Measuring Gravity using Gravimeters and MEMS

Paul A.M. Dirac, an English physicist famous for his work in quantum mechanics, once stated “Pick a flower on Earth and you move the farthest star.”  This statement may be hard to understand from our everyday life, but very small changes in gravity can have profound effects.  If you can measure small variations in gravity, it can also be quite valuable.   For example, measurements of small gravitational changes can be used to identify hidden hydrocarbon reserves, predict volcanic eruptions, find invisible subterranean features, and monitor the Earth’s tides[1].

What is a Gravimeter?

It is possible to detect very small changes in gravity using an instrument known as a gravimeter.  A gravimeter can measure the gravitational field of the Earth at a specified location by measuring the acceleration of a fixed mass, expressed in units called “Gal”.   There are two types of gravimeters: absolute and relative.

Absolute gravimeters measure gravity by looking at the acceleration of an object in free fall over a set distance. For example, a cold atom gravimeter measures gravity by calculating the acceleration of falling atoms that have been cooled to near absolute zero temperature.  Absolute gravimeters offer a direct measurement of gravity and are very accurate, but are also bulky and expensive. This type of gravimeter is useful in scientific research, such as measuring gravity during space exploration, but it is not as practical for certain terrestrial applications.

Relative gravimeters measure the deflection of a mass on a spring, which changes as gravity varies. Relative gravimeters can be made smaller and are less expensive than absolute gravimeters, but they are not as accurate since their accuracy is a function of the spring constant of the spring (a measure of the stiffness of the spring).  The spring constant can change with temperature and other variations in environmental conditions, and affect the accuracy of gravity measurement.

Relative gravimeters are essentially mass-spring systems where sensitivity is maximized if the mass is large compared to the spring constant (or stiffness of the spring).  The resonant frequency of the mass-spring system is used to measure the sensitivity of the device, and is proportional to the square root of the spring constant divided by the mass.   The smaller the resonance frequency, the higher the sensitivity of the gravimeter.

An Introduction to MEMS-Based Gravimeters

Relative gravimeters can be built using MEMS technology, where the mechanical components are micron-scale in size.  MEMS-based gravimeters have the advantage of small size and low power requirements, making them very suitable for remote use and in Internet of Things (IoT) applications.   Unfortunately, achieving a low resonance frequency is not easy in a MEMS-based gravimeter.   Due to the small mass of the micron scale device, the stiffness of the MEMS spring (or its spring constant) must be very low in order to achieve high gravimeter sensitivity.

Prior MEMS gravimeters have achieved low stiffness and high sensitivity by combining two kinds of mechanical springs [2-3], one of which has negative stiffness.  Unfortunately, the mechanical springs are susceptible to process variability and temperature dependence. These problems can be reduced by using an electrostatic spring, in lieu of the mechanical negative spring, combined with electrical feedback control circuitry.

Waseda University MEMS-Based Gravimeter Design

Coventor recently worked with a MEMS research group at Waseda University on a study to model a highly sensitive gravimeter with electrically tunable stiffness [4]. The gravimeter’s resonance frequency was set at 1 Hz. The system uses 2 feedback loops.  One feedback loop is used to maintain a low resonant frequency and ensure high gravimeter measurement sensitivity.  The other feedback loop uses electrostatic force to maintain the mass position, while improving dynamic range and robustness to mechanical shocks. To enhance the gravimeter’s response, the resonant frequency is monitored  ”off-resonance” and in a real-time manner.   This monitoring function helps eliminate process variation and temperature dependence effects that could negatively impact the performance of the gravimeter.

A cross-section of the manufactured MEMS structure is shown in Figure 1. The structure is comprised of a DRIE etched silicon layer which is anodically-bonded to a partially etched glass wafer.

Figure 1: The MEMS-based gravimeter is comprised of a DRIE etched silicon layer which is anodically-bonded to a partially etched glass wafer

The MEMS-based gravimeter was modeled using MEMS+® to better understand the temperature dependence of the device.  Figure 2 displays a representation of the MEMS-based gravimeter device model.  This MEMS+ model was embedded in a Simulink® closed-loop system model (Figure 3), and this larger model was able to accurately predict experimental electrostatic softening and temperature-based performance.

Figure 2:  MEMS-based gravimeter modeled in MEMS+

Figure 3:  Simulink system model including embedded MEMS+ model

The Waseda University gravimeter is a small and low-cost device, with a low resonance frequency, high sensitivity, and good shock and thermal protection.  The small size and low cost of the MEMS-based unit could make it suitable for mounting on a drone for surveying and exploration requirements, or even used to remotely monitor volcanoes.   This device could pave the way for low cost, miniaturized gravimeters suitable in a range of exciting gravity-measurement applications.

References

1. Kumar S, Mineral Exploration: Principles and Applications, 2nd Edition, 2018
2. Tang et al. A high-sensitivity MEMS gravimeter with a large dynamic range, Microsystems & Nanoengineering, 2019, https://doi.org/10.1038/s41378-019-0089-7
3. Middlemiss, R., Samarelli, A., Paul, D. et al. Measurement of the Earth tides with a MEMS gravimeter. Nature 531, 614–617 (2016). https://doi.org/10.1038/nature17397
4. Yi C, Wu J, Maekoba H, Parent A and Ikehashi T, Ultra-Low Resonance Frequency MEMS Gravimeter With  Off-Resonance Closed-Loop Control,  IEEE Inertial Sensors 2022 conference