MODEL DEVELOPMENT FOR THIN BEAMS Ralph C. Smith Department of - - PowerPoint PPT Presentation

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Dec 05, 2023 105 likes •236 views

MODEL DEVELOPMENT FOR THIN BEAMS Ralph C. Smith Department of Mathematics North Carolina State University APPLICATIONS PVDF Polyimide (b) (a) (c) (d) Note: (a) Thin beam with surface-mounted PZT patches employed as a prototype for

SLIDE 1

MODEL DEVELOPMENT FOR THIN BEAMS Ralph C. Smith Department of Mathematics North Carolina State University

SLIDE 2

APPLICATIONS

(c) (d)

PVDF

(b)

Polyimide

(a)

Note: (a) Thin beam with surface-mounted PZT patches employed as a prototype for vibration control. (b) Polymer unimorph comprised of PVDF and polyimide presently considered for pressure sensing and flow control. (c) Curved THUNDER transducer whose width is small compared with the length. d) Electrostrictive MEMs actuator employed as a high speed shutter.

SLIDE 3

FORCE AND MOMENT BALANCING

+∆x (x)

∆ +∆x ( ) M x x + x x+∆x Q( )

Q(x)

x Polyimide PVDF x x w f w f

Force Balance: x+∆x

x

ρ∂2w ∂t2 (t, s)ds = Q(t, x + ∆x) − Q(t, x) + x+∆x

x

f(t, s) − γ∂w

∂t (t, s)

ds
Divide by ∆x and take ∆x → 0 to obtain

ρ∂2w ∂t2 + γ∂w ∂t = ∂Q ∂x + f

SLIDE 4

FORCE AND MOMENT BALANCING

+∆x (x)

∆ +∆x ( ) M x x + x x+∆x Q( )

Q(x)

x Polyimide PVDF x x w f w f

Force Balance: x+∆x

x

ρ∂2w ∂t2 (t, s)ds = Q(t, x + ∆x) − Q(t, x) + x+∆x

x

f(t, s) − γ∂w

∂t (t, s)

ds
Divide by ∆x and take ∆x → 0 to obtain

ρ∂2w ∂t2 + γ∂w ∂t = ∂Q ∂x + f Moment Balance: M(t, x + ∆x) − M(t, x) − Q(t, x + ∆x)∆x + x+∆x

x

f(t, s)(s − x)dx = 0

Note: Limit yields Q = ∂M

∂x

Beam Model:

ρ∂2w ∂t2 + γ∂w ∂t − ∂2M ∂x2 = f

SLIDE 5

MOMENT RELATION Heuristic: M(t, x) = −α∂2y ∂x2 α = Y h3b 12

SLIDE 6

MOMENT RELATION Heuristic: M(t, x) = −α∂2y ∂x2 α = Y h3b 12 More Generally: M(t, x) = −Y I(x)∂2y ∂x2 − cI(x) ∂3y ∂x2∂t

SLIDE 7

BEAM MODEL: STRONG FORMULATION Strong Formulation: ρ∂2w ∂t2 + γ∂w ∂t − ∂2M ∂x2 = f(t, x) w(t, 0) = ∂w ∂x(t, 0) = 0 M(t, ℓ) = ∂M ∂x (t, ℓ) = 0 w(0, x) = w0(x) , ∂w ∂t (0, x) = w1(x) where M(t, x) = −Y I(x)∂2y ∂x2 − cI(x) ∂3y ∂x2∂t

SLIDE 8

BEAM MODEL: WEAK FORMULATION Weak Formulation: ℓ ρ∂2w ∂t2 φdx + ℓ γ∂w ∂t φdx − ℓ M d2φ dx2dx = ℓ fφdx

ℓ ρ∂2w ∂t2 φ dx + ℓ γ∂w ∂t φ dx + ℓ Y I∂2w ∂x2 d2φ dx2 dx + ℓ cI ∂3w ∂x2∂t d2φ dx2 dx = ℓ fφ dx for all appropriate test functions φ.

SLIDE 9

FINITE DIMENSIONAL APPROXIMATION Basis and Approximate Solution:

Basis: {φj(x)}

xj xj−1 xj−2 xj+2 xj+1 1 4

Approximate Solution: wN(t, x) =

N+1

wj(t)φj(x)

Second-order Matrix System

M ¨ w + Q ˙ w + Kw = f where w = [w1(t) , . . . , wN+1(t)]T and [M]ij = ℓ ρφiφjdx [Q]ij = ℓ

γφiφj + cIφ′′

i φ′′ j

[K]ij = ℓ Y Iφ′′

i φ′′ jdx

SLIDE 10

BEAM APPROXIMATION: CONSTANT PARAMETERS Time Domain:

0.5 1 1.5 2 2.5 3 3.5 −60 −40 −20 20 40 60 time (s) displacement (µm) model response measured data

Frequency Domain:

20 40 60 80 100 1 2 3 4 5 6 7 x 10

−8

frequency (Hz) power model response measured data

SLIDE 11

BEAM APPROXIMATION: PIECEWISE-CONSTANT PARAMETERS Time Domain:

0.5 1 1.5 2 2.5 3 3.5 −60 −40 −20 20 40 60 time (s) displacement (µm) model response measured data

Frequency Domain:

20 40 60 80 100 1 2 3 4 5 6 7 x 10

−8

frequency (Hz) power model response measured data