A question about MATLAB library "vibration of rotary machinery"

My question is about vibration of rotary machinery:

https://au.mathworks.com/help/signal/ug/vibration-analysis-of-rotating-machinery.html

The author claims the wavforms correponding to input shaft, output shaft, and gear mesh measured by sensors can be reported as:

vfIn = 0.4*sin(2*pi*fPin*t); % Pinion waveform

vfOut = 0.2*sin(2*pi*fGear*t); % Gear waveform

vMesh = sin(2*pi*fMesh*t); % Gear-mesh waveform

A1 sensor can measure vfOut.

A2 sensor can measure vMesh.

But how can you measure vfIn?

[dpb]

"Two accelerometers, A1 and A2, are placed on the bearing and gearbox housings, respectively."

It is presumed there's enough energy content in both/either A1 and/or A2 transmitted through the mountings of the shaft and gearbox case to be able to find the components' contributions in the overall signals.

In real life(*), this is totally dependent upon construction of the gearbox and quality of accelerometer mountings. Generally it does work at least well enough, but is easy-enough to observe whether the known frequencies are actually contained in the returned waveforms.

That's not to say it is always a trivial exercise by any means. It sometimes isn't all that easy in practice to be able to get sensors in the places they would need to be ideally, and even if is physically possible to get to the location, conditions at that point may be far from ideal for instrumentation or getting a signal from somewhere remote in the bowels of a nearly mile-long paper rolling mill back to a collection point can be a challenge. While at CSi we introduced the first wireless accelerometer...

In the example they're there because the author simulated an ideal gearbox; it would have been instructive to have followed that up with actual data from a real gearbox of the same type -- what we did routinely at CSi to test new instruments, algorithms, etc., etc., ... amongst the lab gear was included the test rig called "The Alligator" that was precisely a set of gears, bearings, and all with known and introducible faults of all kinds.

(*) In former life I was employed by CSi, the leading vendor/pioneer firm in the field...now a part of Emerson.

https://www.emerson.com/documents/automation/article-tuscon-electric-power-builds-award-winning-predictive-maintenance-ams-en-38068.pdf

[Si So]

This is not an answer to my short question.

[Si So]

However, I found the answer... VfIn is the input acceleration applied by the engineer

[dpb]

"VfIn is the input acceleration applied by the engineer"

No.

As you posted above

vfIn = 0.4*sin(2*pi*fPin*t);    % Pinion waveform  

From the text the author writes:

"The pinion rotates at a rate fPinion = 22.5 Hz or 1350 rpm."

fPin = 22.5;        % Pinion (Input) shaft frequency (Hz)

So, vfIn is also an assumed (calculated) voltage from the accelerometer owing to the pinion gear fundamental frequency.

It is NOT an applied acceleration; the pinion shaft is the input (driving) shaft of the hypothetical gearbox and by being so is the one that determines what the operating rpm is, but it is simply being rotated by the driver at that rpm; the engineer does not apply the resulting acceleration--that again is what is being measured.

The composite signal generated and analyzed doesn't really try to simulate the responses that would be expected from the two hypothetical accelerometers A1 and A2 at all; it just is a waveform that contains what are known to be the fundamental frequencies and an added hypothetical synchronous fault noise in order to demonstrate the basic ideas in preventive maintenance diagnostics.

[Si So]

True...

One question is whether the superposition of different acceleration signals is meaningful for descibing the system acceleration.

[dpb]

That's what you get out of an accelerometer in real life, yes.

It responds to whatever happens to it at its base (up to its response frequency, of course); what that resultant waveform is all depends upon the mechanical characteristics of the particular gearbox at the mounting location and, as noted above, the quality of the mount one can manage.

The amplitude of the various parts of the overall response is dependent upon the effective coupling through the mounting components, bearings, lubricants, etc., etc., etc., all as amplified/attenuated by the shape and composition of the components themselves.

"The composite signal generated and analyzed doesn't really try to simulate the responses that would be expected from the two hypothetical accelerometers A1 and A2"

By the above, I didn't mean to imply the signal of the hypothetical accelerometers would not contain these frequencies, I meant that the example didn't try to reproduce a real accelerometer response by modeling the gearbox itself to try to predict the above effects on amplitude and phase of the accelerometers themselves.

This is an idealized example, for sure, but the the basic ideas are there.