  We study the dynamics of MEMS microbeams undergoing electrostatic pull-in. At
DC voltages close to the pull-in voltage, experiments and numerical simulations
have reported `bottleneck' behaviour in which the transient dynamics slow down
considerably. This slowing down is highly sensitive to external forces, and so
has widespread potential for applications that use pull-in time as a sensing
mechanism, including high-resolution accelerometers and pressure sensors.
Previously, the bottleneck phenomenon has only been understood using lumped
mass-spring models that do not account for effects such as variable residual
stress and different boundary conditions. We extend these studies to
incorporate the beam geometry, developing an asymptotic method to analyse the
pull-in dynamics. We attribute bottleneck behaviour to critical slowing down
near the pull-in transition, and we obtain a simple expression for the pull-in
time in terms of the beam parameters and external damping coefficient. This
expression is found to agree well with previous experiments and numerical
simulations that incorporate more realistic models of squeeze film damping, and
so provides a useful design rule for sensing applications. We also consider the
accuracy of a single-mode approximation of the microbeam equations --- an
approach that is commonly used to make analytical progress, without systematic
investigation of its accuracy. By comparing to our bottleneck analysis, we
identify the factors that control the error of this approach, and we
demonstrate that this error can indeed be very small.
