T2-4 2. Misalignment-Compensating Couplings. Such couplings are required for connecting two members of a power-trans- mission or motion-transmission system that are not perfectly aligned. "Misalignment" means that components that are co- axial by design are not actually coaxial, due either to assembly errors or to deformations of subunits and/or their foundations, Figure 2. The latter factor is of substantial importance for transmission systems on nonrigid foundations. If the misaligned shafts are rigidly connected, this leads to elastic deformations of the shafts, and thus to dynamic loads on bearings, vibrations, increased friction losses in power transmission systems, and unwanted friction forces in motion transmission, especially in control systems. Misalignment-compensating couplings are used to reduce the effects of imperfect alignment by allowing nonrestricted or partially  restricted  motion  between  the  connected  shaft  ends.  Similar  coupling  designs  are  sometimes  used  to  change bending natural frequencies/modes of long shafts. When only misalignment compensation is required, rigidity in torsional direction is usually a positive factor, otherwise the dynamic characteristics of the transmission system might be distorted. To achieve high torsional rigidity together with high mobility/compliance in misalignment directions (radial or parallel offset, axial, angular), torsional and misalignment-compen- sating displacements in the coupling have to be separated by using an intermediate compensating member. Frequently, torsionally rigid "misalignment-compensating" couplings, such as gear couplings, are referred to in the trade literature as "flexible" couplings. 3.  Torsionally Flexible Couplings. Such couplings are used to change the dynamic characteristics of a transmission system, such as natural frequency, damping and character/degree of nonlinearity. The change is desirable or necessary when severe torsional vibrations are likely to develop in the transmission system, leading to dynamic overloads in power- transmission systems. Torsionally flexible couplings usually demonstrate high torsional compliance to enhance their influence on transmission dynamics. 4. Combination Purpose Couplings are required to possess both compensating ability and torsional flexibility. The major- ity of the commercially available connecting couplings belong to this group. 3.1    Rigid Couplings Typical rigid couplings are shown in Figure 3. Usually, such a coupling comprises a sleeve fitting snugly on the con- nected shafts and positively connected with each shaft by pins, Figure 3a, or by keys, Figure 3b. Sometimes two sleeves are used, each positively attached to one of the shafts and connected between themselves using flanges, Figure 3c. Yet another popular embodiment is the design in Figure 3d wherein the sleeve is split longitudinally and "cradles" the connected shafts. Figure 3    Examples of Rigid Couplings (d) (c) (b) (a)