### Understanding resonance is essential for solving problems of increased vibration.

Resonance is a condition that can occur in mechanical structures and can be described as sensitivity to a certain vibration frequency. Resonance occurs when a natural frequency is at or close to a forcing frequency, such as rotor speed. For machinery—such as pumps, turbines and electric motors—resonance can amplify the small vibratory forces from machine operation, and severe vibration levels can result. Such problems often develop after a speed change has been implemented, as with retrofitting a machine with an adjustable-speed drive (ASD) or operating a 50-hertz motor on 60 hertz of power.

The solution to these problems frequently depends on distinguishing between structural resonance and a rotor critical speed. Structural resonance refers to excessive vibrations of non-rotating components, usually machine components or supporting structures. Rotor critical speed refers to a condition in which the speed of the rotating element of the machine matches the rotor’s natural frequency.

## Structural Resonance or Rotor Critical Speed?

Structural resonance is the more common resonant condition because of the complex design of the casing and supporting members. Most often the structure supporting a machine or a non-rotating machine component is resonant at or near the rotating speed of the machine. Even slight vibratory forces from residual unbalance and misalignment effects of the machine can excite the resonant base structure, resulting in severe vibration. A good example of structural resonance is the reed frequency vibration that often occurs with vertical turbine pumps that have a motor mounted on top of the discharge elbow. The machine components can also be resonant. There are many examples of two-pole electric motors where a resonant end bracket caused very high axial vibration at 1 x rpm or 2 x rpm.

A rotor critical speed exists when the resonant component is the rotating element of the machine. This is common with centrifugal pumps; gas and steam turbines; and large, two-pole electric motors. While the result is similar to structural resonance (high vibration when a certain operating speed is reached), rotor critical speed is a more complex phenomenon because of speed sensitive components, such as bearings. When the operating speed reaches the resonant frequency of the rotating element, the rotating element distorts, and the vibratory forces increase significantly.

It is important to properly distinguish between structural resonance and rotor critical speed. The term “critical speed” (without the word “rotor”) is somewhat ambiguous. Technically, a critical speed could be either a structural resonance or a rotor critical speed. For the sake of clarity it is best to avoid using that term. The simple term “resonance” can be applied to both conditions to avoid confusion.

## The Characteristics of Resonance

As described above, the most notable characteristic of resonance is increased vibration when a certain operating speed is reached. Also, as the operating speed is increased beyond the resonant frequency, the vibration amplitude will decrease somewhat. The Bode plot in Figure 1 shows the operating speed versus the amplitude. For the sake of illustration, assume that the exciting force is residual unbalance of the rotor at the rotating speed.

The formula for calculating the natural frequency is:

Where “K” is the stiffness of the resonant structure or component, and “W” is the weight (mass). Note that at the core of this formula is:

Increased stiffness will, therefore, raise the natural frequency, and increased mass will lower it. That is logical since stiffness creates a force that is always directed against motion, while mass has inertia, which is a force always directed with motion. Resonance is what happens when these two opposing forces are equal. They cancel each other out, allowing vibration to increase.

Figure 1. Bode plot of resonance |

## The Damping Factor

A third force, damping, is at work throughout the speed range. Damping absorbs vibratory energy, converting it to heat. In doing so, damping reduces the maximum amplitude of the vibration at resonance and increases the width of the amplification zone (see Figure 2). A common example of damping is shock absorbers on a vehicle. Machines with sleeve bearings may have significant damping that can even mask critical speeds. On machinery bases, concrete and grouting add significant damping to a base structure. These forces (stiffness, mass and damping) determine the characteristics of resonance and are important to the distinction between structural resonance and rotor critical speeds.