# Spring Critical Frequency: Shigley vs. MERM



## ezzieyguywuf (Sep 14, 2017)

Shigley's chapter 10, on page 527, describes the fundamental frequency and weight of a spring as follows:




In the MERM, in table 60.1, the same equation is given for the linear frequency of a mass-and-spring system except there is a factor of pi in the denominator of the 1/2. This equation is (I believe) used to derive formula 54.18. I can derive this same formula using the following formula for the weight W of a spring:




My problem is that factor of pi that is in equation 54.18 in the MERM. I'm working on problem 58 in the 6 minute Solutions for Mechanical Systems and Materials, and their solution seems to also include the factor of pi.

Does anyone know why the MERM includes this factor and Shigley does not? Shigley specifically states that this equation is for "a spring placed between two flat and parallel plates". Could this be the reason? I don't see any clarification regarding the end conditions in the MERM. Any help would be greatly appreciated.


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## ezzieyguywuf (Sep 14, 2017)

ezzieyguywuf said:


> Shigley's chapter 10, on page 527, describes the fundamental frequency and weight of a spring as follows:


excuse my typo.


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## JHW 3d (Sep 14, 2017)

The 1/2Pi converts rad/s to Hz.

The 1/2 for critical freq of spring may have to do with the fact that the effective mass of the spring is distributed over the length of the spring. This is a guess.


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## starquest (Sep 15, 2017)

I found this same discrepancy when I was studying back in early 2016.   Which edition of Shigley are you utilizing?  I used the 5th edition.  

Cutting to the chase, you need the 'pi' factor.   It's an error in Shigley's.  Not sure if they corrected it in later editions.


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## JHW 3d (Sep 15, 2017)

starquest said:


> It's an error in Shigley's.


:banhim:

How dare you question Shigley!

Joking aside... *it is not an error*. It is derived from a first order solution to the *wave equation*. Quoted from 9th edition:

_If a wave is created by a disturbance at one end of a swimming pool, this wave will
travel down the length of the pool, be reflected back at the far end, and continue in
this back-and-forth motion until it is finally damped out. The same effect occurs in
helical springs, and it is called spring surge. If one end of a compression spring is held
against a flat surface and the other end is disturbed, a compression wave is created that
travels back and forth from one end to the other exactly like the swimming-pool wave.
Spring manufacturers have taken slow-motion movies of automotive valve-spring
surge. These pictures show a very violent surging, with the spring actually jumping
out of contact with the end plates._

It then goes on to show the wave equation, and then jumps to the solution. The solution of the angular frequency has a *π* term in the numerator, so when you divide by *2π* it cancels out.



ezzieyguywuf said:


> In the MERM, in table 60.1, the same equation is given for the linear frequency of a mass-and-spring system except there is a factor of pi in the denominator of the 1/2. This equation is (I believe) used to derive formula 54.18.


This is an incorrect statement. See the treatise above.


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## starquest (Sep 18, 2017)

I stand corrected, haha.


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## ezzieyguywuf (Oct 11, 2017)

JHW 3d said:


> This is an incorrect statement. See the treatise above.


Thank you for providing your treatise. I am still confused as to equation 54.18 in MERM though. How is this equation derived? I asked mostly because I don't like taking equations like this at face value - I'd rather understand where they come from, as in my experience this makes it easier to adjust/modify the equation if needed for particular problems.


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## JHW 3d (Oct 12, 2017)

It is a solution to the wave equation. See my previous reply. It consists of a family of harmonic solutions to a partial differential equation. If you want more detail on how to go from the wave equation to the solution presented in shigley, Wikipedia may be a good place to start:

https://en.wikipedia.org/wiki/Wave_equation


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