## 1 Introduction  
An analytical model for squeeze film damping is presented by [PERSON] _et al_. in [2] for different venting conditions. Based upon the solution of the linearized Reynolds equations, series representations of the resulting reaction forces are presented for different air venting boundary conditions; in particular, for ideally vented rectangular vibrating plates. In [1], based upon the model of [PERSON] _et al_. [2], an optimal sensor design problem is studied. 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 [1], two kinds of boundary conditions appear for rectangular plates: Plates with two opposite edges closed and two opposite edges venting and plates with two adjacent edges venting and two adjacent edges closed. For the former, we give a closed form solution of the corresponding spring and damping constants that are given as infinite series in [1] and [2]. This model has also been used in [3] to model squeeze film effects. For the latter case, we present an approximation of the resulting reaction forces solution that can be evaluated efficiently. We also give a bound for the approximation error. 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. \(\mathrm{P_{A}}\) is the ambient pressure and g\({}_{0}\) is the nominal gap. Let \(\mathrm{\mu}\) denote the viscosity of the trapped gas. Define \(\mathrm{k_{n}=n\pi/b}\) and the constant  
\[a^{2}=12\mathrm{\mu}/(\mathrm{g_{0}}^{2}\mathrm{P_{a}})\]  
For the vibration frequency \(\omega\), according to [PERSON]'s model we have the spring constant  
\[\mathrm{k=(8\omega^{2}AP_{A}/(\pi^{2}g_{0}))[1/[\omega^{2}+(k_{1}/\omega)^{4} ]+1/[3^{2}(\omega^{2}+(k_{3}/\omega)^{4})]+1/[5^{2}(\omega^{2}+(k_{5}/\omega)^{ 4})]+...]\]  
\[=(8\omega^{2}AP_{A}/(\pi^{2}g_{0}))(a^{4}b^{4}/\pi^{4})[1/[\alpha^{4}b^{4} \omega^{2}/\pi^{4}+1]+1/[3^{2}(\alpha^{4}b^{4}\omega^{2}/\pi^{4}+3^{4})]+1/[5 ^{2}(\alpha^{4}b^{4}\omega^{2}/\pi^{4}+5^{4})]+...]\]  
and the damping constant  
\[\beta=(8 AP_{A}/(\pi^{2}\alpha^{2}g_{0}))[{k_{1}}^{2}/[\omega^{2}+(k_{1}/\omega )^{4}]+{k_{3}}^{2}/[3^{2}(\omega^{2}+(k_{3}/\omega)^{4})]+{k_{5}}^{2}/[5^{2}( \omega^{2}+(k_{5}/\omega)^{4})]+...]\]  
\[=(8 AP_{A}/(\pi^{2}\alpha^{2}g_{0}))(a^{4}b^{2}/\pi^{2})\;[1/[\alpha^{4}b^{4} \omega^{2}/\pi^{4}+1^{4}]+1/[(\alpha^{4}b^{4}\omega^{2}/\pi^{4}+3^{4})]+1/( \alpha^{4}b^{4}\omega^{2}/\pi^{4}+5^{4})+...]\]  
In the sequel, we will give compact representations for k and \(\beta\) that do not require the evaluation of infinite series. In order to do this, we introduce some auxiliary functions in the next section. **2. Closed form Representations for Some Infinite Series**  
In this section we define some auxiliary functions that appear in [PERSON]'s model as infinite series. In Section 4, we will show how stiffness and damping can be expressed using these series. For these series we present closed form representations that can be evaluated efficiently and thus are useful for the solution of the optimal design problem.