Coefficients of basic equations

In order to predict behavior of the device under dynamic acceleration, dynamic model has to be constructed. Basic equations governing this model are following:

Where mass and specific spring constants were found in static model analysis:

Other coefficients have to be found.

 and  are moments of inertia around axes X and Y respectively. Because of symmetric design of proof mass these moments are equal to each other.

Ratio of total area of etching holes to area of proof mass is only about 0.6%, therefore, influence of holes on moment of inertia is neglected. The different density of materials added during MOSIS process is also neglected. So, to calculate moment of inertia we will use the same equation as for solid box.

Where a and b are dimensions of box in plane which is perpendicular to axis of rotation. To calculate it, it is needed to calculate thickness of proof mass first. Thickness of added alumina layer is

Then the total thickness of proof mass is

Now, moment of inertia can be calculated

Next step is to find damping constants. For normal motion only they can be found from damping force

Whose solution

is known from linearized Reynolds equation. Solution with subscript “0” represents action of gas between moving plates when frequency of motion is low (small squeeze number). In that case it acts as pure damper. At higher frequencies solution “1” becomes dominant and gas film acts as spring. Such behavior of film is not desirable. Therefore, accelerometer should be used under acceleration whose frequency is less then certain value. This so called cut-off frequency will be estimated later. Now, only solution F₀ will be considered.

Damping force can be approximated by neglecting the у term in series solution as follows

Where it is used that moving plate has square shape and constant 0.42 is correction coefficient due to its unit aspect ratio.

Finally, the damping constant of normal motion is

For tilt motion expression of angular momentum is also known in form of series solution. According to frequency of acceleration it can act as damper or spring. And we again consider only damping behavior.

In equation above it is applied that aspect ratio is unit. Now, substituting expression for у and treating  as angular velocity, we can obtain damping of tilt motion

The series converges rather fast, therefore only first term will be calculated for tilt motion damping estimation. Also last term in denominator will be neglected.

Damping coefficients of tilt motion around X and Y axes are equal because of symmetry of proof mass.

Now, all nine coefficients of basic equations are know and system of differential equations can be solved.

Natural frequencies

For normal motion natural frequency is

Natural frequencies of rotation around X and Y axes are again the same because of symmetry of proof mass:

Damping ratios

From damping coefficients we can calculate damping ratios for normal motion

and for tilt motion

Where subscript  represents that tilt for tilt motion does not matter which axis we will choose for calculation.