CAD Practice: Stress/Modal Simulation

This weekend I wanted to explore the simulation options within Fusion 360 since I had never used them before. I decided to take the layout from a previous class HW question since I already had the model in Fusion.

Folded-flexure comb drive resonator.

Above is a folded-flexure comb drive MEMS resonator, although modeled in Fusion in millimeters instead of microns. It's a very well studied design, where the resonance frequency is mainly determined by the shuttle mass and the spring constants of the folded-flexure. I though this would be a good test since I know the equations for the spring constants of these beams and have done the analysis for these devices before.

How these devices normally work is that the center moving mass is set to a certain DC voltage, and then an AC signal is applied to one of the side combs. When the AC signal's frequency matches the resonance frequency of the center mass, the center mass will begin to oscillate significantly and the teeth in the combs will slide past each other, causing a change in capacitance at the other side (the output port). The change in capacitance will induce a current at the output comb that can be sensed, and it will have maximum amplitude at the resonant frequency.

I didn't change any of the default material parameters for these devices and just left them as steel with the following properties:

Stress Simulation:

The setup for this was very simple: I fixed the left and right combs and the center two anchors for the beams. Then for the force I applied a 20 N force to one side of the center mass. After running the simulation we get this:

Stress simulation displacement.

We can calculate the effective spring constant:

$$k = \frac{F}{x} = \frac{20 \text{ N}}{0.9586 \times 10^{-3} \text{ m}} = \boxed{20864  \text{ N/m}}$$

Now to check the theory: the beams are 60mm long by 2mm wide by 2mm tall. The spring constant of one beam (equivalent to two half-length cantilevers facing each other) is:

$$k_{beam} = \frac{1}{2}\left(\frac{Ea^3b}{4L_{half}^3}\right)$$

$$k_{beam} = \frac{1}{2}\left(\frac{(210\text{ GPa}(2\text{ mm})^3(2\text{ mm})}{4(30\text{ mm})^3}\right) = 1.56\times10^4 \text{ N/m}$$

On one side of the structure we have two springs in parallel that add, which are then in series with the single beam to the anchor (which is like impedances in parallel). Double this to get the total spring constant:

$$k = 2(2k_{beam}||k_{beam}) = \frac{4}{3}k_{beam} = \frac{4}{3}(1.56\times10^4  \text{ N/m}) = \boxed{20741 \text{ N/m}}$$

The theory checks out; \(20864 \text{ N/m}\) is very close to \(20741 \text{ N/m}\) (0.5% difference). We can also see from the color scale that the side masses that all three beams connect to displace about 66%, or 2/3rds the total displacement of the main shuttle. This is inline with what we expect since the stiffness from the anchor to the side is half the stiffness from the side to the main shuttle.

Modal Frequency Simulation:

Using the same anchor setup, I ran a modal frequency simulation to find out what frequencies this device will resonate at.

Along with the primary resonance (the side to side motion in the direction of the combs) I was looking for, I saw some other interesting modes too. Below is a picture of the primary resonance at 57.19 Hz:

Side-to-side resonance.

An animation of the motion:

Once again we should check theory:

$$f = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$$

Assuming that the mass of the shuttle dominates relative to the beams and side masses, we can calculate it's mass from the volume and density:

$$m = \rho Ah = (7.85\times10^{-6}\text{ kg/mm}^3)(9538.4\text{ mm}^2)(2\text{ mm}) = 0.15 \text{ kg}$$

$$f = \frac{1}{2\pi}\sqrt{\frac{20741 \text{ N/m}}{0.15\text{ kg}}} = \boxed{59.2 \text{ Hz}}$$

As expected, the theory checks out once again: \(57.19 \text{ Hz}\) vs \(59.2 \text{ Hz}\). :) 

The theoretical value is faster as expected since we didn't factor in things like the effective mass of the beams and the side shuttle. Also if the device was simulated at a micron scale like a real MEMS device instead of millimeter the resonant frequency would be 1000 times higher. This is because the spring constant is linearly proportional to a scale factor \(s\) (if we scaled all dimensions by s) but mass is proportional to \(s^3\).

$$f \propto \sqrt{\frac{k}{m}} \propto \sqrt{\frac{s}{s^3}} = \frac{1}{s}$$

Here are some of the other fun vibration modes that are out of plane: