# Understanding of resonance essential for solving vibration problems

## It's no secret that severe vibration can destroy bearings, ruin shafts and potentially disrupt production. What's less well known is that resonant machine components and supporting structures can magnify even small vibration problems enough to damage connected equipment or cause catastrophic machine failure.

09/17/2013

It's no secret that severe vibration can destroy bearings, ruin shafts and potentially disrupt production. What's less well known is that resonant machine components and supporting structures can magnify even small vibration problems enough to damage connected equipment or cause catastrophic machine failure. To solve a vibration issue quickly and avoid such undesirable outcomes, an important first step is to determine if the source of the increased vibration is resonance in the rotating equipment or in a supporting structure.

Resonant vibration in mechanical structures such as pumps, turbines and motors occurs when a natural frequency is at or close to a forcing frequency such as rotor speed. When present, this condition can cause severe vibration levels by amplifying small vibratory forces from machine operation. 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 Hz motor on 60 Hz power. The solution frequently depends on the ability to distinguish between structural resonance and a rotor critical speed.

Structural resonance: Structural resonance refers to excessive vibrations of non-rotating components, usually machine parts or supporting structures. Due to the complexity of these components, it is the more common resonant condition and usually occurs 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 this is the reed frequency vibration that often occurs with vertical turbine pumps that have a motor mounted on top of the discharge elbow. 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.

Rotor critical speed: A rotor critical speed exists when a machine's rotating element is the resonant component and its speed matches the natural frequency of the rotor. 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 at a certain operating speed), a rotor critical speed is a more complex phenomenon. 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's 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. It will also be observed that 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's 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.

### 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 (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 in the distinction between structural resonance and rotor critical speeds.

With structural resonance, the machine is operating very close to a resonant frequency. It is most noticeable when damping is low, since very high vibration amplitude results. There are two rigid modes  which can be described as “bouncing” and “rocking”. Solutions include changing the resonant frequency to move it away from the operating speed by modifying stiffness or mass and increasing damping to directly reduce the amplitude. (The various methods for implementing these corrective measures are topics for another article. The objective here is a comparison to rotor critical speeds.)

With a rotor critical speed, the problem is quite different. First, the stiffness, mass and damping of rotors mounted on rolling element bearings can almost never be effectively changed, and damping is typically very low. (Note: Mounted rotor natural frequencies of large journal bearing machines typically can be changed to some degree by modifying the bearing dynamics.) Second, no rotor is ever intentionally designed to have a critical speed close to its operating speed. The problem in this case is not that the operating speed is close to resonance, but that at the rotor critical speed the rotor distorts and non-linear effects cause excessive vibration. At that point it becomes a “flexible rotor” rather than a “rigid rotor.”

A rigid rotor operates below the first rotor critical speed and may have numerous unbalance forces distributed along its axis. The sum of these unbalance forces can be corrected in any two planes with common two-plane dynamic balancing methods. In these rigid modes the rotor may flex slightly, but the motions at the bearings accurately represent the unbalance condition. However, once the rotor becomes flexible, above the first rotor critical speed, the distribution of unbalance forces will distort the rotor, causing an unbalanced condition that was not present in the rigid modes. This flexible mode unbalance causes increased vibration that persists at higher speeds.

With structural resonance, the force is constant while the vibratory response of the structure changes with speed. With a rotor critical speed, the force changes as the rotor distorts to conform to unbalance forces distributed along the axis of the rotor. The solution to a rotor critical speed is to eliminate the unbalance forces in the various planes along the axis of the rotor. Usually it is not possible to detect where the unbalance forces are with the rotor in the rigid mode, so the rotor must be operated above the rotor critical speed (in the flexible mode) to detect the effects of the unbalance.

### Bending modes

As the speed of a rotor increases it will go through a series of bending modes: first bending mode; second bending mode, third bending mode, and so forth.

Rotors for multistage pumps and gas and steam turbines may operate above the first or second rotor critical speed, and generators sometimes operate above the third rotor critical speed. Rotors for large, two-pole electric motors may operate above the first rotor critical speed but seldom above the second. Rotors that are designed for such “flexible rotor” operation have provisions for additional balancing planes to accommodate dynamic balancing procedures that eliminate the residual unbalance forces that cause flexible rotor distortion. These dynamic balancing procedures require the rotor to spin at operating speed, which can only be done safely with specially designed balancing machines in a spin pit. Alternately, the individual components of flexible rotors, such as impellers, can be balanced before assembly.

Understanding the difference between structural resonance and rotor critical speeds will help clarify the discussion for maintenance and service personnel, especially when the topic is multistage pumps, turbines or large, two-pole motors.

Eugene Vogel is a pump and vibration specialist at the Electrical Apparatus Service Association, Inc. (EASA).

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