Effects of Harmonic Vibration on Cycloidal Rotor

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Proceedings of the ASME 2018 International Mechanical Engineering Congress and Exposition IMECE2018 November 9-15, 2018, Pittsburgh, PA, USA

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IMECE2018-87103 EFFECTS OF HARMONIC VIBRATION ON CYCLOIDAL ROTOR PERFORMANCE Jakson Augusto Leger Monteiro Faculdade de Ciências e Tecnologias Universidade de Cabo Verde Praia, Cabo Verde [email protected]

J. Páscoa C-MAST, Center for Mechanical and Aerospace Sciences and Technology Universidade da Beira Interior Covilhã, Portugal [email protected]

ABSTRACT Cycloidal rotors have the inherent ability to provide vectorized thrust with fast reaction times. However, their present efficiency levels restricts their routinely use as propulsion elements for air-vehicles. Efforts have been made to improve the performance of cycloidal rotors through the optimal combination of its geometric parameters. In the present work the performance improvement of cycloidal rotors is demonstrated using a different approach, namely by imposing an unsteady change on the dynamics and structure of the vortices developed around the blades. This required change on the flow field, around the blades, was applied by adding an harmonic vibration to the traditional cycloidal movement of the blades, thus causing the blades to vibrate as they describe their oscillating pitch movement. This research on the effect of harmonic vibration, on lift and drag coefficients, was done first for a single blade profile, and later for a full cycloidal rotor, and is based on the Takens reconstruction theorem and Poincaré map. Therefore, diverse test cases and conditions were considered: a single static airfoil, an oscillating blade profile, and a complete cycloidal rotor. We concluded that the optimal combination of harmonic vibration parameters, specifically; amplitude, phase angle and vibration frequency, under adequately tuned design conditions, can have a beneficial effect on cycloidal rotor performance.

𝑙 Length of the control rod. 𝑅 Rotor radius. 𝑡 Time. 𝛼𝑜 (𝑡) Instantaneous amplitude angle of the oscillating airfoil. 𝜃 Pitch angle. 𝜈 Kinematic viscosity of fluid. 𝛹 Azimuth angle. 𝛺 Rotational speed of rotor. 𝜔 Angular velocity around pivot point. INTRODUCTION The cycloidal rotor is a turbomachine that allows to convert energy, either in propulsive mode or as generator. In propulsive mode, the fluid is accelerated by the rotor blades as they rotate about an axis perpendicular to the flight direction and parallel to the blade span direction, Fig. 1. Since the blade describes a cycloidal path, this device is called a cycloidal rotor [1]. This device generates aerodynamic force by combining rotation and oscillation motion. Each cycloidal rotor blade oscillates and rotates simultaneously around a fixed point and around the center of the rotor, respectively, Fig. 2. The combined motion around these two points causes each blade to change its slope angle. Thus, at each rotation, the blades cyclically vary the respective angle of attack [2-4]. Since the movement is cyclic, it is characterized by its frequency, amplitude and the phase angle. The variation in the phase angle and amplitude of movement allows, respectively, changes in the thrust direction and thrust magnitude [5-8]. This remarkable feature allows the use of the cycloidal rotor for propulsion, lift, and air vehicles control in both military and civilian applications [9-12]. Thus, an air vehicle whose main source of propulsion and lift is the cycloidal rotor, can take off and land vertically, hover, and may move in any perpendicular direction to the rotating axis [13-17].

NOMENCLATURE 𝑎 Distance from the pivot point to eccentricity point. 𝑐 Chord of airfoil. 𝐶𝐷𝑎 Drag coefficient of the airfoil in oscillation motion. 𝐶𝐿𝑎 Lift coefficient of the airfoil in oscillation motion. 𝑑 𝑒𝑎 𝑒ℎ 𝑓 𝑘

Distance from the pivot point to the connection point. Airfoil performance without harmonic vibration. Airfoil performance with harmonic vibration. Frequency domain of a function. Reduced frequency of blade oscillation, c⁄2R.

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Copyright © 2018 ASME

FIGURE 1. Three-dimensional cycloidal rotor.

representation

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Helicopters have shown tremendous progress over time, in particular on operational performance, ease of maneuverability, and comfort. However, the high level of vibration and noise limits the application of these vehicles in various circumstances, such as in secret missions and in certain public environments. In order to solve these issues, many technologies for the active control of the helicopter rotor have been considered [27]. One of these technologies is called Higher Harmonic Control (HHC). HHC is an active control technology to reduce helicopter vibration levels. The control system detects vibrations caused by the aerodynamic forces acting on the rotor blades, and applies on the blades a high-frequency harmonic pitch motion with small angles. HHC is applied so that its effect suppresses or reduces aerodynamic excitation and consequently reduces the vibration and noise levels [28, 29]. Also some studies have shown that HHC offers other benefits, namely better aerodynamic performance and greater robustness under unsteady flight conditions [30]. Although this technology has been used for the active control of helicopter rotors, it has never been used on cycloidal rotors. In the present work, the flow field around the blades is changed by the addition of a second motion. This second motion corresponds to a harmonic vibration. Thus, the blade vibrates as it describes the oscillating pitch motion. To study the effect of the harmonic motion on the cycloidal rotor performance, two-dimensional numerical models based on CFD are developed. After being properly validated, the models are used to study the vortex dynamics behavior, first around a single profile in oscillating motion and later around the cycloidal rotor. In the next section it is described the set of equations that rule the blade pitching mechanism. Then, it is presented a very detailed description of the numerical grid for the numerical analyses performed on the flow. Finally, the obtained results are presented and discussed.

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One of the topics of greatest interest in the aeronautical field relates to VTOL (Vertical Take-Off and Landing) vehicles. Similarly, efforts have been made to develop rotary-wings unmanned aerial vehicles (UAV) due to the ability to achieve hover flight state as well as VTOL capability [18]. However, vehicles with these configurations are intrinsically limited in advance flight since their performance is lower than the fixedwing air vehicles for cruise conditions. Compared to helicopters rotors, cycloidal rotor have not only excellent characteristics in hovering flight, but also a good ability to perform a cruise flight [19-21, they can thus be the solution for flight performance flexibility increase. Several authors have performed research on the efficiency of cycloidal rotors [22, 23]. Benedict et al. [24] studied the efficiency of cycloidal rotors and compared it to that of conventional rotors. He has also shown that the optimized cycloidal rotor has significantly higher power loading (thrust generated per unit of power) compared to a conventional rotor, when operating at the same disk loading (thrust generated per unit of cycloidal rotor rectangular projected area) [25].

NUMERICAL MODELLING The investigation of the harmonic vibration effect is performed first for the case of a single profile and later for the cycloidal rotor. The analysis starts by considering both the profile and the cycloidal rotor without harmonic vibration. After validation and before imposing the harmonic vibration, the flow behavior in both cases was studied through Takens Reconstruction Theorem and Poincare map. The profile case and the cycloid rotor case were validated through the experimental data obtained in Lee [31] and using the IAT21 rotor [32], respectively. The Oscillating Profile Case The computational study comprised several numerical domains with the purpose of studding the sensitivity of results to the diverse meshes and to ensure the independence of the final solution. Thus, four numerical domains with different cell numbers were considered: Mesh-0, Mesh-1, Mesh-2, and Mesh3 with, respectively, 127489, 230688, 522064 and 1176179 cells. In order to avoid the external borders influence, in this case on the flow behavior around the profile, each numerical domain has a circular shape with a radius equal of 33.33 × c. Figures 3-a) and b) show the numerical domain considering mesh-1. With O-type structured meshes, two regions were generated: one outer region and a very refined region for the blade zone. The blade region is a circular domain with the center located at 35% of the chord length. The motion described by this region is given by,

FIGURE 2. Angular path described by the blades.

Many experimental studies and computational fluid dynamics (CFD) computations have also been performed in order to analyze the generated thrust, and also the power required for hovering and forward flight conditions [1, 16]. Xisto et al. [26] studied the geometrical effects on the performance of a cycloidal rotor. They concluded that the rotor efficiency and thrust capability increase with blade thickness. However, they found that the efficiency decreases as the number of blades increases.

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a)

a)

b)

b)

c) FIGURE 4. a) Overview of the computational mesh used for the study of harmonic vibration effect. b) Blade region. c) The boundary layer region.

c) FIGURE 3. a) Overview of one of the computational meshes (mesh-1). b) Blade region. c) The boundary layer region around the blade.

𝛼𝑜 (𝑡) = 𝛼𝑚 + ∆𝛼𝑠𝑖𝑛(𝜔𝑡) .

The Cycloidal Rotor Case The cycloidal rotor numerical domain has a circular shape with radius approximately 111 times bigger than the chord blade and a number of cells equals to 721812 cells. To compute the IAT21 GmbH L3 rotor, it was necessary to generate three regions both with the structured O-type grid: a very refined region for the blades, the rotor region, as exhibited in Fig. 5-a), and the environment region. Each circular blade region comprises 376992 cells that include a boundary layer mesh type, see Figs. 5-b) and c).

(1)

The blade region exchanges information with the outer region through a sliding mesh interface. The blade region includes a boundary layer region designed for that the computation of the surface spacing on the wall a 𝑦 + < 1 (0.5), Fig. 3–c). A study regarding time step independence was also performed for each of the meshes. Thus, based on the oscillation period, three different values for the time step were considered: 𝑑𝑡1 = 𝑇𝑐 ⁄500, 𝑑𝑡2 = 𝑇𝑐 ⁄1000 e 𝑑𝑡3 = 𝑇𝑐 ⁄2000. Each case was computed on twenty complete cycles of oscillation. In order to capture the effects of harmonic vibration, a larger and more refined numerical domain than the previous one was then considered, Figs. 4-a), b) and c). The outer region and the blade region have 252444 and 312399 cells, respectively. The region of the wing profile describes the movement according to the following rule,

a) 𝛼𝑜ℎ (𝑡) = 𝛼𝑜 (𝑡) + ∆𝛼ℎ 𝑠𝑖𝑛(𝜔ℎ 𝑡 + 𝜑ℎ ).

(2)

A compromise between computational load and accuracy was defined, as such a URANS type computation was performed. The blade surface was defined as a solid wall by imposing a non-slip wall boundary condition. However, the outer domain boundaries were defined as velocity-inlet and pressure-outlet boundary conditions. In order to compute the Navier-Stokes equations for the mean flow a pressure-based solver scheme was selected. Time accuracy and space accuracy were considered using a second order implicit interpolation scheme and a second order linear-upwind scheme, respectively. To model the turbulent flow the SST k-ω was selected. The considered time-step should allow to capture the unsteady flow. Therefore, to analyze the harmonic vibration effect a time step of 0.0004167 s was used.

b)

c)

FIGURE 5. a) Rotor region. b) Global view of the blade region. c) Detail view of the boundary layer.

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This thin layer was devised by assuming that to obtain accurate computations a normal to surface spacing 𝑦 + of less than 1 (0.5) must be used. All circular blade regions have their center located at 35% of the chord length and describe an oscillatory motion relatively to the rotor region shown in Fig. 3-b). The rotor region with 166020 cells, in turn, is rotating with a constant angular velocity Ω relatively to the environment region with 178800 cells. This last region, conversely to the blade and Copyright © 2018 ASME

time step on the results is irrelevant because the curves are practically coincident, as can be seen from Figs. 7 and 8. However, if the profile is in the upper phase of the movement and in the downward path, there is a slight difference between the results obtained with the time step 𝑑𝑡1 and those obtained with the time steps 𝑑𝑡2 and 𝑑𝑡3, see Figs. 9 and 10. In the intermediate phase of the movement with the profile moving upward, the curves almost overlap. This good approximation does not occur in the peak region and in the region corresponding to the lower phase of the movement, where the results show a greater discrepancy.

rotor domains, is always motionless (Ω = 0 rad⁄s). The blade numerical domain exchange information with the rotor domain through a sliding mesh as well as the rotor and the environment domains. Each blade region moves according to Eq. (3) [33] which describes the blade pitch angle variation [𝜃] imposed by the mechanical system shown in Fig. 6. 𝜃=

𝜋 𝑒𝑅 − 𝑠𝑖𝑛−1 [ 𝑐𝑜𝑠(𝛹 + 𝜀)] 2 𝑎 (𝑎2 + 𝑑 2 − 𝑙 2 ) − 𝑐𝑜𝑠 −1 [ ] 2𝑎𝑑

(3)

Where, 𝜋

𝑎2 = 𝑒𝑅 2 + 𝑅2 − 2𝑒𝑅 𝑅 𝑐𝑜𝑠 (𝛹 + 𝜀 + ). 2

(4)

FIGURE 7. Profile lift coefficient, at different time steps, using mesh-0.

FIGURE 6. Four-bar-linkage mechanical system.

For solving the incompressible Navier-Stokes equations a coupled pressure-based solver, with second order fully implicit time discretization, was employed. For spacevariables interpolation the second order linear upwind differencing scheme was used. Turbulence was modeled by the SST k-ω model. Regarding the boundary conditions a nonslip boundary condition was imposed in the walls and all the variables are extrapolated at the outlet. NUMERICAL RESULTS The conditions of the experimental study for the isolated profile involved the following values: 𝛼𝑚 = 100 , ∆𝛼 = 150 e 𝜔 = 18.67𝑟𝑎𝑑/𝑠, for a reduced frequency 𝑘 = 𝜔𝑐 ⁄2𝑈∞ = 0.1. The profile used was NACA0012, and the mean free-speed [𝑈∞ ] was equal to 14𝑚/𝑠, corresponding to a Reynolds number along the chord, 𝑅𝑒 = 𝑈∞ 𝑐 ⁄𝜈 = 1.35x105 . Figures 7 - 12 show the comparison between experimental data and numerical results considering, in each mesh, different values of the time step. For the cases with meshes 0 and 1, the influence of the


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FIGURE 11. Detailed analysis of the lift coefficient values in the peak zone with time step 𝒅𝒕𝟏.


FIGURE 12. Drag coefficient values with time step 𝒅𝒕𝟐.

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meshes and time steps considered, an easily perceptible aspect is that, with respect to numerical results, the stall phenomenon occurs earlier in comparison to the experimental case. Moreover, the decrease in the lift coefficient is accompanied by an increase in the drag coefficient, as can be observed in Fig. 12, which also shows the effect of the complexity of the flow in the peak region. The visualization of the flow field, Fig. 13, helps to understand the cause of such behavior. However, and before proceeding with the imposition of harmonic vibration, it was studied the dynamics vortices behavior around the profile. Thus, by using the Takens Reconstruction Theorem and Poincaré map, Figs. 14 and 15 were obtained.

FIGURE 14. Phase diagram obtained through the Takens reconstruction theorem for the oscillating profile.

FIGURE 15. Poincare section of the orbits obtained in figure 14.

FIGURE 13. Flow field visualization through the vorticity in the dimensionless form.

However, as can be seen in Fig. 11 the meshes 2 and 3 present in the peak region a good agreement, providing better results than the meshes 0 and the 1. It can also be seen from Fig. 11 that, for the same time step, the meshes influence becomes much smaller as we consider increasingly refined meshes. Regardless of the

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The purpose of analyzing the vortex system dynamics is simply to ascertain whether the flow behavior, without harmonic vibration, is chaotic or not. In case that the system behavior is not chaotic, not random, it becomes much easier to predict the system in time and, consequently, allows the imposition of the harmonic vibration effect to act on the vortex shedding. A random system can never be controlled because one never knows the vortex behavior in the following instants. However, it must be understood that, even if it is a chaotic or even random system, it does not mean that the harmonic effect does not exist or, if it Copyright © 2018 ASME

exists, it should not be understood that this effect is always negative or positive. What should be understood is that, being chaotic or random, it is not possible to guarantee a certain future state of the system. And so, in this case, there may be a gain in performance in an instant and loss in the next few instants. Analyzing Figs. 14 and 15 it is clear that the answer is neither chaotic nor random. The phase portrait and collection of points on the Poincaré map should be analyzed on the same scale. Analyzing Fig. 14, it can be seen that these are orbits that close in space. So the answer is periodic. The points on the Poincaré map, Fig. 15, are practically coincident, forming isolated points on the map, which also demonstrates the periodic character.

amplitude, and frequency values equal to, respectively, -180°, 2°, and 10 Hz, table 2. For this scenario a performance gain (𝑒ℎ − 𝑒𝑎 )⁄𝑒𝑎 of approximately 8% was obtained. Figures 18 and 20 show the values of one of the worst scenarios. This scenario was obtained for the values of phase, amplitude, and frequency equal to, respectively, 90°, 1.5° and 10 Hz, table 1. For this scenario a performance reduction of approximately 15.5% was obtained. Analyzing Figs. 17 and 18 it is clear that the main cause of performance improvement was the decrease in the drag coefficient due to harmonic vibration. However, the main cause of performance reduction, Figs. 19 and 20, was a considerable decrease in the lift coefficient. For a better analysis of the results presented in tables 1 and 2, some graphs were constructed from these results. This analysis is shown in Figs. 21-24. Analyzing Figs. 21-24, it can be concluded that the frequencies of 1 Hz and 10 Hz are the ones that most affect the profile performance. These frequencies are also more sensitive to phase change.

TABLE 1. Analysis of the lift and drag coefficients for the oscillating profile, with and without harmonic vibration. Phase angle 𝟎° and 𝟗𝟎°.

FIGURE 16. Fast Fourier Transform of the lift coefficient values for the oscillating profile.

The analysis in the frequency domain of the lift coefficient allows to decompose it in terms of amplitude and frequency, see Fig. 16. However, in order to impose the harmonic vibration effect, 60 combinations of frequency, phase and amplitude were considered. For each of the combinations considered, the performances were computed and compared to the case without the harmonic vibration. The results can be seen in tables 1 and 2. The variables 𝐶𝐿𝑎𝑚 and 𝐶𝐷𝑎𝑚 represent the mean values of the lift and drag coefficients for the oscillating profile, without harmonic vibration. The variables 𝐶𝐿ℎ𝑚 and 𝐶𝐷ℎ𝑚 represent the mean value of lift and drag coefficients with harmonic vibration. 𝐶 The variable 𝑒𝑎 = 𝐿𝑎𝑚 = 2.4822 represents the aerodynamic 𝐶𝐷𝑎𝑚

performance of the oscillating profile, without harmonic 𝐶 vibration. The variable 𝑒ℎ = 𝐿ℎ𝑚 represents the aerodynamic 𝐶𝐷ℎ𝑚

performance with harmonic vibration. Four different values were considered for the harmonic vibration phase: 0°, 90°, 180° and −180°. For each of the phases five different values for the amplitude of oscillation were considered: 0.1°, 0.5°, 1°, 1.5° and 2°. For each phase and amplitude combination, three different frequency values were considered: 1 Hz, 10 Hz and 100 Hz. Analyzing tables 1 and 2 it can be seen that the harmonic vibration influences positively, and also negatively, the flow in terms of aerodynamic performance. However, to better understand these results it was considered one of the worst and one of the best scenarios. Figures 17 and 18 show the values of one of the best scenarios. This scenario was obtained for phase,

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TABLE 2. Analysis of the lift and drag coefficients for the oscillating profile, with and without harmonic vibration. Phase angle 𝟏𝟖𝟎° and −𝟏𝟖𝟎°.

FIGURE 18. Drag coefficient of the oscillating profile. Phase −𝟏𝟖𝟎°, amplitude 2°.

One possible reason is that these frequency values are those closest to the values identified by the Fast Fourier Transform, see Fig. 16. For the phase 0°, the cases with 1 Hz and 10 Hz present, respectively, the best and the worst performance, Fig. 21. However, for the phase values of 180° and -180° the situation is reversed, Figs. 23 and 24. For the phase 90° there is no aerodynamic performance gain, Fig. 22.

FIGURE 19. Lift coefficient of the oscillating profile. Phase 90°, amplitude 1.5°.

FIGURE 17. Lift coefficient of the oscillating profile. Phase −𝟏𝟖𝟎°, amplitude 2°.

FIGURE 20. Drag coefficient of the oscillating profile. Phase 90°, amplitude 1.5°.

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FIGURE 24. Analysis of the profile performance for a harmonic vibration phase of -180°.

FIGURE 21. Analysis of the profile performance for a harmonic vibration phase of 0°.

For the phase 0° the optimum harmonic vibration amplitude is 0.5°. Another aspect to be pointed out is that the harmonic vibration amplitude of 1.5° is the optimum one for the phase 180°. However, this same amplitude is less indicated for the phase 90°, Fig. 22. This shows the aerodynamic performance sensitivity relative to the harmonic vibration amplitude. However, it is considered that it is necessary to make a deeper analysis with turbulence models capable of providing more precise results, as for example the LES. The study of the harmonic vibration effect on the cycloidal rotor performance was done considering the IAT21 L3 test case, whose parameters are described in detail in table 3: TABLE 3. IAT21 rotor parameters. FIGURE 22. Analysis of the profile performance for a harmonic vibration phase of 90°.

Variable

Value

Profile Rotor diameter (m) Span (m)

NACA 0016 1.2 1.2

Chord (m) Pitching axis (m)

0.3 35% da corda

Number of blade Length of control rod (m)

6 0.61

Magnitude of eccentricity (m) Eccentricity phase angle (degree)

0.072 0

d (m) Maximum and minimum pitch angle (degree)

0.12 36.1; -39.1

The unsteady solution was computed for a rotor rotation of 0.50 per time step over 30 cycles. The time-dependent thrust and power were estimated trough the force and torque coefficients, which are computed for each blade and then summed up for providing the overall quantities. Figures 25 and 26 show the results found considering two – dimensional analyses. They also exhibit the experimental data for the IAT21 GmbH L3 rotor configuration that can be used for the validation proposes. Since, the numerical CFD model provides acceptable results, it can be

FIGURE 23. Analysis of the profile performance for a harmonic vibration phase of 180°.

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used to test the harmonic motion around the traditional cycloidal curve for several amplitudes.

FIGURE 27. Phase diagram from the Takens reconstruction theorem. FIGURE 25. Cycloidal rotor thrust as function of rotating speed.

FIGURE 28. Lift coefficient of the cycloidal rotor blade, without and with harmonic vibration of frequency 80 hz, phase 0° and amplitude 1°.

FIGURE 26. Cycloidal rotor power as function of rotating speed.

The cycloidal rotor was considered to operate at a rotational speed of 200 RPM, corresponding to 3.33 Hz, and with two harmonics. The amplitude and phase of the harmonics are respectively 1° and 0°. The harmonic frequencies are approximately 80 Hz and 160 Hz. However, before proceeding with the imposition of the harmonic vibration, the behavior of the vortex dynamics around the cycloidal rotor blade was studied. For this study it was resorted again to the phase space reconstruction by the Takens Reconstruction Theorem. Analyzing the reconstructed attractor shown in Fig. 27, it was concluded that the dynamic behavior of the flow around the cycloidal rotor blade is periodic. The orbits are closed curves in the phase space. This shows a behavior that is neither chaotic nor random. Figures 28 and 29 show the values of the blade lift coefficient, as function of time, for each of the cases mentioned above. The comparison between the results, regarding the generated force and the power consumed by the rotor is made in table 4.

FIGURE 29. Lift coefficient of the cycloidal rotor blade, without and with harmonic vibration of frequency 160 Hz, phase 0° and amplitude 1°.

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TABLE 4. Analysis of the cycloidal rotor aerodynamic performance.

Force

Power

Power

generated

consumed

loading

(N)

(W)

(N⁄kW)

Without harmonic

97.3

574.9

169.2

with harmonic

98.1

575.5

170.4

106.6

643.5

165.7

Cases

values may have different effect, that is to say, instead of noise amplification, a reduction of it can be obtained. Figure 34, which corresponds to only one blade, shows identical features.

(80 Hz) with harmonic (160 Hz) Analyzing Figs. 28 and 29 it is found that the harmonic vibration produces a significant effect on the aerodynamic force coefficients. This effect can be more or less intense depending on the harmonic vibration parameters. However, although there is a change in the lift coefficient values due to harmonic vibration, this change occurs around the lift curve of the rotor case operating without harmonic vibration, Figs. 28 and 29. Therefore, it is concluded that the harmonic vibration around the blade pitch curve causes the oscillation of the lift coefficient values around the corresponding values of the case without the harmonic vibration. For the considered cases, it is verified that the harmonic vibration increases the generated force produced by cycloidal rotor, table 4. However, the consumed power also increases. This increase of generated force and consumed power is not proportional. Thus, the cycloidal rotor power loading can increase or decrease. The results show that although the higher harmonic vibration frequency causes a greater increase of the generated force, the power loading decreases. Thus, the harmonic vibration of 80 Hz frequency provides better results since in this case a slight increase of both the generated force and the power loading is verified in comparison to the case without the harmonic vibration. The reason behind those changes, as seen in the considered cases, can be visualized through Figs. 30-32. The contours correspond to the flow vorticity intensity around the cycloidal rotor. Analyzing Figs. 30-32, it can be clearly seen a difference in the flow field between the case without harmonic and the case with harmonic vibration. Harmonic vibration clearly induces more vortices in the flow. This change in the formation and convection of the vortex structure modifies the interaction between them, and also their interaction with the cycloidal rotor blades. This occurrence causes a change in the cycloidal rotor behavior and consequently a change on the generated force and on the consumed power values. In addition to the effect on the aerodynamic performance, the harmonic vibration can introduce or reduce the noise in the cycloidal system. This study was done by analyzing the values of the Power Spectral Density (PSD). Figure 33 shows the noise analysis considering the entire cycloidal rotor. Figure 34 shows a noise analysis considering only the effect produced by a single cycloidal rotor blade. Analyzing Fig. 33, it can be seen that for high frequency noise values the harmonic of 80 Hz causes a noise decrease. However, the harmonic of 160Hz causes a noise increase. Therefore, the harmonic of 80 Hz has a beneficial effect on high frequency noise reduction. Another aspect to be pointed out is the occurrence of peaks at a frequency close to the harmonic frequency. This suggests that other frequency and/or phase

FIGURE 30. Contours of the vorticity intensity for the cycloidal rotor at 200 RPM and without harmonic vibration.

FIGURE 31. Contours of the vorticity intensity for the cycloidal rotor at 200 RPM and with harmonic vibration (80 Hz).

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CONCLUSIONS With this work, we investigated the possibility of increasing the cycloidal rotors performance from the change in the flow field around the blades. Thus, we considered the situation where the cycloidal rotor blades vibrate as they describe the cycloidal curve. However, the imposition of a harmonic vibration favorable to the cycloidal rotor performance imposes a throughout computation of the flow dynamics around the profiles. In this way, the flowfield behavior was analyzed from the fast Fourier transform and also from a Poincaré map perspective. The analysis of the harmonic vibration effect as performed herein, provided new insights on the cyclorotor performance. Accordingly, the harmonic vibration effect showed positive and negative effects on the performance of the oscillating profile. These effects are strongly sensitive to a combinations of phase, frequency, and amplitude values of the harmonic vibration. Therefore, it was demonstrated that harmonic vibration increases the cycloidal rotor performance, but must be adequately tuned to work adequately. The matching can be made ex-ante using CFD, as demonstrated, in order to reduce operational costs.

FIGURE 32. Contours of the vorticity intensity for the cycloidal rotor at 200 RPM and with harmonic vibration (160 Hz).

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FIGURE 33. Analysis of harmonic vibration effect on the noise of the cycloidal rotor operation.

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FIGURE 34. Analysis of harmonic vibration effect on the noise of the only one cycloidal rotor blade.

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