Graphs in ANSI S2.19 and ISO 1940/1 also provide eccentricity (e, millimeters) based on the balance grade chosen and applicable operating speed. These graphs provide eccentricity values that correspond to the relationships in Equation 4.

Balance grade “G” number = (e)

Equation 4: [(2Π) (rpm) / (60)]

Rearranging and converting the units of e to English customary units result in Equation 5, with G in millimeters/second and e in inches.

e = [(60) (balance grade “G” number)] / [(2Π) (rpm) (25.4)]

OR

Equation 5: e = [(0.376) (balance grade “G” number)] / (rpm)

## Sample Studies

The input parameters required to produce resultant values of top-of-motor vibration are:

- (m/M), mass ratio
- balance grade “G”
- rpm
- ζ, damping factor
- (ω/ωn), frequency ratio

As seen in the discussion of the balance grade, balance grade “G” and rpm result in the required parameter “e.” From Equation 1 through 5, if the universal values of mass ratio (m/M), balance grade “G,” rpm, damping factor and frequency ratio can be identified, then calculated vibration results that can reasonably be expected across a broad range of design operating speeds (and corresponding pump sizes) may be determined.

The September 2012 article provided the derivation for a calculation of the structure reed frequency prior to the actual structure reed frequency characteristics known by testing. This is useful for obtaining a satisfactory separation between the operating speed and the structure reed frequency and an acceptable limit to the frequency ratio. No specific guidance to the frequency ratio was provided. In Figure 2, as the values of ω/ωn diverge from 1.0, increasingly diminished benefit is gained from greater divergence. At some values of ω/ωn removed from 1.0, the cost of implementation and practicality factors may become prohibitive. Conversely, if the frequency ratio ω/ωn is not adequate, the vibration response is magnified and unacceptable.

Relative to an acceptable limit to the frequency ratio, the Hydraulic Institute will soon have a new publication, ANSI/HI 9.6.8 *Guideline for Dynamics of Pumping Machinery*, which provides guidance on this topic and on other topics related to avoiding resonance and performing dynamic analyses on different types of rotodynamic pumping equipment and structures. This guideline states that a frequency separation margin of 10 percent—obtained in the field by test on the installed equipment—is typically satisfactory for avoiding unacceptable vibration. Recognize that the frequency ratio conveys the same information as frequency separation margin, only expressed in a different way.

For an analytical study, a field installation situation is considered with an assumed frequency separation margin as determined by test of 10 percent. The other required input values are:

(m/M) mass ratio = 0.25 to 0.375; use 0.33 for the example study

Balance grade “G” = 2.5 to 6.3; dictated by the motor vendor’s standard or purchasers’ specifications; use these values for comparison

rpm = 3,600 rpm to 600 rpm; synchronous speeds used for simplicity

Damping factor (ζ) = 0 to 0.03; use 0.02 or 2 percent for the example study

Frequency ratio (ω/ωn) = 0.9 or 1.0; use 0.9 for the example study, or ωn 10 percent higher than the pump speed, ω

Note that several studies using alternatively conservative or aggressive assumptions provide an envelope of vibration levels that may be reasonably expected with the mathematical model presented. Tables 1 and 2 depict the results from two example studies using the input information provided above. Table 1 depicts results for G6.3 residual imbalance. Table 2 depicts results for G2.5 residual imbalance with all other inputs the same.