Last of Two Parts

I received interesting (and challenging) feedback from our readers on “Resonant Frequencies, Part 1” from the July issue of Pumps & Systems. Relating mechanical resonance to the electromagnetic effects in biological systems—including humans, bacteria and pathogens—is becoming a new technology similar to what initially might seem like a far removed field. However, many similarities can be discovered upon closer analysis.

Thanks for your interesting July article, “Resonant Frequencies,” which discussed vertical pump shaft, human and soft tissue resonant frequencies. In regard to the other factors that might be considered before using a simplified formula analysis (for example, Equation 1), the following might be considered:

A significant difference in the surrounding temperature (from ambient) could affect the elasticity modulus, E.

Closely spaced shaft sections (between stages) may affect the end condition, βn, of an adjacent, long length of shafting (such as the use of a clamped-end condition instead of simply supported at that location). Hanging weight and hydraulic axial thrust, F (added tension), acting on the end of a length of vertical pump shafting can affect the frequencies.

ωn2 = βn4 x EI / (m/L) [Equation 1]

The following is a modification of Equation 1 to approximately account for the effect of added axial shaft tension:

ωn2 = βn4 x E I / (m/L), can be rewritten as

ωn2 = (βn2 x E I) x βn2 / (m/L)

ωn2 = (βn2 x E I + F) x βn2 / (m/L)

This can then be written as:

ωn2 = ωn(0)4 + βn2 x F / (m/L)

Where:

ωn(o) is the frequency without axial loading from the original Equation 1.

Under “Examples” in the July article, a possible problematic shaft diameter between 1.8 and 2.4 inches is mentioned. However, the natural frequency for the existing 1.8 diameter shaft in Case B is only about 8 percent from the operating speed. I probably would want to see a larger margin to avoid potential lateral shaft vibration. Since the calculated natural frequencies for both the 1.44 and 1.8 diameter shaft sections are below the operating rpm, the pump will pass through these resonant speeds for a short duration while starting up or shutting down. Of course, variable speed applications could add another dimension to consider.

In the article, the natural frequency for humans and soft tissue seems to be based on simply supported boundaries. I probably would have also looked at clamped-free for a standing human and free-free for soft tissue. The Smithsonian/NASA Astrophysics Data System reports vertical (standing) whole-body human resonant frequency between 5 to 10 hertz. http://adsabs.harvard.edu/abs/2001SPIE.4317..469B.

From a different article, “In 1992, Bruce Taino of Taino Technology, an independent division of Eastern State University in Cheny, Wash., built the first frequency monitor in the world. Taino has determined that the average frequency of a healthy human body during the daytime is 62 to 68 hertz.” Perhaps this frequency is when lying down? http://cellphonesafety.wordpress.com/2006/09/17/the-frequency-of-the-hum....