Trying to maintain smooth, precise motion can be an insurmountable challenge in the presence of vibration, which causes unwanted machine movements, loss of register, and improper operation of electronic controls.

Typically, vibration is reduced by balancing the rotating component (rotor) of the driving or driven machine, detuning system natural frequencies to avoid operating speeds, or adding dampers (flexible machine mounts). But, vibration can also be controlled by redirecting machine misalignment so that coupling forces counteract shaft motions. This technique can be more economical than disassembling and rebalancing the equipment.

Unbalance starts it all

Any rotating component (rotor) of a machine has some residual unbalance, even after balancing is performed. This unbalance produces vibration inside bearings within the bearing clearances. These clearances range up to 1 mill (0.001 in.) for rolling-element bearings and 3 to 8 mills in plain bearings, depending on shaft size.

Essentially, couplings influence machine vibrations in two ways: by design, and through misalignment.

Design influence

For analysis purposes, a machine shaft can be divided into two parts: the portion between bearings, and the portion that protrudes outside the machine housing. Computer analysis of most machine shafts show that the influence of couplings on vibrations of the portion between bearings is negligible, particularly when the two portions have different diameters.

Most machine systems (two machines connected by a coupling) are designed to operate at least 20% away from any resonant frequency. This reduces the chance of operating at resonance speed, even with adjustable-speed equipment, and reduces the machine’s sensitivity to unbalance.

To ensure smooth operation, machinery manufacturers perform precision balancing of machine rotors. But, these manufacturers have little control over the portion of the shaft that protrudes from the housing. The responsibility of coupling selection and vibration analysis for the protruding shaft rests with the packager — the shop that connects two or more machines through couplings. Unfortunately, few packagers give sufficient attention to this aspect of installation; few couplings are even balanced.

In a typical coupling application, Figure 1, the resonant frequency of the protruding shaft is estimated by:

f = (0.4 × 106 × d2)÷√WL3

where
f = Resonant frequency, cpm
d = Shaft diameter, in.
W= Coupling weight, lb
L = Overhung distance between coupling cg and bearing centerline, in.

Because coupling weight rests on the shaft through a flexing element (gear mesh, disk pack, or rubber element), it is safe to assume the cg is at the flexing-element centerline.

When the calculated frequency falls too close to the machine speed, designers can select a different coupling, which is either lighter or has a cg closer to the bearing.

Problems can occur later if the selected coupling is replaced with a different type without considering the possible change in resonant frequencies. Consider two similar couplings, Figure 2, which have flexible disk-pack elements. The overhung distance, L1, differs between the two because in (a), the disk pack is placed on the hub, whereas in (b), it is outside the hub. Replacing coupling (a) with coupling (b) lowers the resonant frequency of the shaft, and could make the machine more sensitive to rotor or coupling unbalance. Users that do not recognize this problem might spend (unnecessarily) a lot of time and money rebalancing the machine to restore smooth operation.

Coupling reactions

Misalignment produces two effects in couplings:

• Reaction forces that try to restore alignment. For most non-lubricated couplings, restoring force is proportional to the misalignment. Reactions also depend on the type of flexible element (rubber, urethane, or metal disk) and coupling size. In lubricated couplings, reactions are torque dependent, but can also vary with the lubricant type and coupling wear. In either case, restoring forces are resisted by the machine bearings.

• Heat. In lubricated couplings, misalignment causes the hub teeth to rub on other surfaces, which generates heat. A larger misalignment increases the sliding distance, thereby generating more heat. In elastomeric couplings, misalignment causes flexing, and the inherent damping of elastomers produces heat. Larger misalignment increase flexing, thus generating more heat. High temperature of the flexing elements has a negative effect on the ability of all couplings to transmit torque: heat softens elastomers and degrades the lubricating properties of grease.

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Misalignment influence

Vibrations occur as cyclic flexions of a shaft, either at the machine rotational frequency or a different frequency. Machine speed is measured in revolutions per minute (rpm), whereas vibration is measured in cycles per minute (cpm). The two should not be confused: vibration may, or may not, coincide with machine speed. For example, vibrations can occur once per revolution, twice, or more.

Figure 3a illustrates a flexing machine shaft that vibrates in its “first mode.” This is a theoretical condition in which there is no play, thus no flexing motion, in the bearings. In reality, the radial play or looseness in either plain or rolling-element bearings causes shafts to flex and move radially inside the bearing, Figure 3b.

These shaft displacements are seldom linear. The centerline of the shaft travels in an elliptical or randomly shaped path. This motion is called the orbit, and it can be seen by displaying the signals of proximity sensors on an oscilloscope screen.

Misalignment influences vibration by altering the displacement of the shaft inside the bearing.

Case 1. The solid elliptical-shaped line in Figure 4 illustrates a typical orbit of a well-balanced machine shaft in a lightly loaded bearing. The vertical motion is smaller than the horizontal motion when rotor weight is the only bearing load because gravity opposes the vertical (upward) displacement.

If the two shafts have a horizontal offset, coupling reaction forces tend to bring the shafts into alignment. This generates a force in the bearing that opposes shaft vibration in the horizontal direction, just as the rotor weight opposes vertical shaft vibration. Hence, misalignment forces in the coupling reduce the horizontal amplitude of vibration, as shown by the dotted line in Figure 4.

To reduce vibration amplitude through misalignment, you must determine the direction of the orbit’s long axis. This can be obtained by displaying the signal from two vibration sensors (in horizontal and vertical directions) on an oscilloscope. The shaft offset should then be positioned in the long-axis direction.

The amount of offset required depends on the type of coupling. It is best to start with a small offset, then increase it if required. In no case should the offset exceed the value specified in the coupling catalog.

Case 2. The shaft orbit in Figure 5 represents a not-so-well balanced pinion shaft in a bearing. The forces in gearbox bearings are a combination of gear weight and tooth-mesh reactions. The orbit is an ellipse, as in Case 1, but its shape is more elongated because of a larger unbalance, and its long axis is at an angle that corresponds to the gear-tooth pressure angle.

If the two shafts have a vertical offset, coupling reaction forces tend to bring the shafts into alignment, thereby generating an additional vertical force in the bearing. This force opposes vertical shaft vibration and distorts the ellipse into a banana shape, as shown by the dotted line. The shaft’s centerline moves up and down twice per revolution.

These vibrations were caused by the rotor’s residual unbalance. Misalignment only altered the shape of the orbit, but did not create vibration! In this case, its only influence was to add a two-timesper- revolution frequency in the vertical direction.

Shaft-to-shaft distance

Shaft offset, Figure 6, is used to measure misalignment, but, it should not be confused with misalignment. Rather, misalignment (of a coupling) is the angle between the centerline of either shaft and the centerline of the coupling spacer. To avoid misunderstanding, the term angular misalignment should be used in reference to couplings. Angular misalignment differs between the two ends of a coupling; it can be the same only if the two shafts are parallel.

Though angular misalignment is measured in degrees, millwrights prefer to measure misalignment by the tangent of the angle, expressed in mills (thousands of an inch) of offset per inch of shaft separation (distance between shaft ends). For example, a misalignment of 1/10 deg corresponds to 1.7 mills/in; therefore, if the shaft separation is 8 in., the parallel offset is 0.014 in. For more accurate values, the distance between flexible elements (L2 in Figure 2), and not between shaft ends, should be used.

Based on the above, forces generated by misalignment are a function of the separation between shaft ends. The process of machinery alignment becomes easier, and restoring forces become smaller, if the shaft-to-shaft distance is large. Close-coupled machines save real estate, but are difficult to align, and can cause machinery vibration and short bearing life.

The best rule of thumb for selecting an optimum shaft-to-shaft distance is to set the distance equal to the outside diameter of the coupling. Most coupling types will permit the use of this rule.

Summing up — misalignment and vibration

Several conclusions can be drawn about the relationship between coupling misalignment and shaft vibrations:
• Misalignment influences, but does not generate vibration.
• Properly directed, misalignment reduces the amplitude of vibration. When improperly directed, it distorts the shaft orbit, which can cause a twotimes- per-revolution vibration component.
• Misalignment can generate damaging heat in lubricated or elastomeric couplings, particularly if air flow around couplings is restricted by guards.
• Forces caused by misalignment can be reduced, and the alignment process made easier, if the shaft-toshaft distance is properly selected.
• Sensitivity to machine unbalance can be reduced by operating away from vibration resonances. In turn, resonances can be moved away from the operating speed by selecting a lighter-weight coupling or one with a cg that is closer to machine bearings.

Michael M. Calistrat is a consultant on power transmission design and failure analysis and owner of Michael Calistrat & Associates, Missouri City, Texas.

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