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Calculus Free Differentiation

December 26, 2017

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# Calculus Free Differentiation

December 26, 2017

For those ‘mathematically challenged’ like your humble author, Integration and Differentiation do not need to be some sort of mystical, black art.  I am comfortable divulging that I escaped flunking calculus by a hair’s-breadth, I am honestly astounded that I managed to pass the course.  In the mathematical theory environment, I was totally lost.  I did well in Geometry, Physics, and Trig as the equations were easily applied to physical examples that I could visualize.  Calculus was just one big fur-ball of numbers and brackets which made no logical sense to me.  Perhaps it was the instructor, perhaps only me… but after escaping Calculus Class, I avoided Integration and Differentiation like discarded, infected wound dressings!

Imagine my ‘concern’ when 20 plus years after this humbling experience, I was first introduced to vibration!  Thankfully, my instructors were not overly concerned with ‘theory’, but rather were extremely pragmatic guys who were concerned with physical characteristic of machinery motion.

As a refresher, the relationship between Acceleration, Velocity, and Displacement are ‘mathematically’ Integration and Differentiation (do NOT panic!).  If you Differentiate Displacement, the result is Velocity; Differentiating Velocity results in Acceleration.   Conversely, Integrating Acceleration results in Velocity, and Integrating Velocity yields Displacement.

Let’s deal with ‘Painless Differentiation’ first.  Differentiation is really pretty simple, it is a calculation of rate of change.  Considering Displacement, if there is no movement, the rate of change is zero.  If moving identical distance for each time increment, the speed is constant.  The ‘basics’ are obvious from the units we use to express Velocity: Miles per Hour, inches per second, etc.  These units exactly state the equation: V = d/t.  However, what about a situation where position and velocity are NOT constant, but rather, are constantly changing?

To keep this as simple as possible, we will limit this discussion to Harmonic Motion.  Harmonic Motion is most easily grasped by the pendulum analogy, a mass on the end of a string.  Gravitational force attempts to keep the mass ‘at rest’ at the bottom of the arc: location “O” in the diagram.

If we move the mass to position ‘A’ its Displacement is at Maximum, but it is not initially moving, its speed is ZERO.  When we release the mass, gravity attempts to pull it downward, but being restrained by the string, it arcs toward its ‘rest’ position (‘O’).  The mass moves continuously faster as it reached the bottom of its arc, but as soon as it passes ‘O’ gravity is now working AGAINST the movement, and it begins slowing down.  This speed continues to slow until the mass movement reaches its opposite maximum (‘B’). At this instant, the movement STOPS and then REVERSES back toward ‘O’.  Plotting Position (Displacement} and Speed (Velocity) over time shows this relationship clearly. Note that Velocity ‘lags’ Displacement by 90 degrees.

Now consider Acceleration.  By definition, Acceleration is the rate of change of Velocity, as clearly shown by a little unit-analysis: inches/second/second (in/s/s, m/s/s, etc.)  When Velocity is constant, there is no ‘change’ in speed, thus Acceleration is zero.  Conversely, when Velocity is changing most rapidly, Acceleration will be at maximum.  Considering the velocity curve in the plot above, where is there no ‘change’ in speed?
Yes, it is at the peaks of the Velocity curve, where Velocity has reached maximum in either direction. This occurs at point “O” in the pendulum diagram; zero displacement.  As the mass passes zero displacement, it begins to slow down.  Acceleration becomes ‘deceleration’ (negative ‘increase’ on the plot) as velocity begins to slow.

Therefore, Acceleration ‘lags’ velocity by 90 degrees, and displacement by 180 degrees.  Note the inverse relationship of Displacement and acceleration in the plot.

The mathematical inverse of Differentiation is Integration.  This allows us to work ‘backward’, converting acceleration into velocity and displacement.

This is fairly simple to wrap your head around, however we have to consider one additional parameter which forces us into the dreaded realm of Calculus - Frequency.  Frequency is how rapidly the movement repeats; how many cycles occur in a given time.  Units for Frequency are typically Hz (cycles per second) or CPM (cycles per minute).  Things immediately get more complicated.

Consider this scenario, assuming that the ‘peak’ velocity remains constant in the above graph, but the time required to complete one cycle is cut in half (frequency doubled).  What happens to the total movement?  If you guessed that it will be roughly 50% less, you are correct.   Scenario 2: if the total movement remains constant, but the time per cycle is halved, what happens to velocity?  Yes, it must DOUBLE in order to attain the same amount of movement!

Machine Condition evaluation rarely deals with single-frequency, harmonic motion.  Multiple frequencies are present in our machine data.  Thus, the integration or differentiation must be performed for ALL frequencies contained in the data! This is most easily accomplished in the frequency domain, utilizing data processed through an FFT into discrete frequency ‘bins’, and performing the math on each bin individually.

Thankfully, all the Calculus required is embedded in our tools.  Converting between Acceleration, Velocity, or Displacement is as simple as clicking an icon on a spectrum or time waveform plot.