Static and dynamic reactions of flexible couplings:
case study with a gear coupling

How flexible are gear couplings?

Ideally one would like such couplings to transmit torque and only torque from one shaft end to another without adding either shear reactions or bending moments. In order to assess the importance of these reactions, let us distinguish two cases: static and dynamic reactions.

Static reactions

Static reactions of flexible couplings may over- or un-load nearby bearings whose selection is often based on the weight of the rotor they support. When the static reaction of a flexible coupling nears the value of the rotor weight, one starts to worry about it.
Never think that gear couplings do not react on the shaft lines they connect! This is illustrated by the table below and further down when dealing with gear coupling reactions.

	You may align shaft ends properly at first. Due to foundation sagging, it may happen that one gets a rather severe radial misalignment. You can measure it at light loads via such standard methods as the reverse dial or with more advanced methods like OPTALIGN.
	Trouble is when the torque transmitted by the gear coupling increases to the coupling nominal rating. See what happens! At high values of torque, the gear coupling behaves like a rigid coupling. It bends the shaft ends to bring them in line and causes extra bearing loads. Is that the kind of flexible coupling you like?

Dynamic reactions

Dynamic reactions from flexible couplings cause vibrations. The most common cause for vibrations originates from rotor unbalances. Balancing norms specify the amount of residual unbalance that is admissible on rotors. These depend on specific balancing grades for different types of applications, the rotor weight and maximum rotational speed. To a given admissible residual unbalance corresponds a centrifugal force. The dynamic reactions of the couplings become a nuisance when they exceed these centrifugal forces.

Experimental setup to measure reactions of flexible couplings

Click to view a better resolution of the test rig (94k)

For gear and some other types of flexible couplings relying on sliding surfaces, reactions depend on the torque transmitted and the misalignment of the shaft ends. The power delivered by the coupling is the product of the torque transmitted and the rotational speed in rad/sec. It can be huge for a small test rig. For example, assume you want to transmit a 1600 (1592 to be precise) NXM torque at 1500 rpm. You need 250 KW and , therefore, a rather expensive test rig, unless you use torsion rods.

Another way to get such high torque is to feed a smaller torque through a speed reductor. In the test rig above, back-to-back 1.1 KW electrical motors rotating at roughly 1500 rpm with 1.1 kW power ratings and connected to the input shafts of 260:1 reductors can almost produce 1800 NXM at the level of the gear coupling being tested. At the coupling level, the rotational speed is approximately 5.7 rpm. It can vary around this value depending on the slip of the induction motor that depends on the power it delivers or retrieves.

Such an approach has many advantages and one drawback.

The advantages are:

very small electrical powers to handle and thus reduced expenses.
one can afford embarking lots of sensors like strain gauges on shaft ends and proximity sensors playing the role of electronic clocks. Slip ring contacts can power these sensors and retrieve their signals without expensive telemetry systems.
measuring strains on shaft ends unambiguously yields the coupling reactions.
proximity sensors unambiguously supply the misalignment of the shaft ends near the coupling, both its static component and its variations.
one need not bother about reactions caused by high centrifugal forces and thus focus on the reactions of the coupling alone.

A drawback is the oil films may not get renewed as well as at higher speeds. This concerns only flexible couplings relying on lubricants. This could somewhat influence coupling reactions via somewhat higher friction coefficients between teeth.

Measured gear coupling reactions and local misalignments

Let the shaft ends be misaligned as shown in Fig. 1. One notices that the misalignment of shaft ends near the coupling vary with the torque transmitted. In studying coupling reactions, what matters is the misalignment it undergoes locally.

Fig.1a. Vi describing the relative positions of the shaft ends (in red Rx and Ry radial misalignemnt and Thetax and y for angular misalignment), the shear coupling reactions in daN (Tx and Ty) and the bending moment reactions (Mx and My) in daNXM. The torque transmitted is -480 NXM. Fig.1b. Same as above except that the torque is at the nominal rating of the coupling (1800 NXM) and the reactions are way up whereas the local misalignment of the shaft ends near the coupling approaches zero.	Coupling reactions may bend the shaft over its whole span depending on bearing span, the diameters of the shafts, etc.. Therefore, it is quite normal that the local misalignment of the shaft ends near the coupling would vary despite the fact that bearing pedestals would remained fastened on the foundation. This behavior required to measure coupling misalignments locally to characterize their evolution with respect to coupling reactions. This contrasts with relying on the positions of the bearing pedestals to assess coupling local misalignments. This vi was programmed by P. Ripak (new Email address being modified) based on LABVIEW Picture Control Kit. It runs on-line and allows an immediate visualization how coupling reactions and the subsequent local coupling misalignment vary with the torque transmitted by the coupling. To be fair, shaft ends should be represented as bent ones in the diagrams. They are represented as straight chords instead. These chords connect the beam centers in the cross-sections targeted by the eddy probes playing the role of the clocks in the reverse dial or indicator method.

Fig.1a. Vi describing the relative positions of the shaft ends (in red Rx and Ry radial
misalignemnt and Thetax and y for angular misalignment),
the shear coupling reactions in daN (Tx and Ty)
and the bending moment reactions (Mx and My) in daNXM.
The torque transmitted is -480 NXM.

Fig.1b. Same as above except that the torque is at the nominal rating of the coupling
(1800 NXM) and the reactions are way up whereas the local misalignment of the
shaft ends near the coupling approaches zero.

Coupling reactions may bend the shaft over its whole span depending on bearing span, the diameters of the shafts, etc.. Therefore, it is quite normal that the local misalignment of the shaft ends near the coupling would vary despite the fact that bearing pedestals would remained fastened on the foundation. This behavior required to measure coupling misalignments locally to characterize their evolution with respect to coupling reactions. This contrasts with relying on the positions of the bearing pedestals to assess coupling local misalignments.
This vi was programmed by P. Ripak (new Email address being modified) based on LABVIEW Picture Control Kit. It runs on-line and allows an immediate visualization how coupling reactions and the subsequent local coupling misalignment vary with the torque transmitted by the coupling.
To be fair, shaft ends should be represented as bent ones in the diagrams. They are represented as straight chords instead. These chords connect the beam centers in the cross-sections targeted by the eddy probes playing the role of the clocks in the reverse dial or indicator method.

Besides varying with torque, the misalignment also vary around its static value for a constant torque.

Measurement setup to get coupling reactions and local misalignments

The table below describes the measurement setup in more detail (click on the picture to get a good zoom with comments)

Fig.2. Top left: Measurement setup to get misalignment from embarked eddy probes acting like electronic clocks in reverse dial method.
Top right: embarked strain gauges with conditioning to measure bending moment along the shaft end and finally coupling reactions.
Bottom: Frequency analysis of coupling reactions: shear and bending moment and ditto for misalignment: radial and angular. Abscissa for frequency analyses are double-sided expressed in harmonic order of the rotating speed. Spectra are obtained for two different values of the torque transmitted by the gear coupling. These are shown in the double-sided (full in Bently Nevada terminology) frequency spectra of Fig. 2 (bottom) for two different values of the torque: Like in Fig.1, the static misalignment in the test is rather severe: ca 75/100 mm radial at quasi zero torque. It drops to 15/100 mm at full load near the coupling.
More interesting is the frequency analysis of coupling reactions. Another full frequency spectrum describes the shear and bending moment reactions of the gear coupling. At full torque, the dc (static) contribution amounts to 416 KgF or 4160 N.
There is also a significant 35 KgF or 350 N contribution at the harmonic +2 of the rotational frequency. This corresponds to a force rotating in the same direction as the shaft at double the speed thereof. If experience with all sorts of gear couplings torques and misalignment confirms this trend, there is very little hope that radial vibrations caused by these reactions could indicate in which directions the misalignment occurs.
Results can be compared to those of Fig.1b where the reaction is somewhat higher due to an initially larger radial misalignment at no load.
For data processing buffs, this kind of analysis is rather elegant. In two different cross-sections of an overhung shaft end next to the coupling, one installs pairs of embarked strain gauges measuring bending in two perpendicular directions thus rotating with the shaft. At time zero these directions coincide with the horizontal and vertical respectively. One performs the sampling at equal rotation interval over a shaft rev or a power of 2 number of revs. Then one combines the bridges responses as "strain orbits" to which one applies double-sided Fourier Transform. Quite a twisted way to use these! Then one applies the conventional material strength equations of beams (overhung shaft ends are none others than rotating beams) to these double-sided Fourier Transforms in order to retrieve the coupling reactions spatially and in the frequency domain. This requires a simple coordinate transform from a rotating to a fixed coordinates in space.
In the same vein, one can decompose local misalignment spatially and in the frequency domain by carefully examining the kinematics of the reverse dial method and what it corresponds to in terms of the clock readings mimicked by the embarked eddy probes..

How bad are those coupling reactions?

Static reaction of the coupling and bearing loads

The diagrams below tell the whole story for gear couplings. They were obtained by running the test rig and collecting the vi outputs like in Fig.1a & b for increasing torques and a given initial radial misalignment of the coupling. The figures were then entered in an EXCEL table to obtain the evolution of coupling reactions and local misalignments.

Fig.3a. Increasing the torque causes the gear coupling shear to rise substantially. This trend holds on up to 900 NXM. From there on, the reactions level off because the local coupling misalignment is almost zero (see below). At high values of torques, the gear coupling behaves like a rigid one..

Fig. 3b. The initial misalignment at zero torque is 37/100 mm radial horizontal. The final coupling local misalignment hovers around -3/100 mm at high values of torque. Since bearings do not move, this means that overhung shaft ends bend. In between, some radial vertical misalignment builds up. It is due to the friction forces developing between the coupling teeth. A good model for gear coupling can be found in "Couplings and Shaft Alignment" by M. Neale, P. Needham and R. Horrell , Mechanical Engineering Publication, London. Based on this model, the friction coefficient between teeth is around 0.12.

Case of an induction motor with a gear coupling

The rotor of a 250 KW 1500 rpm induction motor approximately weighs 5000 N. Suppose that it loads its two bearings evenly, i.e. 2500N. The coupling reaction to the load of the nearest bearing exceed 2000 N. For larger initial misalignments it can reach twice as much, i.e. almost twice the contribution of the rotor weight to each bearing load!! This can literally lift the rotor in its bearings. In the above reference this fact is mentioned.

Dynamic reactions of the coupling and vibrations

Dynamic coupling reactions are time-varying loads that generate vibrations. In the present case, are they severe enough to worry about?

Fig.3. Charts for centrifugal forces caused by admissible residual unbalances
according to ISO 1940/1 grade 6.3.

According to ISO balancing norm 1940/1, residual unbalances with grade 6.3 cause centrifugal forces that are plotted in KgF (10N) below assuming the rotors rigid and thus not operating close to critical speeds.

A 500 Kg rotor balanced for 1500 rpm according to ISO 1941/1 grade 6.3 causes at most a centrifugal force equal to 24 Kgf of 240 N at the rotational frequency. Again it can distribute evenly on the bearings, i.e. 120N each, whereas the dynamic reaction of the couplings loads the nearest bearing with more than 350 N at double the rotational frequency. One would then expect the contribution of the coupling reactions to this bearing vibrations to be at least 3 times as severe as that caused by the largest unbalance according to ISO norm 1940/1 G6.3 norm!!!

Another well-known vibration problem with gear couplings occurs at small values of torques. The spacer bridging the two shafts tend to float on the pinions of shaft ends. This introduces an unbalance that can vary with the load.

Conclusions for gear couplings

A few preliminary tests on gear couplings indicates that the flexibility of some so-called flexible couplings may sometimes be a myth. When misaligned and transmitting torques close to their maximum rating, such couplings my alter the bearing loads considerably and cause vibration levels higher than those from admissible unbalances resulting from balancing norms. In other words, when misalignment looms, balancing may not do to bring vibration levels to admissible levels despite the presence of "flexible" couplings.

This study confirms what many practitioners and vibration specialists have long claimed about the incidence of misalignments on vibrations. It does it with hard figures based on loads and not displacements and vibrations that depend on the frequency response of the structure at hand. It establishes a methodology to evaluate coupling reactions and characterize them. The approach relies on somewhat advanced data processing techniques. These have been applied to other types of flexible couplings like disc couplings and rubber couplings with jaws.

If you wish to know what happens with other couplings (disc, croset alias rubber block or jaw),
click here.

Will contact the ULB specialist on this topic?

Static and dynamic reactions of flexible couplings: case study with a gear coupling