INFLUENCE OF SUPPORT CHARACTERISTICS

Figure 5.1[1] compares stiffness of a uniform beam of length l loaded with a concentrated force P or distributed forces of the same overall magnitude and with a uniform intensity q⫽ P/l. Maximum deformation f and stiffness k are

f⫽ Pl³/aEI; k⫽ P/f ⫽ aEI/I³ (5.1)

where E⫽ Young’s modulus of the beam material, I ⫽ cross-sectional moment of inertia, and a⫽ coefficient determined by the supporting conditions. The canti-lever beam (cases 5 and 6) is the least stiff and the double built-in beam (cases 3 and 4) is the stiffest, 64 times stiffer than the cantilever beam for the case of concentrated force loading and 48 times stiffer for the case of distributed loading.

Such simple cases are not always easy to identify in real life systems because of many complicating factors.

Figure 5.2[1] compares 14 designs of joint areas between piston, piston pin, and connecting rod of a large diesel engine. The numbers indicate relative values of deformation f of the piston pin under force P and of maximum stressσin the pin as fractions of deformation fl⫽ Pl³/48EI and stressσ1⫽ Pl/4W in the pin considered as a double-supported beam (the first case). It can be seen that seem-ingly minor changes of the piston and the connecting rod in the areas of interac-tion (essentially, design and posiinterac-tioning of stiffening ribs and/or bosses) may result in changes of deformation in the range of ⬃125:1 and of stresses in the range of⬃12:1

However, it has to be noted that models in Fig. 5.2 are oversimplified since they do not consider clearances in the connections.Figure 5.3illustrates the

in-Figure 5.1 Influence of loading/supporting schematic on stiffness.

Figure 5.2 Influence of assembly design on stiffness.

fluence of the clearances. When the connection is assembled with tight fits (Fig.

5.3a) the loading of the pin model can be described as a double-built-in beam.

A small clearance (Fig. 5.3b) results in a very significant change of the loading conditions caused by deformations of the pin and by the subsequent nonunifor-mity of contacts between the pin and the piston walls and between the pin and the bore of the connecting rod. For a larger clearance (Fig. 5.3c), the contact

Figure 5.3 Influence of fit between interacting parts on loading schematic.

areas become edge contacts and the effective loading pattern (thus, deformations and stresses in the system) is very different from the patterns inFigs. 5.3aand b. The meaning of the terms ‘‘small’’ and ‘‘large’’ clearance depends not only on the absolute magnitude of the clearance, but also on its correlation with defor-mation of the pin as well as of two other components.

The substantial influence of the supporting conditions on deformations in mechanical systems gives a designer a powerful tool for controlling the deforma-tions. Figure 5.4 compares three designs of a shaft carrying a power transmission gear. The system is loaded by force P acting on the gear from its counterpart gear. In Figs. 5.4a and b, the shaft is supported by ball bearings. Since the ball bearings do not provide angular restraint, the shaft can be considered as a double-supported beam. The cantilever location of the gear in Fig 5.4a results in exces-sive magnitudes of the bending moment M, deflection fmax, and angular deforma-tion of the gear (φ⫽ Pt²/3EI ). The latter is especially objectionable since it leds to distortion of the mesh conditions and to significant stress concentrations in the mesh. Location of the gear between the supports as in Fig. 5.4b greatly reduces M and fmax and, especially, φ. In Fig. 5.4c, the same shaft is supported by roller bearings in the same configuration as in Fig. 5.4b. The roller bearings resist angu-lar deformations and, in the first approximation, can be modeled as a built-in support, thus further reducing deflection fmaxby the factor of four while reducing the maximum bending moment Mmax(and thus maximum stresses) by the factor of two. In both cases of Figs. 5.4b and c, due to symmetrical positioning of the gear, angular deflection or the gear is zero and thus the meshing process is not distorted. There would be some, relatively minor, angular deflection and distor-tion of the meshing process if the gear were not symmetrical.

Large angular deflections of gears mounted on cantilever shafts result from

Figure 5.4 Influence of selection and configuration of supporting elements.

Figure 5.5 Reduction of reaction forces in supporting elements by design.

deformations of the double-supported shaft, but also due to contact deformations of the bearings (see Chaps. 4, 6) caused by reaction forces. A very important feature of cantilever designs, as in Fig. 5.5a, is the fact that the reaction forces, especially Nl, can be of a substantially higher magnitude than the acting force P as illustrated by Fig. 5.6a. In Fig. 5.5a, the greatest deformation is in the front bearing 1 accommodating a large reaction force Nl. As can be seen from Fig.

5.6a, the reaction forces are especially high when the span L between the supports (bearings) is less than 2l (l⫽ length of the overhang). In addition to high

magni-Figure 5.6 Reaction forces at supports of a double-supported beam with (a) out-of-span (cantilever) loading and (b) in-out-of-span loading.

tudes, the reaction forces for the model in Fig. 5.6ahave opposite directions, thus further increasing the angular deflection of the shaft inFig. 5.5a.The reaction forces in cases when the external force is acting within the span are substantially less, as shown in Fig 5.6b.

In some cases the force-generating component, such as a gear, can not be mounted within the span between the supports. Sometimes, such components can be reshaped so that the force is shifted to act within the span, while the component is attached outside the span, like the gear in Fig. 5.5b (‘‘inverse cantilever’’).

Even better performance of power transmission gears and pulleys, as well as precision shafts (e.g., the spindle of a machine tool) can be achieved by total separation of forces acting in the mesh, belt preloading forces, etc., and torques transmitted by the power transmission components. Such design is shown in Fig.

5.5c where the gear is supported by its own bearings mounted on the special embossment of the shaft housing (gear-box, headstock). The shaft also has its own bearings, but the connection between the gear and the shaft is purely torsional via coupling A, and the forces acting in the gear mesh are not acting on the shaft.

In this case, the gear is maintained in its optimal nondeformed condition, and the shaft is not subjected to bending forces.

The same basic rules related to influence of supporting conditions on deformations/deflections apply to piston–pin–connecting rod assembly in Fig.

5.2and to power transmission gears on shafts inFigs. 5.4and 5.5. They are true also for machine frames and beds mounted on a floor or a foundation, a massive table mounted on a bed, etc. Although these rules have been discussed for beams, they are (at least qualitatively) similar to plates. For example, for round plates,

‘‘built-in’’ support (Fig. 5.7b) reduces deformation 7.7 times as compared with

‘‘simple edge’’ support (Fig. 5.7a).

It is very important to correctly model the support conditions of mechanical components in order to improve their performance. Incorrect modeling may sig-nificantly distort the actual force schematics and lead to very wrong estimations of the actual stiffness.Figure 5.8[1] presents a case of a shaft supported by two sliding hydrodynamic bearings A and B, and loaded in the middle by a radial

Figure 5.7 Influence of design of the support contour on deformation of round plates under distributed (e.g., weight) loading: (a) simple support contour; (b) built-in contour.

Figure 5.8 Loading schematics of a double-supported shaft.

force P also transmitted through a hydrodynamic bearing. This model describes, for example, a ‘‘floating’’ piston pin in an internal combustion engine. The actual pressure distribution along the length of the load-carrying oil film in the bearings is parabolic (left diagrams in Fig. 5.8c). The peak pressures are 2.5–3.0 times higher than the nominal (average) pressures. In transverse cross sections the pres-sure is distributed along a 90–120°arch (right and center diagrams in Fig. 5.8c).

Comparison of actual loading diagrams in Fig. 5.8c with simplified mathe-matical models in Figs. 5.8a and b shows that the schematic in Fig. 5.8a overstates the deformations and stresses, while the schematic in Fig. 5.8b understates them.

None of the models a and b consider transverse components of the loads and associated deformations and stresses. It is important to remember that a more realistic picture of the loading pattern in Fig. 5.8c may change substantially in the real circumstances due to elastic deformations of both the shaft and the bearings, excessive wedge pressures, etc. Design of the front end of a diesel engine crank-shaft in Fig. 5.9a [1] experienced such distortions. While the nominal loading on the front journal was relatively low, the bearing was frequently failing. It was discovered that the hollow journal was deforming and becoming elliptical under load. The elliptical shape of the journal resulted in reduction of the hydrodynamic wedge in the bearing and deterioration of its load-carrying capacity. The design was adequately improved by enhancing stiffness of the journal by using a rein-forcing plug (Fig. 5.9b).

Figure 5.10 illustrates typical support conditions for power transmission shafts supported by sliding bearings and antifriction ball bearings [2]. Although the bearings are usually considered as simple supports, it is reasonably correct only for the case of a single bearing (Fig. 5.10a). For the case of tandem ball

Figure 5.9 Stiffness enhancement of front-end journal of crankshaft.

bearings (Fig. 5.10b), the bulk of the reaction force is accommodated by the bearing located on the side of the loaded span (the inside bearing). The outside bearing is loaded much less, and might even be loaded by an oppositely directed reaction force if there is a distance between two bearings. Thus, it is advisable to place the support in the computational model at the center of the inside bearing or at one-third of the distance between the bearings in one bearing support,

to-Figure 5.10 Computational models of shafts.

wards the inside bearing (Fig. 5.10b).Due to shaft deformations, pressure from sliding bearings onto the shaft is nonsymmetrical (Fig. 5.10c), unless the bearing is self-aligning. Accordingly, the simple support in the computational model should be shifted off-center and located (0.25–0.3) l from the inside end of the bearing (Fig. 5.10c).

Radial loads transmitted to the shafts by gears, pulleys, sprockets, etc., are usually modeled by a single force in the middle of the component’s hub (Fig.

5.10d). However, the actual loading is distributed along the length of the hub, and the hub is, essentially, integrated with the shaft. Thus, it is more appropriate to model the hub-shaft interaction by two forces as shown in Fig. 5.10d. Smaller shifts of the forces P/2 from the ends of the hub are taken for interference fits and/or rigid hubs; larger shifts are taken for loose fits and/or nonrigid hubs.

5.2 RATIONAL LOCATION OF SUPPORTING AND