Dynamic Modeling

7.2 Piezoactuator-Driven Jetting Dispenser

7.2.3 Dynamic Modeling

The dispenser is a multiphysics system of interacting fl uids and structures. A lumped parameter modeling method is adopted to describe the dynamic behavior of the struc- tural parts (the piezostack, the fl exible beam, and the needle part) as well as the fl uid part (adhesive fl ows). Firstly, the dynamic modeling of the structural parts is performed considering dynamic behaviors of the piezostack, the fl exible beam, and the needle part. The dynamics of the piezostack can be mathematically expressed as follows:

ep c p p p p p p p0 p

(m +m y) +b y +k y =k y V−F (7.17) where

m_ep = m_p/3 is the dynamic effective mass of the piezostack

m_p and m_c are the mass of the piezostack and the contacting cap, respectively b_p, k_p, and y_p are the damping coeffi cient, the stiffness, and the displacement of

the piezostack, respectively

F_p is the force acting on the fl exible beam from the piezostack V is the applied voltage to the piezostack

y_p0 is the free displacement of the piezostack due to a unit of the applied voltage In order to model the forced response of the beam, a lumped parameter method is employed. Figure 7.13a shows a simplifi ed structure of the fl exible beam. The beam is fi xed at one end and consists of three parts: the slender, the middle, and the stiff part. The two forces acting on the beam are the force from the piezostack (F_p) and the force from the needle part (F_n). It is noted that, in order to reduce computation load, the fi rst part of the beam is assumed to be fi xed and not included in the simplifi ed structure. This is reasonable because this part is very stiff and clamped to the body.

Using the lumped parameter method proposed by Irvin [31], the free body diagram of the beam is shown in Figure 7.13b. From the free body diagram, the following moment equations can be obtained:

=

− + Δ +

∑

− = = ^…

n 1 1 1 1 1 1

1

( 0.5) ( ) 0; ( 1, , )

i

i j

j

F i l k l i j l m y i n (7.18a)

=

= +

+ − − + − −

⎡ ⎤

+ ⎣ − + + − − ⎦

+ − + Δ = = + +

∑

∑

…

1

n 1 1 2 p 1 2

1 1 1 2 1

1

2 2 2 2 1 1 2

1 1

[ ( 0.5) ] ( 0.5)

( 0.5) ( 0.5)

( ) 0; ( 1, , )

n

j j

i

j i

j n

F L i n l F i n l

n j l i n l m y

i j l m y k l i n n n (7.18b)

= = + +

+

= +

+ + − − − − + − − − + Δ

+ − + + + − − − + −

+ + − + + − − − =

= + + + +

∑ ∑

∑

…

1

1 2

1 2

1

n 1 2 1 2 3 p 2 2 1 3 3 3

1 1 2 1 2 3 1 3

1 1

1 2 2 1 2 3 2

1

1 2 1 2 3

[ ( 0.5)] [ ( 0.5) ]

[( 0.5) ( 0.5) ] ( )

[( 0.5) ( 0.5) ] 0;

( 1, ,

i

n i

j j

j j n n

n n

j j n

F L L i n n l F L i n n l k l

n j l L i n n l m y i j m y

n n j l i n n l m y

i n n n n n ) (7.18c)

where

L₁, L₂, and L₃ are the lengths of the stiff, the middle, and the slender part, respectively

n₁, n₂, and n₃ are the number of lumps of the stiff, the middle, and the slender part, respectively

l₁, l₂, and l₃ are the base lengths of each full triangle of the stiff, the middle, and the slender part, respectively

Fp Stiff part Fn

Slender part

L3 L2 L1

Middle part (a)

Δ_n

1+1 Δ_n

1 Δ₂

y_n

1+n₂+n₃ y_n

1+n₂₊₁ y_n

1+n₂ y_n

1+1 y_n

1 y2

Δ₁

y1 y0

Fn

k1, l1

k1, l1

k1, l1

yp

m1

m1

m1

m2

m2

m2

m3

m3

m3

Δ_n Δ_n 1+2

1+n2

Δ_n

1+n2+1

Δ_n

1+n2+2

Δ_n

1+n2+n³

k3, l3 k3, l3 k3, l3 k2, l2 k2, l2 k2, l2

(b)

FIGURE 7.13 Lumped parameter model of the fl exible beam. (a) Schematic diagram and (b) free body diagram.

l_k = L_k/n_k (i = 1,2,3)

k₁, k₂, and k₃ are the stiffness constants of each lump of the stiff, the middle, and the slender part, respectively

m₁, m₂, and m₃ are the lump masses of the stiff, the middle, and the slender part, respectively

Δj is the net defl ection of the jth lump (a positive value of Δj refl ects an elongation of the spring)

y_j is the neutral axis defl ection of the beam at the midpoint of the jth lump It is assumed that the geometric parameters and material properties of each part of the above beam are constants, thus one has

; 3^k ( 1, 2,3)

k k k k

k

m A l k EI k

= ρ = l = (7.19)

where

ρ and E are density and Young’s modulus of the beam material, respectively I_k and A_k are inertia moments and cross-section areas of the ith part of the beam,

respectively

The net defl ection Δi relates to the neutral axis defl ection y_i as follows:

1 1 1 1

1 1 1 1

1 2 1 2 1 2 1

1 1 1 1 1 2 1 2

2 1

1 0 1 2 1 1

1 2 1 2

2 1

1 1 2

1 2 1 2

3 2

1

2 3 2 3

2 ( 1, 1, , 1);

2 2

2 3 ; 3 ;

2 2

3 ;

2 2

3

i i i i

n n n n

n n n n

n n n n n n n n

y y y i n ,n n n n n

l l

y y y y y y

l l l l

l l

y y y

l l l l

l l

y y y

l l l l

− +

− +

+ + +

+ + − + +

Δ = − + ≠ + + + +

⎛ ⎞

Δ = − + Δ = −⎜⎝ − + ⎟⎠ + +

⎛ ⎞

Δ = + −⎜⎝ − + ⎟⎠ +

⎛ ⎞

Δ = −⎜⎝ − + ⎟⎠ + + ²

1 2 1 2 1 2 1 2

1

3 2

1 1 2

2 3 2 3

;

2 2

3 .

n n n n n n n n

l l

y y y

l l l l

+

+ + + ⎛ ⎞ + + + +

Δ = + −⎜⎝ − + ⎟⎠ +

(7.20)

It is noted that the beam is clamped at one end, then from the free body diagram one has y_{n1+ +n2 n3} =0. If the beam is excited by a harmonic force whose frequency is ω, the term ÿ_i in the above equations can be replaced by −ω² y_i, which results in equivalent simultaneous linear equations of the beam. Thus far, (n₁ + n₂ + n₃) simultaneous equations have been developed for the beam. These equations can be integrated with dynamic equations of the piezostack and the needle part to provide structural governing equations of the dispenser. It is also remarked that by assuming

the piezostack and the needle always contact the beam (this assumption is true when the operating frequency is signifi cantly lower than the resonance frequency of the beam and the needle part), one has

1 1 1 12

p n 0

1 2

n n ;

y l y l

y y y

l l

+ +

= =

+

where y_n is the needle displacement obtained from the dynamics of the needle.

Figure 7.14a shows the schematic diagram of the needle part integrating with the dispensing fl uid part of the dispenser. In the fl uid part, the behaviors of the adhesive in the ball-seat chamber ( f₀), the dispensing fl ow in the nozzle ( f₁), the back/fi ll- ing fl ow in the seat–annular duct ( f₂), the fl ow in the dispensing chamber ( f₃), and the fl ow in the connecting pipe ( f₄) are considered. In this section, an equivalent Bingham model is used to express rheological behavior of the adhesive. Based on the analogy between a fl uid system and a mechanical system, a lumped parameter model is employed to express dynamic behavior of the fl uid part. In the lumped parameter model, the fl uid system is divided into lumps with lumped masses and

(a)

m_{f 21} bf 21

bf 31

b_{f 3N}

b_{f 4N} PsyApipe

b_{f 41} bf 2N

k_{f 11} b_{f 11} b_{f 1N} k_{f 21}

k_{f 41} k_{f 2N}

k_{f 3N}

xf 2N x_{f 1N}

x_{f 3N}

xf 21 x_{f 11}

x_{f 31}

x_{f 41} x_{f 4N}

ΔPA_{f 2}

x_{f 0} k_{f 0} x_ef0

Fτ21 Fτ11

Fτ31

F_τ4n F_τ4N

F_τ3N

F_τ2N F_τ1N

fv2N y·n

f_v3N y·n

f_v21 y·n

f_v31 y·n

xef 2

xef 3

yn

Fn

k_s bnp

Annular duct (f2) Syringe

Dispensing flow (f1) Needle

Ball-seat chamber (f0) Pressurized air

PSy

Pipe flow (f4) Dispensing chamber (f3) Fn

Spring

(b)

m_{f 2N} m_{f 1N}

m_{f 3N}

m_{f 41} m_{f 4N} m_{f 31}

m_{f 0} mnp

m_{f 11}

FIGURE 7.14 Lumped parameter model for the needle and the fl uid part. (a) Schematic diagram and (b) free body diagram.

average parameters such as velocity and pressure. The system elements are obtained by applying conservation of mass and Newton’s law to the lumps of the fl uid. This method was fi rst proposed by Doebelin [32] for Newtonian fl uid in a pipe. Nguyen et al. [26] developed the lumped parameter models for Bingham fl uid fl ow in a cir- cular pipe and axial Couette fl ow in an annular duct, and successfully applied them in modeling a jetting dispensing process. Figure 7.14b shows the equivalent lumped model diagram of the needle part and the fl uid part. Noteworthily, in the lumped model, fl uid compressibility is represented via spring elements, fl uid inertia effect is represented via mass elements, and viscous shear force is represented by damping elements. From the diagram, the dynamic equation of the needle part can be math- ematically expressed by

np n np n s n n S0

m y +b y +k y =F −F (7.21)

where

mnp is the mass of the needle part including the transition rod, the connector, and the needle

bnp is the damping coeffi cient of the needle part ks is the stiffness of the actuating spring yn is the displacement of the needle FS0 is the pre-stressed force of the spring

The dynamic equations of the adhesive fl ows can be also obtained from the lumped model diagram as follows:

0 0 0 0 e 0

(mf )xf +kf (xf −xf )=0 (7.22a)

0( 0 e 0) 0

f f f

k x x

− − = (7.22b)

11 11 11 11 11 11 12 ,11

12 12 12 12 11 11 12 12 12 13 ,12

1 1 1 1 1 1 1 1 1 ,1

( )

( ) ( )

( )

f f f f f f f

f f f f f f f f f f

f N f N f N f N f N f N f N N

m x b x k x x F

m x b x k x x k x x F

m x b x k x x F

τ

τ

− − τ

+ + − = −

+ − − + − = −

+ − − = −

(7.22c)

21 21 21 21 21 21 22 ,21 n 2 ,21

22 22 22 22 21 21 22 22 22 23 ,22 n ,21

2 2 2 2 2 1 2 1 2 2 2 e 2 ,2 n ,2

( )

( ) ( )

( ) ( )

f f f f f f f v f

f f f f f f f f f f v

f N f N f N f N f N f N f N f N f N f v N N

m x b x k x x f y PA F

m x b x k x x k x x f y F

m x b x k x x k x x f y F

τ

τ

− − τ

+ + − + = −Δ −

+ − − + − + = −

+ − − + − + = −

(7.22d)

2 ( 2 e 2) 0

f N f N f

k x x

− − = (7.22e)

31 31 31 31 31 31 32 ,31 n ,31

32 32 32 32 31 31 32 32 32 33 ,32 n ,32

3 3 3 3 3 1 3 1 3 3 3 e 3 ,3 n ,3

( )

( ) ( )

( ) ( )

f f f f f f f v

f f f f f f f f f f v

f N f N f N f N f N f N f N f N f N f v N N

m x b x k x x f y F

m x b x k x x k x x f y F

m x b x k x x k x x f y F

τ

τ

− − τ

+ + − + = −

+ − − + − + = −

+ − − + − + = −

(7.22f)

3 ( 3 e 3) 0

f N f N f

k x x

− − = (7.22g)

41 41 41 41 41 41 42 ,41

42 42 42 42 41 41 42 42 42 43 ,42

4 4 4 4 4 1 4 1 4 Sy 4 ,4

( )

( ) ( )

( )

f f f f f f f

f f f f f f f f f f

f N f N f N f N f N f N f N f N

m x b x k x x F

m x b x k x x k x x F

m x b x k x x P A F

τ

τ

− − τ

+ + − = −

+ − − + − = −

+ − − = −

(7.22h)

In the above, Equations 7.22b, 7.22e, and 7.22g are the additional equilibrium equations of the massless lumps x_{ef 0}, x_{ef 2}, and x_{ef 3}, respectively. m_{f 0} and k_{f 0} are the analogous mass and stiffness of the adhesive in the ball-seat chamber. m_f1i, bf1i, k_f1i, m_{f 2i}, b_{f 2i}, k_{f 2i}, m_{f 3i}, b_{f 3i}, k_{f 3}, m_f4i, b_f4i, and k_f4i are the analogous mass, the damping and the stiffness of the ith lumps of the adhesive in the nozzle, the seat–annular duct, the dispensing chamber, and the connecting pipe, respectively.

xf1i, x_{f 2i}, x_{f 3i}, and x_f4i are the displacements of the ith lumps of the adhesive in the nozzle, the seat–annular duct, the dispensing chamber, and the connecting pipe, respectively. A_n, A_f1, A_{f 2}, A_{f 3}, and A_f4 are the effective cross-section area of the needle, the cross-section areas of the nozzle, the seat–annular duct, the dis- pensing chamber, and the connecting pipe, respectively. P_Sy is the pressure at the syringe. F_τ,1i, F_τ,2i, F_τ,3i, and F_τ,4i are the additional forces due to the yield stress of the adhesive. f_v,2i and f_v,3i are the frictional coeffi cients due to the needle motion.

xef 0, x_{ef 2}, and x_{ef 3} are the displacements of the adhesive at the exits of the ball-seat chamber, the seat–annular duct, and the dispensing chamber, respectively. ΔP is the pressure drop due to fl ow contraction and expansion at the needle end, which can be calculated as follows:

2

2 2

Re CE 2 2

con

1 2

f f

P C A x

Δ = ρ ς A (7.23)

The local loss factor of the back/fi lling fl ow due to the contraction and expansion, ςCE, can be calculated as follows: