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Understanding "Resonance"

I'm hoping to initiate a series of articles dealing with various topics in the Machine Condition Monitoring space. The intent is to keep most of these pretty short, and as entertaining as possible.

We all have a very good understanding of resonance, even if we may not appreciate that we have this understanding. We experience this natural phenomena continuously, every waking moment. Whenever we speak, resonance is at work; our vocal cords are set in motion by air forced past them, causing resonance. The cords are tightened or relaxed to modify the frequency of the resonance, our throat, tongue and lips further refine the resonance into speech.

This resonation excites waves of pressure variations that project outward from the source. Everything we hear results from the inner structures of our ear ‘resonating’ in exact response to these pressure waves constantly modulating the void that surrounds us. We perceive this as sound, speech, ‘noise’ or music. Thus, we are all intimately familiar with the basic concept of resonance. Resonance is defined as the RESPONSE of a structure to a FORCE. In our everyday example of speech, the driving force is the movement of air past our vocal cords. In the case of hearing, it is the response of our inner ear to the force of pulsating pressure waves in the air, amplified by the ear drum. The FUNDAMENTAL FREQUENCY (fn) of resonance is strictly determined by only two parameters – MASS (m) and STIFFNESS (k). Increasing mass will lower resonant frequency, lowering mass will increase frequency. Conversely, increasing stiffness will raise frequency, and decreasing stiffness will lower frequency. The exact mathematical relationship is shown in the following equation:

Math geeks will notice that old 2*pi factor thrown in there to confuse mere mortals and those mathematically challenged, but this makes perfect sense when you think about it for a bit. Vibration is of necessity something that is intrinsically cyclic or circular in nature – it begins at one point, extends to its maximum, and then returns to the initial point, repeating over and over in the same manner. Also note that the natural frequency is determined by the square root of the quotient of stiffness divided by mass. Thus, doubling mass does not directly halve resonance frequency, it will be reduced by ~30%. Doubling stiffness results in increase in frequency of ~550%! Or, to think of it from another angle, resonance frequency is more ‘sensitive’ to changes in stiffness, than to changes of mass.

OK, now that we have completely confused the concept by actually throwing a little math at my esteemed audience, let’s get back to more logical exploration of the phenomenon as it relates to machinery! Dave’s Laws of Vibration: Rule One – EVERYTHING vibrates! We tend to ignore this, or have become so accustomed to this that we fail to recognize it as we plod through our daily routine. What the MCM guys are concerned with is what is ‘normal’ for a particular machine they are responsible for. We have established that everything has a natural resonant frequency, determined by its mass, and its stiffness. There is not much we can do about this. Every part of a machine, each individual component, has a natural frequency at which it will vibrate, the ‘problem’ is when the operating speed of a machine happens to align with one or more of these natural frequencies! Natural frequencies are easily ‘excited’ and can go completely crazy if there are no damping forces to help limit or control the response of the part. The results can be completely catastrophic! For this reason it is important to know what the predominant resonant frequencies are in a given machine, and to insure that the machine is designed and operated in such a manner that these natural resonant frequencies are not excited.

We’ll investigate this in future discussion of Resonance Testing.

Reference: “Understanding √k/m” John G. Winterton, P. E. Math Geeks will greatly enjoy the insert contained within this article contributed by Dr. Neville Rieger, “The Historical Development of √k/m”

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