Complex representations

The steady-state response Eq. (2.9) may be written in the equivalent phase angle form as

) sin(

)

( t = A wt − φ

x

_d (2.12) from which, and substituting the actual values of M and N

2 )

output lags the input.

Introducing the nondimensional frequency﹕

wn

2.2.2 Complex Representations

Although representing a system’s response as Eq. (2.7) is physically attractive, it is a bit awkward mathematically. Our

solution to this problem will be to represent as complex

The steady-state solution is given by

e

iwt

2.3 Analysis of the Simplified Micropump for the Flow Rate

The diaphragm structure acts as a “piston” which transfers energy to the fluid in the pump chamber, when driving voltage is sine wave, in one period, the fluid flows into the pump chamber in [0, T/4] and [3T/4, T]. The fluid flows out the pump chamber in [T/4, 3T/4]. So it is just necessary to calculate the flow rate in [T/4, 3T/4]. The Mach number is the ratio between the flow velocity u and the speed of sound a and is given by:

a

M

_a

= V

(2.19)

The Mach number is a measure of the compressibility of a gas and can be thought of as the ratio of inertial forces to elastic forces. For the Mach-number in the pump chamber is approximately 5.25e-5 (less than 0.3). Under the conditions the density of a gas will not change significantly while it is flowing through a system. The flow is then considered to be incompressible even though fluid, a gas, is still considered compressible. The compression could be neglected when calculating the pump’s flow rate. The analysis of an incompressible gas flow is greatly simplified as it can be treated with the same versions of the governing equations that will be derived for liquid flows.

Fig. 9 The schematic of a diaphragm micropump for analysis of flow rate

Assume h_v is the distance between diaphragm and the bottom of pump chamber as the figure 9. It could be expressed as[28]﹕

wt h

h

_v

= ′ + δ

_dia

sin 2 π

(2.20) where δ_dia is the centerline displacement of the interior surface of the membrane, i.e. the vibration amplitude ﹔ h’ is the diaphragm’s original distance between diaphragm and pump chamber’s bottom surface﹔w is vibration frequency

Introducing a parameter to correct Eq. (2.20), The membrane shape factor

γ

﹕ it was also determined based on the static membrane displacement ﹕

Not considering the compression and leakage, the instantaneous volume (V) of the pump chamber in [T/4，3T/4] is:

BL wt h

V = ( ′ + γδ

_dia

sin 2 π )

(2.22) where L and B is the length and width of the pump chamber. The change ratio of the instantaneous volume is equal to the

instantaneous flow Q, we have:

wt BL

w V

Q = ′ = 2 π γ δ

_dia

cos 2 π

(2.23) As we known, the pump’s instantaneous flow also could be

expressed as:

Q 2 = AV

A = Bh = B ( h ′ + γδ

_dia

sin 2 π wt )

(2.24) where A and v are the instantaneous micro-diaphragm and chamber cross-section area and the flow rate in pump chamber length

direction (x direction), respectively; From the eq. (2.23), and eq.

(2.24), we have:

The eq. (2.26) can be simplified to:

dia Assume

η

is leakage factor, the mean flow rate considered the leakage is as follow:

The fundamental relation for the piezo effect will be described by the following equation:

t

Here ^Δl_l is the relative expansion of the piezoelectric material achieved by applying a voltage U between the diaphragm and the upper-electrode (t: thickness of the piezo ceramic). The

electromechanical ransformation depends on the piezoelectrical coefficient of the ceramic material, described by d₃₁, on the stiffness s and on the mechanical load T

This is especially noteworthy in the case of a combination of a rectangular piezo disk and rectangular Pyrex glass diaphragm, the load T cannot be deduced by analytic equations. For a approach, the calculation of the pump diaphragm displacement is possible assuming circular plates with a radius R_p instead of square plates for the diaphragm and piezo disk. As an essential result of this model the deflection w(r) of the pump diaphragm can be calculated by

⎟ ⎟

For r=0 the maximal deflection w(r) can be calculated to

max ³¹₂ ²

From the Eq. (2.16), (2.29) and (2.31), the diaphragm’s amplitude could be expressed as

2

Substitute Eq.(2.32) and f=1/T into Eq.(2.28):

U

This is the expression of the diaphragm pump’s mean flow rate which relates with the amplitude and frequency of the voltage, pump’s chamber length, bimorph’s parameters, leakage factor, the membrane shape factor etc..

2.4 The Added Mass and Added Damping

A primary concern in the design of the peristaltic micropump is damping effects on the frequency response. In certain simplification cases, the vibration problems can be reduced to the case of a system with one degree of freedom. In addition, the mass of the spring, beam or membrane can be neglected in comparison with the mass of the load weight . In our case, therefore, we can regard the load weight as excitation force from the piezoceramic, PZT. Although these simplifications are accurate enough in many practical cases, there are technical problems in which a detailed consideration of the accuracy of such approximation becomes necessary. With the miniaturization of the size (range of

W W

μm to ) in the microelectromechanical systems (MEMS) technology, the analysis of vibration and damping play important role in mechanical

mm

engineering design. When a micro-diaphragm vibrates in a viscous fluid, the fluid offers resistance to the motion of the diaphragm. The fluid loading can be interpreted as the sum of two forces: an inertial force that is proportional to the acceleration of the diaphragm (the proportionality constant is called added mass), and a viscous or dissipative force that is proportional to the velocity of the diaphragm (the proportionality constant is called added damping or viscous damping coefficient).

In order to determine the effect of such simplification on the frequency of vibration, an approximate method developed by Lord Rayleigh will now be discussed. In applying this method some assumption regarding the configuration of the system during vibration must be made. The frequency of vibration will then be from a consideration of the conservation of energy in the system.

Fig. 10 vibration of a beam of uniform cross section loaded at the middle with a block of weight W

Let wdenote the weight of the spring per unit length. If the weight

wl

of the beam is small in comparison with the load W，it can be assumed with sufficient accuracy that the deflection curve of the beam during vibration has the same shape as the static deflection curve for a concentrated load at the middle. is the displacement at the center of the beam (see Fig.10) and we express the displacement of any element located at s distance x from the support as :

The maximum kinetic energy of the beam itself will be

g

This kinetic energy of the vibrating beam must be added to the energy ^Wy&m2 _2g of the loaded concentrated at the middle in order to

estimate the effect of the weight of the beam on the period of

vibration. In this case the period of vibration will be the same as for a massless beam loaded at the middle at the middle by the weight

W W wl 35 +17

′= (2.34)

Considering a homogeneous beam, which is vibrating in liquid, has thickness h, length L, width B, mass density ρ_dand the mass per unit length m_d =hbρ_d. In such a simple model, the added mass and added damping per unit length of the beam are roughly:

3

and the liquid dynamic viscosity, respectively.

Lc

2.5 Analysis of the Frequency Shift with the interactions exerted on the micro-diaphragm

With a micropump, such diaphragm structure acts as a “piston” to provide power for the handling of microliter-scaled fluid volumes desired in many lab-on-a-chip chemical and biomedical applications.

In the design of the mechanical efficiency, the pump performance hangs on using resonance to generate sufficient motion of the diaphragm. We are concerned with the interactions between a system and its environment (such as the implicit pressure and shear stress at the solid-fluid interface and the force exerted by the PZT actuator) for the influence on output resonance frequency.

The micro-diaphragm (the Piezoelectric disk and Pyrex diaphragm bi-layer) is integrated with surrounding walls; therefore, it can be considered as a flexible double-clamped beam spring-mass-damper system (as Fig.8). However, the various working fluids play different roles in resistance to the diaphragm vibration. Many inputs to physical systems are periodic in nature.

For example, the forces exerted on marine structures by ocean

waves, the acoustic and electric waveforms of music and speech and mechanical vibrations exerted on structures due to unbalanced elements are all inherently cyclic or periodic in nature. Now consider a simple spring-mass-damper system under harmonic excitation in stead of the real input signals (block wave actuation) that will be taken up in the next section (as the Fig.14 on page 50).

Therefore, a second-order system with an input-output differential equation can obtained

M y && + ky = F sin( wt ) − F

_chamber (2.34) where M, k are mass, spring constant, respectively. The spring is the micro-diaphragm.

The actual input driving signals can be closely approximated by sinusoidal waveforms. Any physical periodic phenomena may be represented by an infinite sum of harmonically related sinusoids, and therefore knowledge of the system frequency response to a sinusoidal input provides a basis for determining the response to a broad class of periodic inputs. Now, the core of the problems is how to define the flow resistance exerted on the micro-diaphragm.

We assume viscous liquid at volume flow rate, Q, is pumped through the central diaphragm and the narrow gap between the parallel disks (as the Fig.11) [29]. The flow rate is low, so the flow is laminar and the pressure gradient due to convective acceleration in the gap is negligible compared to the gradient due to viscous forces.

We assume that the velocity profile at any cross section in the gap is the same as for fully developed flow between stationary parallel

plates. Here the flow is axisymmetric and therefore it is most convenient to take the control volume as annular ring. It is of length, , and has circumference,dr 2πr.

Fig. 11 A simplified model for diaphragm micropump

Therefore, the pressure gradient,^dp_dr, as a function of radius is

₀³

And the flow rate Q can be approximate estimated

Q = π ( a

²

− r

²

) &y

(2.36) From (2.11), we know the solution is the sum of a homogeneous component and a particular solution

) (assumed to be distinct) of the characteristic equation, and

C

_i are n

constants to be determined from the initial conditions. The solution to the homogeneous part decays exponentially with time and is only initially significant. The particular solution is a steady state oscillation of the same frequency as the excitation and it can be assumed to be of the form, w the angular frequency (rad/sec)

y = A

_d

sin( wt − φ )

_,

y wA & =

_d

cos( ) wt

(2.38) Now substitute the result (2.36) into Eq. (2.35) and integrate to find the pressure distribution

⎟

The force acted on the micro-diaphragm is given by

0

( )2

a

Fch =

∫

p r

π

rdr Thus the flow resistance can be evaluated

⎭ ⎬

The viscous damping constant is proportion to the micro-diaphragm velocity,

wA

_d

cos( ) wt

; therefore we can find the damping constant.

(2.41) Thus, the main four parameters： the fluid viscosityμ, the chamber height , the micro-diaphragm length a, the geometric constants h₀

) ln( ⁰

a

r are necessary to determine the performance of a simplified

model for diaphragm micropump on the damping effects.

Substitute the Eq. (2.41) into (2.34) can obtain

M y && + & c y + ky = F sin( wt ) − π a

²

p

₀ (2.42)

We know that the magnitude of the transfer function for a forced-excited system is given by

2 To determine the frequency at which our amplitude-response curve is

a maximum(which we will call the damped oscillation frequency, )，we need only differentiate with respect to and set the result equal to zero . Thus, we obtain [30]

w

d

w

= M is the natural frequency of the micro-diaphragm in

vacuum and 2

C

ζ

= Mk is the damping factor which is drawn on to modify the natural frequency of the actuating diaphragm in vacuum.

Hence, a damped frequency of the micro-diaphragm can be obtained with varying damping factor for light damping )

2

(ς ≤ 1 . From Eq.

(2.44), the resonance peak occurs at a frequency less than the

undamped natural frequency , with the shift increasing toward zero as the damping is increased. We define the difference between the damped and nature frequency as the frequency shift. In the following chapter “fabrication and test “I shall be examining the phenomena of frequency shift through experiments.

w

n

Fig. 12 The illustration of frequency shift

2.6 The System Dynamic for a Peristaltic Micropump

The coupled fluid–structure–electric interaction problem may be considered as a three field problem, i.e. fluid flow, structural deformation and the electric field (see Fig. 14). The effectiveness of a system is intrinsically related to its dynamic behavior. The governing equations are: Piezoelectric constitutive equation, spring-mass-damper systems and Navier–Stokes equation.

Fig. 13 The general environment for the system

2.6.1 Electrical Field

We introduce the phase durations for j = 1. . . 6, as defined in figure 13. The actuation sequences of a 3, 4, and 6 driving phase can refer to figure 5. The cycle duration T is the sum of all . We

also introduce the relative duration of each phase

Tj j j

t =T T . The

driving scheme is therefore defined by all and the total duration T.

The driving frequency is defined as

t

j

f = 1 T

^.

Fig. 14 Definition of the 3, 4 and6 phase durations.

By comparison with the driving signals of the valveless rectification micropump (diffuser/nozzle), we design the so called

“2-phase sequence” driving signals. The three chambers have no signals phase lag. We will discuss the resonance frequency peak shift due to different phase sequence later.

Fig. 15 Actuation sequences of a 2-phase peristaltic micropump

Fig. 16 Definition of the 2-phase durations.

Table 2 The comparison between cycle frequency and driving sequence

To investigate the pump performance of micropump, the pump was actuated with the operation frequency, i.e. the phase frequency.

Furthermore, the cycle frequency means the oscillation times of the micro-diaphragm through a period. Table 2 illustrates the value of cycle frequency at different driving sequence. As a example of the operation (phase) frequency 120Hz, the cycle frequency of 2, 3, 4, and 6 phase sequence are 60 , 40, 30 and 20 times respectively. It reveals the oscillation frequency of 3-phase sequence is higher than those by the 4-phase and 6-phase sequences during the same time interval. Since sinusoidal waveforms are used as the basis for representing other periodic and transient waveforms through the process of Fourier synthesis, it seems reasonable to draw an analogy between 2-phase sequence and sinusoidal waveforms for explaining the movement of micro-diaphragm. Consequently, the cycle frequency factor

α

presented in Table 3 is drawn on to modify 3, 4, and 6 phase sequence.

Table 3 The cycle frequency factor

Therefore, Using Eq. (2.38), (2.41), (2.44) and Table.3, we should consider the cycle frequency factor

α

to correct Eq. (2.41) into (2.45)

4 0 2 0 2

3 4 0

0

12 1 1 1 1

( ) ln( ) ( )

4 2 2 8

r r

c a a

h a a a

α πμ

_{2 2}

⎧ r ⎫

⋅ ⎡ ⎤ ⎡ ⎤

= ⎨ ⎩ ⎢ ⎣ + ⎥ ⎢ ⎦ ⎣ − ⎥ ⎦ − − ⎬ ⎭

^(2.45)

Where

α

_3−phase

= 0.444

_,

α

_4−phase

= 0.25

_and

α

_6−phase

= 0.167

2.6.2 Mechanical Field

In the section, the geometrical dimensions of the actuator and the diaphragm and the chamber height are design which can produce more displacement without complicated fabrication, leading to a higher expansion/compression ratio.

Alternating voltage causes the PZT component to expand and contract along the horizontal direction. This induces a bending stress on the diaphragm, which in turn pumps the fluid through the

chamber. From the simulations below, we turn up the interesting facts that the maximum stresses occur near the edge of the diaphragm where the largest bending moments exist (Fig. 19 and 20).

) 012 . 0 01

. 0

( <r< m

To assure pump reliability for high cycle fatigue, it is, therefore, necessary to design this pump so that the maximum stress level is kept lower than the stress endurance limit of the diaphragm material.

This requirement is vital for many types of micro devices considering the role micro pumps play in sustaining the reliability of MEMS for biomedical applications, such as lab-on-a-chip devices. Consequently, we can take the concept of reliability analysis (lifetime, operating time and electric load of a piezoelectric actuator micropump) into consideration further in the future.

Fig. 17 A top view of the stresses obtained from FEM analysis in the 1^st mode frequency

Fig. 18 The stress distributed at different position in the 1^st mode

2.6.3 Governing Equations

A piezoelectric patch is utilized as the actuator. PZT-5H, a special type of piezoelectric, is used in the simulation. Properties of PZT-5H are given in table1. The coupled electro-mechanical

constitutive equation for the actuator is

(2.46) where

ε

_kl is the mechanical strain tensor,

σ

_ij is the mechanical stress tensor, E_k is the electric field vector, e_ijk is the piezoelectric constant tensor; is the elastic stiffness constant tensor at constant electric field and it is a 6 × 6 symmetric tensor.

E

C

ijkl

Table 4 Properties of PZT-5H [31]

The Pyrex diaphragm in the peristaltic micropump is integrated with surrounding walls; therefore, it can be considered as a clamped plate as the Figure 19. The governing equation of forced vibration of a thin clamped plate is

P t f

h W W

D

_Pyrex

=

_e

−

∂ + ∂

∇

⁴

ρ

² ₂ (2.47)

where

(2.48)

and f_e is the periodic actuating force, which can be solved from equation (2.46), E is the elastic modulus of the Pyrex diaphragm,

λ

is Poisson’s ratio of the Pyrex diaphragm, ρPyrex is the density of the Pyrex diaphragm, h is the thickness of the Pyrex diaphragm, and P is the dynamic pressure exerted on the diaphragm by the liquid. To solve equation (2.47), P should be solved from the Nervier–Stokes equations at every time step. Since we assume the diaphragm to be a clamped plate, displacements, curvatures and velocities of the clamped plate at edges should be zero. Therefore, the boundary

conditions for a clamped plate can be written mathematically as

(2.49)

(2.50)

(2.51)

Fig. 19 a. Schematic of the cross section of a piezo-Pyrex-diaphragm bi-layer b. Deflection of the pump diaphragm D₁: diameter of PZT; D₂: diameter of Pyrex; S:

distance between Pyrex and PZT edges; diaphragm thickness t1 =191μm and t2 = 150 μm

Properties of the fluid used in this simulation are listed in table 5.

Since the characteristic length of the micropump is of the order of 10⁻⁶ and the Reynolds number is very low, the flow can be considered as an incompressible laminar flow, which can be described using the Navier–Stokes equations (2.52) and the mass continuity equation (2.53)

(2.52)

(2.53)

Table 5 Fluid properties

In this study, the commercial software CFD-ACE+ is used to solve the piezo-diaphragm-fluid coupling for multidisciplinary analysis.

The solution algorithm for full coupling is shown in Fig.21.The broad approach is conventional in that the fluid and the structure are solved sequentially. The model considered is shown in Fig. 22. with three layers: a square PZT stack which is easier to manufacture than a corresponding circular multi-layer stack, silicon diaphragm and chamber with two outlets. In the design of the model, the silicon diaphragm is fully covered by a PZT layer and the chamber is fully covered by the PZT–silicon diaphragm bi-layer. Fig. 23.(a)(b) indicate the dynamic analysis result of the vertical displacement and pressure of a point centered on diaphragm-fluid interface respectively. The deflections at different actuating frequencies are presented in Table 6.

Solve Velocity, Pressure Computer Geometry

Update Mesh

Computer Mesh velocity At t=t, Prescribe Initial Flow Field

Geometry, Mesh, Velocity Field

Repeat for Each Solution Iteration(until

solution stops changing) At t=t, Prescribe Initial Flow Field

Geometry, Mesh, Velocity Field

Repeat for Each Solution Iteration(until

solution stops changing)

Fig. 20 The solution algorithm for piezo-diaphragm-fluid coupled solver

Fig. 21 The three layers model under 100Hz sinusoidal excitation (water, 100V). Grids at the center area of each layer are denser.

Fig. 22 (a) Transient behavior of displacement (b) Transient behavior of chamber pressure

Fig. 23 shows the time history of the vertical displacement of a point on the top of the PZT piezoelectric component. This displacement is directly related to the vertical motion corresponding to the 1^st mode. The effect of resonance becomes apparent under the nature frequency. In the transient analysis, a dynamic maximum displacement of 0.9μm is reached after 3 cycles a 90k Hz.

Table 1 The deflections at different actuating frequencies (water)

a photograph (as Fig.24) of the finished micropump with the size of 20 mm by 20 mm. diaphragm thickness t1 =191μm and t2 = 300 μm

Fig. 23 The photo of the finished diffuser/nozzle micropump

Fem Solver: Eigen Values and frequencies

Fem Solver: Eigen Values and frequencies

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