What does an accelerometer measure? The obvious answer is acceleration, but that's not really true. An accelerometer actually measures normal force or restoring force which we equate to acceleration using the formula, F=ma. This article will explain the fundamental operating principles of accelerometers and investigate the capabilities and drawbacks of accelerometers in certain applications.

What types of things are accelerometers good at measuring?

- Acceleration
- Vibrations
- Tilt angle relative to gravity
- Centripetal force and acceleration
- Collisions
- Fast movements

What types of things are accelerometers bad at measuring?

- Velocity
- Position
- Projectile motion

Why if we can measure acceleration with an accelerometer can't we accurately calculate velocity and position by doing an integration? For many reasons.

- Many types of errors in the accelerometer data will propagate and grow when integrated.
- Acceleration measurements due to rotation cannot be separated from acceleration due to linear motion without extra sensor data.
- Vibrations of the accelerometer will accumulate error and cause drift in any velocity or position calculations which is called vibration rectification.

How does an accelerometer work?

At the core all accelerometers contain five essential elements:

- A mass that is able to move around, called a proof mass
- A spring suspension to hold the proof mass
- Some damping either due to designed damper or due to air resistance
- A displacement sensor to measure the proof mass movement
- A frame to connect the accelerometer to the thing you want to measure

The important parameter to measure in this accelerometer model is the difference between the position of the frame X and the position of the proof mass x which we define as variable Z, such that Z = X - x.

Summing the forces on the proof mass and rearranging the typical second order characteristic equation, we get the expression for Z being equal to the equation below:

Where, m is the mass of the proof mass, w is the frequency of motion, k is the spring constant, b is the damping coefficient, and the i term shows that are solution is complex with possible imaginary terms. For a second order system we can calculate the natural frequency w_0 as

We can examine the transfer function for a few values of the terms to understand what happens to our system.

- For the cases where damping, b is 0 or close to 0, when the frequency w equals the natural frequency of the system, w_o, the amplitude of Z will go to infinity.
- When the frequency of the acceleration signal is below the natural frequency, the displacement of the proof mass Z will be proportional to acceleration. This is the regime where an accelerometer operates.
- When the frequency of the acceleration signal is above the natural frequency, the displacement of proof mass Z will be proportional to the displacement X of the frame. This is the regime where a seismometer operates.

The plot below shows how the proof mass displacement, Z, changes vs. frequency for various amounts of damping.

In order to measure the displacement of the proof mass, most modern accelerometers use a capacitance measurement method. The proof mass will have a movable capacitor plate and the frame of the accelerometer will have a stationary capacitor plate. As the proof mass moves, the gap between the capacitor plates changes which changes the capacitance. Special electronic circuitry converts the change in capacitance to a change in voltage which then can easily be read by other digital electronics.

Modern accelerometer designs have many rows of the capacitor plates that function in parallel to increase the amount of capacitance change per unit of displacement of the proof mass. The scanning electron microscope image shows the proof mass and capacitive plates of an ST Micro accelerometer that is commonly used in smart phones and wearable devices.

For more information about accelerometers check out the presentation attached as a .pdf file to this article. Go science!