## Abstract

There are many reasons why an engineer would document the distance their product traveled, especially during field testing. They might want to determine how far the test item deviated from the origin or how quickly it moved from point a to point b. Distance measures the path traveled in units of length (cm, m, km, etc.).

Displacement denotes the change in the position of an object from the initial point to the final. It does not consider the path of the test item, only the item’s overall change in position. As displacement is a vector quantity, its magnitude can equal distance if the test item moves in a straight line.

### Question

A Vibration Research customer inquired why their accelerometer measurement did not produce an acceptable displacement measurement after double-integration. They mounted an accelerometer to the test item to calculate its displacement when it traveled in a straight line from point a to point b. Why didn’t this method work?

### Answer

An accelerometer measures the change in velocity over time, and we can estimate displacement from this measurement using double integration. When the test item is not accelerating or decelerating (constant speed), its velocity is static. This presents a problem because if the velocity is constant, the acceleration is zero. Accelerometer measurements of static velocity will have noise that, when double integrated, leads to very large errors. Overall, the double-integration method will be inaccurate with this setup unless you use a very short timeframe of data. To measure the distance traveled, we’d recommend using a tachometer instead.

#### Explanation

Displacement is related to acceleration and velocity as functions of time. In vibration testing, these are standard measurements of a waveform. Usually, engineers mount an accelerometer to the test item to measure the acceleration waveform of its vibration.

Accelerometers are manufactured for dynamic measurement, which is constantly changing. The first issue with our customer’s test setup is the static measurement of an item moving at a constant speed. If the test item is not accelerating or decelerating, then its velocity is static, and its acceleration is zero. At a constant speed, there is no acceleration measurement value to integrate.

#### Errors with Double Integration

We can treat the curve of a plot as a function representative of a vibration signal. Engineers apply the calculus processes of integration or differentiation to an acceleration, velocity, or displacement curve to convert them to each other.

An integration filter converts acceleration to velocity and velocity to displacement. The area between a function and the x-axis (area “under the curve”) is the integral of the function.

Double integration alone introduces distortion into the signal due to DC offset from the transducer, linear drift, or low-frequency noise. The original signal must be filtered or processed to remove the potential for error.

Accelerometer measurements of static velocity will have noise that, when double integrated, leads to insurmountable errors. The noise floor will hide the signal if the sensor does not produce a strong enough output signal without amplifier gain, and signal gain also amplifies the noise levels. Overall, the double-integration method will be inaccurate with this setup unless the customer uses a short timeframe of data.

#### Solution

To measure the distance traveled, we’d recommend using a tachometer rather than an accelerometer. Tachometers measure the rotation speed of a revolving component, usually in rotations per minute (RPM). As the customer’s test item was traveling on rotating components, they could use a tachometer to measure the wheel’s RPM and convert it to the distance traveled using the wheel’s radius.

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