electricpete - While reading the Den Hartog book I was again puzzled by the question with which this thread started. In 4th ed, page 88, last line, he says "It was seen however that addition of an absorber has not much reason unless the original system is in resonance or at least near it." Again on page 92 he gives example of electric hair clipper where he is trying to design a tuned vibration absorber for 120Hz forced frequency of electromagnet. That is : he is targeting forced vibration which has nothing to do with resonance of hair clipper body on which he mounted the absorber
That's a $64,000 question.
A few thoughts to consider:
Based on the proofs above, the undamped dynamic absorber ideally works to remove vibration at the absorber-tuned frequency, regardless of the attached system (so it is not limited to a system where the forcing frequency that we are tuning for corresponds to resonance of the original system). So it's not so much a question of whether we can be successful at reducing vibration (as long as we're using big enough absorber), as it is a question of whether application of absorber to reduce vibration will accomplish anything useful.
Does it make sense to add an absorber to reduce forced single-frequency vibration in a non-resonant system?
I'd say it depends.
Let’s say I have a 1200rpm 50hp horizontal rolling bearing motor with 1.0 ips horizontal, 0.5ips vertical running speed vibration measured on the bearing housings. Let’s also say the machine is operating far below resonance both directions, although a little further below in vertical direction than horizontal direction since stiffer vertical direction has higher resonance (all of that seems typical for this description machine to me). Let’s say I know or suspect that the cause is unbalance of the rotor. Obviously the ideal solution is to balance it but that maybe the machine needs to continue running and that’s not practical. For this scneario: Should I use absorber to reduce the vibration at the bearing housing?
The system resembles single degree of freedom system far below resonance. The SDOF mass is roughly the rotor mass. The bearing and frame transmit the force to the base. Since far below resonance, the system is spring dominated, the amount of unbalance force that goes into mass acceleration is small, almost the entire unbalance foce is trnasmitted through the bearing and frame to the foundation.
I could could attach a dynamic absorber to the frame near the bearing in such a manner to drive the horizontal vibration towards zero.
If my concern is the force transmitted through bearing, I have done nothing to address it because the force transmitted through bearing is the same (the unbalance force). Adding the ideal dynamic absorber in this case is equivalent to adding an ideally infinitely-stiff brace to the frame near the bearing to drive vibration at that location to zero. We can easily see in the case of the brace that we have only further increased the resonant frequency of the system, it remains very far below resonance (Even moreso than before), so the force transmitted through the bearing is still the unbalance force in this far-below-resonance (spring controlled) system. The result for the dynamic absorber is the same as for the infiitely-stiff brace... I just thought it was a little easier to explain using the infinitely-stiff brace. I hope in this situation it is clear the dynamic absorber does nothing if my goal is to reduce load on the bearing.. . On the other hand if machine were operating near resonance, then the resonance could magnify the unbalance force such that the force seen at the bearing could be many times higher than the unbalance force. In this case, it is helpful (reduce bearing load) to add a brace or to add a dynamic absorber.
Consideration of the above scenarios may be part of the basic thought process by which Den Hartog implies dynamic absorber is generally not real helpful unless used near resonance.
So can we come up to an exception to the "rule" that absorber is only useful in resonant system?
Sure. Den Hartog gave one that you mentioned (hair clipper).
I’ll give another one: let’s go back to the machine above and put it back far below resonance. But let’s say there are cracks on the foundation below the machine running perpendicular to the shaft under the coupling. We suspect the vibration is aiding the crack growth. We also suspect the vibration crack is causing misalignment between motor and load which is causing our 1x vibration (new scenario... not unbalance anymore). Now should I add an absorber to the machine frame? Yes, it would reduce the force transmitted from frame to vibration and reduce the crack growth and help break the cycle. Maybe not a permanent solution but dynamic absorber should help. And it will help even if you are not near resonance.
I had to stretch to come up with scenarios that were simple enough to yield an obvious yes/no answer to the question: should we add a dynamic absorber” (although the cracked foundation is pretty close to a scenario at our plant where we have installed an absorber). In most cases I suspect the answer is not that clear.
Certainly the insight from people like Den Hartog and our own Greg Locock will be valuable as a starting point.
By the way, the particular sentence you quoted at bottom of page 88 occurs in the context of a seguee into a discussion of the particular behavior of dynamic absorber applied to a resonant system. It is Den Hartog’s way to start general and wander to many particular special cases. Maybe he is just justifying to small extent why it is worth to examine that special case (resonant) more closely than the others.
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(2B)+(2B)' ?
Earlier my concept was wrong. I was thinking that the optimum damping equation which you wrote refers to the structural damping of the added masses in tuned vibration absorber. So I was creating such phrases! In fact Den Hartog explains so nicely about 'Optimum tuning' and 'Optimum damping'. What electricpete wrote simply in two steps is summary of that. First we do optimum tuning and then optimum damping. And the formula was actually derived to be a non-dimensional. So it looks like that.
