A novel point absorber wave energy converter with a tuned inertial mass (TIM), which is capable of signiﬁcantly increasing the energy absorption and broadening the eﬀective bandwidth, is proposed in this paper. The mechanism of the TIM has originally been introduced in the ﬁeld of civil engineering as a passive energy absorber for structures subjected to external loadings such as earthquakes. It re- lies on attaching an additional tuning spring and a rotational inertial mass to the primary system, to improve the energy absorption performance by amplifying the displacement of the damper. Thus, considering typical point absorbers modeled as a mass-spring-dashpot system similar way to civil structures, the application of the TIM to wave energy converters can be expected to have a signiﬁcant ef- fect. In this paper, numerical investigation on the power generation performance of a point absorber with the TIM is conducted under random sea waves. The am- plitude response and power generation performance are compared with the con- ventional point absorber, considering both non-resonant and resonant buoy cases. It is shown that by properly designing the tuning spring sti ﬀness and generator damping, the rotation of the generator can be ampliﬁed compared to the buoy, increasing the power absorption drastically. Keywords: Wave energy converter, Point absorber, Tuned inertial mass, Random sea waves 1. Introduction First stimulated by the oil crisis 40 years ago, harvesting energy from ocean waves is now considered as an alternative approach to meet renewable energy targets [1, 2]. The enormous wave power potential has attracted engineers world- wide with more than 1,000 patents in Japan, North America, and Europe [3], Preprint submitted to Energy April 19, 2020 and among them are oscillating water column, oscillating body, and overtopping devices [4, 5, 6, 7, 8, 9, 10]. Out of these, the point absorber classiﬁed as the os- cillating body, is considered one of the most promising concepts which make up a great proportion of existing full-scale prototypes. It consists of a ﬂoater with di- mensions much smaller than the incident wave length, which converts the motion of ocean waves to produce power. Compared to other concepts, point absorbers can exploit more powerful wave regimes available in deep o ﬀshore regions, and absorbs energy from all directions. Moreover, the buoys can be combined in ar- rays to produce su ﬃcient energy for commercial exploitation. Until now, a great amount of e ﬀort has been devoted to improving the e ﬃ- ciency of point absorbers based on mathematical models [11]. It is well known that a point absorber in resonance with the incoming wave has signiﬁcantly high power absorption due to its enhanced amplitude [12]. However, in practice, this frequency matching meets with serious di ﬃculty because real waves are not sin- gle frequency, and the natural frequency of the buoy tends to be much higher than typical ocean wave frequencies for a reasonable size buoy [4, 13]. Therefore, methods to force the system into resonance via active control have been consid- ered. Latching the point absorber at ﬁxed positions to achieve phase control was ﬁrst proposed by Budal and Falnes [14] and the phase control strategies includ- ing latching control and reactive control continue to be an active area of research [15, 16, 17, 18, 19, 20]. This approach requires the prediction of the incoming wave some time in the future, along with solving the complicated control problem to determine the latched time intervals. On the other hand, methods to simply engineer the frequency response of the point absorber have also been considered. Shadmen et al. [21] presented a geometrical optimization method based on tuning the system to oscillate in the range of predominant sea states. Engstrom et al. [22] presented phase control by using a supplementary submerged body to shift the natural frequency of the buoy to coincide with typical sea states. Both of these studies discussed that enhanced amplitude response of the buoy leads to a narrow resonance bandwidth which is in conﬂict with the desire to correspond to the wide spectrum of ocean waves. Thus, the compromise between maximum power ab- sorption and broad resonance bandwidth has always been an issue associated with the optimization of point absorbers. In this study, a point absorber with a tuned inertial mass (TIM) is proposed to increase the power absorption and broaden the e ﬀective wave frequency range. The TIM has a di ﬀerent conﬁguration from the traditional tuned mass damper (TMD) [23] and the mechanism has originally been introduced in the ﬁeld of civil engineering as a passive vibration control device for structures subjected to exter- 2 nal loadings such as earthquakes [24, 25, 26]. It relies on attaching an additional tuning spring and a rotational inertial mass to the primary system, to amplify the displacement of the viscous damper by taking advantage of the resonance e ﬀect. In the theory of point absorbers, it is typical to model the power take-o ﬀ(PTO) system as a linear damper [27], resulting in a mass-spring-dashpot system. Hence, the application of the TIM to point absorbers can be expected to have a signiﬁcant eﬀect in terms of energy absorption. In contrast to previous studies which couple the PTO with the buoy’s oscillation, this study inserts the tuning spring which de- couples the PTO from the buoy’s oscillation and couples with the oscillation of the rotational inertial mass. Therefore, by appropriately designing the tuning spring, rotational inertial mass, and damping coe ﬃcient, the rotation number of the rota- tional inertial mass can be ampliﬁed compared to the buoy, which increases the displacement of the PTO to achieve substantial power absorption. The objective of this paper is to verify the e ﬀectiveness of applying the TIM to a point absorber. First, in Section 2, the theoretical models of the conventional system and proposed system with the TIM are developed, following the modeling of stochastic sea states using the JONSWAP spectrum. Then, in Section 3, the state-space representation is given to evaluate the amplitude response along with the power generation. Finally, in Section 4, numerical studies are carried out fo- cussing on the optimization of design parameters, amplitude response, and power generation of the proposed system compared with the conventional system. 2. Modeling 2.1. Point absorber with tuned inertial mass We begin by reviewing the conventional point absorber illustrated in Fig. 1(a). A semi-submerged cylinder with diameter Dand draft Lis considered as the wave interacting part of a point absorber. The generator is anchored rigidly to the ocean ﬂoor, and the supporting spring with sti ﬀness ksis used to prevent the wire from sagging. For simplicity, we consider the heave motion only as this becomes dom- inant for the power extraction of wave energy [28]. Letzbe the vertical displacement of the buoy from its equilibrium state. Since the displacement of the buoy is coupled with the rotation of the generator, the con- ventional point absorber can be modeled as a single-degree-of freedom (SDOF) system with the following equation of motion: (M+ms)¨z+cs˙z+ksz=fwu (1) 3 ks (a) (b)csmsM CPTOLD Generator ksSupporting spring fw zFigure 1: Conventional point absorber (SDOF system): (a) Schematic illustration, (b) Model. kt (b) LMD kscsmsTuning spring (a)kskt Rotational CPTO inertial massfw z zs Figure 2: Point absorber with tuned inertial mass (TIM system): (a) Schematic illustration, (b) Model. where Mis the buoy mass, msis the inertial mass of the generator and pulley, cs is the mechanical damping including friction, uis the electromagnetic damping force which acts as a control force given by u=CPTO˙z (2) andfwis the hydrodynamic force acting on the cylinder. The corresponding model is shown in Fig. 1(b). Based on linear potential wave theory, the hydrodynamic force fwis described by fw=fa+fb+fc (3) 4 where fais the excitation force, fbis the hydrostatic force due to buoyancy, and fcis the radiation force. The relation between the excitation force faand the amplitude of the incident wave ais given in the frequency domain using a transfer function Fa(!) as ˆfa(!)=Fa(!)ˆa(!) (4) Note that ˆf(!) denotes the Fourier transform of a function f(t). The hydrostatic force becomes a linear function of zgiven as fb=Kwz (5) where Kwis given with the gravity acceleration gand sea water density by Kw=g(D 2)2 (6) and the radiation force fcis given as ˆfc(!)=(j!ma(!)+cr(!))ˆ˙z (7) where ma(!) is the added mass and cr(!) is the radiation damping. Next, to increase the power absorption and widen the e ﬀective bandwidth, a point absorber with TIM is proposed. As illustrated in Fig. 2(a), a tuning spring is installed between the buoy and generator, and a relatively small phys- ical mass which rotates with the generator is attached. When the vertical motion is transformed into rotary motion, the small physical mass can produce an ampli- ﬁed equivalent mass e ﬀect on the order of a thousandfold due to the rotary inertia [24]. As shown in Fig. 2(b), the proposed system is modeled as a two-degree- of-freedom (2DOF) system, noting that the rotational displacement of the inertial mass is coupled with the generator. Deﬁning z,zs, and ztas the displacement of the buoy, rotational inertial mass, and the elongation of the tuning spring, respectively, the relationship among these variables are given by z=zs+zt (8) The equations of motion for the proposed TIM system are derived as M¨z=fwft (9) ms¨zs+cs˙zs+kszs=ftu (10) 5 where ftis the force from the tuning spring and the the electromagnetic damping force is deﬁned by u=CPTO˙zs (11) in this case. Also, denoting the sti ﬀness of the tuning spring as kt, this force becomes proportional to the relative displacement given as ft=ktzt=kt(zzs) (12) 2.2. Power take-o ﬀsystem In this study, the generator is assumed to be a three-phase permanent mag- net synchronous machine, interfaced with a centralized DC power bus. For this assumption, it is reasonable to assume that the quadrature ﬁeld is explicitly con- trolled to regulate power conversion while the direct stator ﬁeld is maintained at zero. Thus, the three phase voltage and current vectors are transformed to ”quadrature components”, i.e., e ﬀective scalar quantities are used for the back- EMF vand corresponding current i. More details can be found in [29, 30]. As- suming an ideal generator with linear behavior and minimal core loss, results in linearity between the back-EMF and the velocity coupled with the generator. Therefore, the equation is deﬁned separately for the conventional system and TIM system, given as v=Ke˙z;v=Ke˙zs (13) respectively, where Keis a constant associated with the back-EMF of the gen- erator. By reciprocity, the electromagnetic force and generator current has the following linear relationship u=Kei (14) In the study of wave energy converters (WECs), it is common to assume vis- cous dampers in place of the PTO system. This constitutes the imposition of a feedback law i=Yv (15) where Yis the admittance of the generator. Applying this to Eq. (14) with Eq. (13) yields u=YK2 e˙z;u=YK2 e˙zs (16) for the individual systems. Thus the generator damping CPTOfor both cases can be given as CPTO=YK2 e (17) expressing how the generator damping, i.e., the electromagnetic damping force u is controlled by the admittance Y. 6 2.3. Stochastic sea state model In simple theoretical models of WECs, it is typical to assume the incident waves to be regular. For a more realistic model, irregular waves are used with time-domain analysis which requires much more computing time [4]. An alter- native method with less computation for modeling true sea states is the stochastic modeling. We assume the wave amplitude a(t) to be a stationary stochastic pro- cess with spectral density Sa(!) and variance given by a2=1 2∫1 1Sa(!)d! (18) Characterizing Sa(!) by the JONSWAP spectrum [31] with peak wave period Tp, signiﬁcant wave height Hs, and peak enhancement factor derives the following Sa(!)=310H2 s T4 p!5exp[944 T4 p!4] Y(19) where Y=exp26666640BBBB@0:191!Tp1p 2ϕ1CCCCA23777775 (20) and ϕ=8>><>>:0:07 : !Tp5:24 0:09 : !Tp>5:24(21) The peak enhancement factor is constrained by 1  3:3, with =1 describ- ing a fully developed sea. To model the wave amplitude, we ﬁnd a ﬁnite-dimensional noise ﬁlter Fw[AwBw Cw0] (22) such that its power spectrum is close to the JONSWAP spectrum, i.e., Sa(!) jFw(!)j2, for a unit intensity white noise input. It should be noted that we make use of the short-hand G[AB CD] to imply G(s)=C[sIA]1B+Din this article. According to the simpliﬁed procedure advocated by Spanos [32], Fwcan be approximated by a fourth-order controllable canonical form of Aw=2666666666666640 1 0 0 0 0 1 0 0 0 0 1 a1a2a3a4377777777777775;Bw=2666666666666640 0 0 1377777777777775;Cw=[ 0 0 c30] (23) 7 0 1 2 3 400.10.20.30.40.5 ω (rad/sec) S ( ω) (m s) a2Figure 3: Example of Jonswap spectrum with Tp=6 s,Hs=1 m, =1 (solid) and the spectrum of the fourth order ﬁnite dimensional approximate system (dashed). where the ﬁlter parameters a1,a2,a3,a4, and c3are chosen to minimize the mean- square error∫1 1(Sa(!) jFw(!)j2)2d!, while constraining a1through a4so that the system poles are in the open left half plane. Fig. 3 shows an example of a JONSWAP spectrum for Tp=6 s,Hs=1 m, =1 and its fourth-order ﬁnite- dimensional approximate system. We can conﬁrm in the ﬁgure that the fourth- order Fwestimates the JONSWAP spectrum very well. 3. State-space representation In this section, to evaluate the amplitude response and assess the power gener- ation for stochastic sea states, a state-space form of the proposed device with TIM is developed. The derivation of the state-space form for the conventional point absorber can be found in [33], which is not delved into in this paper. 3.1. Amplitude response of buoy and rotational inertial mass In the following, the equations of motion for the buoy and rotational inertial mass are expressed in state-space form separately, and then augmented to assess the amplitude response relating the wave elevation. 8 Taking into account the hydrodynamics acting on the buoy, we take the Fourier transform of Eq. (9) with Eqs. (3), (4), (5), and (7) inserted, which gives the transfer function relating aandfttozas ˆz=Ga(!)ˆa+Gf(!)ˆft (24) where Ga(!)=Fa(!) !2(M+ma(!))+i!cr(!)+Kw(25) Gf(!)=1 !2(M+ma(!))+i!cr(!)+Kw(26) The inﬁnite-dimensional systems GaandGfare approximated into ﬁnite-dimensional systems, i.e., Ga[AaBa Ca0] ;Gf[AfBf Cf0] (27) We must note, the function Fa(!) in Eq. (25) is noncausal which will be problem- atic when approximating Ga(!) by a ﬁnite-dimensional state-space. Therefore the technique of spatial delay proposed by Falnes [34] is used, deﬁning aas the wave amplitude at a distance of din front of the buoy. Once Eq. (27) is obtained, the identiﬁed systems are augmented as [ GaGf] [AmBmEm Cm0 0] (28) where the augmented matrices Am,Bm,Em, and Cmare Am=[Aa0 0 A f] ;Bm=[Ba 0] ;Em=[0 Bf] ;Cm=[ CaCf] (29) and the equivalent state-space representation is ˙xm=Amxm+Bma+Emft (30) z=Cmxm (31) On the other hand, the equation of motion for the rotational inertial mass in Eq. (10) can be transformed into state-space form directly. Deﬁning the state vector as xn=[ zs˙zs]Tgives ˙xn=Anxn+Bni+Enft (32) 9 where An=[0 1 ks mscs ms] ;Bn=[0 Ke ms] ;En=[0 1 ms] (33) Now that the state-space form for the buoy and rotational inertial mass are determined, Eqs. (30) and (32) can be augmented by deﬁning xh=[ xT mxT n]Tas ˙xh(t)=Ahxh+Bhi+Ehft+Gha (34) where Ah=[Am0 0 A n] ;Bh=[0 Bn] ;Eh=[Em En] ;Gh=[Bm 0] (35) The generator current and the force from the supporting spring can be expressed using the state variable xhas i=Yv=YChxh (36) ft=kt(zzs)=ktThxh (37) where Ch=[ 0Ke] ;Th=[ Cm1 0] (38) Thus, when the design parameters Yandktare determined, the amplitude response of the buoy and rotational inertial mass to the wave elevation can be assessed by deﬁning a transfer function for each. Substituting Eqs. (36) and (37) into Eq. (34) yields the closed loop system ˙xh=(AYBhCh+ktEhTh)xh+Gha (39) z=Cbxh (40) zs=Crxh (41) where Cb=[ Cm0] ;Cr=[ 01 0] (42) Therefore, the transfer function from the wave elevation to the buoy displacement is given as Hb(s)=Cb[sIA+YBhChktEhTh]1Gh (43) Similarly, the transfer function from the wave elevation to the rotational displace- ment of the inertial mass is given as Hr(s)=Cr[sIA+YBhChktEhTh]1Gh (44) 10 3.2. Power generation The stochastic sea state model in Eq. (22) combined with the WEC dy- namics in Eq. (34) yields the following augmented system, with state vector x=[ xT hxT w]Tas ˙x=Ax+Bi+Fft+Gw (45) where A=[AhGhCw 0 A w] ;B=[Bh 0] ;F=[Eh 0] ;G=[0 Bw] (46) andw(t), the input to the noise ﬁlter, is white noise with unit intensity. Eqs. (36) and (37) can be rewritten as i=Yv=YCx (47) ft=kt(zzs)=ktTx (48) where C=[ Ch0] ;T=[ Th0] (49) In this paper, the total power generation is deﬁned as the extracted power minus the electrical loss [29], i.e., Pg=ivPd (50) where Pdis the electrical loss dominated by conductive dissipation. While the expression for Pddepends on many parameters of the electronic hardware, we make the simplifying assumption that the current-dependent loss is resistive, i.e., Pd=Ri2(51) where Rincludes the stator coil resistance of the generator and an approximate transmission resistance for the drive. From v=Cxand Eq. (51), Eq. (50) would be Pg=fiv+Ri2g =[x i]T[01 2CT 1 2C R] [x i] (52) To maximize the averaged value of Eq. (52) denoted by ¯Pg, the admittance Yand the tuning spring sti ﬀness ktare treated as control gains subjected to a white noise 11 input as in [35]. Substituting Eqs. (47) and (48) into Eq. (45) yields the closed loop system ˙x=(AYBC+ktFT)x+Gw (53) In this study, the static admittance control strategy introduced in [33] is utilized for the current i. Thus both Yandktare treated as constant values. Therefore in stationary stochastic response, the average power generation ¯Pgis given by [36] ¯Pg=GTSG (54) where the covariance matrix S="[xxT] is computed as the solution to the Lya- punov equation (AYBC+ktFT)TS+S(AYBC+ktFT)+CT(Y+Y2R)C=0 (55) For an ideal system, the admittance value to control the input current to the gen- erated is restricted by Y2[0;1=R] (56) to assure the deﬁniteness of S. Thus, the objective becomes the optimization of Y andktto maximize ¯Pg. 4. Numerical simulations To validate the power generation performance of the point absorber with TIM proposed in Fig. 2, numerical studies are carried out and compared with the con- ventional SDOF system illustrated in Fig. 1. A reasonable size cylindrical buoy with diameter 5 m, draft 1 m, in water depth 30 m is considered. The hydro- dynamic coe ﬃcients are calculated with the software WAMIT [37] which uses boundary element method based on linear potential theory. The resulting added mass ma(!), radiation damping cr(!), and the magnitude and phase of the transfer function Fa(!) deﬁned by Eq. (4) are shown in Fig. 4. The parameters for the generator and buoy used in this study are summarized in Table 1. Fig. 5 shows the frequency response data GaandGfgiven by Eq. (25) and (26), respectively, and the ﬁnite dimensional approximations expressed by Eq. (27). For accuracy, Gais approximated with 6 zeros and 7 poles, and Gfis ap- proximated with 3 zeros and 4 poles. The wave amplitude ais taken to be the wave amplitude d=10 m ahead of the buoy in the propagation direction. It can be seen that the approximated model matches the frequency response data very well over the dominant wave frequencies. 12 Table 1: Parameter values used for numerical simulation studies Parameters Values Buoy mass M 4000 kg Buoy diameter D 5 m Buoy height H 2 m Buoy draft L 1 m Mechanical damping cs 50 N s /m Supporting spring sti ﬀness ks 1:0103N/m Electromechanical coupling Ke 500 V s /m Resistance R 25Ω Sea water density  1027 kg =m3 Previous studies which consider irregular waves typically select some number of sea states as representatives of a deployment site. While, in this study, a more comprehensive analysis is carried out, focussing on the variation of peak wave periods since the system’s amplitude response strongly depends on this character- istic parameter. Therefore, stochastic sea state models are determined for each sea state with peak wave period ranging from 2 s to 12 s and Hs=1 m, =1 for the JONSWAP spectrum deﬁned by Eq. (19). 4.1. TIM system 4.1.1. Design For the proposed TIM system, the design parameters Yandktmust be decided for the rotational inertial mass employed. In this study, a rotational inertial mass of ms=8264 kg is examined which corresponds to 30% of the e ﬀective oscillating mass M+m1where m1denotes the inﬁnite frequency added mass having the value of 23,547 kg. This large inertial mass can be realized by a relatively small physical mass due to the rotational amplifying e ﬀect [24]. It is well understood that the damping coe ﬃcient CPTOstrongly a ﬀects the magnitude of the generated power. This can be adjusted through the admittance Yof the electronics, thus the damping can be optimized for each sea state. On the other hand, adjusting the sti ﬀness of the tuning spring after deployment is regarded as unpractical. Therefore, the value for ktis decided so that the power generation is maximized for the predominant sea state assumed as JONSWAP spectrum with Tp=6 s. The optimized value of ktis obtained through the contour plot shown in 13 0 1 2 3 4010,00020,00030,00040,00050,000ma(kg) 0 1 2 3 4010,00020,00030,000cr(N s/m) 0 1 2 3 4050,000100,000150,000200,000| F a | (N/m) 0 1 2 3 400.40.81.2Fa (rad) (a) (b) (c) (d)ω (rad/sec) ω (rad/sec) ω (rad/sec) ω (rad/sec) Figure 4: Hydrodynamic parameters for the heave mode of a circular cylinder with radius 2.5 m, draft 1 m, in water depth 30 m: (a) Added mass, (b) Radiation damping, (c) Magnitude of Fa(!), (d) Phase of Fa(!). Fig. 6, which demonstrates how the admittance and tuning spring sti ﬀness relate to the power generation. The upper limit of ktis set to 1 :0105N/m which is large enough to consider the spring to be rigid, while the admittance is constrained by Eq. (56). The contour plot shows a clear peak at kt=1:72104N/m and Y= 0.0044 Ω1giving maximum power of ¯Pg=1795 W. Hence, the optimum tuning spring sti ﬀness is set to k t=1:72104N/m and held ﬁxed for the varying sea states. In the following simulation, the e ﬀect of adjusting the rotational inertial mass after deployment is also explored. Therefore, two more values, ms=5509 kg 14 012 0 1 2 3 4-20-10001210 -5 0 1 2 3 4024× ω (rad/sec) ω (rad/sec) Ga (rad) | G |a Gf (rad) | G |f (a) (b)Figure 5: Frequency domain data (solid) and ﬁnite-dimensional approximation (dashed) with spa- tial delay of d=10 m: (a) Ga(!), (b) Gf(!). and 11019 kg corresponding to 20% and 40% of the e ﬀective oscillating mass are considered. Note, the value of mscan be adjusted by either changing the attached mass or the radius in which it rotates. 4.1.2. Result The magnitudes of the transfer functions HbandHrdeﬁned by Eqs. (43) and (44) with the obtained optimum value k t=1:72104N/m are compared in Fig. 7. The damping coe ﬃcient is set to the optimum value for Tp=6 s in each plot. For the SDOF system, an inertial mass of ms=50 kg caused by the generator and pulley is assumed. Fig. 7 shows that the amplitude response of the rotational inertial mass is signiﬁcantly higher than the buoy response, indicating that the tuning spring ampliﬁes the oscillation of the rotational inertial mass compared to the buoy. This leads to increased amplitude and velocity of the generator move- ment due to its rigid connection with the rotational inertial mass. Moreover, the frequency for optimum amplitude response of the TIM system decreases with in- creasing ms. This points out that even when the buoy’s natural frequency is apart from the dominant wave frequency, enhanced amplitude response can be gained, and shifting of the frequency is possible through the value of the rotational inertial mass. Fig. 8 shows the power generation for various sea states, with the optimum damping coe ﬃcient given below. It can be clearly seen that all three TIM systems 15 0 0.04500 0.03 10 1,000 81,500 0.02 6 k (N/m)t10 42,000 40.012 0ʷ Y ( )-1 P (W) g_Figure 6: Contour plot of power generation showing the e ﬀects of the tuning spring sti ﬀness kt and admittance Y. The plots are for ms=8264 kg with sea state Tp=6 s,Hs=1 m, =1. perform better than the SDOF system, due to the increased amplitude and velocity of the rotational inertial mass and generator. This results in lowering the optimum damping coe ﬃcient, opening up the possibility to have small generator units. Fur- thermore, when the value of msis changed, the peak of the power curve is shifted, which indicates that adjusting the rotational inertial mass allows the TIM system to correspond to the variability of Tp. 4.2. Phase controlled TIM system 4.2.1. Design Considering the signiﬁcant e ﬀect of the TIM system observed from the am- plitude response, a superior e ﬀect can be expected for when the oscillation of the buoy itself is large, i.e., when the buoy resonates with the wave. In the following example, the e ﬀectiveness of the TIM system is validated for when the buoy’s natural frequency coincides with the dominant wave period. To shift the heave os- cillating period, we assume phase control is applied by adding a fully submerged body [22, 38]. The supplementary inertia due to the submerged body mass and added mass allows the buoy to resonate with the dominating sea state. By placing the submerged body at su ﬃcient depth, the radiation damping and excitation force remains the same with the case of the single cylinder, which is explored before. 16 0 1 2 3 40510 15 TIM-s, ms = 5509 kg TIM-s, m s = 8264 kg TIM-s, ms = 11019 kg SDOF-s 0 1 2 3 40510 15 | H | b | H | r (a) (b)ω (rad/sec) ω (rad/sec) Figure 7: The amplitude of the transfer functions when the tuning spring sti ﬀness is k t=1:72104 N/m and the damping coe ﬃcient CPTOis set to the optimum value for Tp=6 s, (a) Hb(!), (b) Hr(!). The performance of the TIM system is compared with the SDOF system, hav- ing the buoy for both systems to be in resonance with the wave motion. Thus, the supplementary inertia due to the submerged body mass and added mass is set to 200103kg to make the buoy’s oscillation tuned to the sea state of Tp=6 s. Fig. 9 shows the ﬁnite dimensional approximation of the frequency response data GaandGfwith the same orders, displaying a distinct peak at 0.9 rad /s due to the resonance of the buoy. For the rotational inertial mass, the same three values (i.e., ms=5509 kg, ms=8264 kg, and ms=11019 kg) are considered. The optimiza- tion of the design parameters are carried out in a similar process, following the amplitude response and power generation. The sti ﬀness of the tuning spring ktis optimized through the contour plot given in Fig. 10. A sharper peak can be observed at kt=6:06103N/m and Y=0:0036 Ω1giving maximum power of ¯Pg=17:1 kW. Hence, the optimum tuning spring stiﬀness is set to k t=6:06103N/m. 4.2.2. Result The amplitude response of the buoy and the rotational inertial mass are shown in Fig. 11. The solid line in Fig. 11(a) shows that the amplitude response of 17 010002000300040005000 TIM-s, ms = 5509 kg TIM-s, ms = 8264 kg TIM-s, ms = 11019 kg SDOF-s 2 3 4 5 6 7 8 9 10 11 12 010002000300040005000CPTO (N s/m) ˎ T (sec)pP (W) g_Figure 8: Power generation with optimum damping for each sea state. The tuning spring for the TIM system is set to k t=1:72104N/m. the SDOF buoy is greatly enhanced due to phase control, while the bandwidth has become narrow. This has always been an issue associated with the design of point absorbers since a compromise must be made between maximum amplitude response and wide resonance bandwidth over the dominant wave frequencies. On the other hand, the signiﬁcant e ﬀect of the TIM system is that enhanced ampli- tude response and widened bandwidth is gained at the same time as shown in Fig. 11(b). Especially for the TIM system with ms=8264 kg in which the tuning spring was optimized, the beneﬁt of 2DOF is gained, considerably widening the bandwidth. We must emphasize that for the proposed TIM system, increased am- plitude response of the rotational inertial mass is desired since the generator is coupled to its displacement. While the buoy’s amplitude response is reduced for 18 012310 - 4 0 1 2 3 401234× ω (rad/sec) ω (rad/sec) (a) (b)Gf (rad) | G |f 020 40 60 0 1 2 3 4-20-100Ga (rad) | G | aFigure 9: Frequency domain data (solid) and ﬁnite-dimensional approximation (dashed) with spa- tial delay of d=10 m: (a) Ga(!), (b) Gf(!) the TIM system, the amplitude response of the rotational inertial mass is signiﬁ- cantly increased. This indicates that a large amount of energy from the wave to the buoy is successfully transferred to the rotational inertial mass for the TIM system. The power generation of the two systems with phase control is compared in Fig. 12. A similar feature with the amplitude response can be observed, with the TIM system having increased power generation and widened bandwidth. Particu- larly for the sea state in which resonance is gained, the power generation for the TIM system is increased to more than two times the power of the SDOF system. Contrary to the non-resonant case, the e ﬀect of adjusting the rotational inertial mass can not be observed, which indicates that designing ktfor a particular ms value is su ﬃcient to gain the beneﬁt of the TIM system. As with previous results, a decrease in optimum damping coe ﬃcient can be observed for the TIM system, while the optimum damping is less sensitive to the variation of the sea states. 5. Conclusions A theoretical model for a point absorber with tuned inertial mass has been pro- posed. Numerical simulations in stochastic sea states using the JONSWAP spec- trum were carried out to compare the power generation of the proposed device with the conventional SDOF point absorber. For the TIM system with the buoy’s natural frequency apart from the dominant wave frequency, increased power gen- 19 0 0.045,000 0.03 10 10,000 815,000 0.026 10 420,000 4 0.0120ʷ Y ( )-1 k (N/m)tP (W) g_Figure 10: Contour plot of power generation showing the e ﬀects of the tuning spring sti ﬀness kt and admittance Y. The plots are for TIM system applied to a resonant buoy with ms=8264 kg and supplementary inertia of 200 t. Sea state is Tp=6 s,Hs=1 m, =1. eration was gained along with the shifting of the power curve depending on the rotational inertial mass value. On the other hand, for the TIM system with a res- onant buoy, a signiﬁcantly high amplitude response of the rotational inertial mass was gained, while the response of the buoy was reduced. This indicated that sub- stantial wave energy was transferred to the rotational inertial mass, leading to in- crease the power generation to more than two times the power of the conventional system at the resonance frequency. For future work, methods to determine the optimum rotational inertial mass should be established, along with its sensitivity to the wave spectrum. In addition, algorithms to control the sti ﬀness and damping of the PTO system with regard to the change of the sea state are desired. Furthermore, experimental investigation [39] to assess the feasibility of the device under real sea condition is required. This includes the consideration of locking the PTO system in survival mode and understanding the buoy response to highly non-linear extreme waves. 20 0.4 0.6 0.8 1 1.2 1.4 1.6010 20 30 40 TIM-s, ms = 5509 kg TIM-s, ms = 8264 kg TIM-s, ms = 11019 kg SDOF-s 0.4 0.6 0.8 1 1.2 1.4 1.6010 20 30 40 | H | r| H | b (a) (b)ω (rad/sec) ω (rad/sec) Figure 11: The amplitude of the transfer functions when the tuning spring sti ﬀness is k t=6:06 103N/m and the damping coe ﬃcient CPTOis set to the optimum value for Tp=6 s: (a) Hb(!), (b) Hr(!). Acknowledgement This research was partially supported by JSPS KAKENHI Grant number 18K189808 and Fundamental Research Developing Association for Shipbuilding and O ﬀ- shore. The authors would like to thank these ﬁnancial support. In addition, the authors would like to express their appreciation to Prof. Harumichi Kyotoh at University of Tsukuba for the encouragement and valuable advice.