"Top-of-Motor Vibration” in the September 2012 issue of *Pumps & Systems* provided a mathematical model for estimating vibration at the top bearing of a motor used on a vertical, wet-pit, column type pump (see Figure 1)—a topic that continues to interest vertical pump users, specifiers and manufacturers. The simplified cantilever beam model aids in understanding vertical pumps’ vibration and allows comparisons to empirical data for the development of standard vibration acceptance criteria at the top motor bearing, for which no guidance currently exists. Part One of this series discusses insights that may be gained by analyzing the mathematical model.

The structure consists of a motor mounted on a pump discharge head, above the baseplate. As background information, a list of the simplifying assumptions for the mathematical model is provided below:

- The structure of interest is above the baseplate of a vertical, wet-pit, column type pump.
- The focus is vibration at the top motor bearing because of motor imbalance.
- The imbalance acts in a single horizontal plane located at the motor rotor center of gravity (CG) of location.
- The CG of the motor is in the same horizontal plane as the motor rotor CG.
- Only filtered vibration at the operating speed is considered.
- Only the motor mass is considered.
- An infinitely rigid foundation was implied because no foundation effects were considered in the model.
- The natural frequency considered is the reed frequency of the structure above the baseplate. As previously discussed, this is different from the motor reed frequency that is always a higher value than the structure reed frequency.
- Other relevant modes of vibration corresponding natural frequencies and excitation sources that may be experienced are not included within the scope of this series. Only rotating imbalance is considered.

Within these limitations, in the September 2012 article, an example using the characteristics corresponding to one installation was provided showing the calculated vibration response for that specific installation. The model can provide insights that apply generally to installations of this type.

Some questions pertaining to top-of-motor vibration include:

- Should the height of the top motor bearing from the foundation be a consideration for a vibration acceptance criterion at that bearing?
- How does rigidity of the structure factor in?
- What is the effect of motor balance grade?
- How does the foundation rigidity impact vibration?

From the mathematical model, the amplitude (X) vibration at the motor CG is shown in Equation 1.

Equation 1:

Where:

X = Amplitude of displacement of the structure at the motor CG (inches); this is the peak displacement and is doubled to obtain a peak-to-peak value.

e = Eccentricity of the rotating mass related to the residual imbalance (inches) and operating speed. This is determined by the balance grade of the motor. Refer to ANSI S2.19 or ISO 1940/1.

m/M = Dimensionless ratio of the rotating motor rotor weight (m) to the motor weight (M).

ω/ωn = Dimensionless ratio of operating speed (ω, rpm) to the structure reed frequency (ωn, cycles per minute—CPM), whether determined by calculation (as discussed in the September 2012 article) or by test.

ζ = Damping factor (dimensionless); damping factors for these types of structures in pump installations have been observed to be between 0 and 0.03 (0 to 3 percent).

The dimensionless vibration response may be depicted graphically in Figure 2.

The peak-to-peak displacement at the motor CG is 2X (inches). In the example from the September 2012 article, the vibration response at the top motor bearing was twice the vibration at the motor CG. This is fairly typical for such installations. The peak-to-peak vibration at the top-of-motor bearing is 4X (inches). To express in mils, the vibration at the top-of-motor bearing is (4X/1,000).

Because the excitation results from the motor imbalance that occurs at the operating speed, this is filtered vibration that may be directly converted to other units of vibration. For example, converting mils peak-to-peak displacement to velocity can be calculated using Equation 2.

Equation 2: V = (Mils) (CPM) / 19,099

Where:

V = Velocity, inches/second (peak)

Units of root-mean-squared (RMS) velocity are used commonly for vibration. Units of RMS velocity may be calculated using Equation 3.

Equation 3: Vrms = (0.707) (V)

## Balance Grade Decoded

For the purpose of this article, an explanation of America National Standards Institute (ANSI) and International Organization for Standardization (ISO) standard balance grades is necessary. Motor manufacturers’ standard balance grades correspond to a balance grade from ANSI S2.19 or ISO 1940/1, such as G2.5, G6.3 or some interim grade (G, millimeters/second). From ANSI S2.19 or ISO 1940/1, the grade number G refers to a value of constant velocity in units of millimeters/second. For example, G6.3 refers to 6.3 millimeters/second residual imbalance.