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Compressible squeeze film damping is a phenomenon of great importance for micromachines. For example, for the optimal design of an electrostatically actuated micro-cantilever mass sensor that operates in air, it is essential to have a model for the system behavior that can be evaluated efficiently. An analytical model that is based upon a solution of the linearized Reynolds equation has been given by R.B. Darling. In this paper we explain how some infinite sums that appear in Darling’s model can be evaluated analytically. As an example of applications of these closed form representations, we compute an approximation for the critical frequency where the spring component of the reaction force on the microplate, due to the motion through the air, is equal to a certain given multiple of the damping component. We also show how some double series that appear in the model can be reduced to a single infinite series that can be approximated efficiently.

An analytical model for squeeze film damping is presented by Darling

For optimal sensor design problems, the aim is to find system parameters for which a certain real valued function that measures the sensor’s performance is maximized. In order to determine such parameters that solve the optimal design problems, it is useful to have a compact analytical model for the system behavior.

In [

Let us start with a short presentation of the spring and damping constants for a rectangular plate with two opposite edges venting. Let b denote the plate dimension in the direction of venting and let A denote the plate area. P_{A} is the ambient pressure and g_{0} is the nominal gap. Let μ denote the viscosity of the trapped gas. Define k_{n} = n

α^{2} = 12μ/(g_{0}² P_{a})

For the vibration frequency ω, according to Darling’s model we have the spring constant

k = (8ω²AP_{A}/(_{0}))[1/[ω² + (k1/α)4] + 1/[32(ω² + (k3/α)4)] + 1/[52(ω² + (k5/α)4)] + ...]

= (8ω²AP_{A}/(_{0}))(α^{4}b^{4}/^{4})[1/[α4b4ω²/π4 + 1] + 1/[32(α4b4ω²/π4 + 34)] + 1/[52(α4b⁴ω²/π⁴+54)] + ...]

and the damping constant

β = (8AP_{A}/(^{2}g_{0}))[k_{1}^{2}/[ω² + (k_{1}/α)^{4}] + k_{3}^{2}/[32(ω² + (k3/α)4)] + k52/[52(ω² + (k5/α)4)] + ...]

= (8AP_{A}/(^{2}g_{0}))(α^{4}b²/^{4}b^{4}ω²/^{4} + 3^{4})] + 1/(α⁴b⁴ω²/^{4}) + ...]\

In the sequel, we will give compact representations for k and β that do not require the evaluation of infinite series. In order to do this, we introduce some auxiliary functions in the next section.

In this section we define some auxiliary functions that appear in Darling’s model as infinite series. In

F(x) = 1/(1 + x^{4}) + 1/(3^{4}+ x^{4}) + 1/(5^{4}+ x^{4}) + 1/(7^{4}+ x^{4}) + ... (1)

then for all x unequal to zero we have the representation (with w(x)= ^{1/2})

F(x) = 2^{1/2}

and F(0)= ^{4}

As a second real function, define

B(x) = 1/[1 + x⁴] + 1/[32(3⁴ + x⁴)] + 1/[52 (5⁴ + x⁴)] + 1/[72(7⁴ + x4)] + ... (3)

then we have for all x unequal to zero

B(x)= ^{1/2}

and B(0) =

For a complex number z with −z^{2 }not equal to the square of any odd integer, let

H(z) = 1/[1 + z²] + 1/[3²(3² + z²)] + 1/[5²(5² + z²)] + 1/[7²(7² + z²)] + ... (5)

then for all z unequal to zero we have the representation

H(z) =

and H(0) = F(0) =

To prove Equation (6), the series Equation (5) can be interpreted as the Laplace-transform of the corresponding Fourier series. The sum of the Fourier series can be determined explicitly (see for example in reference [^{3} + sin(5x)/5^{3} + ... = πx(π − x)/8). Then H(z) can be obtained as the Laplace transform of the 2

H(z) = F(|ω|^{1/2}) ^{1/2})

that is F(|ω|^{1/2}) is the real part of H(z) and the imaginary part of H(z) is given by −ωB(|ω|^{1/2}). This allows deriving Equations (4) and (2) from (6).

For certain boundary conditions, in Darling’s model infinite double series appear. We give approximations for these double series that can be evaluated efficiently. We also provide bounds for the approximation error. Let the parameter λ greater than or equal to 1 be given.

For a complex number z, let D denote the double series

D(z) = Σ_{nm}d(n,m) = d(1,1) + d(1,2) + d(2,1) + d(3,1) + d(2,2) + d(1,3) + ... (7)

with

d(n,m) = 1/[(2n − 1)²(2m − 1)²(λ²(2n − 1)² + (2m − 1)² + z²)] (8)

Using the definition of H, we obtain the equation

D(z) = H((λ² + z² )^{1/2}) + 1/3²H((λ²3² + z²)^{1/2}) + 1/5²H((λ²5² + z² )^{1/2}) + ... (9)

Define the auxiliary function G(z) = tanh(0.5

D(z) = ^{1/2})/(λ² + z²) + G((λ²3² + z²)^{1/2})/(3²(λ²3² + z²))

+ G((λ²5² + z²)^{1/2})/(5²(λ²5² + z²)) + …] (10)

Thus we have reduced the double series to a standard infinite series. As an approximation for D, we propose to use the function A defined as

A(z) = ^{1/2})/(λ² + z²) + G((λ²3² + z²)^{1/2})/(3²(λ²3² + z²))

+ G((λ²5² + z²)^{1/2})/(5²(λ²5² + z²))+ … + G((λ²(2k − 1)² + z²)^{1/2})/((2k − 1)²(λ²(2k − 1)² + z²))] (11)

with a natural number k. Then we have D(z) = A(z) - R(z,k) with the remainder term R(z,k).

If z^{2 }is purely imaginary, we have |λ²(2n + 1)² + z²| ≥ λ²(2n + 1)² > 1.

Moreover, |1 + exp(−π(λ²(2n + 1)² + z²)1/2| > 0.5. Since tanh(0.5πz)= −1 + 2/(1 + exp(−πz)) this implies

|G((λ²(2n + 1)² + z²)^{1/2})| = |[−1 + 2/(1 + exp(−^{1/2}))]/(λ²(2n + 1)² + z²)^{1/2}| < 5/(λ(2n+1))

and hence

|R(z,k)| < 5^{3})[1/(2k + 1)5 + 1/(2k + 3)5 + 1/(2k + 5)5 + 1/(2k + 7)5 + …]

For k = 7, we obtain |R(z,7)| < 2 × 10^{−5}/λ^{3 }. Thus for k = 7 and purely imaginary z^{2}, A approximates D with a uniform bound that is less than 2 × 10^{−5}.

The spring and damping constants for a plate with two opposite edges venting can be expressed in terms of the functions F and B that have been introduced in

k = (8ω²AP_{A}/(_{0}))(α⁴b⁴/^{4})B(αbω^{1/2}/

with the notation introduced in

β = (8AP_{A}/(_{0 }))(α⁴b²/^{1/2}/

The dimension in Equation (13) is N s/m. Using Equations (4) and (2), these functions can easily be implemented for numerical evaluation.

For a sensor design, Li and Miller [

We will give a suitable approximation in

Force = C_{0}jωH((αbω^{1/2}/^{1/2}) (14)

where C_{0 }is a suitably chosen real constant, j^{2} = −1 and H can be computed using Equation (6).

Our representations allow the approximate computation of the frequency for which k =

x⁴B(x) = CF(x) (15)

with x = αbω^{1/2} /

cosh(w(x)) + cos(w(x)) = (2^{1/2}/

If we let w = w(x) and x = (2^{1/2}/

cosh(w) + cos(w) − sinh(w)/w − sin(w)/w = (

If we replace the functions by the corresponding power series, this yields the equation

w⁴/15 + w⁸/22680 + …. =(

By considering only the constant term and the terms with w^{4} we obtain the approximation

w = [^{1/4} (19)

For Equation (15) this yields the approximate solution

x = (2^{1/2}/^{1/4} (20)

If we replace the denominator (0.4 −

ω = [2π/(α²b²)][C/0.4]^{1/2} (21)

Example With C = 2 × 10^{−7}, Equation (17) has the solution w = 0.0471321702447414... The approximation w_{0} = [π²C/0.4]^{1/4} yields the value w_{0} = π^{1/2}/(2^{1/4}10^{1/2})= 0.0471321702139757... Moreover, we have w_{1} = [π²C/(0.4 − π²C/840)]^{1/4} = 0.04713217028319835....

According to Darling’s model, the normalized reaction force on a rectangular plate with two adjacent edges venting can be expressed as a double series. Define

k_{mn} = ((2m − 1)^{2}^{2}^{2}) + (2n − 1)^{2}^{2}^{2}))^{1/2}

then from Equation (19) in [

F(t)/(abP_{a}) = −64jωexp(jωt)(H'η/π^{4})[u(1,1) + u(1,2) + u(2,1) + u(3,1) + u(2,2) + u(1,3) + u(1,4) + u(2,3) + u(3,2) + u(4,1)+ …]

where u(m,n) = {1/[(2m − 1)^{2}(2n − 1)^{2}]}{1/[jω + (k_{mn}/α)^{2}]}. Here H' is a constant giving the normalized amplitude of the plate vibration. For an isothermal process η = 1, while for an adiabatic process η is the quotient of the specific heats. From Equation (7) we get

F(t)/(abP_{a}) = −64jωexp(j t)(H'η/π^{4})(4a^{2}α²/^{2})D(2aα(jω)^{1/2}/π) with λ = a/b

for the argument z = 2aα(jω)^{1/2}/π of D the number z^{2} is purely imaginary. Therefore for a numerical approximation of F(t)/(abP_{a}), the function

Force_{approx}(t) = −64jωexp(jωt)(H'η/π^{4})(4a^{2}α²/^{2})A(2aα(jω)^{1/2}/π) (22)

(with λ = a/b assuming that λ ≥ 1) can be used. For k = 7, the approximation error is uniformly bounded by 2 × 10^{−5}. Since for the evaluation of H that appears in the definition of A, the representation given in Equation (6) can be used, a direct numerical evaluation of the function Force_{approx}(t) is possible.

The air between two parallel microplates can compress to store energy or vent to dampen energy. This compressible squeeze film damping is a phenomenon of great importance for micromachines. In reference [