Modeling an Actual Foucault’s Pendulum – Design Implications for MEMS Gyroscopes

Introduction

Conventional MEMS gyroscopes are miniaturized devices that measure angular velocity. The performance of current MEMS gyroscopes limits them to stabilization applications, due to bias, scale errors and noise.  If these errors could be minimized or improved, MEMS gyroscopes could be used in more advanced navigation applications, such as north finding or dead reckoning navigation. Whole-angle mode MEMS gyroscopes, which can directly measure angular position, are currently being explored in navigation applications. Direct measurement of angular position can deliver more accurate location information than conventional angular velocity measurements, since angular velocity measurements are prone to larger errors over time.

Whole-angle mode gyroscopes are also referred to as Foucault’s Pendulum gyroscopes because their operating principle is similar to that demonstrated in a famous experiment by Léon Foucault which attempted to prove the Earth’s rotation (Figure 1). A pendulum is a suspended weight that can swing freely.   The swing of a pendulum remains fixed relative to an inertial frame of reference (such as the Earth). Thus, a pendulum can be used to demonstrate the rotation of the Earth.   Foucault’s pendulum was comprised of a 28 kg weight on a 67-meter-long wire.  The apparatus was suspended to a dome at the Panthéon in Paris. The rotation of the pendulum’s swing plane over time could be used to calculate the Earth’s rotation angle.  At the latitude of Paris,  a full rotation of the pendulum takes 31.8 hours.  At the north or south pole, the pendulum takes 24 hours to rotate, while at the equator no rotation would be detected [2].

Modeling a Foucault Pendulum in MEMS+

MEMS-based whole-angle gyroscopes can be modeled and studied using CoventorMP, a software platform used to model miniaturized MEMS devices (Figure 2).   As an experiment, we tried something challenging in CoventorMP.  We used the MEMS+ portion of CoventorMP to model an actual Foucault Pendulum.

While challenging, it’s possible to assemble a somewhat idealistic model of a Foucault Pendulum comprised of a 28 kg mass and a 67-meter wire in MEMS+ (Figure 3 – left), if we assume that the wire is rigid and massless.  We have also assumed that the structure can only rotate around the x- and y-axes (all other degree of freedom are fixed).  Additionally, two reference frame inputs are considered: 1) acceleration along the z-axis and 2) angular velocity around the z-axis. They respectively allow us to apply the Earth’s gravity and rotation to our model.

Oscillators are usually comprised of a mass and a spring. In a pendulum, gravity acts like a spring and pushes the mass back to the resting position (with nonlinear behavior observed at large swing angles). Using MEMS+, we included the force of gravity by creating a non-linear operating point of the pendulum swing centered around the rest position (using a DC analysis). Subsequent analyses were then undertaken from this operating point.   When we ran a modal analysis, the results displayed 2 degenerate modes (2 vertical rotations) which corresponded to the pendulum being free to oscillate in any vertical plane. The oscillation period of the pendulum was 16.5 seconds, which is the correct experimentally observed period (and can also be easily obtained using the equation  [1], where l is the length of the pendulum and g is the acceleration due to gravity).

In addition, we ran an interesting transient simulation of the pendulum oscillation, starting from various tilted positions.  We plotted the portrait phase (angular speed versus angular position [3]), to highlight the nonlinear behavior of the oscillation when the pendulum starting angle was large (Figure 3 – right).

Next, we introduced the Earth’s rotation into our model (in the form of a z-axis angular velocity input of 1 turn per day, or ~7.2 10^-5 rad/s). The oscillation force dissipation (entered into the model using Rayleigh Damping) was set to a low value so that we could track the rotation of the swing plane over a long period of time (since the decay of the oscillation amplitude is small over time). By plotting the rotation of the pendulum around the Y versus X plane coordinates, we calculated that the swing plane rotated by a quarter of a turn for a simulation time corresponding to a quarter of a day (Figure 4).

A simulation of the 3D pendulum rotation is shown in the following MEMS+ 3D animation (see below).

 

 

 

Conclusion: Implications for Dead Reckoning Gyroscopes

In this study, we demonstrated that it is possible to model a Foucault’s pendulum in CoventorMP and MEMS+.  We were able to accurately match the oscillation period and swing plane rotation originally discovered by Foucault in his pendulum experiments.

This study highlights the ability of CoventorMP to accurately model and predict the results of angular position-based devices.   We made this demonstration not only for the fun of it, but also to establish the capability of CoventorMP and MEMS+ to study MEMS gyroscopes that work in a whole-angle mode.  This type of modeling could be highly useful in the development of future MEMS-based dead reckoning gyroscopes.

References:

[1] Wikipedia: https://en.wikipedia.org/wiki/Foucault_pendulum

[2] Igor P. Prikhodko, Sergei A. Zotov, Alexander A. Trusov, Andrei M. Shkel, Foucault pendulum on a chip: Rate integrating silicon MEMS gyroscope, Sensors and Actuators A: Physical, 2012

[3] Wikipedia: https://en.wikipedia.org/wiki/Phase_portrait