Analyzing Non-Linear Dynamical Systems

Excerpts from my forthcoming book entitled:

Solving Complex Problems through Vibration Analysis.


An Introduction to understanding Non-Linear Dynamical Systems.


Notice I wrote ‘solving problems through vibration analysis.’ I did not write the usual way of describing, such as: ‘Solving Vibration Problems.’ That way of saying makes no sense since we are essentially using a tool, namely vibration analysis, to solve intricate and complex problems. Similarly, to term some problem as a ‘vibration problem’ makes little sense since any machine, in fact, anything in the entire universe, vibrates. So, is the entire universe a problem? Certainly not.


An Approach.


Since we would primarily deal with complex problems of machines it would be appropriate to understand a possible approach to deal with such problems.


To quote Feynmen, it is simply for the pleasure of finding out things. He wrote a book with the similar name: ‘The Pleasure of Finding Things Out.’


On pages 103 and 104 of this book (published by Penguin), he writes the following: (I am quoting only relevant portions for our purpose of investigating problems with a scientific attitude. Moreover, I shall put my notes for emphasis of certain points by using third brackets).


The other aspects of science that are important and that have some problem of a relation to society, beside the application and actual facts that are discovered, are the ideas and techniques of scientific investigation: the means, if you will. Because I think that it is hard to understand why the discovery of these means, which seem so self-evident and obvious, weren’t discovered earlier; simple ideas- which, if you just try them, you see what happens and so forth. ….


The first is a the matter of judging evidence – well, the first thing really is, before you begin you must not know the answer. [So important. If we start off with the attitude that the answer is known, we are doomed right from the beginning. The real answer would delude us and we would either keep going in circles or be put on hold till we let go of our pre-formed judgements and opinions or what we think to be an answer. A very difficult practice.] So you begin by being uncertain as to what the answer is. This is very, very important, so important that I would like to delay that aspect and talk about that still further along in my speech. The question of doubt and uncertainty is what is necessary to begin; for if you already know the answer there is no need to gather any evidence about it. [Worth remembering that vibration records in various forms taken in different directions and modes from machines act as evidence only of the how the machine is behaving and how one part or element of a machine interacts with another.] Well being uncertain, the next thing is to look for evidence, and the scientific method is to begin with trials. But another way and a very important one that should not be neglected and that is very vital is to put together ideas to try to enforce a logical consistency among the various things that you know. It is a very valuable thing to try to connect this, what you know, with that, that you know, and try to find out if they are consistent. And the more activity in the direction, the better it is.


After we look for the evidence we have to judge the evidence. There are the usual rules about the judging the evidence; it’s not right to pick only what you like, but to take all of the evidence, to try to maintain some objectivity about the thing — enough to keep the thing going — not to ultimately depend upon authority. Authority may be a hint as to what the truth is, but is not the source of information. As long as it’s possible, we should disregard authority whenever the observations disagree with it. And finally the recording of the results should be done in a disinterested way. That’s a very funny phrase which always bothers me — because it means that after the guy’s all done with the thing, he doesn’t give a darn about the results, but that isn’t the point. Disinterest here means that they are not reported in such a way as to try to influence the reader into an idea that’s different than what the evidence indicates. [This approach and scientific movement was started by Galileo. He demonstrated the power of his ways of looking at things].


Just to give a practical feel of how we come to understand complex problems through vibration analysis, let me show you one of my reports as an example of the application of the scientific approach and attitude, we just discussed, and elaborate further on that with reference to understanding complex systems.


Final Report of Investigation into Problems for Autogenous Mill.


Client: yyy, xxxz (I have removed the name of the client for ethical reasons)


Consultant: Reliability Management Consultant Pvt. Ltd (RMCPL)


Dates of Investigation: 20th and 21st May 2014


Investigators from RMCPL: a) Dibyendu De, Director b) Animesh Ray, Deputy Director


Problem description in brief:


The Autogenous Mill, is a roller crusher that is loosely supported on four rollers, which in turn are supported on concrete columns, connected to adjacent columns by beams; suffers from high and rather abnormal vibrations that possibly results in frequent failures of Roller Shaft no. 4, which remains unexplained for the past three years.


Methodology of the Study:


In an equipment, (let us call it a system) failures are bound to happen. These are engineering problems or failures that affect Reliability, Availability and Performance of any manufacturing system. At times a gear-box cracks. At times, a shaft fails. Yet at another time, chains fail. All happening in the same system. Seen at the micro level, failures not only cost a lot of money and effort but also leads to wastage of energy, money and material. Seen from the macro level, failures do eat up a lot of national wealth and human happiness.


The question is how do we respond to such failures? The usual way is to treat such failures independently and try to arrive at an independent root cause for each failure. And this process can go on forever. Quite frustrating indeed without a desirable state of operation in sight.


The other way is to consider all failures simultaneously. In order to do so, each failure has to be related to one another and not treated independently. . The guiding principle is — if something happens in a system it must affect something else.


However, to do so, we need to understand the dynamics of the machine — i.e. how movement of one part tells upon the movement of another part.


So, the process of understanding failures would be something like this:


0. Start with a zero mind — no answers; a beginner’s mind; only curiosity.

1. Notice the movement of each part.


2. Notice how movement of one part affects other parts or elements in the system.


3. Determine the forces that act and estimate their nature, direction and magnitude.


4. Determine the interactions and the design criteria and limits for such interactions.


5. Try to relate the dynamics of the system to the failures that are observed.


6. Build the hypothesis that relates all failures based on our findings.


7. Look for evidences through data obtained from dynamic analysis tools such as vibration analysis, thermal imaging, wear debris analysis, lubricant analysis etc.


8. Relate the evidences to the hypothesis and make it more cogent and logically foolproof so that the hypothesis turns a theory that not only explains how the failures happen so as to increase the time span of failure free operation but also provide us tools to predict the conditions under which the failures take place (just before they take place) and the degree of maintenance and care we need to take in the future.


9. We now have, in a relatively short time, a great explanation of all the failures happening in the system along with the predictive and maintenance tools.


10. With the theory in place, it is easy to design the countermeasures (minimal interventions) that would not only prevent the failures but also prolong the working life of the system.


11. The context specific theory, thus obtained, would also help us design the necessary monitoring activities — a) to catch changes (of conditions) in the system, in time, to prevent a breakdown b) improve upon the maintenance plan.


12. By doing this we have now designed in-built resilience and flexibility in the complex system that we are dealing with.


Things that were investigated:


1. Interviews with plant personnel at different levels


2. Physical Inspection of the system at site.


3. Inspection of the vibration records as available at site. (overall displacement values and displacement values with phase readings)


4. Inspection of vibration data, at site, as per additional recommendation by RMCPL (vibration values of displacement, velocity, acceleration in different directions, FFT spectrum, Time waveform)


5. Deeper Investigation and analysis into the problems. The idea is to develop a contextual theory that explains all the failures.


6. Explanation and discussion of the findings and analysis with concerned engineers and managers engaged with the issue.


Observations: (the main points)


1. The failure of the shaft of roller no. 4 is a combination of fatigue failure, tensile loading and energy absorption through repeated impacts forces (impulses).


2. The system’s natural frequency is quite close to the driving frequency.


3. The system is insufficiently damped. In fact, the system is negatively damped.


4. The roller crusher, loosely placed on rollers, automatically moves towards the discharge end


5. Presence of self sustaining vibration seen.


6. The system is subjected to rocking and twisting motion, which is pronounced at roller 4 and motor and gear-box foundation.


7. The system runs in under-loaded condition. That is — it is operated much below its designed capacity.


8. Total frictional force is inadequate to constrain motion in undesirable direction.


Recommended Solutions (viable):


Note: All solutions, whether placed in Short Term, Mid Term or Long Term, are important. In total 18 suggested solutions are placed for consideration in this interim draft report.


Short Term Solutions: (those actions that might be immediately carried out)


1. Change the driving chain along with the pair of sprockets. The looseness of the chain was causing the impacts and tension force that pulls the system towards one direction. Note: It does not help improve the system by either changing the chain or the sprockets.


2. Stop greasing the driving chain as presently done. It not only accelerates wear but also induces slip-stick phenomenon. Keep the chain clear of dust. Spray lubricate the roller joints once a week.


3. Improve the dust collection system and make it highly efficient.


4. Remove the shims placed to level the system in static condition. It makes the situation worse in dynamic condition.


5. Change bearing grease from EP 2 grade to EP 3 grade.


6. Place graphite blocks on friction area of supporting rollers instead of placing those on the Mill tyre.


7. Replace the fluid coupling by non-linear rubber coupling. Design if necessary. This would stop the excitation of the natural frequency of the system and the system would relatively unaffected by the impact forces operating under negative damping.


Mid Term Solutions: (Actions that might be carried out within next 15 to 30 days).


1. Change shaft material of roller from EN 24 forged to EN 36 B (forged).


2. Keep surface roughness of the contact surface within +/- 6 microns (CLA – Center Line Average). This is because endurance limit of high strength steels turns out to be no higher than that of ordinary steels if machined roughly.


3. Chamfer the shoulder areas of the shaft to a fillet radius of R15.


4. Strain harden the above fillet area to raise the load carrying capacity of shafts by 50 – 60%


5. Check for proper interference fit. Shaft Tolerance = n6. Housing bore tolerance = K7 (recommended).


6. Restore the original number of flights within the Mill. And maintain them from time to time so as not to run the mill with lesser number of flights than originally designed.


7. If possible, try to run the mill at higher loads than present.


Long Term Solutions (those actions that might be taken within the next 180 days)


1. Incorporate Duplex Chain drive, silent type.


2. Cross brace the civil structure that includes the four concrete pillars/columns and the independent column on which the gear box rests. The outcome should be such that deflection in any direction should not be more than 50 microns, measured at the top position

of the pillars.


3. Get help of foundation (soil mechanics) and structural specialist as required, to check the adequacy or integrity of soil foundation and civil structure.


4. Ground off the excess tire material from the roller crusher.


Records: (of vibration data)


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