In this study, the eﬀectiveness of an oscillating-body WEC with a tuned in- erter (TI) proposed by the authors is shown through wave ume testing. The TI mechanism consisting of a tuning spring, a rotational inertial mass, and a viscous damping component is able to increase energy absorption capabil- ity by taking advantage of the resonance eﬀect of the rotational mass. This mechanism has been recently introduced for civil structures subjected to ex- ternal loadings such as earthquakes and winds to decay vibration response immediately. The authors applied this mechanism to oscillating-body WECs and showed that the proposed WEC increased the power generation perfor- mance and broadened the eﬀective frequency range without increasing the mass of the buoy itself through numerical simulation studies. To verify the validity of the proposed WEC experimentally, a small-scale prototype of the proposed device is designed and wave ume testing is carried out with various regular wave inputs of diﬀerent frequencies. The results show that the WEC with the properly adjusted TI mechanism demonstrates better power gener- ation performance compared to the conventional WEC over a wide range of wave frequencies. Keywords: Wave energy converter, Tuned inerter, Wave ume testing, Renewable energy, Energy harvesting 1. Introduction The public interest in renewable energy has risen over the last decades. In this social situation, ocean wave energy has been considered to be one Preprint submitted to Applied Ocean Research March 23, 2020 of the promising renewable energy sources because of its tremendous energy potential [1] and many researchers have been devoting considerable eﬀort to improve the power take-oﬀ (PTO) system from the ocean wave [2, 3, 4, 5, 6]. Generally, the PTO systems for wave energy converters (WECs) can be clas- si ed into three categories, i.e., oscillating water columns, oscillating bodies, and overtopping devices. Among these WECs, the oscillating-body type which consists of a oating buoy has been expected to be a promising strat- egy because of its availability in deep oﬀshore regions and its extensibility by arraying many buoys. It is well-known that the power generation eﬃcacy of oscillating-body WECs is improved when the buoy is in resonance with the incoming wave. While, typically, the natural frequency of the buoy with reasonable size tends to be much higher than the frequencies of ocean waves [7]. Therefore, to take advantage of the full potential of oscillating-body WECs, the natural frequency of the buoy needs to be shifted closer to the ocean wave frequency. Toward that end, researchers and engineers have been trying to increase the mass of the buoy part, leading to the idea of a system of two objects consisting of a semi-submerged buoy and a lower submerged object which provides an additional inertia to the buoy part [8, 9, 10, 11, 12, 13]. This strategy has achieved certain results, however, the problem seems to lie in the fact that a huge additional mass needs to be submerged to adjust the natural frequency to the ocean wave frequency, which is not practical from the viewpoint of cost and workability. To address this problem, the authors proposed the oscillating-body WEC with a tuned inerter (TI) [14]. The concept of the inerter was originally proposed by Smith [15] and the inerter generates force proportional to the relative acceleration by using gears or ywheels. Also, WECs with the in- erter have been already proposed so far [16, 17, 18, 19, 20]. Based on the concept of inerter, the TI mechanism was proposed to protect civil struc- tures subjected to external loadings such as earthquakes and strong winds [21]. In this mechanism, a rotational inertial mass which produces inerter is combined with a tuning spring and a viscous damping component. Due to the tuning spring, the inerter part is decoupled from the primary struc- ture. Then, the tuning spring stiﬀness and the rotational inertial mass are designed so that the rotational inertial mass, not the mass of the primary structure, resonates with the external loading, leading to the improvement of the energy dissipation by the viscous damping component mounted in paral- lel with the rotational mass. Moreover, the advantage of the system is that 2 a signi cantly ampli ed mass eﬀect can be achieved easily from a relatively small physical mass due to a ball screw mechanism. Originally, in this TI mechanism, uid viscous dampers, which convert vibratory energy to heat, were supposed to be used. While, the authors utilized a motor as the vis- cous damping component, proposing the tuned inertial mass electromagnetic transducer (TIMET) [22, 23] to convert vibratory energy to electrical energy. So far, applications of this device not only to WECs but also to civil struc- tures [24, 25] have been investigated and the eﬀectiveness was shown through numerical simulation studies. The main objective of the present study is experimental veri cation of the TI WEC proposed by the authors. In particular, a small-scale prototype of the device is designed and wave ume testing is carried out with various regular wave inputs of diﬀerent frequencies. First, to explain the diﬀerence, the conventional oscillating-body WEC introduced in the literature and the TI WEC are described. Next, the analytical models are introduced and the equations of motion for both systems are developed. Then, the conditions of the wave ume testing are described and the obtained results are shown. In addition, numerical simulation studies are implemented to examine the validity of the developed models for the small-scale prototypes. Finally, con- clusions gained from this study follow. 2. Mechanism In this section, the mechanism for the proposed TI WEC is introduced by using the small-scale model used for the wave ume testing and its equation of motion is derived. For simplicity, the model is developed to restrict the motion to the heave direction only. 2.1. SDOF WEC Firstly, the mechanism of the conventional WEC [26] is reviewed brie y. When only the heave direction is considered, the conventional WEC can be modeled as a single-degree-freedom (SDOF) system. Therefore, the conven- tional WEC is called a SDOF WEC hereafter in the article. As illustrated in Figure 1(a), the SDOF WEC considered here consists of a oating cylindrical buoy, a three-phase motor, and a constant-force spring. The cylindrical buoy whose diameter is Dis interfaced with the motor through a tether which spools around a pulley of radius rpkeyed to the motor shaft. For practical reasons of conducting wave ume testing, the motor is installed above the 3 r z rpRLRLRLMotor BuoyConstant-force spring (a) (b)Mmscm2 z fw cm1 CPTO Figure 1: SDOF WEC: (a) Schematic illustration, (b) Model. water in this research as shown in Figure 1(a). However, even when the buoy is connected through a tether from the bottom of the buoy to the motor sub- merged under water and anchored to the ocean oor as in [26], the equation of motion developed below still holds. For practical reasons of conducting wave ume testing, the motor is in- stalled above the water. However, the equation of motion developed below holds for when the motor is submerged and anchored to the ocean oor, al- lowing the device to be installed oﬀshore. The constant-force spring, which exerts constant-force Fcfregardless of its deformation, is used to generate the tension of the tether to prevent the tether from sagging and slipping and to cancel part of the downward gravity force acted on the buoy. Therefore the magnitude of the constant-force spring aﬀects the equilibrium position of the buoy, however, it does not contribute to the equation of motion and the conceptual model developed below for the case of cylinder-shaped buoy. The SDOF WEC can be modeled as in Figure 1(b). Let zbe the dis- placement of the buoy. The upward direction is taken as positive and the static equilibrium state is the origin. Then, the equation of motion of the buoy would be (M+ms)z+ (cm1+cm2+CPTO) _z=fw (1) where Mis the mass of the buoy, msis the inertial mass including the motor and the pulley, CPTOis the damping coeﬃcient caused by the motor, and cm1 4 andcm2are the unavoidable mechanical dampings caused in the device. Even though the dampings cm1andcm2can be merged for the SDOF WEC model, these dampings are considered separately here because of the TI WEC model introduced next. In addition, fwis the hydrodynamic force, which is described by fw=fa+fb+fc (2) where fais the excitation force, fbis the hydrostatic force due to buoyancy, andfcis the radiation force. Let the Fourier transforms of faandfcbe denoted ^faand ^fc, respectively. The relation between the excitation force fa and the amplitude of the incident wave ais given in the frequency domain using a transfer function Fa(!) as ^fa(!) =Fa(!)^a(!) (3) where ^ a(!) is the Fourier transform of a. The hydrostatic force becomes a linear function of zexpressed as fb=Kwz (4) where Kwis given with the gravity acceleration gand water density by Kw=g(D 2)2 (5) and the radiation force fcis given as ^fc(!) =(j!m a(!) +cr(!))^_z (6) where ma(!) and cr(!) are the added mass and the radiation damping, re- spectively and are frequency dependent. Note that jis the imaginary unit satisfying j2=1. 2.2. WEC with tuned inerter Next, the mechanism of the proposed TI WEC illustrated in Figure 2(a) is introduced. In addition to the components of the SDOF WEC, the TI WEC has the tuning spring installed between the buoy and the pulley and the intentionally large rotational inertial mass added to the motor shaft. Let 5 r k tTuning spring z rp Rotatinal massRLRLRLMotor Buoyzs Constant-force spring (a) (b) Mcm2 z fwkt ms cm1 z sCPTO ftFigure 2: WEC with TI: (a) Schematic illustration, (b) Model. ftbe the tuning spring force, then the equation of motion of the buoy for the proposed WEC can be written as Mz+cm1_z=fwft (7) where fwis the hydrodynamic force given by Eq. (2). Let zsbe the displace- ment of the rotational inertial mass, then in the TI mechanism, the equation of the equilibrium given by mszs+ (cm2+CPTO) _zs=ft (8) must hold. Because the deformation of the tuning spring becomes zzs, the tuning spring force ftcan be expressed with the tuning spring stiﬀness ktby ft=kt(zzs) (9) Therefore, the proposed WEC with the TI becomes a two-degree-of-freedom (2DOF) system and the power generation by the motor can be improved by designing the tuning spring stiﬀness and the rotational mass appropriately based on the frequency of the external wave input [14]. 2.3. Power Finally, the damping coeﬃcient and the power generated by the three- phase motor is de ned here. As shown in Figs. 1(a) and 2(a), we assume 6 that the motor used here is a three-phase motor which is connected to three resistive loads of resistance RLin star con guration. Let RandKebe the internal or coil resistance and the back-EMF constant of the motor, respec- tively, then the damping coeﬃcient by the motor can be approximated as CPTO=3K2 e 2r2 p(R+RL)(10) The detailed derivation can be found in [27]. And in this research, the generated power Pgis de ned by the summation of the dissipated power by the three resistive loads, i.e., Pg(t) =v2 a+v2 b+v2 c RL(11) where va,vb, and vcare the voltages across the resistive loads. Thus the average power from time 0 to tfis de ned as Pg=1 tf∫tf 0Pg(t)dt (12) 3. Wave ume testing Wave ume testing is carried out to evaluate the power generation per- formance of the proposed WEC in this section. First, the experimental setup is introduced. Next, two preliminary tests are implemented to estimate the parameter values for the mechanical damping coeﬃcients cm1andcm2and the optimal resistive loads RLfor the small-scale models developed for this research. Then, the results obtained from the wave ume testing are shown. 3.1. Experimental setup To validate the eﬀectiveness of the proposed TI WEC, laboratory exper- iments of a 1:36 scale model are carried out in the wave ume located at the Center for Research in Isotopes and Environmental Dynamics (CRiED), University of Tsukuba, Japan. The wave ume is 21 m long, 0.5 m wide and 0.5 m deep and the water depth is 0.3 m. The WEC prototype device is placed at a distance 6 m from the wave generator. In addition, a wave gauge is mounted 0.5 m ahead of the device in the direction of propagation. The experimental setup is shown in Figure 3. 7 (a) (b)rMotor Constant-force springStainless steel disk Wave generatorTuning spring 0.5 m 6 m0.5 m0.5 m Linear bushingShaft holder 0.3 m 0.5 m Hollow shaftRectangular plate Wave guageFigure 3: Experimental setup: (a) Schematic illustration, (b) Photograph. The photograph of the small-scale buoy developed for this research is shown in Figure 4(a). A vertical cylinder used as the oating body has a diameter of 140 mm and a height of 220 mm. The buoy draft is set to 136 mm and a constant-force spring whose output force is 0.98 N is employed, thus some glass sand is added in the cylinder so that the mass including a linear bushing becomes 2.1 kg. The linear bushing is inserted in the coaxial bore of the cylinder to restrict the buoy motion to the heave direction along a hollow shaft of diameter 13 mm. The hollow shaft is erected on a horizon- tal rectangular plate xed to the structure of the wave ume using a shaft holder as illustrated schematically in Figure 3(a). Thus the hollow shaft does not vibrate during the tests. As explained in the previous section, the cylin- der is mechanically coupled with the motor through the tether, which goes through the hollow shaft from the bottom of the cylinder and spools around the pulley keyed to the motor shaft. A three-phase permanent-magnet syn- chronous motor used in the prototype is a YASKAWA SGMMV-A1A [28]. The back-EMF constant of the motor stated on the product speci cation is 0.0516 Nm/Arm, which has been transformed for practical convenience by 0:0516p 1:5p 3= 0:0243 V/rad/sec. Also, the constant-force spring which is em- ployed to prevent the tether from sagging is installed at the other end of the tether. The parameters used in the experiment appear in Table 1. It should 8 (a) (b) (c) Figure 4: Photographs: (a) Buoy, (b) Motor part, (c) Stainless steel disks. Table 1: Parameter values used for the wave ume testing Parameters Values Buoy mass M 2.1 kg Buoy diameter D 140 mm Buoy height H 220 mm Buoy draft L 136 mm Constant-force spring Fcf 0.98 N Mechanical damping cm1 2.35 Ns/m Mechanical damping cm2 0.45 Ns/m Back-EMF constant Ke 0.0243 V/rad/s Internal or coil resistance R 5.25 Ω Water density  1000 kg/m3 be noted that for the small-scale model of 1:36, the diameter of 140 mm and the mass of 2.1 kg corresponds to 5.04 m and 98.0 t for a full-scale WEC from the Froude scaling laws based on the assumption that the most dominating phenomena of the model test is governed by potential ow [29]. Figure 4(b) and (c) are the photographs of the motor part and the stain- less steel disks to add rotational inertial mass for the TI WEC. The pulley is made by a 3D printer and its radius rpis 18 mm. A tuning spring whose stiﬀness 10 N/m is installed between the cylinder and the pulley for the TI 9 Table 2: Experimental types and parameters Type Mechanism rp(mm) kt(N/m) ms(g) RL(Ω) A-R1 SDOF 18 N/A 5.3 5 A-R2 SDOF 18 N/A 5.3 13 B1-R1 TI 18 10 156.3 5 B1-R2 TI 18 10 156.3 13 B2-R1 TI 18 10 283.3 5 B2-R2 TI 18 10 283.3 13 B3-R1 TI 18 10 386.3 5 B3-R2 TI 18 10 386.3 13 WEC. The rotational inertial mass msis de ned as ms=I r2 p(13) where Iis the moment of inertia including the stainless steel disk, the motor, and the shaft. Thus, even the SDOF WEC without the stainless steel disk has a small inertial mass of 5.3 g. While, for the TI WEC, the inertial mass is increased to 156.3 g, 283.3 g, and 386.3 g by adding the stainless steel disks. These three inertial masses are employed to conduct a comparative study and investigate how the eﬀective frequency range is changed. Hereafter, in this article, the SDOF WEC is denoted by Type A and the TI WECs are denoted by Type B1, B2, and B3 depending on the inertial mss values as summarized in Table 2. 3.2. Preliminary tests The power generation performance is highly dependent on the resistive loads RLwhich are diﬃcult to predict due to the existence of the energy loss caused in the devices. Thus, the mechanical damping coeﬃcients cm1and cm2in the models and the optimal resistive loads RLfor the devices in Figs. 1 and 2 are determined experimentally here. 3.2.1. Decay tests The mechanical damping coeﬃcients cm1andcm2existed in the developed devices are estimated for accurate modeling rst. For simplicity, in this research, these mechanical damping coeﬃcients are assumed to be constant 10 (a) (b)rMotor Constant-force spring PulleyFigure 5: Decay test setup: (a) Buoy with the linear bushing, (b) SDOF WEC. -50050 Buoy disp. (mm) 0 2 4 6 8 10 12 14 Time (s)-50050 Buoy disp. (mm) Q P Q P(a) (b) Figure 6: Decay test results: (a) Buoy with the linear bushing, (b) SDOF WEC. regardless of the vibration frequency and the values are determined based on the free vibration decay test under still water as follows. We assume that the SDOF and the TI WECs have the same mechanical damping coeﬃcients. We also assume that cm1comes from the friction between the linear bushing and the hollow shaft and that cm2is attributed mainly to the motor and the constant-force spring. Thus, to obtain cm1andcm2, two decay tests are carried out on the conditions illustrated in Figure 5. 11 For the rst decay test, the buoy is disconnected from the motor as shown in Figure 5(a). Then the buoy with the linear bushing is submerged manually and released gently. The dynamical displacement of the buoy recorded by a laser displacement sensor is shown in Figure 6(a). From the logarithmic decrement method [30], the damping factor is given with the data of points P and Q in Figure 6(a) by =1 2lnuP uQ(14) where uPis the amplitude at the beginning of a cycle and uQis the amplitude at the end of the cycle. In the rst decay test, the recorded values for uP anduQare 40.9 mm and 25.5 mm, respectively, then, from Eq. (14), is calculated to be 0.075. Also, the damped natural frequency !dis given by !d=2 tQtP(15) where tPis the time at the beginning of the cycle and tQis the time at the end of the cycle. The values measured in the test are tP= 8:06 s and tQ= 8:92 s, thus the damped natural frequency !dbecomes 7.31 rad/s from Eq. (15). The equation of motion of the buoy for the rst decay test considering the hydrodynamic force at !=!d= 7:31 rad/s becomes (M+ma(!d))z+ (cm1+cr(!d)) _z+Kwz= 0 (16) Thus, the summation of the damping coeﬃcients including the radiation damping cr(!d) can be expressed as cm1+cr(!d) = 2 √ Kw(M+ma(!d)) (17) where the values for cr(!d) and ma(!d) are provided as 0.71 Ns/m and 0.64 kg, respectively, by the software WAMIT [31]. In this way, the mechanical damping cm1is estimated as 2.35 Ns/m. Note that the values for ma(!d) obtained from the approximated relationship about the damped natural fre- quency !2 dKw M+ma(!d)(18) is 0.73 kg, which is relatively close to the values calculated by the WAMIT. 12 To estimate the value for cm2, the second decay test is conducted in the same way as the rst one. The buoy shown in Figure 5(b) which is the setup for the SDOF WEC is used. In the second decay test, uP= 40 :8 mm, uQ= 26:0 mm are measured, leading to = 0:086 by Eq. (14). Also, !d= 7:22 rad/s is obtained with tP= 6:81 s, tQ= 7:68 s by Eq. (15). The values for cr(!d) and ma(!d) provided by the WAMIT at !=!d= 7:22 rad/s are 0.72 Ns/m and 0.64 kg, respectively. Thus considering the equation of motion for the SDOF WEC given by Eq. (1), the relationship cm1+cm2+cr(!d) = 2 √ Kw(M+ms+ma(!d)) (19) holds. Considering ms= 5:3 g for the model of the SDOF WEC, we have cm2= 0:45 Ns/m. Also, ma(!d) calculated from Eq. (18) is 0.80 kg, which is somewhat diﬀerent from the value calculated by the WAMIT. This dis- crepancy seems be attributable to the fact that the experimental setup does not satisfy the conditions assumed in the WAMIT, which will be discussed particularly in 4.2. 3.2.2. Optimal resistive load test The optimal resistive loads RLfor the power generation of the SDOF WEC and the TI WEC are sought experimentally as well. Regular wave of 1.0 s period is input to the SDOF WEC and the TI WEC with ms= 156 :3 g. These conditions correspond to Type A and Type B2 in Table 2. To nd the resistive loads which maximize power generation, the resistive loads connected to the motor are changed from 4 Ω to 15 Ω with 1 Ω increment. The average power generated by the diﬀerent resistive loads are compared in Figure 7. As can be seen, the average power of Type A peaks when RL= 5 Ω, while Type B2 maximizes its power generation when RL= 13 Ω. So these results clearly show that the optimal resistive load depends on the conditions of WECs. However, nding the optimal resistive load for Type B1 and B3 or under regular waves of diﬀerent periods makes this comparative study more complex. Besides, the objective of this article is to show the mechanical advantage of the TI WEC experimentally. Therefore, the power generation with the resistive loads of 5 Ω and 13 Ω are investigated for Type A, B1, B2 and B3 in the wave ume testing in this article. It should be noted that CPTO= 0:27 Ns/m when RL= 5 Ω and CPTO= 0:15 Ns/m when RL= 13 Ω from Eq. (10). 13 4 6 8 10 12 14 Resistive load value ( Ω)0123Average power (mW) 4 6 8 10 12 14 Resistive load value ( Ω)0246810 12 Average power (mW) (b) (a)Figure 7: Average power generation vs. resistive load value: (a) SDOF WEC (Type A), (b) TI WEC (Type B2). 3.3. Results Finally, the wave ume testing is conducted. Six kinds of regular waves of diﬀerent dominant periods Tp= 0.8 s, 0.9 s, 1.0 s, 1.1 s, 1.2 s, and 1.3 s produced by the wave generator are input to the eight types of WECs summarized in Table 2. The time histories of the wave elevation of the six time periods are shown in Figure 8. The waves shown in Figure 8 are measured by the wave gauge during the tests for Type A. The wave generator is controlled so that the wave elevations become 40 mm, however, some variations are observed depending on the wave period due to the diﬃculty of controlling the wave generator. Note that from the Froude scaling laws, the investigated wave period range of 0.8 - 1.3 s and wave elevation of 40 mm corresponds to 4.8 - 7.8 s and 1.44 m for a full-scale WEC. The measured average power generations are summarized in Table 3. These values are calculated by Eqs. (11) and (12) using the data for a duration of 100 s after the dynamics of the buoy reaches the steady state. Bar graphs are used to compare the experimental average power in Figure 9, in which the average values obtained from the numerical simulation by the developed models in the next section are indicated by circles and crosses as well. The experimental results clearly show that the proposed TI WEC improves the power generation performance of the SDOF WEC signi cantly. In addition, it is evident from the performances of Type B1, B2, and B3 that the TI WEC is able to change the eﬀective period range depending on its rotational inertial mass value. More precisely, the SDOF WEC is unable 14 -20020 Wave elevation (mm) -20020 Wave elevation (mm) -20020 Wave elevation (mm) -20020 Wave elevation (mm) 40 41 42 43 44 45 Time (s)-20020 Wave elevation (mm) 40 41 42 43 44 45 Time (s)-20020 Wave elevation (mm) (a) (b) (c) (d) (e) (f)Figure 8: Wave elevation: (a) Tp= 0:8 s, (b) Tp= 0:9 s, (c) Tp= 1:0 s, (d) Tp= 1:1 s, (e) Tp= 1:2 s, (f) Tp= 1:3 s,. to adjust to long-period waves, however, the TI WEC with larger rotational inertial mass works better in the long-period band without increasing the mass of the buoy itself. Also, the results con rm that the SDOF WEC performs better with resistive loads of 5 Ω than with resistive loads of 13 Ω. While, as expected, it is observed that the optimum resistive loads for the TI WEC vary according to the rotational inertial mass value and the input wave period. 4. Numerical simulation Numerical simulation studies are conducted using the parameter values introduced in the previous section to compare with the test results and to 15 Table 3: Generated average power (mW) Input wave period ( Tp) Type 0.8 s 0.9 s 1.0 s 1.1 s 1.2 s 1.3 s A-R1 1.97 5.18 2.71 1.42 0.88 0.47 A-R2 1.62 4.16 2.19 1.17 0.72 0.37 B1-R1 3.48 9.98 4.72 1.57 1.14 0.21 B1-R2 3.42 10.73 5.36 1.90 1.14 0.32 B2-R1 3.67 7.84 8.98 3.81 1.79 0.22 B2-R2 4.30 7.34 10.83 4.64 1.64 0.24 B3-R1 2.35 4.37 4.84 4.77 2.99 0.35 B3-R2 2.16 3.57 4.93 5.26 4.11 0.40 A B1 B2 B30510 Average power (mW) A B1 B2 B3 A B1 B2 B3 A B1 B2 B30510 Average power (mW) A B1 B2 B3 A B1 B2 B3R1 exp. R2 exp. R1 sim. R2 sim. (a) (b) (c) (d) (e) (f) Figure 9: Comparisons of generated average power: (a) Tp= 0 :8 s, (b) Tp= 0 :9 s, (c) Tp= 1:0 s, (d) Tp= 1:1 s, (e) Tp= 1:2 s, (f) Tp= 1:3 s,. 16 R ei RLLmFigure 10: Approximate circuitry. discuss the validity of the developed models here. 4.1. Problem formulation First, the power generated in the wave ume testing is simulated using the models developed in the previous section by the method proposed in [14]. In this reference, a DC motor is assumed for generating power, so the three- phase motor employed in the prototype is approximated by the technique introduced in [27, 32] as the circuit shown in Figure 10. And, as stated before, the excitation force Fa(!), added mass ma(!), and radiation damping cr(!) are given by the software WAMIT in this research, then the equations of motion given by Eq. (1) for the SDOF WEC and Eqs. (7) and (8) for the TI WEC are approximated in a state-space form of nite dimensional linear time invariant system by the system identi cation technique as _x=Ax+Ba (20) e=Cx (21) where xis the state vector, eis the voltage generated by the DC motor, and A,B, and Care appropriate matrices. The wave amplitude ais taken to be the wave amplitude measured by the wave gauge placed 0.5 m ahead of the buoy for the approximation of an accurate causal system. Considering the circuit shown in Figure 10 , and assuming that the eﬀect of the inductance Lmcan be ignored, the input current to the motor would be i=e R+RL=Cx R+RL(22) Also, on the assumption of use of a DC motor, the generated power can be de ned as Pg(t) =ieRi2(23) 17 where the second term denotes the energy loss due to the internal or coil resistance of the motor [26]. Thus, once the state-space representation given by Eqs. (20) and (21) are derived, the power generation can be evaluated by Eq. (23). The details of this derivation are not described in this paper. However, the outline of the method can be found in [14]. Additionally, to investigate the dependence of the power generation per- formance on the input wave frequency !, sinusoidal waves emulating the regular waves measured in the wave ume testing, which is given by a(t) =a0sin!t (24) are input into the system expressed by Eqs. (20) and (21) as well. a0is the amplitude of the sinusoidal wave, thus the wave elevation of the sinusoidal wave is considered to be 2 a0. Let the transfer function of this system be G(s), which is de ned as G(s) =C(sIA)1B (25) where sdenotes the Laplace transform variable. Then, the voltage e(t) and the current i(t) are given by e(t) =a0jG(j!)jsin(!t\G(j!)) (26) i(t) =a0jG(j!)j R+RLsin(!t\G(j!)) (27) Substituting Eqs. (26) and (27) into Eq. (23) yields Pg(t) =RLa2 0jG(j!)j2 (R+RL)2sin2(!t\G(j!)) (28) The average power generation for 1 cycle, i.e., from 0 to2 !, can be calculated by Pg=RLa2 0jG(j!)j2 (R+RL)2! 2∫2=! 0sin2(!t\G(j!))dt (29) The integration part in the above equation reduces to !, thus, the average power generation under the sinusoidal waves given by Eq. (24) becomes Pg=RLa2 0jG(j!)j2 2(R+RL)2(30) 18 -50050 Buoy disp. (mm) 0 5 10 15 20 25 Time (s)010 20 (mW) Exp. Sim. P g(a) (b)Figure 11: Comparisons of time histories of the experimental and simulation results for Type A-R1: (a) Buoy displacement, (b) Power generation. 4.2. Results The results obtained from the numerical simulation are compared with those from the experiments and the reliability of the developed models are examined. Figures 11 and 12 show the time histories of the generated power and the displacement of the buoy for A-R1 and B2-R2 cases under the regular wave of Tp= 1:0 s. Also, the average power generations of all cases calculated from numerical simulations are marked by circles for R1 cases ( RL= 5 Ω) and crosses for R2 cases ( RL= 13 Ω) in Figure 9 to compare the experimental results shown by bar. Unfortunately, considerable discrepancies are observed especially in the TI WEC, i.e., Type B1, B2, and B3 and these results imply the diﬃculty of developing accurate analytical models of the 2DOF small- scale prototype under external loadings of various frequencies. The possible causes of these discrepancies are that (1) the conditions for Airy linear wave or Stokes rst-order theory assumed in the WAMIT [31, 33] are not satis ed in the wave ume testing [34]; (2) the drag force which might not be negligible around the natural frequency of the buoy, i.e., 0.8 s or 0.9 s, is ignored in the simulation [35]; (3) all the mechanical energy losses including the friction losses in the prototype and the nonlinear hysteresis damping by the constant- force spring are modeled by constant viscous damping coeﬃcients [35]; and (4) waves re ected from side walls which act on the buoy are not considered in the simulation studies. However, these are common problems seen in experiments using small-scale models, which are diﬃcult to address. 19 -50050 Buoy disp. (mm) 0 5 10 15 20 25 Time (s)010 20 P g (mW) Exp. Sim.(a) (b)Figure 12: Comparisons of time histories of the experimental and simulation results for Type B2-R2: (a) Buoy displacement, (b) Power generation. The average power generated from the sinusoidal waves using Eq. (30) are shown in Figure 13. The amplitude of the sinusoidal waves is set to 20 mm, corresponding to 40 mm wave elevation. In this gure, solid and dotted lines are used for the cases of R1 ( RL= 5 Ω) and R2 ( RL= 13 Ω), respectively. Note that the frequency !of the sinusoidal waves are converted to the period Tpby the relationship !Tp= 2in Figure 13. The results indicate that the eﬀective frequency range of the TI WEC is broader than the conventional WEC. Also, it can be seen how the eﬀective period range changes depending on the rotational inertial mass value ms. In other words, the TI WEC is able to shift the eﬀective period range to the higher period range without increasing the buoy mass M. And it can be observed that the resistive loads generating more power depends on the period of the input wave for the TI WEC cases, i.e., Type B1, B2, and B3. These trends agree with the experimental results shown in Figure 9, some diﬀerences are found in the time period range though. In addition, this gure shows how the average power of the proposed device is improved with respect to each input wave period. 5. Conclusions This article reported the wave ume testing using the small-scale model of the WEC with the TI mechanism proposed by the authors and showed 20 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 T p (s)0246810 12 Average power (mW) B1-R1 B1-R2B2-R1 B2-R2 B3-R1 B3-R2A-R1 A-R2 Figure 13: Average power under sinusoidal waves of 20 mm amplitude. that the mechanism is able to improve the power generation performance of the conventional SDOF WEC signi cantly without increasing the buoy mass. Moreover, the results indicated that the eﬀective frequency range of the TI WEC is adjustable by just changing the rotational inertial mass value. Also, to estimate the power generation performance, the models of the devices were established using the experimental data. However, developing accurate models including the energy loss of the small-scale prototypes was diﬃcult and developed models were not adaptable enough for wide frequency range, which con rmed the importance of the experimental study. For future work, experiments utilizing a full-scale model should be car- ried out in the ocean. In addition, a reliable analytical model to estimate the power generation more accurately should be developed and computational uid dynamics (CFD) analysis is considered of value to improve numerical simulation study. And developing the mechanism to change the rotational inertial mass value simply is desired. Furthermore, algorithms for the feed- back control technology to optimize not only the resistive loads but also the rotational inertial mass value in response to changing wave frequency should be proposed because the results obtained in this article demonstrate great 21 potential in a variable rotational inertial mass system to improve the power generation eﬃcacy of oscillating-body WECs. Acknowledgement This research was partially supported by JSPS KAKENHI Grant number 18K189808 and TEPCO (Tokyo Electric Power Company) Memorial Foun- dation. The wave ume testing in the present study was carried out at the Center for Research in Isotopes and Environmental Dynamics (CRiED), Uni- versity of Tsukuba with the help of Prof. Sekiguchi of University of Tsukuba, which is greatly appreciated. The authors would like to thank Mr. Iidaka, a technician of University of Tsukuba, for his help in the experimental setup as well.