2.1. Experimental set-up

To achieve a very low but stable flow, a rotating channel is designed by referring to the rotating tank in geophysical studies (Griffiths & Hopfinger Reference Griffiths and Hopfinger1986; Afanasyev & Peltier Reference Afanasyev and Peltier1998). As shown in figure 2(a), the rotating channel mainly consists of two concentric cylinders with diameters of 500 mm and 380 mm. The channel is driven by a stepper motor which allows the channel to rotate in an angular speed range of 0–0.306 rad s$^{-1}$. To eliminate the effect of temperature on the viscosity coefficient, the temperature of the environment and the water is strictly controlled at $18 \pm 0.1\,^\circ$C. PIV is utilized to quantify the speed distribution of the flow field in the rotating channel at a rotational speed of $N = 0.102$ rad s$^{-1}$.

Figure 2.

Sketch of (a) the rotating water channel and measurement techniques and (b) the experimental model of the cylinder with a PMC.

A continuous laser provides an illumination plane with a thickness of approximately 1 mm at a distance of 45 mm from the bottom of the water tank. A high-speed camera (Pointgrey GS3-U3-23S6M-C) with $1920 \times 1200$ image resolution is used to capture the particle images. In total, 2000 snapshots were captured at a sample rate of 100 Hz for each single test case. The post-processing to acquire the velocity fields is conducted with an open-source package, PIVlab v2.3.1 (Thielicke & Sonntag Reference Thielicke and Sonntag2021). Meanwhile, the displacements of the cylinder were revolved with the DIC method by tracking the reflected light on the cylinder surface.

Figure 2(b) depicts a schematic diagram of the experimental model set-up. The model in the experiment is an aluminium cylinder with a diameter of $D = 2$ mm and a length of $l = 100$ mm. The top of the cylinder is firmly held by a wooden block. The two sides of the wooden block are connected to the fixed base with polyvinyl chloride (PVC) strips to provide elastic support. A rectangular magnet block is the lumped mass of the PMC. The stiffness of the PMC is provided by a PVC strip. The relative motion between the magnet and the aluminium sheet adhered to the light wooden board can generate eddy currents, providing damping. The frequency of the model and PMC can be changed by adjusting the length of the PVC strip. The damping of the PMC can be changed by adjusting the relative distance between the magnet block and the aluminium sheet. It is noted that, in this study, PMC closure refers to the fixed connection between the PMC and the cylinder, while opening means the release of PMC degrees of freedom.

Figure 3(a) depicts the instantaneous speed distribution of the horizontal cross-section of the channel, and the flow speed is rather uniform. Figure 3(b) shows that the turbulence intensity distribution $I_{\infty }$ is less than 1 % in the centre of the channel. To evaluate the effect of rotation, the Rossby number, defined as $Ros = U/2ND$, is introduced. When $Ros > 1$, the effect of rotation on the flow is negligible (Boyer & Kmetz Reference Boyer and Kmetz1983; Pham et al. Reference Pham, Plourde, Kim and Balachandar2006). In this research, $Ros = 55$, which is substantially larger than 1. Thus, the effect of rotation on the flow can be neglected. Considering the effect of the wake on vortex shedding, the diameter of the cylinder used in the experiment is rather small ($D = 2$ mm), so the wake can be dissipated completely after the channel rotates half a circle. The PIV measurement approximately 10$D$ upstream of the model shows that the incoming flow meets the uniformity condition, so the influence of the wake can be ignored. Lyu et al. (Reference Lyu, Kou and Zhang2022) has conducted a series of tests at different $Re$, and the Strouhal number is in good agreement with previous experiments and numerical simulations.

Figure 3.

Distribution of (a) flow velocity and (b) turbulence intensity in the cross-section of the rotating channel at a rotational speed of $N = 0.102$ rad s$^{-1}$.

Figure 4 shows the maximum transverse displacement and the dimensionless response frequency $F^{*}$ (defined as $F^{*} = F_{v} / F_{y}$) varying with the reduced natural frequency of the cylinder in the experiment and computational fluid dynamics/computational structural dynamics (CFD/CSD) simulation for $Re = 60$. The details of the simulation are discussed in Zhang et al. (Reference Zhang, Li, Ye and Jiang2015) and Luo et al. (Reference Luo, Gao and Zhang2022). Because of the structural damping in the experiment, the vibration amplitude is smaller than that in the CFD/CSD simulation. However, the frequency lock-in is well captured in the experiment, as well as the onset of the frequency lock-in.

Figure 4.

(a) The maximum transverse displacement $Y^{max}$ and (b) the response frequency $F$ varying with the reduced natural frequency of the cylinder $F_{y}$ in the experiment and CFD/CSD simulation for $Re = 60$.

2.2. The ARX-based ROM

In this article, an identification technique via the ARX model is employed to construct the reduced-order unsteady aerodynamic model at $Re = 60$ and $Re = 40$. The work of Cowan, Arena & Gupta (Reference Cowan, Arena and Gupta2001) was one of the earliest to use such a model for unsteady aerodynamic modelling. Thereafter, extensive studies on analysing and explaining the mechanisms of different aeroelastic and flow-induced vibration problems have been conducted by Zhang and his coworkers (Gao & Zhang Reference Gao and Zhang2020; Kou & Zhang Reference Kou and Zhang2021).

The compressible Navier–Stokes equations were utilized to simulate the two-dimensional laminar flow past a circular cylinder at $Re = 40$ and $Re = 60$. The ROM is trained based on the unstable steady-state base flow, which is computed by a time-filtering method. More details have been demonstrated in our prior work (Zhang et al. Reference Zhang, Li, Ye and Jiang2015; Luo et al. Reference Luo, Gao and Zhang2022).

The ARX model provides a general description of linear dynamic systems in the discrete-time domain, which are governed by linear differential equations. The model formulation is as follows:

(2.5)\begin{equation} \boldsymbol{f}(k)=\sum_{i=1}^{n a} \boldsymbol{A}_{i} \,\boldsymbol{f}(k-i)+\sum_{i=0}^{n b-1} \boldsymbol{B}_{i} \boldsymbol{u}(k-i)+\boldsymbol{e}(k), \end{equation}

where $\boldsymbol {f}$ is the output vector of the system and $\boldsymbol {u}$ is the input vector, $\boldsymbol {A}_{i}$ and $\boldsymbol {B}_{i}$ are constant coefficients to be estimated, and na and nb are the delay orders determined by the user. For the current single-input–single-output model, $\boldsymbol {u}=[Y]$ (transverse displacement of the cylinder) and $\boldsymbol {f}= [C_{l}]$ (hydrodynamic coefficient).

To couple the structural equations, the discrete-time state-space form is converted into the continuous-time form. The model in the continuous-time state-space form is constructed as follows:

(2.6)\begin{equation} \left.\begin{array}{c@{}} \dot{\boldsymbol{x}}_{a}(t)=\boldsymbol{A}_{a} \boldsymbol{x}_{a}(t)+\boldsymbol{B}_{a} \boldsymbol{u}(t) \\ \boldsymbol{f}_{a}(t)=\boldsymbol{C}_{a} \boldsymbol{x}_{a}(t)+\boldsymbol{D}_{a} \boldsymbol{u}(t) \end{array}\right\} . \end{equation}

The structural equations in state-space form and the output equations can be expressed as follows:

(2.7)\begin{equation} \left.\begin{array}{c@{}} \dot{\boldsymbol{x}}_{s}(t)=\boldsymbol{A}_{s} \boldsymbol{x}_{s}(t)+q \boldsymbol{B}_{s}\,\boldsymbol{f}_{a}(t) \\ \boldsymbol{u}(t)=\boldsymbol{C}_{s} \boldsymbol{x}_{s}(t)+q \boldsymbol{D}_{s} \,\boldsymbol{f}_{a}(t) \end{array}\right\} . \end{equation}

Coupling the structural state equation (2.7) and the aerodynamic state equation (2.6), the state equations for the aeroelastic system can be obtained, which are

(2.8) \begin{equation} \left\{\begin{array}{@{}c@{}} \dot{\boldsymbol{x}}_{s}(t) \\ \dot{\boldsymbol{x}}_{a}(t) \end{array}\right\}=\boldsymbol{A_{FS}}\cdot\left\{\begin{array}{c} \boldsymbol{x}_{s}(t) \\ \boldsymbol{x}_{a}(t) \end{array}\right\}=\left[\begin{array}{@{}cc@{}} \boldsymbol{A}_{s}+q \cdot \boldsymbol{B}_{s} \boldsymbol{D}_{a} \boldsymbol{C}_{s} & q \cdot \boldsymbol{B}_{s} \boldsymbol{C}_{a} \\ \boldsymbol{B}_{a} \boldsymbol{C}_{s} & \boldsymbol{A}_{a} \end{array}\right] \cdot\left\{\begin{array}{@{}c@{}} \boldsymbol{x}_{s}(t) \\ \boldsymbol{x}_{a}(t) \end{array}\right\}, \end{equation}

where $\boldsymbol {B}_{s} = \boldsymbol {B}_{s1} \boldsymbol {B}_{s2}$ and $q = 2/({\rm \pi} m_{y}^{*})$. The subscript a denotes the aerodynamic force and subscript s denotes the structure. The matrices are

(2.9a–d) \begin{equation} \boldsymbol{A}_{s}=\left[\begin{array}{@{}cc@{}} \boldsymbol{O} & \boldsymbol{I} \\ -\boldsymbol{M}^{{-}1} \boldsymbol{K} & -\boldsymbol{M}^{{-}1} \boldsymbol{G} \end{array}\right], \quad \boldsymbol{B}_{s 1}=\left[\begin{array}{@{}c@{}} \boldsymbol{O} \\ \boldsymbol{M}^{{-}1} \end{array}\right], \quad \boldsymbol{B}_{s 2}=\left[\begin{array}{@{}l@{}} \boldsymbol{I} \\ \boldsymbol{O} \end{array}\right], \quad \boldsymbol{C}_{s}=\left[\begin{array}{@{}ll@{}} \boldsymbol{I} & \boldsymbol{O} \end{array}\right], \end{equation}

where $\boldsymbol {I}$ and $\boldsymbol {O}$ represent the identity matrix and zeros matrix, respectively.

Thus, an analysis model for FSI based on ROM, as represented in (2.8), is conducted. The FSI stability problem is converted into solving for and analysing the eigenvalues of $\boldsymbol {A_{FS}}$. The real part of the eigenvalue represents the growth rate of the eigenmode, whereas the imaginary part corresponds to the circular frequency, which is equal to 2${\rm \pi}$ times the eigenfrequency of the eigenmode. For the given Reynolds number and mass ratio, the natural frequency of the cylinder or the PMC is changed to obtain the root loci of the coupled system. On this basis, the effects of different PMC parameters such as the frequency, damping and mass ratio on the stability of the coupled system can be studied, as well as the coupling among various modes after the introduction of the control mode (CM).

In this study, a chirp signal with a broadband reduced frequency of $[0.04,0.20]$ is used as the training signal. We consider the aerodynamic modelling with delay orders $na = nb = 120$ at $Re = 60$ and $Re = 40$. The identified results are compared with those of direct numerical simulations in figure 5; good agreement is observed between the two sets of results. The modelling error is less than $1\,\%$. Therefore, it can be concluded that the identification method based on the ARX model has high numerical accuracy. Figure 6 shows the eigenvalues of the coupled system varying with the reduced natural frequency of the cylinder ($F_{y}$) for $Re = 60$ and $Re = 40$. Two neutral modes of interest exist, namely the ‘nearly structural mode’ (referred to here as the SM) and the ‘von Kármán mode’ (referred to here as the wake mode, WM). The critical Reynolds number of a rigid cylinder is $Re_{cr} \approx 47$. Therefore, the WM is absolutely unstable for $Re = 60$ while it is stable at $Re = 40$. Meanwhile, the SM becomes unstable at certain frequencies because of the interaction with the WM. The instability of the SM plays an important role in the frequency lock-in of VIV.

Figure 5.

Identified results under the training signal compared with those of direct numerical simulations for (a) $Re = 60$ and (b) $Re = 40$.

Figure 6.

Root loci of the coupled system without control for (a) $Re = 60$ and (b) $Re = 40$.

3.1. Supercritical case: $Re = 60$

Figure 7 shows the maximum transverse displacement $Y^{max}$ and the vibration frequency of the cylinder with and without PMC control for $Re = 60\ {\rm and}\ m_{y}^{*} = 21$. Here, the PMC mass ratio is $\varepsilon = 5.35\,\%$, the damping coefficient is $\lambda _{d} = 5\,\%$ and the frequency is $F_{d} = 0.14$. As seen in figure 7(a), the cylinder undergoes large-amplitude vibration at $0.1 < F_{y} < 0.18$. The maximum displacement reaches approximately $0.5D$. After PMC control, the vibration amplitude of the cylinder can be reduced by two orders of magnitude. Meanwhile, figure 7(b) depicts the lock-in phenomenon for the plain cylinder. Notably, the lock-in phenomenon has been eliminated after the introduction of the PMC. The response frequency is consistent with the frequency of unsteady vortex shedding. Figure 8 shows the comparisons of time responses and power spectral density (PSD) results for the plain cylinder and the cylinder attached to a PMC at $F_{y} = 0.169$. The small oscillation caused by vortex shedding after control is negligible. The PSD analysis shows that the response frequency has shifted before and after control.

Figure 7.

(a) The maximum transverse displacement $Y^{max}$ and (b) the response frequency varying with the reduced natural frequency of the cylinder with and without PMC control for $Re = 60 \text{ and } m_{y}^{*} = 21$. The PMC parameters are as follows: mass ratio $\varepsilon = 5.35\,\%$, damping $\lambda _{d} = 5\,\%$ and frequency $F_{d} = 0.14$.

Figure 8.

Comparisons of time responses and PSD results for (a,c) the plain cylinder ($F_{y} = 0.169$) and (b,d) the cylinder attached to a PMC, where $f = 2.56$ Hz corresponds to the reduced frequency $F = 0.165$ and $f = 2.1$ Hz corresponds to the reduced frequency $F = 0.135$.

The linear stability analysis of VIV control for $Re = 60$ is performed based on the FSI ROM. The PMC parameters are consistent with experimental values where $m_{y}^{*} = 21, \varepsilon = 5\,\%, \lambda _{d} = 5\,\%$ and $F_{d} = 0.14$. Particularly, the structural damping coefficient is assigned a zero value. Figure 9 shows the root loci of the controlled system varying with the reduced natural frequency $F_{y}$, as well as the real and imaginary parts of the root loci. From figure 9(a), it can be seen that the root loci of the WM are still unstable, while the SM is stable at low or high reduced natural frequency conditions under which the interaction between the SM and WM is weak. However, compared with figure 6(a), the root loci of the SM become stable as the reduced natural frequency of the cylinder approaches the PMC frequency. Although the FSI effect is strong at these frequencies, the SM is not tightly coupled with the WM but with the CM. That is the reason PMC can eliminate the frequency lock-in and suppress VIV. The experimental results also indicate that when the SM becomes stable, the frequency lock-in will exit and the large-amplitude vibration will be significantly suppressed. However, it is also observed in the experiment that after introducing PMC control, although frequency lock-in has been eliminated, there are still weak periodic oscillations. From the PSD results, it is found that the vibration frequency is consistent with the vortex shedding frequency. The results of the linear stability analysis also prove that the unstable WM does not change due to the introduction of the PMC. The final weak periodic oscillation is a kind of forced vibration caused by the unsteady vortex shedding. Next, the above viewpoint will be further validated by analysing the flow field snapshots captured in the experiment with the DMD method.

Figure 9.

Root loci of the controlled system for $Re = 60$: (a) root loci versus reduced natural frequency $F_{y}$; (b) real and (c) imaginary parts of the root loci.

For a detailed study of frequency lock-in elimination mechanisms with a PMC, figure 10 depicts the vorticity of the flow field and the dominant mode obtained by DMD in three scenarios (elastic cylinder with frequency $F_{y} = 0.169$ with and without control, and stationary cylinder). Figure 10(a ${-}$c) shows the comparison of the vorticity fields in the three scenarios. The 2S mode (two single vortices formed per cycle) is observed for the elastic cylinder with PMC control and the stationary cylinder, whereas the C(2S) mode is observed for the elastic cylinder without control. The C(2S) mode is very similar to the 2S mode except that the vortices coalesce in the far wake, which is typical evidence of the flutter response (Williamson Reference Williamson1988; Singh & Mittal Reference Singh and Mittal2005; Zhang et al. Reference Zhang, Li, Ye and Jiang2015). In the case of $F_{y} = 0.169$, the wake begins to coalesce from the fifth to the sixth vortices (figure 10a). Figure 10(d ${-}$f) illustrates the dominant mode of flow around the cylinder for the three conditions. It can be seen that the dominant mode for the cylinder without control is the VIV mode. The frequency obtained by DMD is $F_{DMD} = 0.163$, which indicates that the SM is unstable. Meanwhile, the dominant mode for the stationary cylinder and the cylinder with PMC control is the von Kármán mode. The frequency of the dominant mode for the stationary cylinder is $F_{DMD} = 0.132$, whereas the frequency of the dominant mode for the cylinder with PMC control is $F_{DMD} = 0.133$. This means that the stability of the SM is typically improved after PMC control, so the vibration has been greatly suppressed and the frequency lock-in has been eliminated. The results provide evidence that the stability of the SM is the key to VIV control.

Figure 10.

The vorticity of the flow field and the dominant mode obtained by DMD at $Re = 60$ for (a,d) the elastic cylinder without control, (b,e) the elastic cylinder with PMC control and (c,f) the stationary cylinder.

3.2. Subcritical case: $Re = 40$

To further explore the role of the SM in the VIV of a cylinder, an experiment is conducted at a subcritical Reynolds number, $Re = 40$. A reduced natural frequency of the cylinder ($F_{y} = 0.149$) with a larger amplitude in the lock-in region is selected to compare the phenomenon before and after PMC control. Owing to the impact on engineering structures, VIV and associated control strategies have been the subject of a number of studies at higher $Re$, as reviewed, for example, by Williamson & Govardhan (Reference Williamson and Govardhan2004), Choi et al. (Reference Choi, Jeon and Kim2008) and Chen et al. (Reference Chen, Huang, Chen, Yu and Gao2022). In recent years, Boersma et al. (Reference Boersma, Zhao, Rothstein and Modarres-Sadeghi2021) presented experimental evidence of VIV at subcritical $Re$. There are no experimental reports of VIV control at subcritical $Re$ to our knowledge. The study of VIV control at subcritical $Re$ can further enhance our understanding of the mechanism of frequency lock-in and PMC control.

Figure 11(a) shows the vorticity of the flow field around the elastic cylinder at $Re = 40$ for $F_{y} = 0.149$. Modal analysis of the flow around the cylinder is performed via DMD. Figure 11(b) illustrates the dominant mode, which represents the VIV mode. The dominant frequency of the flow is $F_{DMD} = 0.145$ and it is similar to the reduced natural frequency of the cylinder, indicating that the SM is unstable. Meanwhile, figure 12 depicts the comparisons of time responses and PSD results at $Re = 40$ for the elastic cylinder with and without control. The black dashed line represents the displacement of the uncontrolled cylinder with a dimensionless amplitude close to $0.5D$. Through PSD analysis, it can be obtained that the vibration response frequency of the uncontrolled cylinder is 1.52 Hz (figure 12b), corresponding to the reduced frequency $F=0.145$. The result is consistent with the dominant frequency obtained from DMD. The solid red line represents the response after PMC control, and it can be seen that the vibration has almost been completely suppressed. Figure 13 illustrates the vorticity of the flow field at $Re = 40$ for the controlled cylinder and the stationary cylinder. It can be seen from figures 12(a) and 13(a) that after applying PMC control, the vibration of the cylinder has been eliminated and there is no vortex shedding. The vorticity of the flow field for the controlled cylinder is the same as that for the stationary cylinder, indicating that the unstable SM has been successfully suppressed. It is noted that the frequency lock-in at subcritical $Re$ no longer exists when the SM becomes stable. Therefore, the experimental results verify that the instability of the SM is the key factor for the frequency lock-in in VIV.

Figure 11.

(a) The vorticity of the flow field and (b) the dominant mode obtained by DMD at $Re = 40$ for the elastic cylinder without control.

Figure 12.

Comparisons of (a) time responses and (b) PSD results at $Re = 40$ for the elastic cylinder with and without control.

Figure 13.

The vorticity of the flow field at $Re = 40$ for (a) the controlled cylinder and (b) the stationary cylinder.

Moreover, linear stability analyses on the uncontrolled and controlled coupled systems at $Re = 40$ are performed based on the FSI ROM. Figure 14 shows the root loci and the real parts of the root loci for the uncontrolled system (figure 14a,b) and the controlled system with PMC (figure 14c,d). The PMC parameters are set to $\varepsilon = 5\,\%, \lambda _{d} = 5\,\%$ and $\varOmega = 0.9$. It can be seen that the WM is absolutely stable at $Re = 40$. As the reduced natural frequency of the cylinder approaches the WM, the FSI effect becomes strong. Finally, the SM becomes unstable at certain natural frequencies ($0.102 < F_{y} < 0.159$). This is also the fundamental cause of subcritical VIV (Kou et al. Reference Kou, Zhang, Liu and Li2017; Li, Zhang & Gao Reference Li, Zhang and Gao2018). As shown in figure 14(a,b), the root loci of the coupled system undergo significant changes due to the application of the PMC. The instability range of the SM has been narrowed from the uncontrolled $0.102 < F_{y} < 0.159$ to $0.1 < F_{y} < 0.122$. Thus, the SM becomes stable for $F_{y} = 0.149$, and that is why the vibration has been eliminated. It is noted that the WM is stable at $Re = 40$ so there is no unsteady vortex shedding after control, which is also observed in the experiment (figure 13a). The results of the experiments and linear stability analysis indicate that the VIV phenomenon at subcritical Reynolds numbers is a SM instability problem caused by the coupling of the SM and the stable WM. By introducing a dynamic subsystem to adjust the SM to stability, the vibration is eliminated and there is no vortex shedding anymore.

Figure 14.

The root loci and their corresponding real parts at $Re = 40$ for (a,b) the uncontrolled system and (c,d) the controlled system with PMC.

4.1. The effect of PMC frequency

The PMC frequency plays an essential role in determining the performance of VIV suppression. Figure 15 displays the maximum vibration amplitude varying with the reduced natural frequency of the PMC at $Re = 60$ for $F_{y} = 0.163$. The PMC mass ratio is set to $\varepsilon = 5.35\,\%$ and the damping coefficient is $\lambda _{d} = 5\,\%$. The long dashed line in figure 15 represents the vibration amplitude of the plain cylinder without control for $F_{y} = 0.163$. Compared with the uncontrolled cylinder, the vibration amplitude of the controlled cylinder with the PMC has decreased in general. Even in the scenario where the frequency lock-in is not eliminated, the amplitude has decreased by approximately 25 %. However, this is different from the traditional civil engineering perspective, which considers the optimal frequency of PMC to be consistent with the natural frequency of the structure. The experimental results indicate that the frequency range in which the PMC can effectively eliminate lock-in is mainly below the structural frequency, and the range is wide. Our previous work has provided an explanation for why the optimal frequency does not occur at the structural natural frequency by considering the coupling between the SM and WM. Furthermore, the modal analysis method is adopted here to validate the explanation.

Figure 15.

The maximum vibration amplitude varying with the reduced natural frequency of the PMC at $Re = 60$ for $F_{y} = 0.163$.

Figure 16 depicts the root loci and the real parts of the root loci of the controlled system at $Re = 60$. The mass ratio and damping coefficient of the PMC are both set to 5 %. Figure 16(a,b) shows the frequency ratio of the PMC to the cylinder for $\varOmega = 0.6$. The real part of the SM can be observed to cross the imaginary axis at $F_{y}=0.163$, transitioning from unstable to stable by becoming negative. Meanwhile, figure 16(c,d) demonstrates that for $\varOmega =1.1$, the real part of the SM switches from unstable to stable at $F_{y} = 0.163$. To more clearly demonstrate the change in structural modal stability with PMC frequency, $F_{y}$ is kept at 0.163 during the linear stability analysis and the PMC frequency is varied to obtain the root loci and eigenvalue real part change curve of the coupled system, as shown in figure 17(a,b). Figure 17(b) indicates that the real part of the eigenvalue of the SM is stable (less than 0) when $\varOmega$ ranges from 0.6 to 1.1 and unstable (greater than 0) when $\varOmega < 0.6$ or $\varOmega > 1.1$. See also the supplementary movies available at https://doi.org/10.1017/jfm.2024.180. The experimental results demonstrate that the effective frequency range of the PMC in this scenario is $0.67 < \varOmega <1.0$ (seen in figure 17c), which is slightly narrower than the frequency range predicted by the linear stability analysis. The reason for this difference may be that the mass ratio and damping coefficient provided in the ROM do not completely match the actual measurements. Nevertheless, the experimental result indicates that it is reasonable and feasible to design VIV control via modal stability analysis.

Figure 16.

The root loci and their corresponding real parts of the controlled system varying with the reduced natural frequency of the cylinder at $Re = 60$ for (a,b) $\varOmega = 0.6$ and (c,d) $\varOmega = 1.1$.

Figure 17.

(a) The root loci and (b) the real parts of the root loci of the controlled system varying with the reduced natural frequency of the PMC at $Re = 60$ for $F_{y} = 0.163$, compared with (c) the maximum amplitude of the controlled cylinder in the experiment.

The range of $\varOmega = 0.6\unicode{x2013}1.1$ obtained from the linear stability analysis corresponds to the effective frequency range of the PMC under the condition of a single natural frequency of the cylinder at $F_{y} = 0.163$, which can be referred to as ‘local stability’. To obtain the range of $\varOmega$ ensuring that the SM is stable at any natural frequency of the cylinder, further linear stability analysis is performed via the ROM, as shown in figure 18. The real parts of the root loci for the SM will be less than 0 at any natural frequency of the cylinder from $\varOmega = 0.74$ to $\varOmega = 1.08$. This can be referred to as ‘global stability’. Compared with the frequency ratio range required for ‘local stability’, it is noted that the frequency ratio range for achieving ‘global stability’ is more stringent. Therefore, by incorporating the FSI ROM with PMC, appropriate PMC parameters can be determined based on whether the actual problem requires ‘local stability’ at a certain natural frequency of the structure or ‘global stability’ at multiple natural frequencies.

Figure 18.

The root loci and their corresponding real parts of the controlled system varying with the reduced natural frequency of the cylinder at $Re = 60$ for (a,b) $\varOmega = 0.74$ and (c,d) $\varOmega = 1.08$.

4.2. The effect of the PMC damping and mass ratio

The mass ratio and damping of the PMC determine the margin of the improved stability of the SM. Therefore, an experiment is conducted to study the effect of the PMC damping and mass ratio on VIV suppression.

In the experiment, the mass ratio of the PMC is relative to the number of magnet blocks. By adjusting the length of the PVC strips as well as the distance between the magnet and the aluminium sheet, the PMC frequency is fixed at $F_{d} = 0.14$ for each mass ratio. The PMC damping coefficient is approximately 5 %. Figure 19 shows the transverse responses of the cylinder with different PMC mass ratios measured in the experiment over a period of 1 second, as well as the corresponding amplitudes varying with the PMC mass ratios. The results indicate that, within a certain range, the greater the mass ratio of the PMC, the better the control effect. However, it is observed that the lock-in has not been eliminated for $\varepsilon = 2.67\,\%$, although the response amplitude has been reduced by half.

Figure 19.

Time responses of the cylinder at $Re = 60$ for $F_{y} = 0.163$, as well as the maximum vibration amplitude $Y^{max}$ varying with the PMC mass ratio.

The damping of the PMC is the core parameter that determines whether the PMC can eliminate the frequency lock-in. Six PMC damping coefficients ($\lambda _{d} = 0.47\,\%, 1.82\,\%, 5.39\,\%, 10.19\,\%, 29.07\,\%$ and $66.84\,\%$) are investigated in the experiment. The PMC damping can be adjusted by changing the distance between the permanent magnet and the aluminium sheet. The PMC free attenuation response is measured by DIC to extract the damping coefficient. Analogously, the tests are conducted at $Re = 60$ for $F_{y} = 0.163$, and the PMC frequency and mass ratio are set to $F_{d} = 0.14$ and $\varepsilon = 5.35\,\%$. Figure 20 demonstrates the transverse responses of the cylinder for different PMC damping coefficients, as well as the maximum vibration amplitude varying with the PMC damping coefficient. The findings reveal that the PMC with a damping coefficient ranging from 5 % to 10 % can yield substantial control benefits, effectively suppressing VIV and eradicating frequency lock-in. However, when the PMC damping coefficient is too small ($\lambda _{d} = 0.47\,\%, 1.82\,\%$), the amplitude of the cylinder after introducing PMC control is similar to that of an uncontrolled cylinder, indicating that the PMC has almost no effect on VIV control. When the damping coefficient of the PMC increases, or even approaches overdamping, the vibration amplitude of the cylinder decreases but the frequency lock-in is not eliminated. This observation suggests that there is no direct correlation between the dampening of the PMC and the quality of the control effect. Furthermore, it is evident that the elucidation of the control mechanism of the PMC through energy dissipation is hindered.

Figure 20.

Time responses of the cylinder at $Re = 60$ for $F_{y} = 0.163$, as well as the maximum vibration amplitude $Y^{max}$ varying with the PMC damping coefficient.