In response to the need for efficient, smallscale power sources for applications such as ocean observation and navigation, this paper presents the design, modeling, fabrication, testing, and analysis of a compact pointabsorber wave energy converter (PAWEC) equipped with a mechanical directdrive power takeoff (PTO) mechanism. The motivation is to address the mismatch between the natural frequencies of conventional PAWECs and dominant ocean wave frequencies, which limits energy capture. The primary objective is to enhance the efficiency of smallscale wave energy converters (WEC) without increasing the buoy size. To achieve this, we introduce a novel design element: an added mass plate (AMP) attached to the buoy. The AMP is devised to increase the WEC added mass and natural period, thereby aligning its natural frequency with dominant ocean wave frequencies. In our case study of a scaled model (1:2.2), the AMP effectively doubled the added mass of the WEC and increased its natural period by 32%. The WEC incorporates a rack and pinion mechanical motion rectifiertype PTO to convert the heave oscillations of the buoy into unidirectional rotation. The scaled model was tested in a wave basin facility with regular waves at zero angle of incidence. The WEC with AMP achieved a maximum root mean square power of 9.34 W, a nearly 30% increase compared to the conventional configuration without AMP, which produced 7.12 W under similar wave conditions. Numerical analysis using the boundary element method in the frequency domain for regular waves confirmed these findings. Finally, it has been derived that the proposed WEC, equipped with an AMP, offers enhanced efficiency in longer wave periods without the need for a larger buoy, establishing its viability as a power source for navigational buoys. This paper also offers a comprehensive guide to experimental techniques for characterizing a PAWEC in a laboratory setting, contributing valuable insights into the wave energy community.
NOMENCLATURE
Abbreviations
 AM

added mass
 AMP

added mass plate
 BEM

boundary element method
 CWR

capture width ratio
 DOF

degree of freedom
 DMD

direct mechanical drive
 KC

Keulegan–Carpenter
 MMR

mechanical motion rectifier
 OWC

oscillating water column
 PAWEC

point absorber WEC
 PTO

power takeoff
 RAO

response amplitude operator
 RMS

root mean square
 SS

stainless steel
 WEC

wave energy converter
Symbols
 a

added mass (kg)
 A

normalized added mass (m^{3})
 $ A m$

added mass coefficient
 $ A \u221e$

added mass at infinite frequency (kg)
 $ b d$

damping coefficient (kg/s)
 B

normalized damping coefficient (m^{3}/rad)
 Β

phase lag between $\u0307z$ and ζ (rad)
 C

system damping coefficient (kg/s)
 $ c hyd$

hydrodynamic damping coefficient (kg/s)
 $ c PTO$

PTO damping coefficient (kg/s)
 $ F a$

amplitude of force (N)
 $ F damp$

damping force (N)
 $ F Diff$

diffraction force (N)
 $ F e x$

excitation force (N)
 $ F F K$

Froude–Krylov force (N)
 $ F hyddamp$

hydrodynamic damping force (N)
 $ F m$

Mooring force (N)
 $ F PTOdamp$

PTO damping force (N)
 $ F PTOfric$

PTO friction force (N)
 $ F rad$

radiation force (N)
 $ F res$

hydrostatic restoring force (N)
 H

wave height (m)
 K

hydrostatic stiffness (N/m)
 $ m p$

mass of the pinion cage (m)
 $ m s$

mass of standard weights (m)
 N

number of peaks
 Ρ

the density of water (kg/m^{3})
 $ P abs$

absorbed power (W)
 $ P max$

maximum power (W)
 $ P wave$

incident wave power (W)
 $ r 0,\u2009 R 0$

the radius of the hole (m)
 $ R B$

the radius of the buoy (m)
 $ R P$

the radius of the plate (m)
 T

wave period (s)
 V

the volume of the fluid (m^{3})
 W

capture width (m)
 $ x n$

the amplitude of the nth peak (m)
 $ x 0$

initial amplitude (m)
 z

heave displacement in the time domain (m)
 z_{a}

heave amplitude (m)
 $\u0307z$

heave velocity (m/s)
 $\xa8z$

heave acceleration (m/s^{2})
 Δ

logarithmic decrement
 ζ

wave elevation (m)
 $\xi $

damping ratio
 φ_{z}

phase lag between heave force and displacement (rad)
 ω

excitation frequency (rad/s)
 ω_{d}

damped frequency (rad/s)
 ω_{n}

natural frequency (rad/s)
 Ω

angular frequency (rad/s)
I. INTRODUCTION
Among renewable energy systems, marine energy systems are getting noticed after the two energy crises in the 1970s and 2000s. Although extensive efforts are going on to harvest wave energy^{1–5} the system efficiency is still low. Wave energy converter (WEC) devices exhibit diverse designs and functionalities, allowing for varied classification approaches. These classifications can hinge on parameters such as deployment locations, operational principles, modes of operation, and design geometries.^{4} Three primary designs emerge when zeroing in on the geometric aspect: point absorbers (PAWECs), attenuators, and terminators.^{6} Among these wave energy converters (WECs), the PAWECs have a WEC width much less than the wavelength.^{7} Distinctly characterized by relatively small dimensions compared to the incident wavelength, PAWECs stand out for their simplicity in the system structure. This inherent simplicity bestows several advantages onto PAWECs: they tend to be easier to manufacture, ensuring reliability in operations and costeffectiveness during maintenance. Moreover, their compact nature often translates to economic feasibility, making them an appealing choice for smallscale energy harvesting purposes and lowenergy seas.^{8}
The natural frequency is determined by the hydrostatic stiffness (wet line surface area) and the total mass, including physical and added masses. Usually, smallsized buoys tend to have low natural periods.^{10} The period of an ocean wave is typically 6–15 s, much longer than the natural period of small buoys for ocean observation or marine navigation. Based on Falcao's calculation,^{4} the diameter of the submerged hemisphere needs to be 52.4 m to match an incident wave frequency of 0.1 Hz, which is too large to be practical. In the literature, many different techniques were used by WEC developers worldwide to match the frequency. For example, twobody wave energy converters have been proposed, which results in a damped natural frequency of the twobody WECs to match the wave frequency in the optimal condition.^{11}
Similarly, in 2019, Al Shami et al. found that the resonant frequency of a WEC can be reduced by increasing its degrees of freedom.^{12} Latching control^{13} and declutch control^{14} were also developed based on this frequency match condition. Heave plates are frequently used in floating offshore structures because they can contribute additional added mass and damping to the system, which improves its hydrodynamic performance.^{15–18} Also, WECs frequently use submerged heave plates to supply reaction forces between the buoy and the submerged second body.^{19–22} However, the possibility of adding a plate to the floating buoy of a WEC to match its natural frequency with the incoming wave frequency is unexplored. This technique is very different from using a heave plate as a second body in WECs to generate reaction forces, which has been discussed extensively in the literature.
Theoretically, adding a plate to a floating buoy should increase its added mass and reduce its natural frequency in the heave mode.^{23} Based on this reasoning, the performance of a smallsize PAWEC in realistic wave periods can be improved by adding a plate called AMP to its buoy. AMPs attached to a float or a buoy in heaving wave motion will have more added mass, which means it will increase the natural period of the system and assist in frequency matching with waves, resulting in higher amplitude. While increasing the added mass and natural period, AMPs ensure that the volume or size of the physical system remains more or less the same. Such a property makes it very useful in the case of smallscale WECs, where a significant amount of capital and operational expenses can be saved by reducing the buoy size.^{24,25}
WECs can be broadly classified into direct and indirect drives based on the types of PTO systems used. The choice between direct drive and indirect drive PTO significantly impacts the overall design, efficiency, and complexity of the WEC system. A directdrive WEC uses either a linear or a rotary generator.^{26} Direct mechanical drive (DMD) WECs utilize the mechanical energy of wave motion to drive a mechanical generator, like a hydraulic pump, directly. In contrast, direct electrical drive WECs convert wave motion into electricity through a direct coupling with an electric generator, often linear.
While both have merits, for the present PAWEC design, a DMD WEC was chosen due to its ability to effectively handle the high forces and low speeds typical of wave energy, its mechanical simplicity, and potentially higher reliability. A rotary generator was chosen for the current PAWEC over a linear generator due to its higher efficiency, lower cost, and greater availability. Rotary generators are a more mature and widely used technology, benefiting from years of development and refinement, which translates to superior reliability and ease of maintenance in the long run.
Various DMD WEC designs found in the literature exhibit a wide range of characteristics. Certain WECs with Rack and pinionbased PTO mechanisms^{27,28} incorporate a rotary generator coupled with a gearbox. However, these designs lack rectification for the buoy's bidirectional motion. Slider cranktype WECs^{29} present an alternative but may falter in breaking wave conditions. Winch and ropetype PTO systems^{30} offer another design path, but these can only harness power during the buoy's upward motion. Mechanical motion rectifier (MMR)based WECs^{11,31} provide a distinctive solution, employing a unique MMR gearbox to convert the buoy's bidirectional motion into unidirectional rotation, thereby enhancing power generation efficiency. For this reason, a similar MMRbased PTO mechanism is used in the current PAWEC design.
This paper reports the experimental and numerical analysis performed on two different buoy designs (with and without AMP) of a PAWEC equipped with an MMRbased PTO. For both numerical and experimental analyses, the waves considered are linear, regular, and with a zero angle of incidence. A comparative study is done on how various hydrodynamic parameters behave for the two buoy designs. Finally, the results of the analyses were used to adopt the most suitable design that works satisfactorily in real sea conditions. In addition to this, this article contains much information on various experimental methods to characterize a PAWEC, including finding its hydrodynamic parameters, PTO damping, and performance parameters. This information can be beneficial for many wave energy researchers worldwide.
To summarize the significance of this work, this article fundamentally reimagines PAWEC design by introducing the innovative added mass plate (AMP) to the buoy, paving a transformative path for smallscale wave energy conversion. Through rigorous numerical and experimental comparisons against traditional designs, this study underlines the efficacy of the novel AMPintegrated buoy approach. Furthermore, the manuscript meticulously details PAWEC characterization methods elucidate the hydrodynamic advantages of the AMP, and accentuates the costeffectiveness and compactness of this new design in the realm of smallscale WECs.
The structure of this article is organized as follows. Section II presents the design of the proposed PAWEC system, including the design of individual components like the PTO, the buoy, and the spar. Fundamental concepts regarding the hydrodynamics of WECs are discussed in Sec. III. The numerical modeling of the proposed PAWEC system is presented in Sec. IV. Experimental methods to test the PAWEC are explained in Sec. V. The results of various experiments and numerical simulations are discussed in Sec. VI. Finally, the article is concluded in Sec. VII.
II. DESIGN OVERVIEW
The PAWEC consists of a cylindrical buoy, an AMP connected to the buoy, a spar attached to a bottom fixer plate, and a PTO mechanism [see Fig. 1(c)]. The buoy contains a linear bearing that ensures the spar passes through and reciprocates smoothly. A pair of stands welded on the top surface of the buoy holds the racks (or the PTO mechanism) above water. The mechanism was designed in such a way that it reduces the sealing requirement of the PTO unit. One conventional buoy [Buoy A, Fig. 1(a)] is without an AMP, while the proposed buoy connected to an AMP is shown in Fig. 1(b) (Buoy B). The spar is a hollow cylinder that passes through the buoy and supports the PTO unit via the connecting rod. Table I provides the dimensional specifications for the various components of this PAWEC.
Parameter .  Value .  Units .  Parameter .  Value .  Units . 

Spar  Buoy  
Outer diameter  0.048  m  Diameter  0.6  m 
Inner diameter  0.042  m  Height  0.4  m 
Thickness  0.003  m  Wall thickness  0.002  m 
Length  2  m  Weight  35  kg 
Fixer plate  0.1 × 0.1  m^{2}  Buoyancy  >50  kg 
Parameter .  Value .  Units .  Parameter .  Value .  Units . 

Spar  Buoy  
Outer diameter  0.048  m  Diameter  0.6  m 
Inner diameter  0.042  m  Height  0.4  m 
Thickness  0.003  m  Wall thickness  0.002  m 
Length  2  m  Weight  35  kg 
Fixer plate  0.1 × 0.1  m^{2}  Buoyancy  >50  kg 
When the buoy moves up and down by the action of the waves, the spar slides inside the buoy. The spar is attached to the bottom of the wave basin. Figure 1(c) shows the assembly under testing in the wave basin. The other end of the spar is connected to a pair of pinions mounted on an output shaft connected to a generator. Each pinion engages with one rack and is mounted to the shaft using a oneway clutch to function like a freewheel (Fig. 2). At any given point, only one pair of pinion and rack will drive the output shaft. When the buoy faces the wave crest, it moves up, and a set of the engaged rack and pinion gives shaft rotation in one direction. During wave trough, the other set of the engaged rack and pinion gives shaft rotation in the same direction. Hence, the shaft continuously gets unidirectional motion, which is transferred to the generator to produce electricity. The function of this PTO mechanism is similar to that of the rack and pinion MMR proposed by Liang et al.^{32}
Initially, a CAD was developed based on buoyancy calculations, specifications of the test facility, and ease of fabrication. The four main parts of the design are a buoy, a spar, a fixer plate, and a PTO mechanism. The fabrication material was stainless steel (SAE 316). Figure 3 shows the design flow chart.
The stability analysis was performed frequently during manufacturing to ensure the system's stability.^{33} The details of this stability analysis can be seen in Appendix A.
III. HYDRODYNAMICS OF WEC
A. Resonance
B. Equation of motion
1. Hydrodynamic characterization
2. Concept of added mass plate
Figure 4 shows the concept of increasing the added mass of the buoy by adding a plate to the oscillating buoy. The added mass of a fully submerged circular plate (Fig. 4) is approximately equal to the mass of an imaginary sphere that encloses the plate.^{10}
C. Response amplitude operator (RAO), capture width, and capture width ratio (CWR)
IV. NUMERICAL MODELING OF THE PAWEC
The most common methods to model the WECs are frequency domain BEM models, spectral models, and Morrison equation solvers. The selection of an appropriate method depends on the type of work. BEM is often used to derive the hydrodynamic parameters of the bodies,^{20,42,43} which the timedomain models like WECSim require as an input. BEM solvers solve linear potential theory in the frequency domain and are useful in understanding fundamental hydrodynamics. Even though the timedomain codes are based on the linear hydrodynamic theory, they can also take into account slight nonlinearities. Hence, these models are beneficial in analyzing the performance of WEC.
For this particular PAWEC, AQWA is used as the preprocessor to determine the hydrodynamic parameters. Wall effects can be neglected since the wave basin is equipped with wave absorbers on three of the four sides. Hence, the spatial domain for the analysis is taken as infinite. The depth is the same as that of the wave basin, i.e., 3 m. Since the spar is fixed in the model test, the hydrodynamic analysis is only performed for the buoy. The Simulink model of the actual concept and the labtested model are shown in Figs. 5(a) and 5(b), respectively.
A. Mesh sensitivity study
A gridindependent numerical solution of the system was achieved through different numerical experiments. The meshing information of the three different resolutions for the PAWEC model is reported in Table II. The hydrodynamic coefficients for different meshes are compared in Fig. 6. Figure 6(a) shows that the change in mesh resolution has a negligible effect on the magnitude of the added mass. The relative magnitude of radiation damping tends to decrease by a maximum of 3.9% when the resolution is decreased [Fig. 6(b)]. Although both medium and fine meshes gave almost the same values, considering the computation cost, the medium mesh is chosen for subsequent analysis.
Type .  Max. element size (m) .  Number of elements .  Number of nodes . 

Fine  0.0167  21 034  20 885 
Medium  0.0208  14 174  14 316 
Coarse  0.025  10 600  10 733 
Type .  Max. element size (m) .  Number of elements .  Number of nodes . 

Fine  0.0167  21 034  20 885 
Medium  0.0208  14 174  14 316 
Coarse  0.025  10 600  10 733 
B. Dynamic modeling of the PTO mechanism
V. EXPERIMENTAL METHODS
Except for the PTO damping test, all other experiments were done in a wave basin (Table III) at IIT Madras, India. A scaling ratio 1:2.2 is adopted based on the Froude scaling law.^{9} Specifications of all the instruments used for the experimental measurements are shown in Table IV.
Parameter .  Value .  Units . 

Length  30  m 
Width  30  m 
Wavemaker type  Flap  
Maximum water depth  3  m 
Parameter .  Value .  Units . 

Length  30  m 
Width  30  m 
Wavemaker type  Flap  
Maximum water depth  3  m 
Instrument .  Parameter .  Remarks . 

Accelerometer  Acceleration  Range: $\xb1$ 5 g pk 
Sensitivity: 1000 mV/g  
Wavemeter/wave gauge  Wave height  Type: Conductive 
Resolution: <1 mm  
Load cell  Load  Capacity: 20 kg 
Output: 2.005 mV/V  
DAQ  Data acquisition  Sample rate: 9600/s 
Resolution: 16bit 
Instrument .  Parameter .  Remarks . 

Accelerometer  Acceleration  Range: $\xb1$ 5 g pk 
Sensitivity: 1000 mV/g  
Wavemeter/wave gauge  Wave height  Type: Conductive 
Resolution: <1 mm  
Load cell  Load  Capacity: 20 kg 
Output: 2.005 mV/V  
DAQ  Data acquisition  Sample rate: 9600/s 
Resolution: 16bit 
The wave basin's pedals and control unit were periodically calibrated using the MIKE21 wavefield simulation tool to maintain a certain confidence level in generating specific wave conditions (H and T) from the wavemaker. Wave probes continuously monitored the generated wave characteristics during experiments. The sample size and the confidence level of the error band for H and T for regular waves were 30, 95%, ±0.19 cm, and ±0.035 s, respectively. As explained before, the two case studies adopted for this study are one buoy without AMP and another similar buoy with an AMP of an outer diameter of 0.6 m, an inner diameter of 0.15 m, and a thickness of 0.002 m connected to its bottom at a distance of 0.6 m. We chose the plate dimensions based on our numerical analysis of various heave plate diameters within the same PAWEC model.^{44}
A. PTO damping test
The numerical model requires the value of the PTO damping to be entered manually into the code during the analysis. The experiment explained in Appendix C gave the damping values under different load conditions (Fig. 7) that were used for numerical simulations.
B. Determination of hydrodynamic parameters
The hydrodynamic coefficients determined from experiments are compared with those obtained from the BEM results. Since the BEM code does not consider the viscous effects, the value of viscous damping determined experimentally from the free decay and radiation tests is manually added to the code while performing the numerical analysis. In radiation tests, the PAWEC device is forced to oscillate at different frequencies, and the resulting radiation forces are measured. This can provide insight into the frequencydependent radiation damping of the device. On the other hand, free decay tests involve displacing the WEC device from its equilibrium position and then allowing it to oscillate freely. By observing the decay of these oscillations, we can estimate the total damping (including both radiation and viscous damping) of the WEC system. By comparing the results of these tests to the hydrodynamic coefficients predicted by the BEM analysis, we estimated the additional viscous damping that needs to be included in our WECSim model. Incorporating these experimentally derived viscous damping coefficients into our WECSim model will bolster the precision of our simulations and will align them more closely with realworld WEC performance.
1. Decay tests
A decay test was performed with the model, where the buoy is placed at an initial height from its equilibrium position and is released to go in still water. Now, the buoy's position is measured over time until the buoy comes to rest. The curve obtained will show an exponentially decreasing trend, as shown in Fig. 11, and the system's natural frequency is then determined. From the heave response obtained from the decay test, the natural period and the damping ratio ( $\xi $) for heave motion were computed by using logarithmic decrement (δ), like in the case of an underdamped system^{9} (see Appendix D).
2. Radiation tests
The hydrodynamic coefficients are found by performing radiation tests on the system, as shown in Fig. 8. During the tests, heave motions with varying amplitude and a fixed frequency were applied to the buoy using a linear actuator. The displacement, acceleration, and heave force exerted by the actuator were logged. One wave probe was fixed at a distance of 1 m from the buoy to measure the radiated waves (Fig. 9). The forcetime series obtained from the experiment was split into terms of sines and cosines using Fourier expansion. The force terms for the first harmonics were integrated to find the inphase and quadrature components. Finally, the radiation and the added mass coefficients were determined from the inphase and quadrature components using Eqs. (10) and (11), respectively.
C. Diffraction tests
For regular waves (0.10 $\u2264$ H $\u2264$ 0.25 m and T = 1, 1.5, 2, and 2.5 s), wave height, heave displacement, heave force, and power outputs for buoys were measured in a wave basin for regular waves of wave heights ranging from 0.10 to 0.25 m and wave periods 1, 1.5, 2, and 2.5 s. Heave forces were measured by a load cell fixed between the buoy and the clamp once the wave interaction was started (Fig. 10). An accelerometer recorded the buoy acceleration for the incoming waves. The spectral noise density of the accelerometer for the experiment was about $200\u2009\mu g/ H z$. At frequency = 0.4 Hz, the double integration error of the data recorded by the accelerometer was ∼1.5%. There were many practical difficulties that needed to be addressed during the experiments. To maintain the buoy at the fixed location, a fixer plate was fixed at the basin bed instead of a mooring system, as the mooring design was not in the scope of the study. The spar produced frictional loss as lubrication was not allowed in the linear bearings. Minor misalignments during the operation were taken care of by introducing a ball and socket joint between the spar and connecting rod of the PTO. Fixing and recording data from the accelerometer and the load sensor were challenging due to vibrations. Despite these challenges, the desired data were recorded during a small time window, where all systems, including the wave basin, the mechanical and electronic systems, the PAWEC system, and the instruments, worked satisfactorily in tandem.
VI. RESULTS AND DISCUSSION
A. Determination of hydrodynamic coefficients
1. Free decay tests
Free decay tests were performed to determine the damping ratio and the natural period for the heave motion of the buoy. The time history of the buoy movement during decay tests with and without the AMP is shown in Fig. 11. The natural periods of buoy A and buoy B obtained from the BEM code were 1.41 and 1.02 s, respectively. The natural periods of the buoys obtained experimentally and numerically are compared in Table V. The values of other parameters calculated from the free decay experiments using Eqs. (D1)–(D4) are shown in Table VI. Inspecting Tables V and VI, one can infer that the AMP increased the system's natural period by ∼32% while increasing the damping by ∼37.5%.
Parameters .  Heave natural period (s) .  

Experimental .  Numerical (BEM) .  
Buoy A  1.09  1.02 
Buoy B  1.44  1.41 
Parameters .  Heave natural period (s) .  

Experimental .  Numerical (BEM) .  
Buoy A  1.09  1.02 
Buoy B  1.44  1.41 
Parameters .  Units .  Buoy B .  Buoy A . 

Td  s  1.470  1.100 
ω_{d}  rad/s  4.270  5.710 
x_{0}  m  0.058  0.044 
x_{n}  m  0.004  0.006 
δ  ⋯  1.320  0.960 
ξ  ⋯  0.200  0.150 
ω_{n}  rad/s  4.270  5.780 
T_{n}  s  1.440  1.090 
Parameters .  Units .  Buoy B .  Buoy A . 

Td  s  1.470  1.100 
ω_{d}  rad/s  4.270  5.710 
x_{0}  m  0.058  0.044 
x_{n}  m  0.004  0.006 
δ  ⋯  1.320  0.960 
ξ  ⋯  0.200  0.150 
ω_{n}  rad/s  4.270  5.780 
T_{n}  s  1.440  1.090 
2. Radiation tests
Radiation tests are done to determine the normalized radiation damping coefficient and the normalized added mass coefficient of any WEC system. The tests are done for two different amplitudes: 6 and 10 cm, and four different periods: 1.5, 2, 2.2, and 2.5 s. The hydrodynamic coefficients are determined using Eqs. (8)–(14) after substituting the values of required parameters, which have been found experimentally.
a. Radiation coefficient
In terms of normalized radiation damping [B(ω)], the experimental results show some deviations from the numerically derived values (Fig. 12). This might be due to the reflection of waves from the wave paddle region of the wave basin, where wave absorbers were not present. A similar trend was also observed by Bonfiglio in 2011.^{45} Further analysis of the decay and radiation tests revealed that the radiation damping effect is lower than the total damping, a sum of radiation, and viscous damping. The reason might be that the radiation is a surface phenomenon, and the geometry of the proposed WEC at the surface is small. Also, the radiation damping for the excitation amplitude of 10 cm is slightly higher than that for the amplitude of 6 cm by 6%. This can be explained by the findings of Tao and Cai^{46} that heave damping increases with an increase in the Keulegan–Carpenter (KC) number, which depends on the motion amplitude. Again, as evident in Fig. 12(b), due to the reduced stability of buoy A, more fluctuations occurred during the experiments compared to buoy B. Figure 12 shows that the maximum difference between the numerical and the experimental values occurs near resonance. This could be related to the formation of eddies due to increased buoy velocity near the resonance.
b. Added mass
The normalized added mass obtained experimentally and through BEM analysis is shown in Fig. 13. The added mass of buoy B (0.12–0.13 m^{3} s) is much higher than that of buoy A (0.07–0.05 m^{3} s). The addition of the AMP increased the added mass of the system significantly because now the system has to displace more fluid when it oscillates. Also, the normalized added mass obtained from BEM analysis and experiments agree well within allowable error limits (3%–10%). This may be because the potential theory used in the BEM code also represents the inertial forces.
Although the experimental values for different amplitudes show a small difference with and without AMP, the added mass increased as the oscillation amplitude increased in both cases. The observed variations in the added mass may be attributed to its dependency on the KC number, a behavior it shares with the radiation damping coefficient. However, the KC number in the range of 0.2–1.4 has minimal effect on added mass;^{47} this article does not study the variation of added mass with the KC number.
c. Comparison with theoretical added mass
Based on the theoretical approach mentioned in Sec. III B 2, the added mass of the two different buoy configurations can be calculated from the following equations (13) and (14):

The theoretical added mass of the buoy A = $\rho \u2009\pi 2 3 R B 3 \u2212 r 0 2 R B$ = 54.64 kg.

The theoretical added mass of the buoy B = $\rho 8 3 R P 3 \u2212 2 \pi R 0 2 R P + \pi 2 3 R B 3 \u2212 r 0 2 R B$ = 116.01 kg.
Table VII shows the two buoy configurations' theoretical and experimental added mass. The maximum difference (=16.60%) was seen in buoy B for 2.50 s. This difference might be due to the sudden rise in experimental added mass for a ω < 4.5 rad/s or a T > 1.4 s [Fig. 13(a)]. The dependence of added mass on the KC number becomes more significant for the KC number outside the range of 0.2–1.2.^{48} Since the resonant frequency of the system of the buoy with AMP lies in the region with ω_{n} < 4.5 rad/s, velocities in that region will be high, and thus, the values of the KC number will be >1.2. This explains the sudden increase in added mass noticed in Fig. 13. However, the theoretical added mass ignores the effects of the KC number. The difference between the calculated and observed values is more near the resonant region. Similarly, the effect of the KC number on added mass for buoy B increases in the region ω < 6 rad/s since the system's resonant frequency lies here. Since the rise in added mass for buoy A is more than buoy B, the difference between the calculated value and the observed value of added mass for the former will be more significant (see Table VII).
Period (s) .  Buoy B .  Buoy A .  

Experimental added mass (kg) .  Difference (%) .  Experimental added mass (kg) .  Difference (%) .  
1.5  126.79  9.30  56.95  4.22 
2.0  133.24  14.86  67.25  23.07 
2.2  134.01  15.52  68.00  24.45 
2.5  135.34  16.60  67.94  24.34 
Period (s) .  Buoy B .  Buoy A .  

Experimental added mass (kg) .  Difference (%) .  Experimental added mass (kg) .  Difference (%) .  
1.5  126.79  9.30  56.95  4.22 
2.0  133.24  14.86  67.25  23.07 
2.2  134.01  15.52  68.00  24.45 
2.5  135.34  16.60  67.94  24.34 
B. Time domain analysis for monochromatic waves
Time responses of position, velocity, heave force, and power of the buoy configurations were studied experimentally and numerically. Experiments were repeated for wave heights of 0.10, 0.15, and 0.25 m and periods 1, 1.5, 2, and 2.5 s. Due to the constraints imposed by the wave maker's capabilities, achieving the precise resonant periods of the configurations during the experiment was not feasible. Consequently, periods near the resonance were selected as a practical compromise. The value of PTO damping was found separately through the experiment explained in Sec. V A and was inserted into the WECSim code.
1. Motion response
In 2020, Dafnakis et al.^{49} highlighted that the Cummins model overestimates the heave amplitude, attributing this discrepancy to the linear potential theory's overestimation of Froude–Krylov forces or wave excitation loads on submerged buoys. A congruent pattern was noted in our study. The numerical analysis overpredicts the values of heave displacement and velocity by ∼15%. This may also be due to the frictional losses occurring in moving parts like the rack and pinion or the linear bearing between the buoy and fixed spar. The phase lag $\beta $ between the wave elevation and buoy velocity also depends on total damping ξ. As ξ is reduced from 0.20 to 0.15 [see Figs. 14(c) and 14(d)], $\beta $ increased from 81.50° to 85.26°.
a. Linearity test
A linearity test was done to determine the ability of the buoy to be resonant with the incoming wave by examining the variation of the amplitude of the heave motion with the incident wave height. The experimental results were fitted to a straight line passing through the origin. The test was conducted for H = 0.10–0.25 m and T = 1, 1.5, and 2.5 s. Reasonable linearity is expected when the amplitude of the waves is less than the radius of curvature of the floating body.^{48,50} The radii of both the buoy configurations are 0.3 m, and the maximum wave height used for testing is 0.25 m. Hence, the condition for linearity is matched; as seen in Fig. 15, the linearity between the heave amplitude and the wave amplitude was good for investigated frequencies.
The values of ω/ω_{n} for the two different buoy configurations are shown in Table VIII. From Fig. 15, the following observations can be made:
Period (s) .  Freq. (rad/s) .  ω/ω_{n} for buoy A .  ω/ω_{n} for buoy B . 

1.0  6.28  1.09  1.47 
1.5  4.19  0.72  0.98 
2.5  2.51  0.43  0.59 
Period (s) .  Freq. (rad/s) .  ω/ω_{n} for buoy A .  ω/ω_{n} for buoy B . 

1.0  6.28  1.09  1.47 
1.5  4.19  0.72  0.98 
2.5  2.51  0.43  0.59 

For T = 1 s [Fig. 15(a)], the slope of the line corresponding to buoy A (ξ = 0.15) is more compared to that of buoy B (ξ = 0.20). This is because ω/ω_{n} for buoy A is nearer to resonance at this wave frequency when compared to buoy B. Similarly, for T = 1 and 2 s, the slope of the line for buoy B is higher than that for buoy A [Figs. 15(a) and 15(b)].

As explained in Sec. VI B 1, the difference in the magnitude of heave displacement obtained from numerical and experimental analyses is because of the unaccountability of various factors like friction, generator damping, and nonlinearities near the resonant region in the numerical model.
2. Heave excitation force response
The heave excitation forces were measured for both buoys by fixing the buoy while subjecting it to regular waves with varying amplitude and period. Figures 16(a) and 16(b) show the experimental and numerical time response of heave excitation for an incoming wave of H = 0.15 m and T = 1.5 s. The numerical and experimental values closely match because other unpredictable forces associated with motion will be negligible since the system was fixed during the experiment.
Figures 16(a)–16(c) shows that the excitation force is reduced for a corresponding increase in ξ because of the former's dependence on added mass. The heave excitation force comprises two counteracting components, the diffraction force (F_{Diff}) and the Froude–Krylov force (F_{FK}). For buoy B, the total damping will be higher due to increased added mass. As the added mass increases, the diffraction force component increases for a small change in F_{FK}. Since these two components are 180° out of force, the net heave force derived from potential theory will be the magnitude difference between these two components. Thus, the net heave force reduces as the added mass increases, and this phenomenon is the heave cancelation effect.^{51} However, since the dominating component is F_{FK}, this reduction in heave force will be comparatively less and will not significantly affect the power absorption. In Fig. 16(c), for T = 1–2.5 s, the maximum reduction of excitation force by adding AMP occurs for 2 s and has a value of 14%.
3. Power response
The hydrodynamic power responses of the buoys obtained for H = 0.15 m, T = 2 s, and a resistive load of 3 Ω are shown in Fig. 17. A much more refined numerical model of the electric generator is needed to calculate the electric power output more accurately, but that is out of the scope of the present work. Hence, even though Fig. 17 shows a difference of about 19% between the numerically calculated RMS power and the experimentally derived RMS power, most of this difference is accounted for by the frictional power loss and the generator damping loss. The frictional loss was significant as the proposed system used a rack and pinionbased PTO mechanism. Figure 17 shows that the pinion's downward stroke gives more power than the upward stroke. This can be due to the assistance of gravity during the downward stroke. However, the numerical model did not predict this phenomenon. From a series of tests with the waves of T = 1–2.5 s, the maximum value of RMS power generated using buoy A and buoy B obtained numerically was 9.48 and 12.34 W, respectively. The same obtained experimentally were 7.12 and 9.19 W, respectively.
C. Performance parameter analysis
As mentioned in Sec. III C, the performance of a WEC is determined by two main parameters, namely, RAO and CWR. Analyses of these parameters can give an idea of the power absorption efficiency of the proposed system.
1. RAO
The buoys and the PTO mechanism were subjected to regular waves with H = 0.15 m and T = 1–2.5 s. Initially, the RAO increases with T, and after reaching a maximum value, it decreases even if T is increased (Fig. 18). The peak value of RAO for buoy A and buoy B occurred at T ≈ 1.3 and 1.6 s, respectively. Even though the technical limitations of the wave maker did not allow for generating a wave with a period equal to the numerically derived natural period of the buoys, the numerical values agree well with the experimentally derived values for other periods. Hence, it is safe to assume that these two periods (1.3 and 1.6 s) are the new resonant periods of respective buoys after connecting the PTO mechanism. In other words, adding PTO increased the resonant period of buoy A and buoy B from 1.09 to 1.3 s and 1.43 to 1.6 s, respectively.
In the resonance zone, the linear theory fails to predict the position or power absorption.^{52} Hence, the numerical values are a bit higher than the experimental values near the resonance zone. Payne^{53} concluded that the difference could be higher for lesser external damping values if viscous damping is ignored during the numerical analysis. The present work experimentally found the PTO and viscous damping and inserted them into the WECSim code. Therefore, the numerical values near resonance are less compared to other similar works.^{20} The peak RAO of buoy B was observed to be higher than that of buoy A. The reason might be that the resonance of buoy B occurs at a higher period than buoy A, and the wave power is higher for a higher period [Eq. (15)].
Buoy A and buoy B performed well for T > 1.5 s and T < 1.5 s, respectively (Fig. 18). Hence, to improve the performance of the WEC at higher periods, buoy B will be a better option. The numerical model slightly overpredicted the value at resonant peaks because the viscous drag was assumed to be linear in numerical analysis.
2. CWR

As mentioned in Sec. VI B 3, the power calculated from the experiment is the output power generated by the generator. In contrast, the power calculated by the numerical model is the hydrodynamic power absorbed by the buoy multiplied by the generator efficiency. Thus, the generator loss (beyond 10%) is ignored in the numerical results.

As explained in Sec. VI C 1, the higher velocity gives higher viscous drag causing the system to behave nonlinearly at nearresonant zones. Since the viscous drag is assumed to be linear in the numerical model, it overpredicts the output power.

The time series of experimental power (Fig. 17) indicates that the peaks produced in the upward and downward strokes are not the same, reducing the RMS value. WECSim code failed to foresee this behavior and, thus, overpredicted the output power.
Like RAO, the CWR increases with the period and peaks at T ≈ 1.2 and 1.6 s for buoy A and buoy B, respectively (Fig. 19). The CWR for buoy B is higher for T < 1.6 than buoy A. The peak CWR of buoy B is 3% less than that of buoy A. The AMP in buoy B increases the system's damping and reduces the conversion efficiency. Even though a small percentage reduces the peak, the curve obtained for buoy B is wider than that obtained for buoy A. This means buoy B is more efficient for a wider range of frequencies than the other buoy configuration. This is desirable in real sea conditions where the frequencies of sea waves lie over a wider range.
VII. CONCLUSION
This paper proposed and validated using an added mass plate (AMP) to tune the natural frequency of a point absorbertype wave energy converter (PAWEC) to match the exciting wave frequency for small power applications. A 1:2.2 scaled version of an innovative PAWEC was prototyped, numerically analyzed, and tested in the wave tank. The paper also details the experimental characterization of a PAWEC, which can benefit many readers working in wave energy. The main conclusions drawn are as follows:

The AMP attached to the buoy increases the added mass by a factor of ∼2.16 and the natural period of the system by ∼32.10% without increasing the size of the WEC.

The PAWEC with AMP (Buoy B) performed well for longperiod waves, while the buoy without AMP (Buoy A) performed well for shortperiod waves.

The numerical model predicted the heave response of both the buoy configurations well except at the resonance, where it overpredicted the position by a maximum of 15%.

The numerical model estimated the power absorbed with a peak over prediction of 16.66% near the resonance.
Considering the disadvantages of modeling a system operating in its resonance region using linear approximations, the overprediction of results by the numerical model is justifiable. Further works are being carried out to replace the fixed central tube with a floating spar and study the WEC response in irregular waves.
ACKNOWLEDGMENTS
V. Vijayasankar, S. Kumar, and A. Samad thank the financial support from the IP cell, IIT Madras (Project ID: ICS/16–17/831/RR1E/MAHS). V. Vijayasankar and L. Zuo wish to thank the support from U.S. NSF IUCRC No. 1738689.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
Vishnu Vijayasankar: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Methodology (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Suman Kumar: Conceptualization (supporting); Data curation (equal); Methodology (supporting); Visualization (equal); Writing – original draft (supporting); Writing – review & editing (supporting). Abdus Samad: Conceptualization (equal); Methodology (equal); Resources (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). Lei Zuo: Conceptualization (equal); Funding acquisition (equal); Resources (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal).
DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding authors upon reasonable request.
APPENDIX A: STABILITY ANALYSIS
Parameter .  Value .  Units . 

Metacentric radius (BM)  0.09  m 
GB  0.07  m 
GM  0.02  m 
Parameter .  Value .  Units . 

Metacentric radius (BM)  0.09  m 
GB  0.07  m 
GM  0.02  m 
In the fullscale design, the floating spar will have a drag plate instead of fixing it to the seabed and will be moored. The large surface area of a drag plate gives a large drag and creates a relative motion between the spar and the buoy. In that case, stability analysis should be performed for the spar and the buoy. The spar in the present case was fixed to the wave basin floor because of the basin depth limitation (=3 m).