Experimental investigation of nonlinearities of ship responses in head waves

Applied Ocean Research 33 (2011) 60-68

ELSEVIER

Contents lists available at ScienceDirect

Applied Ocean Research

journal homepage: www.elsevier.com/locate/apor

Experimental investigation of nonlinearities of ship responses in head waves

Wen-Chuan Tiao*

Department of Marine Meciianical Engineering, Naval Academy, No.669, Junxiao Rd., Zuoying District, Kaolisiung 813, Taiwan

A R T I C L E I N F O

Article history:

Received 8 March 2010 Received in revised form 26 July 2010

Accepted 22 November 2010 Available online 15 December 2010

Keywords:

Nonlinear pressure Volterra model Coherence spectrum

A B S T R A C T

T h e p r e s e n t s t u d y f o c u s e s on b u i l d i n g a s y s t e m a t i c a p p r o a c h to identify, f r o m e x p e r i m e n t a l results, the n o n l i n e a r i t y i n the d y n a m i c s y s t e m of a h i g h - s p e e d ship. T h e e x p e r i m e n t a l p r o g r a m consists of tests in both r e g u l a r a n d i r r e g u l a r h e a d w a v e s , a n d the m e a s u r e d quantities i n c l u d e d w a v e elevation, v e r t i c a l motions, a n d h u l l p r e s s u r e s . By c o n t r a s t i n g t h e s e results to the q u a s i - l i n e a r b e h a v i o r s of h e a v e m o t i o n , the n o n l i n e a r b e h a v i o r s of p r e s s u r e a r e highlighted and p r e s e n t e d . T h r e e n o n l i n e a r a s s e s s m e n t s , the probability d e n s i t y f u n c d o n , a n d the v a r i a n c e s p e c t r a a r e p r o v i d e d . B a s e d o n t h e s e investigations, w e c o n c l u d e that the p r e s s u r e s , p a r d c u l a r i y , at the ship's b o w c o n t a i n m o r e n o n l i n e a r i d e s t h a n j u s t the h e a v e m o t i o n . T h e y a r e i d e n t i f i e d m a i n l y by the large a m p l i t u d e of the h i g h e r h a r m o n i c s and also by the large a s y m m e t r y in the m e a s u r e d signals. F u r t h e r m o r e , the c o h e r e n c e s p e c t r u m obtained f r o m the t h i r d - o r d e r orthogonal f r e q u e n c y - d o m a i n V o l t e r r a m o d e l provides i n f o r m a t i o n r e g a r d i n g the m a g n i t u d e of e a c h o r d e r at the c o r r e s p o n d i n g f r e q u e n c y , w h i c h s e r v e s as a strategy to s i m p l i f y a c o m p l e x p r o b l e m or to a c h i e v e a balance b e t w e e n r e g u l a r and i r r e g u l a r w a v e s . T h e overall r e s u l t s s h o w that the h i g h e r -order c o m p o n e n t s are s i g n i f i c a n t for t h e p r e s s u r e s y s t e m and t h e o u t c o m e of the p r o p o s e d m o d e l c a n offer c o n s t r u c t i v e feedback, w h i c h c a n l e a d to m o r e practical applications.

© 2010 E l s e v i e r Ltd. All rights r e s e r v e d .

1. Introduction

In ship/wave experiments, it is desirable to model the relation-ship between t w o measured random quantities, which are referred to as the effect and cause, or output and input. In the past, l i n -ear approaches were adopted to analyze such data, but they gen-erally failed to provide significant insight. W i t h increasing ship speed, ships become dynamic systems that exhibit inherent non-linear characteristics, and non-linear methods cannot capture the dy-namic phenomena associated w i t h these nonlinear systems. Even running i n mild sea states, a high-speed ship combined w i t h large induced motions leads seakeeping or loads to become highly non-linear. It is important to establish a systematic procedure and a nonlinear model to identify these nonlinearities. Thus, the objec-tive of the present research was to experimentally verify and clarify some nonlinear aspects of these dynamic responses using identifi-cation procedures and to understand the internal dynamic mech-anism resulting from the proposed Volterra model.

When a ship runs in waves, water pressure is a significant fac-tor in sustaining the ship's dynamics. However, studies of nonlin-ear behaviors about this topic are rarely conducted, particularly for high-speed conditions. General studies [ 1 - 4 ] revealed that a ship's sides suffer fatigue, which mainly results from the external wa-ter pressure. However, linear theories were unable to accurately

* Tel.: +886 7 583 4861; fax: +886 7 583 4861.

E-mail address: wctiao@mail.cna.edu.tw.

0141-1187/S - see front matter © 2010 Elsevier Ltd. All rights reserved, doi:10.1016/j.apor.2010.11.001

predict this dynamic pressure. Conclusively, these reviews [5,6] established the following two main points: first, the major indica-tion of the nonlinearity of pressure near the waterline came f r o m a cut-off signal transmitted by a sensor that alternatively crossed between water and air, and second, if the sensor were perpetually submerged underwater, the nonlinear behavior resulting f r o m the ship's geometry or speed was significant. To effectively identify the nonlinearities of the ship, this paper adopts a technique to high-light and present the nonlinear features of the ship's pressure in contrast to the well-known behaviors of heave motion through the proposed investigations.

To experimentally present the data in nonhnear aspects, sev-eral studies regarding ship dynamics routinely display the mea-sured signal in terms of the third-order model. This method takes the higher-order components into account, and is helpful in ex-pressing the nonlinearities, making the statistical quantities more precise [7]. Furthermore, this model can also be used to explain the variation in energy transfer f r o m various bandwidths [8,9], It has been previously demonstrated that the third-order model is a good approximation of the measured responses and that cubic effects are important and even dominate the responses at the high fre-quency range. To further measure the nonlinear degree, the non-linearities were identified and discussed based on the quantified higher-order terms as well as symmetry levels of the signal in time histories [10,11]. Therefore, i f the responses of the ship in waves are viewed as a single-input/single-output (SISO) system, quantify-ing the transfer functions of each order can be an effective method

W.-C. mo I Applied Ocean Researcii 33 (2011)60-68

to identify the nonhnearities. To conduct the quantification exper-imentally, regular waves or irregular waves must be considered. However, it is time consuming to perform these experiments d i -rectly f r o m regular wave tests because these experiments require the systematic determination of tens or even hundreds of dual- and triple-frequency combinations w i t h i n all bandwidths. This goal can be achieved by using irregular wave tests by repeatedly inciting the system i n a specified bandwidth of the wave. In the present study, the data measured i n both wave conditions is provided; data mea-sured in regular waves is used to carry out nonlinear identifica-rion procedures, whereas the other data measured from irregular waves is used to determine the quantification assessments w i t h i n bandwidths.

One o f t h e useful approaches to the quantification assessment of a physical system is the Volterra model. This model extends the concept of the linear system and provides insights into nonlinear problems by including higher-order series. The third-order orthogonal frequency-domain Volterra (TOFV) model is adopted i n the present study and this model was based on an extension of the work done by Bendat [12] for the multiple input/output linear system identification problems. It has the following advantages: (1) In the third-order model, each order has a physical interpretation. For a harmonic input, the first-order term gives an equivalent linear relationship, the second-order term is obtained by systematically combining any two frequencies to demonstrate the "sum" or "difference" effects of the input contributed to the output in moderately high or low frequencies, respectively, and the third-order term, except for extremely high frequency contribudons, also plays a role in supporting the first order to offset the bias because of coupling w i t h the first-order component [8]. For models higher than the third order, one should consider whether the precision of measurement meets the expected quality for extremely high frequencies; in addition, longer computation time should also be taken into account. (2) The model can use either Gaussian or non-Gaussian input [13]. Most natural signals do not f i t the Gaussian requirements but made it sustainable to simplify calculations. However, this leads to a higher error i n the predicted values [14]. Before discussing the system's inherent nonlinearities, it is necessary to analyze the nonlinearities attributable to the input; the statistical properties of the input are generally discussed by using a degree that is different f r o m the one used i n a Gaussian distribution. All the nonlinearities, including both the ones attributable to the input and the system's inherent ones, are considered in the present study. (3) The TOFV model overcomes the limitation of the nonorthogonal model: the magnitude of interference i n the nonorthogonal model may alternatively become positive or negative because of the phase, which is unfavorable to cleariy determine the contributions and has no specific physical interpretation. The orthogonal model overcomes this limitation, and the resultant coherence spectrum determines the contribution of each order; such a determination is helpful in understanding the model's overall prediction and saves time by avoiding repeated calculations of the processed orders i n future order-upgrade operations [9].

This paper extends the studies of the Volterra model and exper-imentally indicates the global nonlinear features o f t h e measured data. Previous studies experimentally obtained frequency trans-fer functions (FRFs) i n regular waves using the framework of the single-frequency Volterra model. However, when these data were further applied to irregular waves based on the superposition principle, the results were unsatisfactory because the quantification i n -formation for each order was inadequate to effectively achieve a balance between these two waves [15,16]. From the outcome of the present model, this information can be used to improve the computing algorithm of the FRFs obtained in regular waves or to simplify nonlinear problems by decreasing a model's order

61

The arrangement of the paper is as follows. Following the i n -troduction, the proposed Volterra model is described in Section 2. This model uses a platform to determine the coherence spectrum, which is useful in determining the quantification of each order i n the frequency bandwidths. The description of the experimental layout is presented in the next section. In Section 4, several exper-imental analyses identifying nonlinear features are presented and discussed. They include three nonlinear assessments, the probabil-ity densprobabil-ity functions, the variance spectra, and the coherence spec-tra. The conclusions f r o m this work are presented in Section 5.

2. Third-order orthogonal frequency-domain Volterra (TOFV) model

The nonlinear relationship between an input and an output signal may typically be described as a Volterra model by means o f a power series. Assuming that the system was excited and responded in the SISO pattern, the global responses can be expressed in the discrete form of the third-order Volterra model as follows: Y(m) = H,(m)X(m) + ^ H2(i,j)X(i)Xij)

i+j=m

+ YI H 3 ( p , q , r ) X ( p ) X ( q ) X ( r ) + £(m) (1) p+q+r=m

where Y{m), X(m) and e(m) denote the system's responses, input source, and errors, respectively. H, (m), H2(i,j), and HaCp, q, r ) are the first-, second-, and third-order frequency response functions (FRFs) obtained f r o m the Fourier transformation, respectively. Eq. (1) can be rearranged and expressed in matrix form as follows: Y(m) = Y(m) + e(m)

= y, (m) -F 72(m) -t- Y^im) -F e(m)

= H(m)X(m)-FÊ(m) (2)

w i t h H ( m ) = [H,(m),H2(m),H3(m)] and X'(m) = [X\(m), Xjim), X3(m)]. The bold script denotes matrices and superscript "t" refers to matrix transpose.

Since Eq. (2) solves straightforwardly to obtain the FRFs based on using the least-square method, it cannot avoid the problem of ill-conditioning which frequently occurs i n computing the solution of the normal equation. Thus, the orthogonal model is proposed. To carry out the orthogonalization, the matrix X(m) i n Eq. (2) is processed using the Modified Gram-Schmidt (MGS) method to obtain the orthogonal input Z(m).

Y(m) = H(m)X(m)

= K(m)L(m)X(m)

= K(m)Z(m) (3)

where L(m) is a lower triangular transformation matrix. On per-forming orthogonalization using the property (z, (m)Zj^(m)) = 0 i f i ^ j, the interference can be ignored. The matrix K(m) in Eq. (3) can be obtained by applying the least-square method i n terms of the orthogonalized input spectral matrices as follows:

K(m) = {y(m)Z''(m))(Z(m)Z'^(m))"'

= Sy^(m)S-\m) (4) where (•) is an expected value o f a random process and A denotes

the Hermitian of a matrix. Sjiz(m) and Szz(m) on the RHS of Eq. (4) are cross-spectral matrix and auto-spectral matrix, respectively, and the power spectrum of the global response Syy(m) can be ob-tained f r o m

Syy(m) = SyAm)S-\m)S^y(m)

62 W.-C. Tiao /Applied Ocean Researcii 33 (2011) 60-68

where S l ( m ) , S 2 ( m ) , and S3(m) represent the system's linear, second-order, and third-order power spectra, respectively, and can be expressed as:

Si(m) = S,z,.(m)S,-^.(m)S,,,(m) / = 1, 2, 3. (6) Accordingly, the total coherence function yj varies from zero to one, indicating the model's overall prediction quality f r o m very poor to excellent, respectively. It is independently comprised of linear, second-order and third-order components.

rrim) y^(m) + y^(m) + y^(m). (7)

Each coherence function on the RHS of Eq. (7) can be determined as follows: y^(m) • Si(m)S-\m).

3. Experimental setup

C a r r i a g e

-® ® ®

®-The model ship used in the experiments was 1/36 scale, made f r o m fiber reinforced plastic (FRP) material, w i t h a round bilge. The length between perpendiculars, draft, vertical position of CG (above base line), and displacement i n the model scale were 250.0 cm, 10.7 cm, 15.7 cm, and 46.0 N, respectively. Throughout the experiments, the vertical motions are measured by two potential meters mounted at the CG and the lateral and longitudinal movements are restrained. Fig. 1 shows a sketch of the configuration of the experimental setup. The measured responses included the incident wave elevation (hereafter denoted as W I ) , heave and pitch motions, four vertical accelerations along the ship length denoted as A1-A4, and 25 water pressures denoted as P1-P25. Strain-gage-type pressure gages (Honeywell Micro Switch 26PCAFA6G) were used. The diameter of the receiving surface, the natural frequency, and the capacity of the pressure gages were 1.8 mm, 3.3 kHz and 1.0 psi, respectively. The right-handed Cartesian coordinate system was used as a reference for all measurements and sign conventions. The origin is located at the initial position of CG of the model ship resting in still water w i t h zero forward speed. Based on the reference coordinate system, heave motion and wave elevation are defined as positive i n the downward direction, while bow-up i n pitch is defined as positive. The sampling rates were 250 Hz, and the initial voltage of each sensor was zeroed at the above-mentioned initial condition. Due to the strong nonlinearities and varied features of bow pressure particularly at high speeds, the core discussions are focused on three specific vertical locations from the surface to keel; these are: at the waterline (PIO), intermediate (P15), and at the bottom (P16). Moreover, heave motion or pressure at amidship zones (P5) are also included to serve as a contrast set for comparisons w i t h the nonlinear level. Locations of the selected pressure sensors are shown i n Fig. 2.

The experiment program consisted of tests in both regular and irregular head waves. Under these conditions, the model ship was towed at two speeds w i t h Froude numbers (Fn) 0.31 and 0.42 where Fn was defined i n terms of the length between the perpendiculars L

4. Experimental results and discussions

4.1. Nonlinear response in regular waves

For the tests i n regular waves, 10 frequencies were selected, corresponding to wavelength/ship-length ratios 0.50-3.20, and 3-6 different wave steepness values were considered for each wave frequency. At least 10 complete cycles were drawn for each measurement, and then analyzed using Fourier series. In order to systematically identify the nonlinear properties, the typical max-i m u m (^•^'"^), m max-i n max-i m u m (^'"'") and mean (^*°') values averaged

Fig. 1. Experimental setup. P1-P25, pressure sensors: A1-A4, accelerations.

Height ( c m ) 15 10 0 | ^ W . L . _ -/ A y -/ / y W / y / / 0 / / / / V V y 1 1 1 ^ 'j^^^ Breadth ( c m ) -20 -10 0 10 20 Fig. 2. Locations of the selected pressure sensors. W.L., waterline,

f r o m a measured sample and /th-order amplitude ( ^ ' ' ' ) were de-termined and are shown in Figs. 3-6. The nondimensional extreme and mean variations of heave are shown i n Fig. 3 plotted on the basis of encounter frequencies (tWe) and wave steepness (Hw/Lw: Waveheight/Wavelength). Fig. 3(a) indicates that the values ob-tained were directly analyzed f r o m the measured signal and a corner diagram is added to represent the measured steady-run re-sponses varying w i t h Fn, and Fig. 3(b) indicates the residual val-ues obtained from the directly measured valval-ues after subtracting the corresponding steady-run response (superscript * indicates a signal without steady-run response). The objective is to assess the contribution of the steady-run response to the global response in waves and also to analyze the dynamic response due to the waves alone. This reasoning assumes that the steady-run response may be decoupled from the dynamic response. It w i l l be seen that the curves w i t h the steady-run responses removed look consistent and well behaved. Moreover, the three wavelength/ship-length ratios corresponding to the possible resonant frequencies are marked in the figure i n order to observe the nature of response when sub-jected to the intense motions in waves. Figs. 4 - 6 show the bow

pressures; in these figures, subfigure (a) shows the amplitude vari-ations of each order and (b) indicates the same parameters as before. All the values indicated in these figures are nondimension-alized using the wave amplitude ( f o ) .

/4ssessmefit o/dynamic s/ii/f: The measured signal supposedly originates f r o m two factors after the ship hits the waves; namely, the steady-run response in calm water and the oscillatory response in waves. A deviation in this oscillatory response, shifted away f r o m zero, is called the dynamic shift. While Fig. 3(a) shows the data directly measured in waves, it cannot provide insight into the significant features of the response. The results show that a

W.-C. Tiao I Applied Ocean Researcii 33 (2011) 60-68 b 63 E a L./L= 1.50 1.30 1.10

:i1

Fig. 3. Extreme and mean value variations of heave with wave steepness, [(a): direct measured signal, the corner diagram shows steady-run responses, (b); signal without steady-run response; Fn = 0.42.1

Fig. 4. a, b. Amplitude variations (left) and extreme/dynamic shift variations (right) of waterline pressure (PIO) with wave steepness at Fn = 0.42.

slight dynamic shift is present, after the steady-run response is subtracted (Fig. 3(b)), which appears to be significant and can be used to confirm that the heave measured i n the waves can be assumed to develop a quasi-harmonic response relative to the steady-run response f r o m the theoretical point of view. Comparing the left and right graphs, it is clear that the steady-run response, or sinkage, has an effect on the response curves. Figs. 4(b)-6(b) show that the oscillatory response of pressure has a stronger dynamic shift as compared to the heave. The major dynamic shift of PIO and the minor one of P16 around the resonance frequencies reflect the fact that these outcomes are mainly associated w i t h the characteristics of their locations and are approximately independent of the wave steepness. Alternatively, the dynamic shifts of PI 5 decrease w i t h increasing wave amplitudes. From the operation of a third-order model, it is known that the zeroth-order component, namely the dynamic shift, is generally coupled w i t h the second-order component of the signal because both magnitudes depend on a same transfer funcdon [8]. In contrast to the left graphs in Figs. 4-6, the double-frequency energy of PI5 mainly contributes to the dynamic shift, but not the signal

itself because its second-order harmonics seem to lack definite tendencies varying w i t h the wave amplitudes, whereas those of PIO and P16 are systematical, but present inverse tendencies. These inverse tendencies are moderately linked and connected by the outcomes of PI 5 and can be used to explain the mechanism of energy transformation. Given their locations, P15 seems to be a pivotal position necessary for maintaining a continuous mechanism of pressure installed above and below.

Assessmeni: of amplitude/extreme variations: A system in which a consistent linear relationship exists between the input source and the output response is defined as a linear system. Using the property of linearity to measure the nonlinearities shows that the result, i f nondimensionalized by a proper input specific, w i l l remain constant. It has been known that the nondimensional first harmonics of the heave increase w i t h the steepness of the wave, and this increment can be as much as 17%. The first harmonics of the pressure at PIO, as shown i n Fig. 4(a), reduce significantly w i t h wave steepness around the resonant frequencies, and this reduction reaches 30% i n the tested range of wave steepness. A similar pattern occurs w i t h PI5, where the reduction reaches

G4 W.-C. Tiao/Applied Oceau Researcii 33 (2011) 60-68

a

Fig. 5. a, b. Amplitude variations (left) and extreme/dynamic shift variations (right) of intermediate pressure (P15) with wave steepness at Fn = 0.42.

a

2-Fig. 6. a, b. Amplitude variations (left) and extreme/dynamic shift variations (right) of bottom pressure (PI 6) with wave steepness at Fn = 0.42.

34%. These qualitative tendencies are a typical feature for the pressures, which are alternately dry and wet. In contrast, P16 shows an inverse tendency, i.e., the magnitude increases w i t h wave steepness for each wavelength, reaching approximately 30%. This result indicates that the bottom of the bow zone in the area of the high deadrise angle is subject to slight impacts and water pile-up for steeper waves, whereas the bow zone near the waterline tends to emerge f r o m the water surface for steeper waves. In conclusion, the magnitudes of the first harmonics o f t h e heave or pressure seem to vary w i t h the wave steepness, which demonstrates nonlinearity near the resonant frequencies.

W i t h respect to the extremes, the right graph of Fig. 3 shows that around the resonant frequencies, the maximums seem less sensitive to the wave amplitudes, whereas the minimums show slight increases. In the cases of smaller and higher frequencies, the extremes are essentially independent of wave amplitudes. This behavior may be explained by the fact that, during the process of an intense motion, a ship w i l l obtain more energy in the heave-up stage than heave-down. Compared to the heave response, bow pressures show a remarkable feature. Fig. 4(b) indicates that the extremes obviously decrease w i t h increasing wave steepness

for high and moderate frequencies and are independent of wave amplitudes at low frequencies. It can also be observed that the minimum suffers a notable reduction at the steepest wave tested near the resonant frequencies. For increasing wave steepness, this phenomenon is expected to become more significant. A contrasting effect is seen at P16; Fig. 6(b) reveals that the maximums slightly decrease and the minimums significantly increase as the steepness o f t h e wave increases. Near the resonant frequencies, at 8.2 rad/s, the minimum-amid span between any two waves is ten times greater than the maximum-amid span, which reflects the fact that the global pressure i n the steeper waves originating f r o m the ship's bow-up stage is larger than the pressure originating from the bow-down stage. In Fig. 5(b), its extreme variations contain the properties of the other two locations, and are generally similar to the waterline-located trends near the resonant frequencies and the bottom-located trends near the side frequencies.

Assessment of the degree of symmetry: If the input is a harmonic and symmetrical signal, then the output signal, based on the property of a linear system, w i l l have the same characteristics. Heave has approximately symmetrical extremes, but near the résonant frequencies, the natures of the extremes are opposite.

W.-C. Tiao/Applied Ocean Researcii 33 (2011) 60-68 ₆₅

Even though the heave resembles a harmonic signal, its shape is characterized by a round crest and a flat trough, indicating that the ship emerges more than i t submerges because of the vertical asymmetry of its waterplane area. As shown in Figs. 4(b)-6(b), the bow pressures are not symmetrical. Sensor PIO, which is located in a dry and a wet zone, indicates a large positive pressure. Sensor P16, which is subjected to slight flare impact, shows a slow increase in positive pressure during the bow-down stage and a large negative pressure during the bow-up stage. Sensor PIS experiences both effects and reveals a moderate variation in the negative pressure.

In addition, the significant higher-order components involved affect the degree of symmetry of the signal or lead to deformation. As shown in Figs 4(a)-6(a), the second harmonics of PIO decrease w i t h increasing wave amplitudes near the peak frequencies, while the third harmonics show no notable trend. In contrast, the higher-order harmonics of PI6 increase systematically w i t h wave steepness, whereas PI5 lacks an explicit rhythm. As compared w i t h the third-order Stoke waves, the second- and third-order components of the incident waves are 4% and 2% o f t h e first order, respectively, which shows that higher-order components i n waves exist, but they are of an insignificant amount. As shown by these experiments, the nondimensional amplitudes of the higher-order components for heave are not greater than 3%. It can be said that the higher-order harmonic components are quite small for the heave w i t h i n the tested range of wave amplitude. Compared to the motion responses, the higher-order harmonics of the pressure responses are definitely more pronounced. The second harmonic amplitudes may be up to around 100% of the wave amplitude for P16, 60% for PIO, and 20% for P15. These quantities are almost of the equivalent order of the incident wave amplitudes or the corresponding first harmonics of the pressure response. Furthermore, their third harmonic amplitudes are smaller, but may sdll reach a few tenths of the wave amplitudes. The pressure responses in those zones are not harmonic variations, despite the harmonic nature of the incident wave. It is proved that a resultant asymmetry is mostly due to the system's innate characteristics. This fact implies also that the nonlinearity is strong for the pressure responses and w i l l become more significant for higher levels of wave steepness. Thus, the higher-order components of pressure are significant and cannot be ignored i n practical applications.

4.2. Nonlinear response in irregular waves

For the irregular wave condition, the ISSC 1979 wave spectrum was used. It was defined using the parameters of the significant waveheight, H1/3 and the average of the zero up-crossing periods

of the one-third highest waves, TH^^^, as follows.

S ( f ) = 0.257 X X T„,/3 x (TH,„ X f y '

X e x p { - 1 . 0 3 x ( r H , / 3 x / r " ) (8) where ƒ denotes the circular frequency of a component wave.

The spectrum w i t h an H1/3 of 3.2 m and T^y^ of 8.0 s in f u l l -scale corresponding sea state of around five was used as the target spectrum for the experiments. Fifteen runs were carried out and the total recorded time of a single sample used for analysis was 26 s.

In Fig. 7, a single sample record of the selected dme-length responses (on the LHS of the figure) is plotted, together w i t h the corresponding probability density function (PDF) p(x) (on the RHS of the figure). In these diagrams, the measured steady-run responses are indicated by a horizontal dashed line. In order to compare the nonlinear level, the dme histories and the PDFs of the incident wave and heave are also included because these two responses are typical input/output responses of a ship dynamic

system which have been verified to approximately follow the Gaussian rule under the set condidons. It is observed that the time histories of the wave and heave are roughly symmetrical i n calm water and steady-run response respectively; however, pressures are certainly asymmetrical. The time history of the PI O's signal shows constant value when it becomes dry. Before becoming dry, PIO passes through a spray zone and shows even lower pressure. This is a typical pattern for high-speed vessel. Besides, steeper pressure increase can be identified when it reentered the water. A larger negative and deformed pressure of PI 6 created due to the flare impact can also easily be recognized f r o m the time histories. W i t h respect to the PDF of the wave, its peak appears near zero and gradually decays on both sides to form a nearly smooth bell-shaped distribution. The heave resembles wave pattern and it has a similar PDF distribudon w i t h its peak shifting to the steady-run response. In contrast to the wave or heave, three significant PDF features are seen in the pressure; these are a two-peak PDF (e.g., PIO or P15), a skewed PDF (e.g., P16), and a roughly symmetrical PDF (e.g., P5). For the single-peak PDF, the span between the peak and the steady-run response quantifies the system's dynamic shift. The dynamic shift of the wave or the heave exist, but they appear to be minor in comparison to the pressure. In the two-peak PDF, generally for a sensor alternating between water and air, one peak is related to the dynamic shift while the other corresponds to the equivalent static depth. In addidon, the distribution pattern of both tails also reflects the characteristics of the system's behavior near the end o f t h e response. PIO displays a mild distribution near the posidve tail (i.e., experiences deep water pressure) and a steep distribudon near the negative tail (i.e., preparation for air exposure). P16 shows an opposite pattern for tail sides, while the P15 sensor shows a transition. It can be generalized for the skewed or two-peak PDF of the pressure that the near-peak pattern depends significantly on environmental effects such as a change in the medium or the impact effect, whereas the near-tail pattern depends on the deadrise angle of the sensor's location and the asymmetry o f the waterplane area. When considering a SISO system, at least three different mechanisms w i l l converge on the pressure systems.

In order to globally present the experimental results, the vari-ance spectra based on Fourier analysis are plotted in Fig. 8. To obtain this spectrum, the signals of equal time length are con-nected w i t h 50% overiap, applied to the Hanning window, the spec-trum is computed using Fast Fourier Transformadon (EFT) w i t h 2048 points, and the resultant spectrum is obtained by averag-ing the overall spectra. It can be seen i n Fig. 8(a) that energy of the experimental wave spreads across the frequencies in the range 3.0-25.0 rad/s and the spectrum is generated quite well by the wave maker even though some tends to have slightly less energy in the high frequency tail. Thus, the measured wave spectrum, i n -stead o f t h e target wave spectrum serves as a basis for the discus-sion of other responses i n the following statements.

It can be observed f r o m Fig. 8 that the heave bandwidth is narrower than that of the incident wave and its energy is distributed mainly in the frequency range 4.0-13.0 rad/s. The pressure bandwidth is wider and the energy expands to the side frequencies mainly in the range 0.0-35.0 rad/s. Based on this, heave and pressures although incited by an identical source can attribute to two different dynamic systems. Heave belongs to an integral-type system because it does not respond to every input frequency. Pressure is a differendal-type system that has an output w i t h a wider bandwidth than the input. Accordingly, the nonlinear instincts in a differential-type system have greatly diversified features than in an integral-type system. In addition, the peak frequency of pressure is approximately 8.2 rad/s, which is consistent w i t h the incident wave. A small hump of energy appears where the double peak frequency is formed only for PIO and PI6 and not for P15, and energy exists i n triple or higher peak frequency but is insignificant. These phenomena are consistent w i t h those shown i n the regular wave tests.

66 W.-C. Tiao /Applied Ocean Research 33 (2011) 60-68 P(x) 1 1 ' ~ 0 0.2 0.4 4 - 1 - 4 H

Ft

Fig. 7. Time histories of the measured responses (left) with corresponding PDF (right) at Fn = 0.42 and sea state 5. 4.3. Coherence spectra

The coherence spectrum serves as a mechanism for measuring the relation between the input and output time series. In order to determine it, half-overlap segmented data w i t h 1024 points are preprocessed and 189 segments in all are available. The m i n i m u m frequency is approximately 0.244 Hz. The incident wave (input) and motion/pressure (output) adopt 17 and 25 frequency units, and the corresponding bandwidths are 0-4.15 Hz (0.0-26.08 rad/s) and 0-6.10 Hz (0.0-38.33 rad/s), respectively. Under these bandwidths, there are 3651 coefficients to be estimated (26 linear coefficients, 299 quadratic coefficients, and 3326 cubic coefficients). Referring to the variance spectra, the selected bandwidths cover the response characteristics and meet the physical requirements.

The coherence spectra in Fig. 9 are heave and pressure of P16. The linear coherence function of heave is centralized near 8.2 rad/s and fades gradually on both sides, and that of the third order thoroughly dominates at higher frequencies. Pressure P16 is dominated by the second and then third-order components beyond the major coverage of linear coherence function and their peak frequencies are double and triple of the centralized frequency, respectively. The previously mentioned hump-like energy of PI6 that appeared in double peak frequency is most

certainly a contribution of the quadratic term. The results verify the applicability of the present model because a total coherence function higher than 0.8 is considered enough for practical use and covers their bandwidths.

To learn more about the interactions inside the responses, the nonorthogonal model is provided and serves as a reference for explaining some interesting features of the orthogonal model. In Fig. 10, the orthogonal outcome of PIO, similar to that of P16, achieves a total coherence function higher than 0.8 w i t h i n its bandwidths. Results for the nonorthogonal model hardly identify a smooth contribution; however, they show a large swing w i t h i n the linear bandwidths that gradually shrinks at higher frequencies. Besides, the odd-order interactions are relatively more remarkable than other combinations. The orthogonal result of PI 5 is a typical example different f r o m the other two pressures. In Fig. 11(a), higher frequencies beyond the major coverage of linear function avoid the second-order domination and directly proceed to the third order because its quadratic effect of signal mainly contributes to alter the dynamic shift. Moreover, i t is observed that the third-order coherence function suddenly drops at 18.0 rad/s near where the double peak frequency of the PIO or P16 appears, and it is observed in Fig. 11(b) that not only the odd-order auto-coherences but also their cross-auto-coherences have a sudden peak at this frequency. This physical mechanism is unclear but f r o m the

W.-C. Tiao I Applied Ocean Research 33 (20U) 60-68 ₆₇

Fig. 8. Variance spectra for the selected responses at Fn = 0.42 and sea state 5. |(a): incident wave, (b): heave, (c): pressures.]

0 10 20 30 40 0 10 20 30 4 0

Fig. 10. Coherence spectrum of waterline pressure (PI 0). [(a): orthogonal model, (b): nonorthogonal model; i , a , C, and T in the diagram denote the linear, quadratic, cubic, and total coherence functions, respectively, and other combinations are the interference terms; Fn = 0.42, sea state 5.]

68 W.-C. Tiao /Applied Ocean Researcii 33 (2011) 60-68

Fig. 11. Coherence spectrum of intermetiiate pressure (P15). |(a): orthogonal model, (b): nonorthogonal model; L, Q, C, and T in the diagram denote the linear, quadratic, cubic, and total coherence functions, respectively, and other combinations are the interference terms; Fn = 0.42, sea state 5.]

internal algorithm of the present model the linear component w i l l be prioritized by relaxing the third-order component while dealing w i t h odd-order coupling i n order to achieve the model's premium result so that a small linear peak accompanies a large third-order drop shown at this frequency.

5. Conclusions

The present study discussed the nonlinearities of a high-speed ship running in head waves and provided a strategy to achieve a balance between regular and irregular waves f r o m the coherence spectrum by applying the third-order orthogonal frequency-domain Volterra (TOFV) model when dealing w i t h strong nonlinear bow pressure. In contrast to the quasi-linear behaviors of heave motion, the nonlinear behaviors of pressure are highlighted and presented. Based on the assessments of dynamic shift, amplitude/extreme variations, and signal symmetry in waves, particular pressures at ship's bow contain more nonlinearities than the heave motion. They are identified mainly by the large amplitude of the higher harmonics and also by the large asymmetry of the measured signals. Furthermore, the pressures are definitely non-Gaussian and three probability density function (PDF) features generally appeared. These nonlinearities of pressure are mainly related to environmental effects such as medium change or the impact effect and the asymmetry of the waterplane area. We also concluded f r o m the variance spectra that the heave is attributed to the integral-type system because it responds to the incident waves only w i t h i n a specified frequency range and possesses less energy at higher frequencies. In contrast, the bow pressures are attributed to a differential-type system that responds beyond wave bandwidths and demonstrates a strong nonlinear nature, suggesting that higher-order effects are significant and should be accounted for.

Within the effective bandwidths, the coherence spectrum pro-vides sufficient information on the contributions of each order at the corresponding frequencies, which serves as a strategy to sim-p l i f y a comsim-plex sim-problem or to achieve a balance between reg-ular/irregular waves by determining which higher-order term is significant. The pressure located at the bow's waterline or bot-tom is dominated i n turn by each order w i t h i n their correspond-ing bandwidths, which explains the source of the energy at these frequencies. However, an intermediate pressure acts against this tendency and the domination of the quadratic term i n correspond-ing bandwidths is lost because the location of this pressure plays

a pivotal role in maintaining a continuous mechanism of pressure above and below.

Acloiowledgement

The author would like to thank the National Science Council of Taiwan for financial support under Grant Number NSC-97-2218-E-012-001.

References

|1] Hansen FP, Winterstein SR. Fatigue damage in the side shells of ships. Mar Struct 1995;8:631-55.

[2] Folso R. Spectral fatigue damage calculation in the side shells of ships with due account taken of the effect of alternating wet and dry areas. Mar Struct 1998; 11:319-43.

(3) Tanizawa K, Taguchi H, Saruta T, Watanabe I, et al. Experimental study of wave pressure on VLCC running in short waves. J Soc Nav Archit Jpn 1993; 174:233-42.

|4] Ito A, Mizoguchi S. Hydrodynamic pressure on a full ship in short waves. J Soc Nav Archit Jpn 1989;166;251-8.

| 5 | Miyake R, Kinoshita T, Kagemoto H, Zhu T. Ship motions and loads in large waves. In; Proceedings of 23rd ONR Symposium on Naval Hydrodynamics. 2000. p. 48-61.

[6] Qiu W, Peng H, Hsiung C. Validation of time-domain prediction of motion, sea load, and hull pressure o f a frigate in regular waves. In: Proceedings of 23rd ONR Symposium on Naval Hydrodynamics. 2000. p. 34-47.

[7] Hasselmann K. On nonlinear ship motions in irregular waves. J Ship Res 1966; 10:64-8.

|8j Adegeest L. Third-order Volterra modeling of ships responses based on regular wave results. In: Proceedings of 21st ONR Symposium on Naval Hydrodynamics. 1996. p. 141-55.

|9] O'DeaJ, Powers EJ.ZselecskyJ. Expenmental determination of nonlinearities in vertical plane ship motions. In: Proceedings of 19th ONRSymposium on Naval Hydrodynamics. 1992. p. 53-70.

[10] Fonseca N, Guedes SC. Experimental investigation of the nonlinear effects on the vertical motions and loads of a containership in regular waves. J Ship Res 2004;48:118-47.

[11] Fonseca N, Guedes SC. Experimental investigation of the nonlinear effects on the vertical motions and loads o f a containership in irregular waves. J Ship Res 2004;48:148-67.

| 1 2 | Bendat JS. Nonlinear system techniques and applications. Wiley Interscience; 1998.

| 1 3 | Im S, Powers EJ. A sparse third-order orthogonal frequency-domain Volterra-like model. J Franklin Inst 1996;333(B)(3):385-412.

|14] Kim Kl, Powers EJ. A digital methotl of modeling quadratically nonlinear systems with a general random input. IEEE Trans Acoust Speech Signal Process 1988;36:1758-69.

[15] Chiu FC, Tiao WC, Guo Jenhwa. Experimental study on the nonlinear pressure acting on a high-speed vessel in regular waves. J Mar Sci Technol (Tokyo) 2007; 12:203-17.

[16] Chiu FC, Tiao WC, Guo Jenhwa. Experimental study on the nonlinear pressure acting on a high-speetl vessel in irregular waves. J Mar Sci Technol (Tokyo) 2009;14:228-39.