## Abstract

A Marine Hydrokinetic (MHK) cycloturbine vehicle can exploit tidal currents to generate sustainable power and also has the ability to station keep and maneuver. The vehicle consists of four counterrotating cycloturbines, which radiate sound underwater. Acoustic control is important to curtail the vehicle’s vibrations and acoustic signature, potentially preventing harmful effects on aquatic life, as well as to reduce the vehicle’s fatigue for longer deployment. A method of reducing the radiated acoustics of the vehicle is determined for tones at foil passing frequency and multiples, by means of clocking the blades between turbines. Experimental work includes testing of a subscale demonstrator in ARL’s Reverberant Tank facility. Fixing the subscale demonstrator to a reaction frame in the tank provides the ability to measure the generated loads using load cells. These measurements verify the effects of turbine clocking on the radiated acoustics.

## 1 Introduction

A Marine Hydrokinetic (MHK) cycloturbine (Fig. 1) is a renewable electric power generation system used in rivers or tidal environments to address the need for electricity in remote regions. These cycloturbines are oriented perpendicular to the flow and have a nonpitching hydrofoil geometry that generates power regardless of the tide flow direction. The hydrofoils experience unsteady loading due to changing angle-of-attack as they rotate through the crossflow. This leads to both vibrations and radiated acoustics.

Fig. 1
Fig. 1
Close modal

The noise generated by MHK cycloturbines negatively impacts the surrounding marine environment (i.e., man-made noise can be particularly intrusive on marine mammal communication and life [15]). In addition, the vibrations, coupled with corrosive seawater damage and the general harsh marine environment, lead to vehicle fatigue, necessitating regular maintenance for the MHK cycloturbines and adding to the cost of operations. To circumvent the costs and difficulties associated with deployment and repairs, the ORPC and Penn State proposed an ambitious goal to design a cycloturbine system capable of generating power and maneuvering to and from its deployment location.

The maneuverable turbine design (presented in Ref. [6]) uses four turbines with variable pitch hydrofoils, which can be powered by electric motors that are traditionally used for power generation. The feasibility of this concept is studied using a subscale demonstrator (SSD) (Fig. 2). This system has the ability to station keep, maneuver, and anchor itself to the sea floor. Furthermore, the system is able to pitch, roll, and yaw. The vehicle maneuvers by varying the thrust magnitude and direction of each turbine through a prescribed sinusoidal foil pitch schedule [7]. It is expected that this configuration will produce even higher vibrations and noise in maneuvering mode compared to a conventional cycloturbine with nonpitching hydrofoils. The research presented here focuses on noise control of this maneuverable system, although similar concepts could be applied to stationary turbines in the power generation mode.

Fig. 2
Fig. 2
Close modal

In this study, an adaptive noise control approach is employed to reduce the radiated acoustics of the MHK vehicle. Adaptive noise control methods attenuate unwanted noise by using another sound source in the system to cancel out the originating noise. An example of this in axial turbomachines is stator-to-stator or rotor-to-rotor indexing (also referred to as vane clocking) [810]. Stator vane clocking is when the circumferential position of the stator blades relative to a downstream stator are shifted. By optimizing the mutual circumferential position between two stators in a stator–rotor–stator interaction in a turbomachine, the tonal noise can be decreased [10]. This method is extended for co-rotating fans [11], as well as two-stage rotor-to-rotor clocking [12], which influences noise emission more than stator vane clocking. Auman [11] investigates the acoustic effect of a slowly co-rotating upstream rotor on a downstream rotor, specifically to reduce blade rate tones. Experimental results from this study validate that for some cases slow co-rotation does reduce sound pressure levels by about 5 dB. Alternatively, Blaszczak analyzes the contribution of two-stage rotor-to-rotor circumferential clocking to noise reduction. A 10 dB reduction of the sound pressure levels is found for rotor-to-rotor indexing, for identical rotor geometry. Applying the indexing effect to modern gas turbines with varying rotor-to-rotor geometry would be a challenge. In addition, optimal stator-to-stator or rotor-to-rotor indexing is determined prior to turbomachine operation and is not performed in situ.

While this method has been applied to axial turbomachines, an example of application to a cycloturbine was not found in the open literature. Therefore, tonal noise reduction control using turbine clocking on a cycloturbine vehicle is assumed to be novel. As with axial flow turbomachinery, crossflow cycloturbines could use clocking by adjusting the rotation rates of the various turbines used in the system to attenuate tonal noise generated by the turbines. If such an approach were feasible, it could be readily applied to a maneuverable MHK turbine design that has independent control of multiple cycloturbines. This approach could be applied to both maneuvering and power generation modes.

This article develops a novel tonal noise reduction method for cycloturbine systems. A vehicle acoustics model, developed using theory for sound radiation from a concentrated hydrodynamic force, is used to assess the effect of turbine clocking on the radiated acoustics, specifically in regards to blade rate noise. Experimental work on a subscale demonstrator verifies that turbine clocking has an impact on noise reduction although not explicitly quantified in the experiment.

## 2 Sound Radiation From a Concentrated Hydrodynamic Force

To reduce vibrations and radiated acoustics at blade rate frequency and multiples, the radiated pressure field from the MHK vehicle must first be understood. The series of steps used to obtain the compact solution will be summarized here.

Aerodynamic noise theory begins with Lighthill’s formulation of the wave equation for a concentrated region of turbulent fluid motion. Lighthill’s work [1315] is unique because it considers the region as an acoustic source that drives the surrounding fluid. Lighthill’s equation can be reformulated using Kirchhoff’s equation for the fluctuating fluid density to include the acoustic phase interactions that occur between sources in a control volume when solid surfaces are not present. Curle [16] extends this work to account for the effect of reflecting boundaries. From Curle’s result, it is found that the sound pressure at each hydrofoil in the MHK vehicle is the resultant of three contributions [17]:

1. Radiation from the turbulent domain.

2. Contribution from the acceleration of the body in a direction normal to its surface.

3. Radiation from a distribution of forces acting on the region.

With respect to the turbulent radiation, Curle argues from the dimensional analysis that the turbulent fluctuations compared to the surface terms scale as the square of the Mach number [18]. Therefore, this contribution to the sound pressure may be neglected for the hydrodynamic work here (since the vehicle operates at low Mach numbers). Hence, the radiated sound from the pitching foils is mainly due to direct motion of the foils and the local pressure field induced by the fluid reacting to the vibrations.

By assuming a constant fluid density in the source region, Curle’s equation can be linearized. Koopmann [19,20] transforms Curle’s equation and simplifies it for rigid bodies whose dimension is less than an acoustic wavelength (compact source). Then, the radiated acoustic pressure can be related to the hydrodynamic force of the hydrodynamic structure by
$p−p0=−xi4πR2[Fi˙+ρ0VUi¨a0+Fi+ρ0VUi˙R]$
(1)
where the raised dot represents the time derivative [18]. The first term in this free-field equation is the far-field contribution to the radiated acoustic pressure, and the second term is the near-field contribution. The far-field region begins at least one acoustic wavelength away from the source. It is the region where pressure and particle velocity are in phase, and energy is being radiated away from the source in the form of sound waves.
The following analysis focuses on the far-field radiation and assumes that the hydrodynamic forces dominate the direct motion terms
$p^=−xi4πR2Fi˙a0$
(2)
to obtain a simpler expression for the radiated pressure field. This equation is an equivalent form of the classical aerodynamic dipole sound from a compact source (see Fig. 3) [17]. By using this definition for pressure, the time-averaged intensity of the radiated pressure field is
$I=|p^|22ρ0a0$
(3)
because the intensity in the far-field is radial [21]. This far-field sound intensity can be integrated over an arbitrary spherical surface that completely encloses the multipole to determine the radiated sound power. Sound power is a more useful measure of the effectiveness of sound radiation than maximum intensity [22].
Fig. 3
Fig. 3
Close modal
The acoustic directivity of the multipole directly contributes to the radiated sound power. The directivity is affected by the alignment and polarity of the sources. This can be demonstrated for a set of sources radiating at the same frequency within a compact volume, using multipole expansions for Green’s functions [21]:
$p^=SeikRR−(D⋅∇)eikRR+∑μ,ν=13Qμν∂2∂xμ∂xνeikRR−∑μ,ν,ζ=13Oμνζ∂3∂xμ∂xν∂xζeikRR+…$
(4)

In this expression, it is clear that the acoustic pressure field appears as a superposition of a monopole field plus a dipole field, plus a quadrupole field, plus an octopole field, etc. Depending on the alignment and polarity of the sources, different acoustic field terms become more dominant, and the contribution of their directivity affects the radiated sound power (see Fig. 4).

Fig. 4
Fig. 4
Close modal

## 3 Model of Vehicle Acoustics

The vehicle dynamics are simulated to assist in the design of controllers for maneuver [6]. This detailed simulation solves the six degrees-of-freedom rigid body equations of motion for the maneuvering MHK system subject to the hydrodynamic lift and drag forces, hydrostatic forces, and the propulsive forces from the turbines. The dynamics model relies on a simplified hydrodynamic analysis of the propulsive forces generated by the rotors as a function of the foil pitch schedule and vehicle state. The prescribed foil pitch schedule is sinusoidal, with a maximum pitching angle βmax and phase angle Φ (see Fig. 5). The phase angle locates where the maximum pitch occurs.

Fig. 5
Fig. 5
Close modal

The lift and drag from each foil sum together to produce a total turbine thrust. The turbine propulsive forces and moments are calculated as the time-averaged foil forces over one turbine revolution using an azimuthally averaged blade element approach. The fundamental approach is derived from simulations used to model trochoidal propellers [23,24]. In addition to producing the steady thrust magnitude and direction, the turbine force model provides the time-varying foil forces in the vehicle frame. The unsteady forces generated by the cycloturbines contribute to the radiated acoustics.

An example of the turbine forces over one revolution produced at a 107 rpm is shown in Fig. 6. The upper and lower turbine forces in the X-direction are in-phase with each other, while the Z-direction forces are 180 deg out-of-phase. This is a result of counterrotation between the top and bottom turbines. In addition, there is a phase shift between the X-direction forces and the Z-direction forces for each foil. As the turbine rotates, each of the foils experiences cyclic variation in its angle of attack, resulting in cyclic variation of the lift and drag forces. In addition, the total X- and Z-direction forces have contributions from both lift and drag, and the magnitudes of these contributions vary based on rotation of the foils. The total force produced from each foil is a sum of the steady force plus a fluctuating component. The fluctuating component of the forces contributes to the radiated acoustic pressure.

Fig. 6
Fig. 6
Close modal

Since the objective of this research is to specifically reduce tonal noise of the vehicle, it is desired to obtain the blade rate contribution to the fluctuating force component. The time-varying forces from each foil can be summed together in the vehicle frame, accounting for the 120 deg circumferential shift between blades. It should be noted that the sound pressure radiation at blade rate frequency and multiples produced for each turbine is not affected by performing this force summation before calculation of the radiated pressure. In other words, computing the far-field pressure radiation from each turbine blade separately and accounting for their relative distances to the far-field produces the same result as summing the blade forces and calculating the radiated pressure from the center of the turbine. This is because each turbine is a compact source as assumed in Sec. 2, i.e., the acoustic wavelength is larger than the turbine diameter, for the lower frequencies of interest.

The steady component of the forces is removed, and the fluctuating forces are converted from the time domain to the frequency domain using fast Fourier transforms (FFTs). The fluctuating force magnitudes at blade rate frequency (5.35 Hz for a three-bladed turbine rotating at 107 rpm) and multiples are obtained. The phase angles for the fluctuating forces are also obtained from the FFT. The signals are then reconstructed with just the contribution from blade rate:
$Fi=∑n=−7,n≠07F~inej(2πnfbpt+φn)$
(5)
the time derivative of which is
$Fi˙=∑n=−7,n≠07j(2πnfbp)F~inej(2πnfbpt+φn)$
(6)
This summation is limited to an upper value of seven times blade rate because the force output from the turbine propulsive model is negligible beyond this value. By using this expression, the radiated pressure can be determined from Eq. (2):
$p^=∑s=14−xis4πRs2F˙isa0$
(7)
for blade rate frequencies and all four turbines. The receiver location is a spherical sweep around the vehicle, centered in the vehicle:
$r**=xe^x+ye^y+ze^zx=Rcosθcosϕy=Rsinθcosϕz=Rsinϕ$
(8)
where the distance R is large enough for the receiver to be in the far field (kR ≫ 1). The receiver location is expressed in terms of an azimuthal angle θ from −180 deg to 180 deg, and an elevation angle ϕ from −90 deg to 90 deg (see Fig. 7).
Fig. 7
Fig. 7
Close modal
Because the turbines are offset from the center of the vehicle, the distance to the receiver is different for each cycloturbine:
$Rs=(x−x0,s)2+(y−y0,s)2+(z−z0,s)2$
(9)
where x0,s, y0,s, and z0,s are constants representing the offsets between the turbine sources and the center of the vehicle in the X-, Y-, and Z-directions. The magnitudes of x0,s, y0,s, and z0,s for the subscale demonstrator are 0 mm, 1429 mm, and 386 mm, respectively.

The time-invariant magnitude of the radiated pressure is used to compute the intensity and the sound power. As stated in Sec. 2, the alignment and the polarity of the sources affects the radiated sound power. The polarity of each turbine source can be changed by adding an additional phase effect into the time derivative of the fluctuating forces (for both the X- and Z-directions):

$Fi˙=∑n=−7,n≠07j(2πnfbp)F~inej(2πnfbpt+φn+λ)$
(10)
where λ is a constant angle between 0 and 180 deg. This phase effect is implemented physically by changing the relative rotational angle between turbines. Changing the rotational angle of adjacent turbines from hereon will be referred to as “turbine clocking.” Fig. 8 shows an example of turbine clocking. Iteration over all clocking options between turbines determines which clocking will minimize the sound power for a specified maneuver.
Fig. 8
Fig. 8
Close modal

It is not only important for an acoustic controller to have information on what clocking is required to reduce the sound power but also important to understand how much is gained by correcting the turbine to that clocking. For example, if the sound power in the far-field will be reduced only by 2 dB by adjusting the turbine clocking, then control may not be necessary.

The sensitivity of the sound power to turbine clocking is evaluated for a range of maneuvers and conditions [25], but two cases are important for the experimental validation in this article:

1. The two starboard turbines thrust forward together at 107 rpm. The maximum pitching angle of the turbines is 30 deg.

2. The two top turbines thrust forward together at 107 rpm. The maximum pitching angle of the turbines is 30 deg.

By using the theory presented, maximum noise reduction occurs when the top turbines are clocked 180 deg out-of-phase with each other, while the ideal clocking for the starboard turbines is to be in-phase with each other. The polarity of the dipoles for ideal clocking is shown in Fig. 9. The orientation of the dipoles are related to the thrust direction [25]. For the case of the top two turbines, the sound power can be reduced by up to 30 dB re 1e-12W (compared to a nonideal clocking case) in the far-field. While still significant, the sound power can be reduced only by 8 dB re 1e-12W for the case involving the starboard turbines. This is due to the radiation difference between a lateral quadrupole and a longitudinal quadrupole.

Fig. 9
Fig. 9
Close modal

## 4 Experimental Work

The effects of turbine clocking to reduce tonal noise at blade rate frequency and multiples can be verified experimentally. A small-scale device, 1/5.56 scale of the full scale MHK system, called the SSD is designed and built. It consists of four cycloturbines. The SSD has a span, width, and height of 3960 mm, 106 mm, and 1464 mm, respectively, and is shown in Fig. 10.

Fig. 10
Fig. 10
Close modal

The Ocean Renewable Power Company worked jointly with the Penn State Applied Research Laboratory to design the structure of the vehicle and a thrust vectoring mechanism. The outer nacelles and central frame supply rigidity for the structure. In addition, ducted turbines provide an increase in power generation by modifying flow conditions into the rotor [26,27]. The vehicle is designed to generate power with a nominal current of 2 m/s.

The turbine characteristics for the SSD are listed in Table 1. The SSD moments of inertia along with total mass and volume are reported in Table 2 as found from swing tests and weight measurements.

Table 1

Turbine characteristics

DescriptionValue
Foil span900 mm
Foil chord95 mm
Foil number3
Foil profileNACA0018
Turbine diameter450 mm
DescriptionValue
Foil span900 mm
Foil chord95 mm
Foil number3
Foil profileNACA0018
Turbine diameter450 mm
Table 2

SSD mass properties

PropertyValue
Total mass1225 kg
Total volume0.93 m3
Pitching moment of inertia153.5 kg/m2
Rolling moment of inertia1884 kg/m2
Yawing moment of inertia1734 kg/m2
Product of inertia (estimated)0
PropertyValue
Total mass1225 kg
Total volume0.93 m3
Pitching moment of inertia153.5 kg/m2
Rolling moment of inertia1884 kg/m2
Yawing moment of inertia1734 kg/m2
Product of inertia (estimated)0

The SSD is tested in the ARL Reverberant Tank (see Fig. 11). The tank has two movable platforms that span the width of the tank. The SSD is fixed to a steel support frame that is clamped to the platforms. The platforms are additionally clamped to the side rails of the tank to reduce vibrations. Constraining the SSD provides the ability to measure the generated loads; a dynamometry system with six strain load cells is used to measure all force and moment components produced from the vehicle. Further discussion of this system is found in Sec. 4.3. In addition to load cell measurements, stepper motor angle and main motor shaft rate are recorded.

Fig. 11
Fig. 11
Close modal

To provide good average values, data are taken for 300 s for each test condition. The load cells are sampled at 2000 Hz. The motors are stopped and the water in the tank is allowed to settle for at least 1 min between runs. The load cells are also re-zeroed between runs.

### 4.1 Sound Radiation From Cycloturbines.

To assess the effectiveness of turbine clocking on the unsteady loads, and by extension the radiated acoustics, data are collected for two scenarios: one where the two starboard turbines are thrusting in the X-direction together and one where the top two turbines are thrusting in the X-direction together, both at a commanded 107 rpm. For both cases, the maximum pitching angle is set to 30 deg. According to the theory presented in Sec. 3, it is expected that the sound power can be reduced significantly by clocking the turbines. The theory relies on a model of the turbine unsteady forces, which is compared to the experimental data presented in Fig. 12. The X-direction force mean and standard deviation of 18 measurements for a single turbine thrusting forward are shown at blade rate frequency and multiples. The predicted unsteady X-direction force from one turbine is shown for the same case. The model reliably predicts the unsteady force magnitude at 1x and 2x blade rate frequency and loses fidelity at higher order blade rates. This is likely due to the contribution of other noise sources at higher frequencies not captured in the model, such as tonal and broadband noise from the structure. However, the forces at the first two multiples of blade rate frequency contribute the most to the total sound power.

Fig. 12
Fig. 12
Close modal

Implementation of the clocking control in the experiment is not possible due to the lack of waterproof encoders to measure the turbine angular displacement. Furthermore, the experiments show significant slippage in the induction motors under load and that the rpm standard variation is on the order of 5%. However, the experiment provides data to help verify the models presented in Sec. 3. A modulation model is fit to the experimental data to show that turbine clocking effects occur in the test. The following analysis provides some validation of the noise reduction theory presented in Sec. 3.

The forces over time for the measured data are filtered using a third-order Butterworth filter for blade rate frequencies. This filter is chosen because it is designed to have a flat frequency response in the passband. The order is selected as a compromise between computation time and frequency response character (see Fig. 13). The cut-on and cut-off frequencies account for the fluctuations in motor rpm. Figure 14 shows the mean square of the forces in the Xdirection over time, filtering out data outside the region around the first blade rate frequency. A 5-s moving average filter is applied to produce the result shown.

Fig. 13
Fig. 13
Close modal
Fig. 14
Fig. 14
Close modal

There are significant variations in the mean square X-direction force when the two starboard turbines are thrusting together. The cause of this fluctuation could be due to a clocking effect; the variation of the rpm between turbines would affect the phasing between the turbines over the course of the 5 min run. In contrast to the forces produced by the starboard turbines, the force in the X-direction from the top two turbines thrusting together shows smaller fluctuations over time.

To understand this result, consider two phasors of magnitudes $F~a$ and $F~b$ that rotate at slightly different speeds (see Fig. 15). The total unsteady force magnitude $F~c$ varies with time according to the law of cosines
$F~c2=F~a2+F~b2−2F~aF~bcosω′t$
(11)
where ω′ is the difference in rotation rate between the phasors. The ratio of maximum to minimum force is
$20log(FmaxFmin)=10log(F~a+F~b)2(F~a−F~b)2$
(12)
on a decibel scale. This model is used to determine the difference in the unsteady force magnitude between turbines as well as the difference in rotation rate.
Fig. 15
Fig. 15
Close modal

The ratio between $F~a$ and $F~b$, as well as the rotation rate difference ω′, is found by fitting to experimental data, as shown in Figs. 16 and 17 for the first blade rate frequency. The values for $F~a$ and $F~b$ are found by averaging the values of experimental data peak points and valley points, respectively. The value for the rotation rate difference is found by iterating over a large range of rpms, with a small 0.01 rpm resolution and using the least squares method to minimize the residual between the experimental data and the model.

Fig. 16
Fig. 16
Close modal
Fig. 17
Fig. 17
Close modal

Tables 3 and 4 present the modulation model predictions of the unsteady force magnitude ratio and the rotation rate difference between turbines, as a function of blade rate frequency.

Table 3

Modulation model results for Fx mean square data, starboard turbines case

Multiple of BR$F~b/F~a$rpm Difference
11.731.82
22.581.79
32.241.99
Multiple of BR$F~b/F~a$rpm Difference
11.731.82
22.581.79
32.241.99
Table 4

Modulation model results for Fx mean square data, top turbines case

Multiple of BR$F~b/F~a$rpm Difference
13.610.30
23.860.16
33.740.50
Multiple of BR$F~b/F~a$rpm Difference
13.610.30
23.860.16
33.740.50

For the first blade rate frequency, the model shows that the difference in rpm between the starboard turbines is 1.8 rpm. This difference is consistent with the model prediction at 2x blade rate frequency for the same run. For the top turbine case, the model shows a difference of 0.3 rpm for the 1x blade rate filtered data. The fit is not as strong at higher blade rate frequencies, which is why there is some variation in the predicted rpm difference. The rpm variation has an effect on when the turbines are clocked in-phase or out-of-phase with each other; the rpm difference explains the time between valleys in the force measurement.

Another aspect of the modulation model is the ratio between the unsteady force magnitudes $F~a$ and $F~b$. For the filtered data at 1x blade rate frequency, there is a difference in the model between the ratio $F~b/F~a$ for the case where the top turbines thrust together versus the case where the starboard turbines thrust together. For the starboard turbines, the model predicts that the ratio between the unsteady force magnitudes is 1.7 at 1x blade rate frequency, while for the top turbines, the model predicts a ratio of 3.6. In the acoustic model discussed in Sec. 3, it is assumed that the unsteady force magnitudes are the same between turbines for the cases where the top and starboard turbines thrust together. When the unsteady forces are of the same magnitude, and perfectly out-of-phase with each other, the reduction in sound power is maximized (30 dB sound power reduction from clocking the top two turbines, and an 8 dB reduction by clocking the starboard turbines). If the unsteady forces are out-of-phase, but of different magnitudes, there is not as much of a reduction in sound power.

This is more clearly demonstrated by applying the unsteady force magnitude ratios found experimentally to the acoustic model. Figures 18 and 19 show a map of the sound power reduction for combinations of thrust vectoring at 107 rpm for the starboard and top turbines, respectively. Turbine clocking can reduce the sound power for the top turbines forward thrust case by roughly 4.6 dB re 1e-12W, while reducing the sound power for the starboard turbines case 6.4 dB re 1e-12W. For some combinations of thrust vectoring between the starboard turbines, turbine clocking may reduce the sound power by as much as 19 dB re 1e-12W.

Fig. 18
Fig. 18
Close modal
Fig. 19
Fig. 19
Close modal

In summary, the closer the ratio $F~b/F~a$ is to 1, the greater the effect on minimizing sound power. Because the ratio $F~b/F~a$ is close to 1 for the experimental case where the starboard turbines thrust together, there are larger fluctuations in the mean force. There are smaller magnitude fluctuations in the mean force for the top turbines case because the unsteady force magnitude ratio is 3.6.

The difference in the unsteady force magnitude ratio between the two cases can be attributed to flow differences in the reverberant tank. The top turbines have a larger separation distance than the starboard turbines (2860 mm versus 770 mm) and likely have different inflows.

### 4.2 Sound Radiation by Structural Vibration.

Sound radiation from vibrating structures is of importance in this analysis. Understanding the mechanisms of sound radiation can improve the quality of the MHK vehicle. In the following analysis, the structural modes of the vehicle are assessed, the contribution of nacelle vibration to the radiated acoustics is quantified, and the dynamometry system used for the experimental work is analyzed.

#### 4.2.1 Contribution of Vehicle Vibration.

It is important to quantify the structural modes of the vehicle and their corresponding frequencies to understand how the structural vibration will radiate sound. Simulation of a finite element (FE) model of the SSD in-air predicts a bending mode at 14.7 Hz, a torsional mode at 16.4 Hz, and a longitudinal mode at 19.1 Hz. At 286 rpm, which is the target operating rate of the turbines, the first blade rate frequency falls at 14.3 Hz. This is very close to the finite element model’s predicted bending frequency of 14.7 Hz. It is possible that with all four turbines operating at this rpm, at some thrust direction, the first bending mode could be excited. Long-term operation at this condition would increase vehicle fatigue, if not lead to vehicle failure.

To check the reliability of the finite element model, which does not include the mass and stiffness from the motors, as well as any nonlinear damping from cables, a modal tap test is performed on the SSD. The tap test is performed in air, with the vehicle modes adjusted in analysis for the submerged case. The SSD is suspended from a crane with straps for the test (see Fig. 20). The stiffness of the straps is assumed to be much smaller than the stiffness of the SSD and so simulates a free-free scenario for the modal test.

Fig. 20
Fig. 20
Close modal

A grid of 78 points (shown in Fig. 21) is demarcated on the SSD. These grid points are chosen to capture the bending and torsional modes that could be potentially excited by the blade rate. A large force hammer with a rubber tip is used to hit the points and induce vibrations on the SSD. Five accelerometers are used to capture the response of the vehicle to the input force. Each grid point is struck three times for an averaged response.

Fig. 21
Fig. 21
Close modal

ARL’s poly-reference estimation code (APEC) is used to identify the modes excited from the experimental data. APEC is a system-identification procedure that uses a poly-reference least-squares complex frequency domain algorithm to determine natural frequencies, loss factors, and mode shapes of the experimental modal analysis data. The results of the SSD modal analysis are summarized in Table 5 and compared to a finite element analysis (which is simulated up to 50 Hz). The last column shows the predicted modal frequencies if the SSD is underwater.

Table 5

Mode shapes and natural frequencies of subscale demonstrator

FE expected naturalMeasured naturalExpected natural
Mode shapefrequency in-air (Hz)frequency in-air (Hz)frequency underwater (Hz)
Rigid body (heaving)4.83.8
Rigid body (bobbing)12.19.7
First bending14.719.415.5
First torsion16.423.618.9
Second torsion44.225.020.0
Second bending23.329.323.4
Third bending36.529.2
Third torsion38.831.0
FE expected naturalMeasured naturalExpected natural
Mode shapefrequency in-air (Hz)frequency in-air (Hz)frequency underwater (Hz)
Rigid body (heaving)4.83.8
Rigid body (bobbing)12.19.7
First bending14.719.415.5
First torsion16.423.618.9
Second torsion44.225.020.0
Second bending23.329.323.4
Third bending36.529.2
Third torsion38.831.0

The natural frequencies of a vibrating structure are decreased by heavy external fluids such as water. This is due to the extra force required to accelerate the fluid’s inertia [28]. The frequency shifts are a function of material, thickness, and aspect ratio, and it can be approximated that the modal frequencies underwater will be 80% of the frequencies in air [29]. The first two frequencies are rigid body modes from the SSD suspension from the crane. The first bending mode is anticipated to occur underwater at 15.5 Hz. The first torsion mode is predicted to occur at 18.9 Hz.

These frequencies are higher than predicted by the finite element simulation and also higher than the target blade rate frequency. This suggests that when operating at the target rpm of 286, acoustic radiation from structural vibrations is not a primary concern.

#### 4.2.2 Contribution of Vibrating Nacelles.

Another consideration is sound radiation from the vibrating nacelles. The nacelles on the SSD are the outermost orange plates. Their dimensions are 1205 mm by 560 mm, with a thickness of 60 mm. It is important to quantify this vibration to ensure that the experimental load cell measurements reflect the unsteady forces produced by the turbines and are not contaminated by sound radiation from the structure.

The power output of a single nacelle is computed for a specified 1 m/s velocity in the direction normal to its surface assuming it radiates as an unbaffled piston. The computations are performed using the equivalent source techniques described by Koopmann and Fahnline [30] with the plate modeled as infinitely thin, dipole sources alone used for basis functions, and a surface mesh of 28 × 60 uniformly spaced rectangular elements. This can be considered an upper bound for the power output because at low frequencies, where kL ≪ 1, any phase variation in the velocity will lead to cancellation. For the computations shown, the maximum kL is calculated as follows:
$kL=2π(30Hz)a0(0.56m)=0.07$
(13)
which is much less than 1. The frequency is chosen to be around 6x the blade rate frequency of a turbine operating at 107 rpm.
The reference velocity for the power output is adjusted using data from the modal tap test performed in-air. From the APEC analysis, the mobility (velocity/force) of each grid point is found. The corner nacelle hit point at the largest X, Y, and Z position (refer to Fig. 21) has the largest mobility on the nacelle, particularly in the vehicle bending mode. This frequency-dependent mobility is converted to velocity with a defined input force. The input force used for this calculation is the measured unsteady force in the X-direction produced by the turbine as it thrusts forward: it is assumed that the unsteady forces transfer through the turbine shaft into the nacelle. The velocity is shifted in frequency to correct for the surrounding water. The power output is corrected to this predicted velocity
$Lw,new=Lw+20log(V1m/s)$
(14)
and doubled for a second nacelle (a 6 dB increase). Figure 22 shows the acoustic power output as a function of frequency for the vibrating nacelles. For comparison, the acoustic model predictions of the power output for the experimental cases are shown. The predictions include the variation in the unsteady force magnitude between turbines found from the experimental data, as discussed in Sec. 4.1. The upper and lower bounds on the predictions show the effect of using turbine clocking to minimize the sound power.
Fig. 22
Fig. 22
Close modal

At 1x and 2x blade rate frequency, the sound power produced by the unsteady turbine forces are much higher than the power output from the nacelles. There is a low risk of structural contamination on the measurements at these frequencies. At 3x blade rate frequency, the structural vibration is a larger contributor to the power output.

The hydrodynamic loads of the SSD are measured within ARL’s Reverberant Tank. A dynamometry system consisting of a frame of links containing strain load cells is shown in Fig. 23. This analysis provides suitability of the dynamometry system, provides a transfer matrix between the link loads and the SSD principal coordinate system, and determines the rigid-body natural frequencies.

Fig. 23
Fig. 23
Close modal

It is assumed that the dynamometry system behaves as a truss where only axial loads are allowed through each link. It is further assumed that the node displacements are small and that the SSD itself acts as a rigid body. Finally, it is assumed that the back of the back support of the dynamometry is held rigidly.

Figure 23 shows the nodes (in boxes) and the links in black. The coordinate system shown corresponds to the center of gravity of the SSD. Only nodes 1 through 4 are allowed to be displaced, and their position is dictated by
$x=Φq(15)$
(15)
where q corresponds to the rigid body translations and rotations. The matrix Φ is a 12 × 6 matrix relating the two systems. The stiffness of the deflections are found from
$Kδ=EALi$
(16)
where E is the Young’s modulus of steel, A is the cross-sectional area of the tubing, and Li is the length of each of the six tubes.
The stiffness matrix is subsequently found by
$Kq=Φ′B′KδBΦ$
(17)
where the matrix B relates the link deflection to the coordinate system. The natural frequencies are found from the eigenvalues of the moment of inertia matrix divided by the stiffness matrix. The moment of inertia matrix is populated with the values determined from swing tests (tabulated in Table 2). The lowest natural frequency occurs at 19.4 Hz.

At the experimental operating rpm of 107, the frequencies of concern are the lower blade rate frequencies at 5.35 Hz, 10.7 Hz, and 16.05 Hz. The natural frequencies of the dynamometry system are above these frequencies. The dynamometry system is not expected to influence the unsteady force measurements at the first few blade rate frequencies.

## 5 Conclusions

An acoustic model of a novel Marine Hydrokinetic cycloturbine vehicle is presented. According to the model, noise is reduced at the blade rate frequency and multipled via a new method of turbine clocking, whereby the phase angle between turbines is varied.

A small-scale vehicle is built and tested in ARL’s Reverberant Tank facility. The unsteady forces are measured using load cells. Data from these tests show fluctuations in the mean square force in the X-direction at blade rate frequencies. It is demonstrated that the fluctuations in mean square force in the thrust direction are a result of the turbines moving in-phase and out-of-phase relative to one another.

The experimental modulations in force data are modeled, quantifying the differences in unsteady force magnitude and rpm between turbines. This modulation model fits well with the measured data, ultimately indicating that the clocking angle between turbines in a Marine Hydrokinetic cycloturbine vehicle has a significant effect on the vehicle forces and by extension the radiated acoustics. While clocking is not actively used for noise reduction in the experiment, these measurements verify the effects of turbine clocking on the radiated acoustics.

Furthermore, the structural vibration of the vehicle and its nacelles is shown to not significantly contribute to the radiated sound power compared to the cycloturbines. The experimental unsteady loads are also not affected by the dynamometry system. The cycloturbines are the main source of the sound radiation at blade rate frequency and multiples.

Turbine clocking can be applied to reduce blade rate noise for systems with more than one cycloturbine, not just for the specific vehicle design shown. It can also be applied to cycloturbine systems that generate power, in addition to maneuvering systems.

In the future work, classical control methods can be used to design an acoustic controller that tracks the relative phase angle between turbines using waterproof encoders. An unsteady dynamometry system comprised force gauges in the cycloturbine shafts could also be used to measure the turbine loads (see Ref. [31]). This sensor data would provide active error measurements of the radiated acoustics. Inclusion of these sensors would allow for further experimental assessment of active clocking for comparison with model predictions.

## Acknowledgment

The authors gratefully acknowledge the support of this work by the U.S. Department of Energy’s Advanced Research Projects Agency-Energy (ARPA-E) as well as the ORPC under Contract Number DE-FOA-0001261. Any opinions, findings, and conclusions or recommendations expressed in this work are those of the author(s) and do not necessarily reflect the views of the U.S. Government. Special thanks go to Michael Prendergast for his hard work helping to assemble the experiment and Eric Trainum for his support of the controls hardware. Additional thanks go to the machinists and technicians that provided support for this experiment: Ron Ayers, Ellis Dunklebarger, Edward Boone, and Brian Kline. The authors also thank Jonathan Bechtel, Tyler Dare, and Ben Beck for their help with computational analysis.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request. The authors attest that all data for this study are included in the paper.

## Nomenclature

• q =

rigid body translations and rotations

•
• f =

frequency

•
• j =

imaginary number

•
• k =

wave number, ω/a0

•
• p =

acoustic pressure

•
• t =

time

•
• x =

•
• A =

cross-sectional area

•
• D =

dipole amplitude

•
• E =

Young’s modulus

•
• F =

hydrodynamic force

•
• I =

sound intensity

•
• K =

stiffness

•
• L =

length

•
• O =

octopole amplitude

•
• Q =

•
• R =

•
• S =

monopole amplitude

•
• U =

volume velocity

•
• V =

volume, inflow velocity

•
• $p^$ =

complex amplitude of acoustic pressure

•
• $F~$ =

fluctuating component of hydrodynamic force

•
• a0 =

speed of sound in water

•
• p0 =

ambient acoustic pressure

•
• fbp =

•
• Lw =

sound power level

•
• VN =

normal inflow velocity

•
• VT =

tangential inflow velocity

•
• α =

angle-of-attack

•
• βmax =

•
• θ =

•
• λ =

clocking angle

•
• ρ0 =

density of water

•
• ϕ =

•
• Φ =

turbine phase angle, transformation matrix

•
• ψ =

phase angle

•
• ω =

angular frequency

• i =

index

•
• n =

•
• s =

source index

•
• δ =

deflection

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