LUMPED ELEMENT MODELING OF PIEZOELECTRIC-DRIVEN ... - Fcla

LUMPED ELEMENT MODELING OF PIEZOELECTRIC-DRIVEN SYNTHETIC JET ACTUATORS FOR ACTIVE FLOW CONTROL

By QUENTIN GALLAS

A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2002

Copyright 2002 By Quentin Gallas

ACKNOWLEDGMENTS Financial support for the research project was provided by a NASA-Langley Research Center Grant and an AFOSR grant. First, I would like to thank my advisor, Dr. Louis N. Cattafesta, for his continual guidance and support, which gave me the motivation and encouragement needed to make this work possible. I would also like to express my gratitude to Dr. Mark Sheplak and Dr. Bhavani Sankar for advising and guiding me with various aspects of the project. I thank all the members of the Interdisciplinary Microsystems group, particularly fellow students Jose Mathew, Ryan Holman, Anurag Kasyap and Steve Horowitz, for their help with my research and their friendship. Finally, special thanks go to my family and friends for always encouraging me to pursue my interests and for making that pursuit possible.

iii

TABLE OF CONTENTS Page ACKNOWLEDGMENTS..................................................................................................iii TABLE OF CONTENTS ................................................................................................... iv LIST OF TABLES .............................................................................................................vi LIST OF FIGURES...........................................................................................................vii ABSTRACT ........................................................................................................................ x CHAPTER 1 INTRODUCTION........................................................................................................... 1 2 LUMPED ELEMENT MODELING .............................................................................. 4 3 EQUIVALENT CIRCUIT ANALYSIS.......................................................................... 8 3.1 3.2

CaC → 0 .................................................... 11 β CaD C Case 2: Rigid Diaphragm Limit β = aD → 0 ................................................. 13 CaC

Case 1: Incompressible Limit

1

=

4 MODEL PARAMETER ESTIMATION ...................................................................... 15 5 MODEL VERIFICATION AND PARAMETER EXTRACTION .............................. 22 5.1 5.2 5.3

Piezoelectric Transduction ................................................................................ 23 Cavity Acoustic Compliance............................................................................. 25 Acoustic Mass and Resistance in Orifice.......................................................... 26

6 COMPARISON BETWEEN MODEL & EXPERIMENTS......................................... 28 6.1 6.2 6.3

Laser Doppler Velocimetry System .................................................................. 29 Data Acquisition................................................................................................ 30 Results ............................................................................................................... 31

7 CONCLUSIONS AND FUTURE WORK ................................................................... 35 REFERENCES.................................................................................................................. 37

iv

APPENDIX A COMPOSITE PIEZOELECTRIC DIAPHRAGM CHARACTERIZATION ............. 40 B SYNTHETIC JET ACTUATORS MEASURMENTS ................................................ 50 C AMPLIFIER CHARACTERIZATION........................................................................ 61 BIOGRAPHICAL SKETCH ............................................................................................ 63

v

LIST OF TABLES Table

page

5-1: Piezoceramic diaphragm characteristics................................................................... 23 5-2: Calculated lumped element model parameters ......................................................... 24 6-1: Specifications of piezoceramic diaphragms. ............................................................ 28 6-2: Calculated lumped parameters of piezoceramic diaphragms. .................................. 29 6-3: Geometry of synthetic jet actuators. ......................................................................... 29 6-4: Uncertainty in maximum velocity at select frequencies for Case II......................... 31 6-5: Comparison between lumped element model and experiment. ................................ 34 A-1: Piezoelectric diaphragm properties.......................................................................... 40 A-2: Experimental values of natural frequency and dc response of tested piezoelectric diaphragms................................................................................ 41 B-1: Geometry of modular synthetic jet actuators. .......................................................... 50

vi

LIST OF FIGURES Figure

page

1–1: Schematic of a piezoelectric-driven synthetic jet....................................................... 1 2–1: Equivalent circuit representation of a piezoelectric-driven synthetic jet................... 6 3–1: Alternative equivalent circuit model.......................................................................... 8 4–1: Variation in velocity profile vs. St = 1, 10, and 30 for oscillatory channel flow in a circular duct. ..................................................................................................... 19 4–2: Ratio of average velocity to centerline velocity vs. St for oscillatory channel flow in a circular duct. ............................................................................................. 20 4–3: Acoustic resistance and reactance normalized by steady value vs. St for oscillatory channel flow in a circular duct............................................................... 21 5–1: Assembly diagram of modular synthetic jet............................................................. 22 5–2: Comparison between predicted and measured response of piezoceramic diaphragms to a sinusoidal excitation voltage at 100 Hz......................................... 25 5–3: Measured acoustic compliance CaC vs. frequency in closed cavity of synthetic jet. ............................................................................................................................ 26 6–1: Laser Doppler Velocimeter system setup for Case I................................................ 30 6–2: Comparison between the lumped element model and experiment for Case I.......... 32 6–3: Comparison between the lumped element model and experiment for Case II. ....... 33 A–1: Frequency response (magnitude, phase and coherence) of the center of two sample piezoceramic diaphragms - Case I. (Amplitude input voltage=1V)............ 42 A–2: Frequency response (magnitude, phase and coherence) of the center of piezoceramic diaphragm - Case I for different boundary conditions. (Amplitude input voltage=1V)................................................................................. 43

vii

A–3: Frequency response (magnitude, phase and coherence) of the center of piezoceramic diaphragm - Case II for different boundary conditions. (Amplitude input voltage=1V)................................................................................. 44 A–4: Frequency response (magnitude, phase and coherence) of the center of piezoceramic diaphragm - Case III for different boundary conditions. (Amplitude input voltage=1V)................................................................................. 45 A–5: Frequency response of piezoceramic diaphragm - Case I. (Amplitude input voltage=1V) ............................................................................................................. 46 A–6: Frequency response of piezoceramic diaphragm - Case II. (Amplitude input voltage=1V) ............................................................................................................. 47 A–7: Frequency response of piezoceramic diaphragm - Case III. (Amplitude input voltage=1V) ............................................................................................................. 47 A–8: Comparison between two boundary condition types applied to piezoceramic diaphragm - Case I driven by a sinusoidal excitation voltage of 1V at 300 Hz. ..... 48 A–9: Comparison between the response of two sample piezoceramic diaphragms Case III driven by a sinusoidal excitation voltage of 1V at 500 Hz. The piezoceramic of sample 1 is not centered on the brass diaphragm. ......................... 49 B–1: Laser Doppler Velocimeter system setup for Case II having cavity volume V3..... 51 B–2: Cavity volume change in Case I (model prediction) ............................................... 52 B–3: Cavity volume change in Case II (model prediction).............................................. 53 B–4: Comparison between the lumped element model and experiment for synthetic jet having diaphragm - CaseII, cavity volume V3, and orifice 1.............................. 54 B–5: Comparison between the lumped element model and experiment for synthetic jet having diaphragm - CaseII, cavity volume V3, and orifice 2.............................. 55 B–6: Comparison between the lumped element model and experiment for synthetic jet having diaphragm - CaseII, cavity volume V3, and orifice 3.............................. 56 B–7: Comparison between the lumped element model and experiment for synthetic jet having diaphragm - CaseII, cavity volume V3, and orifice 4.............................. 57 B–8: Comparison between the lumped element model and experiment for synthetic jet having diaphragm - CaseII, cavity volume V3, and orifice 5.............................. 58 B–9: Comparison between the lumped element model and experiment for synthetic jet having diaphragm - CaseII, cavity volume V3, and orifice 6.............................. 59

viii

B–10: Mean jet velocity response function of input voltage for Case II with cavity volume V3 at f=400Hz ............................................................................................. 60 C–1: Amplifier gain linearity at constant frequency........................................................ 61 C–2: Amplifier gain function of frequency...................................................................... 62

ix

Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science

LUMPED ELEMENT MODELING OF PIEZOELECTRIC-DRIVEN SYNTHETIC JET ACTUATORS FOR ACTIVE FLOW CONTROL

By Quentin Gallas August 2002 Chairman: Dr. Louis N. Cattafesta ΙΙΙ Major Department: Aerospace Engineering, Mechanics, and Engineering Science This thesis presents a lumped element model of a piezoelectric-driven syntheticjet actuator. A synthetic jet, also known as a zero net mass-flux device, uses a vibrating diaphragm to generate an oscillatory flow through a small orifice or slot. In lumped element modeling, the individual components of a synthetic jet are modeled as elements of an equivalent electrical circuit using conjugate power variables.

The frequency

response function of the circuit is derived to obtain an expression for the volume flow rate through the orifice per applied voltage across the piezoceramic. The circuit is analyzed to provide physical insight into the dependence of the device behavior on geometry and material properties.

Methods to estimate the model parameters are

discussed along with pertinent model assumptions, and experimental verification is presented of the lumped parameter models. In addition, two prototypical synthetic jet

x

actuators are built and tested. Very good agreement is obtained between the predicted and measured frequency response functions.

xi

CHAPTER 1 INTRODUCTION Synthetic jet actuators have been the focus of significant research activity for the past decade [1]. The interest in synthetic jets is primarily due to their utility in flow control applications, such as separation control, mixing enhancement, etc. [2-6].

A

schematic of a synthetic jet actuator is shown in Figure 1–1. A typical synthetic jet, also known as a zero net mass-flux device, uses a vibrating diaphragm to drive oscillatory flow through a small orifice or slot. Although there is no flow source, a mean jet flow is established a few diameters from the orifice due to the entrained fluid.

net flow orifice

cavity

vibrating piezoceramic composite diaphragm

Figure 1–1: Schematic of a piezoelectric-driven synthetic jet.

1

2 In addition to studies that emphasize applications, there are numerous others that have concentrated on the design, visualization, and/or measurements of synthetic jets [710]. Several computational studies also have focused on fundamental aspects of these devices [11-14]. Crook and Wood [9] emphasize the importance of understanding the scaling and operational characteristics of a synthetic jet. Clearly, this information is required to design an appropriate device for a particular application.

In addition,

feedback control applications require the actuator transfer function that relates the input voltage to the output property of interest (e.g., volumetric flow rate, momentum, etc.) in the control system.

The analysis and design of coupled-energy domain transducer systems are commonly performed using lumped element models [15-17]. The main assumption employed in lumped element modeling (LEM) is that the characteristic length scales of the governing physical phenomena are much larger than the largest geometric dimension.

For example, in an acoustic system, the acoustic

wavelength must be significantly larger than the device itself. If this assumption holds, then the temporal and spatial variations can be decoupled, and the governing partial differential equations for the distributed system can be “lumped” into a set of coupled ordinary differential equations [17]. This approach provides a simple method to estimate the low-frequency dynamic response of a system for design and control-system implementation. The purpose of this thesis is to rigorously study the application of LEM to piezoelectric-driven synthetic jet actuators. To the author’s knowledge, this represents the first application of LEM to piezoelectric-driven synthetic jets. McCormick [18]

3 employed LEM to a speaker-driven synthetic jet, which is an electromagnetic transducer, to relate the voltage input to the output volume velocity of the actuator. Rathnasingham and Breuer [19] were the first to develop a low-order model of a synthetic jet, using a control-volume model for the flow and an empirical model for the structural dynamics of the diaphragm. In this thesis, the various lumped elements for each component of a synthetic jet are theoretically developed. The resulting equivalent circuit is then analyzed to understand the effects of geometry and material properties on important design parameters, such as resonance frequency and volume displacement per applied voltage. The model assumptions and limitations are discussed, along with the results of experiments designed to assess the validity of this modeling approach.

CHAPTER 2 LUMPED ELEMENT MODELING In LEM, the coupling between the various energy domains is realized by using equivalent two-port models of the physical system. An equivalent circuit model is constructed by lumping the distributed energy storage and dissipation into ideal generalized one-port circuit elements. In an electroacoustic system, differential pressure and voltage are effort variables, while current and volumetric flow rate are flow variables. In this thesis, an impedance analogy is employed, in which elements that share a common effort are connected in parallel, while those sharing a common flow are connected in series. For a synthetic jet, three different energy domains are involved, electrical, mechanical, and fluidic/acoustic. The electromechanical actuator consists of an axisymmetric PZT patch bonded to a clamped metal diaphragm. The composite diaphragm is driven into motion via an applied ac voltage. The primary purpose of the piezoelectric composite diaphragm is to produce large volume displacements in order to force fluid into and out of the cavity, which represents a conversion from the mechanical to the acoustic/fluidic energy domain. Consequently, the frequency range of the analysis is limited from dc to somewhat beyond the fundamental frequency of the composite diaphragm but less than the natural frequency of any higher-order modes [20]. Linear composite plate theory is used to obtain the short-circuit pressure-deflection characteristics.

Then, the diaphragm is

lumped into an equivalent acoustic mass and acoustic compliance. The former represents

4

5 stored kinetic energy, and the latter models stored potential energy.

Similarly, the

electromechanical transduction characteristics are determined by the unloaded or “free” voltage-deflection characteristics.

The piezoelectric electromechanical coupling is

lumped into an effective acoustic piezoelectric coefficient. In general, the cavity contains a compressible gas that stores potential energy and is therefore modeled as an acoustic compliance. Viscous effects in the orifice dissipate a portion of the kinetic energy stored in the motion of the oscillating fluid mass. Therefore, there will be an effective acoustic mass and acoustic resistance associated with the orifice neck. Flow through the orifice produces losses associated with the discharge of flow from the jet exit. An acoustic radiation impedance must also be added if the fluid is ejected into a semi-infinite medium. The equivalent circuit representation for the synthetic jet is shown in Figure 2–1. In the notation below, the first subscript denotes the domain (e.g., “a” for acoustic and “e” for electric), and the second subscript describes the element (e.g., “D” for diaphragm). In the electrical domain, Ceb is the blocked electrical capacitance of the piezoelectric diaphragm. The term blocked is used since it is the impedance seen by the source when the diaphragm motion is prevented. Although not shown here, a resistor can be introduced in series or in parallel with Ceb to represent the dielectric loss in the piezoceramic (see Rossi [17], p. 358).

6

C aD

I 1 1: φ a I V ac

I-I 1 C eb

R aD

Q

M aD

R aN

Qc

M aN

Q out R aO

C aC

P

M aRad e le ctrica l d om a in

a co u stic/fluid ic d o m a in

Figure 2–1: Equivalent circuit representation of a piezoelectric-driven synthetic jet. In the acoustic domain, CaD and M aD are the short-circuit acoustic compliance and mass of the piezoceramic composite diaphragm, respectively, and RaD is the structural damping in the acoustic domain.

Also, a radiation impedance must be

included, if the back-side of the diaphragm is radiating into an open medium. CaC is the acoustic compliance of the cavity, while RaN and M aN are the acoustic resistance and mass of the fluid in the neck, respectively. Finally, RaO is the nonlinear resistance associated with the orifice discharge, and M aRad is the acoustic radiation mass of the orifice. In this thesis, it is assumed that the synthetic jet exhausts into a semi-infinite ambient air medium, and that the diaphragm is not subject to a mean differential pressure. If necessary, a vent channel can be used to equilibrate the mean static pressure across the diaphragm, in a manner similar to a microphone [21]. For simplicity, it is assumed that there is no grazing flow, and compressibility effects in the orifice are neglected.

7 The structure of the equivalent circuit is explained as follows. An ac voltage Vac is applied across the piezoceramic to create an effective acoustic pressure that drives the diaphragm into motion. This represents a conversion from the electrical to the acoustic domain (by-passing the mechanical domain via integration to produce a volume displacement) and is accounted for via a transformer possessing a turns ratio φa with units of [ Pa V ]. An ideal transformer converts energy from one domain to another without losses and obeys the relations I1 = φa Q and Vac =

P

φa

.

{1}

In addition, a transformer converts an electrical impedance Z e to an acoustic impedance

Z a via P P φa Vac Q Z = = 2 = a2 . Ze = φa I1 φa Q φa

{2}

The motion of the diaphragm can either compress the fluid in the cavity or can eject/ingest fluid through the orifice. Physically, this is represented as a volume velocity divider, Q = Qc + Qout . The goal of the design is to maximize the magnitude of the volume flow rate through the orifice per applied voltage Qout Vac .

CHAPTER 3 EQUIVALENT CIRCUIT ANALYSIS Before estimating the lumped parameters defined above, it is instructive to analyze the equivalent circuit to obtain the frequency response function Qout ( s ) Vac ( s ) , where s = jω . Using Eq. {2}, the transformer can be eliminated by converting each of the acoustic impedances to their electrical equivalent. The result is depicted in Figure 3– 1, where

1  1  R + sM aD + , 2  aD φa  sCaD  1  1  Z eC ( s ) = 2   , and φa  sCaC 

Z eD ( s ) =

{3}

( RaN + RaO ) + s ( M aN + M aRad )  Z eO ( s ) =  2

φa

are the electric impedance of the diaphragm, cavity, and orifice, respectively.

φaQ I Vac

φaQout

ZeD

φa(Q-Qout)

I-φaQ

ZeO

ZeC

Ceb

Figure 3–1: Alternative equivalent circuit model.

8

9 Substituting in the expressions for Z eD , Z eC , and Z eO and grouping powers of s in the numerator and denominator results in Qout ( s ) Vac ( s )

=

φa CaD s a4 s + a3 s 3 + a2 s 2 + a1s + 1 4

,

{4}

where a1 = CaD ( RaO + RaN + RaD ) + CaC ( RaO + RaN ) ,

a2 = CaD ( M aRad + M aN + M aD ) + CaC ( M aRad + M aN ) + CaC CaD RaD ( RaO + RaN ) , a3 = CaC CaD  M aD ( RaO + RaN ) + ( M aRad + M aN ) RaD  , and

{5}

a4 = CaC CaD M aD ( M aRad + M aN ) .

Although this expression is somewhat complicated, it reveals some important features without having to estimate any of the parameters in Eq. {5}. For the purpose of this discussion, these parameters can be thought of as constants, although in reality some are likely to exhibit frequency and amplitude dependence (i.e., due to nonlinear effects). For a dc voltage ( s = 0 ), the volume velocity is zero. At low frequencies ( s → 0 ), the volume velocity is proportional to d aVacω since the transduction factor is defined as

φa = d a CaD , where d a is an effective acoustic piezoelectric coefficient defined below — see Eq.{22}. This result emphasizes the need to optimize the design of the piezoceramic composite diaphragm [20] and also indicates that Qout and Vac are 90° out of phase at low frequencies. At high frequencies ( s → ∞ ), Eq. {4} becomes

Qout da . = Vac CaC CaD M aD ( M aN + M aRad ) s 3

{6}

10 The output therefore decreases at a rate of 60 dB/decade and is inversely proportional to the product of the masses and compliances in the system. The denominator in Eq. {4} is a 4th order polynomial in s , indicating two resonance frequencies.

A compact analytical expression for the two resonance

frequencies is desirable. The first and second resonance frequencies, f1 and f 2 , are controlled by, but are not identical to, the short-circuit piezoelectric diaphragm natural frequency,

fD =

1 2π

1 , M aD CaD

{7}

( M aN

1 , + M aRad ) CaC

{8}

and the Helmholtz resonator frequency, fH =

1 2π

with the constraint that f1 f 2 = f D f H .

{9}

The measure of the coupling of a system is a function of the mass or compliance ratio [15]. Assuming small damping, equating the denominator in Eq. {4} with a 4thorder system and using the definition given in Eq. {7} and Eq. {8} yields the following relationship between the natural frequencies:

(f

2 2

− f D2 ) + ( f12 − f H2 ) f

2 H

 M + M aRad =  aN M aD 

  =α , 

{10}

where α is defined as the ratio of the orifice masses to the diaphragm mass. Similarly, a compliance ratio β can be defined as

11

(f

2 2

− f D2 ) + ( f12 − f H2 ) f

2 D

=

CaD =β. CaC

{11}

Given the compliance ratio, which is usually easier to estimate, one can obtain the following quadratic formula for ψ = fi 2 :

ψ 2 −  f D2 (1 + β ) + f H2 ψ + f D2 f H2 = 0 .

{12}

The two roots of Eq. {9} are the square of the natural frequencies of the synthetic jet, f12 and f 22 . In the next section, two important cases are examined to gain physical insight into the behavior of the device under different limiting conditions.

3.1

Case 1: Incompressible Limit

1

β

=

CaC →0 CaD

Assuming that the fluid is an isentropic, ideal gas, the acoustic cavity compliance, is obtained from the cavity volume V0 , gas density ρ0 , and the isentropic speed of sound c0 via CaC =

V0 . ρ 0c0 2

{13}

In practice, CaC CaD → 0 is achieved by minimizing the cavity volume or operating in a liquid medium. Since the coefficients a3 and a4 in Eq. {5} are both proportional to CaC , the synthetic jet transfer function reduces to the 2nd-order system,

12

Qout ( s ) Vac ( s )

=

da s a2′

a′ 1 s2 + 1 s + a′ a′ 2

,

{14}

2

where the prime denotes ai in the limit of CaC CaD → 0 . The denominator in Eq. {14} is written in the form of a canonical 2nd-order system, s 2 + 2ζω n s + ω n2 . By inspection, the natural frequency and damping ratio for the

incompressible case are given by

ωincomp =

1  M + M aRad  CaD M aD 1 + aN  M aD  

{15}

and

ζ incomp =

CaD 1 . ( RaO + RaN + RaD ) M aN + M aRad + M aD 2

{16}

If M aD ! M aN + M aRad , then the natural frequency of the synthetic jet actuator equals that of the diaphragm. At resonance, the response is proportional to the effective acoustic piezoelectric coefficient and is limited by the resistances of the circuit and the acoustic compliance of the diaphragm Qout da 1 . = Vac CaD RaO + RaN + RaD

{17}

13

3.2

Case 2: Rigid Diaphragm Limit β =

CaD →0 CaC

As described in Prasad et al. [20], the size of the piezoceramic patch is not negligible compared to the metal diaphragm for high actuation performance. Therefore, the piezoceramic composite diaphragm cannot accurately be modeled as a homogeneous circular plate.

Nonetheless, assuming that the diaphragm is clamped, the acoustic

compliance of a homogeneous clamped circular plate provides insight into the scaling behavior of the diaphragm

CaD =

π a 6 (1 −ν 2 ) 16 Eh3

,

{18}

where a is the radius, E is the elastic modulus, ν is Poisson’s ratio, and h is the thickness. From Eq. {18}, CaD decreases with decreasing the ratio a h and increasing elastic modulus. As in the previous case, the coefficients a3 and a4 in Eq. {5} are zero, and the synthetic jet transfer function reduces to a 2nd-order system. The limit CaD CaC → 0 leads to the following expressions for the natural frequency, damping ratio, and response at resonance:

ω stiff =

ζ stiff =

( M aN

1 , + M aRad ) CaC

CaC 1 , ( RaO + RaN ) M aRad + M aN 2

{19}

{20}

14 and Qout da 1 = . Vac CaC RaO + RaN

{21}

In this case, the natural frequency of the jet corresponds to the resonant frequency of the Helmholtz resonator. At resonance, the response is limited by the orifice flow resistances and the cavity compliance.

By comparing with Eq. {17}, the resonant

response differs for these cases by the ratio of the acoustic compliances, and by the contribution of the acoustic resistance of the diaphragm in the incompressible case.

CHAPTER 4 MODEL PARAMETER ESTIMATION In this section, the methods and assumptions used to estimate each of the quantities in Eq. {5} are outlined. The interested reader is referred to Merhaut [16], Rossi [17], and Beranek [22] for details of the methodology. The details of the two-port model for the piezoceramic plate can be found in Prasad et al. [20]. The piezoelectric diaphragm vibrates in response to both an applied ac voltage and oscillatory differential pressure according to the relation Q = jω ( d aVac + CaD P ) ,

{22}

where d a = ( Q jωVac ) P =0 is defined as the effective acoustic piezoelectric coefficient that relates the volume velocity Q of the diaphragm to the applied voltage Vac , and CaD = ( Q jω P ) V

ac = 0

= ( ∆Volume P ) V

ac = 0

is the short-circuit acoustic compliance that

relates an applied differential pressure to the volume displacement of the diaphragm. The vertical deflection w ( r ) due to an applied differential pressure is lumped into an equivalent acoustic mass M aD by equating the lumped kinetic energy of the vibrating diaphragm to the total kinetic energy using

ρ ′′ ( r ) 1 2 M aD Q 2 = ∫ w" ( r ) 2π rdr , 2 2 0 a

15

{23}

16 where ρ ′′ ( r ) is the distributed mass per unit area, Q is the net volume velocity of the diaphragm, a is the radius of the diaphragm, and w" ( r ) = jω w ( r ) is the distributed vertical velocity.

All of these parameters are calculated via linear composite

piezoelectric plate theory (see Prasad et al. [20] for details). The acoustic resistance RaD represents the losses due to damping effects in the diaphragm and is given by RaD = 2ζ

M aD , CaD

{24}

where ζ is the damping coefficient that is determined experimentally. The blocked electrical capacitance Ceb in Figure 2–1 is related to the free electrical capacitance of the piezoceramic Cef = ε Ap / hp by  d a2  , Ceb = Cef (1 − κ 2 ) = Cef  1 −  C C  ef asD  

{25}

where κ 2 is the electroacoustic coupling factor, ε is the dielectric constant, Ap is the piezoceramic area, and hp is the thickness of the piezoceramic patch. However, Ceb does not appear in Eq. {4} and is therefore not required for the present analysis. However, it should be noted that the piezoceramic dielectric properties play a major role in determining the electric power requirements of the actuator. The acoustic impedance of the cavity is given by [23] Z aC =

ρ0c0 cot ( kD ) jS

,

{26}

17 where D is the depth, k = ω c0 is the acoustic wavenumber, and S is the cross-sectional area of the cavity. Since kD # 1 , the Maclaurin series is truncated after the first term to yield [23] Z aC ≅

ρ0c02 ρ c2 1 = 0 0 = . jω SD jωV0 jω CaC

{27}

At low frequencies, the acoustic resistance of the neck is obtained assuming fullydeveloped laminar pipe flow in the neck of length L and radius a0 (see Beranek [22] p. 135) RaN =

∆Pout 8µ L , = Qout π a0 4

{28}

where µ is the viscosity of the fluid and Qout is the volume flow rate produced by the differential pressure ∆Pout . Using the same assumption of fully-developed pipe flow, the acoustic mass in the neck is obtained by integrating the distributed kinetic energy and equating it to the lumped kinetic energy in the acoustic domain, 2

a0   r 2  1 1 2 2 ρ0 L ∫ u0 1 −    2π rdr = M aN Qout , 2 2   a0   0

{29}

where u0 is the centerline velocity which is related to the volume flow rate by Qout = u0π a02 2 . The solution of Eq {29} yields the effective acoustic mass, M aN =

4 ρ0 L . 3π a0 2

{30}

18 The acoustic radiation mass M aRad can be modeled for ka0 < 0.5 as a piston in an infinite baffle if the circular orifice is mounted in a plate that is much larger in extent than the orifice (see Beranek [22], p. 124) M aRad =

8 ρ0 . 3π 2 a0

{31}

The acoustic resistance associated with the discharge from the orifice can be approximated by modeling the orifice as a generalized Bernoulli flow meter [18, 24],

RaO

1 1 K D ρ 0u K D ρ 0Qout 2 2 , = = π a02 π 2 a04

{32}

where u is the mean velocity, and K D ≈ O (1) is a nondimensional loss coefficient that is a function of orifice geometry, Reynolds number, and frequency. Note that RaO is a function of the volume flow rate Qout through the orifice and thereby represents a nonlinear resistance, necessitating an iterative solution of Eq. {4}. At higher frequencies, the velocity profile in the orifice is modeled as flow in a circular duct driven by an oscillating pressure gradient. The solution is given in White [25] as   ωr2  J0  − j  ν ∆P  u ( r , t ) = j out 1 −  ωρ0 L   ω a02 − J j  0   ν  

    e jω t ,     

{33}

where J 0 is a Bessel function of zero order and ν is the kinematic viscosity. The velocity u is proportional to the pressure gradient and inversely proportional to ρ 0ω .

19 Furthermore, the velocity profile is characterized by the Stokes number St = ω a02 ν , as shown in Figure 4–1. In the limit of St → 0 , the velocity profile asymptotes to Poiseuille flow. As St increases, the thickness of the Stokes layers decreases below a0 , leading to an inviscid core surrounded by a viscous annular region.

Figure 4–1: Variation in velocity profile vs. St = 1, 10, and 30 for oscillatory channel flow in a circular duct. Figure 4–2 shows that the ratio of the average velocity to the centerline velocity, which is 0.5 for Poiseuille flow, is strongly dependant on the Stokes number.

20

1.0

uavg/u0

0.9

0.8

0.7

0.6

0.5 0.1

1

10

100

1000

St Figure 4–2: Ratio of average velocity to centerline velocity vs. St for oscillatory channel flow in a circular duct. The acoustic impedance of the orifice under these assumptions is determined by directly integrating the velocity profile to obtain Qout as a function of ∆Pout . In this case, the real (resistive) and imaginary (reactive) parts of the acoustic impedance are functions of the Stokes number ∆Pout = Z aN = RaN + jX aN = RaN + jω M aN . Qout

{34}

The results shown in Figure 4–3 reveal that, at low frequencies, the acoustic resistance asymptotes to the steady value given in Eq. {28} and increases gradually with frequency. However, the acoustic mass is approximately constant with frequency. The

21 data in Figure 4–3 are used to provide frequency-dependent estimates for the acoustic resistance and mass in Eq. {4}.

Figure 4–3: Acoustic resistance and reactance normalized by steady value vs. St for oscillatory channel flow in a circular duct.

CHAPTER 5 MODEL VERIFICATION AND PARAMETER EXTRACTION A modular piezoelectric-driven synthetic jet was constructed, as shown in Figure 5–1, to perform a series of experiments to test the validity of the lumped element model parameters. The modular design permits a systematic variation of the cavity volume, orifice diameter and length, and piezoelectric diaphragm diameter and thickness. In addition, an access hole is provided for a microphone to monitor the fluctuating pressure inside the cavity.

orifice plate

top plate

diaphragm mount

body plate

clamp plate

access hole for microphone Figure 5–1: Assembly diagram of modular synthetic jet.

22

23 5.1

Piezoelectric Transduction

The first experiment tested the linear composite plate theory that provides estimates for CaD , M aD , and d a . This was accomplished by measuring the velocity of the clamped vibrating diaphragm (excited by Vac ) using a scanning laser vibrometer (Polytec Model PSV-200) and integrating the velocity in the frequency domain to obtain displacement. The clamped circular diaphragm was removed from the synthetic jet apparatus and mounted on an optical table. The test was also performed in a vacuum chamber to eliminate fluid loading effects. Table 5-1 shows the characteristics of the device tested and Table 5-2 shows the parameters predicted by the lumped element model and defined in CHAPTER 4.

Table 5-1: Piezoceramic diaphragm characteristics Shim (Brass)

Elastic Modulus ( Pa )

8.963×1010

Poisson’s Ratio

0.324

Density ( kg m

3

)

Thickness ( mm )

8700 0.2012

Diameter ( mm )

23

Piezoceramic (PZT-5A)


6.3×1010

Poisson’s Ratio

0.31

Density ( kg m3 ) Thickness ( mm )

7700 0.2322

Diameter ( mm )

20.0

Relative Dielectric Constant

1750

d31 ( m V )

-1.75×10-10

Cef ( F )

2.095 ×10-8

24

Table 5-2: Calculated lumped element model parameters Lumped Element Model Parameters

φa ( Pa V )

M aD ( kg m 4 )

CaD ( s 2 ⋅ m 4 kg ) d a ( m3 V )

κ

Ceb ( F ) Short circuit natural frequency (Hz) Center deflection ( µ m V )

139.3 13538 1.4911×10-13 2.0771×10-11 0.117 20.95 3542.4 0.1146

Figure 5–2 shows a comparison between the predicted and measured mode shape of the piezoceramic diaphragm to a sinusoidal excitation voltage at f = 100 Hz for quasi-static case. The agreement between theory and experiment is excellent. The measured natural frequency of the diaphragm was 3505 Hz, while the computed shortcircuit resonance frequency from Eq. {7} was within 1%. Figure 5–2 includes random uncertainty estimates estimated from the measured response at 100 Hz, as outlined in Bendat and Piersol [26]. Key assumptions of the composite plate model, discussed in detail by Prasad et al. [20], include a linear response, negligible bond-layer thickness, full coverage of the piezoceramic surface by a metal electrode, and an ideal clamped boundary condition. Any violation of these assumptions will obviously degrade the accuracy of the model.

25

Figure 5–2: Comparison between predicted and measured response of piezoceramic diaphragms to a sinusoidal excitation voltage at 100 Hz.

5.2

Cavity Acoustic Compliance

The value of the cavity acoustic compliance CaC is obtained from Eq. {13}. The cavity volume can be calculated from the geometry. To test the theory, the orifice was replaced with a solid cap to provide a closed cavity and all leaks were carefully minimized. The piezoceramic was then driven with a nominal 1 V amplitude sinusoid, and the displacement of the vibrating diaphragm was measured with a laser displacement sensor (Micro-Epsilon Model ILD2000-10). A 1/8 in. Brüel & Kjær (B&K) type 4138

26 condenser microphone with B&K type 2669 preamplifier measured the fluctuating pressure in the closed cavity. The frequency response function between the pressure and displacement signal at the diaphragm center w0 was used to measure P w0 at several frequencies and calculate

CaC , as shown in Figure 5–3. Using the average measured value of 9.89 MPa m and the measured mode shape, the cavity volume was determined to be 2.53×10-6 m3 ± 12%. This nominal value is less than 1% of the cavity volume calculated from the geometry.

Figure 5–3: Measured acoustic compliance CaC vs. frequency in closed cavity of synthetic jet.

5.3

Acoustic Mass and Resistance in Orifice

The flow in the neck of the orifice is modeled via Eq. {33} as a steady, fullydeveloped laminar flow driven by an oscillatory pressure gradient in a circular duct of radius a0 and length L . The data in Figure 4–3 are used to provide frequency dependent estimates for the acoustic resistance and mass in the lumped element model. Also, it has

27 been verified that this model is in good agreement with the semi-empirical model given by Ingard [27] for the impedance of a perforated plate. It should also be noted that, depending on the aspect ratio L a0 of the orifice, the fully developed assumption may not be valid. Only for large values of L a0 is the fullydeveloped assumption expected to be reasonable. For small values of L a0 , the orifice dump loss given in Eq. {32} is expected to dominate. The models discussed in this section are simple and neglect potentially significant issues, such as nonlinear effects due to large amplitude pressure oscillations in the cavity [28] and transition to turbulent flow and compressibility effects in the orifice. Grazing flow effects, which are relevant when the synthetic jet interacts with a boundary layer, have also been ignored [29].

CHAPTER 6 COMPARISON BETWEEN MODEL & EXPERIMENTS In this section, the lumped element model is used to predict the frequency response of two synthetic jet actuators. The dimensions and material properties of the piezoceramic diaphragms are summarized in Table 6-1, and the pertinent lumped element parameters are provided in Table 6-2. Table 6-3 lists the geometry of both synthetic jets.

Table 6-1: Specifications of piezoceramic diaphragms. Shim (Brass)

Case I

Case II


8.963×1010

Poisson’s Ratio

0.324

Density ( kg m3 )

8700

Thickness ( mm )

0.20

0.10

Diameter ( mm )

23.5

37


Elastic Modulus ( Pa ) Poisson’s Ratio

6.3×1010 0.31

Density ( kg m3 )

7700

Thickness ( mm )

Diameter ( mm ) Relative Dielectric Constant

d31 ( m V )

0.11

0.10

20.0

25.0 1750

-1.75×10-10

Cef ( F )

4.42 ×10-8 7.453×10-8

28

29 Table 6-2: Calculated lumped parameters of piezoceramic diaphragms. Quantity

Case I

Case II

CaD ( s 2 ⋅ m 4 kg )

6.53×10

2.93×10-11

M aD ( kg m 4 )

8.15×103

2.25×103

88.6

16.06

φa ( Pa V )

-13

Table 6-3: Geometry of synthetic jet actuators. Cavity:

Case I

Volume V0 ( m3 )

Case II

2.50×10

5.50×10-6

0.825

0.42

1.65

0.84

-6

Orifice:

Radius a0 ( mm ) Length L ( mm )

6.1

Laser Doppler Velocimetry System

A Dantec, fiber-optic, two-component, Laser Doppler Velocimetry (LDV) system was used to measure the magnitude of the peak centerline velocity produced by a synthetic jet for varying frequency. Theatrical fog fluid was used to produce particles with a specified mean diameter of 0.35 µm. Figure 6–1 shows a schematic of the synthetic jet device and the relative size of the probe volume. A 400 mm focal length, 60

mm diameter lens was used to create a probe volume size of 0.194 mm x 0.194 mm x 4.095 mm with a beam half-angle of 2.718°. Because of the beam half-angle and the thickness of the synthetic jet device, the minimum distance from the orifice that velocity data could be acquired was 0.3 mm .

30

Top View

2.718° beams measuring the horizontal velocity component

orifice

beams measuring the vertical velocity component

L = 1.65 mm

2.718°

orifice 2a0 = 1.65 mm cavity 3.81 mm 12.7 mm

Side View

Figure 6–1: Laser Doppler Velocimeter system setup for Case I.

6.2

Data Acquisition

Velocity data were acquired phase-locked with the input actuation sinusoidal signal. As shown in Figure 6–1, the probe volume length extended well beyond the orifice. Therefore, the data contained many points with near zero velocity. Fortunately, these data do not affect the envelope of the phase-averaged velocity over one cycle. The maximum value was extracted from the phase-averaged velocity measurements over a range of input frequencies with Vac = 25 V . To verify repeatability of the data, several frequencies were chosen and velocity data was acquired over a period of several days for Case II. Table 6-4 summarizes the repeatability of the velocity measurements for several

31 frequencies. While the uncertainty exceeds 10% near resonance, the data are sufficiently accurate to test the validity of the lumped element model.

Table 6-4: Uncertainty in maximum velocity at select frequencies for Case II. Frequency ( Hz )

Uncertainty in U max ( f ) ( m s )

200

0.3

500

0.7

800

8.5

6.3

Results

As shown in Table 6-3, Case I uses a smaller piezoceramic diaphragm and cavity volume and a larger orifice than Case II. However, both devices use an orifice with the same aspect ratio L a0 = 2 and are driven with Vac = 25 V amplitude sinusoids. Figure 6–2 and Figure 6–3 show the comparison between the experiment and the model developed above. The lumped element model uses Eq. {33} and Figure 4–3 to model the frequency dependence of the velocity profile to determine the relationship between umax and Qout , which is obtained via Figure 4–2. The agreement between the model and experiment in Figure 6–2 is satisfactory. The main difference between the two models proposed occurs at the maximum velocity peaks where the resistance terms dominate the system response. The model accurately predicts the two resonance frequencies using Eq. {12}.

32

Figure 6–2: Comparison between the lumped element model and experiment for Case I. Case II in Figure 6–3 corresponds to a case with a single dominant peak that provides jet velocities in excess of 60 m s . The excitation voltage amplitude of 25 V corresponds to approximately 35% of the coercive or breakdown electric field strength (~1.18 V µ m ) of the piezoceramic. The lumped element model accurately predicts the resonance frequency and maximum velocity and also possesses the proper shape of the frequency response function. It is important to note that two resonance frequencies are predicted by the lumped element model, one near 350 Hz and the other near 900 Hz. However, the lower peak is damped in Figure 6–3 due to the frequency dependent nonlinear orifice resistance term RaO (ω ) . An inspection of Eq. {4} and {5} reveals that

33 this term has a larger effect at lower frequencies. Gallas et al. [30] mistakenly argued that this case corresponds to the situation when a single peak is obtained in the frequency response function f1 = f 2 =

fD fH .

Table 6-5 summaries the comparison between

experiments and the analytical analysis developed above.

Figure 6–3: Comparison between the lumped element model and experiment for Case II. Finally, the peak in the experimental results for Case II, which occurs around 1200 Hz, corresponds to the second resonance frequency of the piezoelectric diaphragm (deducted from experimental tests).

The lumped element model is not valid at

frequencies corresponding to higher-order vibration modes. As a result, the peak near 1200 Hz is not captured by the model.

34

Table 6-5: Comparison between lumped element model and experiment. Quantity

Source

Case I

Case II

fD

( Hz )

Eq. {7}

2180

620

fH

( Hz )

Eq. {8}

940

470

β

Eq. {11}

0.036

0.735

f1 ( Hz )

Eq. {12}

920

330

( Hz )

Eq. {12}

2230

882

f1 ( Hz )

experiment

970

N/A

( Hz )

experiment

2120

850

f2

f2

CHAPTER 7 CONCLUSIONS AND FUTURE WORK A lumped element model of a piezoelectric-driven synthetic jet actuator has been developed and compared with experiment. LEM provides a compact analytical model and valuable physical insight into the dependence of the device behavior on geometry and material properties. The model reveals that a synthetic jet is a 4th-order coupled oscillator. One oscillator is a Helmholtz resonator, and the second is the piezoelectric diaphragm. Simple arguments reveal two important special cases corresponding to single oscillators. One case occurs when the fluid is incompressible, while the second case is similar to that of a rigid piston and occurs when the acoustic compliance of the piezoelectric diaphragm is small compared to that of the cavity.

In this case, the

synthetic jet acts like a driven Helmholtz resonator. For any case with low damping, a simple formula was obtained to estimate the two natural frequencies of the synthetic jet as a function of the Helmholtz and diaphragm natural frequencies and the compliance ratio β . Methods to estimate the parameters of the lumped element model were presented and experiments were performed to isolate different components of the model and evaluate their suitability. The results indicate that the linear composite plate theory is accurate when the model assumptions are realized.

Similarly, the cavity acoustic

compliance model was validated. The details of the flow in the orifice require further careful study. It is this region that dictates the acoustic mass and resistance in the neck.

35

36 Accurate knowledge of the acoustic mass is required to determine the Helmholtz frequency of the synthetic jet, while the resistance limits the achievable velocities near resonance. The model was applied to two prototypical synthetic jets and found to provide very good agreement with the measured performance. The results reveal the power and shortcomings of the model in its present form. The flow in the vicinity of the orifice must be studied further in order to obtain better quantitative estimates of the losses and velocity profile characteristics. Furthermore, the loss coefficient K D in Eq. {32} is treated as a constant but is probably a function of the orifice Reynolds number, Stokes number, and geometry. Furthermore, grazing flow effects remain to be studied in a rigorous fashion. In future work, additional parameters will be varied in the model and accompanying experiments to yield optimal design rules for the synthetic jet. Future testing of the piezoelectric diaphragm will assess the severity of nonlinear effects when the excitation amplitude is increased. For the orifice, emphasis will be placed on the ratio of the orifice length to the hole radius, L a0 . This variable will be systematically varied in concert with the other important nondimensional parameters, such as the orifice Reynolds and the Stokes numbers. Additional velocity measurements with improved spatial resolution will also be performed to map out the spatial variations in the synthetic jet velocity field. investigated.

Finally, an optimization of the overall device efficiency will be

REFERENCES 1.

Smith, B. L. and Glezer, A., “The Formation and Evolution of Synthetic Jets,” Physics of Fluids, Vol. 10, No. 9, pp. 2281-2297, 1998.

2.

Amitay, M., Smith, B. L., and Glezer, A., “Aerodynamic Flow Control Using Synthetic Jet Technology,” AIAA Paper 98-0208, Jan. 1998.

3.

Smith, D. R., Amitay, M., Kibens, V., Parekh, D. E., and Glezer, A., “Modification of Lifting Body Aerodynamics Using Synthetic Jet Actuators,” AIAA Paper 98-0209, Jan. 1998.

4.

Chen, Y., Liang, S., Aung, K., Glezer, A., Jagoda, J., “Enhanced Mixing in a Simulated Combustor Using Synthetic Jet Actuators,” AIAA Paper 99-0449, Jan. 1999.

5.

Honohan, A. M., Amitay, M., and Glezer, A., “Aerodynamic Control Using Synthetic Jets,” AIAA Paper 2000-2401, June 2000.

6.

Chatlynne, E., Rumigny, N., Amitay, M., and Glezer, A., “Virtual Aero-Shaping of a Clark-Y Airfoil Using Synthetic Jet Actuators,” AIAA Paper 2001-0732, Jan. 2001.

7.

Crook, A., Sadri, A. M., and Wood, N. J., “The Development and Implementation of Synthetic Jets for the Control of Separated Flow,” AIAA Paper 99-3176, July 1999.

8.

Chen, F.-J., Yao, C., Beeler, G. B., Bryant, R. G., and Fox, R. L., “Development of Synthetic Jet Actuators for Active Flow Control at NASA Langley,” AIAA Paper 2000-2405, June 2000.

9.

Crook, A. and Wood, N. J., “Measurements and Visualizations of Synthetic Jets,” AIAA Paper 2001-0145, Jan. 2001.

10.

Gilarranz, J. L. and Rediniotis, O. K., “Compact, High-Power Synthetic Jet Actuators for Flow Separation Control,” AIAA Paper 2001-0737, Jan. 2001.

11.

Kral, L. D., Donovan, J. F., Cain, A. B., and Cary, A. W., “Numerical Simulation of Synthetic Jet Actuators,” AIAA Paper 97-1824, June 1997.

37

38 12.

Rizzetta, D. P., Visbal, M. R., and Stanek, M. J., “Numerical Investigation of Synthetic Jet Flowfields,” AIAA 98-2910, June 1998.

13.

Mallinson, S. G., Reizes, J. A., Hong, G., and Haga, H., “The Operation and Application of Synthetic Jet Actuators,” AIAA Paper 2000-2402, June 2000.

14.

Utturkar, Y., Mittal, R., Rampunggoon, P., and Cattafesta, L., “Sensitivity of Synthetic Jets to the Design of the Jet Cavity,” AIAA Paper 2002-0124, Jan. 2002.

15.

Fischer, F. A., Fundamentals of Electroacoustics, Interscience Publishers, Inc., New York, NY, 1955, Chapter III and XI.

16.

Merhaut, J., Theory of Electroacoustics, McGraw-Hill, Inc., New York, NY, 1981, Chapter 6.

17.

Rossi, M., Acoustics and Electroacoustics, Artech House, Norwood, MA, pp. 245-373, 1988.

18.

McCormick D. C, “Boundary Layer Separation Control with Directed Synthetic Jets,” AIAA Paper 2000-0519, Jan. 2000.

19.

Rathnasingham, R. and Breuer, K. S., “Coupled Fluid-Structural Characteristics of Actuators for Flow Control,” AIAA Journal, Vol. 35, No. 5, pp. 832-837, May 1997.

20.

Prasad, S., Horowitz, S., Gallas, Q., Sankar, B., Cattafesta, L., and Sheplak, M., “Two-Port Electroacoustic Model of an Axisymmetric Piezoelectric Composite Plate,” AIAA Paper 2002-1365, April 2002.

21.

Sheplak, M., Schmidt, M.A., and Breuer, K.S., “A Wafer-Bonded, Silicon Nitride Membrane Microphone with Dielectrically-Isolated, Single-Crystal Silicon Piezoresistors,” Technical Digest, Solid-State Sensor and Actuator Workshop, pp. 23-26, Hilton Head, SC, June 1998.

22.

Beranek, L. L. Acoustics, Acoustic Society of America, Woodbury, NY, pp. 4777, 116-143, 1993.

23.

Blackstock, D. T., Fundamentals of Physical Acoustics, John Wiley & Sons, Inc., New York, NY, p. 145, 2000.

24.

White, F. M., Fluid Mechanics, McGraw-Hill, Inc., New York, NY, pp. 377-379, 1979.

25.

White, F. M., Viscous Flow, McGraw-Hill, Inc., New York, NY, pp. 143-148, 1974.

39 26.

Bendat, J. S., and Piersol, A. G., Random Data: Analysis and Measurements Procedures, 3rd Edition, John Wiley & Sons, Inc., New York, NY, p. 341, 2000.

27.

Ingard, K. U., Notes on Sound Absorption Technology, Poughkeepsie, NY: Noise Control Foundation, 1994.

28.

Ingard, U., “Acoustic Nonlinearity of an Orifice,” Journal of Acoustic Society of America, Vol. 42, No. 1, pp. 6-17, 1967.

29.

Mittal, R., Rampuggoon, P., Udaykumar, H. S., “Interaction of a Synthetic Jet with a Flat Plate Boundary Layer,” AIAA Paper 2001-2773, June 2001.

30.

Gallas, Q., Mathew, J., Kaysap, A., Holman, R., Carroll, B., Nishida, T., Sheplak, M., and Cattafesta, L., “Lumped Element Modeling of Piezoelectric-Driven Synthetic Jet Actuators,” AIAA Paper 2002-0125, January 2002.

APPENDIX A COMPOSITE PIEZOELECTRIC DIAPHRAGM CHARACTERIZATION The limitations of the assumptions discussed in Section 5.1 are outlined. Three different piezoelectric diaphragms have been tested. Case I and Case II correspond to the devices used in CHAPTER 6, while Case III corresponds to the device tested in CHAPTER 5. Their specifications are summarized below in Table A-1. The set up described in Section 5.1 has been used.

Table A-1: Piezoelectric diaphragms properties. Shim (Brass)

Case I

Case II


8.963×1010

Poisson’s Ratio

0.324

Density ( kg m3 )

8700

Thickness ( mm ) Diameter ( mm )

Case III

0.20

0.100

0.201

23.5

37

23


Elastic Modulus ( Pa ) Poisson’s Ratio

6.3×1010 0.31

Density ( kg m3 ) Thickness ( mm )

Diameter ( mm ) Relative Dielectric Constant

7700 0.11

0.102

0.232

20.0

25.0

20.0

1750

d31 ( m V ) Cef ( F )

-1.75×10-10 4.42 ×10-8 7.453×10-8 2.095 ×10-8

40

41 Figure A–1 shows the plot of frequency response function magnitude, phase and coherence of 2 sample piezoceramic diaphragms - Case I, while Figure A–2, Figure A–3 and Figure A–4, respectively, show the plot of frequency response function magnitude, phase and coherence of piezoceramic diaphragm - Case I, piezoceramic diaphragm - Case II and piezoceramic diaphragm - Case III.

It can be observed that each sample

piezoeramic diaphragm is unique in the sense that their natural frequencies occur for different values. Also, the type of boundary condition used is important since it may change the magnitude of the diaphragm deflection and the natural frequency. The values of the natural frequency and dc response for each case are tabulated in Table A-2.

Table A-2: Experimental values of natural frequency and dc response of tested piezoelectric diaphragms

Case I

Case II

Case III

Sample 1 (Figure A–1, Figure A–8) Sample 2 (Figure A–1, Figure A–8) Clamped BC (Figure A–2, Figure A–5) Compliant BC (Figure A–2) Clamped BC (Figure A–3, Figure A–6) Compliant BC (Figure A–3) Sample 1 (Figure A–9) Sample 2 (Figure A–9) Clamped BC (Figure A–4, Figure A–7) Compliant BC (Figure A–4)

Natural frequency (Hz)

dc response ( µm V )

2003.9

0.418

1996.1

0.446

1996.1

0.446

2445.3

0.268

762.5

0.718

770.0

0.664

3381.2

0.131

3387.5

0.133

3381.2

0.131

3400.0

0.112

42

1.E-04

Sample 1 Sample 2

Magnitude (m/V)

1.E-05 1.E-06 1.E-07 1.E-08 1.E-09

200 Phase (degrees)

150 100 50 0 -50 0

2000

4000

6000

8000

10000

12000

2000

4000

6000

8000

10000

12000

-100 -150 -200

Coherence

1 0.8 0.6 0.4 0.2 0 0

Frequency (Hz)

Figure A–1: Frequency response (magnitude, phase and coherence) of the center of two sample piezoceramic diaphragms - Case I. (Amplitude input voltage=1V)

43

1.E-04

Magnitude (m/V)

Clamped BC Compliant BC

1.E-05

1.E-06

1.E-07

1.E-08 0

2000

4000

6000

8000

10000

12000

200

Phase (degrees)

150 100 50 0 -50

0

2000

4000

6000

0

2000

4000

6000

8000

10000

12000

-100 -150 -200

Coherence

1 0.8 0.6 0.4 0.2 0 8000

10000

12000

Frequency (Hz)

Figure A–2: Frequency response (magnitude, phase and coherence) of the center of piezoceramic diaphragm - Case I for different boundary conditions. (Amplitude input voltage=1V)

44

1.E-04


Magnitude (m/V)

1.E-05

1.E-06

1.E-07

1.E-08

1.E-09 0

500

1000

1500

2000

1500

2000

200 150

Phase (degrees)

100 50 0 0

500

1000

-50 -100 -150 -200

1

Coherence

0.8

0.6

0.4

0.2

0 0

500

1000

1500

2000

Frequency (Hz)

Figure A–3: Frequency response (magnitude, phase and coherence) of the center of piezoceramic diaphragm - Case II for different boundary conditions. (Amplitude input voltage=1V)

45

1.E-05

Magnitude (m/V)


1.E-06

1.E-07

1.E-08 0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

5000

6000

7000

8000

9000

10000

200

Phase (degrees)

150 100 50 0 -50 -100 -150 -200

Coherence

1 0.8 0.6 0.4 0.2 0 0

1000

2000

3000

4000

Frequency (Hz)

Figure A–4: Frequency response (magnitude, phase and coherence) of the center of piezoceramic diaphragm - Case III for different boundary conditions. (Amplitude input voltage=1V)

46 Figure A–5, Figure A–6 and Figure A–7 respectively show the effectiveness of the clamp for piezoceramic diaphragms – Case I, Case II and Case III. The magnitude of the center deflection of the piezoceramic diaphragm and of a point very near the clamp edge are plotted, along with the noise floor of the laser vibrometer. It can be observed that a perfect compliant boundary condition is not easy to achieve.

1.00E-04 Center Deflection Noise Floor Clamp Deflection

Magnitude (m/V)

1.00E-05 1.00E-06 1.00E-07 1.00E-08 1.00E-09 1.00E-10 1.00E-11 0

1000

2000

3000

4000

5000

6000

Frequency (Hz)

Figure A–5: Frequency response of piezoceramic diaphragm - Case I. (Amplitude input voltage=1V)

47

1.E-04 Center Deflection Noise Floor Clamp Delfection

Magnitude (m/V)

1.E-05 1.E-06 1.E-07 1.E-08 1.E-09 1.E-10 1.E-11 0

500

1000 Frequency (Hz)

1500

2000

Figure A–6: Frequency response of piezoceramic diaphragm - Case II. (Amplitude input voltage=1V)

1.E-04

Center Deflection Noise Floor Clamp Deflection

1.E-05

Magnitude (m/V)

1.E-06 1.E-07 1.E-08 1.E-09 1.E-10 1.E-11 1.E-12 0

1000

2000

3000

4000

Frequency (Hz)

Figure A–7: Frequency response of piezoceramic diaphragm - Case III. (Amplitude input voltage=1V)

5000

48 Figure A–8 and Figure A–9 show the mode shape of piezoceramic diaphragms – Case I and Case III respectively, driven by a sinusoidal signal with amplitude of 1V. Figure A–8 enhances the boundary condition problem at low frequency, while Figure A– 9 shows the effect of asymmetry of the ceramic layer on top of the brass diaphragm.

4.5E-07

Clamped BC

Magnitude (m/V)

4.0E-07

Compliant BC

3.5E-07 3.0E-07 2.5E-07 2.0E-07 1.5E-07 1.0E-07 5.0E-08 0.0E+00 -1

-0.5

0

0.5

1

normalized radius

Figure A–8: Comparison between two boundary condition types applied to piezoceramic diaphragm - Case I driven by a sinusoidal excitation voltage of 1V at 300 Hz.

49

1.4E-07 Sample 1

Magnitude (m/V)

1.2E-07

Sample 2

1.0E-07 8.0E-08 6.0E-08 4.0E-08 2.0E-08 0.0E+00 -1

-0.5

0

0.5

1

normalized radius

Figure A–9: Comparison between the response of two sample piezoceramic diaphragms - Case III driven by a sinusoidal excitation voltage of 1V at 500 Hz. The piezoceramic of sample 1 is not centered on the brass diaphragm. It can be observed that it is difficult to achieve a perfect boundary condition, and that each device responds differently from the others. The frequencies at which the natural frequency peaks occur and the deflection magnitude at DC value are affected. The model described in Section 5.1 needs therefore to be consciously and carefully used.

APPENDIX B SYNTHETIC JET ACTUATORS MEASURMENTS As shown in Figure 5–1, a modular synthetic jet has been built in such a manner that one can vary different fundamental aspects of the device, as the cavity volume and the aspect ratio of the orifice. The available geometries are summarized below in Table B-1. Note that Case I and Case II are those previously discussed in CHAPTER 6.

Table B-1: Geometry of modular synthetic jet actuators. Cavity:

Volume V0 ( m3 ) Orifice:

Radius a0

( mm )

Length L ( mm )

V0

V1 (Case I)

V2 (Case II)

V3

4.45×10-6

2.50×10-6

5.00×10-6

7.11×10-6

1 (Case II)

2 (Case I)

3

4

5

6

0.84/2

1.65/2

3/2

1/2

1/2

3/2

0.84

1.65

1

3

5

5

The magnitude of the peak centerline velocity produced by a synthetic jet for varying frequency was measured by the LDV system discussed in Section 6.1. For the results below, a 120 mm focal length, 60 mm diameter lens was used to create a probe volume size of 0.059 mm x 0.058 mm x 0.372 mm with a beam half-angle of 8.997°. The probe volume is therefore much smaller than the one used previously in CHAPTER 6 and is better able to capture the peak centerline velocity of the synthesized jet. Because

50

51 of the much larger beam half-angle and the thickness of the synthetic jet device, the velocity data were acquired at least one and an half diameter above the orifice, while the theoretical model predicts the mean velocity at the exit of the orifice. Below in Figure B1 is a schematic of the LDV set up to measure velocity field of Case II having cavity volume V3. LDV Lens Characteristics Focal Length = 120.0mm Beams measuring the vertical velocity component

Distance from probe volume to orifice surface is ~1.5 orifice diameters

8.997°

Orifice

L = 1.65 mm

2a0 = 0.84 mm

Cavity 4.6mm 14.2 mm Side view of the probe volume over the orifice for Case II with cavity volume V3

Figure B-1: Laser Doppler Velocimeter system setup for Case II having cavity volume V3. Figure B-2 and Figure B-3 are comparisons between model predictions to look at the effect on the sole change in cavity volume in the overall response of the device. They use cavity volume data provided in Table B-1.

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Maximum Velocity (m/s) for an applied voltage (50 Vpp) 30 V0 > V1 25

V1

20

15

10

V0

5

0 0

500

1000

1500

2000

2500

Frequency (Hz)

Figure B-2: Cavity volume change in Case I (model prediction)

3000

53 Maximum Velocity (m/s) for an applied voltage (50 Vpp) 70

60

V2 < V3

50 V2 V3

40

30

20

10

0 0

500

1000

1500

Frequency (Hz)

Figure B-3: Cavity volume change in Case II (model prediction) Below are presented frequency response of synthetic jet actuators for various orifice configurations. Velocity data were acquired phase-locked with the input actuation sinusoidal signal, and the maximum velocity value was extracted for each frequency. All configurations are driven by a 25 Vac amplitude sinusoidal signal. Figure B-4, Figure B5, Figure B-6, Figure B-7, Figure B-8 and Figure B-9 compare experimental data with the model prediction. The data provided in Table B-1 have been used. All device tested used the same diaphragm (piezoceramic diaphragm – CaseII) and only the orifice cap was removed and changed from the apparatus. Thus, a value of the damping coefficient

ζ was chosen to match the first experiment and was kept the same for all cases.

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Maximum Velocity (m/s) 70

L a0 = 2

ζ = 0.028 60

50

40

30

20

10

0 0

500

1000

1500

Frequency (Hz)

Figure B-4: Comparison between the lumped element model and experiment for synthetic jet having diaphragm – Case II, cavity volume V3, and orifice 1. (◊ experiment,  model)

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ζ = 0.028

L a0 = 2

45 40 35 30 25 20 15 10 5 0 0

500

1000

1500

Frequency (Hz)


56


ζ = 0.028

L a0 = 2 3

30

25

20

15

10

5

0 0

500

1000

1500

Frequency (Hz)


57


ζ = 0.028

L a0 = 6

60

50

40

30

20

10

0 0

500

1000

1500

Frequency (Hz)


58


ζ = 0.028

L a0 = 10

60

50

40

30

20

10

0 0

500

1000

1500

Frequency (Hz)


59


ζ = 0.028

L a0 = 10 3

30

25

20

15

10

5

0 0

500

1000

1500

Frequency (Hz)

Figure B-9: Comparison between the lumped element model and experiment for synthetic jet having diaphragm – Case II, cavity volume V3, and orifice 6. (◊ experiment,  model) It can be observed that, although the velocity magnitude does not always match with the experiments, the expected global shape of the frequency response does match. One possibility to explain this discrepancy comes from that the experimental results are obtained with a probe volume far from the orifice while the model predicts the velocity magnitude right at the orifice. And in this set up the probe volume is much smaller than the one used in Section 6.1, therefore mostly capturing velocity points with finite velocity while in Section 6.1 the data contained many points having near zero velocity. Another cause of discrepancy between experimental data and model prediction may come from the orifice model used, which is accurate for fully-developed orifice flow. Clearly, for

60 some of the orifice used (as orifice 3 and orifice 6), the flow does not have the time to become fully-developed.

Better model prediction of the flow inside the orifice is

therefore needed. Response of a synthetic jet actuator function of the input voltage at a single frequency is shown in Figure B-10. The synthetic jet used has diaphragm - Case II, cavity volume V3, and orifice 1. It is driven by a sinusoidal signal at 400 Hz, and phaselocked to the input signal. The working frequency of 400 Hz has been chosen since the device is expected to respond without much fluctuation around this frequency.

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Mean velocity (m/s)

60 50 40 30 20 10 0 0

20

40

60

80

100

120

140

160

180

200

220

240

260

Vpp

Figure B-10: Mean jet velocity response function of input voltage for Case II with cavity volume V3 at f = 400 Hz It should be noted that while before experiments were conducted for a single input voltage of 50 Volt peak-to-peak, the above plot shows that at 400 Hz one can expect a velocity of almost 60 m/s at the maximum voltage tolerated by the ceramic. The last four points on the plot indicate that the piezoceramic might have been depolarized.

APPENDIX C AMPLIFIER CHARACTERIZATION A PCB amplifier, model 790A06, was used to drive the synthetic jet actuators. A function generator (HP 33120A) supplied the input voltage signal, and the output voltage signal was measured via an oscilloscope. The gain of the amplifier was set to 30 and the plots below show the severity of the non-linearity of the amplifier gain, first for a specific frequency (Figure C–1), then function of the frequency (Figure C–2).

Amplifier output voltage (Vpp)

300 250 200 150 exp. 2 (400 Hz) exp. 1 (600 Hz) f(x)=30x

100 50 0 0

1

2

3

4

5

6

Amplifier input voltage (Vpp) Figure C–1: Amplifier gain linearity at constant frequency.

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7

8

62 In Figure C–1, exp. 1 and exp. 2 correspond to different devices (two sample piezoelectric diaphragms - case II) driven at two different frequencies. The last four points of exp. 2 occurred when the piezoceramic had been depolarized.

31

Gain (Vout / Vin)

30.5 30 29.5 29 28.5 28 27.5 27 0

500

1000

1500

2000

Frequency (Hz)

Figure C–2: Amplifier gain function of frequency. It can be observed that this amplifier has a constant gain when used at a single frequency, but the frequency response of its gain presents some fluctuations.

BIOGRAPHICAL SKETCH Quentin Gallas was born on September 28th, 1977, in Orange, located in the south of France. He graduated from l’Ecole des Pupilles de l’Air, Grenoble, France, in 1995, specialized in Sciences (major in mathematics and physics). He entered the Université de Versailles--St Quentin-en-Yvelines in Versailles, and earned his undergraduate degree in mathematics, informatics and science applications in June 1998. He moved to Lyon and earned in fall 2001 the degree of Engineering from the Institute of Sciences and Techniques of Lyon, majoring in mechanics.

While finishing his third year of

mechanical engineering studies in Lyon, he entered the University of Florida in spring 2001 with a graduate research assistantship and is currently pursuing his master's degree in aerospace engineering. He intends to continue in the field of fluid dynamics and experimentation for his doctoral degree.

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