Ch 7 Shafts

Definitions 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.

Shaft Sizing Stress Analysis Deflection and Slope In design it is usually possible to locate the critical areas, size these to meet the strength requirements, and then size the rest of the shaft to meet the requirements of the shaft-supported elements. Deflection and Slope They are a function of inertia. Inertia is a function of Geometry. For this reason, shaft design allows a consideration of stress first. Then, after tentative values for the shaft dimensions have been established, the determination of the deflections and slopes can be made.

Shaft Materials Deflection is not affected by strength, but rather by stiffness as represented by the modulus of elasticity, which is essentially constant for all steels. For that reason, rigidity cannot be controlled by material decisions, but only by geometric decisions. Modulus of Elasticity For most steels 200 GPa

Shaft Materials A good practice for material selection: Start with an inexpensive, low or medium carbon steel for the first time through the design calculations. If strength considerations turn out to dominate over deflection, then a higher strength material should be tried, allowing the shaft sizes to be reduced until excess deflection becomes an issue. The cost of the material and its processing must be weighed against the need for smaller shaft diameters.

Manufacturing of Shafts For low production, turning is the usual primary shaping process. An economic viewpoint may require removing the least material. High production may permit a volume conservative shaping method (hot or cold forming, casting), and minimum material in the shaft can become a design goal.

Shaft Layout Shaft, Bearings, Gears, Shoulder

Movie 4:18

Shaft Layout In most cases, Only two bearings should be used in most cases. Load bearing components should be placed next to the bearings to minimize the bending due to large forces. Shafts should be kept short to minimize bending and deflection.

Shaft layout Shoulder It allows precise positioning Support to minimize deflection. In cases where the loads are small, positioning is not very important, shoulders can be eliminated.

Mounting gears, pulleys, etc… Press fits Shrink fits Keyway Spline shafts Rings

Keyway

Spline

Spline Shaft

Flange

Shaft Design for Stress It is not necessary to evaluate the stresses in a shaft at every point; a few potentially critical locations will suffice. Critical locations will usually be on the outer surface. Possible Critical Locations, axial locations where: 1- The bending moment is large and/or 2- The torque is present, and/or 3- Stress concentrations exist.

Torque Torque causes shear stress. Shear due to torque is max at the outer surface

Bending Moment Bending stress is determined from shear force and bending moment diagrams. In most cases forces on the shaft exist in two planes. As a result, moments are added as vectors.

Shaft Stresses The fluctuating stresses are: For a solid round shaft:

Shaft Stresses According to Von Mises:

Factor of Safety Modified Goodman

Goodman

Gerber

ASME Elliptic

Soderberg

First Cycle yielding Von Mises first cycle

Example

Notch sensitivity

Estimating Stress Concentration Initially, the shoulders and keyways dimensions are unknown. In this case, we pick a standard size and consider it as a first guess, then we fine-tune the dimensions. Shoulders for bearing and gear support should match the catalogue recommendation. Typical bearing calls for the ratio of D/d to be between 1.2 and 1.5. with r/d typically ranging from around 0.02 to 0.06

Ways to Minimize Stress Concentration

Worst Case scenario for shoulders 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.

Worst Case scenario for grooves

Grooves By examining typical retaining ring specifications in vendor catalogs, it can be seen that the groove width is typically slightly greater than the groove depth, and the radius at the bottom of the groove is around 1/10 of the groove width. From Figs. A–15–16 and A–15–17, stress concentration factors for typical retaining ring dimensions are around 5 for bending and axial, and 3 for torsion. Fortunately, the small radius will often lead to a smaller notch sensitivity, reducing Kf .

First Iteration Estimation for Shaft design

Where to start from???? Large bending moment and stress concentration

Then we repeat our analysis for the exact dimensions

Possible critical position (Keyway)

Possible critical location (K)

Deflection Consideration Allowable deflections will depend on many factors, and bearing and gear catalogs should be used for guidance on allowable misalignment for specific bearings and gears.

Deflection Consideration