# Damping Mechanisms

Full analysis of a MEMS design requires simulating energy loss mechanisms such as gas damping, thermo-elastic damping and anchor losses. Whether simulating the transient response of an accelerometer or estimating the Q factor of a resonator, getting the damping right is crucial. MEMS designers have traditionally relied on simple analytical formulae or experimental data to estimate damping coefficients. With CoventorWare, it’s possible to simulate energy loss mechanisms and accurately predict damping coefficients.

## Gas Damping

Moving MEMS devices transfer energy to surrounding air or gas through their motion. The resulting “gas damping” plays a critical role in some devices, such as accelerometers, microphones, display mirrors and switches and contributes to the signal-to-noise ratio. While it is possible, in principle, to simulate gas damping with a general-purpose fluid dynamics field solver, such a brute-force approach is generally not practical. CoventorWare’s DampingMM module includes specialized models and solvers that can be used separately or in combination to accurately and efficiently simulate gas damping.

The graphic shows damping force magnitude about the x axis for two axis micro-mirror computed with the General Motion Stokes Solver.

A theoretical model for a sheared thin film of fluid between parallel surfaces with motion parallel to the gap. The solution is in closed form.

The Squeezed-Film solver uses a finite-element representation of the linearized Reynolds equation for a given input geometry, to compute damping and spring forces over a user defined frequency range. A typical solution will be dominated by damping at low frequencies where most of the force response is due to the fluid is from viscous effects of the fluid being squeezed out of the holes and edges. On the other hand, at high frequencies, the motion is too fast for the fluid to flow out, and the reaction is a spring force similar to squeezing a balloon. In complicated regions of the geometry, including edges and perforations the solver employs flow resistance models developed from extensive simulations using the full Navier-Stokes equations. The solver also automatically adjusts for high Knudsen number effects and takes into account mode shapes for systems that have a modal deformation displacement (such as beams and membranes).

This is a solver for fluid damping due to the general motion of bodies with unrestricted geometries. The numerical model is in the form of an integral representation of the Stokes equation (a low Reynolds number derivative of Navier-Stokes). The 3D solution is computed by a boundary elements applied on the surface of all parts and the solution is accelerated with a fast-multipole algorithm.

## Anchor Losses

MEMS devices such as gyroscopes and resonators that rely on continuous vibration for operation lose energy through their anchors, commonly referred to as anchor loss, anchor damping, support loss, clamping loss, or attachment loss. In fact, anchor losses may be the dominant energy loss mechanism in devices that are packaged in high vacuum where gas damping is negligible. CoventorWare’s Mechanical solver includes a “quiet” boundary condition that eliminates the reflection of the elastic waves impinging on the substrate boundary, which can be used to predict elastic energy that is transmitted via the anchor to the substrate.

The graphic shows unbalanced resonator coupling energy to substrate leading to elevated anchor damping.

## Thermoelastic Damping

A vibrating structure generates heat as the material is alternately compressed and tensioned. Dissipation of this heat is known as thermo-elastic damping (TED). For hermetically packaged MEMS that rely on bulk acoustic modes for operation, TED may compete with anchor losses as the dominant energy loss mechanism. TED can be reduced by careful design and placement of perforations in the vibrating devices, but such design can only be done with the help of accurate TED simulations.

Simulation: thermal field generated due to TED

No perforations Q_{TED} = 26550

Slot perforations Q_{TED} = 33083

Square Perforations Q_{TED }= 26800