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NONLINEAR ROTARY TRANSLATIONAL ENERGY HARVESTER FOR WIND AND WAVE POWER

GENERATION

Abstract

This research presents a method to utilize the pitch-roll nonlinear energy transfer phenomenon in vessels for off-shore wave power generation. The power generator is a horizontal pendulum on board a vessel. The surface waves result in rocking of the ship, which in turn results in rotation of the pendulum. The pendulum is connected to a DC generator to produce power. The design in this paper utilizes the nonlinear coupling between the roll and pitch motions of the generator vessel. The natural frequency of the roll motion of the vessel is tuned to be twice the fundamental frequency of the pitch motion. In this specific condition when the vessel is excited in the pitch direction, the energy is nonlinearly transferred to the roll mode and the vessel oscillates in the roll direction. It is shown that the combination of the pitch and roll motion is far superior to the pitch motion since the combination results in full rotations of pendulum. A pendulum in full rotations generates orders of magnitude more power compared to a locally oscillating pendulum.

Project Description

In recent years, many researchers have focused on the development of ocean wave energy generators. The majority of the wave power generators are anchored to the sea floor or riverbed. This makes these technologies incompatible with offshore wave power generation. A pendulum generator that is manufactured by the Wello company [1] contains a pendulum rotating in the horizontal plane, which is placed inside a buoy. The modes considered for buoy in [1] are the pitch and heave motions. The prototype generated about one mega-watt of power. Ref.[2] considered a design similar to the Wello system and generated power using a single frequency ocean wave. They considered the pitch and heave modes of the buoy to be uncoupled and presented extensive numerical results of the rotation of pendulum. Ref.[3-5] improved the power generation in [2] by developing a control algorithm that could actively control the dynamics of pendulum in order to increase the magnitude of generated power.

It has been shown that a ship with the pitch/roll frequency ratio of 2:1 rocks significantly in response to the ocean waves and is thus very unpleasant to ride. In this research we use this phenomena to enhance the motion of the ocean wave generator and improves its performance. Effects of the nonlinear coupling between pitch and roll modes on the motions of vessel was first modeled in Ref. [6]. They studied the relation between the pitch/roll frequency ratio and the energy transfer between the roll and pitch modes of motion. Ref. [7] showed the responses for the vessel model in [6] for various sub-harmonic resonance cases.

Fig. 1: The wave generator

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The schematic of the vessel-pendulum system is shown in Fig. 1. A horizontal pendulum is connected to a DC generator to produce power from the rotations of pendulum. The motion of the vessel is assumed small compared to the wave height. We first modeled the vessel, the rotating pendulum and the DC generator to assess if the proposed wave generator is feasible. At the same time we started to develop experimental setups for experimental investigation of the generator. The results of the analytical and experimental investigations will be discussed in the next two sections:

Modeling Results

We have derived the governing differential equation of the nonlinear wave generator as:

𝐼!𝛼 + 𝜇!𝛼 + 𝑘!𝛼 −𝑘!𝛽!

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+𝑚𝑙! 𝛼 − 𝑠𝑖𝑛 2𝜑 𝑠𝑖𝑛 𝛽  𝛽2

!

− 𝛼 (𝑐𝑜𝑠 𝛽)! + 𝛼 (𝑐𝑜𝑠 𝛽)! + 𝛼 (𝑐𝑜𝑠𝜑)! (𝑐𝑜𝑠 𝛽)! − 𝑐𝑜𝑠 𝛽 𝑐𝑜𝑠𝜑 𝑠𝑖𝑛𝜑 𝛽

+ 2 (𝑐𝑜𝑠𝜑)! 𝑐𝑜𝑠 𝛽 𝛽𝜑 − 2 (𝑐𝑜𝑠𝜑)! 𝑐𝑜𝑠 𝛽 𝑠𝑖𝑛 𝛽 𝛼𝛽 − 2 (𝑐𝑜𝑠 𝛽)! 𝑐𝑜𝑠𝜑 𝑠𝑖𝑛𝜑  𝛼𝜑

+𝑚ℎ! 𝛼 (𝑐𝑜𝑠 𝛽)! − 𝑠𝑖𝑛 2𝛽  𝛼𝛽+𝑚𝑙ℎ − 𝑐𝑜𝑠 𝛽 𝑐𝑜𝑠𝜑 𝜑! + 2 𝑠𝑖𝑛𝜑 𝛼𝛽 + 𝑐𝑜𝑠𝜑 𝑠𝑖𝑛 𝛽 𝛽 − 𝑐𝑜𝑠 𝛽 𝑠𝑖𝑛𝜑 𝜑 + 𝑐𝑜𝑠 𝛽 𝑐𝑜𝑠𝜑 𝛽!

− 2 𝑐𝑜𝑠 𝛽 𝑠𝑖𝑛 𝛽 𝑠𝑖𝑛𝜑 𝛼 − 4 (𝑐𝑜𝑠 𝛽)! 𝑠𝑖𝑛𝜑 𝛼𝛽 + 2 𝑐𝑜𝑠 𝛽 𝑐𝑜𝑠𝜑 𝑠𝑖𝑛 𝛽  𝛼𝜑+𝑚𝑔𝑙 𝑐𝑜𝑠𝜑 𝑐𝑜𝑠 𝛼 − 𝑠𝑖𝑛 𝛼 𝑠𝑖𝑛 𝛽 𝑠𝑖𝑛𝜑 +𝑚𝑔ℎ(𝑐𝑜𝑠 𝛽 𝑠𝑖𝑛 𝛼) = 𝑀!𝑐𝑜𝑠  (𝜔!𝑡)

𝐼!𝛽 + 𝜇!𝛽 + 𝑘!𝛽 − 𝑘!𝛼𝛽

+𝑚𝑙! 𝛽 − 𝑠𝑖𝑛 2𝛽 (𝑐𝑜𝑠𝜑)!  𝛼2

!

− 𝛽 (𝑐𝑜𝑠𝜑)! + 2 cos𝛽 𝛼𝜑 + sin 2𝜑  𝛽𝜑 + cos𝛽 cos𝜑 sin𝜑  𝛼

+ 2  cos𝛽 (𝑐𝑜𝑠𝜑)! 𝑠𝑖𝑛 𝛽 𝛼𝜑 +𝑚ℎ! 𝛽 + 𝑠𝑖𝑛 𝛽 𝑐𝑜𝑠 𝛽 𝛼!

+𝑚𝑙ℎ −𝑠𝑖𝑛𝜑 𝜑!−𝑠𝑖𝑛𝜑 𝛼! + cos𝜑 𝜑 + 𝑐𝑜𝑠𝜑 𝑠𝑖𝑛 𝛽 𝛼 + 2 (𝑐𝑜𝑠 𝛽)! 𝑐𝑜𝑠𝜑 𝛼! +𝑚𝑔𝑙 𝑠𝑖𝑛𝜑 𝑐𝑜𝑠 𝛼 cos𝛽+𝑚𝑔ℎ(𝑐𝑜𝑠 𝛼 𝑠𝑖𝑛 𝛽) = 𝑀! cos 𝜔!𝑡

𝑚𝑙! 𝜑 − sin 2𝜑𝛽2

!

− 𝛼 sin𝛽 − 2 cos𝛽  𝛼  𝛽 + (cos𝛽)! sin𝜑 cos𝜑 𝛼!    + 𝑐𝜑 +𝑚𝑙ℎ(−𝛼 sin𝜑 cos𝛽

+ cos𝜑  𝛽 + cos  β  sin𝛽 cos𝜑 𝛼! + 2 sin𝛽 sin𝜑 𝛼𝛽  ) −𝑚𝑔𝑙 sin𝜑 sin 𝛼 − sin𝛽 cos𝛼 cos𝜑 = 𝑘!𝑖

𝐿!𝑑𝑖𝑑𝑡+ 𝑅𝑖 =  −𝑘!𝜑

where 𝜶,  𝜷 are the pitch and roll angles of the ship respectively,  𝝋 is the angle of the pendulum. m is the mass of the pendulum, h is height of the pivot point of the pendulum from the surface of

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the ship, l is the length of the pendulum,  𝑳𝒅 is the inductance,  𝒌𝒆 is the electro-mechanical coupling coefficient, R is the resistance of the DC Generator and i is the current. The above equations have numerous nonlinear terms and since the pendulum rotates continuously or oscillates with large angles, perturbation methods will not be accurate. The governing equations are solved numerically using fourth order Runge-Kutta method.

In the presented case study, the frequency of the ocean wave is assumed to be closer to the natural frequency of the pitch mode of the vessel. Fig. 2 shows why the proposed design is advantageous. It shows when the vessel oscillates only in the pitch direction, the generating pendulum oscillates locally and generates very small electric power. As soon as the nonlinear coupling results in the roll motion, the pendulum exhibits full rotations (Fig. 2.b and 2.e). The power generated during the full rotations of the pendulum is significantly larger than the power generated during the local oscillations and therefore nonlinear coupling enhances the power generation of the vessel by orders of magnitude.

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 2: Frequency response of vessel, rate of rotation of pendulum and the average power; (a) Pitch and roll for A1=0.03 rad/sec2; Circles indicate pitch response and triangles indicate roll response of vessel; (b) 𝝋    for

A1=0.03 rad/sec2; (c) Pav for A1=0.03 rad/ sec2 (d) Pitch and roll for A1=0.09 rad/sec2; (e) 𝝋  for A1=0.09 rad/sec2 (f) Average Power for A1=0.09 rad/ sec2

Experimental Results

An important part of the experimental setup is the design of an effect wave generator to produce dependable waves to excite the wave energy generator. After investigating commercially available wave generators, it became clear the generator would need to be custom built to reduce costs and fit the water flume available. The water flume is an approximately 2’ x 2’ x 30’ channel located in University at Buffalo Jarvis Hall and is used primarily study water flow. Instead, the flume is filled to a height of at least 1ft and sealed to create a large tank. The wave generator was designed to fit securely in the tank and produce sinusoidal waves at variable frequencies and amplitudes. Fig. 3 shows the wave generator schematics and the fabricated experimental setup.

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Fig. 3: Experimental wave generation setup

Fig. 4 shows the fabricated pendulum fixture. Surrounding the pendulum is a clear cover to provided additional support and protection. At the top is a black joint whose purpose is to secure the ship to an overhanging fixture. The first generation of the wave generator (Fig. 5) is a simple trapezoidal vessel. The vessel was fabricated from transparent polycarbonate plates to allow observation of the pendulum. The power generator pendulum is fixed at the center of the vessel. The first generation vessel was easy to fabricate but its simplistic shape resulted in very small power generation. In the second generation of the wave generator (Fig. 6) we use a commercial Remote Controlled ship. The hydrodynamic shape of the vessel has significantly enhanced the motion of the ship and consequently the generated power.

Fig. 4: Generator pendulum

Fig. 5: First generation vessel

Fig. 6: second generation

vessel The motion and orientation of the ship is one of the most critical pieces of information in this experimental trial. The ship motion acts directly as the base excitation for the pendulum energy harvesters on board the ship. Understanding the motion of the ship is a key component to determining how the pendulum energy harvesters will behave. The sensor that was used in this experimental model was X-IO Technologies’ “x-IMU” with a plastic housing and battery. An IMU is an Inertial Measurement Unit and is typically an array of sensors used to estimate the orientation of a body relative to earth or another referenced frame. shows the x-IMU. The rotations of the pendulum and the generated power are measured using Arduino microcontroller systems (Fig. 8)

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The developed models predict that the nonlinear roll-pitch energy transfer, which can increase the power generation of the wave generator, is sensitive to shape of the incident waves. The current challenge faced in the project is to generate pure sinusoidal waves using the inexpensive wave generation mechanism. The roll and pitch natural frequencies of the vessel have to be precisely tuned. We have achieved the tuning by placing magnetic masses on the vessel.

Abstracts, publications and invention disclosures

We have published two conference papers from this project so far:

1. K. Yerrapragada and M. A. Karami, “Utilization of Nonlinear Resonance of Vessels for Ocean Wave Power Generation”, in ASME 2015 International Design Engineering Technical Conferences (IDETC), Boston, MA, 2015

2. B. Kuch and M. A. Karami, “powering pacemakers with a nonlinear hybrid rotary-translational energy harvester” in ASME 2014 International Design Engineering Technical Conferences (IDETC), Buffalo, NY, 2014

Both papers are being improved to be submitted as journal publications.

Impact and the future plan

We are very thankful to the SUNY sustainability program for this grant. The page limit of this report did not allow us to elaborate the extent of our modeling achievement. So far we have succeeded in developing a model to be able to assess the feasibility of the idea. The model helps us in development of experimental test setups by identifying the optimal dimensions of the wave generator. We have created the setup to generate wave in an existing flume. We also have developed two generations of functional wave generating vessels.

The PI intends to submit an Early Career Research grant to the Department of Energy on the idea investigated in this project. The proposed wave generator is scalable and can be to power small-scale floating sensors. The PI is planning on utilizing this technology for monitoring of the great lakes and for powering the ocean based sensor networks required for precision weather predictions.

Fig. 7: : x-IMU without casing and

battery

Fig. 8: Arduino data logger

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References:

[1]   (2013).  Wello  Direct  Conversion.  Available:  http://www.wello.eu/penguin.php  [2]   J.   G.   Bretl,   A   time   domain   model   for   wave   induced   motions   coupled   to   energy  

extraction:  ProQuest,  2009.  [3]   B.  C.  Boren,  "On  the  Modeling  and  Control  of  Horizontal  Pendulum  Wave  Energy,",  

Oregon  State  University.  [4]   B.   C.   Boren,   B.   A.   Batten,   and   R.   K.   Paasch,   "Model   Predictive   and   Integral   Error  

Tracking  Control  of  a  Vertical  Axis  Pendulum  Wave  Energy  Converter,"  2014.  [5]   B.  C.  Boren,  B.  A.  Batten,  and  R.  K.  Paasch,  "Active  control  of  a  vertical  axis  pendulum  

wave   energy   converter,"   in   American   Control   Conference   (ACC),   2014,   2014,   pp.  1033-­‐1038.  

[6]   A.  H.  Nayfeh,  D.  T.  Mook,   and  L.  R.  Marshall,   "Nonlinear   coupling  of   pitch   and   roll  modes  in  ship  motions,"  Journal  of  Hydronautics,  vol.  7,  pp.  145-­‐152,  1973.  

[7]   D.   T.   Mook,   L.   R.   Marshall,   and   A.   H.   Nayfeh,   "Subharmonic   and   superharmonic  resonances  in  the  pitch  and  roll  modes  of  ship  motions,"  Journal  of  Hydronautics,  vol.  8,  pp.  32-­‐40,  1974.