Time Domain Simulation of Data Buoy Motion

Transcription

1 Proc. Natl. Sci. Counc. ROC(A) Vol. 22, No. 6, pp Time Domain Simulation of Data Buoy Motion MIN-CHIH HUANG Department of Naval Architecture and Marine Engineering National Cheng Kung University Tainan, Taiwan, R.O.C. (Received January 13, 1998; Accepted May 25, 1998) ABSTRACT The measured heave motion in a data buoy system is influenced by the type of accelerometer used (hull-fixed or vertically stabilized), the position of the accelerometer, the pitch response of the buoy itself, electronic noise and digitized error in analysis. These various effects introduce spurious energy into the acceleration spectrum, particularly at low frequencies. In this study, numerical time domain simulation of data buoy motions in regular and irregular waves was conducted to study the various effects of nonlinearity on measured heave acceleration at low frequencies. The results of detailed studies indicate that in low sea-states, nonlinear pitch spectral transfer is the main mechanism which causes fixed accelerometer measurements to contain low-frequency spurious energy. For more severe sea-states, both nonlinear pitch and heave spectral transfer can introduce lowfrequency spurious energy into stabilized accelerometer measurements. Simulation results also indicate that the stabilized accelerometer is preferred over the fixed accelerometer to reduce spurious energy at low frequencies. This spurious energy induced by nonlinear buoy motions can be effectively corrected by an empirical noise correction function which varies linearly with the wave frequency. The slope of this noise correction equation and the frequency range of correction can be found from detailed numerical simulations and field measurements. Key Words: data buoy, numerical simulation, wave tank experiment I. Introduction Longuet-Higgins et al. (1963) showed how estimates of the directional wave spectrum can be constructed from time series records of the water surface displacement and its two slopes, which are assumed to be identical to the vertical, pitch and roll motions of a small pitch-roll buoy. However the quantities actually measured by the buoy are the time series records of the buoy acceleration, pitch and roll motions. The buoy displacement analog can then be obtained by twice integrating the buoy acceleration with respect to time either analytically, numerically or electronically through a double integrator. The processed buoy heave displacement is, of course, different from the sea surface displacement. Therefore, for a pitch-roll buoy to produce accurate data, a number of factors have to be taken into account (Steele et al., 1985). These factors include amplitude and phase alternations due to the hull/mooring effect, sensor/electronics effect, and noise. An overview of different methods for processing time series records of buoy measurements was provided by Steele et al. (1992), and a new procedure for directional wave analysis from pitch-roll buoy measurements was later provided by Huang and Chen (1998). In most wave-measuring payloads, the vertical heave accelerometer is either fixed (i.e., strapped-down) or vertically stabilized in the buoy hull. A problem for all these accelerometer measurement systems is that the buoy continuously tilts due to wave motion. Although a vertical reference can be continuously monitored by means of a gyroscope, this approach is seldom used due to expense or power requirements. When the accelerometer is hull-fixed, the measured acceleration is a sea-state dependent time varying combination of buoy vertical and horizontal accelerations, and gravitational acceleration due to the tilting motion of the buoy. It is well known from earlier works done by the National Data Buoy Center (NDBC) that, due to the mechanism of nonlinear spectral transfer, spurious energy is introduced into the acceleration spectrum (Earle and Bush, 1982; Earle et al., 1984). A nonlinear equation for strapped-down accelerometer measurements was derived by Earle and Bush (1982), and mathematical simulations were performed to calculate the spurious energy. At low frequencies, where there is little or no real wave motion, the spurious energy acquired is significantly amplified when divided by the frequency to a power of four in order to convert acceleration to displacement spectra. Various 820

2 Simulation of Buoy Motion frequency-dependent and sea-state dependent noise correction functions were subsequently derived by NDBC to remove spurious energy from the measured acceleration spectra (Earle and Bush, 1982; Earle et al., 1984). When the accelerometer is vertically stabilized, the measured acceleration should ideally be along a vertical axis. However, some accelerometers, such as HIPPY sensors, may not be perfectly stabilized (Steele et al., 1985). These accelerometers are also not ideal for operational use since they are bulky, heavy, costly, and require special handling. The use of a HIPPY sensor with a built-in electronic double integrator also produces spurious energy at low frequencies, which was found by Steele et al. (1985) to grow at an increasing rate with significant wave height. The cause of the spurious energy was not understood but was attributed to either the HIPPY sensor not being perfectly stabilized, to hull horizontal motion, or to both. Various noise correction functions were also derived by NDBC to remove spurious energy from the measured heave spectra (Steele et al., 1985, 1992). Therefore, in reality, buoy acceleration along an approximate vertical axis is either modified in the frequency domain or double integrated in the time domain to provide sea surface displacement information. A noise correction function is then used to remove spurious energy from the measured acceleration or heave spectra. This function has usually been chosen to decline linearly versus frequency to a small value at a prescribed wave frequency (e.g., 0.12 Hz), above which a constant small energy level has been used to compensate for electronic noise. In previous studies, the measured heave motion was found to be influenced by the type of accelerometer used, the position of the accelerometer, the pitch response of the buoy itself, electronic noise and digitized error in analysis. These various effects introduce spurious energy into the acceleration spectra, particularly at low frequencies. Based on our own experience, the energy levels of electronic noise and digitized error are usually very small and can be neglected (Huang, 1993). The effect of the accelerometer position was found to be small in practical applications (Chen et al., 1997). The spurious energy produced by the nonlinear equation for strapped-down accelerometer measurements was found to increase with the pitch response of the buoy, which in turn became larger for higher seastates (Earle and Bush, 1982). One factor not considered with regard to spurious energy in all these previous studies is the possibility of nonlinear sub-harmonic resonance of buoy motions. As will be shown later, buoy motions in waves are governed by quadratic nonlinear differential equations Fig m discus buoy configuration. with variable coefficients. A general feature of these nonlinear systems with multi-frequency excitations is that both primary resonance and secondary resonance can happen (Nayfeh, 1993). Various types of secondary resonance, which include the sub-harmonic, super-harmonic and combination types, can contribute spurious energy to the measured spectra. In this study, we were mainly concerned with spurious energy at low frequencies, so the possibility of sub-harmonic buoy resonance could then be examined through detailed simulations of buoy motions in regular waves with multi-frequency excitations and in irregular waves. A two-dimensional time domain model was previously developed by Huang (1985) to analyze the dynamic response of an ocean buoy. The model was later extended to an ocean buoy-tether-anchor system by Huang and Baur (1990), in which a lumped parameter method was used to model each segment of the cable. A time domain numerical model of similar features was later developed to study the effect of excess anchor cable on the buoy and mooring dynamics (Teng and Wang, 1995), and to study the motion transfer function for a slack-moored wave-following buoy ( Teng and Taft, 1996 ). It is noted that the buoy in their model was simulated with a spheroid, and that low frequency buoy motions were not considered at all. A 3-m discus buoy as shown in Fig. 1 was used in this study. The basic physical parameters of this buoy are: weight W=1295 Kg, pitching moment of inertia I=1085 kgm-m 2, and center of gravity from bottom = cm. Laboratory experiments were 821

3 M.C. Huang conducted to investigate certain response characteristics of this buoy. Motion transfer functions obtained from the time domain model of Huang (1985) were compared with the results of the frequency domain model (Teng et al., 1989). The time domain model of Huang (1985) was then extended to calculate the motions of the buoy in regular waves with two-frequency excitations and in irregular waves simulated based on Pierson-Moskowitz (P-M) spectra. The possibility of sub-harmonic buoy resonance was studied, and spurious energy at low frequencies in the calculated buoy acceleration was modified in the frequency domain to provide sea surface displacement information. II. Equations of Buoy Motion It is well known that the effects of mooring on buoy dynamics vary for different mooring configurations. These effects are also sea-state dependent for each buoy station. Although they can be determined for a particular set of environmental conditions, they can not be used universally in all situations and, therefore, must be determined directly from measured data (Huang and Chen, 1998). Discus buoys with excess length of anchor chains are known to be very good wave-following pitch-roll buoys; therefore, we will not consider the effects of mooring in this study for the sake of simplicity. Since a discus buoy is symmetrical with respect to the vertical axis, the buoy motion can be considered in a two-dimensional plane. The twodimensional time domain model of Huang (1985) is modified to model a discus buoy shape and briefly summarized below. The buoy is described in terms of its components, where all possible conditions of water submergence can be specified to calculate the external forces acting on the buoy. In the following analysis, these external forces include the buoy weight, buoyancy, nonlinear viscous fluid drag, linear wave-making resistance and fluid added mass force. In order to describe the external forces acting on the buoy and its three-degree-of-freedom responses ( surge, heave, and pitch ), it is necessary to adopt two sets of coordinate systems. The spatial variations of waves and current loading are described with respect to a global coordinate system (X,Z,θ), with Z measured positively upwards from the still water level. An uniform water depth (h) is assumed. A local coordinate system (x,z,θ) with an origin at the buoy center of gravity (cg) is also established where a positive pitch angle (θ) with a clockwise rotation is assumed. The transformations of the buoy acceleration, velocity and displacement between the global and local coordinate systems are: X = xcosθ + zsinθ, Z = xsinθ + zcosθ (1) X = xcosθ + zsinθ, Z = xsinθ + zcosθ (2) X=X cg +xcosθ+zsinθ, Z=Z cg xsinθ+zcosθ. (3) The following three equations of motion are established for the buoy in the local coordinate system: (m + C axi ρv i )x +( C axi ρv i z Vi )θ =(W ρgv i ) sinθ + ρ 2 C dxia xi (u i x i ) u i x i + C fx ( ρgv i ) (u x ) + ρ(c gd axi +1)V i u i (m + ρc az V z )z =( W + ρgv i ) cosθ + ρ 2 C dza z ( w z) w z (4) + C fz ( ρgv i ) ( w z) + ρ(c gd az +1)V z w (5) ( C axi ρv i z Vi )x +(I + λ C axi ρv i z2 Vi )θ = ( ρgv i )(sinθ sinθ W )GM + β ρ 2 C dxia xi (u i x i ) u i x i z Ai + C fx ( ρgv i ) (u x ) gd z f + (C axi +1)ρV i u i z Vi. (6) Due to the pitch response of the buoy, each buoy component has a different surge velocity and acceleration with respect to the corresponding values at the buoy center of gravity, given by: x i = x + θz Vi, x i = x + θz Vi. (7) Therefore, the fluid forces can be calculated for each submerged component of the buoy and summed together (i=1,..., ) according to the state of water submergence. The notations in Eqs. (4)-(7) are defined as: m= the buoy mass; I= the buoy pitching moment of inertia; W= the buoy weight; ρ= the mass density of the water; g= the gravitational acceleration; V i = the submerged volume of the buoy component; V z = the spherical volume of buoy in the heave direction; A xi = the proected area of the buoy component in the surge direction; A z = the proected area of the buoy in the heave direction; d= the buoy diameter at the water 822

4 Simulation of Buoy Motion surface intersection; C dxi, C dz = the nonlinear viscous drag coefficients in the surge and heave, respectively; C axi, C az = the added mass force coefficients in the surge and heave, respectively; C fx, C fz = the linear wavemaking damping coefficients in the surge and heave, respectively; θ w = the wave slope; GM = the metacenter height; z Ai, z f, z Vi = the moment arms of the viscous drag force, wave-making resistance and added mass force, respectively; λ= the empirical multiplier for added pitching moment of inertia; β= the empirical multiplier for pitching moment due to viscous drag force; u i = the fluid velocity in the surge direction; w = the average fluid velocity in the heave direction; u i = the fluid acceleration in the surge direction; w = the average fluid acceleration in the heave direction; x i = the surge velocity of the buoy component; and z= the heave velocity of the buoy. The fluid forces are first described with respect to the global coordinate system, then transformed into a local coordinate system and substituted into Eqs. (4)- (6). The buoy accelerations in Eqs. (4)-(6) are first calculated in the local coordinate system and then transformed back into the global coordinate system for subsequent time domain integration. A detailed description of the numerical algorithm can be found in Huang (1985). It is noted that two empirical multipliers, λ and β, are included to keep the pitch impulse response of the discus-type buoy properly simulated. In contrast, only one multiplier, λ, is required for the spar buoy previously studied in Huang (1985). III. Laboratory Experiments Although there is an abundance of force coefficient data available in the literature, e.g., Hoerner (1958), none corresponds to the shape or component of a discus buoy. Therefore, laboratory experiments were conducted to investigate the impulse response characteristics of the buoy in order to obtain quantitative approximations of the force coefficients in Eqs. (4)- (6). It is noted that these force coefficients are only average values since they are, in general, functions of the buoy geometry, Reynolds number, motion amplitude and frequency etc. Three types of buoy impulse tests were conducted in a 100 m long wave tank. In the surge impulse tests, the buoy was pushed in still water at a constant speed by the carriage of a towing tank, the carriage was stopped suddenly and the buoy was let go to surge freely. In heave impulse tests, the buoy was lifted to an initial position from its equilibrium position in still water. In pitch impulse tests, the buoy was inclined at an initial pitch angle in still water. The response curves of the buoy after release from the initial posi- Fig. 2. Numerical and experimental surge impulse response curves. tions were obtained by video recording and measurements of various sensors (Huang, 1985; Carpenter et al., 1995). The force coefficients in Eqs. (4)-(6) were then obtained using the following procedures: (1) Numerical response curves were simulated using a set of assumed coefficients. (2) Comparisons were made between the laboratory and numerical response curves. (3) The assumed coefficients in the numerical model were then adusted in order to match the measured response characteristics. The surge impulse tests on the buoy were conducted to obtain the nonlinear drag coefficients in the surge, C dxi. The assumed coefficients were adusted in order to match the measured and calculated surge distances. Since numerical simulations were performed to obtain only the nonlinear drag coefficients in the surge, a modified form of Eq. (4), in which the surge motion was uncoupled from the pitch motion, was used in the calculations. Due to the length limit of the wave tank, these tests were conducted with a set of five initial surge velocities: 0.2, 0.4, 0.6, 0.8 and 1.0 m/s. The experimental results are illustrated in Fig. 2, and it can be seen that they compare favorably with the results of numerical simulations based on C dxi =0.18. The heave impulse tests on the buoy were conducted to obtain the following coefficients: C fz, C dz and C az. The assumed coefficients were adusted in order to match the measured and calculated response characteristics of the damped period, peak response, and logarithm decrement. The heave impulse tests were conducted with an initial displacement of the buoy of from 2 cm to 30 cm. Since the heave motion was uncoupled from surge-pitch motions, experimental data with excess pitch angles were discarded from the comparisons. The average damped period of the buoy was 1.5 s, and only two cycles of heave oscillation 823

5 M.C. Huang Fig. 3. Numerical and experimental heave impulse response curves. mounted in the buoy, and the buoy was released to pitch about one of the inclinometer s axes during the tests. Experimental data with excess roll angle, which is an indication of the wave reflection effect due to the tank walls, were then discarded from the comparisons. The average damped period of the buoy was s, and about four cycles of pitch oscillation could be used. An example of experimental and numerical simulation results based on C axi =0.08, C fx =0.02, λ=80, β=2000 and C dxi =0.18 is illustrated in Fig. 4, and it can be seen that the results compare favorably with each other. The assumed added mass and linear damping coefficients (C axi, C fx ) were, in fact, calculated using a finite element model from buoy diffraction theory (Huang, 1983); therefore, only the multipliers (λ,β) were adusted in the numerical calculations to account for the large added moment of inertia and nonlinear pitch damping moment associated with the discus buoy. The calculated added moment of inertia coefficient was about 0.396, which compares favorably with the experimental value for an Naval Civil Engineering Laboratory (NCEL) discus buoy (Jenkins et al., 1995). IV. Motion Transfer Functions Fig. 4. Numerical and experimental pitch impulse response curves. could be used since wave reflection from the tank walls would affect the buoy motion. An example of experimental and numerical simulation results based on C fz =0.9, C dz =2.5 and C az =1.5 is shown in Fig. 3, and it can be seen that the results compare favorably with each other. The assumed added mass and linear damping coefficients (C az, C fz ) were in fact calculated using a finite element model from buoy diffraction theory (Huang, 1983); therefore, only the nonlinear drag coefficient C dz was adusted in the numerical calculations. The pitch impulse tests on the buoy were conducted to obtain the following coefficients: C axi, C fx, λ and β. The assumed coefficients were adusted in order to match the measured and calculated pitch response characteristics of the damped period, peak response, and logarithm decrement. The pitch impulse tests were conducted with an initial pitch angle of the buoy of from 7 to 21. A two-axis inclinometer was Motion transfer functions for the previously described discus buoy were then calculated for various monochromatic wave runs using the time domain model. These motion transfer functions can be expressed in terms of the response amplitude operator (RAO) and phase lag for each wave frequency. This phase lag information is essential for directional wave analysis but is not required in our present two-dimensional model. For comparison, the frequency domain model FREQ was also used to calculate these motion transfer Fig. 5. Heave and pitch RAO from both the time and frequency domain models. 824

6 Simulation of Buoy Motion functions (Teng et al., 1989). Figure 5 illustrates the heave and pitch RAOs calculated using both the time and frequency domain models. Both values of the heave RAO from the time domain and frequency domain models are very close to unity in the meaningful wave frequency ranging up to 0.4 Hz, and close agreement between the two models is also observed. Values of the pitch RAO from the time domain model are fairly close to those obtained using the frequency domain model except for higher-frequency waves which are near to the natural damped period of the buoy. In summary, the 3-m discus buoy used in this study is a very good wave-following pitch-roll buoy. The heave RAOs calculated using the time domain model will be used in a subsequent section, where the measured buoy acceleration is modified in the frequency domain to provide sea surface displacement information. V. Accelerometer Measurements An accelerometer is usually displaced a vertical distance, δ, from the buoy center of gravity due to physical constraints. The position of the accelerometer is given by: X a =X cg +δ sinθ, Z a =Z cg +δ cosθ. (8) Differentiation of Eq. (8) with respect to time twice then yields the surge and heave accelerations at the accelerometer position as: X a = X cg δ sinθθ 2 + δ cosθθ, Z a = Z cg δ cosθθ 2 δ sinθθ. (9) A vertically stabilized accelerometer would then measure the acceleration Z a along an ideally vertical axis. It would also measure a constant gravitational acceleration, g, which is not related to the buoy motion and can be neglected in the analysis. Instead, a hullfixed accelerometer would measure the component of the buoy acceleration and gravitational acceleration perpendicular to the buoy deck as a result of buoy pitch motion. This measurement can be obtained by transformation of axes: vertically stabilized and hull-fixed accelerometer measurements only differed from the actual vertical acceleration in their higher-harmonics components. For clarity, the terms heave acceleration, stabilized heave acceleration, hull-fixed heave acceleration, are often used in the following to represent these three accelerations, respectively. VI. Buoy Motions in Two-Frequency Excitations Since buoy motions in waves are governed by quadratic nonlinear differential equations as shown in Eqs. (4)-(6), it is possible for sub-harmonic resonance to occur with multi-frequency excitations (Nayfeh, 1993). Since analytical solutions can not be obtained for these differential equations with variable coefficients, numerical simulations of buoy motions in regular waves with two-frequency excitations were conducted to study this possibility. For simplicity, numerical simulations were done with two waves of equal amplitude, which were increased systematically, and slightly different frequencies. Motion spectra of Z, θ, Z cg, Z a and z m were then calculated to see if there was spurious energy at low frequencies. The results of an example calculation are shown in Figs The pitch spectrum plotted in Fig. 6 clearly indicates that there exists sub-harmonic and super-harmonic pitch resonance. This is accompanied by low-frequency spurious energy measurements obtained using a hull-fixed accelerometer as shown in Fig. 7. In contrast, the spectrum of vertically stabilized acceleration plotted in Fig. 8 does not contain any lowfrequency spurious energy. The systematic analyses indicate that, in general, the three spectra of heave, heave acceleration and stabilized acceleration do not contain low-frequency energy z m = X a sinθ +(Z a g) cosθ + g. (10) Time series and spectra analysis of three accelerations, Z cg, Z a and z m, were then calculated for various accelerometer positions under monochromatic wave runs using the time domain model. The effect of accelerometer position was found to be small in practical applications (Chen et al., 1997), and both Fig. 6. Pitch spectrum for two-frequency excitations. 825

7 M.C. Huang Fig. 7. Hull-fixed heave acceleration spectrum for two-frequency excitations. Fig. 8. Stabilized heave acceleration spectrum for two-frequency excitations. even when there sub-harmonic pitch resonance exists, and that use of a hull-fixed accelerometer always causes spurious energy measurements when there is sub-harmonic pitch resonance. The spurious energy produced by the hull-fixed accelerometer measurements is also found to increase with increasing pitch response of the buoy, which results from higher-amplitude waves. irregular waves were further conducted to study this possibility. These irregular waves were simulated based on P-M spectra with 128 components, whose frequencies were equally spaced from 0 to Hz. Component wave phases were randomly distributed between 0 and 360. Results of numerical simulations of buoy motions in irregular waves with a significant wave height of less than 1.5 m do not reveal any low-frequency energy in the three spectra of heave, heave acceleration and stabilized heave acceleration. The two spectra of pitch and hull-fixed heave acceleration do contain the expected spurious energy at low frequencies. Areas (or variances) of heave spectrum and the theoretical P-M spectrum are very close to each other for these low seastates, which is an indication that the 3-m discus buoy is a very good wave-following buoy. The shape and area of the heave acceleration spectrum are also fairly close to those of the stabilized heave acceleration spectrum while they are slightly different from those of the hull-fixed heave acceleration spectrum, mainly at low frequencies. Numerical simulations were then conducted with the significant wave height of the P-M spectrum increasing systematically in order to study the effect of sea-states. The results corresponding to a significant wave height of 3 m are shown in Figs The theoretical P-M wave spectrum and the raw spectrum of the heave motion are plotted in Fig. 9. The areas of the two spectra are ft 2 and ft 2, respectively. The pitch spectrum plotted in Fig. 10 clearly indicates severe sub-harmonic pitch resonance and super-harmonic motions, and this is always accompanied by low-frequency spurious energy measurements obtained using a hull-fixed accelerometer as shown in Fig. 11. Small amounts of low-frequency spurious energy can be seen in the stabilized heave acceleration VII. Buoy Motions in Irregular Waves The possibility of sub-harmonic pitch resonance of the buoy under multi-frequency excitations of irregular waves can now be inferred from the calculations described in the previous section. However, it is not clear whether sub-harmonic heave resonance of the buoy exists under which spurious energy can be produced by a vertically stabilized accelerometer. Therefore, numerical calculations of buoy motions in Fig. 9. P-M wave spectrum and heave spectrum. 826

8 Simulation of Buoy Motion Fig. 10. Pitch spectrum. We have clearly demonstrated that nonlinear pitch spectral transfer is the mechanism by which fixed accelerometer measurements contain spurious energy at low frequencies under low sea-states. For more severe sea-states, both nonlinear pitch and heave spectral transfer are the mechanisms by which stabilized accelerometer measurements contain spurious energy at low frequencies. At low frequencies, where there is little or no real wave motion, the spurious energy acquired is significantly amplified when divided by the frequency to a power of four in order to convert acceleration to displacement spectra. This spurious energy at low frequencies induced by nonlinear buoy motions must be corrected using a noise correction equation. The NDBC has used a noise correction function that is a linear function of the wave frequency since This linear function does a good ob of removing the low-frequency spurious energy in most cases (Earle and Bush, 1982; Earle et al., 1984; Steele et al., 1985, 1992). Therefore, in this study, we followed their approach by defining an empirical noise correction function as: NC(f)=S nc (f u f) C m 11 (max) (11) C 11 (f)= C m 11 (f) NC(f); if C m 11 (f) NC(f) =0; if C m 11 (f)<nc(f), (12) Fig. 11. Hull-fixed heave acceleration spectrum. spectrum shown in Fig. 12, which clearly illustrates that sub-harmonic heave resonance can occur in more severe sea-states. This phenomenon indicates that even when the accelerometer is perfectly stabilized vertically, it can still produce spurious energy at low frequencies. As previously pointed out by Steele et al. (1985), this spurious energy would grow at an increasing rate with significant wave height since the stabilized accelerometer measurement is also a function of pitch motion as shown in Eq. (9). in which S nc = the slope of the noise correction function, f= the wave frequency, f u = the upper bound of lowfrequency range, C m 11 (f)= the heave acceleration spectrum, C m 11 (max)= the maximum value of the heave acceleration spectrum below Hz, and C 11 (f)= the corrected value of the heave acceleration spectrum. The slope of this noise correction equation and the frequency range of correction can be found from detailed numerical simulations and field measurements. A procedure for the determination of S nc and f u was provided by Lang (1987). In this procedure, a data set can be constructed either from a large number of simu- VIII. Wave Spectra from Acceleration Measurements Fig. 12. Stabilized heave acceleration spectrum. 827

9 M.C. Huang Fig. 13. P-M wave spectrum and transformed spectrum from the stabilized accelerometer. lations based on a given input spectrum for numerical study, or from long-term in-situ measurements for field study. For each simulation or measurement, the ratio of C m 11 (f)/ C m 11 (max) is calculated for each frequency; then the maximum value of this ratio for each frequency, Max{ C m 11 (f)/ C m 11 (max)}, is found from the whole data set. A linear regression line S nc (f u f) is then calculated to best fit the maximum values of lowfrequency noise up to a limit, f 1, to be determined by looking at the data (Lang, 1987). It is noted that a typical value of f 1 =0.08 Hz was used by Lang (1987) to determine the linear regression lines for several buoy types. Since different regression lines can be calculated from different selections of this frequency limit f 1, further study is required to determine the optimum value of f 1 for each data set. The transformed wave spectrum S(f) can then be obtained from 2.3%. On the other hand, the wave spectra transformed from hull-fixed accelerometer measurements using various values of S nc and f u do not compare very well with the P-M spectrum since the spurious energy level, as shown in Fig. 11, is greater in this case. Comparisons of the P-M spectrum and the wave spectra transformed from hull-fixed accelerometer measurements using S nc =60, f u =0.12 Hz and 0.18 Hz are illustrated in Fig. 14. In general, a higher value of S nc is required to efficiently remove the low-frequency spurious energy induced by the hull-fixed accelerometer. It is noted that the wave spectra will contain excess low-frequency energy in the order of to the wave energy if they are transformed without noise correction procedure. Finally, it is noted that the 3-m discus buoy was actually deployed with an all-chain slack mooring off the coast of Hsin-chu at a site with a water depth of 18.5 m beginning in May, Both hull-fixed and gimbaled (i.e., partially-stabilized) accelerometers were employed in the sensor package. The results of long term field measurements with waves ranging from low to severe sea-states, indicated that f u =0.11 Hz and S nc =20~40, were appropriate for that particular site. Varying the value of S nc only yielded a small difference in the measured wave statistics (Huang and Chen, 1998). IX. Conclusions The theory of and procedure for constructing directional wave spectra from time series records of heave acceleration, pitch and roll motions of a data buoy have been established and modified over the last three decades. It is well known that the measured heave motion is influenced by the pitch response of the buoy, the type of accelerometer used, the position of the S( f )= C 11 ( f ) (R hh ) 2 (2π f ) 4, (13) in which R hh is the heave response amplitude operator (heave RAO). Numerical examples of wave spectra transformations from acceleration measurements corresponding to a significant wave height of 3 m are shown in Figs. 13 and 14. A 15-point moving average method (Otnes and Enochson, 1978) was used to calculate these spectra from raw spectra S(f). Figure 13 illustrates the theoretical P-M wave spectrum and the wave spectra transformed from stabilized accelerometer measurements using S nc =30, f u =0.12 Hz and 0.18 Hz, respectively. It is seen that the transformed spectrum with S nc =30, f u =0.18 Hz compares very well with the P-M spectrum, where the relative error of spectral area is Fig. 14. P-M wave spectrum and transformed spectrum from the hull-fixed accelerometer. 828

10 Simulation of Buoy Motion accelerometer, electronic noise and digitized error in analysis. These various effects can introduce lowfrequency spurious energy in the acceleration and transformed wave spectra. Since buoy motions in waves are governed by quadratic nonlinear differential equations with variable coefficients, both primary resonance and secondary resonance can happen in multifrequency wave excitations. However, this mechanism of nonlinear sub-harmonic resonance of buoy motions producing low-frequency spurious energy has not been studied before; therefore, the main obective of this study was to conduct numerical two-dimensional timedomain simulations of data buoy motions in waves to study this factor. The results of detailed studies indicate that in low sea-states, nonlinear pitch spectral transfer is the main mechanism by which fixed accelerometer measurements contain low-frequency spurious energy. For more severe sea-states, both nonlinear pitch and heave spectral transfer can introduce low-frequency spurious energy in the stabilized accelerometer measurements. Simulation results also indicate that the stabilized accelerometer is preferred over the fixed accelerometer in reducing the amount of spurious energy at low frequencies. These spurious energy induced by nonlinear buoy motions can be effectively corrected by an empirical noise correction function which varies linearly with the wave frequency. The slope of this noise correction equation and the frequency range of correction can be found from detailed numerical simulations and field measurements. Extension of the present study is suggested in the following area: the effects of mooring and wave directionality on sub-harmonic resonance of buoy motions. Acknowledgment Financial support provided by the National Science Council, R.O.C., under grant no. NSC E is gratefully acknowledged. References Carpenter, E. B., J. W. 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