Analysis on Flow Induced Motion of Cylinders with Different Cross Sections and the Potential Capacity of Energy Transference from the Flow

The energy in flow induced motion (FIM) was harnessed in recent years. In this study, the energy transfer ratio was derived to estimate the energy transference from the flow to the FIM.Then the FIM characteristics and energy transference of cylinders with different cross sections were experimentally investigated.Themain findings are listed as follows. (a) Circular cylinders and diamond prisms both present a self-limited motion. The maximum amplitude ratio of circular cylinder is around 1∼1.2 which is higher than that of diamond prism (0.4∼0.5). (b) Triangle prisms and right square prisms present a self-unlimited motion. For triangle prism, amplitude ratio increases over 1.8; for right square prisms, amplitude ratio reaches 1.2. (c) The maximum transfer ratios of circular cylinder and triangle prism are 80% and 57%, respectively, which are much higher than those of other prisms, indicating that circular cylinder and triangle prism have better performances in energy transference. (d) The transfer ratio is strongly dependent on the damping and mass; higher damping or mass will promote a higher transfer ratio. (e) Beyond the critical transfer ratios, amplitude variation coefficients are around 10%∼30% resulting in a better performance in stationarity.


Introduction
Flow induced vibration (FIV) is a common phenomenon which can be observed in many engineering fields.Many structures such as long-spanned bridges [1], marine risers [2], wind turbine towers, and power-transmission cables have been damaged mainly because of FIM.Vortex-Induced Vibration (VIV), as one of the most common FIV phenomena, was firstly observed by Leonardo da Vinci in a simple way, in circa 1500 AD.When the incoming flow was at an appropriate velocity, vortices alternately shed from either side of the bluff body leading the different distribution of pressure on the surface of the body which resulted in the transverse motion [3].In the past fifty years, many scholars have devoted themselves to the studies of VIV.Feng [4] firstly used experimental method to investigate the oscillating displacement, the vortex shedding frequency, and the phase angle in a wind tunnel by using 3-inch diameter circular and D-section cylinders.Similar experiments were then performed in a water tunnel by Bearman [5], and a mathematical model was established to predict the VIV performance of the circular cylinder.Sarpkaya [6] and Williamson and Govardhan [7] used reduced velocities   = /(  ⋅ ) to predict the synchronization or lock-in region and used the evolution of wake mode to explain the self-limited feature in VIV, where   is the natural frequency, and  is the diameter of the cylinder.Their further studies [8][9][10] reported that the VIV performance was strongly dependent on the combined parameter  * , where  * is the mass ratio (the ratio of oscillating mass to displaced fluid mass,  * =  osc /  ) and  is the system damping ratio.At low  *  (the cylinder is usually immersed in water) four branches were observed in the amplitude response.They were the initial branch, upper branch, lower branch, and desynchronization branch.Recently, Raghavan and Bernitsas [11] reported that the Reynolds number had great impacts on the peak of amplitude, the lower branch, and the desynchronization branch for circular cylinders with larger diameters.Except for the Shock and Vibration circular cylinder, other cylinders or prisms with symmetric cross sections can also produce VIV.Nemes and Zhao and et al. [12,13] reported that the incident angle had a significant influence on the performance of square cylinders.When the incident angle was 45 degrees (diamond cross section), the cylinder presented a self-limited motion.
Galloping, as another common FIV phenomenon, is defined as a motion with large amplitude and low frequency [14].The transverse galloping usually occurs in the motion of cylinders with sharp sections such as triangle prisms or rectangular prisms immersed in steady flow due to the periodic variation of attack angle [15].Den Hartog [14] first used the quasi-steady hypothesis to predict the performance of galloping and fluid forces.Parkinson [16] reported that once galloping started some nonlinear features such as the hysteretic behavior were observed.Contrary to VIV, galloping is not a self-limited motion.As the velocity increases, the amplitude monotonically increases until the bluff body reaches the limitation and is damaged.Because of the huge destruction from the galloping to the structures, which is much larger than that of VIV, many researchers [17][18][19][20] focused on the studies of stability and influential factors to predict and control galloping in the past few decades.
Contrary to suppressing the FIM, utilizing the energy from the FIM was a new idea in recent years.To power electronic components, the piezoelectric transducer using aeroelastic oscillation energy was invented for replacing small batteries [21,22].Mehmood et al. [23] suggested that the load resistance had a significant effect on the oscillation amplitude, lift coefficient, voltage output, and harvested power in the piezoelectric transducer.In addition to the piezoelectric transducer, another interesting hydrokinetic energy convertor named VIVACE (Vortex-Induced Vibration for Aquatic Clean Energy) convertor was developed by Bernitsas et al. [24,25] and coworkers based on a similar concept.This convertor, which contains advantages of high energy density, low operable flow velocities, and low maintenance cost, harnessed the hydroelastic oscillation power of the circular cylinder in FIM with PTC [26] immersed in steady fluid flow.In order to accurately investigate the influences of damping, mass, and stiffness, a virtual damper-spring system (  ) was built by Lee et al. [27] and further improved by Sun et al. [28].The measured results showed an excellent agreement between the   and the real springs and dampers.
The majority of researchers used theoretical [29][30][31] or numerical methods [23,32,33] to investigate the FIM characteristics and the energy extracting.Meanwhile, most FIM experiments were performed in wind tunnels [17][18][19][20]34], and large numbers of them only focus on a cylinder with a certain cross section [23][24][25][26][35][36][37].Only limited studies used experimental methods to study the FIM characteristics and the mechanical energy of cylinders with different cross sections.In fact, FIM usually occurs in a combined mode of VIV and galloping.At certain velocities, VIV may occur; at some velocities, galloping may occur; at other velocities, the transmission of VIV and galloping may occur.
This study focuses on the characteristics of FIM and the potential capacity of energy transfer from the incoming flow for cylinders with different cross sections using experimental methods.In Section 2, the energy transfer ratio is derived based on the motion of the single-degree freedom system, and the influential factors are analyzed.In Section 3, the experimental setup, the physical models, the measurements, and the verifications are described.In Section 4, the characteristics of FIM for different cylinders are compared and analyzed.In Section 5, the energy transfer ratio and the capacity of energy transference are discussed.The effects of the influential factors on the transfer ratio are analyzed.In Section 6, the deviation ratio of oscillating amplitude is introduced to estimate the stationarity of oscillation.Then the relationship between the energy transfer ratio and the stationarity is discussed.The conclusions are summarized at the end.

The Motion of a Cylinder.
A simplified configuration of a spring-mounted cylinder, with any cross section sharp prone to FIM under the action of an incoming flow in the transverse direction, is considered, as shown in Figure 1.The motion of the cylinder in the -direction is modeled by a second-order linear equation as follows: where  osc is the oscillating mass of the system, which is the sum of cylinder mass  cyl , transmission mass  tra , and onethird of the spring mass  spr ;  is the mechanical damping coefficient;  is the stiffness of the oscillating system; ÿ , ẏ , and  are the acceleration, the velocity, and the displacement of the oscillator, respectively; ŷ is the unit normal vector in the transverse direction; and  fluid is the force exerted by the fluid on the body in the transverse direction.Generally,  fluid can be decomposed into viscous and inviscid components as follows: where   is the added mass (for circular cylinder,   is approximately equal to 1);   is the instantaneous lift force coefficient;  is the water density;  is the free-stream flow velocity;  is the characteristic dimension of the body normal to the flow (for circular cylinder,  is the diameter of the cylinder); and  is the length of cylinder.Then the equation of motion could be derived as Bearman [5] reported that FIM could be regarded as a sinusoidal motion, and the lift force acted on the bluff bodies can also be regarded as a sinusoidal force.Then the transverse displacement and the lift force coefficient can be obtained as the functions of the time .
where  is the amplitude of the oscillation displacement;  osc is the dominant frequency of oscillation;   is the maximum lift force coefficient;  fluid is the dominant frequency of lift force; and  is the difference of phase angle between the lift force and the displacement.For stable oscillation,  fluid is approximately equal to  osc ,  fluid =  osc .

The Mechanical Power of Oscillator.
In order to harness the energy in FIM, the flow energy must be transformed through two steps.Firstly, the kinetic energy in the incoming flow should be transferred to the oscillating mechanical energy.Then only a part of mechanical energy is extracted by the convertor, and the rest is dissipated due to the mechanical damping of the transmission part and the convertor.The mechanical energy is the base of the harnessed energy, and the transfer ratio, which is the ratio of the mechanical energy to the flow energy, will affect the conversion ratio directly.Consequently, the expression of mechanical energy in FIM is derived in this section.The mechanical energy of the oscillator can be obtained by integrating the left hand side of (3) after multiplying the instantaneous velocity.Then the mean mechanical power  Mech in a period  osc can be written as The displacement  of FIM is assumed as a sinusoidal function of time  (see (4a)).Then  Mech can be simplified as Then  Mech can be further derived as Equation (7) shows that the damping coefficient of the oscillating system , the dominant frequency of oscillation  osc , and the amplitude of the displacement  have great impacts on the mechanical power.

The Energy Transfer Ratio.
The mechanical energy of oscillating cylinders originally comes from the kinetic energy in flow.Until now, previous studies have different opinions on calculating the flow energy   .Some researchers believe that   is the kinetic energy of the flow over the projected area of the cylinder  (as seen in Figure 2(a)).Others believe that   is the kinetic energy of the flow over the projected area of the cylinder and the oscillation ( + 2) (as seen in Figure 2(b)).The expressions are presented, respectively, as follows: Or Then the transfer ratio can be derived according to (7), (8a), and (8b).Here, we use the symbol "" to represent the transfer ratio.Then the expressions can be written as Based on the transfer ratio, the capacity of energy transference from the flow can be efficiently estimated.Then the damping ratio  and the natural frequency   of the oscillating system are introduced as follows: The energy transfer ratio can be further derived as Then the displaced fluid masses   for cylinders or prisms with different cross sections are introduced as Flow  where  represent different cross sections.For the circular cylinder,  is /2; for the regular triangle prism,  is √ 3/4; for right square or diamond prism,  is 1.
Then the transfer ratio could be nondimensionalized as follows: where  * is the amplitude ratio,  * = /;  * is the frequency ratio,  * =  osc /  ; and   is the reduced velocity of flow,   = /( ⋅   ).
As we all know, the performances of the amplitude and the frequency are strongly dependent on the cross section, the mass, the damping, and the velocity.Therefore, the performances of transfer ratio  are subsequently dependent on the cross section, the mass, the damping, and the velocity according to (14a) and (14b).
Based on the factors above, an experimental setup was established and verified in Section 3. Then the characteristics of FIM and the energy transfer ratio were analyzed in Sections 4 and 5.The physical model consists of the frame, the guide ways, the transmission part, the oscillators, and the springs.The frame is made of iron located on the top of the channel.Two linear guide ways are fixed on the frame in the vertical direction to the incoming flow.The transmission part is forced to move on the guide ways in the transverse direction by using four linear bearings.Both end sides of the oscillator are fixed on the foot of the transmission section.The top and bottom of the springs are fixed on the transmission section and the frame, respectively.Then the oscillators can have an alternately linear motion in the transverse direction to the incoming flow.A magnetic induction displacement transducer is fixed on the middle of the frame in the transverse
VIV initial branch Galloping  direction to measure the oscillating displacement as shown in Figure 3.The measurement range is from 0 to 800 mm.The accuracy of the displacement transducer is within ±0.05%, giving a displacement measurement resolution of 0.001% FS.
The rod is fixed on the top of the transmission section.The flow velocity measurement is performed using the Pitot tube with a differential pressure transmitter, which is positioned about 1 m in front of the prism.The accuracy of the differential pressure transmitter is within ±0.1% of the 6 kPa linear range available, and its resolution is within 0.1% FS.All data are collected at a 50 Hz sampling rate.Other details had been introduced by Zhang et al. [38].

Oscillators.
Four typical cylinders and prisms with different cross sections are selected in the present experiments.They are circular cylinders, regular triangle prisms, and right square prisms, as shown in Figure 4.The characteristic dimensions  of the oscillators are 0.06 m; the lengths  of the oscillators are approximately 0.5 m.The oscillators are composed of Plexiglas.Two end plates are fixed on the two end sides of the oscillators, respectively (Figure 4), and the gaps between the end plates and the walls of the channel are less than around 1 cm.Then the boundary effects can be reduced significantly [39][40][41].
For circular cylinders, different incident angles hardly affect the performance of the motion due to the axisymmetric cross section.But for other prisms the different incident angles strongly affect the performances.Some studies reported that the triangle prisms can achieve fully developed transverse galloping with high amplitude and stable frequency when the base of the triangle is vertical to the coming flow (the incident angle is zero).But in other configurations (the incident angle is not zero) the triangle prisms cannot achieve it.Thus, only one configuration of triangle prism, in which the base is vertical to the coming flow, is selected in present experiments (as seen in Figure 5(b)).In addition to the triangle prisms, the performances of the square prisms are more complicated while the incident angles are changing [12,13].Two typical configurations for square prism are selected in present experiment.They are right square prisms (the incident angle is zero) and diamond prisms (the incident angle is 45 degree) shown in Figures 5(c) and 5(d).

Mass and Stiffness.
Inside the oscillators, there is room to insert additional weight  add to change the oscillating mass  osc to control the mass ratio  * of oscillators.For the circular cylinders, the right square prisms, and the diamond prisms, two mass ratios are selected, 2.2 and 4.2.For the regular triangle prisms, the mass ratio of regular triangle prism cannot be decreased to 2.2 due to the limitations of the experimental setup.Only the configuration with  * = 4.2 is selected for the regular triangle prism in present experiment.The details of mass of each oscillator are illustrated in Table 1.
A linear stiffness system is needed to investigate free oscillation.In the present study, the linear stiffness system is composed of different physical extension springs, shown in Figure 6.Then different system stiffness can be modeled by using different groups of springs.The stiffness of a spring group with  springs is the sum of each spring stiffness  spr, in the group.
Six spring groups are selected in our experiment.The range of stiffness is varied from around 100 N/m to 600 N/m.The details are presented in Table 2.

Free Decay Tests and Validation.
The damping ratio  is another key factor affecting the transfer ratio.In this section, the values of the damping ratios in different configurations were obtained by performing free decay tests.In order to guarantee the accuracy of results, every configuration was repeated four times in this experiment.The final results were determined using a simple average method.
Depending on the test results, the damping  could be calculated using the logarithmic decrement method, which is expressed as where   denotes the amplitudes in the th peak, which were measured in the free decay tests.Meanwhile, the natural frequencies of the system in different configurations were extracted using FFT from the time histories of the displacements measured in the free decay tests.The natural frequencies were then calculated by theoretical method and compared with the measured results to further verify the dynamic reliability of the experimental setup.
When a cylinder is immersed in air, its added mass can be negligible due to the low density and viscosity of air.When a cylinder is immersed in water, the added mass effect should be taken into account.Then the calculated natural frequency in air and in water can be expressed as where  and  osc were based on the results in Tables 1 and 2,   is added mass, and   is added mass coefficient.
For circular cylinder,   could be regarded as 1 (  ≈ 1).However, for triangle prism and square prism,  , and   are very complicated.Until now, different researchers had different opinions on the value of   for noncircular cylinders.Our group had tried to reveal  , and   of triangle prism for different system stiffness  [42].But the results showed that the measured  , did not agree with the calculated  , .It is difficult to analyze the added mass effect.In order to ensure that the experiment results for different cylinders are compared under the same conditions, we selected the natural frequency in air   to calculate the reduced velocity (  = /(  )) and frequency ratio ( * =  osc /  ).
The complete results in the free decay tests are listed in Table 3.  ,cal represents the natural frequency calculated using (17a);  ,mea represents the natural frequency measured in each free decay test; and the expression of error is Error =   | ,cal − ,mea |/ ,mea .It is observed that the measured and the calculated natural frequencies have an excellent agreement, and the maximum error is as low as 3.5%, which indicates that the test setup and method are reliable.
The system damping ratio is strongly dependent on the damping oscillating mass, and system stiffness according to (10).In our experimental setup, there was no positive method or device to change the damping coefficient.The change of damping ratio was mainly dependent on the change of system stiffness.Therefore, the increase of stiffness meant the decrease of the damping ratio.
In the present experiment, we used the established oscillation system, which had been introduced in Zhang et al. 's experiment [38].The details of validation had also been described in Zhang et al. 's research [38].

Experimental Results and Discussion
In this section, the responses of the amplitude and the frequency for circular cylinders, regular triangle prisms, right square prisms, and diamond prisms are discussed, respectively.The variations of  * and  * against the reduced velocity   for those cylinders or prisms are plotted in Figures 7, 8, 9, and 10.In the figures,  * is the ratio of amplitude and diameter (/), and  is the average of all amplitudes in time histories of displacements measured over 60 seconds. * is the ratio of the dominant oscillation frequency and the natural frequency,  * =  osc /  , and  osc are also obtained from the time histories of displacements by FFT method.  is the reduced velocity, and its expression is   = /( ⋅   ).
The indexes above are all nondimensional.

The Response of Circular Cylinder.
As the reduced velocity   increasing, three branches, which are the initial branch, the upper branch, and the lower branch or the desynchronization, can be observed clearly for all cases (different damping ratios and different mass ratios) in Figure 7.The details of characteristics in each branch for circular cylinder can be summarized as follows.
(a) In the initial branch, the amplitude ratio  * and the frequency ratio  * both increase rapidly from small values.
The motion of the circular cylinder initiates at around   = 3∼4.
(b) As the reduced velocity   increases to the range of upper branch, the amplitude ratio  * and the frequency ratio  * both present a jump.Then  * stays at a high level of approximately 1;  * increases slowly around 1, too.It indicates that the oscillation is in the lock-in range.The natural frequency and the frequency of lift force are synchronized well.
(c) As the reduced velocity   further increases to move away from the range of upper branch (lock-in range), the response of circular cylinder enters the lower branch or desynchronization.In this branch, the frequency ratio  * increases rapidly again.The frequency of lift force and the natural frequency cannot be synchronized well any more.Therefore, the amplitude ratio  * collapses rapidly.
(d) Different damping ratios and different mass ratios cause different performances of the oscillation.It is found Shock and Vibration that  * increases with the increase of the damping ratio.In addition, as the mass ratio increases, the ranges of lock-in are also changed.When  * = 2.2, the lock-in range of reduced velocity is about 4 to 5; when  * = 4.2, it is about 5 to 6.
The analysis of the results above reveals that the circular cylinder performs a "self-excited" and "self-limited" VIV in the present experiment.This performance is consistent with the results reported by Khalak and Williamson [8,9], Govardhan and Williamson [10], and Raghavan and Bernitsas [11].

The Response of Regular Triangle Prism.
Compared with the performance of the circular cylinder (as shown in Figure 7), the lower branch (VIV) is not observed in the response of the regular triangle prism (as shown in Figure 8).Instead, a galloping branch is observed after the reduced velocity exceeds   = 6.The amplitude ratio  * of the prism monotonically increases as the reduced velocity   increases.Even when   is over 7, the trend of amplitude ratio  * still shows no sign of decrease.This performance is "selfexcited" but not "self-limited," which is different from the circular cylinder.Similar results of regular triangle prisms were reported by Alonso and Meseguer [15] and Ding et al. [33].
The details of characteristics for regular triangle prism in each branch can be described as follows.
(a) The motion of the regular triangle prism initiates at around   = 3.Then it goes into the VIV initial branch.In this branch, the frequency ratio  * ≪ 1 due to the bad synchronization between the frequency of flow lift force and the frequency of oscillation and results in the low value of  * .Additionally, as   increases,  * increases rapidly due to the gradual strengthening of the synchronization, which results in the rapid increase of the value of  * as well.These characteristics are similar to the circular cylinder, which indicates that the motion could be excited by the shedding vortices in rear wake.
(b) As the reduced velocity   exceeds 4.5, the motion goes into the VIV upper branch.In this range, the frequency ratio  * jumps to a high level and approximately maintains a constant.It indicates that oscillation is in the lock-in range, and the natural frequency and the frequency of lift force are synchronized well.The amplitude ratio  * stays at a high level and continues increasing.Nevertheless, although the performances above are similar to the circular cylinder in this range, there still exist some differences from the circular cylinder.For the triangle prism,  * is around 0.7, which is less than the  * of the circular cylinder.In addition,  * still dramatically increases, and  * stays constant; but for the circular cylinder  * stays as a constant, and  * increases slowly.
(c) As the reduced velocity   exceeds 6, the prism enters the galloping branch.In this range, the amplitude ratio  * exceeds 1.8 and continues increasing, but the frequency ratio suddenly drops from 0.7 to 0.6 as shown in Figure 8(a).It implies that the force of shedding vortices would not be the main cause of the oscillation in the galloping branch.Contrarily, the instability of the lift force acting on the prism, which results from the large change of the angle of attack due to the shape of the cross section, the high velocity, and the high amplitude, might be the main cause of the oscillation in this branch.It is noted that  * max = 1.8 is limited due to our present experimental setup.If there is no limitation, higher amplitudes could be achieved.
(d) The high damping ratio (or low system stiffness) causes the disappearance of the VIV initial branch.In our test, the damping ratio is mainly dependent on the system stiffness , according to the expression  = /2√ osc .And the reduced velocity   is also dependent on the system stiffness , according to the expressions   = /(  ⋅ ) and   = 1/2√/ osc .If  osc , , and   maintain their values, the flow velocity  with a high damping ratio must be lower than those with a low damping ratio.The lower flow velocity  (or Reynolds number Re) results in a lower lift force acting on the prism.This is a reasonable explanation for the disappearance of the VIV initial branch for the high damping ratio.

The Response of Right Square
Prism.The right square orientation (the incident angle is 0 degrees) is one of the symmetrical configurations of the square prism.It can be observed that the performances of the right square prism are also "self-excited" but not "self-limited," which are different from the classical VIV performances of the circular cylinder (Figure 7) but are similar to the performances of the regular triangle prism (Figure 9).
As the velocity increases, the response of the amplitude has a monotonic increase trend ( * reaches 1.2 for   > 14), the value of frequency is very low (0.4∼0.7).In 1964, a linear response of amplitude for right square prism was predicted by quasi-steady theory of Den Hartog [14] and Parkinson [16].In the present experiment, the results match well with the classical galloping response, which indicates that the main performance of the right square prism, especially at high reduced velocity, is galloping.In addition, there is a slight deviation in the amplitude trend at   = 10∼12.It probably is the "kink" which was observed by Bearman in wind tunnel experiments.
It is noted that the distinction of each branch is not clear.It is still difficult to judge the initial reduced velocity of galloping even by the analysis of the variation of frequency ratio (as shown in Figures 9(b-1) and 9(b-2)).This feature is significantly different from the galloping performance of the regular triangle prism, which is probably caused by the complex vortex formation or vortex reattachment due to the cross-sectional shape.
In addition, as the damping ratio or mass ratio increases, the initial branch of vibration gradually disappears.This feature is the same as the feature of the regular triangle prism, and the reason is also the difference of lift force resulting from the change of the system stiffness.

The Response of Diamond
Prism.Diamond orientation (the incident angle is 45 degrees) is another symmetrical configuration of the square prism.In Figure 10, the initial branch, the upper branch, and the lower branch or the desynchronization can be clearly observed as the reduced velocity   increases for all cases.These features are comparable with those of the circular cylinder.It indicates that the diamond prism undergoes a "self-excited" and "selflimited" performance.Similar results of diamond prisms were presented by Nemes et al. [12], Zhao et al. [13], and Ding et al. [33].
The details of characteristics in each branch for the diamond prism are summarized as follows.
(a) The frequency of the lift force is low because the low flow velocity is in the initial branch.Then the bad synchronization between the lift force and oscillation causes the low value of the frequency ratio ( * < 1), which results in the very low value of the amplitude ( * = 0.2∼0.3).However, as the reduced velocity   increases, both  * and  * increase rapidly due to the strengthening of the synchronization.These features are the same as the circular cylinder.
(b) When   increases to the upper branch, the resonant response is observed clearly.In this branch, the frequency ratio  * is significantly close to 1, and the slope of  * against the reduced velocity   presents a slight drop.That means the vibration is locked well on the shedding vortices (or lift force), and it is the reason why the amplitude jumps to and remains at a high value until the end of the branch.The features above are very close to the circular cylinder, but the following features are contrarily different from the circular cylinder.The peaks of  * are obviously smaller (for the diamond prism,  * max = 0.4∼0.5;for the circular cylinder  * max = 1∼1.2),and the range of reduced velocity in the upper branch (lock-in range) is wider.It is probably because of the changes of the lift force and the vortex formation due to the sharp corners in the cross section.
(c) When the reduced velocity   exceeds 7, the diamond prism starts to present a drop in the responses of the amplitude as the reduced velocity moves away from the lockin range and goes into the lower branch or the desynchronization.Meanwhile, the frequency ratio  * increases rapidly again due to the desynchronization of the motion and the lift force, leading to the drop in amplitude.It is noted that these features above are similar to the circular cylinder; there still are two significant differences in the drop of the amplitude between the two cylinders.Firstly, the slope of the drop for the diamond prism is obviously smaller.Secondly, as  * increases, the slope becomes too small to distinguish the upper branch, the lower branch, and the desynchronization.
(d) In addition, influences of the damping ratio and the mass ratio on the oscillation of the diamond prism are the same as the circular cylinder.A higher damping ratio leads to a higher amplitude ratio, while a lower mass ratio leads to a wider lock-in range of reduced velocity.
Therefore, we can conclude that the diamond prism undergoes a special "self-excited" and "self-limited" vibration which is different from the classical VIV of circular cylinder.

The Energy Transfer Ratio and the Influential Factors
In this section, the energy transfer ratios for different cylinders or prisms are calculated, analyzed, and compared.Then the impacts of the damping ratios and the mass ratios on the energy transfer ratios are investigated, respectively.In order to ensure the reliability of the results, the energy transfer ratios were all calculated based on the experimental results.The mechanical power of oscillation  mech was calculated by integration method with a time step of 0.02 s.The displacement  was measured by the magnetic induction displacement transducer; the velocity of oscillation ẏ was the differential of the displacement; and the acceleration of ÿ oscillation was the differential of the velocity ẏ .The mechanical damping coefficient  was calculated by the tested results of damping ratios  presented in Table 3.In addition, the flow energy power   was calculated by (9a).The flow velocity  was measured by the Pitot tube.Finally  was calculated by  =  mech /  .

The Performances of the Transfer Ratio For Different
Oscillators.The variations of the energy transfer ratios with the reduced velocity for different cylinders or prisms are plotted in Figure 11.Then the following characteristics are obtained.
It is observed that the variations of the transfer ratios for the circular cylinder and the right square prism are similar.Prior to a critical reduced velocity, the transfer ratio increases until a maximum value is reached.Then the ratio slowly diminishes to a low value.It can be found that that the critical reduced velocity (  ≈ 5) is actually the boundary to distinguish the VIV initial and the upper branch.These features can be attributed to the "self-excited" and the "self-limited" motion for the two cylinders (or prisms).After reaching the critical reduced velocity (  ≈ 5), the oscillation enters the lock-in range (VIV upper branch).In this range, the amplitude and frequency approximately maintain a constant or increase very slowly.This results in a slow increase of the mechanical power.By contrast, the flow energy power grows very fast due to the increasing flow velocity.Therefore, the transfer ratios collapse after the critical velocity.
The variations of the transfer ratios for the regular triangle and right square prism are different from those of the circular cylinder and diamond prism.For the triangle prism, the transfer ratio asymptotically maintains a constant after reaching the critical reduced velocity, which differs from the circular cylinder.For the right square prism, the transfer ratio continuously increases without any sign of decrease.These features can all be attributed to the "self-excited" and the "self-unlimited" motion (galloping) with a monotonic increase in amplitude.
In addition, the transfer ratios of the diamond and right square prism, which are 6% and 1.8%, are much lower than those of the circular cylinder and the triangle prism, which are 80% and 57%.The reasons can be found from the responses in Figures 9 and 10.For the right square prism, the initiation of oscillation needs a higher flow velocity resulting in a higher value of flow power   .For the diamond prism, the values of amplitude ratios and frequency ratios are much lower than those of other prisms.
The features above demonstrate that the capacities of energy transference for the square prisms are lower than those of circular cylinders and triangle prisms to some degrees.Therefore, it is more reasonable to use a circular cylinder or triangle cylinder in the design of the energy conversion from the FIM.

The Influential Factors.
The roles of mass and damping are investigated in this section.In the majority of previous studies, mass and damping are regarded as a combined parameter to affect the performance of FIM.In this study, the influences of mass and damping are analyzed independently.
Before the analysis of the influence of the damping ratio, the combined parameter ( *  * ) 2 /  3 is introduced first.It could be concluded from the expression of the energy transfer ratio (see (14a)) that if other factors are fixed, the transfer ratio would increase linearly with the combined parameter.Figure 12 plots the relationships between the transfer ratio and the combined parameter for three different damping ratios.Meanwhile, four configurations are compared in this figure, which are the circular cylinder with mass ratio  * = 4.2, the circular cylinder with mass ratio  * = 2.2, the regular triangle prism with mass ratio  * = 4.2, and the diamond prism with mass ratio  * = 4.2.As expected, the transfer ratio shows an approximate linear trend with the combined parameter ( *  * ) 2 /  3 .It soundly proves the conclusion above deduced from the expression of the transfer ratio (see (14a)).
According to Figure 12, the transfer ratios, including the maximum transfer ratios, are strongly dependent on the damping ratios for all configurations.The lower the damping ratio is, the lower the transfer ratio is.This feature also agrees well with the expression of the transfer ratio.It is noted that the increase of the damping ratio is actually caused by the decrease of the system stiffness in the present experiments.In conclusion, the increase of the damping ratio or the decrease of the system stiffness can cause a high mechanical energy of FIM leading to a high capacity of the energy transference.
In order to investigate the influences of the mass ratio, two cases with similar damping ratios but different mass ratios are illustrated in Figure 13.It is hard to maintain the damping ratios at the same value for different mass ratios because of the limitation of the experiment setup.So we compared two configurations with similar damping ratios in all cases.
It is concluded from the discussion of the influences of the damping ratio that a higher damping ratio would cause a higher transfer ratio when the mass ratios are the same.Then the transfer ratio at  = 0.1036 and  * = 4.2 would be higher than that at  = 0.0884 and  * = 4.2.In Figure 13(a), it is clearly observed that the maximum of the transfer ratio at  = 0.0884 and  * = 4.2 is higher than that at  = 0.1036 and  * = 2.2.Then we could conclude that the maximum of transfer ratio at  = 0.1036 and  * = 4.2 would be higher than that at  = 0.1036 and  * = 2.2.A similar conclusion could also be drawn from the Figure 13(b).These results indicate that the maximum of the transfer ratio is strongly dependent on the mass ratio.The higher the mass ratio is, the higher the transfer ratio is.The characteristics of mass ratio can also correlate with the expression of transfer ratio.In conclusion, the increase of the mass ratio will also strengthen the capacity of energy transference from the flow.
In addition, the reduced velocity corresponding to the maximum transfer ratio is dependent on the mass ratio.This result agrees well with the result in Barrero-Gil et al. 's studies [29].

The Stationarity of Oscillation
6.1.The Evaluation of the Stationarity of the Amplitude.Four typical examples of the time histories of displacement in each branch, for the cylinders with different cross sections, are illustrated in Figure 14.It could be found that the circular cylinder experienced a quasiperiodic oscillation in the upper branch (  = 5.03), which means each amplitude in this branch approximately maintains a constant.Similar behaviors can also be observed in the upper branch (  = 4.86) and galloping branch (  = 7.48) for the regular triangle prism and in the upper branch (  = 5.02) for the diamond prism.This feature is mainly attributed to the great synchronization of the oscillation and the lift force.By contrast, the amplitudes in other cases could not maintain a constant in each period due to the bad synchronization of the oscillation and the lift force.Hence, the different configurations (cross sections, velocities, etc.) result in different amplitudes in the time histories of displacement.
As we know, the electrical power quality is strongly dependent on the steady output of the convertor.For the energy extracted from the FIM, the steady output is strongly dependent on the steady mechanical power of the oscillator, and the steady mechanical power is strongly dependent on the steady performances of the amplitudes.Consequently, it is necessary to obtain the stationarity of the amplitude to estimate the quality of the mechanical power.
The variation coefficient of amplitude, CV, is introduced to estimate the stationarity of the amplitude, which is expressed as where  denotes the standard deviation of the amplitudes,  is the average of the amplitudes,  is the number of the amplitudes, and   is the th amplitude.
According to the definition and the expression of CV, we can conclude that a low value of CV would achieve a good stationarity of the amplitude.

The Relationship between the Stationarity and the Energy
Transfer Ratio.In this section, the relationships between the stationarity of the amplitude and the capacity of energy transference are discussed.CV values were calculated based on the experimental results, and   values were extracted

Figure 1 :
Figure 1: Sketch of flow induced motion of the cylinder in the transverse direction.

Figure 2 :
Figure 2: The projected area for the calculation of power in fluid   .(a) The projected area of cylinder; (b) the projected area of the vibration.

Figure 3 :Figure 4 :
Figure 3: The experimental setup and the physical model.(a) Circular water channel; (b) experimental setup; (c) displacement transducer; (d) Pitot tube and differential pressure transmitter.

Figure 5 :
Figure 5: The incident angle to the flow direction for different cylinders.(a) Circular cylinder; (b) regular triangle prism; (c) right square prism; (d) diamond square prism.

Figure 6 :
Figure 6: The different extension springs used in the experiments.

3. 1 .
Facility and Apparatus.A continuous flow past a cylinder is provided in the circulating water channel in the TJU Hydraulics Laboratory (Figure 3(a)).The channel is 15 m long, 0.6 m wide, and approximately 0.45 m deep.A centrifugal pump with a 45 kW motor is used to drive the water in the channel.A gearbox is used to control the flow speed.The range of flow speed is from 0.20 m/s to 0.75 m/s.The physical model of the oscillation system is installed in the middle of the channel.

Figure 11 :
Figure 11: Variations of the energy transfer ratios with the reduced velocities for cylinders or prisms with different cross sections. * = 4.2.(a) Circular cylinders; (b) regular triangle prisms; (c) right square prisms; (d) diamond prisms.

Figure 15 :
Figure 15: Variations of the variation coefficients of amplitude CV with the energy transfer ratios .(a) Circular cylinders; (b) regular triangle prisms; (c) right square prisms; (d) diamond prisms.

Table 1 :
The physical parameters of different oscillators.

Table 2 :
The different stiffness groups.

Table 3 :
The complete results in free decay test.