Connected Floating Balls Dynamics for Harvesting Energy of Sinusoidal Water Wave

Dynamics of connected balls floating on surface of fluid oscillating sinusoidally has reported. In this work a simple electrical generator is simply modeled based on the change of distance between connected balls from its initial value or Δl. By considering only one-dimensional motion of each ball, all the changes Δl are positive, that makes the system easier to be predicted. Sum of all Δl as function of initial initial distance between balls or l is discussed. Water wave wavelength λ is also varied to see how it influences the sum of all Δl.


Introduction
There are various mechanisms in harvesting mechanical energy, especially in the form of vibration, from inertial-based generator with spring to cantilever-based system [1], where the mechanism similar to the last system has been further transformed into flexible films based on nanowires [2]. Multiple coils and freestanding magnet-based system has also been developed to accomodate all-direction inplane vibration for more efficient energy harvesting [3]. For low-frequency vibration, e.g. impact of human steps while walking, piezoelectric-based energy harvester system is more practical in the implementation [4]. Accompanying electromagnetic and piezoelectric, electrostatic method play also important role in miniaturizing the energy harvesting system [5]. Triboelectric nanogenerator is promising approach since it can harvest vibration energy from a system that produces vibration while operating or has vibration due to environment change, a self-powered system [6], that has been implemented into the buoy ball for harvesting wave energy [7]. Interaction between a buoy ball and sinusoidal surface wave shows interesting motion modes [8], which is neglected in this work for simplicity. Interaction between two buoy balls is only the given connector acting like spring, while the natural attractive force due to inbalance surface tension [9] is not considered.

Model
A ball i with density ρi and diameter Di will have mass of which relates to gravitational force where vertical direction is ŷ . If position of fluid surface is yf and vertical position of ball i is yi then immersed volume of ball i is given by [8]

Using equation (3) the buoyant force on the ball i is simply
with ρf is fluid density. Spring force between ball i and j with normal length lij and spring constant kij is given by ij ij ij r r r   .

(8)
Total force works on ball i is and its acceleration at this time t using Newton's second law of motion. At time t + Δt velocity and position can be calculated with Euler algorithm (12) Position of fluid surface is obtained from a sinusoidal function (13) where Af and ωf are oscillation amplitude and anguler frequency, respectively. The time t in equation (13) will be advanced from tbeg to tend in the simulation according to Notice that yf in equation (3) is different for every ball i, where from equation (13), it is clearer to state that yf,i(t) = yf(x, t) instead of only yf. Figure 1(a) shows the simulation flow chart in using equations (1) - (14), where vectors are written in bold but without arrow. Output is time, position and velocity of all buoy balls in the system, which later can be used to calculate converterd electric power from the vibration as shown in figure 1(b). It can be seen from figure 2 that due to local oscillation of water surface, vertical position of each buoy ball will be altered periodically, which stretches the spring-like connectors (red line) or at least let the connector at its normal length (green line). Length variation l of the connectors will be used to calculate harvestable vibration energy to be converted to electrical one. The conversion function is given in figure 2 as where n is the type of implemented technology.  As the time t is advanced all buoy balls will also be altered in vertical direction, while their horizontal position are assumed to be constant. An illustration for l = 0.15 m is given in figure 4.  It can be seen from figure 4 that the water wave propagates to the right. Distance between two succesive buoy balls will be substrated with the initial distance, that will produce Δl between two connected balls. There are N different values of Δl that the sum as function of l is given in figure 5.  As function of l the sum of Δl is no simply linear but can be said that it increases as l increses as shown in figure 5. There is also influence of λ, where larger value will reduce the gradient of sum of Δl as function of l. By assuming that equation (15) holds for linear relation between U and Δl, then figure 5 can already tell how the obtained energy as function of l and λ. The first is system paramter, while the second is environment parameter. for each water wavelength λ, gives the relation as shown in figure 6, that the gradient m decreases as λ increases. Coefficien of determination for each λ in figure 6 are 0.9702, 0.95759, 0.9674, 0.9494, and 0.9523, which show that the linear fit can still be accepted.

Summary
Model of vibration energy system based on connecting floating balls or buoy balls has been discussed. By assuming that each ball only move in vertical direction under influence of sinusoidal water wave, it has been observed that energy increases as distance between two buoy balls increases, while energy decreases as water wavelength increases.