# Fundamentals of Vibration – Simplified Calculation for Converting Velocity to Displacement Application: Structural Vibration Monitoring in Shale Gas Operations

NOTE: for a quick read, skip ahead to where it says “Here’s How” below…

At KCF, we are passionate about eliminating needless machine failures, and vibration is our best tool.  A practical problem that often comes up is this: how much is something vibrating, in a way I can understand?  Displacement (distance traveled by an object, in time) is something we can all understand: how many inches is my something vibrating, where something can be a pipe, a motor, a pump, a fan, a beam, ceiling, floor, etc.  There are sensors that measure displacement directly, accelerometers, especially wireless accelerometers, make it much easier because we can mount externally on a machine with a magnet.

With software and a simple calculation, you can use the accelerometer measurement to calculate the displacement, and get an accurate estimate of how many inches (or milli-meters for our Canadian and European friends) something is vibrating.  The fundamental equation for vibration, or any oscillating motion is:

In a simple oscillating vibration, A (t) = A1cos(f1*t) + A2cos(f2*t) + …  (where A1 is the amplitude at frequency f1, etc.)

By integration, the velocity, V (t) = A1/f1*sin(f1*t) + A2/f2*sin(f2*t) + …

Integrating again, displacement, D (t) = A1/f21*cos (f1*t) + A2/f22*cos(f2*t) + …

But fortunately, the software does most of the work for you, and it’s much easier to get a rough estimate, especially when you have one dominant vibration frequency.

Here’s how:

1)    Use the software to find the peak velocity at a given frequency (inches/second).  NOTE: The software has already integrated to convert acceleration to velocity.
2)    Identify the frequency of that peak (RPM)

Displacement

Example: we had a customer recently in a well completion operation, and they were seeking to know the amplitude of vibration of a high-pressure pipe.  They were using a practical and proven method, duct-taping a sharpie to the pipe and duct-taping a piece of cardboard to the adjacent structure.  A great practical solution, and it answered the question, but there is an easier way!  Just follow these steps:

1)    Peak Velocity = 15 in/s  (it was a high-amplitude application)
2)    Identify the frequency of the dominant peak = 943 RPM  (15.7 Hz)
3)    Displacement = 10 X 15 / 943 = 0.15 inches.

Answer: the pipe is vibrating a little more than 1/8th of an inch.  It’s not exact, but it’s accurate enough to be very useful and answer a question, and it doesn’t require duct tape!