non-linear spring as mechanism to limit "resonant" response? 4

[electricpete]

We are installing a Bendix coupling in place of gear coupling in a turbine-driven pump application at or plant.

I am not directly involved, but did have a chance to talk it over with the responsible engineer and it brought up some interesting aspects that I wasn't familiar with.

The turbine and pump have thrust bearings. An axial resonant frequency was calculated under assumption that both shafts are fixed and the coupling mass moves on it's own springiness (diagrpham) between them. If each half has stiffness K, the total stiffness of SDOF model would be K/2 since it is connected to "ground" on both sides.

Quasi-physical model:
Ground ===K===Mass===K===Ground

SDOF Model
Ground ===K/2===Mass

The resonant frequency of the coupling as calculated above falls within the operating speed range. But it is not considered a problem because:
1 - There is not a lot of axial excitation at running speed to be found. (open to comment.. I'm thinking misalignment may create an axial forcing function)
2 - The coupling is axially flexible and for low levels of forces and deflections (far below any endurance limits) it enters the non-linear region of the "spring" (diaphram). Apparently a stiffening spring creates a classic response vs frequency profile shaped like a shark's fin (with the pointy part of the fin pointing toward higher frequencies for stiffening spring and lower frequencis for softening spring). Mechanial Vibration's by Rao discusses this in terms of Duffing's equation. Den Hartog also discusses it in a simpler way.

I can see the shape of this non-linear system response curve is much different than a linear resonant SDOF system and I can see the idea of "resonance" doesn't apply to this non-linear SDOF in exactly the same way as a linear system. I can't particularly judge how effective it is in limiting max vibration magnitude for a system with only structural damping (no fluid or elastomeric damping).

Above is a lot of new concepts for me, I'd be interested in any comments. In particular:
Can anyone provide discussion or link of ability of non-linear spring to limit max vib amplitude for a given forcing function magnitude, variable frequency?
If this is an effective means to limit vibration magnitude, why isn't it used more often or discussed within "vibration control" section of textbooks?













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[MikeHalloran]

For the largest turn it will be some f.
For the next smaller turn it will be f minus some small d1.
For the next smaller turn it will be f minus some larger d2.

So a frequency sweep would see the resonance sort of spread out relative to a more common helical spring. I'm not saying there would be no resonance, just that it would have a low Q.




Mike Halloran
Pembroke Pines, FL, USA

 

[GregLocock]

Actually for the electrical dudes, what are the resonant frequencies of the following?

.    |--C--|    |--x*C--|
.----|     |----|       |---
.    |--R--|    |--y*R--|

Cheers

Greg Locock


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[electricpete]

if x and y are positive scalar constants, then it is a cascade of two first order linear systems. No oscillatory response, just two time constants.

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[GregLocock]

Put it in parallel with the source, not in series, sorry.

Cheers

Greg Locock


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[electricpete]

No matter what you do with it, it's an R/C circuit. Poles fall on the negative real axis can include terms like exp(-sigma*t) for single pole on real axis and t*exp(-sigma*t) for double pole. But no sinusoids or exponentially decaying sinusoids (you'd need to add inductance to the circuit to get an oscillatory response). It's a linear circuit, so I don't see the tie to the nonlinear discussion.

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[GregLocock]

At any one overall displacement at one frequency a non linear spring has an 'average' rate and an 'average' effective mass, which gives the resonant frequency aka the surge frequency of the spring. It is not directly relevant to your issue, it is directly relevant to the beehive springs puzzle.

The frequency response of systems with hardening springs (as they tend to call them) is discussed in most vibration texts, but I don't remember seeing a detailed derivation. eg fig 25

http://www.vibrationmounts.com/V100/PDF/VM10017.pdf

This ignores the practical effect that the a hardening spring will generate higher harmonics of the excitation frequency, particulalry 3x.

Cheers

Greg Locock


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[electricpete]

I like the vibrationmounts presentation.

Figure 25 resembles slide 2 that I posted. The difference is that my curves represent undamped Duffing oscillator, so my curves continue toward an asymptote forever. Figure 25 would be damped stiffening spring, so the curves intersect the asymptote line at a finite value. Figure 26 shows that the system can jump between an upper curve 1/2 and a lower curve 3/4, depending on which direction the frequency is swept. (the curve 2-3 is unstable solution, so increasing frequency past 2 results in jump to 4....and decreasing frequency past 3 results in jump to 1.

The multiple curves correspond to varying forcing function and the center / dashed curve shown on Figure 25 would correspond to zero forcing. So it is a natural frequency in terms of unforced response, which would persist if the system was undamped or die down along the dotted line to zero amplitude if system was damped. I guess my last post 30 Sep 13 20:14 on that subject should be disregarded.

The curves in my presentation do give an idea how the stiffening spring can be effective in limiting forced response imo. In the undamped non-linear/stiffening case, we could predict a max response (for a given force and system) based on max frequency (the higher the max frequency, the higher the max response). For damped non-linear/stiffening case at high frequecies (above frequency of point 2 in Figure 25), response is lower than undamped case. As we increase the force, the comparison between max amplitude of linear system and max amplitude of stiffening system becomes even more favorable… the linear system response increases proportionately to force while the non-linear/stiffening system response increases much less than the force.


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[electricpete]

The curves in my presentation do give an idea how the stiffening spring can be effective in limiting forced response imo. In the undamped non-linear/stiffening case, we could predict a max response (for a given force and system) based on max frequency (the higher the max frequency, the higher the max response). For damped non-linear/stiffening case at high frequecies (above frequency of point 2 in Figure 25), response is lower than undamped case. As we increase the force, the comparison between max amplitude of linear system and max amplitude of stiffening system becomes even more favorable… the linear system response increases proportionately to force while the non-linear/stiffening system response increases much less than the force.

Another property of mesh mounts is demonstrated by Figure 26 [5]. As can be seen, in practice there is a sudden sharp drop from the resonant point, ensuring that isolation is achieved almost immediately. However, it is again safer to assume that isolation does not begin until sqrt(2) Fn is achieved.

I wonder where the sqrt(2) came from. If it was empircal/experimental value then I might expect them to say 1.4. Sqrt(2) leads to suspect maybe some theoretical basis, but I have no idea what that would be.


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