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Aerodynamic damping of sidewall bounded oscillating cantilevers
Journal of Fluids and Structures ] (]]]]) ]]]–]]]
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Aerodynamic damping of sidewall bounded oscillating cantilevers Andrew Eastman, Mark L. Kimber n Department of Mechanical Engineering and Materials Science, University of Pittsburgh, 206 Benedum Hall, 3700 O’Hara Street, Pittsburgh, PA 15261, USA
a r t i c l e i n f o
abstract
Article history: Received 8 September 2013 Accepted 21 July 2014
As a result of their simplicity, low power consumption, and relative ease of implementation, oscillating cantilevers have been investigated for use in multiple applications. However, the in situ operation in many cases, requires oscillating near one or more solid walls. When the separation distance between the vibrating cantilever and the solid wall becomes small, damping from the surrounding fluid is increased, which in turn can increase the power required to maintain certain operational performance characteristics (e.g., vibration amplitude). This increase in damping is a well-studied phenomenon for certain configurations (e.g., microcantilevers in Atomic Force Microscopy, or AFM), but is largely unexplored for a cantilever sweeping across a solid wall, which has direct impact for many macro-based applications including electronics cooling and propulsion. In this paper, we experimentally investigate the aerodynamic damping as a function of the gap between two sidewalls parallel to the oscillating motion of the cantilever. Multiple voltage and frequency inputs are considered in addition to the magnitude of the wall to cantilever gap. Experiments performed across a range of operating conditions reveal that decreasing the distance between the walls and the oscillating cantilever can increase the aerodynamic damping as much as 5 times that of the isolated (i.e., without sidewalls) operation. The resonance frequency is also shown to decrease when the gap spacing is extremely small, suggesting the added mass of the fluid is also sensitive to this variable. However, this change is much smaller ( 0.5%) compared to the change typically observed in damping. The findings in the paper help to quantify the overall effect of solid enclosure walls on the performance of an oscillating cantilever, which will better enable the designer to achieve the maximum operational effectiveness. The experimental findings also suggest viscous damping with sidewalls could be predicted from first principles in a similar manner to well accepted analytical models of a cantilever vibrating above a solid surface. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Aerodynamic damping Piezoelectric fan Oscillating cantilever
1. Introduction Oscillating cantilevers are simple mechanisms that have many applications ranging from atomic force microscopy (AFM) (Binnig et al., 1986) to biological sensors (Gupta et al., 2006) to energy harvesting (Aureli et al., 2010b) and electronics
n
Corresponding author. Tel.: þ1 412 624 8111. E-mail address: [email protected] (M.L. Kimber).
http://dx.doi.org/10.1016/j.jfluidstructs.2014.07.016 0889-9746/& 2014 Elsevier Ltd. All rights reserved.
Please cite this article as: Eastman, A., Kimber, M.L., Aerodynamic damping of sidewall bounded oscillating cantilevers. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.07.016i
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Nomenclature A Amax c ca D Fo I k KC L Lb Lp m ma
Oscillation amplitude maximum amplitude of a single frequency sweep effective structural damping coefficient of the cantilever aerodynamic damping coefficient cantilever width effective force on the beam from the piezoelectric element input current to the piezoelectric fan effective beam stiffness Keulegan–Carpenter number total length of the fan blade length of exposed portion (i.e., not covered by piezoelectric patch) of fan blade length of the piezoelectric patch effective mass of the cantilever mass of the air
tb tp Q Qa Qair Qiso V
β γ δ ν ζ ζair ω ωn ωn,air
thickness of the Mylar blade thickness of the piezoelectric patch structural quality factor aerodynamic quality factor quality factor in air (combined structural and aerodynamic effects) isolated quality factor in air (i.e., when no sidewalls are affecting the fan) input voltage to the piezoelectric fan frequency parameter for oscillating flows dimensionless gap between edge of cantilever and sidewall distance between the wall and side edge of the cantilever kinematic viscosity of the fluid structural damping ratio damping ratio in air oscillation frequency of fan excitation signal structural natural frequency of the fan natural frequency in air
cooling (Acikalin et al., 2004, 2007; Kimber and Garimella, 2009; Wait et al., 2007; Yoo et al., 2000). Their simple structure and straightforward operating requirements make them effective low power solutions with desirable fatigue life characteristics. Although the motion of a cantilever has been a well-understood concept for centuries, the complexity increases many fold when it becomes essential to quantify the coupling between structure and fluid. This is the case for most useful applications (including each of those mentioned above), and as a result, a great deal of literature exists on the topic of cantilevers oscillating within a fluid, but is primarily focused on microscale cantilevers. Perhaps the primary motivation for the vast amount of literature on this topic is the AFM cantilever (Binnig et al., 1986), where the oscillatory motion is employed to quantify certain properties of the surface below the cantilever. For further information on microscale applications, the reader is directed to the seminal work by Sader (1998), who developed an analytical model for the frequency response of an AFM cantilever oscillating in a viscous fluid. This analytical model accounts for the hydrodynamic forces as well as quantifying the quality factor. For the scenario when the microscale beam is vibrating near a solid substrate, many additional studies have been conducted. Green and Sader (2005a) used a boundary integral formulation based on the analysis done by Tuck (1969) to develop analytical models, and found that the impact of the solid surface is very limited unless the separation distance between the cantilever and the surface is less than the width of the cantilever. This and a related study by the same authors (Green and Sader, 2005b), reveal that the surface interaction with the cantilever mainly impacts the viscous dissipation and plays a relatively minor role in changing the inertial load (i.e., added mass) seen by the beam. Other significant contributions to the field of fluid forces present for micro-sized cantilevers include Cho et al. (1994), who investigated the validity of different 2D viscous damping models for laterally oscillating microstructures. They found that a Stokes type fluid motion assumption yields a more accurate quality factor when compared to a Couette type fluid motion. Yum et al. (2004) experimentally determined that the damping ratio increases as a result of decreasing cantilever scale. Findings in the well-explored microscale applications provide a basis from which the macro-scale can be considered, however the magnitude and source of non-negligible forces in macro-sized applications are markedly different. Unlike the fluid forces in microscale studies, the viscous damping caused by the surrounding fluid for macrosized beams is highly dependent on the oscillation amplitude in a non-linear fashion. Modeling of this fluid–structure interaction on this scale includes the high impact study by Jones (2003), where he developed a 2-D theoretical framework designed to predict the flow of an inviscid fluid around a thin oscillating cross section. He was able to capture the flow separation and subsequent vortex formation as the rectangular cross section oscillates in a fluid. Bidkar et al. (2009) continued the work of Jones by integrating this 2-D model along the entire length of a cantilever operating in the fundamental vibration mode. Experiments were also conducted using Mylar and stainless steel cantilever materials, and good agreement was found between predicted and experimentally measured values. As expected, theory predicts that a decrease in aerodynamic damping will lead to an increase in oscillation amplitude. Subsequent studies have extended this analysis by removing the limitation imposed by Jones (2003) and Bidkar et al. (2009), each of which is restricted to inviscid flow. Aureli et al. (2012) expressed the fluid–structure interaction with a complex hydrodynamic function which accounts for behavior expected for small vibration amplitudes, but also includes non-linear terms which are amplitude and frequency dependent. This analysis greatly extends the range of applicability for 2-D flows. Facci and Porfiri (2013) have since extend this work to accommodate three dimensional attributes with a particular emphasis on the thrust production. The result over the past decade has been
Please cite this article as: Eastman, A., Kimber, M.L., Aerodynamic damping of sidewall bounded oscillating cantilevers. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.07.016i
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a substantially enhanced understanding of the flow physics present for cross sections oscillating in an infinitely large viscous medium. One of the major hurdles associated with applications such as thrust generation (Aureli et al., 2010a; Cen and Erturk, 2013; Chung et al., 2008; Eastman et al., 2012) or convection enhancement (Acikalin et al., 2004, 2007; Kimber and Garimella, 2009) is designing an enclosure within which the vibrating beam is mounted, such that the performance is either unaffected or enhanced. Understanding the effect of the enclosure walls on the oscillatory behavior of the beam is an essential first step in generating guidelines for such designs. In terms of flows near stationary surfaces, Kim et al. (2004) looked at the flow field from a wide, sidewall bounded oscillating cantilever using a particle image velocimetry (PIV) system. The oscillating cantilever was found to produce a downstream cycle averaged velocity nearly three times the maximum tip velocity. However, the cantilever employed in that work was very wide in order to simulate a quasi-two-dimensional flow between the two sidewalls. Other research (Eastman et al., 2012) shows that an unbounded cantilever produces a cycle averaged velocity roughly two times the maximum tip velocity, which qualitatively agrees with results from Facci and Porfiri (2013), where they found the thrust production from a three-dimensional cantilever begins to suffer when the amplitude and width become comparable in magnitude. These results suggests that the presence of sidewalls could actually enhance the flow velocities, or at least bring them back up to levels predicted by a 2-D analysis, by directing the flow otherwise lost from the sides of the fan. It is also important to recognize that locations from where the fan is pulling the flow (i.e., the effective inlet) must remain unobstructed. Therefore, although enclosure walls could potentially enhance the flow, they must be strategically placed, so as not to inhibit essential flow from entering into the operational zone of the cantilever. Kimber et al. (2009b) tested multiple enclosure types and found that the inlet flow is primarily originating from upstream locations as well as above and below the wide face of the cantilever. This was further reinforced by Eastman and Kimber (2014) who performed detailed experimental investigations of the flow field around an unobstructed, oscillating cantilever using PIV. Therefore, walls upstream of the cantilever and those above and below the cantilever are not ideal candidates to help direct the flow and prevent unwanted losses in terms of the outflow. Sidewalls are therefore chosen for further investigation. When applying sidewalls or other boundaries to constrict or potentially guide otherwise lost flow, they naturally introduce higher damping. Although of great concern at a very small scale, viscous damping can become a prominent issue at times in the macro-scale as well. It is therefore the purpose of this paper to determine the impact of sidewalls and their effect on viscous damping as well as other performance characteristics. This allows for a more informed design of enclosure walls to help shape the flow and maintain specified levels of performance. 2. Theory Aerodynamic damping near resonance of a thin, flexible beam, can be quantified by first considering the response from approximating the continuous beam as a single degree of freedom system, which yields an expression for the response of the following form: 2 3 2 !2 2 1=2 Ak 4 ω ω 5 ¼ 1 þ 2ζ ; ð1Þ Fo ωn ωn where ω is the operating frequency, ωn is the natural frequency in a vacuum, A is the oscillation amplitude, k is the cantilever spring constant, Fo is the force from the piezoelectric element on the beam and ζ is the structural damping ratio (in the absence of any surrounding fluid). The effect of the fluid can be manifested both in terms of additional damping and extra mass in the system. Accounting for this in Eq. (1) yields the following (Kimber et al., 2009a): 2 3 !2 2 1=2 Ak 4 ma ω 2 ca ω 5 ¼ 1 1þ þ 2ζ 1 þ ; ð2Þ Fo ωn ωn m c where ma is the added mass from the aerodynamic loading, m is the effective mass of the beam, ca is the aerodynamic damping coefficient and c is the structural damping coefficient. The effect of the fluid becomes more significant as the ratio of aerodynamic to structural mass or aerodynamic to structural damping becomes comparable to unity. In such a case, in order to resolve the coefficients ma and ca, one must first determine their structural counterparts (m and c), requiring frequency response tests to be conducted in a vacuum chamber. In the current work, the magnitude of m, c, ma and ca is not of primary concern. Here, we are more interested in the decrease or increase of damping and added mass for a bounded cantilever with respect to a scenario where walls are far removed from the fan. It should be stressed at this point that Eq. (2) assumes amplitude independence for both ma and ca. Such is obviously not the case given the analysis and results from many studies (Aureli et al., 2012; Bidkar et al., 2009; Facci and Porfiri, 2013; Jones, 2003). However, the results obtained using Eq. (2) nonetheless provide an average estimate for both the damping and added mass. In this light, the primary goal initially put forth is achievable, namely quantifying the relative increase or decrease of these quantities as the distance changes between the cantilever and the solid wall. In order to predict ca from first principles, the interested reader should consult the four studies listed above. Please cite this article as: Eastman, A., Kimber, M.L., Aerodynamic damping of sidewall bounded oscillating cantilevers. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.07.016i
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In this work, we compute the total damping present during a particular frequency sweep in air. This damping (ζair) is a combined effect of both structural and aerodynamic damping related by the following expression: c ζ air ¼ ζ 1 þ a : ð3Þ c The natural frequency is treated in a likewise manner, namely the value found during an experiment in air (ωn,air) can be expressed in terms of the structural and aerodynamic effects as m 1=2 ωn;air ¼ ωn 1 þ a : ð4Þ m Substituting Eqs. (3) and (4) into Eq. (2) yields the equation of motion found in Eq. (5), namely: 2 3 !2 2 1=2 Ak 4 ω 2 ω 5 ¼ 1 þ 2ζ air ; Fo ωn;air ωn;air
ð5Þ
which is similar to Eq. (1), but the two parameters of interest are now ωn,air and ζair, instead of their structural counterparts. The damping can also be expressed as the quality factor defined as Q air ¼
1 ; 2ζ air
ð6Þ
where Qair is the quality factor found from an experiment performed in air and represents the combined effect of both structural and aerodynamic loading. 3. Experimental set-up The oscillating cantilever used in this paper is a commercially available piezoelectric fan from Piezo Systems (RFN1-005). The general form can be seen in Fig. 1and is composed of a thin Mylar cantilever with a piezoelectric patch adhered to one side. The approximate weight is 2.8 g with a piezoelectric capacitance of 15 nF and a maximum power consumption of 30 mW. This is affixed to a small bracket to allow for mounting to a flat rigid surface. The Mylar cantilever has nominal dimensions of 64.9 mm, 12.7 mm and 0.27 mm for the overall length (L), width (D) and thickness (tb) respectively. The driving mechanism is a Lead Zirconate Titanate (PZT) piezoelectric element with dimensions of 32.0 mm and 0.53 mm for length (Lp) and thickness (tp) and is bonded near the base of the Mylar cantilever. The width of the piezoelectric patch is the same as the Mylar substrate, namely 12.7 mm, and leaves a 36.5 mm length (Lb) of the Mylar cantilever exposed (i.e., not covered by the piezoelectric patch). This actuator can accommodate a voltage input as high as 120 VAC, which is created with a function generator (Tektronics AFG 3102) signal, and fed through an amplifier (MIDE QPA200) before connecting to the piezoelectric fan. The oscillation frequency of the cantilever is kept within the first mode of vibration. The fundamental natural frequency of the piezoelectric fan used in this study is 61.0 Hz, although this can change slightly depending on the proximity to surrounding walls, as will be shown. Also provided in Fig. 1 is the adopted frame of reference, with the downstream direction from the fan tip defined as the positive x-direction. The piezoelectric fan is rigidly mounted on a linear stage that traverses in the x-direction. This is mounted on an optical table and positioned near the center of a large enclosure with dimensions 1.22 m 0.61 m 0.61 m. The nearest enclosure wall in this configuration is over 90 mm away from the fan. Two additional rectangular sidewalls, one on either side of the fan in the x–y plane, are positioned on two independent linear stages (one for each sidewall) that traverses in the z-direction. These sidewalls span the entire length of the piezoelectric fan to ensure full coverage (102 mm 152 mm). The distance of the gap between the sidewalls and the fan blade (δ) is varied by actuating each of the two stages. An illustration of this orientation can be seen in Fig. 2. A laser displacement sensor (LKG-G157) is positioned above the cantilever ( 150 mm along the positive y-axis) to capture the vibration amplitude. This allows for displacement monitoring of the cantilever tip while it oscillates in the y-direction. This signal is then captured via a data acquisition unit from Measurement Computing (USB-1608HS). This data acquisition unit is also used to collect the components needed to calculate the power consumption. Both the current (I) and
Fig. 1. General illustration of the piezoelectric fan with important dimensions.
Please cite this article as: Eastman, A., Kimber, M.L., Aerodynamic damping of sidewall bounded oscillating cantilevers. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.07.016i
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Fig. 2. Illustration of the gap distance and orientation of the sidewalls in relation to the fan.
voltage (V) signals are captured at a sample rate of 1000 Hz, significantly higher than the operating frequencies around 60 Hz. The magnitude of the oscillations is extracted using the power spectral density of the raw signals. Dimensionless numbers are employed to better characterize the general behavior and enable direct comparison with other related works. Those numbers most common for oscillating flows include the frequency parameter (β) and the Keulegan–Carpenter number (KC), defined as
β¼
ωD2 ; 2πν
ð7Þ
and A KC ¼ 2π ; D
ð8Þ
where ν is the kinematic viscosity of the surrounding fluid (air with ν ¼15.9 10 6 m2/s). It should be noted that a cantilever with fixed dimensions is used with operating frequencies within a tight window surrounding the fundamental resonance frequency of 61 Hz. As a result, β in the current study only varies between 600 and 650. For the KC number, this is amplitude dependent and varies for each experiment. For reference, this ranges from approximately 0.7 to 2.4. The final parameter of interest is the dimensionless gap (γ), normalized by the beam width according to:
δ γ¼ : D
ð9Þ
Although the configuration in the current study is different than that for an AFM cantilever, this dimensionless gap is defined in a similar fashion to those studies. 4. Procedure To determine the parameters of interest, a frequency sweep is conducted at multiple voltage levels. This allows for a determination of the damping ratio, the maximum amplitude (Amax) and the resonance frequency input signal magnitude. This is linked to other performance factors like power consumption and phase lag between the voltage and current inputs. Additionally, in order to investigate the effect of phase lag (ϕ) and oscillation frequency on power consumption, a set of constant amplitude measurements are taken. For the constant voltage experiments, the piezoelectric element is provided with a sinusoidal RMS voltage (V) from the function generator at three different levels: 350, 500 and 650 mVrms. After a 50 amplification, the voltage seen by the piezoelectric fan is 17.5, 25 and 32.5 Vrms. When performing a constant amplitude test, the input voltage is adjusted until the average oscillation amplitude over a period of 1 s ( 61 cycles of oscillation) is found to be within 0.005 mm of the target amplitude. Once that amplitude is met, the test is continued. Before the sidewalls are introduced, the static displacement (Fo/k) is determined by operating the fan at 5 Hz (well below the fundamental resonance frequency of 61 Hz) and measuring the oscillation amplitude (A). Additional amplitudes are independently recorded for individual driving frequencies. The remainder of the operating frequencies are varied from as large of a range as 58 to 64 Hz in 0.25 Hz increments. This range is designed as an attempt to encompass the bandwidth Please cite this article as: Eastman, A., Kimber, M.L., Aerodynamic damping of sidewall bounded oscillating cantilevers. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.07.016i
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around the resonance frequency (as low as 70% response compared to the maximum displacement at resonance). The amplitude measurements for the frequency sweep are normalized by their corresponding static displacement, providing multiple data points for the left hand side of Eq. (5), at which point ωn,air and ζair can be determined from a least squares curve fit. The sidewall gaps considered are 1, 2, 3, 4, 6, 8, 10, 15, and 20 mm. This yields a range for γ (see Eq. (9)) between 0.079 and 1.58. A further increase in gap (γ 2.0) yields nearly identical results as γ ¼1.58. Therefore, this is considered to be sufficiently large enough to accurately mimic an isolated condition (i.e., one in the absence of sidewalls).
5. Results and discussion To help illustrate the effect of sidewalls on the raw data, time traces of the voltage supplied and current drawn from piezoelectric element are shown in the far left column of Fig. 3 for the two extreme supply voltages (V¼17.5 and 32.5 Vrms) and the two extreme gaps (δ ¼1 and 25 mm). As expected, the supply voltage is constant for the first two rows, since V is fixed at 17.5 Vrms. The current drawn by the piezoelectric element changes by approximately 1% from δ ¼25 mm (top row) where I¼0.126 mArms to δ ¼1 mm (second row) where I ¼0.122 mArms. The phase difference between the voltage and current signals indicates that the load has a nonzero reactance component, behavior that is entirely expected from a piezoelectric element. When calculating the power draw in such a case, the power is calculated according to the following (Jordan et al., 1999): P ¼ VI cos ϕ ; ð10Þ where I is the RMS current to the system and ϕ is the phase difference between voltage and current input signals. Although difficult to see from the plots in Fig. 3, this phase difference changes slightly from δ ¼25 mm where ϕ ¼ 1.24 rad to δ ¼1 mm where ϕ ¼ 1.33 rad. Although this represents roughly 7% change in the phase difference, the impact is much larger from a power consumption point of view since the angle is in the neighborhood of π/2 rad. For the data just discussed (top two rows of Fig. 3), the power changes by over 30%. Similar trends are observed for the larger of the two voltage inputs (bottom two rows of Fig. 3), namely that voltage and current magnitude remain constant, and a slight change in phase difference causes the power consumption to drop when the cantilever is highly confined. The raw displacement signal of the cantilever tip is shown in the center column of Fig. 3 (amplitude measurement is also included for each case), acquired via the laser displacement sensor, while the far right column is a power spectrum analysis of that amplitude signal to illustrate the frequency components present in the amplitude signal. Facci and Porfiri (2012) have shown for polychromatic excitation, the higher harmonics of beam vibrations can be experimentally observed. In particular, as the amplitude (or KC number) increases, they show that the fundamental resonance frequency becomes increasingly damped. Such a behavior is not observed in the power spectrum analysis shown in Fig. 3. Although the higher harmonics are present for all four scenarios shown, and are observed to relatively increase between V¼ 17.5 Vrms (top row, where KC ¼1.59) and V¼32.5 Vrms (third row, where KC ¼2.39), they are still at least two orders of magnitude smaller than that of the fundamental resonance frequency (61 Hz). This is not completely unexpected, since for the current study, we employ monochromatic excitation (single frequency) for each experiment, the frequency of which is in the close neighborhood of the fundamental resonance frequency. Therefore, nearly all of the power is absorbed at that fundamental mode, regardless of the KC number. Note that although the input signal is fixed for the top two rows at 17.5 Vrms, the amplitude suffers greatly when the cantilever is tightly confined by the sidewalls. An amplitude of A¼3.22 mm in an unconfined state (top row) drops to A¼ 2.05 mm for δ ¼1 mm. To fully quantify the effect of the sidewalls on the dynamic response of the cantilever, data across a range of frequencies must be analyzed. However, the trends discussed in Fig. 3 underpin the phenomenon of interest, namely an increase in viscous drag is seen as the distance between the vibrating cantilever and the sidewall becomes small. The frequency response curves for inputs of V¼17.5 Vrms and 32.5 Vrms cases are shown in Fig. 4(a) and (b), respectively. Note that the amplitude has been normalized by its static displacement (Fo/k). For both cases, the sharpness of the resonance curve decreases as the sidewall gap (δ) becomes small. This increase in damping (or decrease in quality factor) suggests the presence of the wall itself creates some amount of additional viscous drag, resulting in a decrease in vibration amplitude for the same input voltage. Also worth noting in Fig. 4(a) and (b) is the fact that the V¼ 17.5 Vrms case yields a larger normalized amplitude response for any given value of δ. This indicates that the overall damping is also dependent on the input signal magnitude itself, where higher damping is seen for larger input magnitudes. This is attributed to the fact that damping increases when larger vibration amplitudes and velocities occur, and is also consistent with conclusions found in Bidkar et al. (2009) and Porfiri and coworkers (Aureli et al., 2012; Facci and Porfiri, 2013), which reveal the same trends for aerodynamic damping without sidewalls. Although the current study only quantifies the average value for damping and does not separately express the structural and aerodynamic quantities, it nonetheless captures the expected trend of amplitude dependent damping. It is also apparent that for very small values of δ, a shift in resonance frequency occurs. This is in the direction of a lower frequency rather than an increase. This trend can be better observed in Fig. 5, which represents the natural frequency extracted from the curve fit (using Eq. (5)) of the frequency response data. This indicates that the effect of added mass (due to the sidewalls) only has a measurable effect when the sidewalls are in very close proximity to the fan. However, the drift in resonance frequency as the wall gap increases is quite small (less than 1%). It should also be noted that although the resonance frequency is slightly different depending on the voltage input (three separate curves in Please cite this article as: Eastman, A., Kimber, M.L., Aerodynamic damping of sidewall bounded oscillating cantilevers. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.07.016i
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Fig. 3. Time traces of voltage, current, and tip displacement for a driving frequency of 61 Hz: V ¼ 17.5 Vrms and δ ¼25 mm (top row), V ¼ 17.5 Vrms, and δ ¼1 mm (second row), V ¼ 32.5 Vrms and δ ¼ 25 mm (third row), and V ¼ 32.5 Vrms and δ ¼ 1 mm (bottom row).
Fig. 5), this difference is less than 0.2%. Therefore, the fluid–structure interaction that takes place can almost exclusively be attributed to a change in viscous damping, or the quality factor. The trend of the quality factor as a function of δ can be seen in Fig. 6 for all three voltage inputs. For an approximate gap of δ ¼10 mm, the effect of the sidewalls is negligible, and no meaningful benefit or drawback is realized when the spacing increases beyond this value. Below this spacing, the general trend is what would be expected as the reduction in gap should cause an increase of damping to the beam. It is interesting to note that although the relative orientation of the beam with Please cite this article as: Eastman, A., Kimber, M.L., Aerodynamic damping of sidewall bounded oscillating cantilevers. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.07.016i
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Fig. 4. Comparison of the normalized amplitude response for the (a) V ¼ 17.5 Vrms and (b) V ¼ 32.5 Vrms case.
Fig. 5. The progression of the natural frequency as the sidewall gap is varied.
respect to the solid wall is unique in the current study, similarities can be drawn from important related works. For example, AFM cantilevers have prompted a large number of analytical, numerical, and experimental studies aimed at quantifying the squeeze film damping effect of the fluid between the beam and the solid surface (Green and Sader, 2005a; Grimaldi et al., 2012 to Please cite this article as: Eastman, A., Kimber, M.L., Aerodynamic damping of sidewall bounded oscillating cantilevers. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.07.016i
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Fig. 6. The quality factor for each sidewall gap distance.
Fig. 7. Curve fit of the quality factor normalized by the isolated quality factor.
name a few). The solid wall in that application is rotated by 901 compared to the present configuration. However, the qualitative trends are the same, namely the viscous damping increases when an oscillating beam approaches any solid surface, regardless of the relative orientation. For the data collected, the quality factor at δ ¼1 mm is approximately 35% of the respective isolated values for V¼17.5, 25, and 32.5 Vrms. One important tradeoff in a number of applications is the benefit of introducing sidewalls in order to limit the loss of fluid from the side edges of the oscillating fan, balanced with the drawback of increased damping which must be overcome. The decrease in quality factor (and thus increase in damping) means that more power would be required to maintain the same vibration amplitude. It should also be noted that the difference in the isolated quality factor (Qair at δ ¼20 mm) between the V¼17.5 and 25 Vrms cases is larger than the difference between the V¼25 and 32.5 Vrms cases. The damping plays a role in limiting the amplitude, as a slightly higher amplitude will also yield a higher oscillation velocity. As it can be gathered from an understanding of drag, as the velocity increases it causes the drag resistance to increase. This has also been demonstrated in other studies using similar systems by Kimber et al. (2009a), Bidkar et al. (2009), and Porfiri and coworkers (Aureli et al., 2012; Facci and Porfiri, 2013). This further reinforces that the non-linear response compared to the linear voltage input is not a product of some unknown systematic error. In order to develop a way to predict the quality factor in this type of system, a curve fit is applied to the quality factor data after being normalized by the isolated quality factor (Qiso is equal to Qair at δ ¼20 mm) in each case. This resulting behavior can be captured with the following exponential relationship: Q δ : ð11Þ ¼ 1 C 1 exp C 2 Q iso D A least squares curve fit of the data yields 0.51 and 4.78 for C1 and C2, respectively. The results of this fit can be seen in Fig. 7 with a maximum absolute error of 6.1%. This result allows for an accurate prediction of the resulting quality factor based on sidewall spacing for a set voltage input. It is stressed that this expression (Eq. (11)) is only applicable for the beam geometry and frequency range considered with 600 o β o650 and 0.7oKCo2.4. Once additional cantilevers are tested that Please cite this article as: Eastman, A., Kimber, M.L., Aerodynamic damping of sidewall bounded oscillating cantilevers. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.07.016i
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cover a larger range of beam geometries, a more comprehensive analysis can be conducted. It is also noted that this provides the effective, or in some sense, the average damping as a function of the dimensionless spacing. Despite these limitations, the current efforts show promise in ultimately predicting the viscous damping in the presence of sidewalls from first principles in a fashion similar as Grimaldi et al. (2012) did for an AFM cantilever-related orientation. As a prevailing interest in this paper is damping and its effect on performance, it is also important to quantify the power consumption, a metric of great concern for many potential applications (e.g., electronics cooling), and is determined using Eq. (10). The power calculations are conducted for all gaps at the three voltage inputs previously considered, but fixing the driving frequency at 61 Hz. The result is shown in Fig. 8, where it can be seen that the power actually decreases for small gap distances at constant voltage inputs. Since the supply voltage remains fixed for a given set of data in Fig. 8, only changes in current and/or phase difference can explain the drop in power consumption. The corresponding RMS current values for this data are shown in Fig. 9, which reveal a trend similar to the power data, but the decrease in current is less than 2.5%, whereas the drop in power consumption can be roughly 30%, compared to their respective isolated (large δ) values. Therefore, a large portion of this drop in power should be explained by a change in the phase angle. As seen in Fig. 10, the phase lag is in the range of 1.24 to 1.35 rad. These values are fairly close to π/2 rad, which means that any small change in phase will result in a much larger change in power due to the fact that the cosine of this phase difference is the quantity used in the power calculations. Essentially, the load of the piezoelectric fan has a large imaginary component or electrical reactance. When this is taken into account, it becomes apparent that a change in phase from a gap of 1–25 mm can result in a roughly 25% change in power consumption if everything else were to remain constant. In fact, ϕ changes quite dramatically and does so in a very similar manner to that of the amplitude response of the cantilever, readily apparent in Fig. 11, which can be directly compared to Fig. 4(a), since voltage input is the same (V¼17.5 Vrms). The peak of the phase curve changes with sidewall gap and appears to occur at the same frequency as the resonance frequencies seen from Fig. 4. Although one might suppose an advantage exists since power consumption is reduced, this comes at a cost of lower vibration amplitudes due to an increase in aerodynamic damping. To illustrate this point, consider the peak amplitude
Fig. 8. The power requirement of the piezoelectric fan at 61 Hz as the gap is varied.
Fig. 9. The current as a function of sidewall gap for set voltage inputs.
Please cite this article as: Eastman, A., Kimber, M.L., Aerodynamic damping of sidewall bounded oscillating cantilevers. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.07.016i
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Fig. 10. The phase difference between the current and voltage input at 61 Hz for a set voltage inputs.
Fig. 11. The phase lag between the voltage and current input as a function of frequency for V ¼17.5 Vrms.
Fig. 12. The maximum oscillation amplitude for each sidewall gap distance.
(Amax) for all 9 gaps and each unique input signal magnitude, which is extracted from the frequency response curves and is shown in Fig. 12. For reference, the corresponding KC numbers for this peak amplitude data range from 1.04 to 1.58 for V¼17.5 Vrms, 1.32 to 2.01 for V¼25.0 Vrms, and 1.59 to 2.40 for V ¼32.5 Vrms. It is useful to compare the amplitude measurements from Fig. 12 and the power consumption from Fig. 8. The trends from both of these figures reveal that both the vibration amplitude and power consumption decrease as δ becomes smaller. In order to further evaluate the power Please cite this article as: Eastman, A., Kimber, M.L., Aerodynamic damping of sidewall bounded oscillating cantilevers. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.07.016i
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Fig. 13. The power requirement to the piezoelectric fan with a fixed amplitude as the frequency is changed within the bandwidth (δ ¼ 25 mm).
consumption trends working both on and off resonance, additional experiments are performed at fixed amplitudes of 4, 5, and 6 mm with δ ¼ 25 mm and varying the input frequency across the bandwidth of the frequency response. The result is shown in Fig. 13, where it is apparent that the change in phase lag (ϕ) when operating off resonance cannot adequately make up for the increased power input required to keep the amplitude consistent across the frequency spectrum. For the 4 mm amplitude, the curve is flatter indicating that the phase does have some effect. However, in order to see a potentially flat power curve, the amplitude would need to be so small that there would be no practical use in this setting. Regardless, this finding demonstrates that operating at resonance is, as one would intuitively expect, the best option in terms of minimizing the power requirements. Further experiments should be conducted where the application-specific performance metrics (e.g., thrust or thermal convection) are evaluated along with the input to truly gauge the tradeoff that must be made between the benefits from sidewalls in terms of preventing flow being lost across the side edges compared to the loss of performance due to a decrease in amplitude and/or the increase in power consumption in order to maintain a specific amplitude.
6. Conclusions Frequency response curves are experimentally measured for a piezoelectric fan vibrating near two sidewalls in order to quantify the effect of enclosure walls on the oscillation characteristics of the fan. Three voltage inputs are considered at multiple sidewall gaps. The natural frequency is shown to decrease with proximity to the side wall suggesting an increase in the added mass from the surrounding fluid. For the damping characteristics, as the proximity of the sidewalls to the fan is decreased, the quality factor is found to also decrease. This means that as one seeks to improve the flow shaping by introducing sidewalls, the inevitable damping must be addressed as it becomes more and more of a factor. A curve fit is applied to the quality factor, showing good agreement with experimental data. This gives a basis for predicting the viscous damping as a function of the sidewall gap, and suggests a more comprehensive study would be worthwhile where additional factors are investigated (e.g., beam width, length, operating frequency, etc.). The effect of the sidewalls on damping is in general much larger than the impact on the resonance frequency. In addition, the damping begins to change from its isolated value at a much larger gap compared to the gap where the resonance frequency begins to shift. It has been demonstrated in this paper that, for a constant voltage input, the change in power requirement is primarily a result of the phase difference between the voltage and current input. This change in the phase difference is driven entirely by the damping conditions imposed by the sidewalls or frequency shift from resonance. However, it does not have a pronounced enough effect to meaningfully limit the power increase needed in order to maintain a specific amplitude. The findings here provide the basis for understanding and predicting the effect of sidewalls on damping and impact on power consumption. This allows for evaluation of the operational characteristics and limitations for an oscillating cantilever when introduced into an enclosed space. Fundamentally, higher damping limits the amplitude for a specific voltage input. Additional studies are currently underway to quantify the effect of other performance metrics (e.g., thrust or thermal performance) in terms of proximity to sidewalls and measuring power consumption while holding the amplitude fixed. References Acikalin, T., Garimella, S.V., Raman, A., Petroski, J., 2007. Characterization and optimization of the thermal performance of miniature piezoelectric fans. International Journal of Heat and Fluid Flow 28, 806–820. Acikalin, T., Wait, S.M., Garimella, S.V., Raman, A., 2004. Experimental investigation of the thermal performance of piezoelectric fans. Heat Transfer Engineering 25, 4–14.
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Aureli, M., Kopman, V., Porfiri, M., 2010a. Free-locomotion of underwater vehicles actuated by ionic polymer metal composites. IEEE/ASME Transactions on Mechatronics 15, 603–614. Aureli, M., Prince, C., Porfiri, M., Peterson, S.D., 2010b. Energy harvesting from a base excitiation of ionic polymer metal composites in fluid environments. Smart Materials and Structures 19, 1–15. Aureli, M., Basaran, M.E., Porfiri, M., 2012. Nonlinear finite amplitude vibrations of sharp-edged beams in viscous fluids. Journal of Sound and Vibration 331, 1624–1654. Bidkar, R., Kimber, M., Bajaj, A., Garimella, S., 2009. Nonlinear aerodynamic damping of sharp-edged flexible beams oscillating at low Keulegan–Carpenter numbers. Journal of Fluid Mechanics 634, 269–289. Binnig, G., Quate, C.F., Gerber, C., 1986. Atomic force microscope. Physical Review Letters 56, 930–933. Cen, L., Erturk, A., 2013. Bio-inspired aquatic robots by untethered piezohydroelastic actuation. Bioinspiration & Biomimetics 8, 1–13. Cho, Y.-H., Pisano, A.P., Howe, R.T., 1994. Viscous damping model for laterally oscillating microstructures. Journal of Microelectromechanical Systems 3, 81–87. Chung, H.-C., Kummari, K.L., Croucher, S.J., Lawson, N.J., Guo, S., Huang, Z., 2008. Coupled piezoelectric fans with two degree of freedom motion for the application of flapping micro aerial vehicles. Sensors and Actuators A 147, 607–612. Eastman, A., Kimber, M., 2014. Analysis of three-dimensional attributes and flow intake for an oscillating cantilever. Experiments in Fluids 55, 1164 (14 pp.) http://dx.doi.org/10.1007/s00348-014-1664-1. Eastman, A., Kimber, M., 2014. Aerodynamic damping of sidewall bounded oscillating cantilevers. Journal of Fluids and Structures. (in preparation). Facci, A.L., Porfiri, M., 2012. Nonlinear hydrodynamic damping of sharp-edged cantilevers in viscous fluids undergoing multi-harmonic base excitation. Journal of Applied Physics 112, 124908. (9 pp.). Facci, A.L., Porfiri, M., 2013. Analysis of three-dimensional effects in oscillating cantilevers immersed in viscous fluids. Journal of Fluids and Structures 38, 205–222. Green, C.P., Sader, J.E., 2005a. Small amplitude oscillations of a thin beam immersed in a viscous fluid near a solid surface. Physics of Fluids 17, 1–12. Green, C.P., Sader, J.E., 2005b. Frequency response of cantilever beams immersed in viscous fluids near a solid surface with applications to the atomic force microscope. Journal of Applied Physics 98, 1–12. Grimaldi, E., Porfiri, M., Soria, L., 2012. Finite amplitude vibrations of a sharp-edged beam immersed in a viscous fluid near a solid surface. Journal of Applied Physics 112, 1–13. Gupta, A.K., Nair, P.R., Akin, D., Ladisch, M.R., Broyles, S., Alam, M.A., Bashir, R., 2006. Anomalous resonance in a nanomechanical biosensor. Proceedings of the National Academy of Sciences of the United States of America 103, 13362–13367. Jones, M.A., 2003. The separated flow of an inviscid fluid around a moving flat plate. Journal of Fluid Mechanics 496, 405–441. Jordan, T., Ounaies, Z., Tripp, J., Tcheng, P., 1999. Electrical Properties and Power Considerations of a Piezoelectric Actuator, MRS Proceedings. Cambridge University Press, Boston, MA. Kim, Y.H., Wereley, S.T., Chun, C.-H., 2004. Phase-resolved flow field produced by a vibrating cantilever plate between two endplates. Physics of Fluids 16, 145–162. Kimber, M., Garimella, S.V., 2009. Measurement and prediction of the cooling characteristics of a generalized vibrating piezoelectric fan. International Journal of Heat and Mass Transfer 52, 4470–4478. Kimber, M., Lonergan, R., Garimella, S.V., 2009a. Experimental study of aerodynamic damping in arrays of vibrating cantilevers. Journal of Fluids and Structures 25, 1334–1347. Kimber, M., Suzuki, K., Kitsunai, N., Seki, K., Garimella, S.V., 2009b. Pressure and flow rate performance of piezoelectric fans. IEEE Transactions on Components and Packaging Technologies 32, 766–775. Sader, J.E., 1998. Frequency response of cantilever beams immersed in viscous fluids with applications t the atomic force microscope. Journal of Applied Physics 84, 64–76. Tuck, E.O., 1969. Calculation of unsteady flows due to small motions of cylinders in a viscous fluid. Journal of Engineering Mathematics 3, 29–44. Wait, S.M., Basak, S., Garimella, S.V., Raman, A., 2007. Piezoelectric fans using higher flexural modes for electronics cooling applications. IEEE Transactions on Components and Packaging Technologies 30, 119–128. Yoo, J.H., Hong, J.I., Cao, W., 2000. Piezoelectric ceramic bimorph coupled to thin metal plate as cooling fan for electronic devices. Sensors and Actuators A 79, 8–12. Yum, K., Wang, Z., Suryavanshi, A.P., Yu, M.-F., 2004. Experiemntal measurement and model analysis of damping effect in nanoscale mechanical beam resonators in air. Journal of Applied Physics 96, 3933–3938.
Please cite this article as: Eastman, A., Kimber, M.L., Aerodynamic damping of sidewall bounded oscillating cantilevers. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.07.016i
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Aerodynamic damping of sidewall bounded oscillating cantilevers Andrew Eastman, Mark L. Kimber n Department of Mechanical Engineering and Materials Science, University of Pittsburgh, 206 Benedum Hall, 3700 O’Hara Street, Pittsburgh, PA 15261, USA
a r t i c l e i n f o
abstract
Article history: Received 8 September 2013 Accepted 21 July 2014
As a result of their simplicity, low power consumption, and relative ease of implementation, oscillating cantilevers have been investigated for use in multiple applications. However, the in situ operation in many cases, requires oscillating near one or more solid walls. When the separation distance between the vibrating cantilever and the solid wall becomes small, damping from the surrounding fluid is increased, which in turn can increase the power required to maintain certain operational performance characteristics (e.g., vibration amplitude). This increase in damping is a well-studied phenomenon for certain configurations (e.g., microcantilevers in Atomic Force Microscopy, or AFM), but is largely unexplored for a cantilever sweeping across a solid wall, which has direct impact for many macro-based applications including electronics cooling and propulsion. In this paper, we experimentally investigate the aerodynamic damping as a function of the gap between two sidewalls parallel to the oscillating motion of the cantilever. Multiple voltage and frequency inputs are considered in addition to the magnitude of the wall to cantilever gap. Experiments performed across a range of operating conditions reveal that decreasing the distance between the walls and the oscillating cantilever can increase the aerodynamic damping as much as 5 times that of the isolated (i.e., without sidewalls) operation. The resonance frequency is also shown to decrease when the gap spacing is extremely small, suggesting the added mass of the fluid is also sensitive to this variable. However, this change is much smaller ( 0.5%) compared to the change typically observed in damping. The findings in the paper help to quantify the overall effect of solid enclosure walls on the performance of an oscillating cantilever, which will better enable the designer to achieve the maximum operational effectiveness. The experimental findings also suggest viscous damping with sidewalls could be predicted from first principles in a similar manner to well accepted analytical models of a cantilever vibrating above a solid surface. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Aerodynamic damping Piezoelectric fan Oscillating cantilever
1. Introduction Oscillating cantilevers are simple mechanisms that have many applications ranging from atomic force microscopy (AFM) (Binnig et al., 1986) to biological sensors (Gupta et al., 2006) to energy harvesting (Aureli et al., 2010b) and electronics
n
Corresponding author. Tel.: þ1 412 624 8111. E-mail address: [email protected] (M.L. Kimber).
http://dx.doi.org/10.1016/j.jfluidstructs.2014.07.016 0889-9746/& 2014 Elsevier Ltd. All rights reserved.
Please cite this article as: Eastman, A., Kimber, M.L., Aerodynamic damping of sidewall bounded oscillating cantilevers. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.07.016i
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Nomenclature A Amax c ca D Fo I k KC L Lb Lp m ma
Oscillation amplitude maximum amplitude of a single frequency sweep effective structural damping coefficient of the cantilever aerodynamic damping coefficient cantilever width effective force on the beam from the piezoelectric element input current to the piezoelectric fan effective beam stiffness Keulegan–Carpenter number total length of the fan blade length of exposed portion (i.e., not covered by piezoelectric patch) of fan blade length of the piezoelectric patch effective mass of the cantilever mass of the air
tb tp Q Qa Qair Qiso V
β γ δ ν ζ ζair ω ωn ωn,air
thickness of the Mylar blade thickness of the piezoelectric patch structural quality factor aerodynamic quality factor quality factor in air (combined structural and aerodynamic effects) isolated quality factor in air (i.e., when no sidewalls are affecting the fan) input voltage to the piezoelectric fan frequency parameter for oscillating flows dimensionless gap between edge of cantilever and sidewall distance between the wall and side edge of the cantilever kinematic viscosity of the fluid structural damping ratio damping ratio in air oscillation frequency of fan excitation signal structural natural frequency of the fan natural frequency in air
cooling (Acikalin et al., 2004, 2007; Kimber and Garimella, 2009; Wait et al., 2007; Yoo et al., 2000). Their simple structure and straightforward operating requirements make them effective low power solutions with desirable fatigue life characteristics. Although the motion of a cantilever has been a well-understood concept for centuries, the complexity increases many fold when it becomes essential to quantify the coupling between structure and fluid. This is the case for most useful applications (including each of those mentioned above), and as a result, a great deal of literature exists on the topic of cantilevers oscillating within a fluid, but is primarily focused on microscale cantilevers. Perhaps the primary motivation for the vast amount of literature on this topic is the AFM cantilever (Binnig et al., 1986), where the oscillatory motion is employed to quantify certain properties of the surface below the cantilever. For further information on microscale applications, the reader is directed to the seminal work by Sader (1998), who developed an analytical model for the frequency response of an AFM cantilever oscillating in a viscous fluid. This analytical model accounts for the hydrodynamic forces as well as quantifying the quality factor. For the scenario when the microscale beam is vibrating near a solid substrate, many additional studies have been conducted. Green and Sader (2005a) used a boundary integral formulation based on the analysis done by Tuck (1969) to develop analytical models, and found that the impact of the solid surface is very limited unless the separation distance between the cantilever and the surface is less than the width of the cantilever. This and a related study by the same authors (Green and Sader, 2005b), reveal that the surface interaction with the cantilever mainly impacts the viscous dissipation and plays a relatively minor role in changing the inertial load (i.e., added mass) seen by the beam. Other significant contributions to the field of fluid forces present for micro-sized cantilevers include Cho et al. (1994), who investigated the validity of different 2D viscous damping models for laterally oscillating microstructures. They found that a Stokes type fluid motion assumption yields a more accurate quality factor when compared to a Couette type fluid motion. Yum et al. (2004) experimentally determined that the damping ratio increases as a result of decreasing cantilever scale. Findings in the well-explored microscale applications provide a basis from which the macro-scale can be considered, however the magnitude and source of non-negligible forces in macro-sized applications are markedly different. Unlike the fluid forces in microscale studies, the viscous damping caused by the surrounding fluid for macrosized beams is highly dependent on the oscillation amplitude in a non-linear fashion. Modeling of this fluid–structure interaction on this scale includes the high impact study by Jones (2003), where he developed a 2-D theoretical framework designed to predict the flow of an inviscid fluid around a thin oscillating cross section. He was able to capture the flow separation and subsequent vortex formation as the rectangular cross section oscillates in a fluid. Bidkar et al. (2009) continued the work of Jones by integrating this 2-D model along the entire length of a cantilever operating in the fundamental vibration mode. Experiments were also conducted using Mylar and stainless steel cantilever materials, and good agreement was found between predicted and experimentally measured values. As expected, theory predicts that a decrease in aerodynamic damping will lead to an increase in oscillation amplitude. Subsequent studies have extended this analysis by removing the limitation imposed by Jones (2003) and Bidkar et al. (2009), each of which is restricted to inviscid flow. Aureli et al. (2012) expressed the fluid–structure interaction with a complex hydrodynamic function which accounts for behavior expected for small vibration amplitudes, but also includes non-linear terms which are amplitude and frequency dependent. This analysis greatly extends the range of applicability for 2-D flows. Facci and Porfiri (2013) have since extend this work to accommodate three dimensional attributes with a particular emphasis on the thrust production. The result over the past decade has been
Please cite this article as: Eastman, A., Kimber, M.L., Aerodynamic damping of sidewall bounded oscillating cantilevers. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.07.016i
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a substantially enhanced understanding of the flow physics present for cross sections oscillating in an infinitely large viscous medium. One of the major hurdles associated with applications such as thrust generation (Aureli et al., 2010a; Cen and Erturk, 2013; Chung et al., 2008; Eastman et al., 2012) or convection enhancement (Acikalin et al., 2004, 2007; Kimber and Garimella, 2009) is designing an enclosure within which the vibrating beam is mounted, such that the performance is either unaffected or enhanced. Understanding the effect of the enclosure walls on the oscillatory behavior of the beam is an essential first step in generating guidelines for such designs. In terms of flows near stationary surfaces, Kim et al. (2004) looked at the flow field from a wide, sidewall bounded oscillating cantilever using a particle image velocimetry (PIV) system. The oscillating cantilever was found to produce a downstream cycle averaged velocity nearly three times the maximum tip velocity. However, the cantilever employed in that work was very wide in order to simulate a quasi-two-dimensional flow between the two sidewalls. Other research (Eastman et al., 2012) shows that an unbounded cantilever produces a cycle averaged velocity roughly two times the maximum tip velocity, which qualitatively agrees with results from Facci and Porfiri (2013), where they found the thrust production from a three-dimensional cantilever begins to suffer when the amplitude and width become comparable in magnitude. These results suggests that the presence of sidewalls could actually enhance the flow velocities, or at least bring them back up to levels predicted by a 2-D analysis, by directing the flow otherwise lost from the sides of the fan. It is also important to recognize that locations from where the fan is pulling the flow (i.e., the effective inlet) must remain unobstructed. Therefore, although enclosure walls could potentially enhance the flow, they must be strategically placed, so as not to inhibit essential flow from entering into the operational zone of the cantilever. Kimber et al. (2009b) tested multiple enclosure types and found that the inlet flow is primarily originating from upstream locations as well as above and below the wide face of the cantilever. This was further reinforced by Eastman and Kimber (2014) who performed detailed experimental investigations of the flow field around an unobstructed, oscillating cantilever using PIV. Therefore, walls upstream of the cantilever and those above and below the cantilever are not ideal candidates to help direct the flow and prevent unwanted losses in terms of the outflow. Sidewalls are therefore chosen for further investigation. When applying sidewalls or other boundaries to constrict or potentially guide otherwise lost flow, they naturally introduce higher damping. Although of great concern at a very small scale, viscous damping can become a prominent issue at times in the macro-scale as well. It is therefore the purpose of this paper to determine the impact of sidewalls and their effect on viscous damping as well as other performance characteristics. This allows for a more informed design of enclosure walls to help shape the flow and maintain specified levels of performance. 2. Theory Aerodynamic damping near resonance of a thin, flexible beam, can be quantified by first considering the response from approximating the continuous beam as a single degree of freedom system, which yields an expression for the response of the following form: 2 3 2 !2 2 1=2 Ak 4 ω ω 5 ¼ 1 þ 2ζ ; ð1Þ Fo ωn ωn where ω is the operating frequency, ωn is the natural frequency in a vacuum, A is the oscillation amplitude, k is the cantilever spring constant, Fo is the force from the piezoelectric element on the beam and ζ is the structural damping ratio (in the absence of any surrounding fluid). The effect of the fluid can be manifested both in terms of additional damping and extra mass in the system. Accounting for this in Eq. (1) yields the following (Kimber et al., 2009a): 2 3 !2 2 1=2 Ak 4 ma ω 2 ca ω 5 ¼ 1 1þ þ 2ζ 1 þ ; ð2Þ Fo ωn ωn m c where ma is the added mass from the aerodynamic loading, m is the effective mass of the beam, ca is the aerodynamic damping coefficient and c is the structural damping coefficient. The effect of the fluid becomes more significant as the ratio of aerodynamic to structural mass or aerodynamic to structural damping becomes comparable to unity. In such a case, in order to resolve the coefficients ma and ca, one must first determine their structural counterparts (m and c), requiring frequency response tests to be conducted in a vacuum chamber. In the current work, the magnitude of m, c, ma and ca is not of primary concern. Here, we are more interested in the decrease or increase of damping and added mass for a bounded cantilever with respect to a scenario where walls are far removed from the fan. It should be stressed at this point that Eq. (2) assumes amplitude independence for both ma and ca. Such is obviously not the case given the analysis and results from many studies (Aureli et al., 2012; Bidkar et al., 2009; Facci and Porfiri, 2013; Jones, 2003). However, the results obtained using Eq. (2) nonetheless provide an average estimate for both the damping and added mass. In this light, the primary goal initially put forth is achievable, namely quantifying the relative increase or decrease of these quantities as the distance changes between the cantilever and the solid wall. In order to predict ca from first principles, the interested reader should consult the four studies listed above. Please cite this article as: Eastman, A., Kimber, M.L., Aerodynamic damping of sidewall bounded oscillating cantilevers. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.07.016i
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In this work, we compute the total damping present during a particular frequency sweep in air. This damping (ζair) is a combined effect of both structural and aerodynamic damping related by the following expression: c ζ air ¼ ζ 1 þ a : ð3Þ c The natural frequency is treated in a likewise manner, namely the value found during an experiment in air (ωn,air) can be expressed in terms of the structural and aerodynamic effects as m 1=2 ωn;air ¼ ωn 1 þ a : ð4Þ m Substituting Eqs. (3) and (4) into Eq. (2) yields the equation of motion found in Eq. (5), namely: 2 3 !2 2 1=2 Ak 4 ω 2 ω 5 ¼ 1 þ 2ζ air ; Fo ωn;air ωn;air
ð5Þ
which is similar to Eq. (1), but the two parameters of interest are now ωn,air and ζair, instead of their structural counterparts. The damping can also be expressed as the quality factor defined as Q air ¼
1 ; 2ζ air
ð6Þ
where Qair is the quality factor found from an experiment performed in air and represents the combined effect of both structural and aerodynamic loading. 3. Experimental set-up The oscillating cantilever used in this paper is a commercially available piezoelectric fan from Piezo Systems (RFN1-005). The general form can be seen in Fig. 1and is composed of a thin Mylar cantilever with a piezoelectric patch adhered to one side. The approximate weight is 2.8 g with a piezoelectric capacitance of 15 nF and a maximum power consumption of 30 mW. This is affixed to a small bracket to allow for mounting to a flat rigid surface. The Mylar cantilever has nominal dimensions of 64.9 mm, 12.7 mm and 0.27 mm for the overall length (L), width (D) and thickness (tb) respectively. The driving mechanism is a Lead Zirconate Titanate (PZT) piezoelectric element with dimensions of 32.0 mm and 0.53 mm for length (Lp) and thickness (tp) and is bonded near the base of the Mylar cantilever. The width of the piezoelectric patch is the same as the Mylar substrate, namely 12.7 mm, and leaves a 36.5 mm length (Lb) of the Mylar cantilever exposed (i.e., not covered by the piezoelectric patch). This actuator can accommodate a voltage input as high as 120 VAC, which is created with a function generator (Tektronics AFG 3102) signal, and fed through an amplifier (MIDE QPA200) before connecting to the piezoelectric fan. The oscillation frequency of the cantilever is kept within the first mode of vibration. The fundamental natural frequency of the piezoelectric fan used in this study is 61.0 Hz, although this can change slightly depending on the proximity to surrounding walls, as will be shown. Also provided in Fig. 1 is the adopted frame of reference, with the downstream direction from the fan tip defined as the positive x-direction. The piezoelectric fan is rigidly mounted on a linear stage that traverses in the x-direction. This is mounted on an optical table and positioned near the center of a large enclosure with dimensions 1.22 m 0.61 m 0.61 m. The nearest enclosure wall in this configuration is over 90 mm away from the fan. Two additional rectangular sidewalls, one on either side of the fan in the x–y plane, are positioned on two independent linear stages (one for each sidewall) that traverses in the z-direction. These sidewalls span the entire length of the piezoelectric fan to ensure full coverage (102 mm 152 mm). The distance of the gap between the sidewalls and the fan blade (δ) is varied by actuating each of the two stages. An illustration of this orientation can be seen in Fig. 2. A laser displacement sensor (LKG-G157) is positioned above the cantilever ( 150 mm along the positive y-axis) to capture the vibration amplitude. This allows for displacement monitoring of the cantilever tip while it oscillates in the y-direction. This signal is then captured via a data acquisition unit from Measurement Computing (USB-1608HS). This data acquisition unit is also used to collect the components needed to calculate the power consumption. Both the current (I) and
Fig. 1. General illustration of the piezoelectric fan with important dimensions.
Please cite this article as: Eastman, A., Kimber, M.L., Aerodynamic damping of sidewall bounded oscillating cantilevers. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.07.016i
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Fig. 2. Illustration of the gap distance and orientation of the sidewalls in relation to the fan.
voltage (V) signals are captured at a sample rate of 1000 Hz, significantly higher than the operating frequencies around 60 Hz. The magnitude of the oscillations is extracted using the power spectral density of the raw signals. Dimensionless numbers are employed to better characterize the general behavior and enable direct comparison with other related works. Those numbers most common for oscillating flows include the frequency parameter (β) and the Keulegan–Carpenter number (KC), defined as
β¼
ωD2 ; 2πν
ð7Þ
and A KC ¼ 2π ; D
ð8Þ
where ν is the kinematic viscosity of the surrounding fluid (air with ν ¼15.9 10 6 m2/s). It should be noted that a cantilever with fixed dimensions is used with operating frequencies within a tight window surrounding the fundamental resonance frequency of 61 Hz. As a result, β in the current study only varies between 600 and 650. For the KC number, this is amplitude dependent and varies for each experiment. For reference, this ranges from approximately 0.7 to 2.4. The final parameter of interest is the dimensionless gap (γ), normalized by the beam width according to:
δ γ¼ : D
ð9Þ
Although the configuration in the current study is different than that for an AFM cantilever, this dimensionless gap is defined in a similar fashion to those studies. 4. Procedure To determine the parameters of interest, a frequency sweep is conducted at multiple voltage levels. This allows for a determination of the damping ratio, the maximum amplitude (Amax) and the resonance frequency input signal magnitude. This is linked to other performance factors like power consumption and phase lag between the voltage and current inputs. Additionally, in order to investigate the effect of phase lag (ϕ) and oscillation frequency on power consumption, a set of constant amplitude measurements are taken. For the constant voltage experiments, the piezoelectric element is provided with a sinusoidal RMS voltage (V) from the function generator at three different levels: 350, 500 and 650 mVrms. After a 50 amplification, the voltage seen by the piezoelectric fan is 17.5, 25 and 32.5 Vrms. When performing a constant amplitude test, the input voltage is adjusted until the average oscillation amplitude over a period of 1 s ( 61 cycles of oscillation) is found to be within 0.005 mm of the target amplitude. Once that amplitude is met, the test is continued. Before the sidewalls are introduced, the static displacement (Fo/k) is determined by operating the fan at 5 Hz (well below the fundamental resonance frequency of 61 Hz) and measuring the oscillation amplitude (A). Additional amplitudes are independently recorded for individual driving frequencies. The remainder of the operating frequencies are varied from as large of a range as 58 to 64 Hz in 0.25 Hz increments. This range is designed as an attempt to encompass the bandwidth Please cite this article as: Eastman, A., Kimber, M.L., Aerodynamic damping of sidewall bounded oscillating cantilevers. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.07.016i
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around the resonance frequency (as low as 70% response compared to the maximum displacement at resonance). The amplitude measurements for the frequency sweep are normalized by their corresponding static displacement, providing multiple data points for the left hand side of Eq. (5), at which point ωn,air and ζair can be determined from a least squares curve fit. The sidewall gaps considered are 1, 2, 3, 4, 6, 8, 10, 15, and 20 mm. This yields a range for γ (see Eq. (9)) between 0.079 and 1.58. A further increase in gap (γ 2.0) yields nearly identical results as γ ¼1.58. Therefore, this is considered to be sufficiently large enough to accurately mimic an isolated condition (i.e., one in the absence of sidewalls).
5. Results and discussion To help illustrate the effect of sidewalls on the raw data, time traces of the voltage supplied and current drawn from piezoelectric element are shown in the far left column of Fig. 3 for the two extreme supply voltages (V¼17.5 and 32.5 Vrms) and the two extreme gaps (δ ¼1 and 25 mm). As expected, the supply voltage is constant for the first two rows, since V is fixed at 17.5 Vrms. The current drawn by the piezoelectric element changes by approximately 1% from δ ¼25 mm (top row) where I¼0.126 mArms to δ ¼1 mm (second row) where I ¼0.122 mArms. The phase difference between the voltage and current signals indicates that the load has a nonzero reactance component, behavior that is entirely expected from a piezoelectric element. When calculating the power draw in such a case, the power is calculated according to the following (Jordan et al., 1999): P ¼ VI cos ϕ ; ð10Þ where I is the RMS current to the system and ϕ is the phase difference between voltage and current input signals. Although difficult to see from the plots in Fig. 3, this phase difference changes slightly from δ ¼25 mm where ϕ ¼ 1.24 rad to δ ¼1 mm where ϕ ¼ 1.33 rad. Although this represents roughly 7% change in the phase difference, the impact is much larger from a power consumption point of view since the angle is in the neighborhood of π/2 rad. For the data just discussed (top two rows of Fig. 3), the power changes by over 30%. Similar trends are observed for the larger of the two voltage inputs (bottom two rows of Fig. 3), namely that voltage and current magnitude remain constant, and a slight change in phase difference causes the power consumption to drop when the cantilever is highly confined. The raw displacement signal of the cantilever tip is shown in the center column of Fig. 3 (amplitude measurement is also included for each case), acquired via the laser displacement sensor, while the far right column is a power spectrum analysis of that amplitude signal to illustrate the frequency components present in the amplitude signal. Facci and Porfiri (2012) have shown for polychromatic excitation, the higher harmonics of beam vibrations can be experimentally observed. In particular, as the amplitude (or KC number) increases, they show that the fundamental resonance frequency becomes increasingly damped. Such a behavior is not observed in the power spectrum analysis shown in Fig. 3. Although the higher harmonics are present for all four scenarios shown, and are observed to relatively increase between V¼ 17.5 Vrms (top row, where KC ¼1.59) and V¼32.5 Vrms (third row, where KC ¼2.39), they are still at least two orders of magnitude smaller than that of the fundamental resonance frequency (61 Hz). This is not completely unexpected, since for the current study, we employ monochromatic excitation (single frequency) for each experiment, the frequency of which is in the close neighborhood of the fundamental resonance frequency. Therefore, nearly all of the power is absorbed at that fundamental mode, regardless of the KC number. Note that although the input signal is fixed for the top two rows at 17.5 Vrms, the amplitude suffers greatly when the cantilever is tightly confined by the sidewalls. An amplitude of A¼3.22 mm in an unconfined state (top row) drops to A¼ 2.05 mm for δ ¼1 mm. To fully quantify the effect of the sidewalls on the dynamic response of the cantilever, data across a range of frequencies must be analyzed. However, the trends discussed in Fig. 3 underpin the phenomenon of interest, namely an increase in viscous drag is seen as the distance between the vibrating cantilever and the sidewall becomes small. The frequency response curves for inputs of V¼17.5 Vrms and 32.5 Vrms cases are shown in Fig. 4(a) and (b), respectively. Note that the amplitude has been normalized by its static displacement (Fo/k). For both cases, the sharpness of the resonance curve decreases as the sidewall gap (δ) becomes small. This increase in damping (or decrease in quality factor) suggests the presence of the wall itself creates some amount of additional viscous drag, resulting in a decrease in vibration amplitude for the same input voltage. Also worth noting in Fig. 4(a) and (b) is the fact that the V¼ 17.5 Vrms case yields a larger normalized amplitude response for any given value of δ. This indicates that the overall damping is also dependent on the input signal magnitude itself, where higher damping is seen for larger input magnitudes. This is attributed to the fact that damping increases when larger vibration amplitudes and velocities occur, and is also consistent with conclusions found in Bidkar et al. (2009) and Porfiri and coworkers (Aureli et al., 2012; Facci and Porfiri, 2013), which reveal the same trends for aerodynamic damping without sidewalls. Although the current study only quantifies the average value for damping and does not separately express the structural and aerodynamic quantities, it nonetheless captures the expected trend of amplitude dependent damping. It is also apparent that for very small values of δ, a shift in resonance frequency occurs. This is in the direction of a lower frequency rather than an increase. This trend can be better observed in Fig. 5, which represents the natural frequency extracted from the curve fit (using Eq. (5)) of the frequency response data. This indicates that the effect of added mass (due to the sidewalls) only has a measurable effect when the sidewalls are in very close proximity to the fan. However, the drift in resonance frequency as the wall gap increases is quite small (less than 1%). It should also be noted that although the resonance frequency is slightly different depending on the voltage input (three separate curves in Please cite this article as: Eastman, A., Kimber, M.L., Aerodynamic damping of sidewall bounded oscillating cantilevers. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.07.016i
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Fig. 3. Time traces of voltage, current, and tip displacement for a driving frequency of 61 Hz: V ¼ 17.5 Vrms and δ ¼25 mm (top row), V ¼ 17.5 Vrms, and δ ¼1 mm (second row), V ¼ 32.5 Vrms and δ ¼ 25 mm (third row), and V ¼ 32.5 Vrms and δ ¼ 1 mm (bottom row).
Fig. 5), this difference is less than 0.2%. Therefore, the fluid–structure interaction that takes place can almost exclusively be attributed to a change in viscous damping, or the quality factor. The trend of the quality factor as a function of δ can be seen in Fig. 6 for all three voltage inputs. For an approximate gap of δ ¼10 mm, the effect of the sidewalls is negligible, and no meaningful benefit or drawback is realized when the spacing increases beyond this value. Below this spacing, the general trend is what would be expected as the reduction in gap should cause an increase of damping to the beam. It is interesting to note that although the relative orientation of the beam with Please cite this article as: Eastman, A., Kimber, M.L., Aerodynamic damping of sidewall bounded oscillating cantilevers. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.07.016i
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Fig. 4. Comparison of the normalized amplitude response for the (a) V ¼ 17.5 Vrms and (b) V ¼ 32.5 Vrms case.
Fig. 5. The progression of the natural frequency as the sidewall gap is varied.
respect to the solid wall is unique in the current study, similarities can be drawn from important related works. For example, AFM cantilevers have prompted a large number of analytical, numerical, and experimental studies aimed at quantifying the squeeze film damping effect of the fluid between the beam and the solid surface (Green and Sader, 2005a; Grimaldi et al., 2012 to Please cite this article as: Eastman, A., Kimber, M.L., Aerodynamic damping of sidewall bounded oscillating cantilevers. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.07.016i
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Fig. 6. The quality factor for each sidewall gap distance.
Fig. 7. Curve fit of the quality factor normalized by the isolated quality factor.
name a few). The solid wall in that application is rotated by 901 compared to the present configuration. However, the qualitative trends are the same, namely the viscous damping increases when an oscillating beam approaches any solid surface, regardless of the relative orientation. For the data collected, the quality factor at δ ¼1 mm is approximately 35% of the respective isolated values for V¼17.5, 25, and 32.5 Vrms. One important tradeoff in a number of applications is the benefit of introducing sidewalls in order to limit the loss of fluid from the side edges of the oscillating fan, balanced with the drawback of increased damping which must be overcome. The decrease in quality factor (and thus increase in damping) means that more power would be required to maintain the same vibration amplitude. It should also be noted that the difference in the isolated quality factor (Qair at δ ¼20 mm) between the V¼17.5 and 25 Vrms cases is larger than the difference between the V¼25 and 32.5 Vrms cases. The damping plays a role in limiting the amplitude, as a slightly higher amplitude will also yield a higher oscillation velocity. As it can be gathered from an understanding of drag, as the velocity increases it causes the drag resistance to increase. This has also been demonstrated in other studies using similar systems by Kimber et al. (2009a), Bidkar et al. (2009), and Porfiri and coworkers (Aureli et al., 2012; Facci and Porfiri, 2013). This further reinforces that the non-linear response compared to the linear voltage input is not a product of some unknown systematic error. In order to develop a way to predict the quality factor in this type of system, a curve fit is applied to the quality factor data after being normalized by the isolated quality factor (Qiso is equal to Qair at δ ¼20 mm) in each case. This resulting behavior can be captured with the following exponential relationship: Q δ : ð11Þ ¼ 1 C 1 exp C 2 Q iso D A least squares curve fit of the data yields 0.51 and 4.78 for C1 and C2, respectively. The results of this fit can be seen in Fig. 7 with a maximum absolute error of 6.1%. This result allows for an accurate prediction of the resulting quality factor based on sidewall spacing for a set voltage input. It is stressed that this expression (Eq. (11)) is only applicable for the beam geometry and frequency range considered with 600 o β o650 and 0.7oKCo2.4. Once additional cantilevers are tested that Please cite this article as: Eastman, A., Kimber, M.L., Aerodynamic damping of sidewall bounded oscillating cantilevers. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.07.016i
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cover a larger range of beam geometries, a more comprehensive analysis can be conducted. It is also noted that this provides the effective, or in some sense, the average damping as a function of the dimensionless spacing. Despite these limitations, the current efforts show promise in ultimately predicting the viscous damping in the presence of sidewalls from first principles in a fashion similar as Grimaldi et al. (2012) did for an AFM cantilever-related orientation. As a prevailing interest in this paper is damping and its effect on performance, it is also important to quantify the power consumption, a metric of great concern for many potential applications (e.g., electronics cooling), and is determined using Eq. (10). The power calculations are conducted for all gaps at the three voltage inputs previously considered, but fixing the driving frequency at 61 Hz. The result is shown in Fig. 8, where it can be seen that the power actually decreases for small gap distances at constant voltage inputs. Since the supply voltage remains fixed for a given set of data in Fig. 8, only changes in current and/or phase difference can explain the drop in power consumption. The corresponding RMS current values for this data are shown in Fig. 9, which reveal a trend similar to the power data, but the decrease in current is less than 2.5%, whereas the drop in power consumption can be roughly 30%, compared to their respective isolated (large δ) values. Therefore, a large portion of this drop in power should be explained by a change in the phase angle. As seen in Fig. 10, the phase lag is in the range of 1.24 to 1.35 rad. These values are fairly close to π/2 rad, which means that any small change in phase will result in a much larger change in power due to the fact that the cosine of this phase difference is the quantity used in the power calculations. Essentially, the load of the piezoelectric fan has a large imaginary component or electrical reactance. When this is taken into account, it becomes apparent that a change in phase from a gap of 1–25 mm can result in a roughly 25% change in power consumption if everything else were to remain constant. In fact, ϕ changes quite dramatically and does so in a very similar manner to that of the amplitude response of the cantilever, readily apparent in Fig. 11, which can be directly compared to Fig. 4(a), since voltage input is the same (V¼17.5 Vrms). The peak of the phase curve changes with sidewall gap and appears to occur at the same frequency as the resonance frequencies seen from Fig. 4. Although one might suppose an advantage exists since power consumption is reduced, this comes at a cost of lower vibration amplitudes due to an increase in aerodynamic damping. To illustrate this point, consider the peak amplitude
Fig. 8. The power requirement of the piezoelectric fan at 61 Hz as the gap is varied.
Fig. 9. The current as a function of sidewall gap for set voltage inputs.
Please cite this article as: Eastman, A., Kimber, M.L., Aerodynamic damping of sidewall bounded oscillating cantilevers. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.07.016i
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Fig. 10. The phase difference between the current and voltage input at 61 Hz for a set voltage inputs.
Fig. 11. The phase lag between the voltage and current input as a function of frequency for V ¼17.5 Vrms.
Fig. 12. The maximum oscillation amplitude for each sidewall gap distance.
(Amax) for all 9 gaps and each unique input signal magnitude, which is extracted from the frequency response curves and is shown in Fig. 12. For reference, the corresponding KC numbers for this peak amplitude data range from 1.04 to 1.58 for V¼17.5 Vrms, 1.32 to 2.01 for V¼25.0 Vrms, and 1.59 to 2.40 for V ¼32.5 Vrms. It is useful to compare the amplitude measurements from Fig. 12 and the power consumption from Fig. 8. The trends from both of these figures reveal that both the vibration amplitude and power consumption decrease as δ becomes smaller. In order to further evaluate the power Please cite this article as: Eastman, A., Kimber, M.L., Aerodynamic damping of sidewall bounded oscillating cantilevers. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.07.016i
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Fig. 13. The power requirement to the piezoelectric fan with a fixed amplitude as the frequency is changed within the bandwidth (δ ¼ 25 mm).
consumption trends working both on and off resonance, additional experiments are performed at fixed amplitudes of 4, 5, and 6 mm with δ ¼ 25 mm and varying the input frequency across the bandwidth of the frequency response. The result is shown in Fig. 13, where it is apparent that the change in phase lag (ϕ) when operating off resonance cannot adequately make up for the increased power input required to keep the amplitude consistent across the frequency spectrum. For the 4 mm amplitude, the curve is flatter indicating that the phase does have some effect. However, in order to see a potentially flat power curve, the amplitude would need to be so small that there would be no practical use in this setting. Regardless, this finding demonstrates that operating at resonance is, as one would intuitively expect, the best option in terms of minimizing the power requirements. Further experiments should be conducted where the application-specific performance metrics (e.g., thrust or thermal convection) are evaluated along with the input to truly gauge the tradeoff that must be made between the benefits from sidewalls in terms of preventing flow being lost across the side edges compared to the loss of performance due to a decrease in amplitude and/or the increase in power consumption in order to maintain a specific amplitude.
6. Conclusions Frequency response curves are experimentally measured for a piezoelectric fan vibrating near two sidewalls in order to quantify the effect of enclosure walls on the oscillation characteristics of the fan. Three voltage inputs are considered at multiple sidewall gaps. The natural frequency is shown to decrease with proximity to the side wall suggesting an increase in the added mass from the surrounding fluid. For the damping characteristics, as the proximity of the sidewalls to the fan is decreased, the quality factor is found to also decrease. This means that as one seeks to improve the flow shaping by introducing sidewalls, the inevitable damping must be addressed as it becomes more and more of a factor. A curve fit is applied to the quality factor, showing good agreement with experimental data. This gives a basis for predicting the viscous damping as a function of the sidewall gap, and suggests a more comprehensive study would be worthwhile where additional factors are investigated (e.g., beam width, length, operating frequency, etc.). The effect of the sidewalls on damping is in general much larger than the impact on the resonance frequency. In addition, the damping begins to change from its isolated value at a much larger gap compared to the gap where the resonance frequency begins to shift. It has been demonstrated in this paper that, for a constant voltage input, the change in power requirement is primarily a result of the phase difference between the voltage and current input. This change in the phase difference is driven entirely by the damping conditions imposed by the sidewalls or frequency shift from resonance. However, it does not have a pronounced enough effect to meaningfully limit the power increase needed in order to maintain a specific amplitude. The findings here provide the basis for understanding and predicting the effect of sidewalls on damping and impact on power consumption. This allows for evaluation of the operational characteristics and limitations for an oscillating cantilever when introduced into an enclosed space. Fundamentally, higher damping limits the amplitude for a specific voltage input. Additional studies are currently underway to quantify the effect of other performance metrics (e.g., thrust or thermal performance) in terms of proximity to sidewalls and measuring power consumption while holding the amplitude fixed. References Acikalin, T., Garimella, S.V., Raman, A., Petroski, J., 2007. Characterization and optimization of the thermal performance of miniature piezoelectric fans. International Journal of Heat and Fluid Flow 28, 806–820. Acikalin, T., Wait, S.M., Garimella, S.V., Raman, A., 2004. Experimental investigation of the thermal performance of piezoelectric fans. Heat Transfer Engineering 25, 4–14.
Please cite this article as: Eastman, A., Kimber, M.L., Aerodynamic damping of sidewall bounded oscillating cantilevers. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.07.016i
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Please cite this article as: Eastman, A., Kimber, M.L., Aerodynamic damping of sidewall bounded oscillating cantilevers. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.07.016i