sensors Article

Steady State Response Analysis of a Tubular Piezoelectric Print Head Jiaqing Chang † , Yaxin Liu † and Bo Huang * Received: 22 November 2015; Accepted: 6 January 2016; Published: 12 January 2016 Academic Editor: Vittorio M. N. Passaro State Key Laboratory of Robotics and System, Harbin Institute of Technology, Harbin 150001, China; [email protected] (J.C.); [email protected] (Y.L.) * Correspondence: [email protected]; Tel.: +86-631-568-3139 † These authors contributed equally to this work.

Abstract: In recent years, inkjet technology has played an important role in industrial materials printing and various sensors fabrication, but the mechanisms of the inkjet print head should be researched more elaborately. The steady state deformation analysis of a tubular piezoelectric print head, which can be classified as a plane strain problem because the radii of the tubes are considerably smaller than the lengths, is discussed in this paper. The geometric structure and the boundary conditions are all axisymmetric, so a one-dimensional mathematical model is constructed. By solving the model, the deformation field and stress field, as well as the electric potential distribution of the piezoelectric tube and glass tube, are obtained. The results show that the deformations are on the nanometer scale, the hoop stress is larger than the radial stress on the whole, and the potential is not linearly distributed along the radial direction. An experiment is designed to validate these computations. A discussion of the effect of the tubes’ thicknesses on the system deformation status is provided. Keywords: tubular piezoelectric print head; piezoelectric tube; glass tube; wall thickness

1. Introduction Ink jet technology has been widely used in the fabrication of integrated circuits (ICs) [1], polymer light emitting displays [2], sample preparation [3], microelectromechanical systems (MEMS) [4], cell printing [5,6], and the fabrication of different sensors [7–9]. Numerous excellent published papers have accounted for various applications of ink jet technology, but few of these papers are related to the action mechanism. To fully understand this physical phenomenon, and to obtain some useful instructions in print head design, it is very necessary to determine the mechanism for ink jet technology. This paper studies the action mechanism of a tubular piezoelectric print head, which is typically represented by the mechanical device designed by Zoltan [10]. Dijksman [11] firstly constructed an important model for the dynamic analysis of the tubular print head. Starting at the vibration amplitude and frequency of the inner glass tube, he successfully obtained the speed of the meniscus at the nozzle outlet in the time and frequency domain. The tubular print head was divided into nine segments of different radii or constraints, and the motion of the inner glass tube was decomposed into 250 terms of sinusoidal vibration through Fourier decomposition. Shin [12] analyzed the velocity and stress distribution. The analytical axial velocity history was used as the initial condition when doing his numerical simulation of the jetting and droplet formation process. Fabrication effects, such as the eccentricity alignment and the configuration of the electrode layer on the calculation results based on the model proposed, was also discussed. However, the stress and deformation distributions in piezoelectric tube and glass tube, as well as the influence of the relative thickness of these two tubes on the deformation status, were not provided in their work. Wijshoff [13] Sensors 2016, 16, 81; doi:10.3390/s16010081

www.mdpi.com/journal/sensors

Sensors 2016, 2016,16, 16,81 81 Sensors

13 22ofof13

were not provided in their work. Wijshoff [13] presented the actuator design of a piezoelectric print presented the actuatorthe design of a piezoelectric print and bonded conducted the piezoelectric deformation element. analysis head and conducted deformation analysis of the head channels to the of the channels bonded to the piezoelectric element. However, the analysis was mainly about However, the analysis was mainly about the piezoelectric print heads of bump or bend modes. the piezoelectric print heads of bump or bend modes. Kim [14] analyzed the pressure waves in a silicon MEMS-fabricated printhead under the Kim [14] thevoltage pressure waves in aNumerical silicon MEMS-fabricated printhead the excitation excitation of aanalyzed trapezoid waveform. simulations and lumpedunder element modeling of a trapezoid voltage waveform. Numerical simulations and lumped element modeling were were conducted, and results show that the Helmholtz mode is the dominant resonance mode acting conducted, and results show Helmholtz mode is the dominant mode actingofona on the flow oscillations at that the the nozzle. Bugdayci [15] discussed theresonance quasi-static motion the flow oscillations at the nozzle. Bugdayci [15] discussed the quasi-static motion of a piezoelectric piezoelectric tube filled with liquid but did not consider the effect of the glass tube. Moreover, tube with liquid did notfluid consider the effect the glass Moreover, considering whenfilled considering the but resulting pressure in theofglass tube,tube. it was assumedwhen that both ends of the fluid pressure in the glass tube, it was assumed thatsituation both endswas of the the resulting glass tube were completely sealed. However, the actual thatglass onetube endwere was completely sealed. However, the actual situation was that one end was connected to a plastic hose, connected to a plastic hose, and the other end was connected to an aperture plate, which made the and the other endbywas connected to an considerably aperture plate, which Larbi made [16] the pressure caused by fluid pressure caused fluid compression smaller. developed a model that compression smaller. of Larbi [16] developed aarbitrarily model that accounted forpiezoceramic for the free accounted forconsiderably for the free vibration a simply-supported thick laminated vibration a simply-supported thick laminated piezoceramic completely cylinder of completely filled witharbitrarily fluid. The piezoelectric layers of thecylinder laminated cylinderfilled are with fluid.inThe layers of the cylinder are polarized in of thea radial polarized thepiezoelectric radial direction. Chen [17]laminated investigated the dynamic process hollowdirection. cylinder Chen investigated the dynamic process a hollow cylinder filled compressible fluid based on filled [17] with compressible fluid based onofthree-dimensional statewith space formulations. The free three-dimensional state space piezoelectric formulations. hollow The freecylinder vibration of the a multi-layered piezoelectric vibration of a multi-layered and wave propagation in an hollow infinite cylinder and the wave propagation in an infinite homogeneous cylinder were therefore obtained. homogeneous cylinder were therefore obtained. Under Under the the excitation excitation of of aa step step voltage, voltage, the the piezoelectric piezoelectric tube tube causes causes the the coupling coupling glass glass tube tube to to deform deform and and squeezes squeezes the the internal internal fluid fluid out out of of the the nozzle nozzle aperture. aperture. Aimed Aimed at at the the above above process, process, this this paper paper analyzes analyzes the the deformation deformation and and stress stress statuses statuses of of piezoelectric piezoelectric and and glass glass tubes tubes after after the the input input of of electric understand thethe action mechanism of the liquid jet electric voltage. voltage. The Theobtained obtainedresults resultscould couldhelp helpusus understand action mechanism of the liquid process, determine the deformation and stress statuses when an electric input, is and guideand us jet process, determine the deformation and stress statuses when an voltage electric is voltage input, to design a proper piezoelectric print head. guide us to design a proper piezoelectric print head. A A tubular tubular piezoelectric piezoelectric print head, as shown in Figure 1, is used to validate these calculations. The The mechanical mechanical size size parameters parameters are are also also shown shown in in Figure Figure 1. 1. This This kind kind of of single single nozzle nozzle piezoelectric piezoelectric print head is made and sold by the Microdrop and MicroFab companies. About the model, print head is made and sold by the Microdrop and MicroFab companies. About the model, some some simplifications simplifications and and uncertainties uncertainties are are stated stated as as follows. follows. The epoxy layer is omitted omitted in in this this model model because often measures 10–25 µm inm thickness (refer to(refer [18] for about the fabrication because this thislayer layer often measures 10–25 in thickness to details [18] for details about the process) andprocess) is relatively Furthermore, steady state in analysis, layer only transmits fabrication and thin. is relatively thin. in Furthermore, steadythis state analysis, this layerforces, only so it does forces, not have on the ultimate and deformation stress statuses. aspect transmits so much it doeseffect not have much effectdeformation on the ultimate andAnother stress statuses. needing is that part of the ispiezoelectric wrapped bytube a thin Another consideration aspect needing consideration that part oftube the ispiezoelectric is electrode wrapped layer, by a and thin the wrapped segment ofwrapped the piezoelectric (about 2 mm intube length) will2not an will electric electrode layer, and the segmenttube of the piezoelectric (about mmgenerate in length) not field and will not have a driving The shear deformation and shear stress are also in generate an electric field and willeffect. not have a driving effect. The shear deformation andneglected shear stress the constructed model. are also neglected in the constructed model.

Figure Figure 1. 1. Configuration Configuration of of the the print print head. head.

2. Mathematical 2. Mathematical Model Model Construction Construction The cross cross section section of of the the tubular tubular piezoelectric piezoelectric print print head head isisshown shown in inFigure Figure2.2. Owing Owing to to its its The geometric shape, shape,aacylindrical cylindricalcoordinate coordinatesystem systemisisadopted. adopted.The Thedeformation deformationand andstress stress statuses geometric statuses ofof a a radially polarizedpiezoelectric piezoelectricceramic ceramictube tubeunder underthe theexcitation excitationof ofstep stepvoltage voltage are are studied studied in in this this radially polarized paper.When Whenperforming performing steady state deformation andanalysis, stress the analysis, the axisymmetric paper. the the steady state deformation and stress axisymmetric conditions conditions enable u  0 , u z  u z (r , z , t ) , ur  ur (r , z , t ) , and   (r , z , t ) . Moreover, because the

Sensors 2016, 16, 81

3 of 13

Sensors 2016, 16, 81

3 of 13

enable uθ “ 0, uz “ uz pr, z, tq, ur “ ur pr, z, tq, and ϕ “ ϕpr, z, tq. Moreover, because the length of length of the piezoelectric tube is larger relatively than the is classify suitableittoasclassify as a the piezoelectric tube is relatively thanlarger the radius, it isradius, suitableit to a planeitstrain  uuz r  ur  ϕ ur (“r , ϕpr, t ) and  obtained.  (r , t ) are plane strain , and obtained. Then, the strain problem, so problem, that uθ “so uz that “ 0, uur “ pr,0tq tq are Then, the strain relationship is simplified as relationship is simplified as γrr “ Bur {Br, γθθ “ ur {r (1)  rr  ur r ,    ur r (1)

Figure 2. The cross section of the tubular piezoelectric print head. Figure 2. The cross section of the tubular piezoelectric print head.

Piezoelectric constitutive equations become Piezoelectric constitutive equations become

u  rr  c33ur , r  c13 ur  e33 , r r σrr “ c33 ur,r ` c13 r ` e33 ϕ,r r u   c13ur , r  c11 ur r  e31, r σθθ “ c13 ur,r ` c11 r ` e31 ϕ,r r uurr e33uur,rr , r` ee31 ´ε3333 ,ϕ DrD“ e33 r  31 r ,r rr The stress stress equation equation is is shown shown as as The

 2Bu2 u 11 rr ,` rrrr ´   ρ 2r 2r σθθ σrr,r   q“ r  pσ rr tBt

(2) (2) (3) (3) (4) (4)

(5) (5)

The steady state response is only studied in this paper, paper, so so the the inertial inertial term term is is discarded. discarded. 11 rr ,`  rrrr ´ σθθq“0 0 σrr,r r  pσ rr

(6) (6)

Since there is no free charge inside the piezoelectric tube, the Gauss equation equation becomes becomes 11 B (rD )  0 r rprDr q “ 0 rr Br

(7) (7)

From that From Equation Equation (7), (7), it it can can be be inferred inferred that

Dr  A / r , ( A is a constant) Dr “ A{r, pA is a constantq From Equation (4), the following equations are deduced From Equation (4), the following equations are deduced e e u A , r  33 ur , r  31 r  e33 e31 u r r 33 rA ϕ,r “ 33ur,r ` 33 ´ ε33 ε33 r ε33 r ur 33   e33ur  e31 ż dr  A ln r  k1 rur dr ´ Alnr ` k ε33 ϕ “ e33 ur ` e31 1 r where k1 is an undetermined constant. where k1 is an undetermined constant. By substituting Equations (2) and (3) into Equation (6),

1 1 1   c33  ur , rr  ur , r   c11 2 ur  e33, rr   e33  e31  , r  0 r r r  

(8) (8)

(9) (9) (10) (10)

(11)

Sensors 2016, 16, 81

4 of 13

By substituting Equations (2) and (3) into Equation (6), ˙ ˆ 1 1 1 c33 ur,rr ` ur,r ´ c11 2 ur ` e33 ϕ,rr ` pe33 ´ e31 q ϕ,r “ 0 r r r

(11)

After substituting Equation (9) into Equation (11), ν2 1 k ur,rr ` ur,r ´ 2 ur “ 24 r r r ` ˘ ` 2 ˘ 2 {ε where ν2 “ c11 {c33 ; k4 “ ´ pe31 {c33 ε33 q A; c11 “ c11 ` e31 33 ; c33 “ c33 ` e33 {ε33 . The solution of Equation (12) is u r “ k 2 r ν ` k 3 r ´ν ´

(12)

k4 ν2

(13)

It is assumed that the electrical potential of the inner electrode for the piezoelectric tube is zero, and the electrical potential of the outer electrode is Vin . The thickness of the electrodes is neglected. Substituting Equation (13) into Equation (10), the electrical potential of the inner electrode is given in Equation (14). ˆ ˙ c33 ε33 e33 k1 ` e33 r2 ν k2 ` e33 r2 ´ν k3 ` lnr2 ´ 2 k4 “ 0 (14) e31 ν In addition, the electrical potential at r “ r3 is given by ˆ e33

k k2 r3 ν ` k3 r3 ´ν ´ 42 ν

˜

˙ ` e31

r´ν r ´ν rν rν k k k2 3 ´ k3 3 ´ 42 lnr3 ´ k2 2 ` k3 2 ` 42 lnr2 ν ν ν ν ν ν

¸ (15)

´Alnr3 ` k1 “ ε33 Vin From Equations (2) and (9), Equation (16) is derived σrr “ c33 ur,r ` c13

ur e33 A ´ r ε33 r

(16)

e31 e33 . ε33 The free boundary condition of the outer surface of the piezoelectric tube makes the stress become zero. ˆ ˙ ´ ¯ ” ı c33 e33 c ´pν`1q ´pν`1q σrr pr3 q “ c33 νr3ν´1 ` c13 r3ν´1 ¨ k2 ` c13 r3 ´ c33 νr3 ¨ k3 ` ´ 213 ¨ k4 “ 0 (17) e31 r3 ν r3

where c13 “ c13 `

The stress on the inner surface of the piezoelectric tube, represented as P2 , is outward because of the deformation resistance from the glass tube. Then, σrr pr2 q

ˆ ˙ ı ´ ¯ ” c33 e33 c ´pν`1q ´pν`1q ¨ k3 ` ´ 213 ¨ k4 “ P1 (18) “ c33 νr2ν´1 ` c13 r2ν´1 ¨ k2 ` c13 r2 ´ c33 νr2 e31 r2 ν r2

To simplify the analysis, the glass tube is viewed as an isotropic material. This is suitable in deformation analysis. Timoshenko described the steady state force equation of an isotropic thick-wall hollow cylinder in Reference [19], and the force diagram is depicted in Figure 3. The following analysis quotes the equations in Reference [19] directly. The radial displacement function of the glass tube is u1r “ k5 r `

k6 r

(19)

Sensors 2016, 16, 81

5 of 13

The stress on the outer surface of the glass tube is negative for it behaves similar to a compressive stress and equals the normal stress on the inner surface of the piezoelectric tube in magnitude. Therefore σ1rr pr2 q

Sensors 2016, 16, 81

« ff 1´µ E k5 p1 ` µq ´ k6 2 “ “ ´P1 1 ´ µ2 r2

(20) 5 of 13

Figure 3. 3. Force Force diagram diagram of of the the glass glass tube. tube. Figure

From Equations Equations (18) (18) and and (20), (20), ititcan canbe bededuced deducedthat that From ( 1) 1cq33r2 ( 1) ´p  kν3`1q ı  c33r21  c13r2¯1   k2  ”c13r2´p ν` c33 νr2ν´1 ` c13 r2ν´1 ¨ k2 `  c13 r2 ´ c33 νr2 ¨ k3 ˆ  c33e33 c˙13  E « 1    ff c33e33  2   k4  E 2  k5 (1  )  k6 21 ´  ´ c13  µ0 “ 0 ` e r  r ¨ k ` 1   k5 p1 ` µq ´ kr62 2 e31 r2 31 2ν2 r2 2  4 1 ´ µ2  r2

´

(21) (21)

As for the inner surface of the glass tube, although fluid inside it would resist the deformation As for the inner surface of the glass tube, although fluid inside it would resist the deformation of of the glass tube, the fluid pressure would eventually go to zero because the ends of the glass tube the glass tube, the fluid pressure would eventually go to zero because the ends of the glass tube are are not sealed. Therefore, the radial stress on the inner surface of the glass tube is zero. not sealed. Therefore, the radial stress on the inner surface of the glass tube is zero. E«  1   k (1  )  k6 2  ff 0 (22) E1   2  5 1r1´ µ k p1 ` µq ´ k “ 0 (22) 5 6 1 ´ µ2 r12 The hoop stress of the glass tube is shown as The hoop stress of the glass tube is shown as E  1   k (1  )  k6 2  ff  (r )  (23) 2« 5 1E   r12 ´µ 1 k5 p1 ` µq ` k6 2 σθθ prq “ (23) 1 ´ µ2 The displacement of the outer surface of the glass tube is r2 k The displacement of the outer surface of the glass tube is ur2  k5 r2  6 (24) r2 k6 u1r2 “ k5 r2 ` (24) Considering the bonding condition between the outer surface of the glass tube and the inner r2 surface of the piezoelectric tube, Equation (25) is consequently obtained. Considering the bonding condition between the outer surface of the glass tube and the inner  surface of the piezoelectric tube, Equation (25) is kconsequently k obtained. k2 r2   k3 r2   42   k5 r2  6   0 (25) r2  ˙  ˆ k k6 k2 r2 ν ` k3 r2 ´ν ´ 42 ´ k5 r2 ` “0 (25) r2 ν 3. Solving Method and Results 3. Solving Method and Results Next, by combining Equations (14), (15), (17), (21), (22) and (25), the following equation is Next, by combining Equations (14), (15), (17), (21), (22) and (25), the following equation is obtained obtained

Y HH¨ KK“ Y where

(26) (26)

Sensors 2016, 16, 81

where » — — — — — H“— — — — — –

6 of 13

c33 ε33 e33 lnr2 ´ 2 e31 ν

e33 r2´ν

e33 r2ν

1

0

pc33 ν ` c13 q r2ν´1

0

r2ν

r2´ν

1 c33 ε33 lnr3 pe31 lnr2 ´ e31 lnr3 ´ e33 q ` ν2 e31 ˆ ˙ c33 e33 c13 1 ´ 2 ν ˙ r3 ˆ e31 c33 e33 c13 1 ´ 2 e31 ν r2 1 ´ 2 ν

0

0

0

0

1 0

e33 r3ν `

e31 ν pr ´ r2ν q ν 3

e33 r3´ν `

e31 ´ν ´ r3´ν q pr ν 2 ´pν`1q

pc33 ν ` c13 q r3ν´1

pc13 ´ c33 νq r3

´pν`1q

pc13 ´ c33 νq r2

” K“

k1

k2

k3

k4

k5

k6

ıT

0

ε33 Vin

0

0

0

0

0

0

0

0

E 1´µ ´r2 E 1´µ

” , Y“

0

0

fi

E p1 ` µqr22 1 ´ r2 E ´ p1 ` µqr12 ´

ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi fl

ıT

The material parameters for piezoelectric material PZT-5H are shown in the appendix of Reference [20]. The elastic modulus and Poisson’s ratio of the Pyrex heat-resistant glass tube is 61.67 GPa (see Reference [21]) and 0.2, respectively. Additionally, the radii of the tubes are: r1 “ 2.5 ˆ 10´4 m, r2 “ 4.0 ˆ 10´4 m, and r3 “ 6.5 ˆ 10´4 m. Simultaneous Equation (26) is calculated and rewritten as » — — — –

1.0 1.0 0 0 0 0

1.9396 ˆ 10´2 2.7 ˆ 10´2 4.344 ˆ 1011 4.534 ˆ 1011 8.346 ˆ 10´4 0

2.7846 ˆ 104 1.5005 ˆ 104 ´8.31 ˆ 1016 ´2.096 ˆ 1017 1.1982 ˆ 103 0

2.559 ˆ 103 2.404 ˆ 103 ´1.048 ˆ 1015 ´1.703 ˆ 1015 ´1.22 0

0 0 0 7.71 ˆ 1010 ´4.0 ˆ 10´4 7.71 ˆ 1010

0 0 0 ´3.212 ˆ 1017 ´2.5 ˆ 103 ´8.224 ˆ 1017

fi » ffi — ffi — ffi — fl –

k1 k2 k3 k4 k5 k6

fi

»

ffi — ffi — ffi “ — fl –

0 1.3 ˆ 10´8 Vin 0 0 0 0

fi ffi ffi ffi (27) fl

The solution is not correct if the above equation is typed into the calculation software directly because of the effective numeral problem. Therefore, it is essential to do the following transformation. x1 “ 108 k1 , x2 “ 106 k2 , x3 “ 1012 k3 , x4 “ 1010 k4 , x5 “ 105 k5 , x6 “ 1012 k6

(28)

Suppose the input voltage is 1.0 V, i.e., Vin “ 1.0. Eventually, the following equation can be obtained such that k1 “ ´1.643 ˆ 10´7 , k2 “ ´2.079 ˆ 10´7 , k3 “ ´2.222 ˆ 10´12 , k4 “ 8.997 ˆ 10´11 , k5 “ ´4.642 ˆ 10´6 , k6 “ ´4.352 ˆ 10´13

(29)

Substituting these obtained coefficients into the aforementioned equations, some meaningful results occur. Utilizing Equations (13) and (19), the displacement function along the thickness direction is obtained, as plotted in Figure 4. It can be found that the displacement of the inner surface of the piezoelectric tube equals the displacement at the outer surface of the glass tube, which shows the correctness of the results. The values are all negative, which means that the tubes are all displaced toward the central axis when the input voltage is positive. As shown in Figure 4, the outer surface of the piezoelectric tube has a minimum displacement of 2.1 nm, and the displacement at the location where the piezoelectric and glass tubes adhere together has a maximum displacement of 2.94 nm. These displacement results seem larger than the usually known values. We cautiously believe that this is because the step voltage signal is different from a train of trapezoid pulses from the perspective of the Rayleigh’s energy. Exploiting Equations (18) and (20), the radial stress distribution along the thickness direction is obtained and is shown in Figure 5. The radial stresses in the glass and piezoelectric tubes are negative and positive, respectively, which means that the radial stresses in the glass tube point inward. For the piezoelectric tube, the radial stresses point outward. The radial stresses at the inner surface of the glass tube and the outer surface of the piezoelectric tube are all zero. The stress at the outer surface of the glass tube equals the stress at the inner surface of the piezoelectric tube, but with an opposite direction. These results all conform to the known conditions and show the correctness of the obtained results.

Sensors 2016, 16, 81 Sensors 2016, 16, 81

7 of 13 7 of 13

Figure 4. Displacement distribution along the thickness direction when the voltage amplitude is 1 V. Figure 4. Displacement distribution along the thickness direction when the voltage amplitude is 1 V.

Exploiting Equations (18) and (20), the radial stress distribution along the thickness direction is Exploiting and (20), the radial radial stress distribution along thickness direction is obtained and isEquations shown in(18) Figure 5. The stresses in the glass andthe piezoelectric tubes are obtained and is shown in Figure 5. The radial stresses in the glass and piezoelectric tubes are negative and positive, respectively, which means that the radial stresses in the glass tube point negative andthe positive, respectively, which that theoutward. radial stresses in the glass at tube inward. For piezoelectric tube, the radialmeans stresses point The radial stresses the point inner inward. For the piezoelectric tube, the radial stresses point outward. The radial stresses at the inner surface of the glass tube and the outer surface of the piezoelectric tube are all zero. The stress at the surface of the of glass outerthe surface are all zero. The stress the outer surface the tube glassand tubethe equals stressof at the the piezoelectric inner surfacetube of the piezoelectric tube, butatwith outer surface of the glass tube equalsall the stress attothe surface of the voltage piezoelectric but with an opposite These results conform theinner known conditions show thetube, correctness Figure 4.direction. Displacement distribution along the thickness direction when the and amplitude is 1 V. of Figure 4. Displacement distribution along the thickness direction when theand voltage amplitude is 1 V. of an opposite direction. These results all conform to the known conditions show the correctness the obtained results. the obtained results. Exploiting Equations (18) and (20), the radial stress distribution along the thickness direction is obtained and is shown in Figure 5. The radial stresses in the glass and piezoelectric tubes are negative and positive, respectively, which means that the radial stresses in the glass tube point inward. For the piezoelectric tube, the radial stresses point outward. The radial stresses at the inner surface of the glass tube and the outer surface of the piezoelectric tube are all zero. The stress at the outer surface of the glass tube equals the stress at the inner surface of the piezoelectric tube, but with an opposite direction. These results all conform to the known conditions and show the correctness of the obtained results.

Figure 5. Radial stress distribution along the thickness direction when the voltage amplitude is 1 V. Figure5.5.Radial Radialstress stressdistribution distributionalong alongthe thethickness thicknessdirection directionwhen whenthe thevoltage voltageamplitude amplitudeisis11V. V. Figure

Figure 5. Radial stress distribution along the thickness direction when the voltage amplitude is 1 V.

Figure 6. Hoop stress distribution along the thickness direction when the voltage amplitude is 1 V. Figure 6. Hoop stress distribution along the thickness direction when the voltage amplitude is 1 V. Figure 6. Hoop stress distribution along the thickness direction when the voltage amplitude is 1 V.

Utilizing Equations (3) and (23), the hoop stress distribution along the thickness direction is obtained and is shown in Figure 6. The hoop stresses in the glass and piezoelectric tubes are all negative. This means that the particles in the tubes in the circumferential direction are all compressed. The hoop stress increases as the radius decreases and reaches the maximum at the inner surface of the Figure 6. Hoop stress distribution along the thickness direction when the voltage amplitude is 1 V.

Sensors 2016, 16, 81

8 of 13

Utilizing Equations (3) and (23), the hoop stress distribution along the thickness direction is obtained and is shown in Figure 6. The hoop stresses in the glass and piezoelectric tubes are all Sensors 2016, 16, 81 8 of 13 negative. This means that the particles in the tubes in the circumferential direction are all compressed. The hoop stress increases as the radius decreases and reaches the maximum at the inner surface of the The hoop stress is not continuous at the whereadhere the two tubes adhere glass tube. Theglass hooptube. stress is not continuous at the place where theplace two tubes together, which together, conforms physical Figure it can bestress found that the hoop is conforms which to physical truth. to In Figure 6 ittruth. can beInfound that6 the hoop is generally larger stress than the generally larger than the radial stress. radial stress. Making direction Making use use of of Equation Equation (10), (10), the the electric electric potential potential distribution distribution along along the the radial direction is is obtained. obtained. As Asshown shownininFigure Figure7,7, the the electric electric potential potential isis not not linearly linearly distributed distributed along along the the thickness thickness direction. direction.

Figure 7. Electric potential distribution along the thickness direction when the voltage amplitude is 1 V. Figure 7. Electric potential distribution along the thickness direction when the voltage amplitude is 1 V.

4. Experimental Validation 4. Experimental Validation Since the displacements of the surfaces of the tubes are on nanometer scales, these Since theare displacements the surfaces ofhead. the tubes are on nanometer these deformations deformations structurallyof inside the print It seems impossible toscales, measure these quantities are structurally inside the print head. It seems impossible to measure these quantities directly. directly. An experiment was designed to validate the above computations indirectly. When the An experiment was designed to validate the above computations indirectly. When the glass is glass tube is deformed, it will squeeze a small amount of water out of the nozzle aperture, tube which deformed, it will squeeze a smallwith amount of water out of the nozzle aperture, whichinformation could be imaged could be imaged by a camera the appropriate magnification. The volume was by a camera with the appropriate magnification. The volume information was calculated calculated by using image processing technology, and the volume was correlatedbytousing the image processing andofthe was correlated to the deformation of the inner surface of deformation of thetechnology, inner surface thevolume glass tube. the glass First,tube. the print head was filled with water by using a syringe. Then, the tip part of the print the print head wasthat filled with by using syringe. Then, theseal tip part of thepart printofhead head First, was placed into a cap was fullwater of water. Theacap would nearly the front the was placed into a cap that was full of water. The cap would nearly seal the front part of the print head. print head. Afterwards, the syringe was removed from the top part of the print head and a cap was Afterwards, theprefilled syringe was print head andfirmly. a cap was usedthe thatprint was used that was withremoved water tofrom seal the thetop toppart partofofthe the print head Finally, prefilled with water to seal the top part of the print head firmly. Finally, the print head was installed head was installed on the experimental platform. on the experimental The platform is platform. shown in Figure 8. A step voltage signal was generated by a signal generator The platformSanta is shown Figure 8. and A step signal generated by a Then, signal itgenerator (Agilent 33220A, Clara,inCA, USA) wasvoltage amplified by was a voltage amplifier. was fed (Agilent 33220A, Santa Clara, CA, USA) and was amplified a voltage amplifier. Then, it was fed to to the print head. Together with a high-speed voltagebyoutput device (PCIE 6711, National the print head. Together with a high-speed voltage output device (PCIE 6711, National Instrument, Instrument, Austin, TX, USA), the computer can generate a signal with a sampling frequency of up Austin, TX, The USA), the computer generate with a sampling of up to 1Basler, MHz. to 1 MHz. generated pulsescan were used atosignal trigger a CCD camerafrequency (scA780-54fm/fc, The generated pulses were usedcontroller to trigger(S4000, a CCDAdvanced camera (scA780-54fm/fc, Basler, Ahrensburg, Ahrensburg, Germany). A LED Illumination, Rochester, MN, USA) Germany). A LED controller (S4000, Advanced Illumination, Rochester, MN, USA) was to set was used to set the LED at continuous illuminating mode because the drop adhering to used the nozzle the LED at continuous illuminating mode because the drop adhering to the nozzle is static. A lens is static. A lens (Moritex-ML-Z07545, Saitama, Japan) was used to properly magnify the image. The (Moritex-ML-Z07545, Saitama, Japan) used to for properly magnify thean image. acquireddevice image acquired image was transported to thewas computer analysis through imageThe acquisition was transported to the computer for analysis through an image acquisition device (PCI-1405, National (PCI-1405, National Instrument, Austin, TX, USA). Instrument, Austin, TX,cap USA). Next, the bottom of the print head was unsealed. Through the imaging system, the print Next, the bottom cap of the print headof was unsealed. Through the imaging system, theofprint head head tip was wetted with a small amount water, as shown in Figure 9a. After the feed voltages tip was wetted with a small amount of water, as shown in Figure 9a. After the feed of voltages of 20 V, of 20 V, 40 V, 60 V, 80 V, 100 V, respectively, and owing to the deformation of the tubes in the print 40 V, 60 V, 80 V, 100 V, respectively, and owing to the deformation of the tubes in the print head, small amounts of water were squeezed out of the print head and imaged by the imaging system as shown in Figure 9b–f, respectively. These images were taken two seconds apart in time.

Sensors 2016, 16, 81 Sensors 2016, 16, 81

9 of 13 9 of 13

head, small amounts of water were squeezed out of the print head and imaged by the imaging head, small amounts of water were squeezed out of the print head and imaged by the ofimaging 2016, 16,in 81Figure 9b–f, respectively. These images were taken two seconds apart9 in 13 systemSensors as shown time. system as shown in Figure 9b–f, respectively. These images were taken two seconds apart in time.

Figure theexperimental experimental platform. Figure 8.Diagram Diagramofof ofthe the experimental platform. Figure 8.8.Diagram platform.

Figure 9. A sequence of images acquired at different voltage excitations. Figure 9. A 9. sequence of of images at different different voltage excitations. Figure A sequence imagesacquired acquired at voltage excitations.

To calculate the volume of the droplet, the droplet adhered to the nozzle end face was To calculate thethe volume of thedroplet, droplet, the adhered to the theface nozzle end face was Totocalculate volume of the the droplet adhered to the along nozzle end was assumed to The assumed be axisymmetric and was sliced into droplet many layers vertical direction. be axisymmetric and was into layers the verticalalong direction. volumedirection. of these The assumed toof be axisymmetric and wasmany sliced into many layers theThe vertical volume these droplets wassliced obtained through analong image processing algorithm. droplets was obtained through an image processing algorithm. volumeAt of the these droplets obtained through image algorithm. same time, was the deformation of thean glass tubeprocessing was assumed to be consistent along the At the same time, the deformation of the glass tube was assumed to be alongalong the At thethat same the deformation of the glass tube to consistent be was consistent the section wastime, surrounded by a piezoelectric tube, andwas the assumed other segment assumed to be section that was surrounded by a piezoelectric tube, and the other segment was assumed to be without section that was surrounded by a piezoelectric tube, and the other segment was assumed to be without deformation. The deformation amount of the glass tube was calculated as deformation. The deformation amount of the glass tube was calculated as without deformation. The deformation amount of the glass tube was calculated as Vinp Vinp V   r u l pie 2 1 1 def r Vde f “ 2πr1 ur1VVinp l pie (30) (30) ref V Vdef  2r1ur1 re f l pie (30) Vref ur1deformation where VinpVinp is isthe voltage,VreVfrefis is reference voltage, is the deformation the inner where theinput input voltage, thethe reference voltage, ur1 is the of the innerof surface of the glass tube under the excitation of a reference reference voltage Vrevoltage where in paper in thethis reference where Vinp theglass input voltage, V the voltage, deformation ofpaper the inner fu surface ofis the tube under the excitation of a reference Vrefthis where the r1 is the ref is

voltage is 1.0 V (as is shown in Figure 4), and l is the length of the piezoelectric tube. Each time a higher voltage was fed into the print head, an additional amount of water was Each time a 1.0 higher voltage wasof into 4), the print an length additional amount water was reference voltage is V (as is volume shown infed Figure l piehead, isvoltage the the piezoelectric tube. obtained by subtracting the the droplet atand the higher by theof volume of theof previous obtained by These subtracting volume of into the droplet at head, the by higher voltage theisvolume of the droplet. data voltage arethe compared with the result obtained Equation (30), by which shown in Each time a higher was fed the print an additional amount of water was Figuredroplet. 10. previous These data are compared with the result obtained by Equation (30), which is shown obtained by subtracting the volume of the droplet at the higher voltage by the volume of in the Figure 10.In Figure 10, it can be seen that the data obtained by theoretical computation and image processing previous droplet. These data are compared with the resultisobtained by Equation (30),computation which is shown in agree reasonably well. obtained experimentally smaller than the theoretical In Figure 10, it canThe bedata seen that the data obtained by theoretical computation and at image Figure 10. large, which may be attributed to the bulk-modulus effect of the internal fluid and the evaporation processing agree reasonably well. The data obtained experimentally is smaller than the theoretical Ineffect Figure 10,meniscus it can due be seen that size. the data obtained by theoretical computation and image of the to its small pie surface of the glass istube the excitation of4),a and reference Vrefof the where in this paper reference voltage 1.0 Vunder (as is shown in Figure l pie is voltage the length piezoelectric tube. the

computation at large, which may be attributed to the bulk-modulus effect of the internal fluid and processing agree reasonably well. The data obtained experimentally is smaller than the theoretical the evaporation effect of the meniscus due to its small size. computation at large, which may be attributed to the bulk-modulus effect of the internal fluid and the evaporation effect of the meniscus due to its small size.

Sensors 2016, 16, 81 Sensors 2016, 16, 81 Sensors 2016, 16, 81

10 of 13 10 of 13 10 of 13

Figure 10. A comparison of the deformation amount by theoretical computation and image processing. Figure by theoretical theoretical computation computationand andimage imageprocessing. processing. Figure10. 10.AAcomparison comparisonof ofthe the deformation deformation amount amount by

5. Further Discussion 5. Further Discussion 5. Further Discussion To study the influence of tube wall thickness on the deformation and stress statuses, it is To study the influence of tube wall thickness on the deformation and stress statuses, it is essential to investigate the displacement field when the deformation radius of each tube is changed. in To study the influence of tube wall thickness on the and stress statuses, As it isshown essential essential to investigate the displacement field when the radius of each tube is changed. As shown in Figure 11, when the innerfield radius of the radius glass tube is changed, the wallAs displacements would to investigate the only displacement when of each tube is changed. shown in Figure 11, Figure 11, when only the inner radius of the glass tube is changed, the wall displacements would changeonly significantly. symbols , andis uchanged, refer to the displacements at the when the inner The radius of the uglass the 11 wall displacements would change r1 , ur 2tube r 3 in Figure change significantly. The symbols ur1 , ur 2 , and ur 3 in Figure 11 refer to the displacements at the significantly. symbols in Figure refer to theIn displacements the surfaces  r2 u, r3 r  r3 11 surfaces withThe a radius of urr1, ru1 r2 , ,rand and respectively. addition, it isatnecessary to with note r  r r  r r  r surfaces with a radius of , , and respectively. In addition, it is necessary to note 1 2 3 athat radius r “ rto r “ displacement r2 , and r “ r3 respectively. In of addition, it is necessary note tube that uisr1changed. refers to 1 , the ur1 of refers after the value the inner radius of thetoglass ur1 refers toafter the displacement after theradius value of ofthe theglass innertube radius of the glass tube isvalue changed. thatdisplacement the the value of the inner is changed. When the of r1 When the value of r1 is small, which means the thickness of the glass tube is relatively large, the When the valuemeans of r1 the is small, which means thickness of the glass is relativelyoflarge, the is small, which thickness of the glassthe tube is relatively large, thetube displacements the tube displacements of the tube walls are all small. The displacements of the inner and outer surfaces of walls are all small. The displacements the inner outer surfaces of the piezoelectric tube share displacements of the tube walls are allofsmall. The and displacements of the inner and outer surfaces of the piezoelectric tube share nearly the same displacement, and they are all larger than the nearly the same displacement, and they all larger than the displacement of all the larger inner surface of the piezoelectric tube share nearly thearesame displacement, and they are than the displacement of the inner surface of the glass tube. When the value of r1 increases above 0.25 mm, the glass tube. of When the value of rof mm, glass becomes above thinner0.25 andmm, the r1 increases displacement the inner surface the glassabove tube. 0.25 When thethe value oftube 1 increases the glass tube becomes thinner and the system rigidity decreases. In addition, the displacement of system rigidity decreases.thinner In addition, thesystem displacement each tube In increases significantly. Moreover, the glass tube becomes and the rigidityofdecreases. addition, the displacement of each tube increases significantly. Moreover, the displacements of the inner surface of the glass tube the the inner surface of the glass tube and the outer of the piezoelectric eachdisplacements tube increasesofsignificantly. Moreover, the displacements of thesurface inner surface of the glass tube tube andnearly the outer surface of the areinner nearly the same, the displacement of the are the same, while thepiezoelectric displacementtube of the surface of thewhile piezoelectric tube is different and the outer surface of the piezoelectric tube are nearly the same, while the displacement of the innerissurface of the piezoelectric tube is different and is the smallest. and the smallest. inner surface of the piezoelectric tube is different and is the smallest. 7 7

ur1 ur1 ur2 ur2 ur3 ur3

6 6 5 5 4 4 3 3 2 2 1 1 0 02 2

2.2 2.2

2.4 2.4

2.6 2.6

2.8 2.8

3 3

Figure11. 11.Displacements Displacementsof ofthe thetube tubesurfaces surfaceswhen whenthe thevalue valueof of rr11 isischanged. Figure changed. Figure 11. Displacements of the tube surfaces when the value of r1 is changed.

of the the outer outer radius radius of of piezoelectric piezoelectric tube is changed, changed, the displacements displacements of When only the value of When only the value of the outer radius of piezoelectric tube is changed, the displacements of tube surfaces surfacesare areshown shownininFigure Figure12. 12.Note Notethat thatur3uris3 the is the displacement of the outer surface of the tube displacement of the outer surface of the the tube surfaces are shown in Figure 12. Note that u is the displacement of the outer surface of the piezoelectric tube after the value of r3 is changed.r 3 It could be found that the displacements of the piezoelectric tube after the value of r is changed. It could be found that the displacements of the inner and outer surfaces of the glass3 tube are nearly the same. The displacement difference the inner and outer surfaces of the glass tube are nearly the same. The displacement difference

Sensors 2016, 16, 81

11 of 13

piezoelectric inner Sensors 2016, 16,tube 81 after the value of r3 is changed. It could be found that the displacements of the 11 of 13 Sensors 2016, 16, 81 of 13 and outer surfaces of the glass tube are nearly the same. The displacement difference between the11inner and outer the surfaces the outer piezoelectric increases as r3 increases, and theas inward displacement of between innerof and surfacestube of the the piezoelectric tube increases increases increases, and the the between the inner and outer surfaces of piezoelectric tube as rr33 increases, and the inner surface of the piezoelectric tube is larger than the outer surface, which means the piezoelectric inward displacement displacement of of the the inner inner surface surface of of the the piezoelectric piezoelectric tube tube is is larger larger than than the the outer outer surface, surface, inward tube is under athe “radially elongated” state. From Figure 12, we can conclude that Figure the drive effect is which means piezoelectric tube is under a “radially elongated” state. From 12, we can can which means the piezoelectric tube is under a “radially elongated” state. From Figure 12, we strengthened thedrive thickness the piezoelectric tubethickness is increased. conclude that thatas the effect of is strengthened strengthened as the the of the the piezoelectric piezoelectric tube tube is is increased. increased. conclude the drive effect is as thickness of 5 5

ur1 ur1 ur2 ur2 ur3 ur3

4.5 4.5 4 4 3.5 3.5 3 3 2.5 2.5 2 2 1.5 1.5

6 6

6.2 6.2

6.4 6.4

6.6 6.6

6.8 6.8

7 7

7.2 7.2

7.4 7.4

Figure 12. Displacements of the tube surfaces when the value of r3 is changed. Figure changed. Figure12. 12.Displacements Displacementsof ofthe thetube tubesurfaces surfaceswhen whenthe thevalue valueof of rr3 isischanged.

Figure 13 13 shows shows the the results results of of the the displacements displacements for the the tube tube surfaces surfaces when when only only the the value value of of Figure Figure 13 shows the results of the displacements forfor the tube surfaces when only the value of r2 is r is changed, that is, the inner radius of the piezoelectric tube and the outer radius of the glass tube r22 is changed, that the inner radius the piezoelectric andouter the outer radius of glass the glass changed, that is, theis,inner radius of theofpiezoelectric tube tube and the radius of the tubetube are r are changed simultaneously. When is relatively small, in other words, when the glass tube 2 r are changed simultaneously. When small, other when words,thewhen tube changed simultaneously. When r2 is relatively small, in otherinwords, glassthe tubeglass becomes 2 is relatively becomes thinner and the the tube piezoelectric tube becomes becomes thicker, alla larger tube walls walls have aa Likely, larger thinner and the piezoelectric becomes tube thicker, all tube thicker, walls have displacement. becomes thinner and piezoelectric all tube have larger displacement. Likely, when becomes larger, larger, dueexcessive to insufficient insufficient drive and excessive rigidity, rigidity, the when r2 becomes larger, due to drive and rigidity, theand displacements of the tube rr22 insufficient displacement. Likely, when becomes due to drive excessive the displacements of the tube surfaces become considerably smaller. surfaces become smaller. displacements ofconsiderably the tube surfaces become considerably smaller. 15 15

ur1 ur1 ur2 ur2 ur3 ur3

10 10

5 5

0 03.5 3.5

3.7 3.7

3.9 3.9

4.1 4.1

4.3 4.3

4.5 4.5

Figure 13. Displacements of the tube surfaces when the value of r2 is changed. Figure changed. Figure13. 13.Displacements Displacementsof ofthe thetube tubesurfaces surfaceswhen whenthe thevalue valueof of rr22 isischanged.

6. Conclusions Conclusions 6. 6. Conclusions The steady steady state state response response of of aa tubular tubular piezoelectric piezoelectric print print head head under under the the excitation excitation of of electric electric The The steady state response of a tubular piezoelectric print head under the excitation of electric voltage, together with the effect of the tubes’ wall thicknesses on their deformation and stress voltage, together togetherwith withthethe effect of tubes’ the tubes’ wall thicknesses ondeformation their deformation and stress voltage, effect of the wall thicknesses on their and stress statuses, statuses, were studied in detail in this paper. The following conclusions were obtained. statuses, wereinstudied inthis detail in this paper. The following conclusions were obtained. were studied detail in paper. The following conclusions were obtained. (1) When the piezoelectric tube is under the excitation of a positive step voltage, the the displacements displacements (1) When the piezoelectric tube is under the excitation of a positive step voltage, of the the glass glass tube tube and and the the piezoelectric piezoelectric tube tube are are inward inward towards towards the the central central axis. axis. However, However, their their of radial stress directions are opposite. The glass tube is compressed and has an inward radial radial stress directions are opposite. The glass tube is compressed and has an inward radial stress, while while the the radial radial stress stress of of the the piezoelectric piezoelectric tube tube is is outwardly outwardly oriented. oriented. stress,

Sensors 2016, 16, 81

12 of 13

(1)

When the piezoelectric tube is under the excitation of a positive step voltage, the displacements of the glass tube and the piezoelectric tube are inward towards the central axis. However, their radial stress directions are opposite. The glass tube is compressed and has an inward radial stress, while the radial stress of the piezoelectric tube is outwardly oriented. (2) For the special tubular piezoelectric print head discussed above, the displacements of the tube walls are obtained at the several or dozens of nanometers level, which depend on the input voltage. (3) The hoop stress is larger than the radial stress on the whole, and the potential is not linearly distributed along the radial direction. (4) When the glass tube becomes thinner, the system has a smaller rigidity so that a larger deformation field of the tubes is obtained. Similarly, when the piezoelectric tube becomes thicker, the system’s drive effect becomes more significant so that a larger deformation field of the tubes is acquired. Acknowledgments: This work is supported by the Fundamental Research Funds for the Central Universities (No. HIT.IBRSEM.2013038) and the National Natural Science Foundation of China (No. 51105116). These supports are gratefully acknowledged. Author Contributions: All authors contributed to the mathematical analysis and design of the experiment. J.C. and Y.L. co-wrote the manuscript, and all authors contributed to the refinement of the paper. Conflicts of Interest: The authors declare no conflict of interest.

Nomenclature c11 ,c13 ,c33 Dr e31 ,e33 E l pie

elastic moduli at constant electric field; electric displacement vector; piezoelectric constants; elastic modulus; length of the piezoelectric tube; radial stress of the inner surface piezoelectric tube; radial coordinate; radii of glass and piezoelectric tubes; stresses;

P1 r r1 ,r2 ,r3 σrr ,σθθ

ur1 Vde f Vin ε33 θ γrr ,γθθ ∆V ϕ µ ur

deformation of the inner surface of glass tube; deformation volume of glass tube; input voltage; permittivity component at constant strain; circumferential coordinate; strain; amount of volume compression; electrical potential; Poisson’s ratio. radial displacement;

Appendix A The property parameters of PZT-5H used in the work is shown as below. » — — — — rcs “ — — — –

12.7 ˆ 1010 8.02 ˆ 1010 8.47 ˆ 1010 0 0 0

8.02 ˆ 1010 12.7 ˆ 1010 8.47 ˆ 1010 0 0 0 »

8.47 ˆ 1010 8.47 ˆ 1010 11.7 ˆ 1010 0 0 0

0 0 0 2.30 ˆ 1010 0 0

0 0 0 0 2.30 ˆ 1010 0

0 0 0 0 0 2.35 ˆ 1010

fi 0 0 0 0 17.034 0 — ffi res “ – C{m2 0 0 0 17.034 0 0 fl ´6.228 ´6.228 23.24 0 0 0 » fi 1.5 ˆ 10´8 0 0 — ffi rεs “ – C2 {N ¨ m2 0 1.5 ˆ 10´8 0 fl ´ 8 0 0 1.3 ˆ 10

fi ffi ffi ffi ffi ffi ffi ffi fl

N{m2

Sensors 2016, 16, 81

13 of 13

References 1. 2. 3. 4. 5. 6. 7. 8. 9.

10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

Gysling, H.J. Nanoinks in inkjet metallization—Evolution of simple additive-type metal patterning. Curr. Opin. Colloid Interface Sci. 2014, 19, 155–162. [CrossRef] Dijksman, J.F.; Duineveld, P.C.; Hack, M.J.J.; Pierik, A.; Rensen, J.; Rubingh, J.E.; Schram, I.; Vernhout, M.M. Precision ink jet printing of polymer light emitting displays. J. Mater. Chem. 2007, 17, 511–522. [CrossRef] Holthoff, E.L.; Farrell, M.E.; Pellegrino, P.M. Standardized sample preparation using a drop-on-demand printing platform. Sensors 2013, 13, 5814–5825. [CrossRef] [PubMed] Haldar, A.; Liao, K.; Curran, S. Fabrication of inkjet printed organic photovoltaics on flexible Ag electrode with additives. Sol. Energy Mater. Sol. Cells 2014, 125, 283–290. [CrossRef] Kim, J.D.; Chio, J.S.; Kim, S.B.; Choi, Y.C.; Cho, Y.W. Piezoelectric inkjet printing of polymers: Stem cell patterning on polymer substrates. Polymer 2010, 51, 2147–2154. [CrossRef] Dijksman, J.F.; Pierik, A. Fluid dynamical analysis of the distribution of inkjet printed biomolecules in microarray substrates for genotyping application. Biomicrofluidics 2008, 2. [CrossRef] [PubMed] Jeranˇce, N.; Vasiljevi´c, D.; Samardži´c, N.; Stojanovi´c, G. A compact inductive position sensor made by inkjet printing technology on a flexible substrate. Sensors 2012, 12, 1288–1298. [CrossRef] [PubMed] Fuchiwaki, Y.; Tanaka, M.; Makita, Y.; Ooie, T. New approach to a practical quartz crystal microbalance sensor utilizing an inkjet printing system. Sensors 2014, 14, 20468–20479. [CrossRef] [PubMed] Haque, R.I.; Ogam, E.; Loussert, C.; Benaben, P.; Boddaert, X. Fabrication of capacitive acoustic resonators combining 3D printing and 2D inkjet printing techniques. Sensors 2015, 15, 26018–26038. [CrossRef] [PubMed] Zoltan, S.I. Pulsed Droplet Ejecting System. U.S. Patent 3683212, 8 August 1972. Dijksman, J.F. Hydrodynamics of small tubular pumps. J. Fluid Mech. 1984, 139, 173–191. [CrossRef] Shin, D.Y.; Grassia, P.; Derby, B. Numerical and experimental comparisons of mass transport rate in a piezoelectric drop-on-demand inkjet print head. Int. J. Mech. Sci. 2004, 46, 181–199. [CrossRef] Wijshoff, H. The dynamics of the piezo inkjet printhead operation. Phys. Rep. 2010, 491, 77–177. [CrossRef] Kim, B.H.; Lee, H.S.; Kim, S.W.; Kang, P.; Park, Y. Hydrodynamic responses of a piezoelectric driven MEMS inkjet print-head. Sens. Actuators A Phys. 2014, 210, 131–140. [CrossRef] Bugdayci, N.; Bogy, D.B.; Talke, F.E. Axisymmetric motion of radially polarized piezoelectric cylinders used in ink jet printing. IBM J. Res. Dev. 1983, 27, 171–180. [CrossRef] Larbi, W.; Deü, J.F. A 3D state-space solution for free-vibration analysis of a radially polarized laminated piezoelectric cylinder filled with fluid. J. Sound Vib. 2011, 330, 162–181. [CrossRef] Chen, W.Q.; Bian, Z.G.; Lv, C.F.; Ding, H.J. 3D free vibration of a functionally graded piezoelectric hollow cylinder filled with compressible fluid. Int. J. Solids Struct. 2004, 41, 947–964. [CrossRef] Lee, E.R. Microdrop Generation; CRC Press: Boca Raton, FL, USA, 2003; pp. 128–185. Timeshenko, S. Strength of Materials, Part II; Van Nostrand Co. Inc.: New York, NY, USA, 1940; pp. 236–241. Peelamedu, S.M.; Kosaraju, C.B.; Dukkipati, R.V.; Naganathan, N.G. Numerical approach for axisymmetric piezoceramic geometries towards fluid control applications. J. Syst. Control Eng. 2000, 214, 87–97. [CrossRef] Radovic, M.; Lara-Curzio, E.; Riester, L. Comparison of different experimental techniques for determination of elastic properties of solids. Mater. Sci. Eng. 2004, 368, 56–70. [CrossRef] © 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons by Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).