# Shafts and axes - Applied Mechanics

## Shafts and axes

The shafts are intended for fastening parts (gears, worms, sprockets, pulleys, half couplings, etc.) to them and transferring the torque. The axes only serve to maintain the rotating parts of the mechanisms and, unlike the shafts, do not transmit the torque. The axes can be rotating and stationary.

By the form of the geometrical axis, the shafts are divided into straight, cranked and flexible. The most common applications are direct shafts (Figure 4.68, a - c). Crankshafts (Figure 4.68, d) are used only in piston machines to convert rotary motion into translational and vice versa (internal combustion engines, pumps, compressors). Flexible shafts with an arbitrary shape of the geometric axis are used to transfer rotation in mechanisms whose nodes change their position during operation, for example, remote control devices, dental drills, etc. Cranked and flexible shafts refer to special-purpose parts and in the course "Machine parts" are not considered.

Straight shafts are divided according to the shape of the outer surface into smooth ones (see Figure 4.68, a) and stepped or shaped (see Figure 4.68, b, о) . Smooth shafts along the entire length are of one nominal size, and the corresponding plantings of various parts are provided with limiting deviations. In power machines, smooth shafts have limited application. They are mainly used in transmissions for the transmission of only torque. More apply -

Fig. 4.68

me Omi were obtained in unloaded small-sized kinematic mechanisms.

Stepped shafts are less technologically efficient in manufacturing, but are more convenient for assembly, especially for complex multistage mechanisms. Each piece freely passes into its place, and on one side its axial fixation is ensured. In addition, the stepped shaft has a smaller mass, since in form it approaches a beam of equal resistance to bending. Hollow shafts (see figure 4.68, c) are more expensive to manufacture than solid ones, and they are used for rigid requirements to the mass of the structure (for example, the mechanisms of aviation and space technology). With a ratio of the inner diameter of the shaft to the outer diameter d/D = 0.6, 0.7 its mass is reduced by 40-50%, and the moment of section resistance to the bend W - 15-25%, which does not cause a sharp decrease in strength. Typically, d/D & lt; 0,75, which is associated with the need for keyways, slots, threads. Hollow shafts are also applied when another part is passed through the shaft, lubricant is supplied, etc.

The design of the stepped shaft is determined by the number and design of the parts that are placed on it, the location of the supports, and the assembly conditions. On the shaft it is possible to distinguish separate elements: end sections; transitional sections between adjacent stages of different diameters; the landing place of bearings, seals and parts transmitting torque.

The input and output shafts of the transfer mechanisms must have cantilever areas for the installation of pulleys, sprockets, gears, half couplings. The end sections are made with cylindrical, less often conical, shapes and sizes determined by the standards. Cylindrical are easier to manufacture, and conical (with a conicity of 1:10) provide high accuracy of basing and centering of mating parts, ease of assembly and disassembly.

In places where the diameter of the shaft is changed, a smooth transition is made - a fillet of constant radius (Figure 4.69, a). To reduce the stress concentration, the difference between the shaft shaft diameters should be minimal and the fillet radius maximum. The ratio r/d is at least 0.1. In order to ensure the thrust of the part mating with the shaft along the shoulder plane, the fillet radius must be less than the facet of the chamfer of the part /, and the height of the shoulder t & gt; 2 /. When transferring large axial forces, the height of the ledge is selected from the condition of the strength of the end face for crushing, and the thickness of the shoulder is from the condition of providing strength to the shear. The height of the collar (or ledge) for the thrust of the inner ring of the bearing should allow the bearing to be removed during dismantling. If the key at the end of the shaft has a tight connection with the shaft, the height of the shoulder t should be greater than the key protruding from the key shaft so that the bearing can be fitted into place without removing the key. Tolerances for the beating of thrust beads of shafts are assigned in the range of 0.01-0.06 mm.

One way to increase the fatigue strength of the shaft is to close the fillet (Figure 4.69, b), which is used to install parts that have a small radius of curvature or a chamfer at the entrance. Axial fixation of the workpiece is carried out using an intermediate ring 1, which allows increasing the radius of the fillet r. Sometimes, to increase the radius, use a fillet with an undercut (Fig. 4.69, c), while decreasing length of the cylindrical shaft part.

If there is a need to grind the seats on the shaft adjacent to the ledge, grooves are provided for the exit of the grinding wheel (Figure 4.69, d). For small diameter shafts, such grooves reduce resistance to bending and torsion, the landing surfaces of such shafts are only possible at high values ​​of safety reserves n & gt; 2.0? 2.5.

Fig. 4.69

The landing surfaces of the axles and shafts are mainly cylindrical. The design of these sections of the paws depends on the type of fitting and the method of transferring the torque. The length of the sections is taken to be less than the hub length , to ensure axial fixation parts. Roughness of surfaces ( ) is assigned depending on the nature of the interface, the quality, the type of the part being mounted, etc.

At the ends of the shafts or intermediate sections, facet chamfers are made to facilitate assembly, to prevent chipping of the edges and cutting the arms of the assembler. Dimensions of the chamfer are assigned depending on the diameter of the shaft mm at mm; mm with mm and mm with mm.

The bearing surfaces of the shaft for bearing when perceiving the radial load are called trunnions or necks for intermediate supports. These sections have a cylindrical shape for rolling bearings, but there may be conical or spherical pivots for plain bearings. The landing diameters for rolling bearings are selected from the standard range of diameters of the rolling bearing bores. When perceiving axial loads, these shaft sections are called heels . The roughness of bearing surfaces for bearings is assigned, depending on the nature of the interface of the bearing with the shaft, the diameter of the journal and the bearing accuracy class. For bearings of zero accuracy class, the roughness of the seats μm, butt ends μm; for bearings of elevated accuracy classes Ra equal to 0.63 and 1.25 microns respectively. Deviations from roundness and cylindrical landing sites should not exceed 0,5 tolerance for diameter, and for bearings of accuracy classes 5,4 and 2 - not more than 0,003-0,018 mm.

The material of shafts and axles is carbonaceous and alloyed steels that have high strength, ability to surface and bulk hardening (to increase fatigue strength and wear resistance) and good workability. The material of the shafts is chosen taking into account the operating conditions of the mechanism. In lightly loaded mechanisms, shafts that do not undergo heat treatment are made of carbon steels of 20, 45A, 50, etc. For medium- and heavily loaded shafts, alloy steels 40X, 40X11.40X112MА, 30ХГСА, etc. are used. Shafts from alloyed steels are subjected to improvement, hardening with high leave; To increase the wear resistance, individual parts of the shafts are subjected to surface hardening of the HDTV. Trunnions of axes and axles under the bearings of sliding mechanisms with a long life for increasing wear resistance are cemented. The choice of the type of heat treatment is carried out in accordance with the steel grade (cemented or allowing nitriding). To increase the wear resistance, chromium-nickel steels are used or the necks of the shafts are chrome plated, while the resource is increased 3-5 times.

Seeding places of high-loaded shafts and axles are re-grinded after turning. Under alternating loading, surface irregularities are stress microconcentrators. Sanding and polishing reduce the amount of unevenness and increase the durability of the shaft. High-tension shafts are grinded over the entire surface.

The calculation of shafts is carried out in three stages.

In the absence of data on the linear dimensions of the shaft and, respectively, the bending moments in the first stage, an approximate value of the shaft diameter in the most loaded section is determined. From the condition of the torsional strength of the shaft, we have

where T - the torque transmitted by the shaft, N • mm; [τ] is the allowable torsional stress, MPa (for steel shafts, take [τ] = 12 ± 20 MPa).

In the second stage, in accordance with the resulting shaft diameter, a structural shape is created that corresponds to the kinematic scheme and reflects the requirements of manufacturability and assembly. As a result, all shaft dimensions are set.

At the third stage, the shaft verification is performed. The main criterion for rotating shafts and axes is the cyclic strength, since the constants in the value and direction of the force cause variable voltages in them. The fixed axes and some shafts are calculated for static strength under the action of large starting moments. Inadequate stiffness of the shafts negatively affects the operation of the associated joints, bearings, gears and other parts; increases wear and tear; reduces fatigue resistance of parts and joints; reduces the accuracy of mechanisms, etc. Calculation of the shaft for rigidity is performed in those cases when these effects are significant and require mandatory accounting.

Calculation of fatigue resistance. The following stages can be identified in the calculation of the shaft: compilation of the calculation scheme; determination of design loads and construction of diagrams of normal forces, bending and twisting moments; calculation of stresses and safety reserves in dangerous sections of the shaft.

For calculation, the rotating shafts and axles are represented as a beam on hinged bearings. The location of the bearings depends on the type of bearing. When installing the shaft in the radial ball or roller bearings, the bearing points are considered the middle of the width of each bearing (Figures 4.70, a, b). When the shaft is mounted in the radial thrust bearings, the supports are offset from the end by an amount a , depending on the angle of contact. For ball bearings (Figures 4.70, c), but for tapered roller bearings (Figure 4.70 , d), where is the coefficient of axial loading, depending on the angle of contact (Table 4.16). When installing two bearings in the bearing, the conditional support is located at a distance of one third from the middle of the inner bearing (Figure 4.70, ∂). Y of the shafts rotating in the plain bearings, conditional

Fig. 4.70

the hinged support is located at a distance (0.254-0.3)/from the end of the bearing (Figure 4.70, e).

The loads acting on the shaft are transmitted from the associated parts, such as cogwheels and worm wheels,

Table 4.16

 Type Bearing Angle contact, α ° Single-row bearings Double-row bearings \$ X Y X Y X Y X Y Radial ball bearings 0 0.014 1 0 0.56 2.30 1 0 0.56 2.30 0.19 0.028 1.99 1.99 0.22 .056 1.71 1.71 0.26 0.084 1.55 1.55 0.28 0.11 1.45 1.45 0.30 0.17 1.31 1.31 0.34 0.28 1.15 1.15 0.38 0.42 1.04 1.04 0.42 .056 1.0 1.0 0.44 Angular contact ball bearings 12 0.014 1 0 0.45 1.81 1 2.08 0.74 2.94 0.30 .029 1.62 1.84 2.63 0.34 .057 1.46 1.60 2.37 0.37 .086 1.34 1.52 2.18 0.41 0.11 1.22 1.39 1.98 0.45 0.17 1.13 1.30 1.84 0.48 0.29 1.04 1.20 1.69 0.52 0.43 1.01 1.16 1.64 0.54 0.57 1.0 1.16 1.62 0.54 18-20 - - - 0.43 1.0 1.09 0.70 1.63 0.57 24-26 0.41 0.87 0.92 0.67 1.44 0.68 30 - - - 0.39 0.76 0.78 0.63 1.24 0.80 35-36 0.37 0.66 0.66 0.60 1.07 0.95 40 - - - 0.35 0.57 0.55 0.57 0.93 1.14 Roller conical 1 0 0.4 0.4 ctgα 1 0.45 ctgα 0.67 0.67 ctgα 1.5 ctgα

pulleys, sprockets, etc. They are determined by the appropriate dependencies of the calculation of gears or experimentally. In shaft calculations, these loads distributed over the contact surface are replaced by concentrated equivalent forces and are applied in the middle of the hub of the part. The loads found are transferred to the axis of the shaft, the corresponding diagrams are constructed.

When calculating for fatigue, the calculated sections are the sections with stress concentrators: gantry transitions, splines, key grooves, transverse holes, threads in which high bending and twisting moments act. In complex shafts it is sometimes difficult to single out one dangerous cross-section and then the calculation is carried out for several sections. For each of the calculated sections, the safety factor coefficients are determined and compared with the allowable value . To ensure reliable operation, there must be . Strength is estimated by the formula

where and - strength stocks for normal and shear stresses:

where and - the endurance limits of a standard sample under a symmetrical stress-change cycle; and amplitude stresses of normal and shear stresses; and - medium voltage cycles; coefficients of reducing the endurance limits of the part; and are the coefficients of the material's sensitivity to the asymmetry of the stress cycle.

For carbon articles for alloy steels . Coefficient of lowering the endurance limit of the detail:

• When calculating for bending

• When calculating for torsion

where and are the effective stress concentration coefficients (depend on the type of stress concentrator); and - the coefficients of the effect of the dimensions of the part; - coefficient that takes into account the increase in the endurance limit for surface hardening; and - the coefficients of the influence of roughness.

Effective coefficients and the stress concentrations for steel in the bending and torsion of the shafts at the location of the annular groove are found from the table. 4.17; in a gradual transition from the fillet - according to Table. 4.18; with bending and torsion of shafts with splines, keyway groove, with thread and transverse hole - but tab. 4.19.

Coefficients - and are shown in Table. 4.20; coefficient -in the table. 4.21.

Values ​​ depending on the roughness parameters Ra and Rz are shown in Fig. 4.71. The value of is determined from the relation

Table 4.17

 Effective concentration co-concentration σ ". MPa r/d 0.01 0.03 0.05 0.1 0.01 0.02 0.03 600 1.98 1.82 1.71 1.52 2.43 2.32 2.22 800 2.09 1.92 1.82 1.59 2.56 2.45 2.35 1000 2.20 2.02 1.93 1.66 2.70 2.58 2.47 1200 2.31 2.12 2.01 1.73 2.84 2.71 2.59 t/r = 1 t/r-3 600 2.21 2.03 1.91 2.56 2.42 800 2.37 2.14 2.03 2.73 2.66 1000 2.15 2.25 2.15 2.90 2.70 1200 2.57 2.36 2.27 - 3.07 2.84 - 600 1.80 1.60 1.46 1.23 - - - 800 2.00 1.75 1.57 1.28 - - - 1000 2.20 1.90 1.69 1.34 - - - 1200 2.40 2.05 1.81 1.40 - -

Fig. 4.71

Table 4.18

 Effective concentration co- efficients r/d 0.01 0.03 0.05 0.1 0.01 0.02 0.05 600 1.38 1.67 1.64 1.50 1.94 2.02 2.03 800 1.41 1.76 1.73 1.61 2.03 2.13 2.16 1000 1.45 1.84 1.83 1.12 2.12 2.25 2.30 1200 1.49 1.92 1.93 1.83 2.21 2.37 2.44 600 1.57 1.88 1.82 - 2.17 2.23 - 800 1.62 1.99 1.95 - 2.28 2.38 - 1000 1.67 2.11 2.07 - 2.39 2.52 - 1200 1.72 2.23 2.19 - 2.50 2.66 - 600 1.29 1.42 1.44 1.39 1.59 1.66 1.68 800 1.30 1.45 1.47 1.43 1.64 1.72 1.74 1000 1.31 1.48 1.51 1.46 1.68 1.79 1.81 1200 1.32 1.52 1.54 1.50 1.73 1.86 1.88 600 1.40 1.57 1.57 - 2.24 2.12 - 800 1.43 1.61 1.62 - 2.37 2.22 - 1000 1.46 1.66 1.68 - 2.48 2.31 - 1200 1.47 1.71 1.74 - 2.60 2.40 -

Table 4.19

 Slots Keyway Groove Thread A transverse hole of diameter d forward flow involute with Steps Private Exit with a smooth Exit 600 1.55 1.76 1.46 1.96 2.05 1.85 800 1.65 2.01 1.62 2.20 2.10 1.90 1000 1.72 2.26 1.77 2.61 2.20 2.00 1200 1.75 2.50 1.92 2.90 2.30 2.10 600 2.36 1.46 1.54 1.54 1.80 800 2.55 1.52 1.88 1.71 1.96 1000 2.70 1.58 2.22 2.22 1.98 1200 2.80 1.60 2.39 2.39 2.00

Table 4.20

 Deformation and material d, mm 15 20 30 40 50 70 100 200 Bending for carbon steels kj a 0.95 0.92 0.88 0.85 0.81 0.76 0.7 0.61 Bending for doped and torsion for all steels kj a , k,. 0.87 0.83 0.77 0.73 0.7 0.65 0.59 0.52

Table 4.21

 Processing Surfaces Core length σв, MPа Coefficient of hardening k v smooth Shafts shafts with stress concentration Hardening with HD heating: d = 10 or 20 mm 600-800 1.5-1.7 1.6-1.7 2.4-2.8 800-1000 1.3-1.5 - - Nitriding 900-1200 1.1-1.25 1.5-1.7 1.7-2.1 Cementation 700-800 1.4-1.5 - - 1000 1200 1.2-1.3 2.0 Shot blasting: d = 8 or 40 mm 600-1500 1.1-1.25 1.5-1.6 1.7 2.1 Rolling with a roller: d = 17 <30 mm - 1.1-1.3 1.3-1.5 1.6-2.0

Calculation of static strength. The static strength is calculated by equivalent stresses. Since the main stresses are bending and torsion, and the stresses from the normal forces are relatively small, the equivalent stresses are determined by the formula

where - the largest stresses, respectively, bending and torsion.

For shafts of solid circular cross-section

Yield Strength by Yield Strength

The allowable safety margin is taken to be 1.2-1.8. The dangerous cross-section in calculating the shaft for static strength is determined by the values ​​of the moments and the size of the cross sections. These values ​​are found after the construction of bending and twisting moments.

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