# Improved VIV energy harvesting with a virtual damper-spring system

Agathe Schmider, Franck Kerhervé, Andreas Spohn, Laurent Cordier

► To cite this version:

Agathe Schmider, Franck Kerhervé, Andreas Spohn, Laurent Cordier. Improved VIV energy harvesting with a virtual damper-spring system. Ocean Engineering, 2024, 293, pp.116668. 10.1016/j.oceaneng.2024.116668. hal-04717022

HAL Id: hal-04717022

https://hal.science/hal-04717022v1

Submitted on 1 Oct 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
<figure><img src="image_2.png" /></figure>

<figure><img src="image_3.png" /></figure>

# Improved VIV energy harvesting with a virtual damper-spring system

Agathe Schmider *, Franck Kerhervé, Andreas Spohn, Laurent Cordier

Institut Pprime, CNRS, Université de Poitiers, ISAE-ENSMA, UPR 3346, France

## ARTICLE INFO

**Keywords:**

Vortex-induced vibrations

Energy harvesting

Virtual damper-spring

Efficiency

## ABSTRACT

In recent years, energy harvesting systems driven by vortex induced vibrations (VIV) emerged as potential alternative to conventional rotor-based installations, in particular for low flow speeds in water. Best energy outputs are obtained when the flow-dependent VIV frequency locks in the natural frequency of the energy harvesting structure. However, placed in tidal currents or rivers with variable flow rates, the VIV frequency changes, leading to frequency mismatch with efficiency losses. In this paper, we present a new mechatronic experimental set-up to study the harvesting efficiency of a VIV driven cylinder under various flow rates. In particular, we explore options to improve harvesting efficiency by adding computer controlled virtual damping and stiffness to the mechanical system. The setup comprises an elastically mounted, single-degree-of-freedom cylinder within a low-speed water tunnel. Virtual damping and stiffness are applied by controlling the force exerted through a mechanically coupled DC motor. The experiments reveal that optimal values for damping and stiffness are consistent with previous results from the literature. The maximum energy conversion efficiency of around 18% is achieved when oscillation and vortex shedding frequencies are synchronized. The explored parameter space highlights in particular that mismatches between oscillation and vortex shedding frequency strongly decrease the achievable efficiency. By contrast, the peak efficiency is less sensitive to an increase of damping due to energy harvesting. All these observations suggest that VIV driven energy harvesting systems need a real-time structural stiffness control to maintain an economically acceptable efficiency under various flow conditions. In the near future, we plan to automate the structural stiffness control under various flow rates by testing Machine Learning algorithms with our mechatronic set-up.

## 1. Introduction

Vortex-induced vibrations (VIV) have long been seen as a phenomenon to be suppressed (Williamson and Govardhan, 2004; Hao et al., 2022; Zhao, 2023), since they can generate very high hydrodynamic forces on bluff bodies. This is particularly true for oil risers and heat exchangers where VIV may cause severe oscillations and provoke eventually structural failure (Zdravkovich, 1981; Krishnakumari et al., 2023; Zhu et al., 2023). Recently, VIV have emerged as a beneficial mechanism to exploit in the context of energy harvesting (Bernitsas et al., 2008; Ma and Zhou, 2022). In that case, the aim is often to enhance oscillation amplitudes to maximize the amount of harvested energy. The long-term objective is to harness energy from the ocean or other water resources to contribute to the global energy challenge. In these applications, flow conditions are naturally variable in time and space. Tidal currents and river flows for instance are inherently unsteady and random. The question of the robustness of the energy harvesting system is then an essential part of the specifications. To achieve this, we aspire to develop an experimental system for which

it is extremely simple to modify the values of the model parameters during an experiment. In this way, the system can adapt, if possible in real time, to changes in operating conditions. More ambitiously, we aim at developing an autonomous energy harvesting system that can perform self-learning and self-optimization. Recently developed Machine Learning techniques (Fan et al., 2019; Rabault et al., 2019) are ideally suited for this purpose. However, these algorithms require in general a huge amount of data before they can be deployed. In addition, learning data must be representative of all system operating conditions. It is therefore essential, whether for the learning phase or the real-time on-site application of optimization strategies, to have a highly flexible system that can easily cover the entire parameter space. This is the challenge of this paper, in which we present the experimental set-up developed and its performance.

This study focuses on the VIV response of a smooth circular cylinder allowed to oscillate only perpendicular to the free-stream. This so-called single-degree-of-freedom (DOF) configuration is particularly interesting for practical applications (Derakhshandeh et al., 2015; Chang

<footnote>* Corresponding author.</footnote>

<footnote>E-mail addresses: Agathe.Schmider@univ-poitiers.fr (A. Schmider), Franck.Kerhervé@univ-poitiers.fr (F. Kerhervé), Andreas.Spohn@ensma.fr (A. Spohn), Laurent.Cordier@univ-poitiers.fr (L. Cordier).</footnote>
<caption>**Table 1**</caption>

Main non-dimensional parameters for vortex induced vibrations of a single-degree-of-freedom circular cylinder. For the range of Reynolds numbers considered in this study, the Strouhal number is assumed constant and equal to $St = 0.2$. The amplitude and frequency responses of VIV are usually presented with reference to the reduced velocity $U^*$ where the characteristic time is the inverse of the natural frequency.

<table><thead><tr><th>Mass ratio</th><th>m<sup>*</sup></th><th>m<sub>s</sub>/m<sub>a</sub></th><th>m<sub>s</sub>/C<sub>a</sub>ρπ(D/2)<sup>2</sup>L<sub>i</sub></th></tr></thead><tbody><tr><td>Reduced amplitude or amplitude ratio</td><td>A<sup>*</sup></td><td>A/D</td><td></td></tr><tr><td>Reduced frequency or frequency ratio</td><td>f<sup>*</sup></td><td>f/f<sub>n</sub></td><td></td></tr><tr><td>Strouhal number</td><td>St</td><td>f<sub>sh</sub>D/U<sub>∞</sub></td><td></td></tr><tr><td>Reynolds number</td><td>Re<sub>D</sub></td><td>U<sub>∞</sub>D/v</td><td></td></tr><tr><td>Reduced velocity or velocity ratio</td><td>U<sup>*</sup></td><td>U<sub>∞</sub>/f<sub>n</sub>D</td><td></td></tr><tr><td>Structural damping ratio</td><td>ζ</td><td>c<sub>s</sub>/2√k<sub>s</sub>(m<sub>s</sub> + m<sub>a</sub>)</td><td></td></tr></tbody></table>

et al., 2022; Zhao, 2023). While only few recent experiments addressed the energetic efficiency of such an installation (Bernitsas et al., 2008; Soti et al., 2018; Gao et al., 2023), studies by Khalak and Williamson (1999), Williamson and Govardhan (2004, 2008), Han et al. (2023) among others, shed some light into the parameters that influence VIV oscillations. This research work, which analysed the vortex-induced oscillations of cylindrical bluff-bodies, was particularly useful for gaining a better understanding of the physical phenomena at play, and for refining our specifications. We recall the essential elements of VIV-induced dynamics in Section 1.2.

## 1.1. Parameters influencing the dynamics

VIV-induced oscillations of a single-degree-of-freedom elastically mounted circular cylinder can be characterized by the non-dimensional amplitude $A^* = A/D$ and frequency $f^* = f/f_n$, where D represents the diameter of the cylinder and $f_n$ the natural frequency of the oscillating system. The response to VIV excitation further depends on

- the mass ratio $m^*$ defined as the ratio between the structural mass $m_s$ and the added mass $m_a$,

- the Strouhal number $St$ defined as the ratio between the vortex shedding frequency in the cylinder wake $f_{sh}$ and the advection frequency in the flow $D/U_\infty$, where $U_\infty$ is the free-stream velocity,

- the Reynolds number $Re_D$, constructed from a reference velocity $U_\infty$, a reference length D and the kinematic viscosity $\nu$,

- the reduced velocity $U^* = U_\infty f_n/D$ and finally

- the damping ratio $\zeta$ defined as the ratio between the structural damping $c_s$ and the natural critical damping which involves $k_s$, the structural stiffness or spring constant.

The added mass $m_a$ is given by the mass of the displaced fluid $m_d = \rho\pi(D/2)^2L_i$ where $\rho$ is the fluid density and $L_i$ the immersed cylinder length, multiplied by $C_a$, the coefficient of added mass, which is close to 1 for the potential flow around a circular cylinder. The main physical non-dimensional parameters of interests are summarized in Table 1.

## 1.2. Dynamics of oscillating systems under VIV

An important question that has been debated for decades in the VIV community is whether a unique parameter can collapse reduced amplitudes $A^*$ for oscillating systems of varying mass ratio and Reynolds number (Kang et al., 2019). Khalak and Williamson (1999) found two distinct types of oscillation modes by using the combined mass-damping ratio $m^*\zeta$ as parameter. For low values of $m^*\zeta$, the evolution of the reduced amplitude $A^*$ versus the reduced velocity $U^*$ (the

so-called “Griffin plot”) shows a three-branch behaviour, denoted as initial branch, upper branch and lower branch, respectively (see later Fig. 6(a)). According to the experiments of Williamson and Roshko (1988), Williamson and Govardhan (2004), each branch is associated with a particular wake vortex pattern. In contrast, for high values of $m^*\zeta$, only the initial and lower branches are observed. This last configuration is referred in the literature as Feng-type or two-branch response (Feng, 1968).

One major difficulty in the analysis of the oscillation amplitudes lay in the important scattering of available experimental data obtained for various mass ratios $m^*$ and Reynolds numbers $Re_D$. To overcome this difficulty, different experimental set-ups with tunable damping have been considered in the literature. The pioneering work of Feng (1968) used an electromagnetic-eddy-current damper to study the effect of damping on VIV of circular and D-shaped cylinders. Since the experiments were carried out in an air flow, the apparatus was constrained to high mass-damping ratios ($m^*\zeta \ge 0.35$). As previously announced, the vibration response exhibited only a low-amplitude two-branch type. An increase in damping was found to result in a decrease of the maximal oscillation amplitude. Simultaneously, the frequency range matching the natural vortex shedding frequency diminished. Later, Khalak and Williamson (1999) conducted a detailed investigation of VIV oscillations for an elastically mounted rigid cylinder at low mass-damping ($0.014 \le m^*\zeta \le 0.019$). The authors showed peak-amplitude data collapse over a wide range of mass ratios when the influence of the added mass is taken into account in the mass-damping parameter. For that, they proposed a modified mass-damping parameter $\alpha = (m^* + C_a)\zeta$. More recently, Klamo et al. (2005) proposed a variable magnetic eddy-current damping system. In these experiments, mass-damping values from 0.01 to 0.2 were achieved. The maximum amplitude $A^*$ was found to be independent of the mass ratio $m^*$ but dependent on the Reynolds number $Re_D$. The authors concluded that this Reynolds number dependency was responsible for the significant scattering of experimental data reported in the literature. In 2006, Govardhan and Williamson tested the previous propositions of Khalak and Williamson (1999) and Klamo et al. (2005). For this, they used a variable damper system based on a spring, whose base is displaced in or out of phase with the oscillating body’s velocity, resulting in an external force proportional to the oscillating mass velocity. It was then possible to vary $\zeta$ by applying negative and positive damping to the oscillating body. This study confirmed the usefulness of $\alpha$ and proposed a quadratic fit of the peak amplitude as a function of $\alpha$ and $Re_D$. A few years ago, Soti et al. (2018) improved an eddy-current damping system by controlling precisely the distance between an immobile conductive plate and a permanent magnet mounted on the oscillating part. The authors used the range of mass-damping ratio $m^*\zeta$ allowed by the experimental setup ($0.006 \le m^*\zeta \le 0.684$) to propose an improved exponential fit of the peak amplitudes as a function of the mass-damping parameter $\alpha$. This partial bibliography shows that many parameters are involved in the system’s response to VIV. Physically setting parameters by means of a mechanical arrangement does not therefore enable the development of an highly automated experimental set-up. The use of a mechatronic approach, combining in a synergistic manner, mechanics, electronics, control engineering and computer science, should undoubtedly surpass the initial mechanical constraints. This is the direction we analyse in the next section.

## 1.3. Virtualization of parameter changes: towards a mechatronic solution

In this paper, we follow the concept of cyber-physical devices, introduced in the late 90’s by Miller (1996). Hover et al. (1997) was one of the first to implement this idea in VIV by implementing a force-feedback controller for virtually impose any arbitrary force. The forces acting on the oscillating cylinder (the physical part) were measured experimentally and used to drive a real-time simulation (the cyber
part) of the one-dimensional motion equation, given arbitrarily stiffness, structural mass and damping. The oscillating cylinder was then forced to follow the estimated trajectory. Later, similar set-ups were developed in various contexts related to VIV oscillations (Mackowski and Williamson, 2011; Derakhshandeh et al., 2015; Onoue et al., 2015). More recently, Riches and Morton (2018) used this approach to study the behaviour of an oscillating cylinder for different values of $m^*$ and $m^*\zeta$ at fixed Reynolds number. However, due to the necessary filtering of the noisy force signal, these force-feedback controllers have the disadvantage of causing a phase lag between the cylinder displacement and the force really acting on the oscillation body. Phase lag has been pointed out to be detrimental for energy harvesting by Bernitsas et al. (2008). To overcome this difficulty, Lee et al. (2011) developed a virtual damper-spring system with a servo motor providing a feedback force based on displacement and velocity measurements. This method avoids any phase lag, but requires compensation for the additional friction and inertial mass effect associated with the mechanical coupling between the servo motor and the moving part.

The results reported in this paper are produced with a similar virtual damper-spring system coupled to an elastically mounted rigid circular cylinder, oscillating in cross-flow. The mechanical parameters of the oscillator are virtually modified by exerting a compensating force calculated in real-time. The latter is the sum of a damping force, proportional to the cylinder velocity, and a stiffness force proportional to the cylinder displacement. This force is applied by a DC motor which acts as motor or generator. The device allows to simulate positive or negative damping values.

The paper is organized as follows. The developed experimental setup is first presented in Section 2. The operating principle of the virtual damper-spring system is then given in Section 3. Section 4 reports on free-decay tests performed in air and water to qualify the developed system experimentally. It is shown that it is possible to calibrate the system's virtual parameters to compensate for the additional frictional force associated with coupling the oscillating system to the motor. The influence of virtual damping and stiffness on natural oscillation frequencies and damping ratios is also investigated. Section 5 first presents the VIV responses of the virtual damper-spring system for the uncoupled and coupled oscillating system, and then studies the influence of virtual stiffness and damping on the oscillation amplitude. In Section 6, we analyse the harvesting efficiency and present the optimum operating parameters. Finally, Section 7 offers some concluding remarks.

## 2. Experimental set-up

In this section, we present the main features of the developed experimental set-up. Fig. 1 shows a side view of the low-speed, closed-loop water channel in which the oscillating cylinder is mounted.

The flow is driven by an axial pump located in the return pipe of the closed-circuit water channel. The test section is 2.1 m long, 0.51 m wide and filled with water up to a height of 0.33 m. The flow can be observed through all four lateral sides of the test section, as well as from behind in the upstream direction. The free-stream velocity can be regulated between 5 and 45 cm s⁻¹ with a precision better than 2%. The flow uniformity all along the test section was checked in different planes by two-dimensional, two-component (2D-2C) Particle Image Velocimetry (PIV). The measured deviation from flow uniformity of the velocity profiles is better than 2%, while flow unsteadiness remains less than 5% over the whole velocity range. Fig. 2 shows the experimental rig with the cylinder mounted on the air bearings system. To reduce the oscillating mass as far as possible, the circular cylinder is hollow and made in aluminium with a wall thickness $e = 1.8$ mm. The outside diameter of the cylinder is $D = 50$ mm and its length is $L = 400$ mm. The cylinder is vertically immersed 400 mm downstream of the beginning of the test section. The blockage ratio due to the presence of the cylinder in the test section is 10%. To allow the cylinder

to move freely, the gap between the cylinder and the channel floor is set to 2 mm. This small gap, together with the chosen aspect ratio of the immersed cylinder ($L_i/D = 6.6$), helps to ensure parallel shedding of the wake vortices (Szeppesy and Bearman, 1992).

The cylinder is mounted on a sliding table made in aluminium. To reduce the mechanical friction in the crossflow direction, this table supports three commercial OAV air bearings supplied with a pressurized air flow at 4 bar. The guide shafts of the air bearings system are made of two high precision stainless steel shafts of 20.00(2) mm diameter. A pair of linear springs (LeeSpring LE014B13S), mounted in parallel on each side of the sliding table, provides the restoring force. The equivalent spring constant of the system is $k_s = 0.022$ N mm⁻¹. A low inertia 24 V DC servo motor with a dual track encoder is coupled to the sliding table with a rack and pinion mechanism. Such arrangement results in additional dry friction due to the mechanical coupling. This arrangement is used for three reasons. First, it allows to measure in real-time, accurately and robustly, the position and velocity of the cylinder. Second, it enables to quantify the energy that can be harvested from the oscillating system by measuring on-line the generated current. Finally, the motor can be used to apply an arbitrary time-dependent forcing on the oscillating mass. In particular, this possibility may compensate for additional sources of friction, such as the one induced by the mechanical coupling of the motor. This type of compensation is introduced in Section 3 and further discussed in Section 4.1. This arrangement allows the cylinder to move in the cross-flow direction over a maximum distance of $\pm 2.5$ D with minimal damping. The overall oscillating mass is $m_s = 2.740$ kg which results in a mass ratio $m^* = m_s/m_a = 4.2$ with $m_a$ the added mass estimated at 0.65 kg.

The encoder of the DC motor (Allied Motion, 30 W, 1600 rev min⁻¹) is directly used to measure both the transverse displacement y and velocity $\dot{y}$ of the cylinder. To this end, the encoder furnishes 500 TTL pulses per revolution. A NI PXIe-6356 I/O acquisition card acquires the TTL signal with a sampling frequency of 500 Hz. In this way, the position of the cylinder is measured with an accuracy of $\pm 0.15$ mm.

## 3. Virtual damper-spring system

The immersed cylinder described in Section 2 behaves like a damped single-degree-of-freedom oscillator under unsteady external forces. Aerodynamic forces $F_y$ are created by pressure and shear stresses caused by vortex structures periodically shed in the cylinder's wake. This resulting aerodynamic transverse force can be expressed via the related aerodynamic coefficient $C_y$, i.e. $F_y(t) = \frac{1}{2}\rho U_\infty^2 D L_i C_y(t)$. The mechanical coupling of the DC motor with the oscillating system introduces an additional friction force $F_\mu$ that must be counterbalanced. This force can be modelled by a Coulomb friction force given by $\mu_c \operatorname{sgn}(\dot{y})$ where sgn is the sign function (Feeny and Liang, 1996). Finally, by driving the DC motor via an unsteady current, an additional force $F_{DC}$ is introduced (the motor is torque controlled). Under these conditions, the equations of motion for the oscillating system can be written as follows

$$ \underbrace{m}_{m_s+m_a} \dot{y}(t) + c_s \dot{y}(t) + k_s y(t) + \underbrace{F_\mu(t)}_{\mu_c \operatorname{sgn}(\dot{y}(t))} = \underbrace{F_y(t)}_{\frac{1}{2} \rho U_\infty^2 D L_i C_y(t)} + F_{DC}(t). \quad (1) $$

The idea behind the “cyber-physical devices” presented in Section 1 is to exploit the possibility of controlling the DC motor to compensate for frictional forces or to modify the system's mechanical parameters $c_s$ and $k_s$. To do this, knowing the cylinder position y and velocity $\dot{y}$, we can estimate the values of control parameters $(\Delta c, \Delta k, \hat{\mu}_c)$ such as

$$ F_{DC}(t) = -\Delta c \dot{y}(t) - \Delta k y(t) + \hat{\mu}_c \operatorname{sgn}(\dot{y}(t)), \quad (2) $$

where $\hat{\mu}_c$ is the estimated value of $\mu_c$ compensating exactly for dry friction $(\Delta c = \Delta k = 0)$. Hereafter, $\Delta c$ and $\Delta k$ are referred to as virtual damping and virtual stiffness, respectively. To simplify the presentation of results, we also refer to $S^u$ as the oscillating system when the motor
<figure><img src="image_4.png" /><figcaption>Fig. 1. Side view of the closed-circuit water loop. The elastically mounted rigid cylinder is represented in the test section.</figcaption></figure>

<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 2. (a) Schematic view and (b) photography of the experimental set-up. In this paper, we treat only the single-degree-of-freedom configuration. For this reason, the DC motor meant for rotating the cylinder is not used.</figcaption></figure>

<table><caption>**Table 2**
Notations used to differentiate the different configurations of interest.</caption><tr><td>S<sup>u</sup></td><td>motor uncoupled (μ<sub>c</sub> = 0)</td></tr><tr><td>S<sup>c</sup></td><td>motor coupled but dry friction<br>not compensated (μ<sub>c</sub> ≠ 0 and μ̂<sub>c</sub> = 0)</td></tr><tr><td>S<sup>cc</sup>(p)</td><td>motor coupled, dry friction compensated<br>(μ̂<sub>c</sub> = μ<sub>c</sub> ≠ 0)</td></tr><tr><td>with p = (Δk/k<sub>s</sub>, Δc/c<sub>s</sub>)</td><td>and k<sub>s</sub> and c<sub>s</sub> modified by Δk and Δc</td></tr></table>

is uncoupled ($μ_c = 0$). By extension, we denote $S^c$, the oscillating system when the motor is coupled and dry friction is not compensated ($μ_c ≠ 0$ and $μ̂_c = 0$). Lastly, we note $S^{cc}(p)$ with<fnref n="1" /> $p = (Δk/k_s, Δc/c_s)$, the oscillating system when the motor is coupled, dry friction is compensated ($μ̂_c = μ_c ≠ 0$) and the structural values of stiffness and damping are modified by the values $Δk$ and $Δc$, respectively. Since the aim of friction compensation is to return to the uncoupled system, we would like to ensure that $S^u ≡ S^{cc}(0)$. For ease of reading of the coming sections, all these notations are listed in Table 2. The parameter $Δc$ can take positive or negative values to account for increased or reduced damping. Similarly, by changing the value of $Δk$, we can vary the stiffness of the set-up without mechanically modifying the spring configuration.

The forcing $F_{DC}$ is applied to the oscillating system by controlling the current of the DC motor in a closed-loop manner. For both damping and stiffness, the range of possible values $Δc$ and $Δk$ are restricted by limitations of the output current furnished by the motor. The signals $y$ and $\dot{y}$ are acquired continuously by a NI PXI-RT board (PXIe-1071), which furnishes in return the command signal for the DC motor.

## 4. Experimental qualification of the virtual damper-spring system

In this section, free decay tests are used to experimentally qualify the virtual system conceptually introduced in Section 3. We show (Section 4.1) that the additional dry friction introduced by coupling the DC motor to the oscillating system can be compensated. In Section 4.2, we determine the influence of virtual stiffness and damping on the natural oscillation frequency and damping ratio.

### 4.1. Additional dry friction compensation

As the coupling of the DC motor to the oscillating system introduces additional dry friction, it is necessary to study its influence on the system dynamics and introduce a method for compensating it. To do this, we are going to carry out free decay experiments on the oscillating system in air and water. These experiments are made with a high-precision CCD Laser Displacement Sensor (Keyence), which replaces the information from the DC motor encoder. We start with the configuration $S^u$, i.e. the case where the oscillating system is disconnected from the motor. These results will be used as a reference.

<footnote n="1">¹ Remember that the first and second argument of $p$ is given in units of $k_s$ and $c_s$, respectively.</footnote>
<table><caption>**Table 3**</caption><caption>Characteristic parameters of the virtual damper-spring system for the configurations S<sup>u</sup>, S<sup>c</sup> and S<sup>cc</sup>(0) with m* = 4.2. We report the values of structural stiffness k<sub>s</sub> and damping c<sub>s</sub>, the natural frequencies f<sub>n</sub><sup>a</sup> and f<sub>n</sub> in air and water, respectively, the values of damping ratios ζ<sup>a</sup> and ζ in air and water, respectively and the mass-damping ratio (m*ζ).</caption><thead><tr><th rowspan="2">Case</th><th>k<sub>s</sub></th><th>c<sub>s</sub></th><th colspan="2">Air</th><th colspan="3">Water</th></tr><tr><th>(N m<sup>-1</sup>)</th><th>(N s m<sup>-1</sup>)</th><th>f<sub>n</sub><sup>a</sup> (Hz)</th><th>ζ<sup>a</sup></th><th>f<sub>n</sub> (Hz)</th><th>ζ</th><th>m*ζ</th></tr></thead><tbody><tr><td>S<sup>u</sup></td><td>0.022</td><td>0.039</td><td>0.463</td><td>0.0027</td><td>0.417</td><td>0.0024</td><td>0.010</td></tr><tr><td>S<sup>c</sup></td><td>0.022</td><td>0.757</td><td>0.459</td><td>0.0528</td><td>0.416</td><td>0.0479</td><td>0.202</td></tr><tr><td>S<sup>cc</sup>(0)</td><td>0.022</td><td>0.049</td><td>0.464</td><td>0.0034</td><td>0.415</td><td>0.0030</td><td>0.013</td></tr></tbody></table>

<figure><img src="image_7.png" /><figcaption>**Fig. 3.** Time histories of the non-dimensional cylinder displacement $y(t^*)/y_0$ for free-decay tests (a) in air and (b) in water obtained for the configurations: S<sup>u</sup>(dashed), S<sup>c</sup>(dot-dashed) and S<sup>cc</sup>(0) (plain). We note $y(0) = y_0$.</figcaption></figure>
The oscillating system can be modelled as a damped single-DOF system which solution is well approximated (see Zhao, 2017, §3.7.2) for underdamped free decay<fnref n="2" /> (0 ≤ ζ < 1) by<fnref n="3" />

$$y(t) = y(0)e^{-\zeta\omega_n t}\cos(\sqrt{1-\zeta^2}\omega_n t), \quad (3)$$

where ζ denotes the damping ratio and ω_n = 2πf_n the corresponding natural angular frequency. The parameters ζ and ω_n in Eq. (3) are identified using the Nelder-Mead method, also known as the downhill simplex method (Powell, 1973). This gradient-free optimization algorithm is simple to implement and known for its efficiency in the case of non-derivable functions. Fig. 3(a) shows that air bearings introduce very low damping (see the results for the configuration S⁰). The values of f_nª and ζª obtained in air are given in Table 3.

When the DC motor is coupled to the oscillating system without compensation for dry friction (configuration Sᶜ), the structural damping coefficient increases, as clearly shown in Fig. 3(a) (see also Table 3 for frequency and damping ratio). While the value of ζª is an order of magnitude higher than the one obtained for the S⁰ configuration, the natural frequency f_nª remains nearly constant. To compensate for the additional dry friction, the current supplied to the DC motor can be increased to create an external friction force represented by the expression F_μ(t) = μ_c sgn(y(t)) (see Section 3). A first guess of the value of μ_c can be obtained by applying the method presented in Feeny and Liang (1996). The method assumes that viscous friction dominates in the large-amplitude oscillations, whereas Coulomb friction dominates low-amplitude oscillations. In the absence of forcing and in still flow, Eq. (1) is piece-wise solvable for ȳ < 0 and ȳ > 0. A recursive relation for the successive peaks and valleys of the free-oscillating response can be used to isolate the viscous effect and then to extract the Coulomb friction term μ_c thanks to a logarithmic decrement. The value estimated in this way recovers almost exactly the damping ratio of the uncoupled system S⁰ (not shown here for conciseness), but a shift of around 10% from the value of the natural frequency is observed. The forcing term is adjusted accordingly, becoming F_μ(t) = μ_c sgn(y(t)) + Δk_c y(t). Thereby, the value pair (μ_c, Δk_c) is optimized via a Nelder-Mead method using experimental data. By convention, the resulting configuration is hereafter referred to as Sᶜ(0), see Table 2. As illustrated in Fig. 3(a), the free oscillation response of the compensated coupled system Sᶜ(0) fully matches the reference case S⁰. The same conclusion is reached by comparing the frequency and damping ratio values in Table 3.

Similar free decay tests were conducted on the oscillating cylinder immersed in still water. In this case, hydrodynamic forces (so-called Morison force described by Blevins 1990) act on the cylinder. The total damping therefore comprises the structural damping with added mass and the fluid damping. In the following, we consider for expression of the damping ratio, the one constantly used by Williamson's group, i.e. (see Zhao, 2017, §3.7):

$$\zeta = \frac{\omega_n}{\omega_n^a} \zeta^a \quad (4)$$

with ω_n the undamped natural angular frequency in still water. Fig. 3(b) shows the corresponding temporal evolution of the non-dimensional amplitude for S⁰, Sᶜ and Sᶜ(0). Due to the higher viscous friction in water than in air, free oscillations die out more rapidly in water. The values (μ_c, Δk_c) previously determined in air are retained. Here again, we find that the configuration Sᶜ(0) behaves similarly to the reference configuration S⁰. The natural frequency and damping ratio values are given in Table 3, showing that configuration Sᶜ(0) leads to a value of m*ζ almost as low as for the uncoupled configuration S⁰.

## 4.2. Influence of virtual stiffness and damping on the natural oscillation frequency and damping ratio

As we showed in Section 4.1, we can calibrate the parameters of an additional friction force F_μ to compensate for the friction introduced by coupling the DC motor to the oscillating system. In this section, we analyse the effect on the system's dynamics of correcting the structural values of damping c_s and stiffness k_s by Δc and Δk, respectively. To do this, we are performing free decay tests in air, as before. With friction compensated, the equation of motion (1) reduces to

$$m\ddot{y} + (c_s + \Delta c)\dot{y} + (k_s + \Delta k)y = 0 \quad (5)$$

or, in a more usual form

$$\ddot{y} + 2\zeta^a \omega_n^a \dot{y} + (\omega_n^a)^2 y = 0 \quad (6)$$

where ω_nª is given by

$$\omega_n^a = \omega_0^2 + \frac{\Delta k}{m} \quad \text{with} \quad \omega_0^2 = \frac{k_s}{m} \quad (7)$$

and ζª is expressed as

$$\zeta^a = \frac{1}{\sqrt{1 + \Delta k / k_s}} \left( \zeta_0 + \frac{\Delta c}{2\sqrt{k_s m}} \right) \quad \text{with} \quad \zeta_0 = \frac{c_s}{2\sqrt{k_s m}} \quad (8)$$

Eqs. (7) and (8) indicate how the values of the natural angular frequency ω_nª and of the damping ratio ζª vary by modifying the values of Δk and Δc. As in the case of friction compensation considered in Section 4.1, we use the Nelder-Mead algorithm to determine the model parameters. In Fig. 4(a), we compare for three different values of Δc, the reduced natural angular frequencies ω_nª/ω_0 obtained by identification with those given by Eq. (7). As expected, we find that the natural angular frequency increases with increasing Δk value and that ω_nª is quite insensitive to changes in the virtual damping coefficient. Furthermore, the variation of ω_nª with Δk is well predicted by Eq. (7). This general behaviour is highly relevant from a physical point of view. Indeed, changing the natural frequency of the system independently of the Reynolds number by modifying the value of the stiffness coefficient is particularly interesting. In this way, it becomes possible to adjust the system's response to maximize the amplitude of oscillations over a wide range of free-stream velocities (Chang et al., 2022; Raghavan and Bernitsas, 2011).

Fig. 4(b) shows the influence of the virtual damping coefficient Δc on the damping ratio ζª for Δk = 0. As expected, an increase in Δc results in an increase in the damping ratio. The measurements confirm the linear dependence of the damping ratio ζª with the virtual damping Δc predicted by Eq. (8). Note that we have only considered positive virtual damping values, as negative ones are of no interest in terms of energy harvesting. As further discussed in Section 6, the possibility of driving the reduced virtual damping is particularly interesting to optimize energy harvesting.

Finally, Fig. 5 shows the influence of the virtual stiffness Δk on the reduced structural damping for different values of virtual damping Δc. The evolution predicted by Eq. (8) is indicated as dashed lines. For positive values of Δk (resulting in an increase of the overall stiffness), the evolution of ζª is consistent with that predicted by Eq. (8). However, for negative values of virtual stiffness, the measurements differ significantly from the predicted values. Repeated experiments performed for Δc = 0 showed that the results were highly reproducible. Note that the same trend was observed for free decay tests performed in still water (not shown here for conciseness). Negative stiffness values are well known to increase structural damping due to increased dissipation. Clearly, it would be necessary to improve the model used in Eq. (5) in order to describe correctly the almost linear increase observed for the reduced structural damping. However, in the context of energy harvesting, negative stiffness values are not desirable, as the resulting energy dissipation would reduce the level of recovered energy. Consequently, we do not further consider the influence of negative stiffness.

<footnote n="2">² This hypothesis is verified a posteriori, as shown by the damping ratios reported in Table 3.</footnote>

<footnote n="3">³ The solution y is valid in both air and water.</footnote>
<figure><img src="image_8.png" /><figcaption>Fig. 4. (a) Variations of the reduced natural angular frequency $\omega_n^a/\omega_0$ as a function of the reduced virtual stiffness $\Delta k/k_s$ for different values of reduced virtual damping $\Delta c/c_s$. The model is given by Eq. (7). (b) Variations of the reduced structural damping $\zeta^a/\zeta_0$ as a function of the reduced virtual damping $\Delta c/c_s$ for $\Delta k = 0$. The model is given by Eq. (8). All values are obtained from free decay tests in air.</figcaption></figure>

## 5. VIV response of the virtual damper-spring system

After experimentally qualifying the virtual damper-spring system in Section 4, we analyse the VIV response of the oscillating cylinder for different values of virtual damping and stiffness. Section 5.1 reports the VIV response for the uncoupled configuration $S^u$. Section 5.2 deals with oscillation for the coupled uncompensated $S^c$ and compensated $S^{cc}(0)$ configurations. Lastly, Sections 5.3 and 5.4 study the influence of virtual stiffness and damping on the reduced oscillation amplitude, respectively.

For all the configurations studied, the cylinder displacement is recorded at a frequency of 500 Hz for 180 s, covering more than 80 oscillation periods. Since the oscillation amplitudes undergo some scatter,

we take as our reference reduced amplitude $A_{10}^*$, which is defined as the mean of the 10% largest amplitudes measured experimentally. As shown in Table 1, the reduced frequency is equal to $f^* = f/f_n$ where f is the frequency of the cylinder oscillations and $f_n$ the natural frequency for the cylinder immersed in water. Reynolds numbers based on D and $U_\infty$ range from 6000 to 16000.

### 5.1. VIV response of the uncoupled virtual damper-spring system

First, we consider the configuration $S^u$, in which the DC motor is decoupled from the oscillating system. We wish to validate the experimental set-up by comparing the oscillations amplitudes with those of the literature. Fig. 6(a) shows the amplitude responses $A_{10}^*$
<figure><img src="image_9.png" /><figcaption>Fig. 5. Reduced structural damping $\zeta^a/\zeta_0$ as a function of reduced virtual stiffness $\Delta k/k_s$ for different values of reduced virtual damping $\Delta c/c_s$. The dashed lines correspond to the analytical expression given by Eq. (8).</figcaption></figure>

as a function of the reduced velocity $U^*$, the Griffin plot. Clearly, the uncoupled configuration $S^u$ exhibits a typical VIV response in three branches: (i) an initial branch for $U^* \le 4.8$ with low oscillation amplitudes, (ii) an upper branch lying in the range $4.8 \le U^* \le 6.6$ where the cylinder experiences large oscillations amplitudes of up to $\approx 0.9$ and (iii) a lower branch within the range $6.6 \le U^* \le 10.2$ with $A_{10}^* \approx 0.6$. Above $U^* \approx 10$, the oscillations persist, but with limited amplitudes $A_{10}^* < 0.10$. The results of Sooti et al. (2018) are also represented in Fig. 6(a) for comparison. Their results were obtained for a damping ratio approximately equal to our own ($\zeta = 0.002$), and for a similar Reynolds number range. Both the number of detected branches and the amplitudes $A_{10}^*$ match remarkably well with our results.

The reduced frequency response for the $S^u$ configuration is shown in Fig. 6(b). The characteristic frequency $f$ is determined as the one which maximizes the power spectral density (PSD) of the cylinder displacement. Along the initial branch, frequency $f$ increases with $U^*$ as the vortex shedding frequency $f_{sh}$ associated to the Strouhal number $St_D = 0.19$. When the value of $U^*$ reaches the range of the upper branch, the oscillation frequency gets locked on the natural frequency $f_n$ with self-excitation of the oscillation. By further increasing $U^*$ up to the limit of the lower branch, we obtain a slight frequency increase of about 10% above the natural frequency. This tendency has been observed in the literature for systems with a mass ratio $m^*$ of order 1. This trend is commonly associated with an apparent added mass effect due to a total transverse fluid force in phase with the body acceleration (Khalak and Williamson, 1999). Finally, for $U^* > 10$, the cylinder's oscillations become desynchronized from the natural frequency $f_n$. The cylinder undergoes small amplitude oscillations at both the vortex shedding frequency $f_{sh}$ and the natural frequency $f_n$. The PSD of the oscillation amplitude clearly exhibits two distinctive peaks at these frequencies (PSD not shown for conciseness). All these measurements agree well with previously reported results for experimental installations with low values of structural damping ratio $m^*\zeta$ (see for example Khalak and Williamson, 1999; Zhao et al., 2014).

## 5.2. VIV response of the coupled virtual damper-spring system without and with compensation

We begin to analyse the configuration obtained by coupling the oscillating system to the DC motor. This configuration, called $S^c$, is characterized by a mass-damping ratio ($m^*\zeta = 0.189$) one order of

magnitude greater than the $S^u$ configuration studied in Section 5.1. Therefore, it is natural to expect changes in oscillation amplitude and frequency response. Fig. 6 shows variations in $A_{10}^*$ and $f^*$ as a function of $U^*$. In contrast to the results of the $S^u$ configuration, frequency synchronization is restricted to $5.5 \le U^* \le 6.5$. Moreover, the maximum value reached by $A_{10}^*$ is only 0.7, compared to 0.9 for the $S^u$ configuration. For $U^* > 6.5$, the cylinder oscillation frequency remains the same as the vortex shedding frequency $f_{sh}$. A similar response has been measured by Feng (1968) with $m^*\zeta = 0.36$.

We now consider the configuration where the dry friction introduced by the coupling of the oscillating system with the DC motor is compensated. The amplitude and frequency responses obtained for this configuration, called $S^{cc}(0)$, are also shown in Fig. 6. Amplitude and frequency variations are nearly identical to those obtained for the reference configuration $S^u$. Even the synchronization region observed for $S^u$ is recovered. The dry friction compensation achieved is therefore very efficient for simulating an oscillating system with a low mass-damping ratio.

## 5.3. Influence of virtual stiffness on reduced oscillation amplitude

Energy harvesting systems in natural flows such as rivers and tidal currents are subject to variable flow conditions. They need therefore to adjust their operating conditions to maximize their efficiency. In the case of an oscillating cylinder, an increase in the incoming velocity leads to a rise in the natural frequency of vortex shedding, making it necessary to adjust the system's stiffness. The developed experimental set-up enables to modify the stiffness of the system by adjusting the virtual stiffness $\Delta k$. Table 4 indicates the virtual stiffness values tested for $\Delta c = 0$, and the characteristic parameters of the oscillating system. In our experiments, the maximum value of virtual stiffness corresponds to twice the structural stiffness $k_s$ of the installed pair of springs.

Fig. 7 compares the evolution of oscillation amplitudes with respect to the Reynolds number (panel a) and the corresponding reduced velocity distribution (panel b). Clearly, Fig. 7(a) shows a shift towards higher Reynolds numbers and higher amplitudes of the Griffin plot as stiffness values increase. Oscillation amplitudes thus increase by 50% as stiffness doubles. A similar behaviour was obtained by Klamo (2007). While an increase in Reynolds number necessarily leads to higher forces, Table 4 indicates that the increase in stiffness is accompanied by a decrease in the structural damping ratio $\zeta$ which is favourable for large amplitude oscillations. Therefore, both the Reynolds number and
<figure><img src="image_10.png" /><figcaption>**Fig. 6.** Variations in reduced amplitude $A_{10}^*$ (a) and reduced frequency $f^*$ (b) as a function of reduced velocity $U^*$ for the three configurations analysed: $S^u$ (○), $S^c$ (□) and $S^{cc}(0)$ (■). For comparison, the results of Soti et al. (2018) are also shown (△). The vertical dashed lines indicate the limits of the initial, upper and lower branches.</figcaption></figure>

<caption>**Table 4**</caption>

Characteristic parameters of the compensated coupled oscillating system $S^{cc}$ for different values of $\Delta k$ and $\Delta c = 0$. We report the total stiffness $k$, the natural frequencies and damping ratios in air and water.

<table><thead><tr><th>Case</th><th>Δk (N m<sup>-1</sup>)</th><th>Δk/k<sub>s</sub></th><th>k (N m<sup>-1</sup>)</th><th>f<sub>n</sub><sup>a</sup> (Hz)</th><th>ζ<sup>a</sup></th><th>f<sub>n</sub> (Hz)</th><th>ζ</th></tr></thead><tbody><tr><td>S<sup>cc</sup>(0, 0)</td><td>0</td><td>0.00</td><td>21.83</td><td>0.464</td><td>0.0034</td><td>0.416</td><td>0.0030</td></tr><tr><td>S<sup>cc</sup>(0.27, 0)</td><td>6.0</td><td>0.27</td><td>28.66</td><td>0.559</td><td>0.0033</td><td>0.477</td><td>0.0028</td></tr><tr><td>S<sup>cc</sup>(0.50, 0)</td><td>11.0</td><td>0.50</td><td>34.27</td><td>0.574</td><td>0.0023</td><td>0.521</td><td>0.0021</td></tr><tr><td>S<sup>cc</sup>(1.00, 0)</td><td>22.0</td><td>1.00</td><td>46.27</td><td>0.683</td><td>0.0011</td><td>0.606</td><td>0.0010</td></tr></tbody></table>

the structural damping ratio affect the maximum oscillation amplitude achieved, which is in good agreement with the work reported by Modir and Goudarzi (2019). However, Fig. 7(b) shows that the three branches do not change their positions with the reduced velocity $U^*$. Since $U^* \propto f_{sh}/f_n$, the synchronization between $f_{sh}$ and $f_n$ plays obviously a major

role to raise $A_{10}^*$ by decreasing $\zeta$. In contrast, changes in stiffness appear to do not affect the unsynchronized flow regimes. The same conclusion can be drawn from Fig. 8, that shows that the frequency lock-in and desynchronization occur at exactly the same reduced velocities $U^*$ regardless of the virtual stiffness imposed.
<figure><img src="image_11.png" /><figcaption>Fig. 7. Variations in reduced amplitude as a function of Reynolds number (a) and reduced velocity (b) for different values of virtual stiffness $\Delta k$ and $\Delta c = 0$.</figcaption></figure>

5.4. Influence of virtual damping on reduced oscillation amplitude

During energy harvesting, the transformation of the oscillating system's mechanical energy into electrical energy causes an increase in damping. Therefore, the reduced amplitude $A_{10}^*$ is modified as a function of the extracted energy, i.e. the damping. In this section, we study variations in oscillation amplitude with increasing virtual damping $\Delta c$ for a constant value of stiffness given by $k_s$. Table 5 gives the values of $\Delta c$ covered in the experiments, together with the characteristic parameters of the coupled oscillator. The latter values can be deduced from measurements of $\zeta^a$, $f_n^a$ and $f_n$ made by free decay tests in air and still water. These data illustrate that the mechatronic system can

virtually multiply the damping ratio $\zeta$ by 40, thus changing the mass-damping ratio of more that one order of magnitude. This should be enough to observe different types of VIV response such as three-branch and two-branch oscillations.

Fig. 9(a) shows the variations in reduced oscillation amplitude $A_{10}^*$ as a function of reduced velocity $U^*$ for different values of reduced virtual damping $\Delta c/c_s$ and $\Delta k = 0$. The corresponding damping ratio values $\zeta$ are also given in the figure for information. As expected, the amplitude $A_{10}^*$ decreases over the whole range of considered $U^*$ with increasing virtual damping, i.e. damping ratio. The maximum amplitude drops from 0.9 for $\Delta c/c_s = 0$ (or $m^*\zeta = 0.013$) to 0.5 for $\Delta c/c_s = 20$ (or $m^*\zeta = 0.261$). Below $m^*\zeta \approx 0.324$, the three-branch response is clearly visible, but for $m^*\zeta > 0.324$ the upper branch disappears, leading to a
<figure><img src="image_12.png" /><figcaption>Fig. 8. Variations in reduced frequency as a function of reduced velocity. The same virtual stiffness values as in Fig. 7 are considered.</figcaption></figure>

<table><caption>**Table 5**

Characteristic parameters of the compensated coupled oscillating system $S^{cc}$ for $\Delta k = 0$ and different values of $\Delta c$. We report the damping ratios in air and water, the reduced structural damping and the mass-damping ratio, all compared with the structural damping of the reference case $S^{cc}(0)$, for which $c_s = 0.049 \text{ N s m}^{-1}$.</caption><thead><tr><th>Case</th><th>Δc (N s m<sup>-1</sup>)</th><th>Δc/c<sub>s</sub></th><th>ζ<sup>a</sup></th><th>ζ</th><th>ζ/ζ<sub>0</sub></th><th>m<sup>*</sup>ζ</th></tr></thead><tbody><tr><td>S<sup>cc</sup>(0,0)</td><td>0.0</td><td>0.00</td><td>0.0034</td><td>0.003</td><td>1.00</td><td>0.013</td></tr><tr><td>S<sup>cc</sup>(0,5.1)</td><td>0.25</td><td>5.1</td><td>0.0195</td><td>0.018</td><td>5.79</td><td>0.074</td></tr><tr><td>S<sup>cc</sup>(0,10.2)</td><td>0.50</td><td>10.2</td><td>0.0356</td><td>0.032</td><td>10.58</td><td>0.136</td></tr><tr><td>S<sup>cc</sup>(0,15.3)</td><td>0.75</td><td>15.3</td><td>0.0517</td><td>0.047</td><td>15.48</td><td>0.199</td></tr><tr><td>S<sup>cc</sup>(0,20.4)</td><td>1.00</td><td>20.4</td><td>0.0678</td><td>0.062</td><td>20.3</td><td>0.261</td></tr><tr><td>S<sup>cc</sup>(0,25.5)</td><td>1.25</td><td>25.5</td><td>0.0839</td><td>0.077</td><td>25.17</td><td>0.324</td></tr><tr><td>S<sup>cc</sup>(0,30.6)</td><td>1.50</td><td>30.6</td><td>0.1000</td><td>0.091</td><td>29.97</td><td>0.386</td></tr><tr><td>S<sup>cc</sup>(0,40.8)</td><td>2.00</td><td>40.8</td><td>0.1322</td><td>0.120</td><td>39.5</td><td>0.508</td></tr></tbody></table>

Feng-type response. At the same time, the extension of the lower branch diminishes with increasing virtual damping value. These observations, in particular the transition between three- and two-branch behaviour, are comparable with those made by Soti et al. (2018), Klamo et al. (2006), among others.

Fig. 9(b) shows the variations in reduced frequency as a function of reduced velocity for the same virtual damping values as in Fig. 9(a). For a mass-damping ratio lower that 0.261, the reduced frequency varies with reduced velocity like the vortex shedding frequency on the initial and upper branches, then locks on the natural frequency for the lower branch. In contrast, for $m^*\zeta > 0.261$, cylinder displacement is essentially dominated by the cylinder's vortex shedding frequency.

All the results presented in this section demonstrate that the behaviours obtained by the virtual-damper system we have developed are largely similar to those discussed in the reference literature. An essential feature of our system is the versatility offered by the use of virtual damping and stiffness.

## 6. Harvesting efficiency

The virtual damper-spring system presented in this study extracts mechanical energy by dissipation and thus simulates energy harvesting. It is therefore relevant at this stage to estimate the energetic efficiency of the oscillating system (Bernitsas et al., 2008; Barrero-Gil et al., 2012), i.e. the ratio between the power dissipated by the damper and that available through the flow region occupied by the cylinder ($\frac{1}{2}\rho U_{\infty}^3 DL_i$). Soti et al. (2017) showed that a simple model with a constant damping ratio predicts the same mean extracted power than

a more sophisticated model taking into account the full electromagnetic power transmission. The instantaneous non-dimensional power extracted by the damper is therefore equal to<fnref n="4" />

$$P(t) = \frac{c\dot{y}(t)^2}{\frac{1}{2}\rho U_{\infty}^3 DL_i}$$

where $c = c_s + \Delta c$ is the total damping. This power can also be written as

$$P(t) = 2\pi^2 \frac{1}{U_{\infty}^*} m^* \zeta \left( \frac{dy^*}{dt}(t) \right)^2$$

with $y^*$, the non-dimensional displacement of the cylinder. The advantage of the latter expression is to clearly show the dependence of the extracted power on the reduced velocity and the damping ratio.

The mean power extracted by the system can then be defined as $\bar{P} = \frac{1}{T} \int_{0}^{T} P(t) dt$ where T represents the time window considered for averaging. The average power can therefore be evaluated as

$$\eta = \bar{P} = \frac{c}{\frac{1}{2}\rho U_{\infty}^3 DL_i} \times \frac{1}{T} \int_{0}^{T} \dot{y}(t)^2 dt, \quad (9)$$

where $\eta$ (in percent) corresponds to the energetic efficiency. In the following, we experimentally estimate the mean power over 70 cylinder oscillation cycles. Assuming that the displacement y follows a sinusoidal law given by Eq. (3), it is possible, as in Soti et al. (2017),

<footnote n="4">⁴ Although this definition has been widely used to estimate the efficiency of VIV systems, we must point out that, in contrast to rotor-based systems, the characteristic cross-section is not constant but tends to increase with the amplitude of oscillation. Therefore, the assumption of constant cross-section leads to an overestimation of $\eta$.</footnote>
<figure><img src="image_13.png" /><figcaption>**Fig. 9.** Variations in reduced amplitude $A_{10}^*$ (a) and reduced frequency $f^*$ (b) as a function of reduced velocity $U^*$ for different values of damping ratios. The corresponding virtual damping values are also reported.</figcaption></figure>

to determine an analytical expression for the mean power. In the following, we do not use this assumption and proceed directly with Eq. (9).

Fig. 10 shows variations in energy harvesting efficiency as a function of reduced velocity for different damping ratio values. Whatever the value of the damping ratio, maximum efficiency is obtained on the upper branch (between the two vertical lines). In this range, $\eta$ values vary between 10 and 18%, with a maximum value for $\zeta = 0.032$. We observe an optimum damping ratio value. Efficiency increases with progressively higher damping ratios, until $\zeta = 0.032$, when maximum efficiency begins to decay. Along the upper branch, the efficiency also depends on the reduced velocity. The efficiency begins to increase with the reduced velocity, up to a value of approximately 5.7. Further increases in reduced velocity lead to a decrease in efficiency. The increase in damping ratio also leads to a shift of the maximum efficiency towards higher reduced velocities, materialized by the grey dashed line in Fig. 10. The maximum efficiency is reached at the very beginning of the

upper branch ($U^* = 4.99$) for the lower damping ratio. For $\zeta = 0.091$, this maximum is reached for $U^* = 6.15$. A similar shift can be observed on the maximum amplitude of oscillation in Fig. 9(a). The maximum efficiency is achieved right at the centre of the upper branch, when the ratio between the shedding frequency and the oscillation frequency is closest to 1 (see Fig. 9(b)). This leads to larger oscillation amplitudes $y$ and displacement velocities $\dot{y}$, resulting in increased power extraction. Conversely, when $\zeta > 0.032$, the greater damping force reduces $y$ and $\dot{y}$, leading to a reduction in $\eta$.

The influence of the reduced velocity on the results can be better understood by rewriting $U^*$ in a form involving the natural frequency of the oscillating system and the vortex shedding frequency. We immediately show (see Table 1) that $U^* = \frac{1}{Si} \frac{f_{sh}}{f_n}$. Now, according to Zdravkovich (1997), the Strouhal number of the system is constant and equal to 0.2 for the Reynolds number range considered in this study ($Re = \Theta(10^4)$). Therefore, the synchronization between the system natural frequency and the vortex shedding frequency can be identified
<figure><img src="image_14.png" /><figcaption>**Fig. 10.** Variations in energy harvesting efficiency (in percent) as a function of the reduced velocity. The same damping ratio values as in Fig. 9 are considered. The vertical dashed lines indicate the limits of the initial and upper branches. The highest efficiencies are obtained for the upper branch. For each damping ratio value, we determine the maximum efficiency $\eta_{max}$ over the entire range of reduced velocities, i.e. $\eta_{max}(\zeta) = \max_{U_*} \eta(U^*, \zeta)$.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>**Fig. 11.** Iso-values of energy harvesting efficiency (in percent) in the plane parametrized by $\delta$, the ratio between the natural and shedding frequencies, and $\zeta$, the damping ratio. The red cross marked at $\zeta = 0.035$ and $\delta = 0.9$ represents the optimal configuration in terms of efficiency.</figcaption></figure>

by a simple intersection between the two dashed lines in Fig. 9(b). To better illustrate the dependence of the efficiency on frequency synchronization, Grouthier et al. (2014) used the frequency ratio $\delta = f_n/f_{sh}$, instead of the reduced velocity. For our experiments, the introduction of the parameter $\delta$ is particularly interesting, as it takes direct account of variations in natural frequency via virtual stiffness. Fig. 11 shows the results of Fig. 10 transposed into the plane $(\zeta, \delta)$. A global maximum efficiency value is observed for optimum damping ratio and frequency ratio of $\zeta = 0.035$ and $\delta = 0.9$ (or $U^* = 5.7$) respectively. The optimum value of $\delta = 0.9$ is an indication that the maximum efficiency locates among the lock-in regime as previously observed in Fig. 10. The overall efficiency map is well consistent with the previous findings of Grouthier et al. (2014) and more recently of Han et al. (2023). The maximum of energy harvesting efficiency from VIV is known to be controlled by both the reduced velocity (or equivalently the frequency ratio $\delta$) and the Reynolds number (Barrero-Gil et al., 2012; Han et al., 2023). For a similar configuration with $m^* = 6.07$ at a Reynolds number of $6 \times 10^3$, Han et al. (2023) find a maximum efficiency of approximately

$0.17$ for a damping ratio of $\zeta = 0.0379$ and a reduced velocity of $U^* = 5.275$, very close to the present results. The iso-values representation highlights that changes in damping ratio are less critical than variations of $\delta$ in maintaining reasonable efficiency. Therefore, changes in vortex shedding frequency due to varying flow regimes must be compensated for by adapting the system stiffness $k$ with a control loop to keep $\delta$ constant.

## 7. Conclusion

A new mechatronic experimental set-up has been introduced to study the efficiency of VIV driven energy harvesting over a wide operation range. The structural stiffness and damping of an elastically mounted cylinder were varied by imposing forces with the help of a computer controlled DC motor. A force component proportional to the oscillator velocity was implemented to simulate for virtual damping, while a component proportional to the oscillator amplitude was used to mimic virtual stiffness. This strategy enables real-time adjustments
to the mechanical response of the oscillator. The experimental set up was proven adequate to simulate any given damped oscillator within a range of twice the original stiffness and up to forty times the original damping ratio. While the coupling of the DC motor introduces additional mechanical friction, the latter is used to compensate it with a feedback force calibrated from free decay tests. A thorough full characterization has been conducted which demonstrates that the device is able to reproduce oscillating responses from VIV regimes ranging from the three-branch response case to the two-branch response case according to the values of the virtual stiffness and damping coefficients.

Both damping and stiffness play an essential role in the energy harvesting process; increased damping leads to dissipation of mechanical energy to simulate energy harvesting and tuning of stiffness allows to maintain high oscillation amplitudes under changing upstream flow conditions. In accordance with results from the litterature, a maximum energy harvesting efficiency of about 18% is achieved for a damping ratio of 0.035 and a natural oscillation frequency of about 0.9 times the vortex shedding frequency. The explored parameter space highlights that mismatches between oscillation and vortex shedding frequencies strongly decrease the achievable efficiency. By contrast, the peak efficiency is less sensitive to increasing damping due to energy harvesting. Natural flows generally present variable flow rates. Consequently, energy harvesting systems driven by VIV need a real-time control of the structural parameters among which stiffness to keep an economically acceptable efficiency under various flow conditions. To this end, the current device shows high potential to elaborate real-time control strategies to optimize energy harvesting by VIVs and to maintain high energy efficiency with regards to the variable incoming flow conditions. In the present study, changes of stiffness and damping are achieved by the control of additional linear virtual damping and stiffness forces. However, the non-linearity of the vortical flow entrained by the cylinder may indicate that non-linear forcing might improve harvesting efficiency. Moreover, to select between the overwhelming amount of possible control laws, we plan in the near future to implement Machine Learning algorithms (Guégan et al., 2021) in our highly automated mechatronic experiment.

### CRediT authorship contribution statement

**Agathe Schmider:** Conceptualization, Data curation, Investigation, Methodology, Software, Writing – original draft, Formal analysis. **Franck Kerhervé:** Funding acquisition, Methodology, Supervision, Writing – review & editing, Conceptualization, Validation. **Andreas Spohn:** Conceptualization, Validation, Writing – review & editing.

**Laurent Cordier:** Supervision, Writing – review & editing.

### Declaration of competing interest

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Agathe Schmider reports financial support was provided by Region Nouvelle-Aquitaine. If there are other authors, they declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Data availability

Data will be made available on request.

### Acknowledgements

Support from the Région Nouvelle-Aquitaine, France Grant n°2018-1R10223 is acknowledged. The authors are particularly grateful to N. Dellinger from ICUBE (Strasbourg, France) for his support in the conception of the device. For the purpose of Open Access, a CC-BY public copyright licence has been applied by the authors to the present document and will be applied to all subsequent versions up to the Author Accepted Manuscript arising from this submission.

### References

Barrero-Gil, A., Pindado, S., Avila, S., 2012. Extracting energy from vortex-induced vibrations: a parametric study. Appl. Math. Model. 36 (7), 3153–3160.

Bernitsas, M., Raghavan, K., Ben-Simon, Y., Garcia, E., 2008. VIVACE (Vortex Induced Vibration Aquatic Clean Energy): A new concept in generation of clean and renewable energy from fluid flow. J. Offshore Mech. Arct. Eng. 130 (4).

Blevins, R.D., 1990. Flow-Induced Vibration, second ed. Van Nostrand Reinhold, New York, NY.

Chang, Q., Fu, Z., Zhang, S., Wang, M., Pan, X., 2022. Experimental investigation of Reynolds number and spring stiffness effects on vortex-induced vibration driven wind energy harvesting triboelectric nanogenerator. Nanomaterials 12 (20), 3595.

Derakhshandeh, J., Arjomandi, M., Cazzolato, B., Dally, B., 2015. Harnessing hydro-kinetic energy from wake-induced vibration using virtual mass spring damper system. Ocean Eng. 108, 115–128.

Fan, D., Jodin, G., Consi, T.R., Bonfiglio, L., Ma, Y., Keyes, L.R., Karniadakis, G.E., Triantafyllou, M.S., 2019. A robotic intelligent towing tank for learning complex fluid-structure dynamics. Science Robotics 4 (36), eay5063.

Feeny, B., Liang, J., 1996. A decrement method for the simultaneous estimation of Coulomb and viscous friction. J. Sound Vib. 195 (1), 149–154.

Feng, C., 1968. The measurement of vortex induced effects in flow past stationary and oscillating circular and D-section cylinders (Ph.D. thesis). University of British Columbia.

Gao, Z., Li, Z., Niu, J., Yin, Q., Liu, K., 2023. Application of nonlinear stiffness mechanism on energy harvesting from vortex-induced vibrations. Front. Mar. Sci.

Govardhan, R., Williamson, C.H., 2006. Defining the ‘modified Griffin plot’ in vortex-induced vibration: revealing the effect of Reynolds number using controlled damping. J. Fluid Mech. 561, 147–180.

Grouthier, C., Michelin, S., Bourguet, R., Modarres-Sadeghi, Y., De Langre, E., 2014. On the efficiency of energy harvesting using vortex-induced vibrations of cables. J. Fluids Struct. 49, 427–440.

Guégan, T., Bucci, M.A., Semeraro, O., Cordier, L., Mathelin, L., 2021. Closed-loop control of complex systems using deep Reinforcement Learning. In: Euromech Colloquium on Machine Learning Methods for Turbulent Separated Flows. Paris, France.

Han, P., Huang, Q., Pan, G., Qin, D., Wang, W., Gonçalves, R.T., Zhao, J., 2023. Optimal energy harvesting efficiency from vortex-induced vibration of a circular cylinder. Ocean Eng. 282, 114869.

Hao, Z., Sun, C., Lu, Y., Bi, K., Zhou, T., 2022. Suppression of vortex-induced vibration and phase-averaged analysis of the wake generated by a circular cylinder covered with helical grooves. Fluids 7 (6), 194.

Hover, F., Miller, S., Triantafyllou, M., 1997. Vortex-induced vibration of marine cables: experiments using force feedback. J. Fluids Struct. 11 (3), 307–326.

Kang, Z., Zhang, C., Chang, R., Ma, G., 2019. A numerical investigation of the effects of Reynolds number on vortex-induced vibration of the cylinders with different mass ratios and frequency ratios. Int. J. Nav. Archit. Ocean Eng. 11 (2), 835–850.

Khalak, A., Williamson, C.H., 1999. Motions, forces and mode transitions in vortex-induced vibrations at low mass-damping. J. Fluids Struct. 13 (7–8), 813–851.

Klamo, J.T., 2007. Effects of Damping and Reynolds Number on Vortex-Induced Vibrations. California Institute of Technology.

Klamo, J., Leonard, A., Roshko, A., 2005. On the maximum amplitude for a freely vibrating cylinder in cross-flow. J. Fluids Struct. 21 (4), 429–434.

Klamo, J., Leonard, A., Roshko, A., 2006. The effects of damping on the amplitude and frequency response of a freely vibrating cylinder in cross-flow. J. Fluids Struct. 22 (6–7), 845–856.

Krishnakumari, A., Saravanan, M., Vishal, S., Krishna, A.A., Kumar, A.S., Lindsay, S., 2023. Modelling and analysis of vortex induced vibration in marine risers. Mater. Today: Proc.

Lee, J., Xiros, N., Bernitsas, M., 2011. Virtual damper-spring system for VIV experiments and hydrokinetic energy conversion. Ocean Eng. 38 (5–6), 732–747.

Ma, X., Zhou, S., 2022. A review of flow-induced vibration energy harvesters. Energy Convers. Manage. 254, 115223.

Mackowski, A.W., Williamson, C.H., 2011. Developing a cyber-physical fluid dynamics facility for fluid-structure interaction studies. J. Fluids Struct. 27 (5–6), 748–757.

Miller, S.N., 1996. Measurement of Vortex-Induced Oscillations of Marine Cables Using Feedback with Explicit Structural Modeling (Ph.D. thesis). Massachusetts Institute of Technology.

Modir, A., Goudarzi, N., 2019. Experimental investigation of Reynolds number and spring stiffness effects on vortex induced vibrations of a rigid circular cylinder. Eur. J. Mech. B Fluids 74, 34–40.

Onoue, K., Song, A., Strom, B., Breuer, K.S., 2015. Large amplitude flow-induced oscillations and energy harvesting using a cyber-physical pitching plate. J. Fluids Struct. 55, 262–275.

Powell, M.J.D., 1973. On search directions for minimization algorithms. Math. Program. 4, 193–201.

Rabault, J., Kuchta, M., Jensen, A., Réglade, U., Cerardi, N., 2019. Artificial neural networks trained through deep reinforcement learning discover control strategies for active flow control. J. Fluid Mech. 865, 281–302.
Raghavan, K., Bernitsas, M., 2011. Experimental investigation of Reynolds number effect on vortex induced vibration of rigid circular cylinder on elastic supports. Ocean Eng. 38 (5-6), 719-731.

Riches, G., Morton, C., 2018. One degree-of-freedom vortex-induced vibrations at constant Reynolds number and mass-damping. Exp. Fluids 59 (10), 1-16.

Soti, A.K., Thompson, M.C., Sheridan, J., Bhardwaj, R., 2017. Harnessing electrical power from vortex-induced vibration of a circular cylinder. J. Fluids Struct. 70, 360-373.

Soti, A.K., Zhao, J., Thompson, M.C., Sheridan, J., Bhardwaj, R., 2018. Damping effects on vortex-induced vibration of a circular cylinder and implications for power extraction. J. Fluids Struct. 81, 289-308.

Szepessy, S., Bearman, P.W., 1992. Aspect ratio and end plate effects on vortex shedding from a circular cylinder. J. Fluid Mech. 234 (-1), 191.

Williamson, C.H., Govardhan, R., 2004. Vortex-induced vibrations. Annu. Rev. Fluid Mech. 36 (1), 413-455.

Williamson, C.H., Govardhan, R., 2008. A brief review of recent results in vortex-induced vibrations. J. Wind Eng. Ind. Aerodyn. 96 (6-7), 713-735.

Williamson, C.H., Roshko, A., 1988. Vortex formation in the wake of an oscillating cylinder. J. Fluids Struct. 2 (4), 355-381.

Zdravkovich, M., 1981. Review and classification of various aerodynamic and hydrodynamic means for suppressing vortex shedding. J. Wind Eng. Ind. Aerodyn. 7 (2), 145-189.

Zdravkovich, M.M., 1997. Flow Around Circular Cylinders: A Comprehensive Guide Through Flow Phenomena, Experiments, Applications, Mathematical Models, and Computer Simulations. In: Oxford Science Publications, Oxford University Press, Oxford ; New York.

Zhao, J., 2017. Flow-Induced Vibration of Circular and Square Cylinder with Low Mass and Damping (Ph.D. thesis). Monash University.

Zhao, M., 2023. A review of recent studies on the control of vortex-induced vibration of circular cylinders. Ocean Eng. 285, 115389.

Zhao, J., Leontini, J.S., Lo Jacono, D., Sheridan, J., 2014. Chaotic vortex induced vibrations. Phys. Fluids 26 (12).

Zhu, H., Liu, W., Gao, Y., Ji, C., Liu, H., Zhou, T., 2023. An experimental investigation on the vortex-induced vibration mode transition and response interaction of a fixed-hinged catenary flexible riser. Phys. Fluids 35 (1).