in Eq. (14). A continuous record of random seas is the function F the measured roll response is used with Eq. (14) to identify the nonlinear function using a neural networks technique. A single hidden layer feedforward neural network, as shown in Fig. 1, is used to identify the nonlinear function in Eq. (14). The input to the network is a vector having the expected value of the measured roll response, the expected value of the roll rate, and a bias. The hidden layer may consist of several neurons. In this paper, a value of 11 neurons gives a sufficient accuracy in the identification process. The output layer consists of one neuron whose output is the identified nonlinear function (see Haddara, 1995; Flood and Kartam, 1994). The input to the hidden layer neuron ith is the weighted sum of the inputs expressed as X Aj ¼ wij li ; i ¼ 1; . . . ; 3 and j ¼ 1; . . . ; 12 ð26Þ i;j
where wij is the synaptic weight of the ith neuron in the input layer to the jth neuron in the hidden layer. l3 is the bias and is equal to 1.0. The weighted sum of the inputs into hidden layer neurons are processed by a sigmoidal function f, expressed as f ðAj Þ ¼
1 1 þ eðAj Þ
ð27Þ
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Then, the nonlinear function F ðy1 ; y2 Þ is equal to the weighted sum of the outputs of the hidden layer neurons expressed as
ðy1 ; y2 Þ ¼ F
12 X bj f ðAj Þ
ð28Þ
i¼1
where bj is the synaptic weight of neuron j in the hidden layer. Using an arbitrary starting set of weights, an initial value of the nonlinear function is obtained. The obtained value of that function is substituted back in Eq. (14), which is integrated numerically using a Runge–Kutta algorithm. The corresponding roll response obtained from the integration is compared to the input roll response to the network. The difference between these two responses is the error. The synaptic weights wij and bj are then updated and the process is repeated in an iterative fashion until the error in the response is minimized. A simple flowchart for the neural networks technique is shown in Fig. 2. The nonlinear damping and restoring parameters e1 and e2, respectively, are then obtained using a multiple linear regression algorithm for Eq. (13) (see Dingman and Sharma, 1997; Montgomery, 1997).
Fig. 2. Neural networks technique.
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3. Validation Numerically generated data and experimental data for the ship rolling motion are used to test the accuracy and the validity of the method. 3.1. Numerically generated data A mathematical model that has been used in the generation of the random roll response is given as (see Wu, 1992): € þ 2fx/ ðU_ þ e1 U_ 3 Þ þ x2 ðU þ e2 U3 Þ ¼ U /
70 X
0:07sinðxi t þ /i Þ
ð29Þ
i¼1
The values of the linear and nonlinear damping and restoring coefficients used are given in Table 1. The wave exciting moment is composed of 70 sinusoidal components of constant amplitude of 0.07 1/s2. The phase angle, /i, between these components is taken as a random variable uniformly distributed between 0 and 2p. The frequency band for the excitation is taken between 2.0 and 5.0 rad/s, with components’ frequencies to ensure that the excitation is a broadband process (see Haddara and Hinchey, 1995; Haddara, 1992). Four case studies are used in this work to test the accuracy of the proposed method in the identification of the linear and nonlinear parameters in the rolling equation. The values of the linear and nonlinear damping and restoring coefficients corresponding to each case are given in Table 1. Close attention has been given to the selection of these values in order to cover the range between the light damping motion and the heavy damping motion cases as shown in the table for case 1 and case 4, respectively. The proposed method in this work has been applied to the data generated using Eq. (29). The results are shown in Table 1. It is obvious in this table that the estimated linear parameters are close to the actual ones while the nonlinear parameters are not. This does not mean that the identification process using the proposed method has failed. This is obvious when the estimated parameters are substituted back in Eq. (29) and the predicted free roll response is obtained and it is found that the predicted response is close to the simulated ones as shown in Figs. 3–6. Table 1 Comparison between the actual and the estimated parameters from the numerical simulation Case
1 2 3 4
Actual
Predicted
f
x/
e1
e2
f
x/
0.0240 0.1500 0.2000 0.2500
3.39 5.00 5.00 5.00
3.204 3.204 3.204 3.204
0.148 0.148 0.148 0.148
0.0270 0.1735 0.2237 0.2754
3.38 5.02 4.92 5.00
e1 0.8336 0.1039 0.0574 3.4521
e2 0.6541 1.9274 0.3190 51.4314
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Fig. 3. Comparison between the simulated and the predicted free response [Case # 1].
The results for the lightly damped motion case are shown in Fig. 3. This figure shows a comparison between the generated free roll response and the predicted one. Figs. 4–6 show similar results for the heavy damping motions. The agreement between the generated and the predicted free roll response for the light damping case is excellent; however, and for the heavy damping motions, it is good. Even when a common excitation is applied for both the actual and the estimated parameters in Eq. (29), it produces an excellent agreement between the generated and predicted roll responses as shown in Figs. 7–10. In these figures the
Fig. 4. Comparison between the simulated and the predicted free response [Case # 2].
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Fig. 5. Comparison between the simulated and the predicted free response [Case # 3].
Fig. 6. Comparison between the simulated and the predicted free response [Case # 4].
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Fig. 7. Comparison between the simulated and the predicted regular response [Case # 1].
Fig. 8. Comparison between the simulated and the predicted regular response [Case # 2].
wave excitation is taken as: F ðtÞ ¼ 0:2sinð2:5t þ pÞ
ð30Þ
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Fig. 9. Comparison between the simulated and the predicted regular response [Case # 3].
In addition, the proposed method highlights that the estimated nonlinear parameters of damping and restoring moments in the rolling equation are no longer to be close to the actual ones.
Fig. 10. Comparison between the simulated and the predicted regular response [Case # 4].
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The predicted free response using the proposed method is excellent as shown in the previous figures. In conclusion, the numerical simulation proves that the accuracy of the proposed method in the identification of the linear and nonlinear parameters in the rolling equation is high even in the case of the heavily damped motions. After testing the accuracy of the proposed method, there is an essential need to validate it. The only way for this to happen is to use an experimental data as shown in the next section. 3.2. Experimental data Experimental data have been obtained by testing a Series 60 Model in the towing tank of Memorial University. Table 2 shows the main particulars of the model. JONSWAP wave spectrum with different significant wave heights and different modal frequencies was used as an exciting moment in the tank. Three main runs were used to validate the proposed method with each being carried out twice. For the first two runs (A5401 and A5601), JONSWAP spectra were taken with constant wave modal frequency of 0.5 Hz and with two significant wave heights of 4 and 6 cm, respectively. For the third run (A7401), the wave modal frequency and the significant wave height were taken to be 0.7 Hz and 4 cm, respectively. In addition, calm water experiments have been carried out to measure the free roll motion response (A0005) for the model. This has been conducted by giving the model an initial angle of heel. After the model is initially inclined, it is then released and the free rolling response is measured. The random and free roll motion responses are measured using the available data acquisition system at the wave tank. Table 2 Hydrostatic particulars for Series 60 Model (without appendages) block coefficient 0.60, and scale = 1:40 Length between perpendiculars Length on water line Waterline beam at midships Draught at midships Draught at maximum section Maximum draught Draught above the keel Maximum section forward of midships Area of midships section Longitudinal center of buoyancy Vertical center of buoyancy above datum (KB) Wetted surface area Volume of displacement Displacement in fresh water Longitudinal center of floatation (LCF) Area of waterline plane Transverse metacentric radius (BM)
3.0480 m 3.0920 m 0.4065 m 0.1625 m 0.1625 m 0.1625 m 0.1625 m 0.0380 m 0.1295 m2 +0.0455 m 0.0870 m 1.5924 m2 0.1206 m3 120.45 kg +0.1155 m 0.8768 m2 0.0767 m
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Table 3 Estimated linear and nonlinear parameters from experiments Run
A5401 A5601 A7401
Actual f
x/
0.0161 0.0196 0.0189
3.38 3.41 3.41
e1 5.8316 0.2646 0.6297
e2 2.2908 0.1828 0.3355
Using only the measured random roll response, the linear and nonlinear parameters in the equation describing the rolling motion of a ship are identified using the proposed method. The results are shown in Table 3. A better way to validate the proposed method is to use the predicted parameters to generate the predicted random and free responses of the system, which can be compared with the measured ones. Since the wave excitation is nonmeasurable, the nonhomogenous equation of the predicted system cannot be integrated to predict the corresponding random response. In addition, the comparison between the measured random decrement, auto-correlation signatures, and the predicted ones cannot be conducted. Therefore, the validation of the proposed method is limited only to the comparison of the measured free response (A0005) and the predicted one. The predicted free motion response is obtained by the numerical integration of Eq. (29) corresponding to the identified parameters for each run. The comparison of the predicted and measured free responses is excellent as shown in Figs. 11–13.
Fig. 11. Comparison between the measured and the predicted free response [A5401].
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Fig. 12. Comparison between the measured and the predicted free response [A5601].
Fig. 13. Comparison between the measured and the predicted free response [A7401].
4. Conclusions It has been shown from the experimental results that the effect of varying a significant wave height on the agreement between the random decrement, auto-corre-
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lation, and free response signatures is not significant. However, varying the wave modal frequency may cause a considerable effect on that agreement. In addition, the proposed method highlights that the estimated nonlinear parameters of damping and restoring moments in the rolling equation are no longer to be closed to the actual ones. The accuracy of the proposed method has been tested using numerical generated data and has been validated using experimental data. The proposed method is shown as an accurate method in the identification of the parameters involved in the equation describing the rolling motion of a ship in a realistic sea using only the measured response at sea. Approximate theoretical basis for estimating the free nonlinear roll response of a ship from its steady state response in random waves has been investigated. It has been proven that the free response for the nonlinear rolling motion can be obtained using the proposed method as well as from either the propagation of the expected value of the rolling motion response in a random sea, or from the calculation of the auto-correlation function of the steady state random rolling response. In all cases the measurement of the wave excitation is not required. The linear and nonlinear damping and restoring moments parameters predicted using the proposed method seem to be more accurate since a good agreement between the predicted free response and the measured one is excellent. The error in the damped natural frequency of rolling motion predicted using the proposed method is less than 1%. The method developed in this paper will serve as a very practical tool for estimating the linear and nonlinear damping and restoring moments parameters and the roll natural frequency for a ship sailing in a realistic sea. This will provide a means for an immediate estimation of the margin against capsizing for a ship sailing in a certain conditions.
Acknowledgements The author is grateful for the financial support from NSERC (the Natural Sciences and Engineering Research Council of Canada) and InCoreTec Incorporated. The author would like to thank Professor M.R. Haddara for his support and guidance, and for providing constructive comments. Also, the author would like to thank Mrs. Mona El-Tahan, President & COO of InCoreTec Inc., and Dr. Hussein El-Tahan, Vice President of InCoreTec Inc. for their support. In addition, the author acknowledges the assistance of Miss Shana Murphy and Mr. David Chafe of InCoreTec for reviewing the paper.
References Cole Jr., H.A., 1973. On-line failure detection and damping measurement of aerospace structures by random decrement signatures. NASA CR-2205. Dingman, S.L., Sharma, K.P., 1997. Statistical development and validation of discharge equations for natural channels. Journal of Hydrology 199, 13–35.
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Flood, I., Kartam, N., 1994. Neural network in civil engineering I: principles and understanding. Journal of Computing in Civil Engineering 8 (2), 131–148. Haddara, M.R., 1992. On the random decrement for nonlinear rolling motion. In: Proceedings of the 12th International Conference on Offshore Mechanics and Arctic Engineering, Calgary, Canada, pp. 283–288. Haddara, M.R., 1995. On the use of neural network techniques for the identification of ship stability parameters at sea. In: Proceedings of the 14th International Conference on Offshore Mechanics and Arctic Engineering, Copenhagen, Denmark, pp. 127–135. Haddara, M.R., Hinchey, M., 1995. On the use of neural network techniques in the analysis of free roll decay curves. International Shipbuilding Progress 42 (430), 166–178. Haddara, M.R., Xu, J., 1997. On the identification of ship coupled heave and pitch motions using neural networks. Ocean Engineering 26 (5), 381–400. Montgomery, D., 1997. Design and Analysis of Experiments. John Wiley & Sons, New York, NY. Roberts, J.B., Dunne, J.F., Debonos, A., 1991. Estimation of ship roll parameters in random waves. Proceedings of the 10th International Conference on Offshore Mechanics and Arctic Engineering, Safety and Reliability, vol. II. Stavanger, Norway, pp. 97–105. Roberts, J.B., Dunne, J.F., Debonos, A., 1994. Stochastic estimation methods for non-linear ship roll motion. Probabilistic Engineering Mechanics 9, 83–93. Wu, X., 1992. Roll motion parameter identification using response in random waves. Master’s Thesis, Faculty of Engineering and Applied Science, Memorial University of Newfoundland.