## Abstract

In this study, we investigated two-degree-of-freedom (2d.f.) vortex-induced vibrations (VIVs) of a circular cylinder with a pinned attachment at its base; it had identical mass ratios and natural frequencies in both streamwise and transverse directions. The cylinder had a mass ratio, *m*^{*} of 0.45, and a mass damping, (*m*^{*}+*C*_{A})*ζ*, equal to 0.0841. Laser-induced fluorescence flow visualization and digital particle image velocimetry experiments were conducted over a Reynolds number range, 820≤*Re*≤6050 (corresponding to the reduced velocity range, 1.1≤*U*^{*}≤8.3). Measurements and visualization studies were made in a fixed plane at the cylinder mid-height, providing a two-dimensional picture of a highly three-dimensional system. However, significant insights can be gained from these experiments and form the basis of this paper. A large transverse amplitude response, (or four diameters peak-to-peak), in the upper branch was observed. The streamwise amplitude response exhibits an even higher peak amplitude, , which is approximately 125% of peak . Results show that there is no lower branch for this system and the transverse upper branch exhibits asymptotic behaviour, i.e. a wide regime of resonance. For *Re*>3000, the Strouhal number for the vortex shedding was 0.16 (±9%). Both the transverse cylinder oscillation and vortex-shedding frequencies, *f*_{OS,Y} and *f*_{VS}, respectively, were virtually identical throughout this range. While the streamwise oscillation frequency is typically twice the transverse oscillation frequency for a 2d.f. system, this is not the case at the lowest reduced velocities where oscillations first occur. Under these conditions the streamwise and transverse oscillation frequencies were identical. Finally, we observed that the cylinder wake exhibits both the P+S vortex-shedding mode and a desynchronized vortex pattern, which are uncommon for flow past a cylinder experiment. Very interestingly, the wide *U*^{*} range over which resonance occurs is dominated by a desynchronized vortex pattern. These results clearly demonstrate the differences that arise in 2d.f. VIV occurring below the critical mass ratio.

## 1. Introduction

Vortex-induced vibrations (VIVs) of a circular cylinder have received considerable attention over the past three decades from scientists and engineers. This is owing to the particular importance in marine problems such as the vibration of offshore structures, submarine towed array cables and ship moorings in water as well as atmospheric problems including smokestacks, cellular towers and buildings. Bluff body VIV is a subset of the broader field of fluid–structure interactions (FSI). Fundamental FSI studies of rigid bodies, both experimental and computational, often address only a single degree of freedom (1d.f.) due to the difficulties and complexities associated particularly with VIV.

The present research is focused on fully coupled, two-degree-of-freedom (2d.f.) VIV of a rigid pivoted circular cylinder. That is, a freely oscillating, surface-piercing circular cylinder was mounted in a water tunnel like an inverted pendulum. This is the 2d.f. extension of the 1d.f. work reported in Benaroya & Wei (2000), Dong *et al*. (2004) and Voorhees *et al*. (2008). In addition, the cylinder used in this investigation had a mass ratio, *m*^{*}, below the critical ratio, , identified by Govardhan & Williamson (2002); note that most VIV studies to date have been done on cylinders above . It is believed that results for a 2d.f. pivoted cylinder below critical mass ratio are shown for the first time.

It should be noted at the outset that a decidedly two-dimensional view has been made of a very three-dimensional system. However, keeping the three-dimensional nature of the problem at the forefront, it is possible to gain significant insights into the FSI at subcritical mass ratios. This is particularly true for this investigation because the cylinder itself is rigid except for the very short pin that anchors the cylinder to its base. In this regard, it is only necessary to track motions in one plane to capture the entire frequency–amplitude response characteristics. For a detailed analysis of three dimensionalities associated with these flows, the reader is referred to Voorhees *et al*. (2008).

The challenge behind studying fully coupled VIV is that the interactions between fluid dynamics and structural motion are typically nonlinear and stochastic. Sarpkaya (1979) aptly put it: ‘VIV is not a small perturbation superimposed on a mean steady motion. VIV is inherently nonlinear, self-governed or self-regulated, multi-degree of freedom phenomenon. It presents unsteady flow characteristics manifested by the existence of large-scale structures, sandwiched between two equally unsteady shear layers’. More traditional decoupled studies have been done either by forcing the structural motion and examining vortex dynamics, or mathematically assuming some form of periodic fluid forcing function(s) and computing the structural response. Indeed, much FSI research is still done in the traditional, i.e. decoupled, way.

The canonical two-dimensional VIV experiment is commonly referred to as the ‘elastically mounted cylinder’. Except for small-scale three-dimensionalities superimposed on the large-scale Kármán vortices (Wei & Smith 1986), flow along the entire length of the cylinder is effectively two dimensional. Low-pressure regions associated with the Kármán vortices induce fluctuating lift and drag forces on the cylinder. Under the right combination of damping, *ζ*, and reduced mass, *m*^{*}, these fluctuating forces will cause the cylinder to undergo periodic transverse oscillations. When VIV occurs, the frequency of vortex shedding, *f*_{VS}, is approximately equal to the oscillation frequency of the cylinder, *f*_{OS}. This condition has been defined as ‘synchronization’. The range of flow speeds over which this occurs is called the ‘synchronization regime’. Note that synchronization and ‘lock-in’ are used interchangeably throughout this paper. For some excellent and extensive reviews on VIV, the reader is referred to Sarpkaya (1979, 2003), Bearman (1984), Chen (1987), Blevins (1990), Naudascher & Rockwell (1994), Williamson (1996), Zdrakovich (1997, 2003) and Williamson & Govardhan (2004), just to name a few.

### (a) Frequency and amplitude response

The frequency and amplitude response of an elastically mounted rigid circular cylinder undergoing transverse VIV have been well characterized. There exist two main types of amplitude response depending on the mass-damping parameter, *m*^{*}*ζ*. Here *m*^{*} is the mass ratio and *ζ* is the ratio of critical damping to structural damping. Cylinders with high *m*^{*}*ζ* undergoing 1d.f. VIV exhibit two response branches, namely the initial and lower branches. This type of response is also often referred to as the classical Feng-type response (Feng 1968). Low *m*^{*}*ζ* cylinders, on the other hand, display three distinct branches: the initial branch, upper branch and lower branch. This is described in Khalak & Williamson (1999).

The initial and lower branches are common to both types of amplitude response. The initial branch is characterized by a vortex-shedding pattern with two single counter-rotating vortices per period. Williamson & Roshko (1988) referred to this as the ‘2S’ mode. This is followed by the lower branch, characterized by constant oscillation amplitude and frequency across a range of reduced velocity, *U*^{*}. In contrast to the initial branch, the upper and lower branches are characterized by a ‘2P’ vortex-shedding mode. That is, there are two pairs of counter-rotating vortices per shedding cycle. The second vortex of each pair in the upper branch is weaker than the first vortex and decays very rapidly.

Khalak & Williamson (1999) demonstrated that for a fixed value of mass damping, *m*^{*}*ζ*, the extent (i.e. range of *U*^{*} values) of the synchronization regime is controlled primarily by *m*^{*}, whereas peak amplitudes are controlled mainly by *m*^{*}*ζ*. For a complete discussion on frequency and amplitude response of elastically mounted low *m*^{*}*ζ* cylinders, please refer to Khalak & Williamson (1999). Amplitude and frequency response characteristics of elastically mounted cylinders can be generalized to pivoted cylinders as well and have been examined by Voorhees *et al*. (2008).

Two-degree-of-freedom, elastically mounted cylinders, with *m*^{*}>6 and low *m*^{*}*ζ*, exhibit transverse amplitude response behaviours similar to those of 1d.f. cylinders oscillating purely in the transverse direction as shown by Jauvtis & Williamson (2004). Responses for transverse cylinders below critical mass ratio are presented by Govardhan & Williamson (2002).

### (b) Two-degree-of-freedom VIV of a circular cylinder

Streamwise and transverse displacements associated with the 2d.f. VIV of a circular cylinder immersed in a uniform free stream can be well represented by(1.1a)(1.1b)In these expressions, *A*_{X} and *A*_{Y} are the oscillation amplitudes in *x*- and *y*-directions, respectively. The oscillation frequency, *ϖ*, is 2*πf*_{OS} and *θ* is the phase angle between the two-component displacements. Figure 1 shows the trajectory shapes (Lissajous figures) of 2d.f. cylinder oscillations as a function of phase angle. Note that the oscillation frequency in the *x*-direction must be twice that of oscillations in the *y*-direction to yield figure-eight type trajectories.

An early experiment on 2d.f. VIV was conducted by Chen & Jendrzejczyk (1979). They worked with a cantilevered circular cylinder in water up to a reduced velocity of *U*^{*}≈10. They found that the cylinder response was divided into several regions. For *U*^{*}>4.5, oscillations were predominantly in the transverse direction. They concluded that flow above this regime could be treated as a quasi-1d.f. problem.

Over a decade later, Moe & Wu (1990) carried out an investigation in which the ratio of natural frequencies in streamwise and transverse directions (i.e. *f*_{n,X}/*f*_{n,Y}) was 2.18. The mass ratios in both directions were also different. In one case, the mass ratio in the transverse direction was twice that in the streamwise direction. Owing to these conditions, they did not identify the different response branches observed for transversely oscillating cylinders.

Sarpkaya (1995) focused on different natural frequency ratios. When *f*_{n,X}/*f*_{n,Y}=1, he showed that the maximum 2d.f. amplitude response was 19% larger than for the 1d.f. case. In addition, peak oscillations occurred at a higher *U*^{*} for the 2d.f. case versus the transverse oscillating case. This meant that the ability to oscillate in 2d.f. had an impact on the system undergoing VIV. Sarpkaya (1995) also showed an amplitude response plot similar to that of Moe & Wu (1990) which had no distinct branches. Note, however, that the mass ratios were somewhat different in those experiments.

Jeon & Gharib (2001) studied cylinders undergoing forced 2d.f. motions. Specific phase angles, *θ*=0° and −45°, were chosen based on the assumption that figure-eight type motions (see figure 1) occur most often in nature. Jeon & Gharib (2001) argued that the transverse motions set the vortex-shedding frequency and that the streamwise motions determined the phase angles.

Jauvtis & Williamson (2004) investigated elastically mounted 2d.f. circular cylinders with low mass and damping. Their experimental set-up permitted exact mass and natural frequencies in both streamwise and cross-stream directions. They defined a moderate mass ratio (*m*^{*}>6) and found that the freedom to oscillate in the free-stream direction had surprisingly little effect on the transverse vibrations. However, there were dramatic changes for mass ratios less than six (which they called small mass ratio). They observed peak-to-peak amplitudes of three diameters in what they called a ‘super-upper’ branch. Correspondingly, they discovered a periodic vortex wake mode, comprising a triplet every half cycle which they refer to as the ‘2T’ mode. Working along the lines of Govardhan & Williamson (2002) for critical mass ratio, Jauvtis & Williamson (2004) found a critical mass ratio, , for 2d.f. elastically mounted cylinders.

Recently, Flemming & Williamson (2005) have studied VIV of a pivoted cylinder in 2d.f. They concluded that for moderate values of inertia damping, *I*^{*}*ζ*, the system exhibited the initial and lower branches, and for low inertia damping, the initial, upper and lower branches appeared, analogous to results of transverse-only oscillating cylinders. They also observed a new vortex formation mode, named the ‘2C’ mode, which comprised two co-rotating vortices that shed each half cycle, for their lightest inertia ratio, *I*^{*}, of 1.03. The critical mass ratio for a pivoted cylinder was experimentally determined to be approximately 0.5.

Numerical simulations of cylinders undergoing 2d.f. VIV have been done by Blackburn & Karniadakis (1993) and Newman & Karniadakis (1996). However, their two-dimensional simulations were performed at relatively low *Re* (i.e. *Re*≈100–200) compared with those investigated experimentally.

This study of 2d.f. VIV is of flow past a subcritical mass ratio circular cylinder which is pinned at one end. Thus, this is a system involving three-dimensional effects. Three-dimensionality has been thoroughly investigated by Voorhees (2002) and Voorhees *et al*. (2008) for a transverse oscillating inverted pendulum case. This experimental system has similar mass ratio and natural frequencies in both streamwise and cross-stream directions.

### (c) On the existence of a critical mass ratio

The mass ratio *m*^{*} is defined as the mass of the structure divided by the mass of the fluid displaced by the structure. That is, *m*^{*}=4*m*/*πρD*^{2}*L*, where *m* is the cylinder mass and *D* and *L* are the cylinder diameter and length, respectively. In general, experiments conducted in air necessarily have higher mass ratios, *m*^{*}=*O*(100), like those of Feng (1968). Experiments in water typically have lower mass ratios, *O*(1)≤*m*^{*}≤*O*(10). Experiments conducted by Williamson's group, e.g. Khalak & Williamson (1999), Govardhan & Williamson (2002) and Jauvtis & Williamson (2004) and the present studies, Benaroya & Wei (2000), Dong *et al*. (2004) and Voorhees *et al*. (2008), fall into this category.

As has been discussed to this point, the literature clearly shows the richness of the dynamics associated across the *m*^{*} parameter space. For large mass ratios, cylinder oscillations are excited only at harmonics of the structure's natural frequency. For moderate to low mass ratios, however, cylinder oscillations occur at frequencies below the structure's natural frequency. This is then followed by the lock-in behaviour at the natural frequency (and higher harmonics) observed for high mass ratio cylinders. The interesting question that naturally arises is whether or not there is a critical mass ratio for VIV of a circular cylinder below which cylinder oscillations occur irrespective of the cylinder's natural frequency.

Govardhan & Williamson (2002) showed that there is, in fact, a critical mass ratio, , for transverse VIV of elastically mounted cylinders. This was found to occur at low mass-damping values, (*m*^{*}+*C*_{A})*ζ*<0.05. In their analysis, Govardhan & Williamson (2002) showed that the cylinder response frequency, defined as *f*^{*}=*f*_{OS}/*f*_{n} (where *f*_{n} is the natural frequency in quiescent water), can be shown to take the form(1.2)Govardhan & Williamson (2002) defined *C*_{A} as the potential added mass coefficient and *C*_{EA} to be the effective added mass coefficient due to wake vortex dynamics. Note that for a circular cylinder, *C*_{A}=1. Govardhan & Williamson (2002) deduced an expression for the frequency of the lower branch, at relatively small mass-damping values, as(1.3)The value *C*_{EA}=−0.54±0.02 was obtained from a curve fit of a range of experimental data obtained for low mass damping. Equation (1.3) has very significant implications. In particular, one can identify a critical mass ratio, , where *f*^{*} will become infinitely large.

For , Govardhan & Williamson (2002) predicted that the lower branch can never be reached and ceases to exist. They hypothesized instead that the upper branch of synchronization will continue in the limit as *U*^{*} approaches infinity. Flemming & Williamson (2005) had experimentally determined that the critical mass ratio for a 2d.f. VIV of a pivoted cylinder is approximately 0.5. The main objective here is to study the dynamics related to cylinders below critical mass ratio.

### (d) Problem statement

In classical cylinder VIV experiments, cylinders oscillate purely in the transverse direction. However, many real engineering applications, such as offshore structures, smokestacks and cellular telephone antennae towers, respond in 2d.f. (or higher) and the structures may be pivoted or elastically mounted. In this investigation, the cylinder is pivoted at one end and free to move on the other in both streamwise and cross-stream directions to simulate real engineering applications. The cylinder has similar mass ratio and natural frequencies in both directions.

What is salient about this experiment is that the mass ratio of the cylinder is well below the critical values identified by Williamson's group. This opens up a whole new paradigm into bluff body VIV research. If there is a dramatic change in the response and body dynamics due to the second degree of freedom below the critical mass ratio, then there is a need to incorporate this understanding into the design and analysis of structures operating within this parameter space. The main goal of this investigation, therefore, is to better understand the dynamics of 2d.f. VIV of a circular cylinder, with a subcritical mass ratio, mounted as an inverted pendulum. The critical path to this objective includes the following:

mapping the frequency and amplitude response of the cylinder over as wide a range of reduced velocities as possible,

identifying the various vibrational modes arising across the reduced velocity range, and

using laser-induced fluorescence (LIF) flow visualization and digital particle image velocimetry (DPIV) to characterize the vortex dynamics associated with each mode of vibration.

### (e) A note on coordinates and non-dimensional groups

In this study, coordinates were chosen such that *x*, *y* and *z* correspond to the stream, transverse or cross-stream and axial directions, respectively. These are indicated in figure 2. The *z*-direction points up towards the free surface opposite the direction of gravity. This coordinate system is consistent with the ones used widely in VIV studies.

Table 1 shows the key non-dimensional groups used in this study. Added mass, *m*_{A}, is given as the drift volume multiplied by the fluid density (i.e. *m*_{A}=*ρ*_{fluid}*V*_{drift}); this was first articulated by Darwin (1953). In this case, the density of fluid is the density of water. Added mass can also be defined as *m*_{A}=*C*_{A}*m*_{d}, where *C*_{A} is the added mass coefficient and *m*_{d} is the displaced mass. As noted previously, *C*_{A}=1 for a circular cylinder.

Since the study of VIV involves a number of different characteristic frequencies, it would be expedient to clear up any ambiguities at the outset. The natural frequency of the cylinder in quiescent water has already been defined as *f*_{n}. Since the natural frequency is independent of direction, i.e. *f*_{n,X} is identical to *f*_{n,Y}, one needs only deal with *f*_{n}. The cylinder oscillation frequency is defined as *f*_{OS}, regardless of whether or not there is synchronization. *f*_{VS} is defined to be the vortex-shedding frequency of the cylinder in motion. The corresponding Strouhal number (sometimes called the Strouhal frequency), *St*, refers to the non-dimensional vortex-shedding frequency when the cylinder is held rigid in the flow. It is commonly accepted, see Schlichting (1979), that *St*≈0.21 for flow around a circular cylinder at *Re*>1000.

## 2. Experimental apparatus and methods

### (a) Flow facility

Experiments were conducted in a large free surface water tunnel facility. This closed-loop facility consists of an upstream end tank and settling chamber, two-dimensional contraction, test section, downstream end tank and two pumps. The test section is 57.2 cm in width×122 cm in depth×610 cm in length. It is constructed entirely from 1.91 cm thick glass panels placed in a welded steel I-beam frame. Flow is driven by two pumps operating in parallel. The maximum flow rate of both pumps combined is 1500 l min^{−1}, which is equivalent to approximately 30.0 cm s^{−1} when the test section is full. The free-stream velocity is uniform to within ±2% across the test section and turbulence intensities are less than 0.1% of the free-stream velocity. Detailed descriptions of this facility can be found in Smith (1992) and Grega *et al*. (1995).

### (b) Cylinder assembly

The circular cylinder used for experiment was designed to allow freedom to oscillate in both *x*- and *y*-directions. This cylinder assembly consisted of a base plate, stainless steel pin, dye injection module and two acrylic cylinder tube sections as shown in figure 2. In this manner, the cylinder was free to oscillate like an inverted pendulum. For details of the components and assembly, the reader is referred to Leong (2005).

The circular cylinder itself was manufactured entirely from thin-walled extruded acrylic tube owing to its optical clarity. The structure was hollow, rigid and of low mass ratio. The outer diameter was 2.54 cm with an inner diameter of 2.22 cm. The overall length was 109.22 cm for an *L*/*D* of 43. Each of the two tube sections and dye injection plug, comprising the length of the cylinder, were 50.8 and 7.6 cm long, respectively.

The dye injection module consisted of a short section of acrylic tube capped at both ends by machined plugs. A thin dividing plate was placed in the tube section and enabled us to create two separate dye chambers and then perform two-colour flow visualization studies. The module was equipped with two narrow dye injection slots located 30° on either side of the nominal forward stagnation line.

The cylinder was mounted to a 122 cm long×57.2 cm wide×1.27 cm thick base plate through a stainless steel pin. The effective length of the pin, i.e. the distance between the base plate and the bottom end of the cylinder, was 2 cm. For small angle deflections, the pin was assumed to act as a linear flexural spring with a flexural spring constant, *k*, of 11.0 N m rad^{−1}. This was measured and verified by applying static lateral forces to the top of the cylinder and measuring the deflection while it was mounted in the water tunnel (and the tunnel was filled with water). The maximum deflection angles caused by VIV in the transverse and streamwise directions were 5.7° and 7.1°, respectively.

The cylinder was immersed in a uniform flow of water, 101.6 cm in depth. The top of the cylinder protruded through the free surface. The mass ratio of the cylinder *m*^{*} was 0.45. The natural frequency of the structure in quiescent water, *f*_{n}, was 1.14 Hz. This was again measured by deflecting and releasing the cylinder mounted in the water tunnel with water, but without flow. The damping ratio *ζ* was 0.058 and the mass-damping parameter, *m*^{*}*ζ*, was 0.026. It is important to note that both the mass and natural frequencies are identical in the *x*- and *y*-directions.

### (c) Experimental methods

Detailed LIF flow visualization experiments and DPIV measurements were conducted over the range of reduced velocities, 1.1≤*U*^{*}≤8.3. This corresponds to a Reynolds number based on cylinder diameter range of 820≤*Re*≤6050. The maximum flow speed, *U*^{*}=8.3, was constrained by the maximum pump speed. All studies were conducted at the mid-height of the cylinder. The distance from the free surface was −43.7 cm (or *z*^{*}=*z*/*D*=−17.2).

In order to identify and characterize the various cylinder vibration modes, the range of flow speeds was subdivided into a large number of increments, Δ*U*^{*}. For *U*^{*}≤4.2, Δ*U*^{*} was approximately 0.2. This was done because many of the changes in vibrational modes occurred at low reduced velocities. The small incremental steps ensured that every mode would be fully captured. For *U*^{*}≥4.2, data were collected at increments of Δ*U*^{*}≈1.0. Each time the speed was changed, a minimum of 30 min were permitted to elapse before measurements commenced. This ensured that any transient effects were no longer present. These measurements were performed for approximately 20 different cases across the speed range.

Fluorescein (yellow-green) and rhodamine (orange-red) fluorescent dyes were used for the two-colour LIF experiments. Fifteen-minute-long S-VHS video sequences were recorded for each case. For each DPIV case, two sets of 225 image pairs (i.e. 15 s per set) were recorded. For lower *U*^{*} values, the interval between images in a pair, Δ*t*, was 0.005 s. At higher speeds, Δ*t* was 0.001 s. An in-house DPIV correlation program was used to determine the velocity field data. This included a two-stage correlation algorithm, zero padding and other features for maximum accuracy. A detailed description of the software, calibration, accuracy and uncertainty of the correlation program can be found in Hsu *et al*. (2000). For this study, the coarse and fine correlation interrogation windows were 128×128 pixels and 64×64 pixels, respectively. The image resolution was 73.62 pixels cm^{−1}.

Cylinder frequency and amplitude response measurements were made using dual photoelectric sensors, one each for the stream and cross-stream directions. These are described in Voorhees (2002) and Leong (2005). Data were sampled at 300 Hz, 10 times the DPIV sampling rate. This provided excellent time-resolved response measurements. Two-minute-long time records were recorded for each reduced velocity. Measurements were made while both incrementing and decrementing the flow speed in order to quantify any hysteresis effects that may be present. It is worth noting that cylinder position measurements were made near the tip of the cylinder, that is, above the free surface. However, the results presented in the following sections have been scaled relative to the cylinder mid-height.

## 3. Results and discussion

### (a) Frequency and amplitude response

A principal driver for this research was understanding the importance of 2d.f. for a pivoted, low mass ratio cylinder. The response of 1d.f. elastically mounted cylinders, such as that of Khalak & Williamson (1999), and pivoted cylinders, as in Dong *et al*. (2004) and Voorhees *et al*. (2008), has been extensively investigated. Govardhan & Williamson (2002) worked on the response of transversely oscillating cylinders below critical mass ratio and recently Jauvtis & Williamson (2004) have studied the response of 2d.f. elastically mounted cylinders.

In this section, the results relevant to a 2d.f. pivoted cylinder with nominally subcritical mass ratio, *m*^{*}=0.45, and mass-damping parameter, (*m*^{*}+*C*_{A})*ζ*=0.0841, will be presented and discussed. Both amplitude and frequency response are plotted against reduced velocity, *U*^{*}, in figure 3. The amplitude response from Jauvtis & Williamson (2004) of a 2d.f. elastically mounted cylinder with small mass ratio and from Govardhan & Williamson (2002) for a 1d.f. (transverse only) elastically mounted cylinder below critical mass ratio are superimposed for comparison. The reduced velocity, *U*^{*}=*U*/*f*_{n}*D*, is defined in table 1 for the free vibration experiment along with other key parameters.

For an elastically mounted cylinder with low but super-critical mass ratio, the amplitude response plot can be divided into three regimes. There is the initial branch (I) at low reduced velocities, characterized by small amplitude oscillations. This is followed by an upper branch (U) in which the oscillation amplitudes reach a maximum. When the Kármán vortex-shedding frequency matches the cylinder natural frequency, there exists an extended range of reduced velocities in which the cylinder will oscillate at a frequency close to *f*_{n}. This is known as lock-in and has come to be known as the lower branch (L).

Observe in figure 3*a* that for the present 2d.f. subcritical case, there is an initial branch, which spans the range 1.1<*U*^{*}<2.6, followed by an upper branch extending beyond *U*^{*}>2.6. Although there does not appear to be a clear amplitude jump distinguishable from the curve, a break in the branch around *U*^{*}≈2.6 has been imposed. The reason for this is that there is a small shift or discontinuity in the frequency curve .

One thing that is particularly noteworthy in figure 3*a* is the comparatively large maximum transverse amplitude in the upper branch. While classical elastically mounted transverse oscillating cylinders and cylinders mounted like inverted pendulums have peak amplitudes , the results of Jauvtis & Williamson (2004) indicated that response characteristics with larger amplitudes (i.e. ) should not be too surprising. What is surprising, however, is the value of peak amplitude, *A*_{Y}∼2, or about four peak-to-peak diameters. Oscillation amplitudes of this magnitude have not yet been observed.

Another important feature of the transverse amplitude response plot, versus *U*^{*}, is the distinct absence of a lower branch. This is consistent with the results of Govardhan & Williamson (2002) for 2d.f. elastically mounted cylinders. For the low, but super-critical mass ratio VIV case, the amplitude response curve will contain a ‘break’ (i.e. from either initial or upper branch to lower branch) at a certain reduced velocity. This typically occurs after . However, for the subcritical mass ratio, inverted pendulum examined in this study, this is clearly not the case. There is no lower branch and the transverse amplitude response approaches an asymptotic value, .

For the classical cylinder experiment, frequency response in the upper branch is flow driven, while in the lower branch it is structure driven. That is, in the upper branch, it is the vortex shedding that excites the cylinder motions. By contrast, for the lower branch, cylinder oscillations at the structure's natural frequency modulate the vortex-shedding mechanisms. In the present case, the cylinder is too ‘light’ to drive the flow. Therefore, the lower branch cannot exist and the upper branch will continue with increasing *U*^{*} indefinitely. The identification of an asymptotic amplitude is entirely consistent with the findings of Govardhan & Williamson (2002). They named this asymptotic amplitude phenomenon ‘resonance forever’.

From the peak of the upper branch, *U*^{*}≈5.2, and beyond, the transverse oscillations become irregular. This can be better understood by examining the sample time trace shown in figure 4 for *U*^{*}=8.25. Note in particular the anomalies at *t*≈15 and 45 s. To the contrary, Jauvtis & Williamson (2004) found that the peak of super-upper branch showed constant transverse amplitude vibration. The reason for the irregularities observed for the present experiment was traced to the onset of a ‘quasi-periodic shift’ in cylinder mean position. That is, the cylinder would oscillate about one deflection angle and then oscillate around a different deflection angle, switching back and forth between these positions.

Turning next to the streamwise amplitude response plot, versus *U*^{*}, in figure 3*b*, one can see that the maximum streamwise amplitude is very large, . This previously undocumented behaviour is approximately 125% of peak . In the 2d.f. elastically mounted cylinder case of Jauvtis & Williamson (2004), the maximum amplitude of across the entire *U*^{*} range was only approximately 0.3. By comparison with the present experiment, this is only approximately 20% of .

Continuing with the streamwise amplitude response plot shown in figure 3*b*, the upper branch can be seen to break around *U*^{*}≈4.4 after reaching its peak amplitude. The amplitude branch begins to increase again with increasing reduced velocity. However, owing to limitations in the top speed of the water tunnel, it was not possible to ascertain whether the streamwise amplitude keeps increasing or if approaches an asymptote like its transverse counterpart, . This streamwise amplitude response is not observed for an elastically mounted 2d.f. system above critical mass ratio (Jauvtis & Williamson 2004), principally because there is a lower branch. At that branch, the streamwise amplitude is very small, albeit visible, and almost constant.

The transverse oscillation frequency response plot shown in figure 3*c* is very similar to that shown by Govardhan & Williamson (2002). The dashed line between 0≤*U*^{*}≤1.5 indicates the Strouhal frequency, *St*=0.21; for vortex shedding from a stationary cylinder, see Schlichting (1979). It may be observed that the cylinder oscillation frequency data, appearing in figure 3*c* as solid circles, vary linearly with reduced velocity for *U*^{*}>2.6. It is expected that this trend will continue as *U*^{*} approaches infinity in light of the absence of a lower branch for subcritical mass ratio cylinders.

For a 2d.f. system, the oscillation frequency in the streamwise direction is generally twice that of the oscillation frequency in the cross-stream direction, i.e. *f*_{OS,X}≈2*f*_{OS,Y}. One would therefore expect to see a streamwise resonance when . However, this is not the case for the present low mass ratio cylinder.

For the *m*^{*}=7 case in Jauvtis & Williamson (2004), the streamwise oscillation amplitude is a maximum for *f*_{OS,X}≈*f*_{n}. For their small mass ratio case, *m*^{*}=2.6; however, the maximum streamwise oscillations coincide with the maximum transverse oscillation amplitudes, i.e. for *f*_{OS,Y}≈*f*_{n}. This is consistent with the present findings for *m*^{*}=0.45. Thus, for low mass ratios, there is a coupling between the streamwise and transverse motions, which does not occur for large mass ratios.

Figure 3*c* shows vortex-shedding frequencies across the reduced velocity range; these appear as open circles. Note that the cylinder transverse oscillation frequency is virtually identical to the vortex-shedding frequency, i.e. *f*_{OS,Y}≈*f*_{VS}. Recently, Dong *et al*. (2004) examined the quasi-periodic beating that occurs when *f*_{OS,Y}≈*f*_{VS}. This phenomenon exists for pivoted or inverted mounted cylinders (Voorhees *et al*. 2008) and has also been shown by Govardhan & Williamson (2002) for transverse elastically mounted cylinders below critical mass ratio. This can also be seen in figure 5 in which transverse oscillation amplitude is plotted as a function of frequency ratio, *f*_{VS}/*f*_{OS,Y}, and overlaid on the Williamson & Roshko (1988) map of vortex modes. Because data in figure 5 indicate that *f*_{VS}/*f*_{OS,Y}≈1, one can predict the cylinder oscillation frequency knowing the vortex-shedding frequency; for a 2d.f. pivoted cylinder below critical mass ratio, the non-dimensional vortex-shedding frequency was found to be *St*=0.16±9% for *Re*>3000 (or *U*^{*}>4.2).

In summary, then, the most significant response features are the high peak transverse and streamwise amplitude values and the lack of a lower branch. Thus, in some respects, this current research can be thought of as a combination of 2d.f. elastically mounted cylinders (Jauvtis & Williamson 2004), and transverse only elastically mounted cylinders below critical mass ratio (Govardhan & Williamson 2002). However, it is critically important to keep in mind that the inverted pendulum will become highly three dimensional as the cylinder deflects in the stream direction. The exact nature of these three dimensionalities is beyond the scope of the present study.

### (b) Vortex dynamics

This section contains a description of visual observations made from a detailed LIF flow visualization investigation. Sample images will be shown while a complete description of these experiments is presented in Leong (2005). To place this work in context, however, it would be helpful to re-examine the amplitude response characteristics in a different way.

An alternative way of plotting transverse and streamwise amplitude response characteristics is to use a different or modified normalized velocity, defined by . This is analogous to *f*_{St}/*f*_{OS,Y}. Using this independent parameter, the transverse amplitude response plot, shown as figure 5 for the present experiment, has been replotted in figure 6 using . The Williamson & Roshko (1988) map of vortex-shedding modes is overlaid for comparison.

One advantage of figure 5 is that it shows a more distinguishable break in the initial and upper response branches at *U*^{*}≈2.6. This provides the justification for imposing the break in figure 3 as discussed in §3*a*. The Williamson & Roshko (1988) map of vortex modes was originally developed for forced transverse oscillations of circular cylinders. And it has been shown to collapse pretty well with numerous different systems including transverse freely oscillating elastically mounted cylinders (Govardhan & Williamson 2000).

Before proceeding, it may be worthwhile to interject two observations about the Williamson & Roshko (1988) map. First of all, the map may not well characterize the vortex-shedding modes for stationary cylinders because *f*_{St}/*f*_{OS,Y} is infinite and the map was developed for frequency ratios up to *f*_{St}/*f*_{OS,Y}=3.

The second and more important observation is that the vortex-shedding frequency used for normalization is the shedding frequency of a cylinder at rest (i.e. Strouhal frequency) and not the actual shedding frequency associated with the VIV. Note that for most systems, the vortex-shedding frequencies of a cylinder undergoing VIV are usually not equal to the vortex-shedding frequency of the same cylinder at rest.

For these reasons, then, the transverse amplitude response was plotted versus the frequency ratio, *f*_{VS}/*f*_{OS,Y}, where *f*_{VS} is the actual vortex-shedding frequency of the cylinder. This is shown in figure 5. The key feature of this plot is that *f*_{VS}/*f*_{OS,Y} is, in fact, unity.

For the freely oscillating elastically mounted cylinder described in Govardhan & Williamson (2000), the initial branch is associated with the classical von Kármán vortex street wake. This has come to be known as the 2S mode, with two single counter-rotating vortices shed from the cylinder per period. The upper branch shows a weaker 2P mode consisting of two pairs of counter-rotating vortices per shedding cycle. In the 2P mode, the second vortex of each pair is weaker than the first. The lower branch exhibits a stronger 2P mode whereby each vortex within each vortex pair possesses almost equal strength.

For the subcritical mass ratio inverted pendulum used in this study, the initial branch is initiated with what will be referred to as a coalescence of 2S mode. This is essentially a Kármán-like vortex street pattern (2S) in which the vortices have just become strong enough, and the frequency is just close enough to the cylinder's natural frequency, so that weak oscillations occur.

At slightly higher reduced velocities, a symmetric vortex-shedding mode develops. This can be envisioned as the periodic shedding of starting vortex pairs, which, in turn, gives rise to in-line oscillations. This ‘in-line vortex-shedding’ mode is not uncommon to 2d.f. and in-line oscillating cylinders but does not exist for transversely vibrating cylinders.

With increasing *U*^{*}, flow returns to a more typical antisymmetric 2S mode. At the high end of the initial branch, vortex shedding begins to exhibit characteristics of a 2P mode. Descriptions of the cylinder motions associated with these different vortex-shedding modes are provided in §3*c*.

Figure 7 shows the coalescence of 2S mode, in-line mode, and 2S mode. Included in the figure are side-by-side comparisons of instantaneous DPIV vector fields and LIF images. In each image, flow is from left to right. While the images in figure 7 are shown in black and white, a complete set in colour may be found in Leong (2005).

Representative vector fields and flow visualization images are also provided from experiments in the upper branch. These are shown in figure 8. The upper branch starts with the 2P mode for *U*^{*}>2.6. Around the maximum of the amplitude response plot, vortex patterns are best described as a P+S mode. This is characterized by a pair of counter-rotating vortices shed from one side of the cylinder followed by and a single vortex shed from the other side. Very interestingly, there does not appear to be any synchronized vortex pattern at the peak of the upper branch.

Neither the P+S mode nor the desynchronization of vortices is observed for classical elastically mounted cylinders. This is probably owing to the much smaller amplitude response. A weaker P+S mode, shown in figure 8*a*, was observed at lower values of *U*^{*}. For these flows, the second vortex of the pair was notably weaker than the first. At larger values of *U*^{*}, a stronger P+S mode was observed. In those cases, both vortices within a pair were approximately equal in strength. This is shown in figure 8*b*.

The ‘desynchronization’ of vortices is so named because the Kármán vortices break up multiple times (at least in a two-dimensional sense). When this occurs, eddy sizes may appear quite large with seemingly random patterns of vorticity within each vortex core. Vorticity magnitudes within these vortices can, however, be very high.

It should be pointed out that the desynchronized vortices occur for VIV (as opposed to forced vibration experiments) when the cylinder oscillation amplitudes are very large, that is, when and *f*_{St}/*f*_{OS,Y}≈1. This occurs only for small mass ratios where the cylinder does not enter into a lock-in type of response. One can conjecture that this desynchronization phenomenon is related to the energy transfer between fluid and structure which drives and sustains the large amplitude oscillations.

In summary, the Williamson & Roshko (1988) map of vortex modes, in general, also holds for 2d.f. pivoted cylinders below critical mass ratio. The key exception is that there does not appear to be a 2P mode in the initial branch. One must keep in mind that the inverted pendulum will necessarily induce three dimensionalities in the flow. This may well result in different vortex-shedding modes along the span of the cylinder. However, in this study experiments were conducted only at the cylinder mid-height; the study of three dimensionality is deferred for later study.

### (c) Flow regimes

The previous two sections were focused on the structural (i.e. frequency and amplitude) response, §3*a*, and fluid dynamics (i.e. vortex dynamics), §3*b*, of the low mass ratio inverted pendulum. In this section, *x*–*y* trajectories traced out by the cylinder are examined. In particular, the phase angle *θ*(*t*) between the instantaneous streamwise *x*(*t*) and transverse *y*(*t*) displacements is examined. Phase angle was defined in equation (1.1*a*).

Figure 9 shows characteristic trajectory shapes for a number of different reduced velocities. Each trajectory was developed using approximately 120 s of position sensor data sampled at 300 Hz. At the lowest speed, *U*^{*}=1.54, the sampling period was sufficiently long to capture 55 complete cylinder oscillation cycles.

The initial branch exhibits two different types of oscillations: (i) unsteady quasi-in-line oscillation and (ii) figure-C like motions. An interesting feature in figure 9 is that the onset of figure-eight motions coincides with the beginning of the upper branch. This result, reported by Leong *et al*. (2003*a*,*b*), provides further justification for imposing a break at *U*^{*}=2.6 in the amplitude response plot, figure 3. It is important to observe, however, that not all figure-eight shapes are identical.

The trajectories shown in figure 9 are notably different from those reported by Jauvtis & Williamson (2004) for a 2d.f. elastically mounted cylinder. While they observed in-line oscillations at low *U*^{*}, they also reported figure-eight type motions in the initial branch. Oscillations in their super-upper branch were also predominantly figure eights. Cylinder motions at the apex of the super-upper branch, however, were described as a figure-C or crescent shape. Finally, Jauvtis & Williamson (2004) reported that oscillations in the lower branch were quasi-transverse reflecting the very low streamwise oscillation amplitudes.

In order to better understand the relationship between streamwise and transverse oscillation frequencies, spectra of the cylinder position data were computed. A unique feature from the spectral analysis is that for all cases except at low *U*^{*} (i.e. *U*^{*}=1.54) when the cylinder oscillations just begin, the ratio of streamwise to transverse oscillation frequency is unity, i.e. *f*_{OS,X}/*f*_{OS,Y}=1. It is generally accepted that in 2d.f. systems, the oscillation frequency in the streamwise direction will be twice the transverse oscillation frequency (i.e. *f*_{OS,X}=2*f*_{OS,Y}). This is simply due to the fact that for a figure-eight or C-shape trajectory, the cylinder must move back and forth twice for every transverse oscillation.

At *U*^{*}=1.54, another significant feature is that there are two competing frequencies for the cross-stream oscillation: the cylinder oscillation frequency and the Strouhal frequency. This has not been observed for 2d.f. elastically mounted cylinders. However, Chen & Jendrzejczyk (1979), who worked on a cantilevered cylinder, showed that the streamwise oscillation frequency was similar to cross-stream frequency for their lowest normalized velocity case. They did not, however, show two competing frequencies for that case.

At *U*^{*}≈1.95, transverse oscillations begin to exhibit quasi-periodic beating. This is also a characteristic of initial branch amplitude response for classical elastically mounted cylinders. At the same time, streamwise oscillations become intermittent.

## 4. Conclusions

The 2d.f. VIV of a subcritical mass ratio cylinder, attached at one end by a small diameter vertical pin, was examined using LIF, DPIV and position sensors. The goal of this study was to assess differences between the inherently three-dimensional FSI of the 2d.f. inverted pendulum and the nominally two-dimensional VIV of an elastically mounted cylinder. Key observations and conclusions drawn from this work are summarized below.

First, significantly high transverse amplitude response, , in the upper branch with an even higher streamwise peak amplitude of has been identified. The present results show that there is no lower branch for this system and the transverse upper branch extends well beyond the maximum reduced velocity of this study, if not to *U*^{*}⇒∞. The vortex-shedding frequency for this system matches the transverse oscillation frequency for *Re*≥3000. The corresponding Strouhal number is 0.16±9%.

The cylinder wake exhibits P+S and desynchronized vortex patterns, which are both uncharacteristic of 2d.f. VIV for classical elastically mounted cylinders. Very interestingly, the wide resonance regime is dominated by desynchronized vortices. In-line vortex shedding, which is characterized by symmetric roll up of vortices on opposite sides of the cylinder, was also observed at low *U*^{*}. In general, the Williamson–Roshko map of vortex-shedding modes described the pinned cylinder. The principal difference may be found in the initial branch where one finds only the coalescence of 2S mode, 2S mode and in-line modes. That is, there is no 2P mode.

There are two modes of oscillation in the initial branch: unsteady quasi-in-line and C-shaped (crescent). Immediately beyond the initial branch, the upper branch is dominated by figure-8-like motions. The quasi-in-line oscillations occur for 1.54<*U*^{*}<1.74, while the C-shaped oscillations were observed in the range 1.95<*U*^{*}<2.16.

In short, there are distinct differences between the pinned cylinder and the classical elastically mounted cylinder. These differences, both structural and fluid dynamics, must be incorporated when modelling 2d.f. VIV of cylinders with subcritical mass ratios.

## Acknowledgments

Support for this research from the Office of Naval Research through Dr Thomas Swean is gratefully acknowledged. The authors would also like to thank Mr John Petrowski for his technical help with the position sensor instrumentation for this investigation.

## Footnotes

- Received August 10, 2007.
- Accepted May 19, 2008.

- © 2008 The Royal Society