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Chaotic motions of nonlinearly moored structures Phadke, Amal C. 1993

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CHAOTIC MOTIONS OF NONLINEARLYMOORED STRUCTURESbyAmal Chandrakant PhadkeB.E., University of Bombay, India, 1989M.Tech., Indian Institute of Technology, Bombay, India, 1991A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF CIVIL ENGINEERINGWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAAugust 1993© Amal C. Phadke, 1993In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at The University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission forextensive copying of this thesis for scholarly purposes may be granted by the Headof my Department or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.Department of Civil EngineeringThe University of British Columbia2324 Main MallVancouver, BC V6T 1Z4CANADA11AbstractA number of studies have described the theoretical and numerical aspects of the chaoticmotions of offshore structures with nonlinear moorings. As one such example, Aoki,Sawaragi and Isaacson (1993) described the numerical simulation of the motions of a singledegree of freedom system with a piecewise-linear restoring force function. However,relatively few laboratory measurements of chaotic motions have been reported, and theprimary aim of the present study is to investigate the corresponding problem experimentally.Thus, the present work describes the measurement of chaotic motions of a floating structurewith nonlinear moorings. The structure is modeled as a rectangular box, and the moorings arerepresented by a nonlinear restoring force - displacement relationship, corresponding to anidealized geometric nonlinearity associated with a slack mooring or a mooring with gaps.The experiments were conducted in the wave flume of the Hydraulics Laboratory of theDepartment of Civil Engineering at the University of British Columbia. The flume is 40 mlong, 0.62 m wide, operates with a nominal water depth of 0.55 m, and is equipped with awave generator capable of producing regular and random waves and controlled by a DECVAXstation-3200 computer.The model structure is 76 cm long x 25 cm wide x 20 cm high. Two vertical Plexiglasplates parallel to the sides of the wave flume were installed so as to limit the structure motionsto three degrees of freedom corresponding to surge, heave and pitch. Ball bearings mountedon the sides of the box are used to minimize friction between the plates and the structure. Twovertical cantilevered beams located at some distance from the each end of the structure wereused to simulate the nonlinear mooring stiffness.111Displacement measurements at three different locations on the body were made usingpotentiometers mounted on a rigid aluminum frame, with a system of strings, pulleys andcounter-weights used to transmit the structure motions to the potentiometers. The measureddisplacements were transformed to provide the surge, heave and pitch motions with respect tothe centre of gravity of the structure.The results are presented in the form of time series, phase portraits, spectra, Poincaremaps, and Lyapunov exponents. The influence of various governing parameters on theresponse is examined. These include a dimensionless wave height, which characterizes themagnitude of the excitation; a relative wave frequency; and gap width and a dimensionlessspring stiffness, which characterize the moorings.Periodic, sub-harmonic and chaotic responses are observed for both monochromatic andbichromatic waves. In general, sub-harmonic and chaotic responses were obtained forbichromatic excitation to a greater extent than for monochromatic excitation. Transient chaoticmotions have also been observed, such that the response initially appears to be very irregular,but eventually settles to a regular periodic motion. Poincart maps of the response exhibit adistinct fractal structure under certain conditions, indicating the presence of chaotic motions.Finally, Lyapunov exponents, which provide a quantitative indication of chaotic motions,have also been computed for each time series, and are used to confirm the presence of chaoticmotions.ivTable of ContentsPageAbstract^ iiTable of Contents^ ivList of Tables viiList of Figures^ viiiAcknowledgment xi1^Introduction^ 11.1 General 11.2 Literature Review^ 31.3 Scope of Present Work 72^Nonlinear Dynamics and Chaos^ 8Introduction^ 82.1 Nonlinear Dynamics^ 82.1.1 Nonlinear Vibration Theory^ 92.2 Identifying Chaotic Vibrations 102.2.1 Nonlinear System Elements^ 122.2.2 Random Inputs^ 122.2.3 Observation of Time Series^ 132.2.4 Phase Portrait^ 13V2.2.5 Fourier Spectrum^ 142.2.6 Poincare Map 142.2.7 System Parameter Variation^ 152.2.8 Lyapunov Exponents 162.2.9 Estimation of Lyapunov Exponents from Experimental Time Series^173^Experimental Investigation^ 19Introduction^ 193.1 Dimensional Analysis^ 193.2 Natural Frequency in Surge 203.3 Location of Centre of Gravity^ 223.4 Measurement of Spring Stiffness 233.5 Experimental Setup^ 243.5.1 General 243.5.2 Wave Flume and Wave Generator^ 243.5.3 GEDAP Software^ 253.6 Experimental Procedure 263.6.1 Calibration of Potentiometers^ 263.6.2 Wave Generator Signal File 263.6.3 The Experiment^ 273.7 Computation of Response 274^Results and Discussion^ 294.1 Time Series, Phase Portraits and Spectra^ 29vi4.2 Effect of Wave Height^ 314.3 Effect of Wave Period 324.4 Effect of Spring Stiffness and Gap Width^ 324.5 Non-dimensional Plots^ 334.6 Poincare Map^ 334.7 Lyapunov Exponent 344.7.1 Fixed Evolution Time Program for X1^ 355^Conclusion and Scope for Further Research 37References^ 39Appendix A 41Appendix B^ 44Tables 51Figures^ 53viiList of TablesTable 3.1 Measured and calculated load-deflection values for spring #4^51Table 3.2 Spring characteristics^ 51Table 4.1 Input parameters and Lyapunov exponents - monochromatic excitation^52Table 4.2 Input parameters and Lyapunov exponents - bichromatic excitation^52List of FiguresviiiIdealization of the classical spring-mass-dashpot oscillatorClassical resonance curves of a linear single degree of freedom systemClassical resonance curve for a nonlinear oscillatorComparison of linear and nonlinear systemsDivergence of nearby orbitsEstimation of Lyapunov exponents from experimental time series535354545555Fig. 2.1Fig. 2.2Fig. 2.3Fig. 2.4Fig. 2.5Fig. 2.6Fig. 3.1Fig. 3.2Fig. 3.3Fig. 3.4Fig. 3.5Fig. 3.6Mathematical model of surge motions of the box^ 56Definition sketch for estimating the location of the centre of gravity^56Definition sketch for calculation of cantilever beam stiffness^57Comparison of measured and calculated values of cantilever beam deflection 57Sketch of experimental setup^ 58Definition sketch - transformation equations^ 58Fig. 4.1 Surge responseH= 13 cm, T =Fig. 4.2 Surge responseH= 15 cm, T =Fig. 4.3 Surge responseH= 14 cm, T =Fig. 4.4 Surge responseH= 13 cm, T =Fig. 4.5 Surge responseH= 13 cm, T =Fig. 4.6for monochromatic2.5 s, a = 0.423, Kfor monochromatic2.5 s, a = 0.423, Kfor monochromatic2.5 s, a = 0.423, Kfor monochromatic2.0 s, a = 0.529, Kfor monochromatic2.2 s, a = 0.481, Kexcitation= 4.25, dg = 11.5 cmexcitation= 4.25, dg = 11.5 cmexcitation= 4.25, dg = 11.5 cmexcitation= 4.25, dg = 11.5 cmexcitation= 4.25, dg = 11.5 cmSurge response for monochromatic excitation5961636567ixH = 13 cM, T = 2.8 s, a = 0.378, K = 4.25, dg = 11.5 CM^69Fig. 4.7 Surge response for monochromatic excitationH= 13 cm, T = 3.0 s, a = 0.353, K = 4.25, dg = 11.5 CM^71Fig. 4.8 Surge response for monochromatic excitationH= 11 cm, T = 2.8 s, = 0.378, K = 4.25, dg = 11.5 CM^73Fig. 4.9 Surge response for monochromatic excitationH= 15 cm, T = 2.8 s, = 0.378, K = 4.25, dg = 11.5 CM^75Fig. 4.10 Surge response for monochromatic excitationH= 11 cm, T = 2.0 s, a = 0.529, K = 4.25, dg = 11.5 cm^77Fig. 4.11 Surge response for monochromatic excitationH= 11 cm, T = 2.2 s, = 0.481, K = 4.25, dg = 11.5 CM^79Fig. 4.12 Surge response for bichromatic excitationH1= 7.0 cm, H2 = 7.0 CM, Ti = 2.3 S, T2 = 2.7 s, a = 0.423, K = 4.25^81Fig. 4.13 Surge response for bichromatic excitationH1= 9.0 cm, H2 = 9.0 Cm, Ti = 2.2 S, T2 = 2.8 S, = 0.423, K = 4.25^83Fig. 4.14 Effect of wave height variation on surge response^ 85Fig. 4.15 Effect of wave period variation on surge response 85Fig. 4.16 Plot of response amplitude vs. gap width^ 86Fig. 4.17 Plot of R/H vs. B/gT2 for K = 4.25, H = 13 cm 86Fig. 4.18 Plot of R/gT2 vs. B/H for K = 4.25, T = 2.5 s^ 87Fig. 4.19 Plot of R/H vs. a for K = 4.25, H = 13 cm 87Fig. 4.20 Poincare map of surge response for bichromatic input(a) fic = 10 (b) = 20^ 88Fig. 4.21 Poincare map of surge response for bichromatic input(a) ti = 30 (b) ti = 40 89Fig. 4.22 Poincare map of surge response for bichromatic input(a) ti = 50 (b) ti = 60^ 90Fig. A-1 Displacement probe #1, Li = original length, L1' = length at time t^91Fig. A-2 Displacement probe #2, L2 = original length, L2 = length at time t^92Fig. A-3 Displacement probe #3, L3 = original length, L3' = length at time t^93Fig. B-1 Definition sketch of a moored two-dimensional floating body^94Fig. B-2 Mathematical model of moored floating object with nonlinear moorings^94xiAcknowledgmentsThe author would like to thank his supervisor Dr. Michael Isaacson for his guidance andencouragement throughout the preparation of this thesis. The author would like to express aspecial word of gratitude to Kurt Nielson of the Civil Engineering Department Workshop atUBC for his help in building the model and other associated facilities. Thanks also to JohnWong for making available the required electronic parts and components and to SundarPrasad for his help with the computing and wave generator facilities. A specialacknowledgment is made to Sundar Prasad, Andrew Kennedy and Dr. Fai Cheung for theiradvice and suggestions during the course of this experimental work. Finally, the financialsupport in the form of a Research Assistantship from the Natural Sciences and EngineeringResearch Council of Canada is gratefully acknowledged.1Chapter 1Introduction1.1 GeneralThe phenomenon of chaotic behaviour in nonlinear dynamical systems has receivedconsiderable attention in recent years. At the turn of the century, the French mathematicianH. Poincare (1890) discovered that certain mechanical systems whose time evolution isgoverned by Hamilton's equation could display chaotic motion. Unfortunately, this wasconsidered by many physicists as a mere curiosity, and it took another 65 years until in 1963meteorologist Edward Lorenz discovered a similar phenomenon. Lorenz (1963), in hisfamous work "Deterministic non-periodic flow", which was published in the Journal ofAtmospheric Science, presents a system of three differential equations which aredeterministic, but show very irregular (also called random-like) behaviour. Lorenz's paper,the general importance of which is recognized today, was not widely appreciated until manyyears after its publication. He discovered one of the first examples of deterministic chaos indissipative systems. A physical system is said to have deterministic time dependence, ifthere exists a well defined governing differential or difference equation for calculating itsfuture behaviour from given initial conditions.The term deterministic chaos denotes an irregular or chaotic motion that is generated bya nonlinear system whose dynamical laws uniquely determine the time evolution of a statefrom the knowledge of its previous history. In recent years, new theoretical results, theavailability of high speed computers, and refined experimental techniques have made it clear2that this phenomenon is widespread in nature and has far reaching consequences in manybranches of science and technology.Examples of nonlinearities in mechanical and electrical systems include the following:• Nonlinear elastic or spring elements• Nonlinear damping such as friction• Backlash, limiters or bilinear springs• Fluid related forces• Nonlinear boundary conditions• Nonlinear feedback control forces in servo systems• Nonlinear resistive, capacitor or inductive circuit elements• Diodes• Many transistors and other active devices• Electric and magnetic forcesIt should however be noted that, a system nonlinearity is a necessary but not a sufficientcondition for the generation of a chaotic motion.Observed chaotic behaviour is not due to external sources of noise, but rather is aproperty of the nonlinear system which results in adjacent trajectories separatingexponentially fast in a bounded region of phase space. Consequently, it becomes virtuallyimpossible to predict the long-term behaviour of such systems, because in practice initialconditions can only be fixed with finite accuracy, and errors increase exponentially fast. Anattempt to solve such a nonlinear system on a computer, gives rise to a result which dependsincreasingly on more decimal digits of the numbers representing the initial conditions.Since, the errors increase exponentially fast, even a very small change in the initialconditions produce an entirely different response.The above results give rise to a number of fundamental questions:• Can one predict whether or not a given system will display deterministic chaos?3• Can one specify the notion of chaotic motion more mathematically and developquantitative measures for chaos?• Does the existence of deterministic chaos imply the end of long-time predictability inphysics for certain nonlinear systems or can one still learn something from a chaoticsignal?These questions have been discussed in great detail in literature. (e.g. Schuster, 1988).1.2 Literature ReviewAlthough there is a vast literature on chaotic behaviour in nonlinear dynamical systems,there has been relatively little work reported on the chaotic motions of floating structureswith nonlinear moorings. Before discussing chaos in nonlinear offshore systems, a briefreview of chaos in nonlinear systems will be presented in paragraphs to follow for the sakeof completeness.As already mentioned, considerable interest in chaotic systems began after Lorenz'(1963) work on the model of atmospheric convection. Many other deterministic equationsshowing chaotic behaviour have been obtained, both as simple, analytical systems and asmodels of real physical, biological or chemical systems. These include both systems ofnonlinear ordinary differential equations and maps.Duffing's equation is perhaps the most widely studied by researchers working in the areaof chaotic dynamics. It is defined asx + ax + bx + cx3 = f(x)^ (1.1)where, an overdot denotes a time derivative.4This equation is important in the theory of nonlinear oscillations because of the largenumber of systems that can be modeled by it. Duffing's equation in a slightly different formwas investigated by Ueda (1979) as applied to certain electrical systems..3 Bx + ax + x = cos t (1.2)This equation is now widely recognized as exhibiting important dynamical properties. Anexcellent coverage of Duffing's equation can be found in Kapitaniak (1991).The concept of chaos is relatively new to ocean engineering. The first reported researchwork dates back to mid to late eighties. In their pioneering work, Bishop and Virgin (1988)used a combined numerical and geometric approach to study the dynamic behaviour of amoored semi-submersible based on solutions of the nonlinear differential equation used tomodel the system. They observed competing steady states, sub-harmonic resonance andchaos as typical responses in regular seas. They used a quantitative overview to classify thecomputer generated results of direct time simulation, with the aim of illustrating theinadequacies and limitations of a linear, analytical approach.Aoki, Sawaragi and Isaacson (1993) have studied motions of a floating body withnonlinear moorings modeled as a single degree of freedom system. Response of the systemto monochromatic and bichromatic excitation with both material and geometricnonlinearities is studied. They have reported existence of jump phenomenon for bothmaterial and geometric nonlinearity cases. They report sub-harmonic and chaotic responsefor geometrically nonlinear system for monochromatic and bichromatic excitation.The damping ratio is a one of the most crucial parameters deciding the systembehaviour. Chaotic behaviour in nonlinear systems is normally found at lower dampingratios. Aoki, Sawaragi and Isaacson (1993) have reported that chaotic behaviour disappearsat high damping ratios. Interestingly, Sumanuskajonkul and Hu (1992) have observed thatin certain dynamical systems chaotic motions may occur at high damping situations even5when periodic motions are found at identical low damping systems. They investigateddynamic responses of bilinear and impacting oscillators subjected to harmonic loading.Instead of adapting the frequency ratio of excitation and structure as the controllingparameter, they used damping and stiffness ratio.Gottlieb et. al. (1990) have reported on a semi-analytic method for predicting localinstability, global bifurcation and the onset of chaotic motion in a multi-point mooringsystem. They considered large geometric nonlinearities and combined periodic waves andregular current. They have shown that a stability analysis based on an approximate solutionof a strongly nonlinear ocean system can serve as an efficient indicator for the nonlinearbehaviour, thus reducing numerical search efforts for global instability and chaotic responseregions.Yim and Lin (1991) investigated chaotic and stochastic dynamics of the rockingresponse of free-standing offshore equipment subjected to horizontal base excitation. Theyused a realistic model to take into account the geometric nonlinearity (finite slendernessratio) of the rocking system. Additional important nonlinear effects including transition ofgoverning equations of motion at impact were examined. It was demonstrated that thenonlinearities associated with the transition of governing equations at impact producedcomplex responses. In addition to the anticipated harmonic and sub-harmonic periodicresponses, two new types of steady state motions - quasi-periodic and chaotic responseswere observed. In this study, it was shown that although the excitations to the rockingsystems were simple and purely deterministic, some stochastic characteristics of the chaoticrocking responses could be detected using Poincard maps and amplitude probabilitydensities.Papoulias and Bernitsas (1988) analyzed dynamic behaviour of a single-point mooringsystem under time-dependent external excitation. They described the time evolution of thecorresponding dynamical system in a six-dimensional phase space. Bifurcation sequences of6state equations were studied and parameter values at which the response of SPM changedrapidly were identified. An analysis of stability and instability domains of the systemrevealed regions of operationally hazardous response. An important conclusion of theirstudy was that an SPM system under time-independent environmental excitation might notstay in a position of static equilibrium.Lyapunov exponents are perhaps the best quantitative estimate of chaotic nature of adeterministic dynamical system. A Lyapunov exponent characterize the properties of anattractor of a dynamical system. Lyapunov exponents are related to the average rates ofconvergence and/or divergence of nearby trajectories in phase space and, therefore, theymeasure how predictable or unpredictable the system is. A considerable amount of work hasbeen reported on the estimation of the Lyapunov exponents. They were introduced to thetheory of dynamical systems by Oseledec (1968). The first numerical visualization of thechaotic motion in phase space trajectory in terms of the divergence of nearby trajectorieswas introduced in Henon & Heiles (1964). It was then developed further by Chirikov(1979), Ford (1975), Wolf et al. (1985), Wolf (1986) and others.Generally, in experimental chaotic dynamics, observations are stored in the form of atime series. The next important step is to assess whether the system behaviour is chaotic orperiodic. The Lyapunov exponent offers a quantitative measure of aperiodicity of thesystem response. Generally Wolf et. al. (1985) are credited with presenting the firstalgorithm to compute Lyapunov exponents from experimental time series. They providetwo useful computer programs to evaluate the two largest Lyapunov exponents. A systemwith one or more positive Lyapunov exponents is defined to be chaotic. Recently Frank(1992) has modified the algorithm of Wolf et. al. for the improved estimation of the largestLyapunov exponent in the case of noisy and small data sets. Wolf's algorithm is useful tocompute the two largest Lyapunov exponents only. A few other algorithms to compute allLyapunov exponents have been proposed in literature (e.g. Parker and Chua, 1989).7However, in spite of its limitation, Wolf's algorithm has been widely used by researchers inthe area of chaotic dynamics.1.3 Scope of Present WorkThe primary aim of the present investigation is to detect chaos experimentally in a nonlinearoffshore structural system. Several numerical studies have been reported on this topicbefore, but there is generally a lack of supporting experimental work. On the basis of asingle degree of freedom model, Aoki, Sawaragi and Isaacson (1993) have reported sub-harmonic and chaotic surge motions in an offshore structural system with geometricallynonlinear moorings and their work is taken as a base for the present experimentalinvestigation. A three degree of freedom system is considered, such that heave and pitchdegrees of freedom are included in addition to surge. Displacement measurements at threedifferent locations on the body are made using potentiometers mounted on a rigid aluminumframe, with a system of strings, pulleys and counter-weights used to transmit the structuremotions to the potentiometers. The measured displacements are transformed to provide thesurge, heave and pitch motions with respect to the centre of gravity of the structure(Appendix A provides a detailed derivation of these transformation equations). The resultsare presented in the form of time series, phase portraits, spectra, Poincar6 maps, andLyapunov exponents. The influence of various governing parameters on the response isexamined. These include a dimensionless wave height, which characterizes the magnitudeof the excitation; a relative wave frequency; and gap width and a dimensionless springstiffness, which characterize the moorings. Finally, the largest Lyapunov exponent iscomputed for each experimental time series using Wolf's algorithm An analysis of theresults shows that chaos is indeed observed in this nonlinear dynamical system. Ananalytical approach to the present problem involves solution of the nonlinear equation ofmotion in time domain. This procedure is described briefly in Appendix B.8Chapter 2Nonlinear Dynamics and ChaosIntroductionThis chapter provides a brief introduction to the theory of nonlinear dynamics and itsapplications to chaotic systems. This provides sufficient theoretical background of chaoticdynamics, and discusses methods employed in identifying and quantifying chaotic motions.The ideas discussed in the present chapter will be incorporated directly in subsequentchapters in order to analyze experimental data.2.1 Nonlinear DynamicsThe spring-mass-dashpot system shown in Fig. 2.1 provides the classic example of adynamic system exhibiting linear vibrations. In the absence of any external excitation, theundamped system (A. = 0) vibrates with a frequency coo that is independent of the amplitudeof vibration.k } 1/2coo = { K4- (2.1)where, M and K are respectively the mass and stiffness. In this state, energy flowsalternately between elastic energy in the spring and kinetic energy in the mass. Thepresence of damping introduces a decay in the free vibration such that the displacementamplitude of the mass exhibits the following time dependence:9x(t) = Ao e-milt cos (03dt - (0)^ (2.2)where, x(t) is the instantaneous displacement response, Ao is the displacement responseamplitude, C is the damping ratio, cod is the damped natural frequency and 4) is the phasedifference between the input excitation and displacement response.One of the classic phenomena of such a linear system is that of resonance underharmonic excitation. For this problem, the differential equation that models the system maybe expressed in the form:i + 2Cconk + con2 x = Fo cos ox^(2.3)where, an overdot represents a time derivative, Fo and co are the forcing function amplitudefrequency respectively.If Fo is fixed and the driving frequency w is varied, the absolute magnitude of the steadystate displacement reaches a maximum which is close to the natural frequency con at thedamped natural frequency cod. This phenomenon is illustrated in Fig. 2.2 as a transferfunction plot. The effect is more pronounced when the damping ratio C is small. With thisbackground, the behaviour of nonlinear systems is now considered.2.1.1 Nonlinear Vibration TheoryA classical example of a nonlinear system is one with a nonlinear spring described by theDuffing equation.X + 2Co)nic + ax + I3x3 = F(t)^ (2.4)where, a and r3 are coefficients of the nonlinear stiffness term.10If the system is acted on by a periodic force, in the classical theory one assumes that theoutput also will be periodic. When the output has the same frequency as the force, theresonance phenomenon for the linear spring is shown in Fig. 2.3. If the amplitude of theforcing function is held constant, there exist a range of forcing frequencies for which threepossible output amplitudes are possible as shown in Fig. 2.3. One can show that the dashedcurve in Fig. 2.3 is unstable so that a hysteretic effect occurs for increasing and decreasingfrequencies. This is called a jump phenomenon. Other periodic solutions can also be foundsuch as sub-harmonic and superharmonic vibrations. Sub-harmonics play an important rolein pre-chaotic vibrations.There exist three classic types of dynamical motion:• Equlibrium• Periodic motion or limit cycle• Quasiperiodic motionThese states are called attractors, since if some form of damping is present the transientsdecay and the system is attracted to one of the above three states. There is another class ofmotions in nonlinear vibrations that is not one of the above classic attractors. This new classof nonlinear motions is chaotic and is known in literature as a strange attractor.2.2 Identifying Chaotic VibrationsIn this section, a set of diagnostic tests are presented to help in identifying chaoticoscillations in physical systems. Engineers often have to diagnose the source of unwantedoscillations in physical systems. The ability to classify the nature of oscillations can providea clue as to how to control them. For example, if the system is thought to be linear, largeperiodic oscillations may be traced to a resonance effect. However, if the system is non-11linear, a limit cycle may be the source of periodic vibration, which in turn may be traced tosome dynamic instability in the system.A checklist to identify chaotic or non-periodic motion is compiled below:Qualitative Methods• Identifying a nonlinear element in the system• Check for sources of random input in the system• Observe the time history of the measured signal• Examine the phase plane history• Examine the Fourier spectrum of the signal• Obtain the Poincare map of the signal.• Vary the system parameters (routes to chaos)Quantitative Method• Compute the Lyapunov exponentsA diagnosis of chaotic vibrations implies that one has a clear definition of such motions.However, as research uncovers more complexities in nonlinear dynamics, rigorousdefinitions seem to be limited to certain classes of mathematical problem. Anexperimentalist may find this rather difficult to achieve, so that one is encouraged to use twoor more tests to obtain a consistent picture of the chaos.Characteristics of Chaotic VibrationsFollowing are some of the important characteristics of chaotic vibrations:• Sensitivity to initial conditions• Broad spectrum of Fourier transform when motion is generated by a single frequency12• Fractal properties of motion in phase space which denote a strange attractor• Increasing complexity of regular motions as some experimental parameter is changed• Transient or intermittent chaotic motion; nonperiodic bursts of irregular motion(intermittency) or initially randomlike motion that eventually settles down into a regularmotionTests to identify chaotic motion can be divided into two categories, viz., qualitative methodsand quantitative methods. Each of these methods will be discussed separately in followingsections.2.2.1 Nonlinear System ElementsA chaotic system must have nonlinear elements or properties. This is a necessary but not asufficient condition. A linear system cannot exhibit chaotic vibrations. In a linear system,periodic inputs produce periodic outputs of the same frequency after transients havedecayed. In mechanical systems, nonlinearities may exist as a result of:• Nonlinear elastic or spring elements• Nonlinear damping• Most systems with fluids• Nonlinear boundary conditionsNonlinear elastic effects can be associated with either material or geometric properties.2.2.2 Random InputsIn chaotic vibrations, the excitation is assumed to be deterministic. By definition, chaoticvibrations arise from nonlinear deterministic physical systems or nonlinear deterministicdifferential or difference equations. It is presumed that large non-periodic signals do not13arise from very small input noise. Thus, a large output signal to input noise ratio is requiredif one is to attribute a non-periodic response to a deterministic system behaviour.2.2.3 Observation of Time SeriesUsually, the first clue that a physical model exhibits chaotic vibrations arises from anobservation of the time series of the output signal. The motion is observed to exhibit nopattern or periodicity. However, this test is not very reliable, since a motion could have along-period behaviour that is not easily detected. Also, some nonlinear systems exhibitquasi-periodic vibrations where two or more incommensurate signals are present. Here thesignal may appear to be non-periodic but can be broken down into the sum of two or moreperiodic signals.2.2.4 Phase PortraitConsider a single degree of freedom system with displacement x(t) and velocity v(t). Thephase plane is defined as the set of points (x,v). When the motion is periodic, the phaseplane orbit traces a closed curve. For example, the forced oscillations of a linear spring-mass-dashpot system exhibit an elliptical orbit. However, a forced nonlinear system withcubic spring element may show an orbit that crosses itself but is still closed correspondingto sub-harmonic oscillations.On the other hand, chaotic motions correspond to orbits that never close or repeat. Thus,the trajectory of the orbits in the phase space will tend to fill up a section of the phase space.Although this wandering of orbits provides a clue to chaos, continuous phase plane plotsprovide relatively little information, and a modified phase plane technique called thePoincare mapping should rather be used.142.2.5 Fourier SpectrumThe appearance of a broad spectrum of frequencies in the output signal is anothercharacteristic of chaotic vibrations. This feature becomes more important if the system is oflow dimension (one to three degrees of freedom). Often, if there is an initial dominantfrequency component con, a precursor to chaos is the appearance of sub-harmonics atfrequencies coo/n in the frequency spectrum. In addition, harmonics of this frequency willalso be present, i.e. at mwo/n.However, it may be erroneous to conclude that multi-harmonic outputs imply chaoticvibrations, since the system in question may turn out to posses many degrees of freedom. Insystems with many degrees-of-freedom, the Fourier spectrum may not be of much value indetecting chaotic vibrations, unless one can observe changes in the spectrum as one variessome parameter such as driving amplitude or frequency.2.2.6 Poincarë MapThe theoretical basis for Poincar6 maps was introduced by Poincare (1898). The recentwidespread use of computers with graphics facilities for examining chaotic behaviour indynamical systems has led to the method of Poincare maps becoming one of the mostpopular and illustrative methods.Consider, the motion of a point with time as displayed in the phase plane (displacement,velocity). Rather than viewing the continuous motion of the point, it is convenient to viewthe point at discrete times so that the motion appears as a sequence of dots in the phaseplane. If xn= x(tn) and Yn =i(tn), this sequence in the phase plane represents a two-dimensional map, which when the sampling time t o is chosen according to certain rules iscalled a Poincare Map. When the exciting motion is periodic with period T, an obvioussampling rule for a Poincar6 map is to choose t o = nT + to, where To is an arbitrarily chosen15time delay. This allows one to distinguish between periodic motions and non-periodicmotions.A Poincarë map enables the study of continuous time systems to be reduced to the studyof an associated discrete time system. The construction of the Poincarë map involveselimination of at least one of the variables of the system, resulting in a lower dimensionalproblem to be studied. In lower dimensional problems numerically computed Poincaremaps provide an insightful display of the global behaviour of the system. Unfortunately,there exists no general method of constructing the Poincare maps associated with arbitraryordinary differential equations, since this construction requires some knowledge of thegeometrical structure of the phase space of the ordinary differential equation.2.2.7 System Parameter VariationIn examining a system for chaotic response, it is useful to vary one or more of the controlparameters of the system, so that one may examine the presence of a steady or periodicresponse for some range of parameter space. In this way one can have confidence that thesystem is deterministic and that there are no hidden inputs and sources of truly randomnoise.In changing a parameter, one looks for a pattern of periodic responses. A responsecharacteristic precursor to chaotic motion is the appearance of a sub-harmonic periodicresponse and by varying the system parameters, such sub-harmonic vibrations may changeinto chaotic motion.162.2.8 Lyapunov ExponentsLyapunov exponents are perhaps the most useful diagnostic tool for detecting a chaoticresponse. Lyapunov exponents measure the mean rate of divergence of adjacenttrajectories. They were introduced in a form adapted to the theory of dynamical systems andto ergodic theory in the late sixties, when Oseledec (1968) published his non-communicativeergodic theorem which provides a general and simple way to compute all Lyapunovexponents. In the general case, there are as many exponents as phase-space dimensions,though a particular Lyapunov exponent is not associated with a unique direction in phasespace. An excellent coverage of Lyapunov exponents may be found in Wolf (1986).Positive Lyapunov exponents indicate divergence and chaos, while negative or zeroLyapunov exponents are characteristic of regular behaviour.Chaos in deterministic systems implies a sensitive dependence on initial conditions.This means that, if two trajectories start close to one another in phase space, they will moveexponentially away from each other for small times on the average. If Lo is the initialdistance between the two starting points, at a small time later the distance changes toL(t) = Lo 2Xt (2.5)where, X is the Lyapunov exponent. The choice of base '2' is arbitrary but convenient.The divergence of chaotic orbits can only be locally exponential, since if the system isbounded, as most physical experiments are, L(t) cannot go to infinity. In order to define thisdivergence of orbits, the exponential growth may be averaged at many points along thetrajectory as shown in Fig. 2.5. The process involves beginning with a reference trajectory(called a fiduciary) and a point on a nearby trajectory, and measuring L(t)/Lo. When L(t)becomes too large (i.e. the growth departs from exponential behaviour), a new nearby17trajectory is chosen and a new L0(t) is defined. Hence, the first Lyapunov exponent isdefined asNL(tk)= lim ,"1_^log2 { 4)}N —>00^`u k=1(2.6)where k is an iteration number. The limit of large N is necessary to obtain a quantity thatboth describes long-term behaviour and is independent of initial conditions. The motion isconsidered chaotic, if X > 0 and regular if A, 0.This procedure can also be used to estimate Lyapunov exponents from an experimentaltime series as described in section 2.2.9.2.2.9 Estimation of Lyapunov exponents from experimental time seriesAs mentioned earlier, there are as many exponents as phase-space dimensions. For anexperimental time series, the number of phase-space dimensions may not be known inadvance. In such a case, the technique of phase-space reconstruction with delay coordinatesmakes it possible to obtain Lyapunov spectrum from discrete time samples of almost anydynamical observable. An m-dimensional phase portrait of a time series x(t) can beconstructed by the delay co-ordinates method. A point on the attractor is given byfx(t),x(t+r),...,x(t+(m-1)t)}, where ti is the almost arbitrarily chosen delay time. The nearestneighbour in the Euclidean sense to the initial point ( x(to), x(to+t), . . x(to+(m-1)t) )ischosen, with the distance between these two points denoted as L(to). After time t1, theinitial length will have evolved to the length L' (t1). The length element is propagatedthrough the attractor for a time short enough so that only a small-scale attractor structure islikely to be examined. If the evolution time is too large L' shrinks as the two trajectories thatdefine its pass through a folding region of the attractor. In this case there is an18underestimation of Xi, so that a new data point must be selected that satisfies the followingtwo criteria:• Its distance from the evolved fiducial point is small• the angular separation between the evolved and replacement elements is small.This procedure is described in Fig. 2.6. When the proper replacement point cannot befound, the points that were being used are retained. This procedure is repeated until thefiducial trajectory has traversed the entire data file, at which point Xi may be estimated:,^1^I- (t k)ln ,1 = 11111 tnt0^I, (t k-1 )n-->°°(2.7)as the maximum Lyapunov exponent.Wolf et. al. (1985) provide a useful computer program to compute the largest Lyapunovexponent from experimental time series. This program has been used in the presentinvestigation.In Chapter 4, the methods discussed above will be applied to the experimentalobservations.19Chapter 3Experimental InvestigationIntroductionAs mentioned earlier, one of the aims of present research is the demonstration of the chaoticresponse of floating structures with nonlinear moorings. Keeping this objective in mind, aseries of experiments were carried out at the Hydraulics Laboratory of the Civil EngineeringDepartment, UBC. This chapter gives a detailed account of this experimental investigation.A numerical approach to the present problem involves solving the nonlinear equation ofmotion in the time domain using appropriate time stepping procedure. Appendix B may bereferred for a brief description of the procedure involved.The floating structure model is a rectangular plywood box. The nonlinear spring actionis simulated by two cantilever beams placed on either side of the box. The cantilever beamis not in immediate contact with the box, but leave some gap between the box and itself.GEDAP software developed by NRC is used for wave generator control and dataacquisition.3.1 Dimensional AnalysisDimensional analysis provides an important preliminary step to any experimentalinvestigation and may be used to identify key non-dimensional parameters of the problem athand. In the present case, the response in any degree of freedom may be expressed as a20function of parameters characterizing the incident wave conditions, structural and fluidproperties. Hence, for a monochromatic wave train we may writeR = f (H, T, d, B, dg, k, K, M, p, g, C)^(3.1)where, H, T and d are the incident wave height, incident wave period and water depthrespectively. Also dg , k, K and M are the gap width, spring stiffness, stiffness ratio andmass respectively; and p, g and C are the water density, gravitational acceleration anddamping ratio respectively.Dimensional analysis yieldsR_ ff H d B^M^rFr Fr H' pH3' con'Similarly, for a bichromatic incident wave train, we can writeR = f (H, T, d, B, dg, k, K, M, C, AT, p, g)where, AT is the difference in the incident wave periods.which givesR { H dB d M co AT rf H H H' 013' (on' T '3.2 Natural Frequency in SurgeFor a linear dynamical system, the undamped natural frequency is constant and is given by(3.2)(3.3)(3.4)(3.5)21In contrast to this, for a nonlinear dynamical system the undamped natural frequency isnot constant, but is a function of the initial displacement. For the present problem involvinga floating box with nonlinear stiffness, an expression for the undamped natural frequency insurge motion may be derived as indicated below.Let the initial compression of the spring K1 be 81. Hence, the potential energy stored inthe spring will bePE = –2101'-1^9 (3.6)Assuming no energy loss, and from the principle of conservation of energy, the potentialenergy PE stored in the spring must be equal to the kinetic energy of the box. Hence, wemay write1^1ICE = –2MV2 = –2 1(812which givesV „\[171—The time required to uncompress spring k1 is Ti/4, where T1 is the natural period ofvibration of the spring kl. As the box loses contact with spring kl, it travels a distance ofdg l+dg2 with a constant velocity V as there is no energy loss. The kinetic energy of the boxis still 1/2 MV2. The box then compresses spring k2 and the maximum compression of thespring k2 is2 =^ -1 v (3.9)(3.7)(3.8)Hence, the total time required to complete one cycle isTi^dgl+dg2 T2Tn = —2 + 2^+ 2V (3.10)22Since, V = col 81, Ti = 27c/coi and T2 = 27t/c02, we may rewrite Eq. (3.10) as[ , (01 , do+dg2 Tn — — -r(01^0)2^81where, Tn is the natural period in surge. Hence, the natural frequency co n in surge is2 col con –[ 1 + ("`) + 2 4grgE(1 2(3.11)(3.12)Eq. (3.12) indicates the dependence of natural frequency on the initial displacement 81. Itmay be noted that natural frequency decreases with an increasing gap width and increasesfor an increasing initial displacement.For a linear system, with k1 = k2 and dg i = dg2 = 0, the natural frequency (o n reduces tocol as expected.3.3 Location of Centre of GravityResponses in the surge, heave and pitch degrees of freedom are defined as the displacementsof the centre of gravity of the system with respect to these three degrees of freedom. Hence,it is important to know the location of the centre of gravity. This may be accomplished asdescribed below.The box is idealized as a set of discrete masses ml..m4 as indicated in Fig. 3.2.Referring to Fig. 3.2, ml, m2 and m3 are the masses of upstream/downstream side pieces,lateral pieces and bottom piece respectively. The "m4" is the external mass added to thesystem.23Due to symmetry, the centre of gravity will be located on a vertical axis passing throughthe centre of the box as shown in Fig. 3.2. Hence, the x coordinate of the centre of gravitywith respect to the lower bottom corner of the box is B/2, where B is the beam length. The ycoordinate can be computed by taking moments of the masses about the bottom edge of thebox. Hence, we may writeGy - { (2m1 + m2) d1 + 2m4 d31LM 1 M2 + M3 + 2 M4 j (3.13)3.4 Measurement of Spring StiffnessAs mentioned earlier, aluminum cantilever beams were used to simulate nonlinear springaction. The load-deflection curve for these aluminum cantilever beams was found to belinear. Fig. 3.3 shows a schematic diagram of the procedure used to obtain the load-deflection curve.A load P is applied at a distance z from the fixed end of the beam. Deflection 8 of thebeam at the point of application of load P is measured. The procedure is repeated fordifferent load values. The load-deflection values are as shown in Table 3.1. The deflectionin colunm 3 is calculated from applied load 'P' as, 8 = Pz 3/3EI. It is evident from Table 3.1that, the load-deflection behaviour is quite linear and there is a good agreement betweenexperimental and calculated values (Fig. 3.4). Hence for calculating w/con and K, thecomputed values of spring stiffness are used.243.5 Experimental Setup3.5.1 GeneralFig. 3.5 shows a schematic sketch of the entire experimental setup. The floating object is ahollow plywood box of size 30" X 10" X 8" (76 cm X 25 cm X 20 cm) suitably coated withwaterproof paint to avoid water seepage. The empty box has a mass of 4.1 kg. Provision ismade to add extra weights to the box so as to vary its mass and mass moment of inertia.These extra weights, each of mass 6.25 kg, are kept at 6 cm from the plane of symmetry ofthe box as shown in the figure. Two Plexiglas plates limit motion of the box to threedegrees of freedom, viz., surge, heave and pitch. Eight ball bearings mounted on longersides minimize friction between Plexiglas plates and the box which otherwise would havehad an adverse effect on the experiments. Two cantilever beams are used to simulate thenonlinear springs, with different size beams used to vary the effective spring constant.Displacement measurements at three different locations on the body are carried out usingdisplacement probes. These probes consist of strings, about 1.5 m long with one endattached to the point of measurement on the box. These strings run over three pulleys fittedwith potentiometers and are mounted on a rigid aluminum frame. Small weights 100 gm)are attached to the free ends of the strings to keep them under tension at all times and toprevent slippage. The potentiometers require a ± 5 V regulated DC supply. The gap widthdg may be varied as required. During the experiments, spurious high frequency springvibrations were initially observed, and were eliminated by the use of rubber bands acting asvibration dampers.3.5.2 Wave Flume and Wave GeneratorThe Hydraulics Laboratory Wave Flume measures 20 m X 0.5 m X 0.75 m. An artificialbeach is located at its downstream end. This beach is an essential component of the wave25flume as it helps in reducing the degree of wave reflection. During the experimentalprogram, the water depth was maintained at 55 cm. (Depths above 65 cm are notrecommended as there is a possibility of water spilling out of the flume.) Waves areproduced by a single paddle wave generator located at upstream end of the flume. It iscapable of generating both regular and random waves. This generator is controlled by aDEC VAXstation-3200 minicomputer using GEDAPt software developed by the NationalResearch Council (NRC) of Canada. This generator is capable of producing waves of heightup to 30 cm and a minimum period of 0.5 s. In the present investigation wave heightsranging from 6 cm to 16 cm are used. The range of wave periods used is 2.0 s to 3.0 s.3.5.3 GEDAP SoftwareThe GEDAP software was used extensively during all stages of the experimentalinvestigation. GEDAP stands for GEneral purpose Data-acquisition and Analysis Program.This is a general purpose software package available on Digital Equipment Corporation'sVAX computers for the analysis and management of laboratory data, including real-timeexperimental control and data acquisition functions. GEDAP is a fully-integrated, modularsystem which is linked together by a common data file structure. GEDAP maintains astandard data file format so that any GEDAP program is able to process data generated byany other GEDAP program. This package also includes an extensive set of data analysisprograms so that most laboratory projects can be handled with little or no project-specificprogramming. The most rewarding feature of GEDAP is its fully-integrated interactivegraphics capability, such that results can be conveniently examined at any stage of the datasynthesis or analysis process. The GEDAP package also includes a vast collection of utilityprograms These consist of data manipulation software routines, frequency domain analysisroutines, and statistical and time-domain analysis routines. The RTC_SIG (Real Timet GEDAP is a registered mark of the National Research Council of Canada.26Control - SIGnal generator) and RTC_DAS (Real Time Control - Data Acquisition System)are two important routines of this software package. The program RTC_SIG generates thecontrol signal necessary to drive the wave paddle, while the routine RTC_DAS reads thedata acquisition unit channels and stores the information in GEDAP binary formatcompatible with other GEDAP utility programs.3.6 Experimental Procedure3.6.1 Calibration of PotentiometersBefore carrying out the experiments, it is necessary to calibrate the potentiometers used tomeasure structural displacements. The calibration is carried out as described below.The diameter of the pulley mounted on the potentiometer is measured. The pulley isthen rotated through precisely 180 degrees and the resulting voltage across the potentiometeris measured. This procedure is repeated for four more steps of 180 degree rotation. A graphof the known displacement vs. measured voltage is then plotted. The slope of straight linefit is stored as the calibration factor for that particular potentiometer. The remaining twopotentiometers are also calibrated in a similar fashion.3.6.2 Wave Generator Signal FileThe wave generator requires a signal file in order to generate waves of the desired waveheight and period and the GEDAP program RWREP2 is used to create the correspondingsignal file. This program requires four main input parameters, viz., wave generatorcalibration file, water depth, desired wave height and period. RWREP2 stores the computeddriving signal in a format readable by the wave generator controller program RTC_SIG.273.6.3 The ExperimentAfter setting up the box in the flume with the springs installed at desired separation, thewater in the flume is allowed to calm down; the Data Acquisition System Channels are theninitialized. After this initialization procedure, the GEDAP wave generation and dataacquisition programs (RTC_SIG and RTC_DAS respectively) are invoked. The programRTC_SIG generates waves of desired height and period using an appropriate driving signalfile, and the data acquisition program RTC_DAS reads the channels assigned todisplacement probes and stores the data in a suitable format. After the desired durationRTC_SIG ramps down the wave generator motion. The RTC_DAS output data file is thende-multiplexed to separate the three wave probe measurements. These measurements arethen transformed into displacements of the centre of gravity of the box using suitabletransformation equations given in section 3.7.3.7 Computation of ResponseSurge, heave and pitch responses of the floating body are defined as the displacements of thecentre of gravity in these three degrees of freedom. Since the wave probes do not measurethe displacement of centre of gravity, a suitable set of transformation equations must beestablished to compute the response from the measurements.Referring to Fig. 3.6, the equations relating the measured displacements di, d2 and d3with surge, heave and pitch response, ug , wg, and 9 respectively, are as follows:1d1 = Ll - { (ug + A sin 9)2 + (L1 + A - wg - A cos 9)2 1 2 (3.15)d2 = L2 - { (ug - B/2 + R2 cos(02-0))2 + (L2 + A - wg - R2 sin(A2-0))2 12 (3.16)d3 = L3 -^(L3 + B/2 - ug + R3 cos(A3-13))2 + (wg - Zpi + R3 sin(03-0))2 1 2 (3.17)28Eqs. (3.5)-(3.7) represent exact transformation equations. However, they are nonlinear andhence difficult to solve. Nevertheless, under certain conditions, these equations may belinearized.If Li, L2, L3 are sufficiently large compared to the measurements d1, d2, d3 and responses u gand wg then eq. (3.15)-(3.17) may be linearized to:di = (wg - A) + A cos 0^ (3.18)Solving for ug, wg, and 0:d2 = (wg - A) + R2 sin(A2-0)d3 = ug + R3 cos(A3 -0)0 = - sin -1 f d2 - di 11^11(3.19)(3.20)(3.21)andwg = d1 + A (1 - cos 0)^(3.23)ug = d3 + I  - R3 cos(A3 - 0) (3.24)wherem = 11(R2 sin A2 - A)2 + (R2 cos A2)2^(3.22)Eqns. (3.21)-(3.24) may now be used to evaluate the pitch, heave and surge responsesrespectively. The linearized surge, heave and pitch responses are found to be quiteacceptable for further analysis. The above equations have been derived in detail inAppendix A.29Chapter 4Results and DiscussionThis chapter describes the results obtained from the experiments. The results have beenpresented in the form of time series, phase portraits and Fourier spectra. Due to theimmense storage requirement, a Poincarë map is drawn for one measured time series only.The largest Lyapunov exponents have been computed for all experimental time series.4.1 Time Series, Phase Portraits and SpectraThe time series, phase portrait and Fourier spectrum are the most important qualitative testsused in detecting chaos in nonlinear systems and these are shown in Fig. 4.1 for the surgeresponse corresponding to T = 2.5 s, H = 13 cm and K = 4.25. The time series shown in Fig.4.1a appears to be chaotic and the corresponding phase portrait shown in Fig. 4.1b alsosuggests that the system behaviour is not periodic. Fig. 4.2 shows corresponding results forthe case in which the incident wave height is increased by 2 cm and indicates that a drasticchange in surge response occurs. In particular, the time series in Fig. 4.2a that, the systembehaviour has changed from non-periodic to periodic; and the corresponding Fourierspectrum shown in Fig. 4.2c is now composed of peaks at integral multiples of incidentwave frequency, confirming the periodic nature of the response.Fig. 4.3 shows results for H = 14 cm, T = 2.5 s, a = 0.423, K = 4.25 and d g = 11.5 cm.These results show an interesting phenomenon in that there is a sudden change in response30characteristics from non-periodic to periodic. Such a phenomenon is called transient chaos.Initially the motion appears to be quite irregular but soon settles down to a periodicresponse.Figs. 4.4 - 4.7 show effect of wave period variation on surge response. The incidentwave period is varied while maintaining the other controlling parameters constant. Fig. 4.4,which corresponds to H = 13 cm and T = 2.0 s, exhibits some sub-harmonic response.When the incident wave period was increased by 0.2 s, a significant spectra peak at 1/4th ofincident wave frequency appears as shown in Fig. 4.5d. Phenomenon of transient chaos canbe observed for a wave period of 2.8 s. The response is changed from non-periodic toperiodic after about 250 s (Fig. 4.6). Fig. 4.7 shows time series, phase portrait and Fourierspectra for T = 3.0 s. The observed response is periodic. The phase portrait is nearly aclosed loop and the Fourier spectrum is composed of spectral peaks at integral multiples ofincident wave frequency, thus confirming periodic nature of the response.For an incident wave with H = 11 cm and T = 2.8 s very significant sub-harmonicresponse is observed. The time series shown in Fig. 4.8 clearly indicates presence ofmultiple frequency components. In addition to a peak at the incident wave frequency, asignificant second peak is observed at 1/3rd of the incident frequency. A very periodicresponse is observed for H = 15 cm and T = 2.8 s (Fig. 4.9). No sub-harmonic response isobserved in this case.Some sub-harmonic response is observed for H = 11 cm and T = 2.0 s as shown in Fig.4.10. A very significant sub-harmonic spectral peak is observed for H =11 cm and T = 2.2 s(Fig. 4.11).Both chaotic and periodic/sub-harmonic responses were observed for bichromatic waveexcitation. Ten experiments were conducted to study the model's response to bichromaticwaves. The observed response to H1 = H2 = 7 cm, T1 = 2.2 s and T2 = 2.5 s is very non-31periodic (Fig. 4.12). A considerable sub-harmonic response is observed in this case. Thespectra shows a number of sub-harmonic peaks at multiples of the incident wave period.Considerably more sub-harmonic response is observed for bichromatic waves than for themonochromatic wave excitation. Fig. 4.13 shows an example of the periodic response forbichromatic excitation.4.2 Effect of Wave HeightThe wave height appears to be an important governing parameter of the system behaviour.Tests with four different wave heights (11, 12, 13, 14 and 15 cm) were carried out, whilemaintaining a constant wave period (2.5 s) and the other controlling parameters heldconstant. Fig. 4.14 shows the effect of various wave heights on the surge response. Chaoticmotion was observed for wave heights of 12 and 13 cm, whereas for heights of 11 and 14cm the observed motion was periodic, as is evident from Fig. 4.14. For a wave height of 11cm, hardly any spring action was observed. This phenomenon could be attributed to the factthat the incident wave had insufficient energy to produce required surge to cover the gapwidth so that no chaotic motion was observed. For a wave height of 14 cm, the observedresponse was completely periodic, as if the system was linear. This observation may beexplained as follows. Since, the total energy of a regular sinusoidal wave is proportional tothe wave height cubed, the 14 cm wave has more energy by a factor of two than the one with11 cm wave height. Hence, the 14 cm wave produced enough surge to cover the gap width.For wave heights between 11 and 14 cm, intermittent contact with springs was observed andthe resulting response was very aperiodic.324.3 Effect of Wave PeriodWave period was also observed to have a similar effect as the wave height on the surgeresponse. Figure 4.15 shows surge response time series for various values of incident waveperiod T. The wave height (13 cm) and the other system parameters were kept constant.For an incident wave with wave period of 2.0 s, the observed response was relativelyperiodic and hardly any spring action was observed. In this case, the wave period was notsufficient enough to produce horizontal displacement comparable to the gap width. Hence,spring action was absent and so was the chaotic motion. On the other hand, the surgeresponse for wave period of 3.0 s was very periodic with full spring action. The systembehaviour appeared to be very linear. The gap width had hardly any effect on the response.This may also be explained by applying the same logic as in the previous case. The incidentwave in this case induced sufficient horizontal displacement to produce enough restoringforce resulting in periodic system behaviour. In the intermediate case with wave period of2.5 s, the spring action was intermittent and the system behaviour appeared very non-periodic.4.4 Effect of Spring Stiffness and Gap WidthA wider gap width was observed to produce a slight drift in the equilibrium position. At theend of the experiment, the floating box was observed to come to rest not at its initial positionbut a different position in the downstream direction. For example, for a gap with of 14 cm,the observed drift was of the order of 10 cm. Interestingly, no appreciable drift wasobserved for gap width of 7 cm. Fig. 4.16 shows a plot of response amplitude vs. gap width.It can be seen from the graph that, the response amplitude falls sharply with increase in thegap width.33Spring stiffness also was observed to be an important factor in deciding the surgeresponse characteristics. It was observed that for lower values of frequency ratio (a) theresponse tends to be periodic. For higher values of a some sub-harmonic response wasobserved.4.5 Non-dimensional PlotsFig. 4.17 - 4.19 show plots of the non-dimensional response as a function of various non-dimensional input parameters, viz., wave height, wave period, frequency ratio and gapwidth.Fig. 4.17 is a graph of R/H vs. B/gT2 . Chaotic response was observed for B/gT 2between 0.012 and 0.020. For smaller values of B/GT 2 (or larger values of wave period)response was periodic.Fig. 4.18 is a plot of R/gT2 vs. B/H. As can be seen from the plot, chaotic response wasobserved for values of B/H lying between 5.5 and 6.5. Response was periodic beyond thisrange of B/H.Frequency ratio a has been observed to be one of the very important deciding factors ofthe system behaviour. As is clear from fig. 4.19, chaotic response was observed for valuesof a between 0.4 and 0.6. For a below 0.4, the response was periodic. This is inconfirmation with the numerical prediction of Aoki, Sawaragi and Isaacson (1993).4.6 Poincare MapAs indicated in Chapter 2, the Poincare map is a discrete time view of a continuous phaseportrait. In order to observe a fractal-like structure of a strange attractor on a Poincare map,34a very long response time series is needed. In numerical experiments, it is possible to obtainresponse time series of any required length, although in physical tests this may not alwaysbe feasible owing to experimental limitations. For example, in present case, due to immensedisk space requirement, only one time series of 40 min length was measured. The largestLyapunov exponent for this particular time series is positive indicating chaos. Fig. 4.20 -4.22 show Poincare maps for various values of "phase". Strange attractor is visible in Fig.4.20(a). Fig. 4.20(b) shows two distinct patches indicating presence of two dominantfrequency components (Moon, 1987). On the other hand, a Poincare map of periodicresponse lacks any fractal structure. Points tend to concentrate in certain regions of thephase space unlike the case of a Poincar6 map of chaotic response. Finally, positiveLyapunov exponent confirms the chaotic nature of the former response.4.7 Lyapunov ExponentLyapunov exponents are of interest in the study of dynamic systems in order to characterizequantitatively the average exponential divergence or convergence of nearby trajectories.Since, they can be computed either from a mathematical model or from experimental data,they are widely used for the classification of attractors. Negative or zero Lyapunovexponents signal periodic orbits, while at least one positive exponent indicates chaotic orbit.and divergence of initially neighboring trajectories.Computation of all Lyapunov exponents is computationally very demanding. As amatter of fact, one doesn't need all the Lyapunov exponents to decide whether the systembehaviour is periodic or chaotic, because presence of a single positive Lyapunov exponent issufficient enough evidence to declare the motion as chaotic. Hence in reality, one needsonly the largest Lyapunov exponent. Various numerical algorithms have been proposed inliterature to compute the largest Lyapunov exponent. An algorithm developed by Wolf et al.35(1985) is used in the present investigation. Wolf et. al. have also given a computer programbased on their algorithm.4.7.1 Fixed Evolution Time Program for XiA time series of given duration is read from a data file, along with the parameters necessaryto reconstruct the attractor, viz., the dimension of the phase space reconstruction, thereconstruction time delay and the time between data samples, required for normalization ofthe exponent. Three other input parameters are required: length scales that we consider to betoo large and too small and a constant propagation time between replacement attempts. Wealso supply a maximum angular error to be accepted at replacement time, but it is notconsidered as a free parameter as its selection is not likely to have much effect on exponentestimates. It is usually fixed at 0.2 or 0.3 radians.The calculation is initiated by carrying out an exhaustive search of the data file to locatethe nearest neighbor to the first point (also known as the fiducial point), omitting pointscloser than a pre-assigned minimum distance. The main program loop, which carries outrepeated cycles of propagating and replacing the principal axis is now entered. The currentpair of points is propagated through a preset evolution steps through the attractor and itsfinal separation is computed. The logarithm of the ratio of final to initial separation of thispair updates a running average rate of orbital divergence. A replacement step is thenattempted. The distance of each delay coordinate point to the evolved fiducial point is thendetermined. Points closer than certain minimum but further away than certain maximumdistance, are examined to see if the change in angular orientation is less than maximumallowable angular orientation. If more than one candidate point is found, the point definingthe smallest angular change is used for replacement. If no points satisfy these criteria, weloosen the larger distance criterion to accept replacement points as far as twice the maximumallowable distance. If necessary the large distance criterion is relaxed several more times, at36which point we tighten this constraint and relax the angular acceptance criterion. Continuedfailure will eventually result in our keeping the pair of points we had started out with, as thispair results in no change whatsoever in phase space orientation. We now go back to the topof main loop where new points are propagated. This process is repeated until the fiducialtrajectory reaches the end of the data file, by which time we hope to see stationary behaviourof Xi.This computer program was used to compute the largest Lyapunov exponents fromexperimental time series. It has been observed (Wolf et. al., 1985) that attractorsreconstructed using smaller values of dimension (m) often yield reliable Lyapunovexponents. Hence, m was chosen to be equal to 3 in this case. The time series in presentcase is practically noise-free, hence the largest and smallest length scales may be chosenarbitrarily. The reconstruction time delay (t) is chosen neither so small that the attractorstretches out, nor so large that MT is much larger than the orbital period. The reconstructiontime delay in present case is chosen equal to the mean orbital period. Decisions aboutpropagation times and replacement steps depend upon additional input parameters or on theoperator's judgment. Too frequent replacements cause a dramatic loss of phase spaceorientation, and too infrequent replacements allow volume elements to grow overly largeand exhibit folding. It has been recommended that the evolution time in the range of 1/2 to3/2 orbits almost always provides stable exponent estimates. In the present case anevolution time of 1/2 the mean orbital period is used. In order to make sure that theLyapunov exponent computed by the program is reasonable, the program is fed with aperfectly periodic time series having the same period as the observed experimental timeseries and approximately the same average amplitude. A combination of program inputparameters giving zero Lyapunov exponent value is then used as parameters for theexperimentally measured time series. This initialization procedure makes sure that theLyapunov exponent is indeed the one which is being sought for. Tables 4.1 and 4.2 give thelargest Lyapunov exponents computed using the above program.37Chapter 5Conclusion and Scope for Further ResearchAs mentioned earlier, the primary aim of this study was to demonstrate chaos experimentallyin nonlinearly moored offshore structures. Experiments were carried out on a hollowplywood box of size 76 cm x 20 cm x 25 cm. Two aluminum cantilever beams were used tomodel nonlinear spring action. The motion of the model was restricted to three degrees offreedom only, viz., surge, heave and pitch. Displacement measurements at three differentlocations on the body were made using potentiometers mounted on a rigid aluminum frame,with a system of strings, pulleys and counter-weights used to transmit the structure motionsto potentiometers. Measured displacements were transformed to provide surge, heave andpitch motions with respect to the centre of gravity of the structure. From the analysis of theexperimental data, it appears that the goal has been achieved.Wave height, wave period, spring stiffness and gap width dictate the behaviour of thesystem. For a certain range of wave height and period, the observed response is chaotic.The effect of gap width is in general coupled with the other parameters. If the other threequantities are held constant, a wider gap width is more likely to generate chaotic responsethan a narrower one, as a reduction in the gap width causes a corresponding decrease in thesystem's nonlinearity. In addition to chaotic response, a sub-harmonic response wasobserved in some cases. Interestingly, for particular threshold values of wave height andperiod, a sudden transition from chaotic to periodic response is observed. This phenomenonhas been referred to in literature as transient chaos. It would be of interest to investigate this38transition phenomenon in more detail. Further research effort in this area may lead to abetter understanding of the parameters responsible for occurrence of this phenomenon.Lyapunov exponents appear to offer the most suitable quantitative estimate of chaossuch that presence of even a single positive Lyapunov exponent is sufficient evidence todeclare the motion as chaotic. As mentioned earlier, Wolf's computer program forestimating the largest Lyapunov exponent has been tested with a perfectly periodic timeseries, and was then used with the experimental time series. The presence of positiveLyapunov exponents in some of these tests confirm the chaotic response.In the previous numerical study conducted by Aoki, Sawaragi and Isaacson (1993), asingle degree of freedom model (surge) was developed, whereas the study reported heredeals with a three degrees of freedom model (surge, heave and pitch). It would be useful usethe mathematical formulation developed (Appendix B) to compare the numerical results tothe experiments and to study the effects of the other two degrees of freedom on the surgeresponse by comparing such results with those of the single degree of freedom model. Thepresent work deals only with geometrical nonlinearity. As an extension to this work, onecould study the effect of material nonlinearity on response of the floating box which has notbeen studied here.Application of chaotic dynamics to ocean structures is still in a developmental stage.Much can be done in this new area of ocean engineering. The present case is a veryidealized case of geometric nonlinearity. Geometric stiffness characteristics of catenarymoorings are somewhat different than the case considered here. Careful numerical andprototype experiments performed taking into account the catenary effects may shedadditional light on this phenomenon.39ReferencesAoki, S. I., Sawaragi, H., and Isaacson, M. (1993) "Wave-Induced Low FrequencyMotions of Compliant Offshore Structures with Nonlinear Moorings", Proceedings of the3rd International Offshore and Polar Engineering Conference, Singapore, June, 1, pp. 369-376.Bishop, S. R. and Virgin, L. N. (1987) "The Onset of Chaotic Motions of a Moored Semi-Submersible, Proceedings of 6th International Offshore Mechanics and Arctic EngineeringConference, Houston, 2, pp. 319-323.Bishop, S. R. and Virgin, L. N. (1988) "Catchment Regions of Multiple DynamicResponses in Nonlinear Problems of Offshore Mechanics", Proceedings of 7th InternationalOffshore Mechanics and Arctic Engineering Conference, Houston, 2, pp. 15-22.Chirikov, B. V. (1979) "A Universal Instability of Many Dimensional Oscillator Systems",Phys. Reps., 52, pp. 263-379.Ford, J. (1975) "The Statistical Mechanics of Classical Analytic Dynamics", InFundamental Problems in Statistical Mechanics, E. G. D. Cohen (ed), North-Holland,Amsterdam, 3, pp. 215-255.Gottlieb, 0. and Yim, S. C. S. (1990) "Onset of Chaos in a Multi-point Mooring System",Proceedings of the First European Offshore Mechanics Symposium, Trondheim, Norway,pp. 6-12.Kapitaniak, T. (1991) Chaotic Oscillations in Mechanical Systems, Manchester UniversityPress, Manchester, U.K.Lau, S. L.; Ji, Z. and Ng, C. 0. (1990) "Dynamics of an Elastically Moored FloatingBody by the Three-Dimensional Infinite Element Method", Ocean Engineering, 17, No. 5,pp. 499-516.Lorenz, E. N. (1963) "Deterministic Nonperiodic Flow", Journal of Atmospheric Science,20, 130-41.Miles, M. D. (1989) "Guide to Using GEDAP on VAX/VMS", Hydraulics Laboratory,National Research Council of Canada, Ottawa, Canada.Moon, F. C. (1987) Chaotic Vibrations: An Introduction for Scientists and Engineers, JohnWiley & Sons, New York.40Oseledec, V. I. (1968) "A Multiple Ergodic Theorem: Lyapunov Characteristics forDynamical Systems", Transactions of Moscow Mathematical Society, 19, pp. 197-231.Papoulias, F. A. and Bernitsas, M. M. (1988) "Autonomous Oscillations, Bifurcations,and Chaotic Response of Moored Vessels", Journal of Ship Research, 32, No. 3, pp. 220-228.Parker, T. S., and Chua, L. 0. (1989) Practical Numerical Algorithms for ChaoticSystems, Springer-Verlag, New York.Poincare, H. (1890) "Sur les equations de la dynamique et le probleme de trois corps.",Acta Math., 13, 1-270Poincare, H. (1898) Les Methodes Nouvelles de la Michanique Celeste, 1-3, Gauthier-Villars, Paris.Sumanuskajonkul, S and Hu, S. James (1992) "Responses of Bilinear and ImpactingSystems Subjected to Regular Waves", Proceedings of 9th Conference on EngineeringMechanics, ASCE, NY, pp. 196-199.Sarpkaya, T. and Isaacson, M. (1981) Mechanics of Wave Forces on Offshore Structures,Van Nostrand Reinhold Company, New York.Schuster, Heinz G. (1988) Deterministic Chaos: An Introduction, VCH VerlaggesellschaftmbH, Weinheim, FRG.Ueda, Y. (1979) "Randomly transitional phenomenon in the system governed by Duffing'sequation", Journal of Statistical Physics, 20, pp. 181 - 196.Wolf, A., Swift, J. B., Swinney, H. L., and Vastano, J. A. (1985) "Determining LyapunovExponents from a Time Series", Physica, 16D, North-Holland, Amsterdam, pp. 285-317.Wolf, A. (1986) "Quantifying Chaos with Lyapunov Exponents", Nonlinear Science:Theory and Applications, ed. A. V. Holden, Manchester University Press, pp. 273-290Yim, S. C. S. and Lin, H. (1991) "Chaotic and Stochastic Dynamics of OffshoreEquipment", Proceedings of the First International Offshore and Polar EngineeringConference, Edinburgh, U.K., 3, pp. 420-427.41Appendix - ATransformation EquationsThe displacement probes measure displacement of three different points on the floatingbody. The surge, heave and pitch responses of the floating body are defined as thedisplacements of its centre of gravity in these three degrees of freedom. The presentexperimental setup does not allow the surge, heave and pitch response to be measureddirectly. Hence it is necessary to transform these displacements from their present points ofmeasurement to the centre of gravity. These equations may be derived as shown below.Let di be the potentiometer reading at time t corresponding to displacement probe #1.Referring to Fig. A-1, we can express di in terms of L1 and L1' as,di = Li - Li'^ (Al)L1' may itself be expressed in terms of ug, wg and 0 asL1' = "q(ug + A sin 0)2 + (Li + A - wg - A cos 0)2^(A2)Substituting in equation (Al):di = Li - 11(ug + A sin 0)2 + (Li + A - wg - A cos 0)2^(A3)similarly, referring to Figs. (A2) and (A3), corresponding expressions for d2 and d3 ared2 = L2 - -v [ug - B/2 +R2 cos (02 - 0)] 2 + [L2 + A - wg - R2 sin (02 - 0)]2^(A4)42andd3 = L3 - -‘1[Wg Zpi +R3 cos (A3 - 0)[ 2 + [L3 + B/2 - ug - R3 sin (A3 - 0)]2^(A5)Equations (A3), (A4) and (A5) in their present form are difficult to solve due to the presenceof nonlinearity. Attempts to solve these equations failed as the solution procedure producedvery unstable results. A closed-form solution of linearized equations is a possibility.Following assumptions are involved in the linearization procedure.• lengths L1, L2 and L3 are sufficiently large compared to ug and wg• Angle 0 is small compared to A2 and A3Using above assumptions, equations (A3), (A4) and (A5) may be linearized to:di = Li - (wg - A) + A cos 0^(A6)d2 = L2 - (wg - A) + R2 sin (A2 - 0)^(A7)d3 = ug - R3 sin (A3 - 0) - B (A8)Equations (A6)-(A8) can be solved for u g , wg and 0. Hence we get,0^sin_i d2 - dil^ (A9)where, p. =^(R2 sin A2 - A)2 + (R2 cos A2)2 and similarly ,wg = di + A - A cos 0^ (A10)ug = d3 + - R3 sin (A3 - 0) (All)In order to assess appropriateness of linearization, the linearized values of u g, wg and 0 weresubstituted back into the exact equations and new values of di, d2 and d3 were obtained.The new values were then compared with the original di, d2 and d3 values as shown in Fig.43(A-4), (A-5) and (A-6). Evidently, the comparison is quite good especially for thedisplacement probe #3. Hence, we can conclude that, ug , wg and 0 obtained from thelinearized equations can be used in subsequent development of the problem withoutsacrificing accuracy.44Appendix BMathematical Formulation of Hydrodynamic CoefficientsA numerical approach to the present problem requires an initial calculation of added mass,hydrodynamic damping coefficients and wave forces. Once these hydrodynamiccoefficients are known, the response may then be obtained by solving the nonlinearequations of motion in the time domain using an appropriate time-stepping procedure. Thefollowing paragraphs briefly describe the procedure involved.Fig. B-1 shows a definition sketch of a moored two-dimensional floating body ofarbitrary shape, while Fig. B-2 shows a corresponding mathematical model of this body.Two coordinate systems are defined. O-X-Z and G-X'-Z are fixed and moving coordinatesystems respectively, as indicated in Fig. B-1. The origin 0 of the fixed system is defined asthe point of intersection of SWL and the vertical through G when the body is in itsequilibrium position. The origin G of the moving system is the centre of gravity of thebody. Note that in the equilibrium position Z and Z' axes overlap.The equation of motion for the moored two-dimensional body may be written as[M] + + [K(4)] (4) = F(t) (B-1)Where [M], [X] and [K(4)] are the mass, damping and stiffness matrix respectively. It maybe noted that, equation (B-1) is a nonlinear equation since the stiffness matrix is a functionof the displacement {4). The excitation F(t) may be computed from linear diffractiontheory. The system properties, viz., the added mass [g], damping matrix [2d, mass matrix[M], and the stiffness matrix [K] can be individually evaluated.45The evaluation of incident wave force F(t) is quite classical. This incident wave forcemay be easily computed using linear diffraction theory. Since the problem is linear, thevelocity potential can be represented as a sum of three separate components.= + + (B-2)where, 4)w , 4)s, and of are the incident, scattered and forced velocity potentials respectively.These three components separately satisfy the Laplace equation, together with the bottomand free surface boundary conditions, and 4)s, Of must also satisfy the radiation condition.The boundary condition at the body surface must account for the velocity of the bodyitself and is given by,aow aos aof „an +^= v n (B-3)where Vn is the velocity of the body surface in the direction 'n' normal to itself. Since, themotions are small, this condition is applied at the equilibrium surface S o taken at the restposition, rather than at the instantaneous position. Equation (B-3) may be broken down intotwo equationsas in a fixed body case, together with,aow . a4s = 0an an aOf ,an^v nat So^(B-4)at So^(B-5)The problem defining 4)s is identical to that for the fixed body case and 4) s thus may bedetermined in exactly the same way as the fixed body case. Many a times, 4) s is not neededexplicitly since it will be possible to express the wave forces directly in terms of Of.46The velocity Vn is the velocity of the body surface in the direction. Hence, we maywrite,3aofVn = E, njj=1(B-6)where, nj is given as, ni = n x, n2 = nz and n3 = xnz - znx. nx, nz are the direction cosines ofthe normal to the surface. In order to apply the boundary conditions, it is convenient todecompose 4)f into three components associated with each degree of freedom and eachproportional to displacement amplitude j. Hence, we may write3= yi4;J.,(B-7)The coefficients 4)Cf) are generally complex. This representation enables the body surfaceboundary conditions to be written in terms of 4)r and independent of kj as,a — ion^on so4):.)J.1,..3^(B-8)The right-hand-sides of these equations are known and the three functions may be found inthe same manner as is the scattered potential 4) s.The forces and moments associated with 4) w and 4)s comprise the exciting force F(e) onthe body. Application of Green's theorem makes it possible to express the exciting forcedirectly in terms of the incident and forced potentials. Such expressions are called Haskindrelations. They may be given in the form(f) aOwFr p fso [4)w^- - an ]dSan (B-9)47Hence, one can evaluate F e) from the knowledge of Of° calculated to obtain the added-massand damping coefficients.There are three components IP corresponding to each mode motion, and each of thesemay be written as (Sarpkaya and Isaacson, 1981),3aofFfo = p f ni dS = -i(OP^Uso^ni dS )jSo^ j.1(B-10)Above equation may be decomposed into components in phase with the velocity and theacceleration of each mode and we put3F(f) = -^ra^PAI)1^at2^kJ at )J= 1i=1,..3^(B-11)where the coefficients gij and Xij are taken as real and are called the added mass anddamping coefficient respectively. Hence the rearranged equation of motion may be writtenas(mij + 1100 + a ij PP. + Kij 4j = F e) (B-12)In the above equation Xij represents only the hydrodynamic damping. In cases wherestructural damping or viscous damping are important, these would need to be included inadditional terms alongside the ?4j terms. Likewise Ffe) represents only the force due to wavefield, and if external forces are present these would need to be included alongside Fr.Application of the body surface boundary condition gives following explicit expressions forthe added mass and damping coefficientsgij =^So^JIm[0:1).(0 ] ni dS^(B-13)48kij = -pi so Reel ni dS^(B-14)Now a consequence of Green's theorem is that the added mass and damping coefficients aresymmetric. Finally, it should be noted that both the added mass and damping coefficientsare frequency dependent.Solution of the linearized diffraction problem always results in the wave elevation 11 ofthe form11(t) = A cos(cot)^ (B-15)where, A is the wave amplitude. Hence, the exciting force corresponding to 1(t) may bewritten asF(t) = A F* (t) cos(cot+S(co))^(B-16)where, F*(a)) is a transfer function of the first order exciting force and 5(0)) is the phase lagbetween 1(t) and F(t), both of which may be obtained using linear diffraction theory.For a bichromatic wave train with frequency components coi and 0)2, the free surfaceelevation my be expressed as11(t) = Al cos(0)1t) + A2 cos(0)2t + E)^(B-17)where, A1 and A2 are the two component amplitudes, col and 0)2 are the componentfrequencies and E is the phase difference between the two components. The correspondingexciting force F(t) may be written in the form*^ *F(t) = Al F 1 cos(wit +13(0)1)) + A2 F 1 (0)2) cos(0)2t + e + 8(w2)) +1^(0)1+0)2AiA2 f pg Cd^2 ) cos ( (col - co2)t - E + &L(DI, 0)2))^(B-18)49Where Cd(co) is the steady drift coefficient which may be obtained from the results of lineardiffraction theory. In a particular case of component wave having same amplitude andphase, Al = A2 = A and e = 0, the wave exciting force F(t) given by Eq. (B-18) may besimplified to:F(t) = 2AF*(co) cos ( t°—)t cos(cot + 8(co)) + pgA2 Cd(co) cos (Acot)^(B-19)where, co = (col + (02)/2 as before and Aco = (01 — (02.The mass matrix for the problem can be easily set by lumping masses corresponding tothe three degrees of freedom, viz., surge, heave and pitch respectively. Hence we can write,[ m 0 0[m] =^0 m 00 0 I(B-20)Here, m is the mass of the floating body and I is the mass moment of inertia of the bodyabout the axis passing through the centre of gravity. The total mass matrix is the sum of realand added mass matrix. Hence, we may write [M] = [m] + [p.]The restoring force on the body is developed as a result of the combined effect ofmooring stiffness and hydrostatic effect. Stiffness characteristic of the spring is as shown inFig. B-3. Note that, the spring action becomes effective only after certain displacement d sof the body. Hence, the spring stiffness matrix Ks(4) may be written as,^k i^5_ dg^KA) = 0^-dg^dg^ 2^4?. dg}(B-21)Hence, the total stiffness matrix [K] due to combined contribution from spring effects andhydrostatics may be given as:50LK] = KM + 8(4) [KS]^ (B-22)Where, 5 is the well known Kronecker delta function defined as0^if 4^cis1^if > ds(B-23)A detailed derivation of [Kg] and [KS] may be found in Lau et. al. (1990). It shouldhowever be noted that, due to presence of nonlinearity in the stiffness term, a compatibilitycondition must be satisfied at all times. Equation (B-1) may now be solved using a suitabletime stepping procedure to obtain the response (4).51Load P^Deflection^Calculated(N)^(cm)^Deflection^12.29^0.8^0.7524.58^1.5^1.536.88^2.1^2.349.17^2.8^3.0Table 3.1 - Measured and calculated load-deflection values for spring #4I Spring #^1 Width (mm)^0 Thickness (mm) I MI (mm4) Stiffness (N/m) I1 50 6 900.00 251.882 63 6 1134.00 317.373 76 6 1386.00 382.864 63 9 3827.25 1071.12Table 3.2 - Spring characteristicsTestNo. _^(cm)^II- I^T^I^a^I^K(sec)I^dgI^(cm) 1I LyapunovExponent1 13 2.5 0.423 4.25 -^11.5 -0.45672 11 2.5 0.423 4.25 11.5 -0.04993 12 2.5 0.423 4.25 11.5 -0.55704 14 2.5 0.423 4.25 11.5 0.06225 15 2.5 0.423 4.25 11.5 0.12766 13 2.0 0.529 4.25 11.5 -0.12937 13 2.2 0.481 4.25 11.5 -0.97458 13 2.8 0.378 4.25 11.5 -0.01439 13 3.0 0.353 4.25 11.5 -0.803410 11 2.8 0.378 4.25 r^11.5 -0.179511 15 2.8 0.378 4.25 11.5 -0.098512 11 3.0 0.353 4.25 11.5 -0.501613 11 2.0 0.529 4.25 11.5 -0.233014 15 2.0 0.529 4.25 11.5 0.118015 11 2.2 0.481 4.25 11.5 0.599816 15 2.2 0.481 4.25 11.5 0.418517 13 2.5 0.708 1.00 11.5 0.244818 13 2.5 0.778 1.00 11.5 -0.395619 13 2.5 0.873 1.00 11.5 -0.272420 13 2.5 0.423 1.00 11.5 -0.048721 13 2.5 0.423 4.25 7.0 -0.174022 13 2.5 0.423 4.25 14.0 0.0271Table 4.1 - Input parameters and Lyapunov exponents - monochromatic excitationTestNo.H1(cm)I^H2[ (cm)T1^T2I^1[^(s)^I^(s)la1K R^Lyapunov^IH^Exponent^I1 7.0 7.0 2.3 2.7 0.423 4.25 0.8802 7.0 7.0 2.0 3.0 0.423 4.25 0.5193 8.0 8.0 2.0 3.0 0.423 4.25 -0.5574 8.0 8.0 2.2 2.8 0.423 4.25 -0.4515 9.0 9.0 2.2 2.8 0.423 4.25 -6 7.0 7.0 2.3 2.7 0.778 1.0 0.0287 7.0 7.0 2.0 3.0 0.778 1.0 0.6168 8.0 8.0 2.0 3.0 0.778 1.0 -0.3589 8.0 8.0 2.2 2.8 0.778 1.0 0.206Table 4.2 - Input parameters and Lyapunov exponents - bichromatic excitation52Fig. 2.1 Idealization of the classical spring-mass-dashpot oscillatoroilo)nFig. 2.2 Classical resonance curves of a linear single degree of freedom system53Frequency coFig. 2.3 Classical resonance curve for a nonlinear oscillator54Input (harmonic) OutputLinearSystemOutput("\.}^>^PeriodicSubharmonicChaoticInput (harmonic)NonlinearSystemFig. 2.4 Comparison of linear and nonlinear systemsFig. 2.5 Divergence of nearby orbits.55YL(to)Fig 2.6 Calculation of the largest Lyapunov exponent from time series [Kapitaniak (1991)]56d g2+.dg 1 -f_K1 K2Box4d3m2n14M3 i M4d2^d2m1 m1Fig. 3.1 Mathematical model of surge motions of the boxFig. 3.2 Definition sketch for estimating the location of the centre of gravity. 57zFig. 3.3 Definition sketch for calculation of cantilever beam stiffness3.532.521.510.510^20^30^40^50Load (N)Fig. 3.4 Comparison of measured and calculated values of cantilever beam deflectionNomenclature: B - Beam = 76 cmD - Draft = 10.5 cmd - Water Depth = 55 cmW - Small weight attached to the stringk2 - Spring Stiffnessa, 13, y — Pulleys fitted with Potentiometersdg - Gap Widthk1Wave FlumeBeams Acting as SzingsCrlA Rubber 71151W W , BandTop Edge^IaW^0SWL—d —+^0^d^ B—>1Floating Body ( 30" X 8" X 10" )Wave Flume Bed4fse Z de de de se sr se se se de se 4' I' dr Z Z se se dr dr sr dr se de dr de se se se dr sr se de dr dr de Z Z se de Z se se de dr de de de de de se se ,,Inextensible String58TW1-21ZpiB/2›c)3SWLFig. 3.5 Sketch of experimental setupFig. 3.6 Definition sketch - transformation equations0.450.30.154.)^0.00- 0. 15-0.3(b)-0.45-0.24Displacement (m)Fig 4.1 - Surge response for monochromatic excitation(a) Time series (b) Phase portrait (c) Spectrum - logarithmic (d) Spectrum - linearH= 13 cm, T= 2.5 s, = 0.423, K = 4.25, d g= 11.5 cm-0.18 -0.12 -0.06 0.0 0.06 0.12 0.18 0.24••••^•••137.5^150.0^162.5^175.0•Time (sec)(a) 0.240.16•A 0.080a)em0 . 00La •-0.08 •I-0.16•112.5^125.0100.0II^I187.5^200.059• ••IL orN(C)0.1r• 0.010.001 h0.0001._ra 0.00001rDs▪ 0.1E-05• 0.1E-06 Ea)0.1E-07 Eal• 0.1E-05 r*-;C.) 0.1E-09 rca 0.1E-10 r0.1E-11 r0.1E-12 ^0 0 0.25^0.5 0.75^1.0 1.25^1.5 1.75^2.060Frequency (Hz)00^0.125^0.25^0.975^0.5^0.625^0.75Frequency (Hz)(b)(a) 0.30.2I^I^I11 1 1^I^'^,^I^1^I^I^1^I^1^I^1^I^1^1^i^i^1^1 1^i^$^14.,..._.•00 0.10arna)iza)ea0.0N -0.10tO.- 0.2 iIIIIIIIIIIIIII111111111111111111 t i ll ! !!- 0.3 100.0 112.5 125.0 137.5 150.0 162.5 175.0 187.5 200.0Time (sec)611 i 1 1Displacement (m)Fig 4.2 - Surge response for monochromatic excitation(a) Time series (b) Phase portrait (c) Spectrum - logarithmic (d) Spectrum - linearH= 15 cm, T = 2.5 s, a = 0.423, K = 4.25, dg= 11.5 cmrn 0.00010.00001rg 0.1E-05r0.1E-06 Nr.0.1B-070.1E-08 ra) —0.12-tor0.25 0.50.1E-110.0 0.75^1.0 1.25^1.5 1.75^2.0in . 1E-09 r62Frequency (Hz)00^0.125^0.25^0.975^0.5^0.625^0.75Frequency (Hz)(a)..-..0.30.2I 4^I40m 0.1006ala) 0.01:4a)ea.IRI -0.10 IM.-0.2I-0.3.•Itiiii^IIIIIIIIII I4' ' I lIiiiimilllim1H175.0 187.5I I I I 1 I I200.0 212.5 225.0 237.5 250.0 282.5 275.0631 10.01 1I0.121-0.06 0.06 0.18^0.24Time (sec)(b) 0.450.300-,mi 0.150PI4_, 0.0..../C.,0.--,as›. -0.15-0.31-0.45-0 24 -0.18Displacement (m)Fig 4.3 - Surge response for monochromatic excitation(a) Time series (b) Phase portrait (c) Spectrum - logarithmic (d) Spectrum - linearH= 14 cm, T = 2.5 s, a= 0.423, K= 4.25, dg= 11.5 cm1 I I I I I I I0.25 0.5 0.75 1.0 1.25 1.5 1.75 2.00.010.0010.00010.000010.1E-050.1E-060.1E-070.1E-080.1E-09164Frequency (Hz)00^0.125^0.25^0.375^0.5^0.625^0.75Frequency (Hz)Displacement (m)Fig 4.4 - Surge response for monochromatic excitation(a) Time series (b) Phase portrait (c) Spectrum - lograrithmic (d) Spectrum - linearH = 13 cm, T = 2.0 s, a = 0.529, K = 4.25, dg= 11.5 cm(b) 0.3I^I 1-0 24 -0.18^-0.12I0.0610 . 0I-0.06I0.12I0.240.160•a• a4a 4• 0 •0 aaf •IIE•• •II ••I1 165(a) 0.18.--.. 0.120 40740 0.060 •04ina)1:4a)bC0.0140rla-0.06 I-0.12100.0• ••I I I 1 I I I I112.5 125.0 137.5 150.0 162.5 175.0 187.5 200.0Time (sec)0.2-0.2-0.3(C)^1.0rN 0.1 r0.010.0014-4• 0.0001 6-ra^=-ph 0.00001 F.1▪ 22 0.1E-05 F• 0.1E-06co- 0.1E-07 irO▪ 0.1E-08 r4.1)gh 0.1E-00 r0.1E—ior(d)N00^0.125^0.25^0.975^0.5^0.625^0.7566Frequency (Hz)67(a) 0.10I.-.a0.12444.■To • #aOat.nioI:40.061# acoum#..0.0II0in-0.061I^1-0.12100.0^112.5^125.0#I4t11141aII41137.5^150.0 187.5^200.0III144t1#a1tttI.(b) 0.9I-0.0610.00.2-0.2- 0.3-024 -0.18^-0.12i I I 0.18^0.24I0.06I0.12Time (sec)Displacement (m)Fig 4.5 - Surge response for monochromatic excitation(a) Time series (b) Phase portrait (c) Spectrum - lograrithmic (d) Spectrum - linearH= 13 cm, T= 2.2 s, a= 0.481, K= 4.25, dg= 11.5 cm4.1• 0.1E-07 r=UU^—a  0.1E-08 r.-.V1^—0.1E-09 r0.001......g•-•.....•La 0.0001 rPs.4-3 0.00001r.,-.1VI —ct 0.1E-05 1.-—AI■1 0.1E-06 r=as0 68W 4I 1 I I 1 1 I I0.25 0.5 0.75 1.0 1.25 1.5 1.75 2.0Frequency (Hz)N=%.%N0.975....-..II-,•■•■■••ICI0. 000^0.125^0.25^0.975^0.5^0.625^0.75Frequency (Hz)0 . 0 0.06-0.18 -0.12^-0.06 0.18 0.240.12Displacement (m)0.460.30.16• 0.0• 400• -0.15-0.3-0.45-0 24(b)Time (sec)111'111111iV I II1t^1^1^1^t0111111I^I^I^I^I^I^I^I237.5^250.0^282.6^275.0^287.5^300.0^312.5^325.0 0.240.160O 0.08004rn 0.00P40eak -0.08ra(a)-0.16-0.24225.0Fig 4.6 - Surge response for monochromatic excitation(a) Time series (b) Phase portrait (c) Spectrum - lograrithmic (d) Spectrum - linearH= 13 cm, T = 2.8 s, = 0.378, K = 4.25, d g= 11.5 cm691 I I I I I I 10.26 0.5 0.75 1.0 1.25 1.5 1.75 2.0 Mt'Op......0.001P-. 0.00014-1....•132 0.00001Aci) 0.1E-05 r.--4 0.1S-060k 0.15-07 E.40^_0 0.15-oe r1:4Ea 0.1K-09 r0.1E-10 r0.1K-110.070Frequency (Hz)00^0.125^0.25^0.375^0.5^0.625^0.75Frequency (Hz)137.5^150.0 162.5^175.0 187.5^200.0112.5^125.0111 ,^'(a) 0.240.16O 0.080In•^0 . 0P40be• -0.080T71.1111111 11 111 111 1 1111111111111111-0.16Time (sec)(b) 0.60.40.2AJ 0 . 000-0.2-0.4-0.6-0 24Fig 4.7 - Surge response for monochromatic excitation(a) Time series (b) Phase portrait (c) Spectrum - lograrithmic (d) Spectrum - linearH= 13 cm, T = 3.0 s, a = 0.353, K = 4.25, dg= 11.5 cm710.12 0.18^0.24I I^I I^I I^I I-0.24100.0I)N,.«./ 0.00001....■TaO 0.1E-05= 0.1E-06 r.-.^:as7... 0.1E-07rO 0.1E-08 rta.ca  0.t6-09 r:...0.1E-10 r......0rn 0.0001I I 1 I I I I I0.25 0.5 0.75 1.0 1.25 1.5 1.75 2.0720.1E-110.0Frequency (Hz)00^0.125^0.25^0.375^0.5^0.625^0.75Frequency (Hz)(a)(b) 0.3750.26..---.in0.1260PI••-)^0.0..-•C)0.--eP.4) -0.125-0.25-0.375—-0.2473I I I I I I I I-0.18 -0.12 -0.08 0.0 0.08 0.12 0.18 0.24Displacement ( m )Fig 4.8 - Surge response for monochromatic Excitation(a) Time series (b) Phase portrait (c) Spectrum - logarithmic (d) Spectrum - linearH = 11 cm, T = 2.8 s, a =0.378, K = 4.25, dg= 11.5 cm^0.0 1 '^- ^0.125^0.25 0.975^0.5-0.625^0.75J MI(c)NN1.00.10.010.0010.00010.000010.1E-050.1E-080.1E-070.1E-080.1E-000.1B-100.1E-1174 0.1E-120 0^0.25^0.5^0.75^1.0^1.25^1.5^1.75^2.0Frequency (Hz)0.8Frequency (Hz)(a)(b) 0.4-0 24 -0.18^-0.12 -0.06^0.0 0.06^0.12 0.18^0.241^1^1^1^1^1^1^1^1^1 11 ^1^1^1^1^1^1^1^1^1^1^1^1^1^1^a^1^1^11 1 I 1 I I I 1 1 I 1 1 I 1 I 1 1 I II I I I I I 1112.5 125.0 137.5 150.0 162.5 175.0 187.5Time (sec)I200.0Displacement (m)Fig 4.9 - Surge response for monochromatic Excitation(a) Time series (b) Phase portrait (c) Spectrum - logarithmic (d) Spectrum - linearH= 15 cm, T = 2.8 s, a = 0.378, K = 4.25, dg= 11.5 cm1 1 1 1 I 1 1 10.2- 0.4- 0.6i I 1 1 I 1 I I 1 1 I 1 1 I 1 1I175-0.24100.0-0.15Frequency (Hz)iN(C)^10.0g-N=i.or04▪ oAr0^0.014.1.4■■•••^0.001TZ1C., 0.0001*.aTO 0.00001a)• 0.1E-05co▪ 0.1E-067.44.)O 0.1E-07• 0.1E-08Ca0 0.125 0.25 0.375 0.5 0.625 0.75(d) 4.0N9.0U11>12.0■-■0(1)ca0.00Frequency (Hz)0.1E-000.1E-10 ^0 0 1.75^2.00.75^1.00.25^0.5 1.25^1.676(b) 0.3 -0.2 --0.3-0 24^-0.18I0.08I0.1210.0I^I0.18^0.24I^I-0.12^-0.08ITime (sec)(a) 0.160.12 1 I II I 1.--,. I • IIIWva 0.08 -I-0 I004ma) 0.04 o 1.eII *$ . . I 1 i • •04tiOk 0.0 -Ii•I•I ■e t I 1 10 I Q IVI-0.04I I-0.08 I^I^I^I^I^I^I^I100.0^112.5^126.0^137.6^150.0^162.5^175.0^187.5^200.077Displacement (m)Fig 4.10 - Surge response for monochromatic Excitation(a) Time series (b) Phase portrait (c) Spectrum - logarithmic (d) Spectrum - linearH= 11 cm, T = 2.0 s, a = 0.529, K = 4.25, d g= 11.5 cm00 0.125 0.25 0.975 0.5 0.625 0.75(d)NLow(c)N 0.1r• 0.010.0014.4 0.0001 rz-taDs 0.00001132 0.1E-05 rAa) 0.1z-06 IN-• 0.1E-07 irO 0.1E-05 e-gat 0.1E-00 rCO.0.1E-10 r 784#1f0.1E-11 ^00 0.25^0.5^0.75^1.0^1.25^1.5^1.76 2.0Frequency (Hz)Frequency (Hz)I II 41oI 41 4 FI IiI sI1I 10I I(a) 0.15I,-.. 0.1 I I0..._..Osylg00.05 T I 4astaas1 I=o 0.0beFr4 I i0rn I-0.051^I-0.1I 4IIl100.0^112.5 125.0^137.5I^I^I^I^I150.0^162.5^175.0^187.5^200.079II 1 1 I I I-0.06 0.0 0.08 0.12 0.18 0.24Displacement (m)Time (sec)(b)or....la...„0.3 -0.20 . 1--E.....Pos.0.).-.0.0 -VO..as -0.1 -- 0.2 --0.9I I-0 24 -0.18 -0.12Fig 4.11 - Surge response for monochromatic Excitation(a) Time series (b) Phase portrait (c) Spectrum - logarithmic (d) Spectrum - linearH= 11 cm, T= 2.2 s, a= 0.481, K= 4.25, dg= 11.5 cm(c)N=■.,...N 10.0 k-A_..^. A IV0 0 0.1251.00.10.010.0010.00010.000010.1E-050.1E-060.1E-070.1E-080.1K-09801 1 I^I I I I0.5 0.75 1.0^1.25 1.5 1.75 2.0Frequency (Hz)A^1 1^-I %.^I I I0.25 0.375 0.5 0.625 0.75Frequency (Hz)(a) 0.24,00.18i0W0 0.080Ofalla)P40tillr.,00.0DI-0.08 r-0.161I,i •i,,/rts1I1 i I^I I^I100.0 112.5^125.0 137.5^150.0I^I187.5^200.0162.5^175.0811 I I1 1 1 1-0.12 -0.06^0.0 0.06^0.12 0.18 0.24(b) 0.450.9,-...tai 0.150P.1I-) 0.0•.-•00.--.0P- -0.15-0.9-0.45 1-0 24 -0.18Time (sec)Displacement (m)Fig. 4.12 Surge response for bi-chromatic excitation(a) Time series (b) Phase portrait (c) Spectrum - logarithmic (d) Spectrum - linearH1 = 7.0 cm, H2 = 7.0 cm, T1 = 2.3 s, T2 = 2.7 s, a= 0.423, K = 4.25(c)1.00.10.01 RI.10.001 II0.00010.00001 -0.1E-050.1E-050.1E-070.1E-050.1E-090.1Z-10820.1E-11^I -^I^I^I^I^I^I^I^I0. ^0.26^0.5^0.76^1.0^1.25^1.5^1.75^2.0Frequency (Hz)(b) 0.760.50.265P10.0C)0-0.25-0.5-0.75-0 2410.06-0.18^-0.12^-0.06^0.0Displacement (m)100.0 112.5831^1125.0^137.51150.01^1162.6^175.01^1187.5^200.0ITime (sec)A•0.1810.12 0.240.30.2-0.2-0.3I(a)Fig. 4.13 Surge response for bi-chromatic excitation(a) Time series (b) Phase portrait (c) Spectrum - logarithmic (d) Spectrum - linearH1 = 9.0 cm, 1-12 = 9.0 cm, T1 = 2.2 s, T2 = 2.8 s, a= 0.423, K = 4.25(C)^10.0N^1.0=■.,^0.1eg0.011,.., 0.001...—...TM.0.0001Ds4J 0.00001....1TO0 0.1E-055)A 0.1K-05p.,650.1E-07&■4.)0 0.1E-085.)/34 0.1E-09MI0.1E-101.25^1.5^1.75^2.084Frequency (Hz)850.3750.250 0.126004Tn^0.00E., - 0.125-0.26 ^ H=12 cm- - -^H=13 cm^ H=14 cmI-0.375^ I^I^I^I^I^I^I^I^I2 50.0^255.0^280.0^285.0^270.0^275.0^280.0^285.0^290.0^295.0^300.0Time (sec)Fig. 4.14 Effect of wave height variation on surge response0.3750.250 0.1260i22^0.0a,L4 -0.125rn.-0.26^ T-2.0 Oec- - - - - T=2.5 orec^ T=3.0 ilecI-0.375^ I^I^I^I^I^I^I^I^I260.0^255.0^260.0^285.0^270.0^275.0^280.0^285.0^290.0^295.0^300.0Time (sec)Fig. 4.15 Effect of wave period variation on surge response860.250.20.150.10.056^8^10^12^14^16Gap width (cm)Fig. 4.16 Plot of surge response amplitude vs. gap width+ Periodic/Quasi-Periodic• ChaoticFig. 4.17 Plot of R/H vs. B/gT2 for K = 4.25, H = 13 cm87+ Periodic/Quasi-periodic♦ ChaoticB/HFig. 4.18 Plot of R/gT2 vs. B/H for K = 4.25, t = 2.5 s+ Periodic/Quasi-Periodic• Chaotic3.002.502.001.500.30^0.40^050^0.60^0.70aFig. 4.19 Plot of R/H vs. a for K = 4.25, H = 13 cm(a)(b)1;30.600.400.200.00-0.20-0.40-0.200.20 ^0.100.00-0.10-0.20-0.30-0.4088-0.10^0.00^0.10^0.20Displacement (m)-0.15 -0.10 -0.05^0.00^0.05^0.10^0.15Displacement (m)Fig. 4.20 Poincare map of surge response for bichromatic input(a) ti = 10 (b) ti = 2089(a)Displacement (m)Displacement (m)Fig. 4.21 Poincaré map of surge response for bichromatic input(a) t = 30 (b) t = 4090(a)(b)Fig. 4.22 Poincaré map of surge response for bichromatic input(a) t = 50 (b) t = 6091Fig. A-1 Displacement Probe #1, L 1 - original length, L 1 ' - length at time t92Fig. A-2 Displacement Probe #2, L2 - original length, 1,2 - length at time tFig. A-3 Displacement Probe #3, L3 - original length, 143 - length at time t94Z, Z'Fig. B-1 Definition sketch of moored two-dimensional floating bodyFig. B-2 Mathematical model of moored floating object with nonlinear moorings

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