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As a review, see Figure 1 indicating the modal shapes of truly resonating parts. Note that the circles with the phase marks allow you to determine which direction the part is moving. The major part of the “curl” is called the antinode. The place of zero amplitude is called the node.

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Figure 1: Resonance modal shapes showing antinodes and nodes

Note that the antinode portion with the largest amplitude in the “curl” shape is not where the metal cracks. Instead, see Figure 2 showing the resonance mode shape of a part when vibrating at its second resonance frequency (or a higher resonance frequency).

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Figure 2: Typical locations for obtaining amplitudes for plotting resonance mode shapes

Note that the location of the smallest or almost zero amplitude is called the node. At one side of the node, the metal at the top surface is momentarily in tension. The bottom is in compression. At only a one-half vibration cycle later, the situation reverses. The surface that was in tension is now in compression. The surface in compression reverses to go into tension. Back and forth at the frequency of the vibration caused the resonance modal shape.

To get an idea of what is happening, hold in your hand a portion of the steel in an ordinary paper clip. Flex it back and forth with reversible stresses at the zero point. Within a minute or so, the steel will break at the node.

I have had many vibration consulting experiences where the problem was a part cracking about every two weeks or so. In every situation encountered, the resonating part modal shape was graphed showing curling at the antinodes and almost zero amplitudes at the nodes. Cracking, or “fatigue” as some would call it, always occurred at a node.

Never have I experienced cracks forming in situations where there was no resonance with its antinodes and nodes.

For example, the resonant sides of a gearbox would have at least one antinode. Update International instructors refer to this vibrating modal shape as curling. Assume that the sides of the gearbox are supporting the bearings that also support the gear’s shaft. As the sides vibrate with the curling motion, the gear’s shaft also moves in a motion that results in improper gear mesh. This increases the amplitude of the gear mesh vibration, causing the analyst to assume the gear was improperly machined or improperly assembled (yet, a slow roll of the same shaft would indicate proper mesh).

Nodes due to a resonance mode shape are also easily visualized on, for example, a cover plate, beam, length of shaft, etc. It is also known that fatigue cracks form at the nodes. However, not all analysts are familiar with how nodes and antinodes also form on a circular disc-shaped part, such as a spur gear.

Nodes and Antinodes on a Gear’s Main Body

At the conclusion of a seminar, I was stopped by the operations manager of a large chemical company, who indicated one of the spur gears in a gearbox driven by a large steam turbine repeatedly fractured. Fracture and total machine shutdown occurred at an average of every 90 to 100 days. Each time, a new replacement gear would crack in about the same length of time (there were actually several turbines and gearboxes where this occurred).

I was reminded of an article written by Dr. Neville Rieger. The diagram in Figure 3 supplied by Dr. Rieger’s article says it all. Note the equally spaced radial nodes. His diagrams recorded only the frequencies at which the nodes formed. The frequencies were very high and must have originated from the different gear mesh frequencies that resulted at different gear running speeds (it is assumed that at frequencies in between those indicated that the nodes and antinodes disappeared). The gear’s RPM was gradually increased, thereby increasing much higher gear mesh frequencies. The four equally spaced nodes occurred at the lowest resonance frequency. The next, at higher frequency, showed a circular node. As the frequencies increased, more nodes and antinodes resulted. Dr. Rieger concluded that when a gear’s rotational speed at the time of resonance remained constant, cracking formed at the nodes.

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Figure 3: Nodes forming on a gear plate when increasing RPM and specific gear mesh frequencies cause resonances

As with antinodes on a pipe or flat plate, such as the top or end of a steel base, antinodes and nodes also form on a round plate, such as a circular steel plate that supports fan blades or pump impeller blades. Instead of visualizing equally spaced nodes along the straight line of a pipe, for example, imagine the line with antinodes and nodes bent around into a circle. The spacing would be equal. The higher the resonance frequencies, the greater the number of nodes and antinodes. In the situation described by the operations manager, the turbine driver gear’s rotational RPM created the specific gear mesh frequency to resonate the gear with several radial nodes. Finally, a complete fracture would occur at one of the nodes. Partially developed cracks at the other nodes revealed that they too were equally spaced.

In another gear that was very large, the gear would fracture several times a year, resulting in extremely high production as well as rebuilding costs. Calculations indicated that the gear’s strength was several hundred percent greater than what was required. Yet fracturing started when the machine’s operating speed was increased less than 10 percent. The gear’s operating speed was under 500 RPM. However, there were over 100 gear teeth per gear. The very small RPM increase resulted in a gear mesh increase of well over 1000 cpm. The new gear mesh amplitude/frequency resonated in the angle between the rim and gear’s main disc, resulting in a node that finally resulted in a crack. The crack at the node kept lengthening until the rim was weak enough to break off several inches. This shut down the whole paper machine. (See Figures 4-6 of the portion of the gear’s outer rim with the gear teeth that broke off and stopped production.)

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Figure 4: A sheet of 8 ½ x 11 inch paper to show gear’s size

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Figure 5: Crack forming at the resonant portion’s node

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Figure 6: Portion of the gear’s outer rim that would eventually break off, starting at the node

Symptoms of Nodes Developed by Resonances Due to Torsion:

Not to be overlooked are nodes formed from torsional resonance, which sometimes occurs in the rotor shaft itself. For myself, I found it only once in a Lamson blower rotor. The problem presented to me was a crack that developed just a couple of inches aft of the bearing next to the overhung flywheel with gear teeth around its periphery. The other cracks that developed were at the axial center of the rotor. The rotor’s crack and complete severance at its axial center had its own unique shape. In other situations, the crack at the resonance rotor’s node would break straight across the shaft, square with the rotor shafts diameter. However, the blower’s torsional resonance would cause a node, whereby the torsional motion in one radial direction would be followed by a twisting motion in the opposite direction. Finally, the repeated torsional vibration would cause the rotor’s torsional node to crack and result in a completely broken shaft.

As there were other Lamson blowers in the same section of the plant, I used a variable vibration speed vibrator to torsionally resonate another shaft. All resulting situations followed similar instances of vibration amplitudes becoming larger at the torsional antinodes and almost zero amplitudes at the torsional nodes. The main difference between the more common resonance breaks at the nodes and the less common torsional vibration amplitudes was the torsional cracks were not straight across the shaft (not square), but instead were breaking at a 45 degree angle.

The 45 degree angle break reminded me of the breaks I experienced as a 15-year-old student at Brooklyn Technical High School’s Strength of Materials Laboratory. At that time, we saw all types of breaks called ultimate strength breaks. The usual break would be straight across, but the torsional ultimate strength breaks would be a 45 degree angle. That’s how I put the two thoughts together and realized the break on the Lamson blower’s axial center of the shaft was due to torsional resonance. It was then easily prevented by replacing the overhung flywheel with smaller and a different number of gear teeth. This produced a different torsional frequency that no longer resonated the rotor shaft’s torsional resonance and therefore, did not result in a break at a torsional node.

From the information presented in this article, it could easily be assumed that the best solution to prevent cracks from forming at a resonating part node would be to search for the parts that are prone to resonate or are already resonating. To change the resonance frequency of those parts by changing their rigid bracing, adding weight, etc., would work, but may not be the easiest or lowest cost solution. In my experiences, I have found that a relatively large percentage of machinery rotor and structural parts, such as pipes, skids, gears, gearbox covers, etc., are resonant to vibration frequencies originating in other nearby machines. To eliminate the resonance in a resonating part, it is often easier to reduce the vibration amplitude that is originating at another nearby machine.

While these are only a few case histories, I have never been involved in a tough consulting problem where a crack formed was not due to a resonating part. Not a one.

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