# Interdisciplinary Applied Mathematics

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(17.5)

1 (<Jt(x,y)6A)2    = 1    <Jt(x,y)2

2c06A/(d + wt(x)) 2 e0/(d + wt(x))

where at(x,y) denotes the surface electric charge density as a function of position. The total differential of equation (17.5) is given by

SU

8U    dU _

+ я—owt. da t    dwt

The voltage at time instant t, vt (see the discussion leading to equation (17.2)), can be expressed as

_ dU_ _    &t(x,y)

dat e0/(d+wt(x))

Similarly, the mechanical pressure acting along the surface pt can be expressed as

_ dU_ _ at(x,y)2 Pt dwt    2c0

Similar to the derivation of equations (17.3), (17.4), the expressions for v =    Avt    and p = Apt    in terms    of w =    Awt    and a = Aat    are    given    by

d + w0    v0    vo

v =а ——:-w and p = — —a + U • w.

eo    d + wo    d + wo

The operating point is indicated by the subscript 0, and the coefficient “0” arises because the stiffness properties of the beam are considered to be external to the transducer (Woodson and Melcher, 1968). Rewriting the above equations, the expressions for a and p are given by

a

eo

eovo

7    ^    /7    NO

d + wo    (d + wo)2

and p = —

eovo

tv +

eov2

(d + wo )2    (d + wo )3

In the absence of elastic stiffness, the exterior mechanical pressure p is completely counterbalanced by the electrostatic pressure pe, and hence