Vibration levels at the top of a vertical motor in a typical pump-motor structure (see Figure 1) remain an area of interest in the pump industry. This article provides a mathematical model for estimating the vibration at the top of the motor to compare to empirical data and provide insight into how parameters can impact the vibration level produced. This insight may help those in the industry understand vertical pump vibration and assist in the development of standard vibration acceptance criteria.

**Figure 1. Typical vertical pump-motor structure**

Current vibration standards provide acceptance criteria for measurement locations at the top of the motor support (near the bottom of the motor) for vertical pumps. Guidance on acceptance levels for the top of the motor is not provided. General consensus among standards-producing organizations is that measurements obtained near the motor bottom are best suited for pump acceptance.

Provided the structure is not in resonance, this is a fair approach. For a fixed vibration acceptance criterion near the bottom of a motor, the vibration response at the top of the motor results from the motor properties that can vary considerably. Figures 2 and 3 show a sample of observed motor reed frequencies and the derived stiffness using data from different manufacturers’ motors. Motors with such varying properties may be expected to produce diverse vibration responses at the top of the motor for the same level of vibration near the bottom of the motor.

**Figure 2. Observed typical values of stated motor reed frequency versus torque**

**Figure 3. Observed typical values of motor stiffness versus torque based on stated motor reed frequency values**

The pump head that supports a motor can also vary. Structure heights and weights are different, as are the mechanical design and material properties. Consider that the modulus of elasticity that relates to deflection (and vibration) for cast iron is about half that of steel, but conversely, the damping capacity of gray cast iron is considerably greater than that of steel. To top it off, bearing manufacturers’ research studies regarding the impact of vibration levels on bearing life are lacking. Such information would help the industry weigh the costs of lower vibration levels versus the benefits. This comparison is important because providing low vibration levels in a vertical pump-motor structure may significantly impact the equipment’s initial costs, and higher vibration levels can impact long-term maintenance costs.

These issues represent a few challenges faced when considering vibration acceptance criteria at the top of the motor. A mathematical model may help provide insight and is examined in this article.

This material is presented with the following simplifying assumptions because the pump-motor structure’s vibration can be complex:

- The primary focus is the vibration at the operating speed due to residual imbalance of a motor used on a vertically suspended pump.
- The structure of interest is the area above the baseplate.
- Of necessity, the natural frequency of the vertical structure must be discussed. This discussion is limited to the reed frequency—the first bending mode lateral natural frequency above the baseplate.
- In a particular structure, other relevant modes of vibration, corresponding natural frequencies and other relevant excitation sources may be experienced, but these are not included within the scope of this article.

## Modeling the Motor Properties

Examining the reed frequency of the vertical pump-motor structure begins with knowing the reed frequency of the motor. National Electrical Manufacturers Association (NEMA) MG 1 references the motor properties that are typically supplied by the motor manufacturer:

- The reed frequency
- Motor weight
- Distance to the motor center of gravity from the motor mounting flange face
- The static deflection of the motor center of gravity (CG)

The motor static deflection is the deflection of the CG when considered mounted to a rigid foundation, turned horizontally and deflecting under its own weight. As stated by NEMA, the motor reed frequency (f_{n})—in cycles per minute (CPM)—and the motor static deflection (Δ_{s}, in inches) are related as shown in Equation 1.

**Equation 1**

Where g = 1,389,600 in./min.^{2}

The effective motor frame stiffness is obtained from the classic spring stiffness equation (see Equation 2).

**Equation 2**

Where:

k = The effective stiffness of the motor frame

W = Weight of the motor (pounds)

Equation 2 was used to determine the values of k shown in Figure 3. Note that the deflection of the motor CG location may also be represented by a simple cantilever beam model as (see Equation 3).

**Equation 3**