Non-linear behaviour of an idealised submergedWEC with Coulomb damping

Jana Orszaghova∗, Hugh A. Wolgamot∗, Scott Draper∗ and Ashkan Rafiee†∗Faculty of Engineering and Mathematical Sciences

University of Western Australia, Crawley, WA 6009, AustraliaE-mail: jana.orszaghova@uwa.edu.au, hugh.wolgamot@uwa.edu.au, scott.draper@uwa.edu.au

†Carnegie Clean Energy, Perth, Western AustraliaE-mail: arafiee@carnegiece.com

Abstract—Wave energy converters (WECs) with Coulomb-typedamping are relatively common as this behaviour is typical ofhydraulic power take-off (PTO) systems. This study investigatesmotion response of a mass-spring system under combined linearand Coulomb damping, as a simplified representation of a WECwith a Coulomb damping force in the PTO. Motivation forthis work stems from observed heave motions, of a submergedmodel-scale WEC with Coulomb PTO damping, which in somecases exhibited large offsets/drifts from the mean position. Forsubmerged devices any restoring force must be engineered, as thehydrostatic restoring force is zero. Given a choice of restoringforce and Coulomb coefficients, we are interested in how tomake the optimum choice to minimise displacements in largewaves. In this paper a simple model is used to investigate thisquestion. In the range of parameters considered, the extremedisplacements are minimised by adopting negligible Coulombdamping force (relative to the exciting forces) and a weak detunedspring coefficient. The low-frequency drifting behaviour is seento be important for both mean and extreme behaviour of theoscillator.

Index Terms—wave energy, Coulomb damping, nonlinearity,submerged, extremes

I. INTRODUCTION

Hydraulic power take-off systems are often proposed forwave energy devices and provide a damping force which isapproximately of Coulomb damping form. There is a largeliterature on the behaviour of classical mass-spring oscillatorswith Coulomb damping (for example [1] and [2]). The litera-ture highlights the obvious fact that, if the Coulomb force islarger than the other forces at work, and the mass is at rest, itwill remain at rest until the situation changes. Hence motionwith ‘stops’ is observed in the literature.

Studies of power absorption by simple WEC systems withsuch damping have been conducted by [3] for example, whoinvestigated a floating heaving body. Power absorption withhydraulic PTO was also investigated by [4] and [5] whomodified the hydraulic PTO to have a ‘stick-slip’ like frictionbehaviour and ‘on-off switch’ respectively.

In wave energy applications, like those above, Coulombdamping has traditionally been considered in the context ofa floating WEC. In this case, the hydrostatic restoring forceis usually significant. For a submerged device this vanishes,and so stiffness applied is determined by the designer, andcan conceivably be varied over a wide range. In operational

conditions it is accepted that choosing a stiffness to achieveresonance at a frequency aligned with incident waves isdesirable. In extreme cases, where the objective is to minimisemotions so as to reduce the possibility of end-stop hits(amongst other unwanted effects), it is desirable to move thenatural frequency outside the range of linear excitation, as istraditionally done with oil and gas infrastructure. Recent workon extreme responses of submerged devices ( [6], [7]) hasfocussed on predicting maximum excursion with linear PTO.As hydraulic PTO is more commonly implemented than a truelinear system, it is of interest to consider the excursions of asubmerged WEC with hydraulic PTO.

We are motivated in this endeavour by the results shownin Figure 1 which displays the experimentally derived heavemotion of a WEC with Coulomb damping applied in ratherlarge irregular waves of peak period Tp. The model-scaledevice consisted of a submerged cylindrical buoy, which wastethered to the basin floor. The tether length was controlledby a winch system, which was programmable, and actedas a model-scale PTO, with Coulomb damping imposed inthis case. From the figure, it is apparent that in addition tothe wave-frequency oscillation there are relatively large low-frequency oscillations which we refer to as an offset of themean position. We are interested in what combination of springand Coulomb damping force would minimise the extremeexcursion for given excitation.

450 460 470 480 490 500 510 520 530 540 550

t/Tp

-0.5

0

0.5

Z/m

ax(Z

)

Fig. 1: Example timeseries of measured heave motion undersevere wave conditions (peak period Tp) with Coulomb damp-ing applied in the PTO.

II. MASS-SPRING OSCILLATOR WITH COULOMB ANDLINEAR DAMPING

We consider a single degree-of-freedom mass-spring oscil-lator with both Coulomb and linear damping, excited by asum of harmonic forcing components. The linear dampingis present to roughly represent hydrodynamic damping. Thegoverning equation is

mZ +KZ +BZ + Csign(Z) =∑i

Fi cos(ωit+ φi) (1)

where m is mass, K and B are the linear spring stiffness andlinear damping coefficients, C is the amplitude of the Coulombdamping force; and Fi, ωi and φi are the amplitude, angularfrequency and phase of the ith component of the excitationforce. The applied Coulomb damping always opposes themotion of the device, and as such the Coulomb force is givenby sign(Z)C, where the device displacement is Z and ˙denotesdifferentiation with respect to time.

The closed form analytical solution of Equation 1, with thesign of the Coulomb force either positive or negative, is wellknown. By considering homogeneous and non-homogeneousversions of Equation 1, due to linearity, the solution isa super-position of a free decay motion (Z1), steady-stateharmonically-forced vibration (Z2) and a time-invariant termdue to a constant force (Z3). The solution is presented below.

The free decay term is

Z1 = e−B2m t(A1 cos(ωdt) +A2 sin(ωdt)

)(2)

where ωn and ωd = ωn

√1−

(BBc

)2are the undamped and

damped natural frequencies, and Bc = 2√Km denotes the

critical damping. Coefficients A1 and A2 are to be determined.The steady-state harmonic response is

Z2 =∑i

Fi

ωi

√B2 + (ωim− K

ωi)2

cos(ωit+ θi) (3)

where θi are phases given by θi = φi − π2 − tan−1((ωim −

Kωi)/B).Finally the constant term is given by

Z3 =

{−CK for Z > 0

+CK for Z < 0

(4)

Using the full solution satisfying the initial conditionsZ(t = 0) = Z0 and Z(t = 0) = Z0, the coefficients A1

and A2 are given as below.

A1 = Z0 ±C

K−∑i

Fi

ωi

√B2 + (ωim− K

ωi)2

cos(θi) (5)

A2 =Z0

ωd+

B

2mωdA1 +

∑i

Fi

ωd

√B2 + (ωim− K

ωi)2

sin(θi) (6)

It is possible to construct a semi-analytical solution ofEquation 1 by applying solutions 2-4, while keeping trackof the sign of the velocity Z. The final piece-wise solutionis semi-analytical, as the instances when velocity goes to 0

need to be determined numerically. Care has to be taken atany such instance, since if the Coulomb force is larger thanthe combined spring and excitation forces, the device willremain motionless until the damping force is overcome. Thecalculation algorithm can be summarised as follows. Similarsteps are proposed by [8] and [9].

1) Calculate Z and Z using Equations 2 - 4.2) Solve Z = 0 numerically to find t∗.3) Check whether ‘stop’ occurs using

Fm = | −KZ +∑i

Fi cos(ωit∗ + φi)|. (7)

If Fm ≥ C, ‘stop’ does not occur. Go to 4)If Fm < C, ‘stop’ occurs. Solve Fm = C numericallyto find new t∗.

4) Go back to 1) with updated initial conditions Z0 =Z(t∗) and Z0 = 0 and new updated force phasesωit

∗ + φi.Works such as [1] and [2] consider a mass-spring system

with both linear and Coulomb damping under monochromaticexcitation, and derive an analytical periodic steady-state so-lution which undergoes no ‘stops’. The bounds for such asolution to exist give rise to a diagram with three potentialmotion regimes. As shown in Figure 2, under small Coulombdamping and large waves/excitation forces, the device will bemoving at all times, stopping only instantaneously. As the ratioof the Coulomb and driving forces increases, motions with‘stops’ occur. The threshold/boundary values of CF , which varywith the non-dimensional frequency, are shown in colouredlines in Figure 2 for different ratios of the linear and Coulombdamping. Naturally, with increased linear damping, the regionwith no ‘stops’ is smaller. If the Coulomb force is larger thanthe excitation force amplitude, a permanent ‘stop’ occurs andas such the device remains at rest.

0 0.5 1 1.5 2

-=!n / !

0

0.2

0.4

0.6

0.8

1

1.2

C/F

no "stops" motion

motion with "stops"

permanent "stop" B/Bc=0

B/Bc=0.2

B/Bc=0.4

B/Bc=0.8

Fig. 2: Three solution regimes diagram.

III. REGULAR EXCITATION

We first investigate motion with Coulomb damping undermonochromatic forcing, representative of regular waves. Cho-

sen examples are shown with triangular and square markersin the regimes diagram in Figure 2. Calculated semi-analyticalsolutions are shown in Figures 3 and 4. Note that the plottedcurves have been non-dimensionalised such that Z and Z havebeen scaled with F

K and FB respectively.

A numerical solution (calculated with a fourth order Runge-Kutta solver) has also been computed for reference, andresults plotted with black dashed lines. It is noted that thenumerical computation can be considerably slower than thesemi-analytical solution, particularly for cases with stops. Thefast semi-analytical solver is well-suited for long durationsimulations with multiple realisations.

The calculated solutions exhibit an early transient, followedby a steady state solution. The steady state solution is periodic,but is not harmonic. The dominant Fourier component is at thedriving frequency ω, but odd order higher harmonics (3ω, 5ω,...) are also present. These higher harmonics become moreimportant as the duration of ‘stops’ becomes longer. Figure5 shows the calculated spectrum with the higher harmonicsclearly visible.

0 1 2 3 4 5t/T

-0.4

-0.2

0

0.2

0.4

Z_Z

Fig. 3: Calculated non-dimensionalised solution without‘stops’ for β = 0.5, CF = 0.25, B

Bc= 0.2.

0 1 2 3 4 5t/T

-0.05

0

0.05

Z_Z

Fig. 4: Calculated non-dimensionalised solution with ‘stops’for β = 0.5, CF = 0.75, B

Bc= 0.2.

As displacement and velocity are approximately π2 out

of phase, the ‘stops’ occur at the maximum displacement.However, solutions with more than 2 stops per period are alsopossible, as shown in [9].

Unsurprisingly, under larger Coulomb damping, the re-sponse reduces. The amplitude of the steady-state responseis bound by 0, which applies for C

F ≥ 1, and by the harmonicsolution from Equation 3, which applies for C = 0. Similarly,

0 1 2 3 4 5 6 7 8f/(1/T)

10-5

100

PSD

Fig. 5: Power spectral density of the calculated motion shownin Figure 4.

the response diminishes as the forcing frequency departsfurther from the natural frequency of the device (results notshown).

IV. IRREGULAR EXCITATION

Next we consider irregular forcing. A matrix of differentconditions has been selected to investigate the response undervarying Coulomb damping and spring stiffness. All solutionsare calculated via the semi-analytical method. The followingparameter space has been explored:

• m = 1000 kg• B = 100 Ns/m• K: 10, 40, 70, 100, 150, 200, 300, 400, 500, 700 and

1000 N/m• C: 100, 200, 300, 400, 500, 600, 700 and 800 N• top-hat force spectrum with Fm0 = 3500 N and frequen-

cies spanning 0.4 Hz < f < 0.6 Hz.The above fixed values of mass, damping and frequency werechosen to be representative of model-scale conditions for asubmerged WEC. Note that the rather large mass containsa substantial contribution approximately representing heaveadded mass (for the frequency range investigated) predictedby linear potential flow theory for the device considered.

Stiffness values are relatively low, to ensure that the naturalfrequency is very dissimilar to (always below) the excitationfrequencies. The Coulomb values vary across an interestingrange, from situations where stops are infrequent to simula-tions where stops occur over a considerable proportion of thetime (larger forces likely being unrealistic in any case).

Each simulation is run for approximately 2600 s, such thatover 1000 waves are simulated in each run, while also allowingfor any initial transient behaviour to be discarded from furtheranalysis. 25 different sets of simulations were completed, eachset/realisation comprising all combinations of K and C valuesrun with an identical forcing timeseries (i.e. the same phasesφi were used in one set, but varied across different sets). Thetotal number of these long-duration irregular simulations wasthus 11× 8× 25 = 2200.

A. Stopping times

We now consider the proportion of each simulation inwhich ‘stops’ occur. The excitation force timeseries, from one

realisation/set, is shown in Figure 6. For a particular run,the force value F = C which marks the permanent ‘stop’solution regime is shown in red. During small ‘waves’ whichfall below this line, the friction cannot be overcome, andthe device will be at rest (for the duration of such waves).Also shown, in green, is the calculated threshold force valueF = F ∗, which marks the solution regime with ‘stops’ (basedon a representative monochromatic case, i.e. the mean drivingfrequency). For any waves between these two lines, partialstopping will occur. To illustrate this behaviour, zoomed-intimeseries of the excitation forcing and the calculated motionexhibiting ‘stops’ are shown in Figure 7. Both partial ‘stops’(for example around t = 1888 s) and full/permanent ‘stops’(for example around 1904 < t < 1906 s) can be seen.

0 500 1000 1500 2000 2500

t (s)

-2000

0

2000

F (

N)

F=C

F=F*

Fig. 6: Excitation force timeseries (particular realisation). Forrun C = 300 N and K = 100 N/m, two threshold forcevalues marking the permanent ‘stops’ regime (red line) andthe motion with ‘stops’ regime (green line).

-1000

-500

0

500

1000

F (

N)

F=C

F=F*

1880 1890 1900 1910 1920 1930 1940

t (s)

-0.2

-0.1

0

0.1

0.2

Z (

m)

Fig. 7: Zoomed-in excitation and response timeseries for runC = 300 N and K = 100 N/m (particular realisation).

Using Rayleigh statistics (assuming the force spectrum issufficiently narrow-banded), we can calculate the probabilitiesp(F < C) and p(F < F ∗). These can be used as an estimateof a lower and an upper bound on the proportion of the timethe device is at rest. For the case presented, the probabilitiesare 6% and 18%, and the actual total ‘stop’ time calculated

for this realisation by solving the governing equations semi-analytically was approximately 212 s (out of 2600 s), makingit 8%. The threshold force values C and F ∗, and as suchthe associated probabilities, vary across different runs. In allbut a few of the 2200 simulations performed, the actual ‘stop’times calculated fall within the two bounds. For the cases withthe highest Coulomb damping (C = 800 N), the calculatedprobability bounds are approximately 34% and 77%, with theactual calculated proportion of ‘stop’ time being around 46%.The device will be subject to ‘stops’ very often. On the otherhand, for the cases with the least Coulomb damping (C =100 N), approximately only 0.5-1% of the total time is spentin ‘stops’ (which in the majority of runs falls between thepredicted bounds of around 0.7% and 2%).

B. Long-period offset motions

Motion timeseries from four runs, which lie on the cornersof our test matrix (i.e. all four combinations of maximumand minimum stiffness and Coulomb damping), are shown inFigure 8. Although these simulations have the same underlyingexcitation force sequence, large differences in the responsecan be seen. The least stiff cases exhibit long-period offsetsfrom the mean position (blue and yellow curves), and theseare much more pronounced under smaller Coulomb damping(blue curve). We also note that the simulated responses withlarge Coulomb damping comprise prolonged periods with nomotion, and these occur irrespective of the stiffness coefficient(yellow and purple curves).

2200 2250 2300 2350 2400 2450 2500

t (s)

-0.4

-0.2

0

0.2

0.4

Z (

m)

K=10 C=100 K=1000 C=100 K=10 C=800 K=1000 C=800

Fig. 8: Calculated motion timeseries for four runs at thecorners of the test matrix.

Figures 9 and 10 display the motion spectra from onerealisation for all runs (all combinations of K and C withidentical force timeseries). In Figure 9 all runs with the samestiffness K are shown in one colour. The response at thedriving frequencies is clearly seen. Large variations can beseen in the low-frequency responses. These are offsets we areinterested in investigating. From the figure it appears that theselong-period drifting motions are primarily governed by thespring stiffness. Blue, red and yellow lines correspond to thethree smallest spring coefficients tested respectively, and blackand grey lines are used for the two largest spring coefficients.Weak springs lead to large offsets.

Although the natural frequencies (and hence free decaysolution components Z1) are in the low-frequency range due

to the small values of the spring coefficient, we do not seesubstantial response around the natural frequency - rather,the response increases towards zero frequency. We note thatonly for cases with no Coulomb damping (C = 0 N) wouldthe response at the natural frequency be apparent (an initialtransient at the very beginning of a simulation), as it wouldbe the only long-period response component.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

f (Hz)

10-4

10-2

100

PS

D (

m2/H

z)

K=10K=40K=70K=100K=150K=200K=300K=400K=500K=700K=1000

Fig. 9: Calculated motion spectra for all runs from onerealisation (colour-coded by stiffness K). Blue denotes thesmallest K tested, and black denotes the largest K tested.

In Figure 10 all runs with the same Coulomb value C areshown in one colour. The plots are zoomed in on the responseat the excitation frequencies, and we can see that theseresponses are principally governed by the Coulomb damping.Large Coulomb damping leads to diminished responses in thislinear frequency range. We also note that stiffness values fur-ther away from optimal stiffness (for resonance to occur) resultin a smaller response, which agrees with findings presented formono-chromatic forcing. In the plot, the theoretical responsespectra under no Coulomb damping (C = 0 N), calculatedaccording to Equation 3, are also shown for reference withdashed lines. The upper line corresponds to K = 1000 N/m,while the lower line corresponds to K = 10 N/m.

We note in passing that small responses around three timesthe mean frequency are also visible on the spectra, thoughare not shown below, as they are smaller than the othercontributions.

Figure 11 displays the phase difference between the exci-tation force and the calculated response, zoomed in on thedriving frequencies. For small Coulomb values, the responseis close to being out of phase with the external force. AsCoulomb values are increased, the phase difference gradu-ally becomes smaller. The theoretical phase angle for zeroCoulomb damping is also shown in the plot using dashed lines.Two lines, for K = 10 and K = 1000 N/m, have been plotted,though they are almost indiscernible.

In Figures 10 and 11 we see smooth spectral shapes withoutlarge peaks and slowly-varying phase plots without changeof sign. This is to be expected, as all the cases consideredrepresent situations far from resonance, due to the weakstiffness coefficients chosen.

C. Experimental resultsHeave spectra from a handful of model-scale laboratory

experiments of a submerged point absorber WEC are presented

0.4 0.45 0.5 0.55 0.6

f (Hz)

0

0.05

0.1

0.15

PS

D (

m2/H

z)

C=100C=200C=300C=400C=500C=600C=700C=800

Fig. 10: Calculated motion spectra for all runs from onerealisation (colour-coded by Coulomb damping C). The blackdashed lines denote the theoretically calculated response spec-tral densities for C = 0 N, with the lower and upper curvescorresponding to K = 10 and K = 1000 N/m respectively.

0.4 0.45 0.5 0.55 0.6

f (Hz)

90

120

150

180

phas

e an

gle

(deg

rees

)

C=100C=200C=300C=400C=500C=600C=700C=800

Fig. 11: Phase difference between the excitation force andthe calculated device motion for all runs from one realisation(colour-coded by Coulomb damping C). The black dashedlines denote the theoretically calculated phase angle for C = 0N, with the lower and upper curves corresponding to K =1000 and K = 10 N/m respectively (though the two lines aredifficult to discern at the shown scale).

in Figures 12 and 13, whereby a linear spring and a Coulombdamping were applied in the PTO. For ease of comparison,the frequency axis has been normalised by the peak frequencyfp as the values were varied in the experiments. Similarlythe applied stiffness and damping coefficients have been non-

dimensionalised via√K/m

ω = ωn

ω and CFm0

respectively, wherem denotes the combined device mass and the heave addedmass (from linear potential flow theory) and ω = 2πfp denotesto the peak angular frequency. For each laboratory experiment,the values of significant force Fm0 are calculated from thetheoretical input heave force spectra. The heave spectra aregiven by the input wave spectra pre-multiplied by a squareof the heave excitation force RAO (from linear potentialflow theory). We note that the experiments were carried outfor approximately 800 waves, apart from a shorter run ofapproximately 200 waves (purple curve).

Figure 12 shows the measured response at the excitationfrequencies (around f

fp= 1), as well as the low-frequency

offsets. The offsets are largest under low spring coefficients(blue, red and yellow curves), and much reduced under stiffersprings (green curve). This trend of weak springs leading tolarge offsets agrees with our numerical results (see Figure 9).

For completeness, Figure 13 displays the measured heavespectra zoomed in on the wave excitation frequency range.The effect of Coulomb damping is difficult to ascertain, asthe tested values span a very narrow range. However, we notethat stiffness values closer to resonance give rise to higherlinear responses (green curve), as expected. It is encouragingto see qualitative agreement between the experimental data andthe results from our simplified numerical model, especiallyconsidering the extent of physics not accounted for in themodel.

0 0.5 1 1.5 2f/fp

10-2

PS

D (

m2/H

z)

!n/!=0.18

C/Fm0

=0.07

!n/!=0.21

C/Fm0

=0.08

!n/!=0.25

C/Fm0

=0.10

!n/!=0.40

C/Fm0

=0.08

!n/!=0.55

C/Fm0

=0.08

Fig. 12: Experimental heave spectra from model scale labora-tory experiments with Coulomb damping applied in the PTO.

0.5 1 1.5 2f/fp

0

0.005

0.01

0.015

0.02

PS

D (

m2/H

z)

!n/!=0.18

C/Fm0

=0.07

!n/!=0.21

C/Fm0

=0.08

!n/!=0.25

C/Fm0

=0.10

!n/!=0.40

C/Fm0

=0.08

!n/!=0.55

C/Fm0

=0.08

Fig. 13: Experimental heave spectra from model scale labora-tory experiments with Coulomb damping applied in the PTO,zoomed in on the wave frequency range.

D. Largest responses and response statistics

We are interested in studying maximum responses. Intu-itively based on Figures 9 and 10, a worst-case combinationmight be of a weak spring with limited Coulomb damping,as under such conditions both large offsets and large linearresponses occur. By the same logic, a stiffer spring togetherwith strong damping might seem preferable for limiting themaximum responses. Though under such conditions the overallmechanical forces might be higher than desirable.

In order to study dependence of the largest motion on Kand C, the maximum motion (positive or negative, since thereis no difference in our model) is identified in each simulation.Statistics of these are plotted in non-dimensionalised (K,C)parameter space in Figures 14 and 15, where the x-axis and

y-axis have been scaled via√K/m

ω and CFm0

respectively. Thetested values are shown with black dot markers. Note thatwe have also included results from cases with C = 0 N,which have been calculated with the present semi-analytical

solver (though these could easily be computed via an RAOin the frequency domain). Figure 14 shows a contour plot ofthe overall maximum response recorded from the 25 differentrealisations (for each K and C combination tested), and Figure15 shows the averaged (across the 25 realisations) maximumresponses.

0.05 0.1 0.15 0.2 0.25 0.3

!n / !

0

0.05

0.1

0.15

0.2

C/F

m0

0.45

0.5

0.55

0.6

0.65

0.7

0.75

10

40

70

100

150

200

300

400

500

700

1000

K (N/m)

0

100

200

300

400

500

600

700

800

C (

N)

Fig. 14: Contour plot of the overall maximum response am-plitudes (from 25 realisations).

0.05 0.1 0.15 0.2 0.25 0.3

!n / !

0

0.05

0.1

0.15

0.2

C/F

m0

0.4

0.45

0.5

0.55

0.6

0.6510

40

70

100

150

200

300

400

500

700

1000

K (N/m)

0

100

200

300

400

500

600

700

800

C (

N)

Fig. 15: Contour plot of the averaged (across 25 realisations)maximum response amplitudes.

Additionally, Figures 16 and 17 display the maximum am-plitudes of high-pass and low-pass filtered response timeseries,with the cut-off frequency being just below the minimum ex-citation frequency of 0.4 Hz. These two contour plots presentthe dependence on K and C of the largest (primarily) linearmotions and the largest long-period offsets respectively.

Using Figures 14 and 15, the maximum responses are notminimal under large spring and large Coulomb damping, aswe have reasoned above. It appears that the largest responsesare least severe either under very low values of C and low tomedium values of K or under high values of C and mediumvalues of K. From the design point of view, the first option

would be favourable, as increased mechanical forces are moreexpensive. On the other hand, the most extreme responsesoccur under small stiffness and low to medium Coulombdamping.

0.05 0.1 0.15 0.2 0.25 0.3

!n / !

0

0.05

0.1

0.15

0.2

C/F

m0

0.34

0.35

0.36

0.37

0.38

0.39

0.4

0.41

0.4210

40

70

100

150

200

300

400

500

700

1000

K (N/m)

0

100

200

300

400

500

600

700

800

C (

N)

Fig. 16: Contour plot of the maximum amplitudes of high-passfiltered response (averaged across 25 realisations).

0.05 0.1 0.15 0.2 0.25 0.3

!n / !

0

0.05

0.1

0.15

0.2

C/F

m0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

10

40

70

100

150

200

300

400

500

700

1000

K (N/m)

0

100

200

300

400

500

600

700

800

C (

N)

Fig. 17: Contour plot of the maximum amplitudes of low-passfiltered response (averaged across 25 realisations).

We can offer partial explanations of the trends observed asfollows.

• Under no (or very minimal) Coulomb damping, the de-vice does not experience stopping behaviour and thereforecannot build up the long-period offsets (see Figure 17).As such, the bulk of the response is in the wave fre-quency range, which is minimised under lower stiffnesscoefficients (where the natural frequency is further fromthe input frequency range, meaning that the ‘RAO’ in theinput frequency range is lower).

• When a small amount of Coulomb damping is included,increased stiffness is needed to reduce the offsets andthus minimise the largest responses. However, there isan optimal stiffness range, as increasing the stiffness

further would be detrimental due to the linear responsesincreasing (see Figure 16).

• When Coulomb damping is larger, and stiffness valuesare kept low, the long-period offsets contribute signifi-cantly to the largest responses. For the range of stiffnesscoefficients tested and the chosen force spectrum, themaximum response amplitudes occur around C

Fm0= 0.06

and√K/m

ω < 0.05.• However, as Coulomb damping is further increased, the

maximum motions gradually drop off. Both the excitationfrequency responses and the low-frequency responsesdiminish under larger Coulomb damping. Note in passingthat in the limit as Coulomb damping becomes very large,the device would remain motionless, as the dampingcould not be overcome by the excitation forces throughoutthe whole storm/simulation duration.

Statistics other than those related to the maximum responsescan be calculated. Figure 18 show the mean of the absolutevalue of the response signal |Z(t)| . Apart from very weaksprings, increased Coulomb damping leads to reduced meanresponses. This is rather different to the dependence observedfor maximum responses in Figures 14 and 15.

0.05 0.1 0.15 0.2 0.25 0.3

!n / !

0

0.05

0.1

0.15

0.2

C/F

m0

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

0.1210

40

70

100

150

200

300

400

500

700

1000

K (N/m)

0

100

200

300

400

500

600

700

800

C (

N)

Fig. 18: Contour plot of the mean of the absolute value of theresponse timeseries (averaged across 25 realisations).

The probability density functions (pdf) of the response,under different stiffness and Coulomb values, are displayedin Figures 19 and 20. Note that we have analysed the absolutevalue of the response (from all 25 realisations), as in oursimplified model there is no difference between positive andnegative responses. The bulk of the pdfs are shown in Figure19, and the larger responses and their associated probabilitiesare displayed on a logarithmic scale in Figure 20. For the caseswith no Coulomb damping, the theoretical folded normal pdfis applicable (with 0 mean and variance given by the zero-order moment of the response spectra calculated via an RAOfrom Equation 3). The theoretical pdf is shown in thick greylines, and is seen to align with the calculated pdfs under

0

2

4

6

8

10pd

fK=10 a)

0

5

10

15

20

25

30

pdf

K=200 b)

0 0.05 0.1 0.15 0.2 0.25

Z (m)

0

10

20

30

40

pdf

K=1000 c)

C=0C=200C=400C=600C=800

Fig. 19: Probability density function of the absolute value ofthe response signal |Z(t)|. Each sub-plot is for a given stiffnesscoefficient, with the thick grey line denoting the theoreticalfolded normal distribution.

no Coulomb damping (red curves), as expected. From thetwo top sub-plots, Figures 19a) and 20a), we see that undervery low stiffness values, the mean responses can be eitherlarger or smaller, than the situation with no Coulomb damping.However, the largest responses are more likely under non-zeroCoulomb damping. From the other four sub-plots it followsthat increasing Coulomb damping significantly reduces themean responses. However, the tails of the pdfs suggest thatthe extreme responses do not necessarily diminish also. Thiswould suggest that Coulomb damping appears more efficientin reducing mean responses than in reducing maximum re-sponses. Similar behaviour was observed by [10].

V. DISCUSSION

In the previous section it was shown that, in the rangeconsidered, it does not appear to be necessary, or desirable, tomaximise spring stiffness and Coulomb damping to minimise

10-4

10-2

100

pdf

K=10 a)

10-4

10-2

100

pdf

K=200 b)

0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6

Z (m)

10-4

10-2

100

pdf

K=1000 c)

Fig. 20: Probability density function of the absolute value ofthe response signal |Z(t)|, zoomed in on the function tails.Each sub-plot is for a given stiffness coefficient, with the thickgrey line denoting the theoretical folded normal distribution.

the largest displacements. For a given spring stiffness itappears that the maximum displacements first increase as C isincreased over some range, and then decrease (and in the limitthere is no motion at all). The lowest maximum responsesare under no/very minimal Coulomb damping and a (weak)detuned spring. In addition, the largest motions can also bereduced under very large Coulomb damping, which, however,might be impractical due to the associated large mechanicalforces.

The solution with no Coulomb damping gives an upperbound on the responses in the excitation frequency range. Inthis work we have not been able to identify a correspondingbound on the long-period offset responses.

The model used in this paper is a very simplified rep-resentation of a submerged WEC. Numerous extensions areplanned, including introducing linear hydrodynamics such thata memory effect is present and the exciting force is based

on a realistic spectrum and force transfer function. Nonlineareffects could also be included - for example, a simple depthdependent exciting force as implemented by [11] in their studyof vertical instability. Ultimately highly nonlinear phenomenacould also be included, but it then becomes very difficult toisolate the effects of each parameter. Extension to a 3 degrees-of-freedom model (for surge, heave and pitch) is also intended.

VI. CONCLUSION

In this study behaviour of a mass-spring system underCoulomb and linear damping is explored. The investigationis motivated by experimentally observed heave motions of asubmerged WEC with Coulomb damping force in the PTO,which exhibit large offsets from the mean position. First, asemi-analytical solution is presented, which is composed ofclosed form expressions. Three regimes of motion are foundto be possible. The solution is used to compute responsesunder harmonic excitation, and then excitation by a range offrequencies is pursued. A large parameter space of differentstiffness coefficients and Coulomb damping values is explored,though only a single value of linear damping coefficient isconsidered. Multiple realisations are carried out with the fastsemi-analytical solver to collect response statistics. We inves-tigate the duration of each simulation during which ‘stops’occur, frequency filter the motions to isolate the low-frequencyand wave-frequency behaviours, and ultimately consider thelargest and the mean excursions. The low-frequency offsetsare most prevalent under low stiffness coefficients, and areseen to influence both the extreme and the mean behaviour.For the range of parameters studied, the maximum excursionsoccur under no/very minimal Coulomb damping and a weakspring, under which the low-frequency offsets are minimal.Of course, in the limit as Coulomb damping is very large,responses will also be minimised, but this might not bepractical. Qualitative agreement with the trends derived fromthe numerical simulations is provided from a limited range ofexperimental data.

ACKNOWLEDGMENT

This research was supported under Australian ResearchCouncil’s Linkage Project scheme (LP150100598). Provisionof the experimental data from Carnegie Clean Energy isgratefully acknowledged.

REFERENCES

[1] J. D. Hartog, “Lxxiii. Forced vibrations with combined viscous andCoulomb damping,” The London, Edinburgh, and Dublin PhilosophicalMagazine and Journal of Science, vol. 9, no. 59, pp. 801–817, 1930.[Online]. Available: http://dx.doi.org/10.1080/14786443008565051

[2] H.-K. Hong and C.-S. Liu, “Non-sticking oscillation formulae forCoulomb friction under harmonic loading,” Journal of Sound andVibration, vol. 244, no. 5, pp. 883 – 898, 2001. [Online]. Available:http://www.sciencedirect.com/science/article/pii/S0022460X01935199

[3] P. C. Vicente, A. F. de O Falcao, and P. A. Justino, “Nonlineardynamics of a tightly moored point-absorber wave energy converter,”Ocean Engineering, vol. 59, pp. 20 – 36, 2013. [Online]. Available:http://www.sciencedirect.com/science/article/pii/S0029801812004179

[4] A. F. de O Falcao, “Phase control through load control of oscillating-body wave energy converters with hydraulic PTO system,” OceanEngineering, vol. 35, no. 3, pp. 358–366, 2008.

[5] A. Babarit, M. Guglielmi, and A. H. Clement, “Declutching control of awave energy converter,” Ocean Engineering, vol. 36, no. 12, pp. 1015–1024, 2009.

[6] J. Orszaghova, A. Rafiee, H. Wolgamot, S. Draper, and P. H. Taylor, “Ex-perimental study of extreme responses of a point absorber wave energyconverter,” in Australasian Fluid Mechanics Conference (AFMC2016),2016.

[7] A. Rafiee, H. Wolgamot, S. Draper, J. Orszaghova, J. Fievez, andT. Sawyer, “Identifying the design wave group for the extreme responseof a point absorber wave energy converter,” in Asian Wave and TidalEnergy Conference (AWTEC2016), 2016.

[8] S. Shaw, “On the dynamic response of a system with dry friction,”Journal of Sound and Vibration, vol. 108, no. 2, pp. 305 – 325,1986. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S0022460X8680058X

[9] H.-K. Hong and C.-S. Liu, “Coulomb friction oscillator: Modelling andresponses to harmonic loads and base excitations,” Journal of Sound andVibration, vol. 229, no. 5, pp. 1171 – 1192, 2000. [Online]. Available:http://www.sciencedirect.com/science/article/pii/S0022460X99925944

[10] J. M. Johnsen and A. Naess, “The effect of Coulomb damping onthe response statistics of a SDOF oscillator subjected to white noiseexcitation,” in EURODYN93 - Structural Dynamics, 1993, pp. 735 – 42.

[11] R. Eatock Taylor and J. Knoop, “Dynamic instability of a freely floatingplatform in waves,” in Design for Dynamic Loading - The Use of ModelAnalysis, 1981, pp. 57–62.