Interdisciplinary Applied Mathematics

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(18.16)


where V is    the    voltage    drop    through    the    element    (analogous    to    velocity


in mechanics and represented by the flux term -||). At steady state (all signals are sinusoidal having a single angular frequency ш), the current and velocity (flux term) can be expressed as


m,nIm,n exp(jшt),    u    U exp(jшt),


where I and U are complex coefficients. Putting this in equation (18.16), we get



I



Ju



Rm



+ juL m,n



u,



and hence    the    total    current    in    the    squeezed    film    equivalent circuit    that


corresponds to    the    total    force is    the    sum    of    all    currents    of    the parallel


sections



Is    ^ ^    Im,n


m,n(odd)



U



E


m,n(odd)



ju



Rm



+ juLm


The imaginary and the real parts of the ratio Is/U are


T i Is    Rm,nu


Im[V)= E R2 +lv2L2


m,n(odd) Rmn +    m



( Is A    L,m,nu


6 D7 / E Д2 _|_ w2£2


U    m,n(odd) Rmn +    m


which satisfy the frequency dependency specified in equations (18.14), (18.15), respectively. This requires that Im(jj) = Fq/x and Re ( jf ) = Fi/ x. This gives


Lm



(mn)



4


2 n 9


64Apo ’



Rm



(mn)2(m2 + c2n2)



n6g3



768AW 2ge


The components Lm n and Rm,n depend on the distance g (the static gap). If the displacement is large, the component values will also vary with the displacement and hence are nonlinear in nature. The equivalent circuit of squeezed film damping is connected in parallel with the MEM device circuit, and the whole system can be solved using any standard circuit simulator. For more details on circuit modeling of squeezed film damping, see (Veijola, 2001; Turowski et al., 1998).

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