Criteria for Power Transmission Couplings

Eugene I. Rivin

Introduction
Power transmission couplings are widely used for modification of stiffness and damping in power transmission systems, both in torsion and in other directions (misalignment compensation). Technical literature on connecting couplings is scarce and is dominated by trade publications and commercial coupling catalogs. Many coupling designs use elastomers in complex loading modes; some couplings have joints with limited travel distances between the joint components accommodated by friction; often, couplings have severe limitations on size and rotational inertia, etc. These factors make a good coupling design a very difficult task, which can be helped by a clearer understanding of the coupling's functions. Stiffness values of couplings in both torsional and misalignment directions, as well as damping of couplings in the torsional direction, have a substantial, often determining, effect on the drive system dynamics. Torsionally flexible couplings are often used for tuning dynamic characteristics (natural frequencies and/or damping) of the drive/transmission by intentional change of their stiffness and damping.

The purpose of this article is to distinctly formulate various couplings' roles in machine transmissions, as well as to formulate criteria for comparative assessment, optimization and selection of coupling designs. To achieve these goals, a classification of connecting couplings is given and comparative analyses of commercially available couplings are proposed.

General Classification of Couplings According to their role in transmissions, couplings can be divided into four classes:

3. Torsionally Flexible Couplings. Such couplings are used to change dynamic characteristics (natural frequency, damping and character/degree of nonlinearity) of a transmission system. The changes are desirable or necessary when severe torsional vibrations are likely to develop in the transmission system, leading to dynamic overloads. Designs of torsionally flexible couplings usually arenot conducive to compensating misalignments.

4. Comhination Purpose Couplings. These combine significant compensating ability with significant torsional flexibility. The majority of the commercially available connecting couplings belong to this group. Since the torsional deformations and deformations due to misalignments are not separated/decoupled by design, changes in torsional stiffness may result in changes in misalignment-compensating stiffness, and vice versa. These couplings will be discussed in detail in Part II of this article.

3. Torsionally Flexible Couplings. Such couplings are used to change dynamic characteristics (natural frequency, damping and character/degree of nonlinearity) of a transmission system. The changes are desirable or necessary when severe torsional vibrations are likely to develop in the transmission system, leading to dynamic overloads. Designs of torsionally flexible couplings usually are not conducive to compensating misalignments.

4. Combination Purpose Couplings. These combine significant compensating ability with significant torsional flexibility. The majority of the commercially available connecting couplings belong to this group. Since the torsional deformations and deformations due to misalignments are not separated/decoupled by design, changes in torsional stiffness may result in changes in misalignment-compensating stiffness, and vice versa. These couplings will be discussed in detail in Part II of this article.

Figure 3a shows a compensating (Oldham) coupling, which-at least theoretically-allows the connection of shafts with a parallel misalignment between their axes without inducing nonuniformity of rotation of the driven shaft and without exerting high loads on the shaft bearings. The coupling comprises two hubs, (1) and (2), connected to the respective shafts and an intermediate disc (3). The torque is transmitted between driving member (1) and intermediate member (3), and between intermediate member (3) and driven member (2), by means of two orthogonal sliding connections, a-b and c-d. By decomposition of the misalignment vector into two orthogonal components, this coupling theoretically assures ideal radial compensation while being torsionally rigid. The latter feature may also lead to high torque-to-weight ratios. However, this ingenious design finds only an infrequent use, usually for noncritical, low-speed applications. Some reasons for this are as follows.

Its direction changes abruptly four times per revolution. Similar effects occur in gear couplings. An experimental Oldham coupling (Tr = 150 N-m, external diameter Dex = 0.12 m, e = 1 mm and n = 1,450 rpm) exerted radial force on the connected shafts Fcom = 720 N.

Since a clearance is needed for normal functioning of the sliding connections) the contact stresses are nonuniform with high peak values (Fig. 4). Figure 4a shows stress distribution between the contacting surfaces a and b of hub (1) and intermediate member (3) of the Oldham coupling shown in Figure 3a assembled without clearance. The contact pressure in each contact area is distributed in a triangular mode along the lengrh 0.5(D - e) ≈ 0.5D. However, the clearance is necessary during assembly, and it increases due to inevitable wear of the contact surfaces. Presence of the clearance changes the contact area as shown in Figure 4b, so that the contact length is 1 ≈ 0.3(D - e) ≈ 0.3D (or the contact length is 0.5c(D - e), c ≈ 0.68), thus significantly increasing the peak contact pressures and further increasing the wear rate. This leads to a rapid increase of the backlash, unless the initial (design) contact pressures are greatly reduced. Such non-uniform contact loading also results in very poor lubrication conditions in the stick-slip motion. As a result, friction coefficients in gear and Oldham couplings are quite high, especially in the latter.

Oldham couplings and u-joints with elasticconnections (Subclass B). The basic disadvantages of conventional Oldham and gear couplings (high radial forces, jumps in the radial force direction) energy losses) backlash, nonperformance at small misalignments, noise) are all associated with reciprocal, short travel, poorly lubricated sliding motion between the connected components. There are several known techniques of changing friction conditions. Rolling friction bearings in u-joints greatly reduce fricrion forces. However, they do not perform well for small-amplitude reciprocal motions. In many applications, shafts connected by u-joints are installed with an artificial 2-3° initial misalignment to prevent jamming of the rolling bodies.

Another possible option is using hydrostatic lubrication. This technique is widely used for rectilinear guide ways, journal and thrust bearings, screw and worm mechanisms, etc. However, this technique seems impractical for rotating systems with high loading intensity (and thus high required oil pressures).

Replacement of sliding and rolling friction by elastically deformable connections can resolve the above problems. Figure 5 shows a "K" or "Kudnaverz" coupling made from a strong, flexible material (polyurethane) connected with the hubs by two "tongues" each. If the connected shafts have a radial offset, it is compensated by bending of the tongues, thus behaving like an Oldham coupling with elastic connections between the intermediate member and the hubs. If the connected shafts each have an angular misalignment, the middle membrane behaves as a "cross," while twisting deformations of the tongues create kinematics of a u-joint. As a result, this coupling can compensate large radial misalignments (~2.4 mm for a coupling with external diameter of D = 55 mm), as well as angular misalignment s of , ±10 degrees. The torque ratings of such couplings are obviously quite low; e.g., a coupling with outside diameter D = 55 mm has a rated torque Tr = 4.5 N-m.

Since displacements in the sliding connections a-b and c-d in Figure 3a are small (equal to the magnitude of the shaft misalignment), the Oldham coupling is a good candidate for application of the thin-layered rubber metal laminates (Ref. 1). Some of the advantages of using laminates are their very high compressive strength, up to 100,000 psi (700 MPa), and insensitivity of their shear stiffness to the compressive force. This property allows the preloading of the flexible element without increasing deformation losses.

Figure 6 (from Ref 2) shows such an application. Hubs 101 and 102 have slots 106 and 107, respectively, whose axes are orthogonal. The intermediate disc can be assembled from two identical halves 103a and 103b. Slots 108a and 108b in the respective halves are also orthogonally oriented. Holders 105 are fastened to slots 108 in the intermediate disc and are connected to slots 106 and 107 via thin-layered rubber-metal laminated elements 111 and 112 as detailed in Figure 6b. These elements are preloaded by sides 125 of holders 105, which spread out by moving preloading roller 118 radially toward the center.

This design provides for all kinematic advantages of the Oldham coupling without creating the above-listed problems associated with conventional Oldham couplings. The backlash is totally eliminated since the coupling is preloaded. For the same rated torque, the coupling is much smaller than the conventional one due to the high load carrying capacity of the laminates and the absence of stress concentrations like ones shown in Figure 4b. The intermediate disc (the heaviest part of the coupling) can be made from a light, strong material, such as aluminum, since it is not exposed to contact loading. This makes the coupling suitable for high speed applications.

The misalignment compensation stiffness and the rated torque can be varied by proportioning the laminated elements (their overall dimensions, thickness and number of rubber layers, etc.). The loads on the connected shafts are greatly reduced and are not dependent on the transmined torque since the shear stiffness of the laminate does not depend on the compression load. To derive an expression for efficiency of the Oldham coupling with laminated connections, let the shear stiffness of the connection between one hub and the intermediate member be denoted by ksh and the relative energy dissipation in the rubber for one cycle of shear deformation by Ψ. Then, maximum potential energy in the connection (at maximum shear e) is equal to

and energy dissipated per cycle of deformation is

Each of the two connections experiences two deformation cycles per revolution; thus the total energy dissipated per revolution of the coupling is

Total energy transmitted through the coupling per revolution is equal to

where Pt = T/Dex is tangential force reduced to the external diameter Dex and T is the transmitted torque. Efficiency of a coupling is therefore equal to

where β is the loss factor of th e rubber.

For the experimentally tested coupling (Dex = 0.12 m), the parameters are: a laminate with rubber layers 2 mm in thickness; ψ = 0.2; ksh = 1.8 x 105 N/m; T= 150 N-m; e = 0.001 m; thus η = 1 - (0.2 x 1.8 x 105 x 10-6)/150π = 1 - 0.75 x 10-4 = 0.999925, or the losses at full torque are reduced 200 times as compared to the conventional coupling.

Test results for the conventional and modified Oldham couplings having Dex = 0.12 m have shown th at the maximum transmited torque was the same, but there was a 3.5 times reduction in the radial force transmitted to the shaft bearings with a modified coupling. Actually, the coupling showed the lowest radial force for a given misalignment compared with any commercially available compensating coupling, including couplings with rubber elements. In addition to this, the noise level at the coupling was reduced by 13 dBA to Leq = 83 dBA. Using ultra thin-layered laminates for the same coupling would further increase its rating by at least one order of magnirude, and may even require a redesign of the shafts to accommodate such a high transmitted load in a very small coupling.

The inner surface of the inner layer (40) is made tapered and conforming with the tapered outer surfaces of trunnions 33 and 34. The outer surface of the outer layer (41) is made cylindrical and conforming with the internal cylindrical surface of th e bore in yoke 31. Each bearing sleeve (35) is kept in place by a cover (44) abutting the end surface (43) of the outer metal layer (41). The cover is fastened to the outer metal layer and to the yoke by bolts. A threaded hole is provided in the center of the cover.

To disassemble the connection, bolts attaching the cover to the yoke are removed, and then a bolt is threaded into a hold until contacting the end surface of the trunnion. The further threading of the bolt pushes the outside cover together with the outer metal layer of the bearing sleeve, to which the cover is attached by bolts. The initial movement causes shear deformation in the rubber layers until disassembly protrusions engage with the inner metal layer, thus resulting in a uniform extraction of the bearing sleeve.

It is highly beneficial that u-joints with the rubber-metal laminated bearings do not need sealing devices and are not sensitive to environmental contamination (dirt, etc.).

The efficiency analysis for such u-joints is very similar to the analysis for the modified Oldham coupling. The efficiency of the joint is

where ktor is the torsional stiffness of the connection between the intermediate member and one yoke. It can be compared with efficiency of a conventional u-joint where d is the effective diameter of the trunnion bearing, 2R is the distance between centers of the opposite trunnion bearings, and µ is the friction coefficient in the bearings.

A comparison of Equations 7 and 14 with Equations 12 and 13, respectively, shows that while efficiencies of conventional Oldham couplings and u-joints for a given e, α are constant, efficiency of the modified designs using rubber-metal laminated connections increases with increasing load (when the energy losses are of the greatest importance). The losses in an elastic Oldham coupling and u-joint at the rated torque can be 1-2 decimal orders of magnitude lower than the losses for conventional units. Due to high allowable compression loads on the laminate (in this case, high radial loads), the elastic Oldham couplings and u-joints can be made smaller than the conventional units with sliding or rolling friction bearings for a given rated torque. The laminates are preloaded to eliminate backlash, to enhance uniformity of stress distribution along the load-transmitting areas of the connections, and to increase torsional stiffness. Since there is no acrual sliding between the contacting surfaces, the expensive surface preparation necessary in conventional Oldham couplings and u-joints (heat treatment, high-finish machining, etc.) is not required. The modified Oldham coupling in Figure 6 and the u-joint in Figure 7 can transmit very high rorques while effectively compensating large radial and angular misalignments, respectively, and having no backlash since their laminated flexible elements arepreloaded. However, for small-rated torques, there are very effective and inexpensive alternatives to these designs whose kinematics are similar. These alternatives are also backlash-free.

One is a Kudriavetz coupling, shown in Figure 5. Another alternative is a modified spider or jaw coupling whose cross section by the mid-plane of the six-legged spider is shown in Figure 8 (Ref 4). In this design the elastomeric spider of the conventional jaw coupling shown in Figure 10a is replaced with a rigid spider (9, 11) carrying tubular sleeves or coil springs (10) supported by spider pins (11) and serving as flexible elements radially compressed between cams (6 and 7) protruding from the respective hubs. If the number of spider legs is four, at 900 to each other, then the hubs have relative angular mobility, and the coupling becomes a u-joint with angular mobility greater than 10°, but with much higher rated torque than an equivalent size Kudnavetz coupling.

Purely misalignment compensating couplings described in this section have their torsional and compensating properties decoupled by introduction of the intermediate member. Popular bellows couplings have high torsional stiffness and much lower compensating stiffness, but their torsional and compensating properties are not decoupled, so they are representatives of the "combination purpose couplings" group, which will be discussed in detail in Part II of this article.

Part II

Eugene I. Rivin

Introduction
Part I of this article appeared in the October 2008 issue. It provided an overview and general classifications of power transmission couplings, along with selection and performance criteria for rigid couplings and misalignment-compensating couplings. Part II continues the discussion with selection and performance criteria for torsionally flexible and combination-purpose couplings.

Torsionally Flexible Conplings and Combination Purpose Couplings
Torsionally flexible couplings usually have high torsional compliance (as compared with the torsional compliance of shafts and other transmission components) in order to enhance their influence on transmission dynamics. Figure 9a shows an example of a purely torsionally flexible coupling with an elastomeric flexible element having low stiffness in the torsional direction (shear of rubber ring) and high stiffness in the misalignment-compensation directions (compression of rubber ring). Figure 9b shows a torsionally flexible coupling with a metal flexible element (Bibby-style coupling). The flexible element is a spring steel band wrapped around judiciously shaped teeth on each hub and deforming between the teeth. The deformations become more restrained with increasing transmitted torque; thus the coupling has a strongly nonlinear torsional stiffness characteristic of the hardening type. The lowest stiffness is at zero torque (Fig 9c) increasing towards the rated torque (Fig 9d), becoming very high at the allowed peak torque (Fig. ge) and approaching a rigid condition at an overload/shock torque (Fig 9f). Since some misalignment-compensating ability is desirable for many applications, use of purely torsionally flexible couplings, with combination-purpose couplings being used as torsionally flexible couplings with more or less compensating ability.

For torsionally flexible and combination-purpose couplings, torsional stiffness is usually an indicator of payload capacity. In such cases, the basic design criterion can be formulated as a ratio between the stiffness in the basic misalignment direction and the torsional stiffness. In the following analysis, only radial misalignment is considered. Since couplings are often used as the cheapest connectors between shafts, and since end users often do not have full understanding of what is important for their applications, it is of interest to analyze what design parameters are important for various applications.

Torsional Flexibility
Torsional flexibility is introduced into transmission systems when there is a danger of developing resonance conditions and/or transient dynamic overloads. Their influence on transmission dynamics can be due to one or more of the following factors: torsional compliance, damping or nonlinearity of load-deflection characteristics.

Reduction of torsional stiffness of the transmission and, consequently, shift of its natural frequencies. If a resonance condition occurs before installation (or change) of the coupling, then shifting of natural frequency due to use of a high torsional-compliance coupling can eliminate resonance; thus dynamic loads and torsional vibrations will be substantially reduced. However, in many transmissions (e.g., vehicle transmissions), frequencies of the disturbances acting on the system and natural frequencies (especially in variable speed transmissions) may vary widely. In such instances, a simple shift of the natural frequencies of the drive may lead to a resonance occurring at other working conditions, but the probability of its occurrence is not lessened. A reduction in the natural frequency of a drive, for example, is advisable for the drive of a milling machine only at the highest spindle speeds and may be harmful introduced in the low-speed stages.

A shift of natural frequencies of the drive may be beneficial in transmissions with narrow variations in working conditions. If, however, a drive is operated in the pre-resonance region, an increase in torsional compliance would lead to increased amplitudes of torsional vibrations, and thus to a nonuniform rotation. In some cases excessive torsional compliance may lead to a dynamic instability of the transmission and create intensive self-excited torsional vibrations.

Compensation Ability of Combination-Purpose Couplings.
A huge variety of combination purpose couplings is commercially available. Unfortunately, selection of a coupling type for a specific application is often based not on an assessment of performance characteristics of various couplings, but on the coupling cost or other non-technical considerations. As a result, bearings of the shafts connected by the coupling may need to be more frequently replaced than when an optimized coupling is used; the device might be noisier than it would be with a coupling type optimal for the given application, etc.

Figure 11 shows some popular designs of combination-purpose couplings.

Combination-purpose couplings do not have a compensating member. As a result, compensation of misalignment is accomplished, at least partially, by the same mode(s) of deformation of the flexible element as used for transmitting the payload. To better understand the behavior of combination-purpose couplings, an analysis of the compensating performance of a typical coupling with a spider-like flexible element is helpful. The coupling in Figure 12 shows a schematic of the jaw coupling in Figure 11a. It consists of hubs 1 and 2 connected with a rubber spider 3 having an even number Z = 2n of legs, with "n"legs ("n" might be odd) loaded when hubs arc rotating in the forward direction and the other n legs loaded during the reverse rotation. Deformation of each leg is independent. The radial (compensation) stiffness of the coupling with Z = 4 is

or F is constant and is directed along the misalignment vector.

The ratio kr / kt varies with changing rubber durometer H, and for typical spider proportions, kr / kt = 0.26-0.3 for medium H = 40-50, and kr / kt = 0.4 for hard rubber spiders, H = 70-75.

A spring/tubular spider coupling modification is shown in Figure 8 (Ref 4). In this design, each leg of the spider can be represented by a coil spring loaded radially by the transmitted torque. For the tightly coiled extension spring kt ;

The torsional stiffness ofboth spider coupling designs is

where the effective radius Reff = ~0.75Rex = 0.75(Dex/2). The ratios between torsional and compensation stiffness values are as follows:

Comparative evaluation of the commercially available couplings based on available manufacturer-supplied data on flexible couplings is presented in Figure 13. Plots in 'Figures 13a-d give data on torsional stiffness ktor , radial stiffness krad , external diameter Dex , and design index A.

The "modified spider" coupling in Figure 11b is different from the conventional spider coupling shown schematically in Figure 11a by four features: its legs are tapered, instead of uniform width, and made thicker even in the smallest cross section, at the expense of reduced thickness of protrusions on the hubs; lips on the edges provide additional space for bulging of the rubber when the legs are compressed; and the spider is made of a very soft rubber. These features substantially reduce stiffness values while retaining the small size characteristic of the spider couplings.

References
1. Rivin, E.I. Stiffness and Damping in Mechanical Design, 1999, Marcel Dekker, Inc., NY.
2. "Torsionally Rigid Misalignment Compensating Coupling," U.S. Patent 5,595,540.
3. "Universal Cardan Joint with Elastomeric Bearings,"U.S. Patent 6,926,611.
4. "Spider Coupling," lJ.S. Patent 6,733,393.
5. "Torsional Connection with ,Radially Spaced Multiple :Flexible Elements," U.S. Patent 5,630,758.
6. Rivin, E.I. "Shaped Elastomeric Components for Vibration Control Devices," Sound and Vibration, 1999, Vol. 33, No. 7, pp. 18-23.

This article has been reproduced with the permission of Power Transmission Engineering.