Yildiran - Shafts and Shaft Components

Shafts and Shaft Components Lecture Notes Prepared by H. Orhan YILDIRAN

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MECE 304 Mechanical Machine Elements-Shafts

LECTURE NOTES- MECE 304 Mechanical Machine Elements Chapter 4- Shafts and Shaft Components (Notes from: Chapter 7, Budynas R.G., Nisbett J.K., Shigley’s Mechanical Engineering Design, Mc Graw Hill, 8th Edition and special notes)

Spring Semester 2007/2008 Halil Orhan YILDIRAN, MS 2

MECE 304 Mechanical Machine Elements-Shafts

7-1 Introduction A shaft is a rotating member, usually of circular cross section, used to transmit power or motion. It provides the axis of rotation, or oscillation, of elements such as gears, pulleys, flywheels, cranks, sprockets, and the like and controls the geometry of their motion. Approach to shaft sizing: *Determine stress *Establish tentative size (consider also axial location of elements) *Make deflection and slope analysis 7-1 Shaft materials Common: SAE 1020-1050 steels w/o heat treatment Heat treatment: SAE 1040, 4140, 5140, 8640 steels For surface hardening: SAE 1020, 8620 steels

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MECE 304 Mechanical Machine Elements-Shafts

7-3 Shaft layout *Use shoulders for axial locating *The length of the cantilever should be kept short to minimize the deflection which is made for ease of mounting *Use 2 support bearings *Keep the length as short as possible * Use shoulders, retaining rings, and pins, to transmit the axial load into the shaft *One bearing carry axial load *Common torque-transfer elements are: • Keys • Splines • Setscrews • Press fitting *Consideration should be given to the method of assembling the components onto the shaft, and the shaft assembly into the frame. *For press fitting design a short slide length

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MECE 304 Mechanical Machine Elements-Shafts

Figure 7-3 Tapered roller bearings used in a mowing machine spindle.

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MECE 304 Mechanical Machine Elements-Shafts

Figure 7-4 A bevel-gear drive in which both pinion and gear are straddlemounted.

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MECE 304 Mechanical Machine Elements-Shafts

Figure 7-5 Arrangement showing bearing inner rings press-fitted to shaft while outer rings float in the housing

Figure 7-6 Arrangement as of Fig. 7--5 except that the outer bearing rings are preloaded.

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MECE 304 Mechanical Machine Elements-Shafts

Groove for grinding

Lock nut

Figure 7-7 In this arrangement the inner ring of the left-hand bearing is locked to the shaft between a nut and a shaft shoulder

Figure 7-8 Arrangement is similar to Fig. 7-7 in that the bearing positions the entire shaft assembly.

Floating RS bearing

Retaining ring

Shield

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MECE 304 Mechanical Machine Elements-Shafts

7-4 Shaft design for stress The fluctuating stresses due to bending and torsion are:

For solid shafts with round sections

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MECE 304 Mechanical Machine Elements-Shafts

The von Mises stresses for rotating round, solid shafts, neglecting axial loads, are given by.

Expressions for any of the common failure criteria obtained by substituting the von Mises stresses from Eqs. (7–5) and (7–6) into any of the failure criteria expressed by Eqs. (6–45) through (6–48), p. 298. The resulting equations for several of the commonly used failure curves are summarized below. The names given indicates: First is the significant failure theory, followed by a fatigue failure locus name. Example; DE-Gerber indicates the stresses are combined using the distortion energy (DE) theory, and the Gerber locus is used for the fatigue failure. 10

MECE 304 Mechanical Machine Elements-Shafts

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MECE 304 Mechanical Machine Elements-Shafts

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MECE 304 Mechanical Machine Elements-Shafts

The Gerber and modified Goodman criteria do not guard against yielding, requiring a separate check for yielding. Therefore von Mises maximum stress is calculated as:

Checking for yielding

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MECE 304 Mechanical Machine Elements-Shafts

SAMPLE PROBLEM 7-1 At a machined shaft shoulder the small diameter d is 28 mm, the large diameter D is 42 mm, and the fillet radius is 2.8 mm. The bending moment is 142.4 N.m and the steady torsional moment is 124.3 N.m. The heat-treated steel shaft has an ultimate strength of Sut = 735 MPa and a yield strength of Sy = 574 MPa. The reliability goal is 0.99. (a) Determine the fatigue factor of safety of the design using each of the fatigue failure criteria described in this section. (b) Determine the yielding factor of safety. Estimating Stress Concentration *Finding stress concentration factor is an iterative process *Shoulders for bearing and gear support should match the catalog recommendation for the specific bearing or gear *For the standard shoulder fillet, for estimating Kt, the worst end of the spectrum, with r/d = 0.02 and D/d = 1.5, Kt values from the stress concentration charts for shoulders indicate 2.7 for bending, 2.2 for torsion, and 3.0 for axial.

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MECE 304 Mechanical Machine Elements-Shafts

Figure 7-9 Techniques for reducing stress concentration at a shoulder supporting a bearing with a sharp radius. (a) Large radius undercut into the shoulder. (b) Large radius relief groove into the back of the shoulder. (c) Large radius relief groove into the small diameter

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MECE 304 Mechanical Machine Elements-Shafts

*For keyways estimate the stress concentration for keyways regardless of the actual shaft dimensions by assuming a typical ratio of r/d = 0.02. This gives Kt = 2.2 for bending and Kts = 3.0 for torsion, assuming the key is in place. *Retaining rings. If the groove width is slightly greater than the groove depth, and the radius at the bottom of the groove is around 1/10 of the groove width then from Figs. A–13–16 and A–13–17, stress concentration factors for typical retaining ring dimensions are around; Kt=5 for bending and axial, and Kts=3 for torsion.

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MECE 304 Mechanical Machine Elements-Shafts

Table 7-1 Summary of first Iteration Estimates for Stress Concentration Factors Kt. Do not use if actual dimensions are known!

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MECE 304 Mechanical Machine Elements-Shafts

7-5 Deflection Considerations Deflection of the shaft, both linear and angular, should be checked at gears and bearing locations against allowable values. Table 7-2 Typical Maximum Ranges for Slopes and Transverse Deflections

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MECE 304 Mechanical Machine Elements-Shafts

If any value is larger of the found deflection and slope than the allowable deflection or slope at that point, a new diameter can be found from:

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MECE 304 Mechanical Machine Elements-Shafts

For a stepped shaft with individual cylinder length li and torque Ti , the angular deflection can be estimated from

for a constant torque throughout homogeneous material, from

Torsional stiffness of the shaft k in terms of segment stiffnesses is

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MECE 304 Mechanical Machine Elements-Shafts

7-6 Critical Speeds of Shafts Critical speeds: at certain speeds the shaft is unstable, with deflections increasing without upper bound. Designers seek first critical speeds at least twice the operating speed. The shaft, because of its own mass, has a critical speed. When geometry is simple, as in a shaft of uniform diameter, simply supported, the task is easy. It can be expressed as:

where m is the mass per unit length, A is the cross-sectional area, and γ the specific weight.

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MECE 304 Mechanical Machine Elements-Shafts

For an ensemble of attachments, Rayleigh’s method for lumped masses gives

where wi is the weight of the ith location and yi is the deflection at the ith body location.

Figure 7-12 (a) A uniformdiameter shaft for Eq. (7–22). (b) A segmented uniform-diameter shaft for Eq. (7–23)

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MECE 304 Mechanical Machine Elements-Shafts

Use of influence coeeficients to find crital speeds: An influence coefficient is the transverse deflection at location i on a shaft due to a unit load at location j on the shaft. From Table A–9–6 we obtain, for a simply supported beam with a single unit load as shown in Fig. 7–13.

Figure 7-13 The influence coefficient δi j is the deflection at i due to a unit load at j. 23

MECE 304 Mechanical Machine Elements-Shafts

For three loads the influence coefficients may be displayed as

From the influence coefficients above, one can find the deflections y1, y2, and y3 of Eq. (7–23) as follows (δij = δji ):

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MECE 304 Mechanical Machine Elements-Shafts

Taking Fi=mi ω2 yi and for non trivial solution

Expanding the determinant

The three roots of Eq. (7–27) can be expressed as 1/ω12, 1/ω22, and 1/ω32

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MECE 304 Mechanical Machine Elements-Shafts

Comparing 7-27 and 7-28

Define 1/ω112 = m1 δ11 and so forth

If we order the critical speeds such that ω1 < ω2 < ω3, then 1/ω12» 1/ω2 2 , and 1/ω32. So the first, or fundamental, critical speed ω1 can be approximated by

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MECE 304 Mechanical Machine Elements-Shafts

For n body shaft

Eq. 7-32 is called Dunkerleys equation (speed is below actual) For the load at station 1, placed at the center of span, denoted with the subscript c, the equivalent load is found from:

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MECE 304 Mechanical Machine Elements-Shafts

SAMPLE PROBLEM 7-5 Consider a simply supported steel shaft as depicted in Fig. 7–14, with 25 mm diameter and a 775 mm span between bearings, carrying two gears weighing 175 and 275 N. (a) Find the influence coefficients. (b) Find ∑ wy and ∑ wy2 and the first critical speed using Rayleigh’s equation, Eq. (7–23). (c) From the influence coefficients, find ω11 and ω22. (d) Using Dunkerley’s equation, Eq. (7–32), estimate the first critical speed. (e) Use superposition to estimate the first critical speed. ( f ) Estimate the shaft’s intrinsic critical speed. Suggest a modification to Dunkerley’s equation to include the effect of the shaft’s mass on the first critical speed of the attachments.

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MECE 304 Mechanical Machine Elements-Shafts

Figure 7-14 (a) A 25 mm uniform-diameter shaft for Ex. 7-5 (b) Superposing of equivalent loads at the center of the shaft for the purpose of finding the first critical speed

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MECE 304 Mechanical Machine Elements-Shafts

7-7 Miscellaneous Shaft Components Setscrews

Figure 7-15 Socket setscrews: (a) flat point; (b) cup point; (c) oval point; (d) cone point; (e) half-dog point

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MECE 304 Mechanical Machine Elements-Shafts

Table 7-4 Typical holding force for cup point socket setscrews Size, mm

Seating Torque, N.m

Holding force, N

#0

0.11

222

#1

0.2

289

#2

0.2

378

#3

0.5

534

#4

0.5

712

#5

1.1

890

#6

1.1

1112

#8

2.2

1713

#10

4.0

2403

6

9.8

4450

8

18.6

6675

10

32.8

8900

11

48.6

11125

12

70.0

13350

14

70.0

15575

16

149.7

17800

20

271.2

22250

22

587.6

26700

25

813.6

31150

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MECE 304 Mechanical Machine Elements-Shafts

Keys and pins

Figure 7-16 (a) Square key; (b) round key; (c and d) round pins; (e) taper pin; (f) split tubular spring pin.

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MECE 304 Mechanical Machine Elements-Shafts

Table 7-6 Millimeter dimensions for some standard square and rectangular key applications

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MECE 304 Mechanical Machine Elements-Shafts

Figure 7-17(a) Gib head key, (b) woodruff key

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MECE 304 Mechanical Machine Elements-Shafts

Table 7-7 Dimensions of woodruff key,

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MECE 304 Mechanical Machine Elements-Shafts

Table7-8 Sizes of Woodruff key suitable for various shaft diameter

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MECE 304 Mechanical Machine Elements-Shafts

For fillets cut by standard milling-machine cutters, with a ratio of r/d = 0.02, Peterson’s charts give Kt = 2.14 for bending and Kts = 2.62 for torsion without the key in place, or Kts = 3.0 for torsion with the key in place Retaining rings Is frequently used instead of a shaft shoulder or a sleeve to axially position a component on a shaft or in a housing bore. Appendix Tables A–13–16 and A–13–17 give values for stress concentration factors for flat-bottomed grooves in shafts, suitable for retaining rings. Figure 7-17 Retaining rings: (a,b) external, (c,d) internal

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MECE 304 Mechanical Machine Elements-Shafts

SAMPLE PROBLEM 7-6 A UNS G10350 steel shaft, heat-treated to a minimum yield strength of 525 MPa, has a diameter of 36 mm. The shaft rotates at 600 rev/min and transmits 36 kW through a gear. Select an appropriate key for the gear.

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MECE 304 Mechanical Machine Elements-Shafts

7-8 Limits and Fits The definitions illustrated in Fig. 7–20 are explained as follows: • Basic size is the size to which limits or deviations are assigned and is the same for both members of the fit. • Deviation is the algebraic difference between a size and the corresponding basic size. • Upper deviation is the algebraic difference between the maximum limit and the corresponding basic size. • Lower deviation is the algebraic difference between the minimum limit and the corresponding basic size. • Fundamental deviation is either the upper or the lower deviation, depending on which is closer to the basic size. • Tolerance is the difference between the maximum and minimum size limits of a part. • International tolerance grade numbers (IT) designate groups of tolerances such that the tolerances for a particular IT number have the same relative level of accuracy but vary depending on the basic size. • Hole basis represents a system of fits corresponding to a basic hole size. The fundamental deviation is H. • Shaft basis represents a system of fits corresponding to a basic shaft size. The fundamental deviation is h.

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MECE 304 Mechanical Machine Elements-Shafts

Figure 7-20 Definitions applied to a cylindrical fit

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MECE 304 Mechanical Machine Elements-Shafts

Table 7-9 Preferred Fits

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MECE 304 Mechanical Machine Elements-Shafts

Stress and Torque capacity in Interference Fits The pressure p generated at the interface of the interference fit, is given by

If both member are of the same material

where d is the nominal shaft diameter, di is the inside diameter (if any) of the shaft, do is the outside diameter of the hub, E is Young’s modulus, and v is Poisson’s ratio, with subscripts o and i for the outer member (hub) and inner member (shaft), respectively. δ is the diametral interference between the shaft and hub.

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MECE 304 Mechanical Machine Elements-Shafts

Since there is tolerance on both diameters, the min. and max. interferences will be

Tangential stress at the interference of the shaft and hub

The radial stress at the interference of the shaft and hub

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MECE 304 Mechanical Machine Elements-Shafts

A stress element on the surface of a rotating shaft will experience a completely reversed bending stress in the longitudinal direction, as well as the steady compressive stresses in the tangential and radial directions. This is a three-dimensional stress element. Shear stress due to torsion in shaft may also be present. Since the stresses due to the press fit are compressive, the fatigue situation is usually actually improved. For this reason, it may be acceptable to simplify the shaft analysis by ignoring the steady compressive stresses due to the press fit. There is, however, a stress concentration effect in the shaft bending stress near the ends of the hub, due to the sudden change from compressed to uncompressed material. For first estimates, values are typically not greater than 2. The friction force Ff and the torque capacity of the joint is

Where l is the length of the hub and f is the coefficient of friction. 44

MECE 304 Mechanical Machine Elements-Shafts-Supplements

4-2 Tension, compression and Torsion δ = Fl/AE

4-3

θ = Tl/JG

4-5

4-3 Deflection due to bending q/EI=d^4y/dx^4 V/EI=d^3y/dx^3 M/EI=d^2y/dx^2 θ =dy/dx y=f(x)

4-10 4-11 4-12 4-13 4-14

4-4 Beam deflection methods Equations 4-10 to 4-14 are the basis for relating the intensity of loading q, vertical shear V, bending moment M, slope of neutral surface θ and transverse deflection y. The methods used to solve the integration problem for beam deflection are. *Superposition *The moment area method *Singularity function *Integration (numerical, graphical)

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MECE 304 Mechanical Machine Elements-Shafts-Supplements

4-6 Beam deflection by superposition Superposition resolves the effect of combined loading on a structure by determining the effect of each load separately and adding the results algebraically. Conditions for superposition *each effect is linearly related to the load that produces it *the load does not create a condition that effects the results of another load *the deformation resulting from any specific load are not large enough to appeciably alter the geometric relations of the parts of the structural system (See Sample Problem S_2)

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MECE 304 Mechanical Machine Elements-Shaft-Supplements

ADDITIONAL INFORMATION Figure- Dimensions for some standard square and rectangular key applications

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MECE 304 Mechanical Machine Elements-Shafts-Supplements

Table- Designation and Tolerances for shafts and hubs for keys (see Fig S-47)

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MECE 304 Mechanical Machine Elements-Shafts-Supplements

SPLINES (there are standards for splines as DIN, SAE, TS)

Fig. Parallel side splines

Fig. Involute splines

Fig. Involute serrations

Fig. Parallel side splined shaft with six splines

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MECE 304 Mechanical Machine Elements-Shafts-Supplements

*Parallel side splines may be major or minor dia fit *The torque which an integral multispline shaft can transmit Mt =(1/2) phLi(D- h) Where Mt= Torque capacity N.mm P= bearing pressure, Mpa h=Spline heigth, mm L= Length of spline contact, mm i= number of splines D=Diameter, mm Failure of involute splines may be with shear or bearing pressures. So must be checked for shear and for bearing pressures.

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MECE 304 Mechanical Machine Elements-Shafts-Supplements

Table Proportions of SAE standard parallel side splines

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MECE 304 Mechanical Machine Elements-Shafts-Supplements

The theoretical torque capacity of straight-sided spline with sliding according to SAE Mt = 6.895* 10^6*i*((D+d)/4)* L*h Where Mt= torque in N.m i =number of splines d =inside diameter of spline, m D =pitch diameter of spline, m L =length of spline contact, m h =minimum height of contact in one tooth of spline, m

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