Flow-induced motion (FIM) experiments of a single circular cylinder or multiple cylinders in an array involve several configuration and hydrodynamic parameters, such as diameter, mass, damping, stiffness, spacing, Reynolds number, and flow regime, and deviation from circular cross section. Due to the importance of the FIM both in suppression for structural robustness and in enhancement for hydrokinetic energy conversion, systematic experiments are being conducted since the early 1960s and several more decades of experimentation are required. Change of springs and dampers is time consuming and requires frequent recalibration. Emulating springs and dampers with a controller makes parameter change efficient and accurate. There are two approaches to this problem: The first involves the hydrodynamic force in the closed-loop and is easier to implement. The second called virtual damping and spring (Vck) does not involve the hydrodynamic force in the closed-loop but requires an elaborate system identification (SI) process. Vck was developed in the Marine Renewable Energy Laboratory (MRELab) of the University of Michigan for the first time in 2009 and resulted in extensive data generation. In this paper, the second generation of Vck is developed and validated by comparison of the FIM experiments between a Vck emulated oscillator and an oscillator with physical springs and dampers. The main findings are: (a) the Vck system developed keeps the hydrodynamic force out of the control-loop and, thus, does not bias the FIM, (b) The controller-induced lag is minimal and significantly reduced compared to the first generation of Vck built in the MRELab due to use of an Arduino embedded board to control a servomotor instead of Labview, (c) The SI process revealed a static, third-order, nonlinear viscous model but no need for dynamic terms with memory, and (d) The agreement between real and virtual springs and dampers is excellent in FIM including vortex-induced vibrations (VIVs) and galloping measurements over the entire range of spring constants and velocities tested (16,000 < Re < 140,000).

## Introduction

FIMs present a source of challenge for diverse structures in steady flows such as heat exchangers, bridges, buildings, offshore structures, or power-transmission cables. The most common FIM phenomenon is vortex-induced vibration (VIV), which was first observed by Leonardo da Vinci in 1504. The first mathematical formulation was done by Strouhal in 1878. Comprehensive reviews of VIV have been published by Bearman [1,2], Sarpkaya [3], and Williamson and Govardhan [4].

Galloping is another form of the FIM. It is an aero/hydro-elastic instability, which is characterized by lower frequencies and larger amplitudes, than the VIV and is perpendicular to the flow [5,6]. It is more vigorous and destructive than the VIV, yet not as complex a phenomenon as VIV. Galloping does not depend on the vortex formation, occurs above a critical flow speed, and has much bigger amplitude of oscillation.

For a single rigid cylinder on elastic supports in a cross-flow, FIM is initiated as VIV due to vortex shedding at low speeds, when the reduced velocity reaches the synchronization range starting with the initial branch. This response amplitude increases with the velocity of the flow in the upper branch. It is followed by the lower branch with smaller response, then the desynchronization zone, and then the end of the VIV synchronization range. The various branches differ significantly depending on whether the flow regime is TrSL2 or TrSL3 [7]. Williamson and Govardhan [4] conducted experiments for Re about 3800 in TrSL2 (1000 < Re < 20,000) and response amplitude reached one cylinder diameter. Bernitsas et al. [8] and Raghavan and Bernitsas [9] conducted experiments for 16,000 < Re < 140,000 in the TrSL3 (20,000 < Re < 300,000) flow regime. In TrSL3, the shear layers on the two sides of the cylinder are saturated and vortex roll-up is stronger resulting in shorter formation length of the von Kármán vortices than in the TrSL2 regime. Thus, when vortices shed they are closer to the after-body of the cylinder resulting in high lift and amplitude response that reaches two diameters.

At higher speeds, galloping may be reached depending on the geometric deviation of the cylinder from the circular cross section [10]. Experimentally, galloping reaches the physical limits of the oscillator in the experimental facility or structural failure [11,12].

Multiple cylinders in a school as in Fig. 1 exhibit the two FIMs described above. In addition, interference effects become very significant [13] resulting in an enhanced VIV or galloping and gap-flow induced instabilities. Specifically, depending on the relative location of cylinders, the flow around the first cylinder may go between the two cylinders resulting in enhanced forces due to Bernoulli Effect and instabilities [1416].

Fig. 1
Fig. 1
Close modal

FIMs are suppressed in engineering applications because of their destructive nature. On the other hand, by enhancing and controlling FIM the Marine Renewable Energy Laboratory (MRELab) at the University of Michigan, developed the VIVACE converter [1721] to convert marine hydrokinetic (MHK) energy to electricity. The concept is being advanced continually in Ph.D. thesis [12,2227] and scientific papers. Relevant research on harvesting MHK energy using VIVs has been published in several countries: [28] (TX), [29] (Switzerland and France), [30] (Spain), [31] (Greece), and [32] (China).

The research objective of the MRELab is to study the FIMs and find ways to enhance them in order to design multicylinder Vortex Induced Vibration for Aquatic Clean Energy (VIVACE) converters and optimize their power output for a broad range of velocities. Power envelopes as a function of flow speed and type of FIM including need to be generated. FIMs studied in the process include VIVs, galloping, and gap flow. Generating power envelopes in such a large parametric design space and for different FIM phenomena is equivalent to finding the efficiency of propellers with different number of blades and geometric properties. Further, in open-water propeller tests all blades are in equivalent position while in schools of cylinders the hydrodynamics of each cylinder changes depending on the cylinder relative location in the school.

To enable such intense experimentation, a reliable Vck system is developed, built, and validated in this work. The Vck system emulates the mechanical side of the oscillator only using controls without including the hydrodynamic force in the closed-loop to prevent bias and minimize the added phase lag between force and displacement.

Conducting tests with real springs and dampers requires lengthy preparation for each set of experiments. A more efficient way to conduct experiments faster and accurately is developed based in this work on a controller embedded virtual spring–damping system (Vck) that does not include the hydrodynamic force in the closed-loop. Each oscillator consists of one Vck, one interchangeable cylinder moving on submerged roller blocks and driven by the fluid flow, and connected to the controller through belts and pulleys. It is designed to achieve the desired static/dynamic friction through the Vck. An Arduino-embedded board controls a servomotor with an optical encoder, which enables real-time position/speed measurement. A SI methodology is developed making it possible to identify the damping model of any oscillator, which is typically significantly more complicated than the classical linear-viscous model. Upon completion of the SI process for an oscillator, the controller subtracts the nonlinear damping model leaving the system with zero damping. Then, a mathematically linear viscous damping model is added, thus, resulting in a system with real linear-viscous damping. This process enables changing the spring constant and harnessing damping through the controller instantly. Experiments are then conducted with both real spring dampers and Vck to validate the process. All FIM experiments are conducted in the low turbulence free surface water (LTFSW) channel of the University of Michigan at 16,000 < Re < 140,000.

The drawback of this approach is that it requires extensive SI to identify accurately the spring constant and system damping. The former is relatively easy but the identification of the system damping requires extensive testing with multiple different driving functions and velocity ramps [33]. On the other hand, once the damping is identified it can be subtracted to achieve zero damping and then impose any exact function of viscous damping linear or nonlinear.

The alternative method of measuring the hydrodynamic force and solving the differential equation of motion to find and impose the displacement is much faster but keeps the hydrodynamic force in the control loop. That results in 5–12 deg added phase-lag. This method was developed by Smogeli et al. [34] and adopted more recently by Mackowski and Williamson [35].

The development and validation of the Vck system is presented in this paper. In Sec. 2, the system is described. In Sec. 3, the SI process, which is very elaborate and time consuming, is presented in detail. In Sec. 4, tests are conducted with the Vck system and results are compared to experiments with physical springs and dampers for validation.

## System Description

In this section, the oscillator is described and modeled mathematically including the Vck system. The SI making possible to realize and implement the Vck system is presented in Sec. 3.

### Oscillator Simulated by the Vck System.

The simplest form of the VIVACE converter consists of a single smooth cylinder suspended by springs with a Vck system. Figure 1 shows four such oscillators in an array (school). In early versions of the oscillator in the MRELab, the system was operational on physical springs and dampers. For high-energy conversion, the oscillator needs to have minimal system damping and additional damping for harnessing MHK energy. The total damping cannot exceed certain limit or the FIM—VIV and galloping alike—will be suppressed. If the total damping is above optimal, it will decrease the efficiency of the VIVACE converter. For systematic testing, two issues are important: (a) Changing the physical springs and dampers requires extensive preparation for calibration and alignment and (b) The viscous damping of the system is rarely linear as modeled in classical vibration textbooks. To compensate for these two major challenges, Lee et al. [33] and Lee and Bernitsas [36] introduced and implemented the virtual spring-damping device called Vck system.

Vck is a feedback loop, which uses the servomotor encoder to keep track of the position and velocity of the cylinder and provide the necessary torque to simulate the spring stiffness and provide the linear viscous-damping required. This logic keeps the hydrodynamics out of the control loop making it possible to simulate the mechanical part of the oscillator only, without affecting the hydrodynamic excitation. The Vck end-product provides the experimental setup the flexibility to input the values of spring stiffness and linear viscous damping into the controller and continue the experiments without interruption. This huge advantage comes at the expense of extensive time in Vck system development.

It is critical to compare the accuracy of the Vck system to implement a mechanical oscillator with the desired damping and spring constants. That requires systematic calibration followed by comparison of Vck to an oscillator with physical springs and dampers. The process requires performing repeated experiments in free air and in the LTFSW channel of the MRELab for SI. A nonlinear model for damping is defined after many trials. The resulting damping model is subtracted resulting in a zero damping system. Then, a linear viscous damping model is added producing a perfectly linear viscous damping matching the textbook oscillator models.

The earlier implementation of a virtual system [33,36] used the National Instruments data acquisition system for reading the position and completing the force feedback. This system was a success and showed improvements in the operational efficiency but it had its own disadvantages as it introduced significant amount of phase-lag between force-feedback and displacement into the system. The phase lag was mainly because of the data acquisition system where the analog to digital conversion (ADC) and digital to analog conversion (DAC) should run through a digital filter before performing. The phase-lag was clocked between 9 ms, which was not acceptable for FIM measurements or energy harnessing. The phase-lag was compensated by introducing four dynamic terms with memory to model the hysteresis effects [33].

In the current research, a controller embedded virtual spring–damping system (Vck) for FIM experiments is developed for each oscillator. Each oscillator consists of one interchangeable cylinder, which is designed to reduce the static/dynamic friction, has adjustable mass, and moves on submerged roller blocks. The mechanical part of the oscillator is being simulated by the controller providing the feedback through belts and pulleys. An Arduino embedded board controls the servomotor, which has an optical encoder. Digital signal is chosen, making the system more responsive, theoretically with no lag expect for that due to the central processing unit operation clocked to 10 μs. This approach enables real-time position/speed measurement.

### Physical Vck VIVACE System.

By introducing passive turbulent control (PTC), the MRELab research team was able to reduce VIV [37,38] and at the same time to initiate galloping, and to achieve back-to-back VIV and galloping on a single circular cylinder [39]. The effects of PTC on multicylinder FIM were further studied experimentally by Kim et al. [11] and using CFD by Ding et al. [40,41]. In Kim's study, the oscillatory amplitudes of the multiple cylinders reached the limitation of the facility at flow speed of 1.5 m/s, which is below the maximum flow speed of the water channel. In 2011, in order to investigate further the FIM of cylinders with fewer restrictions, both the channel and the new VIVACE converter were rebuilt. This third generation of test-models was designed and built as shown in Figs. 14. In the new design, the linear motion mechanism was brought under water, which resulted in most of the structural elements being under water. The whole linear motion system, which consists of a sliding block, timing belt, and pulleys, is fitted inside of a 1.5 in. wide slender tube. The sliding block connects the cylinder to the Vck system through the timing belts and pulleys as shown in Fig. 2. The mass ratio can be changed by putting a mass-shaft inside the cylinder, as shown in Fig. 3. There is a shock absorber to protect the cylinder from hitting the plexi-glass at the bottom of the water channel. The center-to-center distance (d/D) between the cylinders in the flow direction can be adjusted by the rail system starting at 5 in. and increased by 0.5 in. The features of the new VIVACE converter are summarized in Table 1.

Fig. 2
Fig. 2
Close modal
Fig. 3
Fig. 3
Close modal
Fig. 4
Fig. 4
Close modal
Table 1

Specifications of the new VIVACE converter

Mass ratioAspect ratioMaximum oscillatory amplitudeLongitudinal distance
0.634 ≤ m* ≤ 2.00L/D = 10.07Amax/D3.5″ = 5.51.429 ≤ d/D3.5″ ≤ 6.0
Mass ratioAspect ratioMaximum oscillatory amplitudeLongitudinal distance
0.634 ≤ m* ≤ 2.00L/D = 10.07Amax/D3.5″ = 5.51.429 ≤ d/D3.5″ ≤ 6.0

### Math Model for a Vck VIVACE System.

The physical model (Fig. 5) is assumed to have structural symmetry and the gravitational force is balance by spring pretension. Thus, the cylinder oscillates around its equilibrium position after the virtual springs and damper have been implemented. Table 2 shows the components in the Vck system.

Fig. 5
Fig. 5
Close modal
Table 2

Components of embedded Vck system

 $θi,i=1,2,3,4$ i = 1: angle of the motor coupling and pulley i = 2: angle of the pulley top-center i = 3: angle of the pulley top-right i = 4: angle of the pulley lower-right Y Displacement of the cylinder Mosc (kg) Equivalent mass of oscillating components Jmotor (kg/m2) Mass moment of inertia of the motor tmotor (N) Nonlinear damping torque of the motor Tmotor (NM) Torque generated by the motor Jpulley (kg/m2) Mass moment of inertia of the pulley tpulley,ii = 1, 2, 3, 4 Nonlinear damping torque of ith pulley rpulley (m) Radius of all pulleys fbearing (N) Nonlinear damping force of all bearings Fi (N) ith tension of the time-belt
 $θi,i=1,2,3,4$ i = 1: angle of the motor coupling and pulley i = 2: angle of the pulley top-center i = 3: angle of the pulley top-right i = 4: angle of the pulley lower-right Y Displacement of the cylinder Mosc (kg) Equivalent mass of oscillating components Jmotor (kg/m2) Mass moment of inertia of the motor tmotor (N) Nonlinear damping torque of the motor Tmotor (NM) Torque generated by the motor Jpulley (kg/m2) Mass moment of inertia of the pulley tpulley,ii = 1, 2, 3, 4 Nonlinear damping torque of ith pulley rpulley (m) Radius of all pulleys fbearing (N) Nonlinear damping force of all bearings Fi (N) ith tension of the time-belt
The equations of motion of the new VIVACE system with the embedded Vck are Eqs. (1)(5)
$(Jmotor+Jpulley)θ¨1=Tmotor−tmotor+rpulley(F−F2)$
(1)
$Jpulleyθ¨2=−tpulley3+rpulley(F1−F2)−2tpulley2$
(2)
$Jpulleyθ¨3=2tpulley2+rpulley(F3−F5)−tpulley3$
(3)
$Jpulleyθ¨4=−tpulley1+rpulley(F5−F4)$
(4)
$MOSCy¨=−fbearing+2(F4−F3)$
(5)
Assuming that the timing-belts are inelastic, the kinematic relations among $θi$ and y are:
$θ1=θ2=θ3=θ$
(6)
$y=rpulleyθ$
(7)
Thus, Eqs. (1)(5) are simplified to Eq. (8), by utilizing Eqs. (6) and (7)
$(Jmotor+6Jpulley+Moscrpulley2)θ¨=Tmotor−tmotor−6tpulley2−3tpulley2−3tpulley3−2tpylley1−fbearingrpulley$
(8.1)
$(Jmotor+6Jpulley+Moscrpulley2)θ¨=Tmotor−(tmotor+2tpylley+6tpulley2+3tpulley3+fbearingrpulley)$
(8.2)
$(Jmotorrpulley2+6Jpulleyrpulley2+MOSC)y¨=Tmotorrpulley−(tmotorrpulley+2tpulleyrpulley+6tpulley2rpulley+3tpulley3rpulley+fbearing)$
(8.3)
Assuming that $Fmotor=Tmotor/rpulley$, the above equation can be simplified for convenience as
$my¨=Fmotor−f$
(9)
where
$m=Jmotorrpulley2+6Jpulleyrpulley2+MOSC$
(10)
$f=tmotorrpulley+2tpulleyrpulley+6tpulley2rpulley+3tpulley3rpulley+fbearing$
(11)

## SI for Vck

In this application, NI-7340 samples data and is clocked at 1000 Hz compared to previous Vck 200 Hz. The servodrive QS1A05AA is connected to the Arduino Due (ARM® Cortex®-M3, 32-bit ARM Cortex M3 Process clocked at 84 MHz) board through Universal Motion Interface UMI-7764. The servodrive operates in torque-command mode and receives via the servomotor in order to produce motion proportional to the command signal. The control scheme for Vck is shown in Fig. 6.

Fig. 6
Fig. 6
Close modal

### Calibration of Virtual Spring.

As shown in Fig. 7, in order to calibrate the torque of motor (Tmotor) and voltage output (Vout), different weights (thus torque) are added to capture the referenced voltage. The calibration results are shown in Fig. 8. The calibration formula derived is

Fig. 7
Fig. 7
Close modal
Fig. 8
Fig. 8
Close modal
$Tmotor=4.965Vout$
(12)

With the current physical system, by considering the pulleys, the force applied to the system can also be plotted as in Fig. 9.

Fig. 9
Fig. 9
Close modal

Based on the motor torque versus voltage calibration, variable spring constants are defined. Due to test-bed limitations, the last five spring constants only used three weights. Weights applied to the system are shown in Table 3. In Fig. 10, tick-marks are experimental k values, and solid lines are calculated k values.

Fig. 10
Fig. 10
Close modal
Table 3

Weight applied to the system

1234
1234

The result shows congruent linear relation between the defined virtual spring constant and the output spring function. The forces tested are sufficient for the FIM experiment. Thus, the virtual spring constant values are sufficient for the experiments.

Repeatability of the virtual spring was validated by adding weights and measuring the resulting displacement. Figure 11 shows the relation between displacement and weight for k = 800.

Fig. 11
Fig. 11
Close modal

### Virtual, Nonlinear, Viscous Damping Model.

The main function of the Vck system is to add linear adjustable damping to the system to harness MHK energy. To develop the sought power envelope, systematic tests need to be conducted. Prior to adding prescribed damping, the physical system damping needs to be identified and subtracted. The latter is due to all moving parts, especially the timing belts and pulleys which bring undesirable/excessive amount of nonlinear damping to Vck. The mechanical equation of the Vck system can be expressed as

Table 4

Identification of nonlinear viscous damping plus kinetic friction model

Velocity $y˙>0$

Velocity $y˙>0$
$y˙$ (m/s)Force (N)$y¨$ (m2/s)$y˙$ (m/s)Force (N)$y¨$ (m2/s)
0.93329.280.618−0.940−28.2810.628
1.29933.641.451−1.303−32.6061.551
1. 28933. 8241.451−1.306−32.6061.651
2. 30336.8154.131−2.030−35.934.331
2.13136.7264.148−2.055−35.7274.248
2.1137. 2975.590−2.337−36.4335.6
2.36737. 6545.597−2.370−37.2225.7
2.5938.1337.151−2.59−38.1537.351
2.6538.2087.147−2.642−39.2947.447
013.50600−12.5060

Velocity $y˙>0$

Velocity $y˙>0$
$y˙$ (m/s)Force (N)$y¨$ (m2/s)$y˙$ (m/s)Force (N)$y¨$ (m2/s)
0.93329.280.618−0.940−28.2810.628
1.29933.641.451−1.303−32.6061.551
1. 28933. 8241.451−1.306−32.6061.651
2. 30336.8154.131−2.030−35.934.331
2.13136.7264.148−2.055−35.7274.248
2.1137. 2975.590−2.337−36.4335.6
2.36737. 6545.597−2.370−37.2225.7
2.5938.1337.151−2.59−38.1537.351
2.6538.2087.147−2.642−39.2947.447
013.50600−12.5060
$my¨+c(y˙)+ky=Ftotal$
(13)
where Ftotal is the total hydrodynamic transverse force acting on the cylinder. So the damping model can be obtained through
$c(y˙)=Ftotal−my¨−ky$
(14)
In order to find the nonlinear viscous damping and static friction, the total damping force f, which contains motor, pulleys, and bearings, can be expressed as
$f=fmotor+fpulley+fbearing$
(15)
which can also be expressed as a function of dy/dt. Here, all the damping force is assumed to be the sum of nonlinear viscous damping of a third-order polynomial friction
$c(y˙)=c3y˙3+c2y˙2+c1y˙+ccsign(y˙)$
(16)
c1, c2, and c3 are the first-, second-, and third-order coefficients, respectively, and cc, is the coulomb friction coefficient.

The first generation of Vck system [33] had a closed-loop of the vibration components and the cylinder could be detached from the belt. Constant forces were applied continuously to the timing belt to achieve sufficient speed. The new Vck system has sliding blocks and timing belts inside the structure and, thus, they cannot be disconnected (Fig. 5). To achieve sufficient speed and acceleration, various single direction forces are fed to the system by the motor. Programed stoppers are set when the cylinder reaches the upper and lower ends of the travel to prevent collisions with the plexi-glass bottom of the LTFSW Channel.

Figure 12 shows the experimental and curve fitting results. Table 4 shows the speed, acceleration, and associated applied forces. The regression curves are shown in the following equations:

Fig. 12
Fig. 12
Close modal
$c(y˙)=0.8793y˙3−7.7678y˙2+23.736y˙+13.439, when y˙>0$
(17)
$c(y˙)=1.1544y˙3+9.2064y˙2+25.992y˙−12.434, when y˙<0$
(18)
The sinusoidal forces were fed to the system in which the damping force was calculated as
$c(y˙)=F sin(ωt+φ)−my¨−ky$
(19)
where F is force amplitude. A low pass filter was applied to attenuate the noise caused by numerical differentiation. Figures 1315 show the damping force versus the nonlinear static (NS) model. The blue line is the damping force, and the red line is the NS model applied by the controller. All the spring constants were set to be 600 N/m.
Fig. 13
Fig. 13
Close modal
Fig. 14
Fig. 14
Close modal
Fig. 15
Fig. 15
Close modal

### Controller Design.

Based on the nonlinear viscous damping model, the controller force can be expressed as
$Fcontroller=Ffluid−cvirtualy˙−kvirtualy$
(20)
where Fcontroller is the force applied by the virtual system, Ffluid is the total hydrodynamic force acting on the cylinder, kvirtual is the virtual spring constant, cvirtual is the virtual linear viscous damping. cvirtual is added after subtracting the nonlinear viscous damping defined by Eqs. (17) and (18).

Free decay tests are also performed to verify the subtraction of the system damping and the application of the linear viscous damping. The spring constants were set to 600 N/m and 800 N/m, respectively. All experiments were performed with gradually increasing the linear viscous damping from 10% to 90% of the system, Fig. 16, Table 5, and Table 6 show the free decay test results. The repeatability was checked five times for each spring constant. Where c is the damping of the system, m* is the mass ratio, Td is the oscillation period, and $ωn$ is natural frequency of the system in vacuum.

Fig. 16
Fig. 16
Close modal
Table 5

Free decay results for kvirtual = 600 N/m

k (N/m)Damping ratioc (Ns/m)mosc (kg)m*Td$ωn$
6000.05376.155.45551.0060.610.487
6000.04535.2045.48741.0120.601510.457
6000.03554.0565.45081.0050.610.492
6000.02763.1615.45811.0060.60110.485
6000.02162.4695.45971.0060.60210.483
6000.01461.6795.47471.0090.60310.469
Stand dev.0.0138630.002650.0011720.01318
k (N/m)Damping ratioc (Ns/m)mosc (kg)m*Td$ωn$
6000.05376.155.45551.0060.610.487
6000.04535.2045.48741.0120.601510.457
6000.03554.0565.45081.0050.610.492
6000.02763.1615.45811.0060.60110.485
6000.02162.4695.45971.0060.60210.483
6000.01461.6795.47471.0090.60310.469
Stand dev.0.0138630.002650.0011720.01318
Table 6

Free decay results for kvirtual = 800 N/m

K (N/m)Damping ratioC (N s/m)mosc (kg)m*Td$ωn$
8000.6899.2225.60661.0330.527311.945
8000.5937.9395.59751.0320.526511.955
8000.05116.8375.5921.0310.525511.961
8000.04596.155.60281.0330.52711.949
8000.03494.6735.61311.0350.52711.938
8000.02163.5015.62411.0370.52711.927
8000.01782.3915.61551.0350.524511.936
Stand dev.0.01210.002220.001030.0128
K (N/m)Damping ratioC (N s/m)mosc (kg)m*Td$ωn$
8000.6899.2225.60661.0330.527311.945
8000.5937.9395.59751.0320.526511.955
8000.05116.8375.5921.0310.525511.961
8000.04596.155.60281.0330.52711.949
8000.03494.6735.61311.0350.52711.938
8000.02163.5015.62411.0370.52711.927
8000.01782.3915.61551.0350.524511.936
Stand dev.0.01210.002220.001030.0128

## Controlled FIM Experiments

Experiments were first designed and performed covering the entire range of VIV and galloping to verify the effectiveness of the Vck system. A real spring was used instead of the Vck and timing belts. Since the inertial mass of VIVACE with Vck is larger than one-third of the real spring mass, additional mass was put inside the oscillator to match the mass ratio of Vck and the physical springs system. Free decay experiments were performed to measure the damping.

Figure 17 shows comparison of amplitude ratio A/D between tests with real springs and with the Vck VIVACE model. The range of velocities covers VIV and galloping until the facility limit is reached. Error bars are shown in the form of one standard deviation. The reduced velocity U* is

Fig. 17
Fig. 17
Close modal
$U*=UfnD$
(21)
where U is the flow speed, fn is the natural frequency in water
$fn=12πkm+ma$
(22)
and ma is added mass. Discussion of modeling of ma can be found in Ref. [33].

The agreement between the real spring system and Vck is very good in VIV and galloping measurements over the entire range of velocities tested. The Vck system tends to initiate galloping sooner than the real spring experiment at the reduced velocity around ten. That is the transition region from VIV to galloping. That can be attributed to additional drag force caused by the spring vibration, which can create slight added turbulence during FIM.

Typical results of FIM experiments performed by Vck are shown in Figs. 1823. They show A/D, the average harnessed power, and frequency of oscillation by processing data over 30 cycles away from transients. The points exhibiting the zero oscillation frequency indicate that the cylinder FIM stops due to high damping. Extensive experiments for different mass ratio will be performed for harness energy in the near future. The spring stiffness k = 400 N/m for Figs. 1820 and 600 N/m for Figs. 2123.

Fig. 18
Fig. 18
Close modal
Fig. 19
Fig. 19
Close modal
Fig. 20
Fig. 20
Close modal
Fig. 21
Fig. 21
Close modal
Fig. 22
Fig. 22
Close modal
Fig. 23
Fig. 23
Close modal

## Conclusions

An embedded virtual spring–damping system Vck, designed for FIM experiments, was developed, verified, and validated in this research by extensive calibration and comparison to physical spring–damper oscillators. In the Vck, the spring and damping constants are programmable. The torque simulating the mechanical part of the oscillator is provided by a servomotor driven by a controller. The hydrodynamic force is not affected, as it is not included in the control loop.

In the new VIVACE oscillators, most of the structural components including the linear motion mechanism consisting of a sliding block, timing belts, and pulleys were brought under water. The new system better models the field-deployed devices compared to previous models. A resistor bank is used to dissipate the generated power.

The SI process was developed for the oscillator that can also be extended to any electrically based single degree-of-freedom oscillator. Upon subtracting the identified nonlinear damping model using the controller, a mathematically linear damping model was added, which results in a system with adjustable, ideally linear, viscous damping. This process enables changing the spring constant k and harnessing damping charness through the controller instantly.

The main improvement of the new Vck system is that it was developed as a standalone application via an embedded controller. Digital signals are used compared to analog signals in the earlier version of Vck. The sample time was reduced by three orders of magnitude, thus, eliminating the need to include dynamic terms with memory in the damping model to achieve no bias in FIM and energy harnessing. Compensating for the system nonlinear damping makes it possible to introduce a theoretically linear viscous damping, which can hardly be achieved in engineering applications; and yet is used in modeling oscillators.

Control experiments were performed to verify the effectiveness of the virtual system using the spring constant of 400 N/m and 600 N/m. The agreement between physical and virtual springs and dampers was excellent in FIM measurements over the entire range of spring constants and velocities tested.

Tests were performed for FIM measurements and energy harnessing for 0.00 < ζharness < 0.24. The embedded Vck system enables systematic experiments with different mass ratios, variable k and charness, and multiple oscillators needed to generate envelopes for the VIVACE converter.

## Acknowledgment

The following supports are gratefully acknowledged: DOE Contract No. DE-EE0006780 to Vortex Hydro Energy with subcontract to the University of Michigan; Ministry of Industry and Information Technology of the P. R. China, Project No. G014614002.

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