Undamped Compression Spring Natural Frequency

[mkenwort]

Hi everyone, bit of a debate going on amongst some mechanicals at my company and I thought I'd fish for some insight online. With regards to the natural frequency of a spring (both ends fixed), I've encountered a company design guide that suggests a spring will resonate longitudinally at: [(2*wire_dia)/(PI*mean_dia^2*active_coils)]*sqrt[(modulus_rigidity*grav_const)/(32*density)]. So far I agree, in fact Shigley echoes this equation except that it makes the substitution of 9 ~= 2*PI*sqrt(2). My issue is that the guide goes on suggest equivalence of this equation to the following: (1/(2*PI))*sqrt(spring_rate/weight_active). I came across the eFunda derivation and it does not include the factor of PI in the denominator. This makes sense to me because a real spring is a distributed mass whereas the standard resonance equation [nat_angular_freq = sqrt(stiffness/mass)] I see in many textbooks assumes a lumped mass and generally the spring mass is negligible relative to the sprung mass.

Does anyone have any additional insight or perhaps an alternate derivation approach? I have modeled a spring in ANSYS and I get a natural frequency consistent with 1/2 rather than 1/2*PI and corroborated this on the test bench. I would like to contact the author of the design practice in order to have this document corrected, but would like to more thoroughly understand it such that I feel comfortable explaining to others. Aside from the eFunda derivation, I was completely unable to find another suitable reference that goes into any level of detail.

http://www.efunda.com/DesignStandards/springs/spring_frequency.cfm

Regards,
Mike

 

[israelkk]

My assumption is that the PI is used when the frequency is in Hz. The PI is ommited when the frequency is in rad/sec. I am almost convinced that ANSYS gives the results in rad/sec.

 

[sbozy25]

israelkk is correct. ANSYS does give frequency in rad/sec. and the PI is used when looking for Hz.

 

[mkenwort]

wouldn't that be 2PI? that's just angular frequency

i did find another textbook that corroborates the "no pi" equation...3rd edition of Machine Design by Norton...