Caterpillar Shaft Design

Chapter 9

Shaft Design Transmission shafts transmit torque from one location to another Spindles are short shafts Axles are non-rotating shafts Figure 9.1 is an example of a shaft with several features. It is a shaft for a Caterpillar tractor transmission 1 .

Figure 9.1: Example of a typical shaft design

1 From

Frederick E. Giesecke, Technical Drawing, Chapter 13.

1

9.1

Shaft Loads

• Torsion due to transmitted torque • Bending from transverse loads (gears, sprockets, pulleys/sheaves)∗ o * a pulley and a sheave are essentially the same thing Steady or Fluctuating Steady transverse-bending load ♦ fully reversing bending stress (fatigue failure)

9.2

Attachments and Stress Concentrations

Steps and shoulders are used to locate attachment (gears, sheaves, sprockets) Keys, snap rings, cross pins (shear pins), tapered pins Use generous radii to reduce stress concentrations Clamp collars Split collar Press fits and shrink fits Bearings may be located by the use of snap rings, but only one bearing is fixed Issues - axial location, disassembly, and element phasing (e.g., alignment of gear teeth for timing)

MACHINE DESIGN

-

An Integrated Approach, 2ed

by Robert L. Norton,

Prentice-Hall 2000

snap ring

clamp collar

taper pin

key hub bearing

hub

shaft

step

bearing

press fit

step step

step

press fit

axial clearance frame

frame sprocket

gear

FIGURE 9-2 Various Methods to Attach Elements to Shafts

Figure 9.2: Example of a shaft with various attachments and details

9.3 • • • •

Shaft Materials

Steel (low to medium-carbon steel) Cast iron Bronze or stainless steel Case hardened steel

2

sheave

9.3.1

Shaft Power

Power is the time rate of change of energy (work). work = Force * distance or Torque * angle, so Power = Torque * angular velocity P wr = T orq ∗ ω

9.4

(9.1)

Shaft Loading, Approaches to Analysis

Most general form - A fluctuating torque and a fluctuating moment, in combination. If there are axial loads, they should be “taken to ground” as close to the load as possible. Given knowledge of the moments and the torques (i.e., mean and alternating components) Use the “Design Steps for Fluctuating Stresses” in Section 6.11 in combination with the multiaxial-stress issues addressed in Section 6.12.

9.5

Shaft Stresses

Bending Stress σalt σmean

Ma c I Mm c = kf m I

= kf

(9.2) (9.3)

Torsional Shear Stress

τalt τmean

9.5.1

9.6 9.6.1

Ta r J Tm r = kf sm J = kf s

(9.4) (9.5)

Shaft Failure in Combined Loading

Shaft Design General Considerations

1. To minimize both deflections and stresses, the shaft length should be kept as short as possible and overhangs minimized. 2. A cantilever beam will have a larger deflection than a simply supported (straddle mounted) one for the same length, load, and cross section, so straddle mounting should be used unless a cantilever shaft is dictated by design constraints. (Figure 9-2 shows a situation in which an overhung section is required for serviceability.) 3. A hollow shaft has a better stiffness/mass ratio (specific stiffness) and higher natural frequencies than a comparably stiff or strong solid shaft, but will be more expensive and larger in diameter. 4. Try to locate stress-raisers away from regions of large bending moment if possible and minimize their effects with generous radii and relief. 5. General low carbon steel is just as good as higher strength steels (since deflection is typical the design limiting issue). 6. Deflections at gears carried on the shaft should not exceed about 0.005 inches and the relative slope between the gears axes should be less than about 0.03 degrees. 3

MACHINE DESIGN

-

An Integrated Approach, 2ed

by Robert L. Norton,

Prentice-Hall 2000

from ref. 2

from ref. 3

σa Se

σa Se

from ref. 3

2

2

 σa   τa   S  +S  =1  e  es 

2

2  τm   σa   S  +S  =1  e  ys 

τa Ses

τm Sys (a) Combined stress fatigue-test data for reversed bending combined with static torsion (from ref. 4)

(b) Combined stress fatigue-test data for reversed bending combined with reversed torsion (from ref. 5)

FIGURE 9-3 Results of Fatigue Tests of Steel Specimens Subjected to Combined Bending and Torsion (From Design of Transmission Shafting, American Society of Mechanical Engineers, New York, ANSI/ASME Standard B106.1M-1985, with permission)

Figure 9.3: Shaft failure in combined loading

4

7. If plain (sleeve) bearings are to be used, the shaft deflection across the bearing length should be less than the oil-film thickness in the bearing. 8. If non-self-aligning rolling element bearings are used, the shaft’s slope at the bearings should be kept to less than about 0.04 degrees. 9. If axial thrust loads are present, they should be taken to ground through a single thrust bearing per load direction. Do not split axial loads between thrust bearings as thermal expansion of the shaft can overload the bearings. 10. The first natural frequency of the shaft should be at least three times the highest forcing frequency expected in service, and preferably much more. (A factor of ten times or more is preferred, but this is often difficult to achieve). Designing for Fully Reversed Bending and Steady Torsion ASME Method (ANSI/ASME Standard for Design of Transmission Shafting B106.1M-1985. Uses the elliptical curve of Figure 9-3. Equations 9.5e and 9.6a,b. 9.6 can be applied only for • constant torque • fully reversed moment. • No axial load v s u u 32Saf etyF actor Ma 2 3 T m 2 3 (kf ) + ( ) d=t π Sf 4 Sy

(9.6)

More general loading cases require Equation 9.8. See Example 9..

9.6.2

Shaft Deflection

Deflection is often the more demanding constraint. Many shafts are well within specification for stress but would exhibit too much deflection to be appropriate.

9.6.3

Keys and Keyways

P

gear d T

a b l

bearings are self-aligning so act as simple supports

FIGURE P9-3

P9-03.pdf

Shaft Design for Problems 9-6, 9-9. 9-11, and 9-12

Figure 9.4: Shaft with overhung gear Example -Homework Problem 9-2

5

9.6.4

Splines

9.6.5

Interference Fits

Components can be attached to a shaft without a key or spline by using an interference fit. There are two methods used to assemble these components: • press fit • shrink (and/or expansion) fit The amount of interference is important The analysis of interference follows from the equations for pressure on thick-walled cylinders. A rule of thumb that is used is one to two thousands of diametral interference per unit of shaft diameter, e.g., a shaft of two inch diameter would have 0.004 inches of interference with an attached gear hub. Machinists use a simplified approach to this – 1/1000 of interference for each inch of diameter. However, there is a formal approach Standards have been developed for these fits. Metric Preferred Metric Limits and Fits — ANSI B4.2-1978. US Customary Preferred Limits and Fits for Cylindrical Parts —ANSI B4.1-1967

9.7 9.7.1

Terms related to Fits and Tolerances ANSI B4.2-1978 Definitions

D – basic size of the hole d – basic size of the shaft δu – upper deviation δl – lower deviation δF – Fundamental deviation ∆D – tolerance grade for the hole ∆d – tolerance grade for the shaft Tolerance – the difference between the maximum and minimum size limits of the dimensions of a part Natural tolerance – a tolerance equal to ± three standard deviations from the mean Clearance – amount of space between an internal and external member Interference – the amount of overlap between an internal and external member 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, i.e., IT 9 Smaller numbers mean tighter tolerances, IT 6 through IT 11 are used for preferred fits. For a 32 mm hole we might use 32H7 • The H establishes the fundamental deviation and the number 7 defines a tolerance grade of IT7. The grade number specifies a tolerance zone. For the mating shaft we might have 32g6

9.7.2

Table of Tolerance Grades

2

Lower and Upper Deviations • For shaft letter codes c, d, f, g, and h 2 Shigley

Table E-11, page 1188.

6

Table 9.1: International Tolerance Grades Basic Sizes All values in mm A