Mathematical formulation of equation in Junker 1969 paper: self-loosening of fastener by vibration 3

[kbad]

Can you help me to understand how the first equation in the paper 'New Criteria for Self-Loosening of Fasteners Under Vibration' by GH Junker, 1969 is derived? I'm not able to understand the mathematical and mechanics formulation as I tried to derive the same equation by myself.
Please refer to below images for reference to the equations and the text in the paper.

 

[IRstuff]

Does the derivation even matter? The Nordlock videos are pretty clear that bare fasteners just don't stay tight.

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[TugboatEng]

I don't agree with that statement. There are plenty of fasteners that don't come loose without fancy washers or retention.

It might be worth noting that some joint configurations are more prone to loosening.

 

[3DDave]

Nordlock is the answer to the question - how do I keep the bolts tight when the joint has failed. The Junkers test specifically forces that to be the case.

Anyway, the equation looks like it is adding the friction under the head of the nut to the friction at the threads. The load on the nut is L. The tan(rho) is the ratio between the normal load and the shear load. The tan(-omega) is the reduction in that ratio due to the adverse slope of the thread, basically the down-hill component tending to turn the nut.

So, if the total force on the nut face and the nut threads become zero or less the nut comes loose.

It's not a particularly useful equation as the variability in all the components are so large that keeping far away from that condition is wise and preventing slip failure in the joint is a better idea than letting it happen and then trying to stop it becoming worse.

 

[IRstuff]

There are plenty of fasteners that don't come loose without fancy washers or retention.

They don't come loose because they're in a sufficiently benign environment. Few cars get to 150,000 miles without squeaking joints and some loose fasteners.

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[TugboatEng]

Few engines in said cars experience loosen of fasteners on their reciprocating assemblies. Loosening of connecting rod bolts is never directly attributed to vibration for example.

Squeaking joints and loose fasteners may be endemic to European vehicles but other manufacturers have it figured out. Nordlock washers were not the solution.

 

[IRstuff]

Nordlock washers were not the solution.

They are the solution for a very specific vibration environment, specifically, the Junkers vibration. Show me what the alternative would be for that environment.

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[TugboatEng]

Obviously body fitted bolts would be the best solution. Dowel pins would work. Grip length always makes things better.

 

[saplanti]

perhaps these explain it better;

https://www.boltscience.com/pages/self-loosening-of-threaded-fasteners.pdf

https://www.fluidsealing.com/wp-content/uploads/Sealing-Sense-Jul-2022.pdf

http://www.perfectlock.net/pdf/junker_testingreport.pdf

Longer but detailed information is here;

https://core.ac.uk/download/pdf/9632852.pdf