Resonant frequency tuning of wave energy converters using variable moment of inertia

2.1 Modeling

A schematic of the yo-yo vibrating system is illustrated in Figure 1. The proof mass and spring are connected to the reels by wires, and the translating motions of the proof mass and spring are converted to rotation of the reels. The radii of the reels are denoted as r_m and r_s , respectively. When the stiffness of the spring is k and the proof mass is m, the impedances of the translating and rotating parts can be respectively given by

Figure 1:

Schematic of the proposed yo-yo vibrational energy harvester with a variable moment of inertia device

where F and V are the excitation force on the proof mass and its velocity, respectively; τ and Ω are the torque applied to the reel axis and its rotational velocity, respectively; c is the damping coefficient; J is the integrated moment of inertia of the rotating part; and ω is the base excitation frequency. Because τ = r_mF and V = r_mΩ, the impedance in Equation (1a) can be expressed as

The total impedance of the system can be obtained from the parallel connection of Equations (1b) and (2), as follows:

When the amplitude of the excitation is Y , the excitation torque at the proof mass is τ = −mr_mYω²sinωt. Substituting it to Equation (3), the rotating speed of the reel can be expressed as follows:

The undamped resonant frequency f in which the rotating speed reaches its maximum value can be obtained from the denominator of Equation (4):

where T is the period and γ_s = r_s⁄r_m. The resonant frequency of the yo-yo vibrating system can be modulated via a geometric parameter γ_s and the stiffness-to-mass ratio.

2.2 VMI

The resonant frequency in Equation (5) can be further modified using the integrated moment of inertia of rotating part J. The VMI device, as shown in Figure 1, can be realized. Three auxiliary masses m_a are attached to the mass reel. When their distances ρ from the center are identical, the integrated moment of inertia can be rewritten as

where n is the number of auxiliary masses and J₀ is the moment of inertia of the reel assembly. The resonant frequency with VMI can then be derived from Equation (5) by substituting Equation (6):

where n is the total number of auxiliary masses and γ_a = ρ⁄r_m. In Equation (7), the resonant frequency can vary with the geometric parameter γ_a. In field applications, it is difficult to change the resonant frequency from the constituent parameters, that is, m, k, and the ratio of the two reels, γ_s. Thus, the introduction of a VMI device can be a good choice for tuning of resonant wave power generators.

2.3 Property

The performance of the VMI device can be measured by using the ratio of the two resonant frequencies of the yo-yo vibrating system with and without VMI. For simplicity, let J₀ = 0, which yields

where $$ \mathrm{\mu }=\frac{\sum _{i=1}^n{m}_{a,i}}{m}$$. Figure 2 illustrates the frequency ratios for μ and γ_a, which show that the resonant frequency can be reduced when both the total mass of the auxiliary masses and distance ρ increase. In Figure 2, the frequency ratio variance of the mass ratio decreases slightly when μ changes from 0.1 to 0.5, whereas that of γ_a increases when γ_a changes from 0 to 1. Thus, geometrical tuning by manipulating the distance between auxiliary masses is more efficient for the VMI device.

3.1 Implementation

A yo-yo vibrating system with a VMI device was fabricated, as shown in Figure 3. Iron wires are wound around the reels and connected to the spring and proof mass, respectively. The two reels are fixed to the shaft and their rotational displacements are recorded by an encoder (AUTONICS E50S8-100-3-T-1). The physical parameters of the experimental setup are listed in Table 1. The implementation of the VMI is shown in Figure 4. Three auxiliary masses are attached to the mass reel at three positions. The distance ratios (γ_a) are 0.32, 0.60, and 0.89. The resonant frequencies were then calculated as 0.65, 0.62, and 0.58 Hz for the three positions of the VMI device based on the parameters in Table 1 using Equation (7). These frequencies correspond to the periods of 1.56, 1.61, and 1.72 s, which are often found in shallow seas. This result also shows that the maximum frequency change is 10.7% of the largest resonant frequency at the lowest position of the VMI device.

Figure 3:

Experimental setup: (a) front view and (b) top view

Table 1:

Physical parameters

Figure 4:

Three positions of auxiliary masses

3.2 Experimental validation

To determine the resonant frequency of the yo-yo vibrating system with VMI, we pulled and released the proof mass. The vibrational motions of the reels were recorded for 30 s by the encoder and PXI system of National Instruments. The experimental results are shown in Figure 5. The measured frequencies obtained from the numbers of oscillations divided by the elapsed times are 0.61, 0.57, and 0.54 Hz, respectively, when the distances are 50, 93, and 137.5 mm. The differences between the numerical calculations are 6, 8, and 7%, respectively. These errors are attributed to the inertias of the neglected parts, for example, wires, and enlargement of the effective radius of the reels owing to the winding. The frequency change range from the experiment is 11.5%, which is similar to that of the numerical calculation.

Figure 5:

Rotation angle responses from free vibration tests: (a) ρ = 50 mm (b) ρ = 93 mm (c) ρ = 137.5 mm

Author Contributions

Conceptualization, S. -J. Jang; Methodology, S. -J. Jang; Formal Analysis, S. -J. Jang; Investigation, S. -J. Jang; Data Curation B. -R. Kim; Writing-Original Draft Preparation, S. -J. Jang; Writing-Review & Editing, S. -J. Jang; Project Administration, S. -J. Jang; Funding Acquisition, S. -J. Jang.