Identification of the excitation and reaction forces on offshore platforms using the random decrement technique

ARTICLE IN PRESS Ocean Engineering 36 (2009) 521–528

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Identification of the excitation and reaction forces on offshore platforms using the random decrement technique Ahmed A. Elshafey a,b, Mahmoud R. Haddara c,, H. Marzouk c a b c

Faculty of Engineering, Minufiya University, Egypt Visiting Scholar, Faculty of Engineering and Applied Science, Memorial University of Newfoundland, St. John’s, Canada Faculty of Engineering and Applied Science, Memorial University of Newfoundland, St. John’s, Canada

a r t i c l e in fo

abstract

Article history: Received 16 May 2007 Accepted 18 February 2009 Available online 3 March 2009

This paper describes a method to identify the parameters of the dynamic model of a fixed offshore platform subjected to wind-generated random waves using its stationary response. The structure is modeled as a single degree of freedom system. The parameters identified are the damping coefficient, the natural frequency, and the excitation. In addition, the moment and force acting on the foundation are also identified. The method uses the random decrement signature as a tool to identify the parameters in the equation of motion. Excellent agreements were obtained between the predicted and actual values of the parameters as well as for the reaction and moment at the platform’s foundation. The method can be applied without any interruption to the operation of the offshore structure. The method is easy to apply, and uses inexpensive motion measurement instruments. The estimated force and moment can be used as a tool for an on-line foundation check. & 2009 Elsevier Ltd. All rights reserved.

Keywords: Parametric identification Offshore structures Random decrement

1. Introduction Early detection of fatigue cracks occurring in offshore platform members is critical to the safe, efficient, and economic operation of the platform. Several approaches have been suggested to rationalize inspection of offshore platforms using reliability-based methods (Onoufriou, 1999; Pillai and Prasad, 2000) and damage detection techniques (Viero and Roitman, 1999). Offshore platforms subjected to random waves are usually modeled as multidegree of freedom systems. The forces caused by the random waves can excite certain vibratory modes corresponding to frequencies near or equal to wave frequencies. A number of vibration-based damage detection techniques have also been suggested (Budipriyanto et al., 2007). Other methods which depend on the measurement of the vibratory response of structures only can also be used e.g. operational modal analysis technique (Brincker et al., 2001). The random decrement (RD) technique has been successfully applied to multi-degree of freedom systems to predict early damage occurrence (Zubaydi et al., 2000). The Random decrement is an averaging technique that can be used to extract the free decaying response of a vibrating body from its random excited stationary response. It was first introduced by Cole (1968) to identify the damping of an aerospace structure using stationary

random response. The Random decrement can be obtained without a prior knowledge of the excitation forces under the assumption that the forces are zero mean, stationary Gaussian random process. Owing to its efficiency and simplicity in processing vibration measurements and the lack of requirements for input excitation measurements, the method is applied extensively to detect damage in civil and offshore structures (Yang et al., 1980, 1984; Zubaydi et al., 2002; Budipriyanto et al., 2007). The method can also be used to identify mode shapes and frequencies of multi-degree of freedom systems (Ibrahim and Mikulcik, 1977). Vandiver et al. (1982) showed that the random decrement can be obtained from the auto-correlation function by multiplying the auto-correlation function by the threshold or triggering level. Haddara (1992) extended the random decrement technique to nonlinear systems. Zubaydi et al. (2000, 2002) and Budipriyanto et al. (2007) used the random decrement signature to identify the damage in the side shell of a ship.

2. Equation of motion The response of a single degree of freedom linear system is governed by the following basic dynamic equation: mx€ ðtÞ þ cx_ ðtÞ þ kxðtÞ ¼ FðtÞ

 Corresponding author.

E-mail address: [email protected] (M.R. Haddara). 0029-8018/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2009.02.005

(1)

where m stands for the total virtual mass, c for the damping, k for the stiffness, t for time, and F(t) is the external force. The total

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Nomenclature

x, x˙, x¨

c fr d F(t) Hs k m ODE P(Y,t|Yo) RD Tz

xs

damping coefficient (N s/m) natural frequency (Hz) logarithmic decrement excitation force (N) significant wave height (m) stiffness (N/m) mass (kg) ordinary differential equation conditional probability density random decrement average zero up crossing period (s)

d z

m1, m2 n11, n22 s2 co

oo, od

virtual mass is the sum of the physical and the hydrodynamic added mass of the system. Eq. (1) can be normalized with respect to the total virtual mass, m as x€ ðtÞ þ

c k FðtÞ x_ ðtÞ þ xðtÞ ¼ m m m

(2)

or x€ ðtÞ þ 2oo zx_ ðtÞ þ o2o xðtÞ ¼ f ðtÞ

(3)

where oo is the natural frequency (rad/s), z is the damping ratio, f (t) is the force per unit total virtual mass, and x(t) is the response of the system. A dot over the derivative indicates differentiation with respect to time. The random excitation f(t) is assumed to satisfy the following conditions: hf ðtÞi ¼ 0 hf ðtÞf ðt þ tÞi ¼ co dðtÞ

(4)

d is the Dirac delta function and co is the variance of the excitation. The following change of variables is used: y1 ¼ x;

y2 ¼ x_

(5)

y_ 1 ¼ y2

m_ 1 ¼ m2 m_ 2 ¼ h2zoo y2 þ o2o y1 i

(8)

where m1 and m2 stand for the mean values of the displacement and the velocity, respectively. Multiplying Eq. (7) by y21, y22 and y1y2, respectively, and integrating the whole equation over the complete domain of the variables y1 and y2, we get (See Appendix A)

n_ 11 ¼ 2n12 n_ 22 ¼ 2h2zoo y22 þ o2o y1 y2 i þ co n_ 12 ¼ n22  h2zoo y1 y2 þ o2o y21 i

(9)

where n11 is the variance of the displacement, n22 is the variance of the velocity, and n12 is the covariance of the displacement and velocity. Eqs. (8) and (9) describe the means and the variances of the displacement and velocity as functions of time. These equations will be used for the identification of the parameters in the equation of motion of the offshore structure. Eq. (8) can be combined in one equation as

m€ þ 2zoo m_ þ o2o m ¼ 0

(10)

(6)

It can be shown (Haddara, 2006) that the conditional probability density function P(Y,t|Yo) governing the vector random ( ) y1 , satisfies the Fokker–Planck equation given process, YðtÞ ¼ y2 by @P @ @ c @2 P ¼ ðy PÞ þ ½y f2zoo y2 þ o2o y1 gP þ o 2 @t @y1 2 @y2 1 2 @y2

(See Appendix A)

where m is the mean value of the displacement. Eq. (10) shows that the free decay motion can be derived from the stationary random response. This is the equation of the random decrement.

Using the change of variables (5) in Eq. (3), one gets

y_ 2 ¼ 2oo zy2  o2o y1 þ f ðtÞ

displacement (m), velocity (m/s) and acceleration (m/ s2), respectively triggering level Dirac delta function non-dimensional damping coefficient mean values of displacement and velocity, respectively the variance of displacement and velocity, respectively variance variance of excitation undamped and damped natural frequencies (rad/s), respectively

(7)

The symbol P is used in place of P(Y,t|Yo). Solution of Eq. (7) subject to the initial condition lim(P(Y,t|Yo) ¼ d(t) as t-0 yields an expression for the conditional probability density function which governs the process Y(t). Instead of solving Eq. (7), one can use it to derive expressions that describe the propagation of the mean and variance of the process, Y(t) as functions of time.

3. Equations of the means and variances By multiplying both sides of Eq. (7) by y1 and y2, respectively, and integrating the whole equation over the complete domain of the variables y1 and y2, it can be shown that

4. Random decrement signature Eq. (10) shows that the random decrement can be used to describe the free decay response of the system. The advantage of this approach is that one can obtain the free response from the stationary random response of the system. To obtain the random decrement from the stationary random response, the response is divided into a number of segments, N, each of length t. All of these segments should have the same initial condition, xi(ti) ¼ xs ¼ constant, i ¼ 1, y, N. The initial condition is called the triggering value. These segments will also have initial slopes with alternating signs. The ensemble average of the N segments yields the random decrement as shown in Fig. 1. This approach can be expressed mathematically using the following equations: xðtÞ ¼

N 1X x ðt þ tÞ N i¼1 i i

(11)

where xi(ti) ¼ xs for i ¼ 1, 2, 3, y, N, x˙i(ti)X0 for i ¼ 1, 3, 5, y, N1, x˙i(ti)p0 for i ¼ 2, 4, 6, y, N.

ARTICLE IN PRESS A.A. Elshafey et al. / Ocean Engineering 36 (2009) 521–528

x (t) xs

x (t)

t1 t 2

523

x (t) xs

xs 

t



Fig. 1. Random decrement approach: (a) random response, (b) RD after two summations, and (c) RD after N summations.

Eq. (11) gives the random decrement signature. It is assumed that using segments with the same initial value and alternating slope sign simulates an initial condition where the initial displacement is constant and the initial velocity is zero. The triggering level used in this paper is equal to the root mean square of the measured parameter. Vandiver et al. (1982) related the random decrement to the auto-correlation function and showed that the random decrement can be obtained from the autocorrelation function by multiplying it by the triggering level. The random decrement approach requires no knowledge of the excitation, f (t) as long as it is a stationary, zero-mean Gaussian random process.

5. Analysis of the random decrement signature The random decrement signature can be used to determine the natural frequency and the damping ratio. The random decrement obtained from the stationary response of a multi-degree of freedom system will contain all the frequencies of that system. In the approach we are presenting here, we are assuming that we can model the system as a single degree of freedom system, represented by its first mode. In order to isolate the first mode shape, the response signal has been passed through a band filter whose range encompasses the fundamental frequency of the structure. Thus, a random decrement is created for the fundamental mode of the structure. This approach was used successfully by Zubaydi et al. (2000) and Budipriyanto et al. (2007). The natural frequency can be identified from the random decrement signature using the measured period of one cycle, as fr ¼

1 Hz T

(12)

2p rad=s T

(13)

decrement equation. The next step is to identify the excitation force f(t). The variance of the excitation force can be obtained using Eq. (9). In the stationary case, n_ 11 ¼ n_ 12 ¼ n_ 22 ¼ 0, then Eq. (9) can be reduced to

co ¼ 4zoo n22

(16)

and

n22 ¼ o2o n11

(17)

n11 and n22 can be obtained from the time history of the response. Eq. (16) can then be used to determine the variance of the exciting force. The time history of the excitation can be obtained using Eq. (3), rewritten as f ðtÞ ¼ x€ þ 2zoo x_ þ o2o x

(18)

The time history of the reaction force at the foundation, R(t) can be obtained as RðtÞ ¼ ð2zoo x_ þ o2o xÞ

(19)

7. Numerical examples Two examples are used to validate the method developed in this work: a simple single degree of freedom system, and an offshore platform. 7.1. The excitation

or

o¼

The damping ratio is calculated using the logarithmic decrement concept. The logarithmic decrement (d) is given by   1  A  (14) d ¼ ln i  n Aiþn where Ai and Ai+n stand for the amplitudes of the ith and (i+n)th cycles, respectively. The modal damping (z) is related to the logarithmic decrement by the equation sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 d (15) z¼ 2 4p2 þ d

The excitation force, f(t) used to generate the response for the system is represented by the sum of N sinusoidal terms, given by f ðtÞ ¼

N X

ai sinðo ¯ i t þ yi Þ

(20)

i¼1

The phase angle, yi is a uniformly distributed random variable that takes values between 0 and 2p. The amplitudes ai are to be determined according to a chosen spectrum. For the first example, the excitation force is assumed to have a wide band spectrum with constant density between the frequencies, o ¯ 1 to o ¯ N . This range encompasses the natural frequency of the system. In the second example, the excitation is assumed to have a Pierson– Moskowitz spectrum. 7.2. Example 1: a single degree of freedom system

6. Identification of the excitation and the reaction In the previous sections, we showed how one can identify the natural frequency and the damping ratio using the random

In this example, we show the results of applying the method to an SDOF system subjected to a broad band excitation. The force amplitudes per unit mass ai are assumed to be 0.0625. The frequency o is assumed to be 10 rad/s, the damping ratio z is assumed to be 0.02. The excitation is represented by five

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Table 2 The variances for the actual and predicted exciting forces.

0.80 0.40

Item

From actual force

From estimated force

Percent error

Variance (s2)

9.714

9.732

0.19%

0.00 -0.40 1.20

-0.80

1.00

2.00 3.00 Time (sec)

5.00

4.00

Fig. 2. Random decrement signature of response.

Table 1 Actual and predicted values of natural frequency and damping ratio. Item

Actual (rad/s)

Predicted (rad/s)

Percent error (%)

Damped natural frequency (od) Damping ratio (z)

9.998 0.02

10.012 0.0203

0.14 1.50

thousands sinusoidal components over a frequency range 5.0–15.0 Hz. The response is calculated using a Runge–Kutta method (Shoup, 1980; Thomson, 1993). Integration was done using a time increment of 0.02 s. Fig. 2 shows a random decrement signature for this system obtained using a triggering level equal to the standard deviation (s ¼ 0.1898). Several triggering levels were tried; it was found that a triggering level of one to one and half the standard deviation gives excellent results. The number of segments used to construct the random decrement shown in Fig. 2, is 936. Good results are usually obtained if the number of segments is more than 100. The time series generated is 500 s long which gives 25,000 sampling points. The period is measured directly from the digitized graph using the average of the first five cycles and found to be 0.6274 s. The natural frequency is calculated using the relation: fr ¼ 1/T Hz. The damping is calculated using the logarithmic decrement as in Eqs. (14) and (15). Table 1, shows the comparison between the actual values and the predicted values.

7.2.1. Prediction of the excitation A time history for the exciting force was generated using Eq. (18). Usually, only the displacement (or acceleration) is known from the measured response, the velocity can be obtained using numerical differentiation of the displacement (or numerical integration of the acceleration). In this work, we assume that we have displacement measurements and we use a Lagrange 5-points central differentiation to calculate the first and second derivatives of the displacement. For the first point a forward method is used and for last point a backward method is used. These quantities are used in Eq. (18) to obtain a predicted record of the excitation force. To compare the actual and predicted values of the excitation, two measures are used: the variance (s2) and the auto-correlation function. The excitation is a zero-mean process. The variance is calculated for a record having a length of 500 s. The variances for the actual and predicted exciting forces are shown in Table 2. The error in the prediction is less than 0.2%. A comparison between the auto-correlation functions for the actual and predicted force is given in Fig. 3. It is evident that the correlation functions for the actual and predicted forces have an excellent agreement.

0.80

Actual force Estimated force

0.40 0.00 -0.40 -0.80 -1.20 0.00

1.00

2.00 Time Lag (sec)

3.00

4.00

Fig. 3. Auto-correlation function of the excitation.

mass = 3000 Kg

20m

0.00

26m

-1.20

Auto-Correlation Function

Normailzed Displacement

1.20

Steel Pipe External diam = 1.50m Wall Thickness = 4cm

Steel Pipe External diam = 2.10m Wall Thickness = 5cm

Mudline

Fig. 4. Offshore platform.

7.3. Example 2: an offshore structure In this example we deal with a simple offshore platform as shown in Fig. 4. The platform consists of a vertical steel pipe standing in 26.0 m water depth. The diameter of the pipe is 2.10 m for the part under water and its wall thickness is 5 cm. The part outside the water is 1.20 m in diameter and has a wall thickness of 4 cm. The platform supports a mass of 3000.00 kg on deck. It is assumed that the platform is subjected to waves having a

ARTICLE IN PRESS A.A. Elshafey et al. / Ocean Engineering 36 (2009) 521–528

Normalized Displacement

1.20 0.80

Item

Actual (rad/s)

Estimated (rad/s)

Percent error

Damping ratio

First natural frequency Second natural frequency

6.553 3.895

6.346 3.950

3.15 3.86

0.0776 0.0470

Actual Reaction (N)

4x105 2x105

0.00

Y = 61875.74 X

0x100 -2x105 -4x105 -8.00

Actual points Fitted -6.00

-4.00 -2.00 0.00 2.00 4.00 Estimated Reaction / Unit mass (N/kg)

6.00

Fig. 7. Relationship between total numerical reaction and estimated reaction per unit virtual mass.

1.20 actual estimated

0.80 0.40 0.00 -0.40 -0.80 -1.20 0.00

0.40

1.00

3.00 2.00 Time-Lag (sec)

4.00

5.00

Fig. 8. Auto-correlation functions for numerical and estimated reaction forces.

-0.40

The period of one cycle was measured directly from the digitized curve and found to be 0.99 s. The total damping ratio is extracted from the logarithmic decrement and found to be 0.0776.

-0.80 -1.20 0.00

1.00

2.00 3.00 Time (sec)

4.00

5.00

Fig. 5. Random decrement signature for the first mode of platform.

1.2 Normailzed Displacement

Table 3 Actual and estimated values for first and second natural frequencies and damping ratios.

Auto-Correlation Function

Pierson–Moskowitz spectrum. The significant wave height (Hs) of the waves is 5.0 m and the average zero-up-crossing period (Tz) is 3.00 s. The response of the platform and the reaction force and moment were calculated using an ANSYS software package. A study was done for choosing a suitable number of elements. The wave input was represented by a series of sine waves in the time domain. The amplitudes and frequencies were taken from the assumed spectral density function of the waves and the phase angle was assumed to be a random variable having a uniform distribution between 0 and 2p. The platform is divided into 46 elements. To allow for water waves, an ANSYS pipe59 element was used. The data was recorded at a rate of 20.00 Hz. The drag and inertia coefficients were assumed to be 1.20 and 2.00, respectively. The top nodal displacement was recorded at a frequency of 20.00 Hz. The responses for the first and second mode of the platform were obtained by passing the response through a band bass filter centered around the first and second natural frequencies, respectively. The auto-correlation functions were used to obtain the random decrements for the first two natural frequencies. These random decrements were assumed to represent the free response of two single degree of freedom systems having natural frequencies equal to the first two natural frequencies of the platform. The two random decrement signatures are shown in Figs 5 and 6, respectively. It can be shown from Table 3, that the RD approach can predict the natural frequencies with sufficient accuracy. The material damping used in the analysis was 0.02; this means that the water hydrodynamic damping is estimated to be 0.0576 for the first natural frequency and 0.027 for the second natural frequency. It should be noted that only the modes in the direction of wave propagation are considered.

525

0.8 0.4 0 -0.4 -0.8 -1.2 0.00

0.50

1.00

1.50 Time (Sec)

2.00

2.50

Fig. 6. Random decrement signature for the second mode of platform.

3.00

7.3.1. Estimation of the reaction force and bending moment For an SDOF system, the reaction force does not depend on the inertia force; it depends only on the damping and restoring forces. In this work, we model the platform as a single degree of freedom system having a natural frequency equal to the fundamental frequency of the platform. The horizontal reaction per unit mass at the foundation can be estimated from Eq. (19). A comparison between the actual reaction and estimated reaction force per unit mass is shown in Fig. 7. By plotting the relationship between the actual reaction and the predicted reaction per unit virtual mass, one is able to determine the value of the total virtual mass of the structure. A straight line passing through the origin is fitted to the points taken from time history records of the reaction forces. The slope of the straight line fit gives an approximation of the total virtual mass. The normalized auto-correlation functions for both of the estimated reaction per unit virtual mass and the total reaction obtained from the finite element analysis are shown in Fig. 8. An excellent agreement between the two auto-correlation functions

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1x107 Actual Values Fitted

Actual My (N.m)

8x106 4x106 0x100 -4x106

Y = 1587797.9 X

-8x106 -1x107 -8.00

0.00 4.00 -4.00 Estimated Reaction / Unit Mass(N/kg)

8.00

Force Location / water depth

2.00

1.50

1.00

0.50

0.00 14

Fig. 9. Relationship between estimated reaction per unit virtual mass and numerical value for the moment.

16

18

20 22 24 Water Depth (m)

Estimated My actual My

0.80

28

30

3.6x103

0.00 -0.40 -0.80 1.00

2.00 3.00 Time-Lag (sec)

4.00

5.00

Fig. 10. Auto-correlation function of moment at base.

Virtual Mass / Water Depth

0.40

-1.20 0.00

26

Fig. 11. Relationship of water depth and location of force producing the bending moment at the foundation.

1.20 Auto-Correlation Function

Actual Fitted

Actual Values Fitted

3.2x103 2.8x103 2.4x103 2.0x103 1.6x103 18

can be observed. It is clear that the method can successfully estimate the reaction force per unit mass successfully. The actual bending moment is plotted against the estimated reaction force per unit virtual mass, the relationship is found to be linear as shown in Fig. 9. The slope of this line gives the product of the virtual mass and the moment lever arm. Using the two relations obtained from Figs. 7 and 9, one can obtain an approximate value for the point of application of the resultant external force on the platform. This point was found to be at a height of 25.66 m above the base, for this example. This is approximately equal to the height of the mean free water surface. The auto-correlation functions of the actual bending moment and the estimated normalized bending moment are compared in Fig. 10, an excellent agreement is shown. This result was further investigated using different water depths and significant wave heights. The results of this investigation are shown in Figs. 11 and 12. Results shown in Fig. 11 indicate that for the cases considered here, the variation in the location of the resultant external force is very small. For water depths between 26.00 and 30.00 m, the point of application of the resultant external force is located at a point very close to the mean free water level. When the depth is decreased to 20.00 m, the point of action of the external resultant force is found to be located at a height of about 0.9 the depth of the platform above the level of the base. When the water depth is further decreased to 16.00 m, the point of action of the force was lowered to a point 0.85 of the depth of water above the base. This is a logical result, since the surface area of the platform subjected to wave motion is approximately equally distributed around the free water surface. The change of the virtual mass with depth is shown in Fig. 12. The results show that as the depth increases the density of the virtual mass decreases. Since the density of the virtual mass is

20

22

24 26 Water Depth (m)

28

30

32

Fig. 12. Relationship between virtual mass and water depth.

calculated as the total virtual mass divided by the water depth, we expect that as the water depth increases the density should decrease.

8. Conclusions In this work, we describe the first stage of work in progress at the Faculty of Engineering and Applied Science at Memorial University of Newfoundland. The ultimate objective of this work is to develop a methodology for the early detection of damage in components of fixed offshore structures. The work reported here describes the validation of the methodology for intact structures. Several significant results are obtained in this work. These include:

 A method for the prediction of the wave exciting force and its location on an offshore structure.

 A method for the prediction of the reaction force at the foundation of an offshore structure.

 A method for the estimation of the total virtual mass of the structure in waves. The random decrement method is a powerful method that can be used to predict the parameters in the equation of motion of vibratory systems using their stationary response in random seas with no prior knowledge of the excitation. This allows the method

ARTICLE IN PRESS A.A. Elshafey et al. / Ocean Engineering 36 (2009) 521–528

to be used without interrupting the normal operation of the system. The method was extended to predict the exciting force and the reaction force at the foundation of a fixed offshore structure. The analysis also allows the determination of the average location of the resultant wave exciting force on the structure. The structure was modeled as a single degree of freedom system and the fundamental mode was used to describe the response of the structure. The error in the prediction of the fundamental natural frequency is less than 5%. The next steps of this work will include an experimental investigation of a model structure in the towing tank of Memorial University and theoretical extension for the method to investigate the effect of adding more modes on the accuracy of the predictions.

527

  Z 1 y1 ¼1 @ y21  ðy2 PÞ dy1 dy2 ¼  y21 y2 P j dy2 @y y 1 1 1 1 ¼1 Z 1Z þ2 y1 y2 P dy1 dy2 ¼ 2n12

Z

Z

1

1

Z

1

Z

y21

1

Z

1

Z

1

Z

1 1

Appendix A

2

dy1 dy2 ¼

Z

co 2

1

y31

1



@P @y2



1

j dy1 ¼ 0

1

y22 PðY; tjY o Þ dy1 dy2 ¼ n22 ðtÞ

1

Z

@y22

y22 PðY; t þ dtjY o Þ dy1 dy2 ¼ n22 ðt þ dtÞ

1

Z

co @2 P

Z

y22

Z

@ ððN þ FÞPÞ dy1 dy2 @y2

1

y2 ¼1

ðy22 ðN þ FÞP j dy1 y2 ¼1 1 Z 1Z  2y2 ðN þ FÞP dy1 dy2 1 Z 1Z ð2ozy22 þ o2 y1 y2 ÞP dy1 dy2 ¼ 2h2ozy22 þ o2 y1 y2 i ¼ 2 ¼

Details of the derivation of Eqs. (8)–(10) are given in this Appendix A. Using the notations N ¼ 2ozy2, F ¼ o2y1, and dP(Y,t|Yo) ¼ P(Y,t+dt|Yo)P(Y,t|Yo) Eq. (7) can be rewritten as " # @ @ c @2 P PðY; t þ dtjY o Þ  PðY; tjY o Þ ¼  ðy2 PÞ þ fðN þ FÞPg þ o 2 dt @y1 @y2 2 @y2 (A1) Multiplying the two sides of Eq. (A1) by y1, y2, and y1y2, respectively, and integrating the equation w.r.t. y1 and y2 from N to N, we get Z 1Z y1 PðY; t þ dtjY o Þ dy1 dy2 ¼ m1 ðt þ dtÞ 1

Z

Z

1

1

Z

1

Z

y22

1

 co

Z

co @2 P 2 1

1

@y22

Z

y2

dy1 dy2 ¼

co

Z

2

1 1

y22

@P dy1 @y2

@P dy1 dy2 ¼ co @y2

In deriving Eqs. (8) and (9) we assumed the following boundary conditions: y1 y2 P

y1 ¼1

j y1 ¼1

¼ ðN þ FÞP

y2 ¼1

j

¼P

y2 ¼1

¼ yi ðN þ FÞP y1 PðY; tjY o Þ dy1 dy2 ¼ m1 ðtÞ

y1 ¼1

j y1 ¼1

y1 ¼1

j y1 ¼1

¼

¼P

y2 ¼1

j y2 ¼1

@P y2 ¼1 j ¼ 0; @y2 y2 ¼1

i ¼ 1; 2.

1

Z

1

Z

1

    Z Z y1 ¼1 @ y1  ðy2 PÞ dy1 dy2 ¼  dy2 y1 y2 P j  y2 P dy2 @y1 y1 ¼1   Z 1 y1 ¼1 ¼ m2  dy2 y1 y2 P j ¼ m2 y1 ¼1

1

Z

1

Z

@ ðN þ FÞP dy1 dy2 ¼ @y2

y1

1

Z

1

Z

co @2 P

y1

2 @y22

1

Z

1

Z



y2 

1

Z

Z

1 1

Z

y2

dy1 dy2 ¼

Z

2

1 1

y1 dy1 1

Z

@ ðy PÞ dy1 dy2 ¼ @y1 2

@ ðN þ FÞP dy1 dy2 @y2 dy1 ðy2 ðN þ FÞP

1

¼ hN þ Fi þ

Z

y2 ¼1

1

y2 ¼1



j

¼0

y2 ¼1



 @P y2 ¼1 j ¼0 @y2 y2 ¼1

 y2 dy2 y2 P

1

Z

y2 ¼1

j

 y1 dy1 ðN þ FÞP

1



1

¼

co

Z

 y2 ¼1 j ¼0 y2 ¼1

1

ðN þ FÞP dy2 Þ



1 y2 ¼1

dy1 ðy2 ðN þ FÞP

j

Þ ¼ h2ozy2 þ o2 y1 i

y2 ¼1

Multiplying the two sides of Eq. (A1) by y21, y22, and y1y2, respectively, and integrating the equation w.r.t. y1 and y2 from N to N, we get Z 1Z y21 PðY; t þ dtjY o Þ dy1 dy2 ¼ n11 ðt þ dtÞ 1

Z

1 1

Z

y21 PðY; tjY o Þ dy1 dy2 ¼ n11 ðtÞ

Eq. (10) can be derived as follows:

m_ 2 ¼ m€ 1 ¼ h2zoo y2 þ o2o y1 i m€ 1 ¼ ð2zoo m_ 1 þ o2o m1 Þ m€ 1 þ 2zoo m_ 1 þ o2o m1 ¼ 0 In Eq. (10), the symbol m is used instead of m1. Where m is used for the mean value of the displacement. References Brincker, R., Zhang, L., Andersen, P., 2001. Modal identification of output-only systems using frequency domain decomposition. Smart Material Structures 10, 441–445. Budipriyanto, A., Haddara, M.R., Swamidas, A.S.J., 2007. Identification of damage on ship’s cross-stiffened plate panels using vibration response. Ocean Engineering 34, 709–716. Cole, H.A., 1968. On the line analysis of random vibration. In: Proceedings of the AIAA/ASME 9th Structures, Structural Dynamics and Materials Conference, Palm Springs, California. Haddara, M.R., 1992. On the random decrement for nonlinear rolling motion. In: Proceedings, 11th. International Conference on Offshore Mechanics and Arctic Engineering, vol. II, Calgary, pp. 321–324. Haddara, M.R., 2006. Complete identification of the roll equation using the stationary random roll response. Journal of Ship Research 50 (4), 388–397. Ibrahim, S.R., Mikulcik, E.C., 1977. A method for the direct identification of vibration parameters from the free response. Shock and Vibration Bulletin 47 (4), 183–198. Onoufriou, T., 1999. Reliability based inspection planning of offshore structures. Marine Structures 12, 521–539. Pillai, T.M.M., Prasad, A.M., 2000. Fatigue reliability analysis in time domain for inspection strategy of fixed offshore structures. Ocean Engineering 27, 167–186.

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Computational methods for offshore structures, special publication, AMD, vol. 37, ASME. Library of congress #80-69180. Yang, J.C.S., Chen, J., Dagalakis, N.G., 1984. Damage detection in offshore structures by random decrement technique. Journal of Energy Resources Technology, Transactions of the ASME 106, 38–42. Zubaydi, A., Haddara, M.R., Swamidas, A.S.J., 2000. Damage identification in stiffened plates using the random decrement technique. Oceanic Engineering International 4 (1), 22–30. Zubaydi, A., Haddara, M.R., Swamidas, A.S.J., 2002. Damage identification in a ship’s structure using neural networks. Ocean Engineering 29 (10), 1187–1200.