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**Shafts and Shaft Components**

Lecture 3 Ali Vatankhah Chapter 7 Shafts and Shaft Components

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Chapter Outline

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SHAFT The term shaft usually refers to a component of circular cross-section that rotates and transmits power from a driving device, such as a motor or engine, through a machine. Shafts can carry gears, pulleys and sprockets to transmit rotary motion and power via mating gears, belts and chains. A shaft may simply connect to another via a coupling. A shaft can be stationary and support a rotating member such as the short shafts which support the non-driven wheels of automobiles, often referred to as spindles. Shigley’s Mechanical Engineering Design

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SHAFT A shaft is a rotating member usually of circular cross-section (solid or hollow), which transmits power and rotational motion. Machine elements such as gears, pulleys (sheaves), flywheels, clutches, and sprockets are mounted on the shaft and are used to transmit power from the driving device (motor or engine) through a machine. Press fit, keys, dowel, pins and splines are used to attach these machine elements on the shaft. The shaft rotates on rolling contact bearings or bush bearings. Various types of retaining rings, thrust bearings, grooves and steps in the shaft are used to take up axial loads and locate the rotating elements. Couplings are used to transmit power from drive shaft (e.g., motor) to the driven shaft (e.g. gearbox, wheels). Shigley’s Mechanical Engineering Design

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**SHAFT Shaft design considerations include:**

1. Size and spacing of components (as on a general assembly drawing), tolerances 2. Material selection and material treatments 3. Deflection and rigidity bending deflection torsional deflection slope at bearings shear deflection 4. Stress and strength static strength fatigue reliability 5. Frequency response 6. Manufacturing constraints Shigley’s Mechanical Engineering Design

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Shaft Design Shigley’s Mechanical Engineering Design

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**Shaft- A rotating member used to transmit power. **

Shaft Types Shaft- A rotating member used to transmit power. Axle- A stationary member used as support for rotating elements such as wheels, idler gears, etc. Spindle- A short shaft or axle (e.g., head-stock spindle of a lathe). Stub shaft- A shaft that is integral with a motor, engine or prime mover and is of a size, shape, and projection as to permit easy connection to other shafts Line shaft- A shaft connected to a prime mover and used to transmit power to one or several machines Jackshaft- (Sometimes called countershaft). A short shaft that connects a prime mover with a line shaft or a machine Flexible shaft- A connector which permits transmission of motion between two members whose axes are at an angle with each other Shigley’s Mechanical Engineering Design

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**Hollow Versus Solid Shafts **

Shaft Design Most shafts are round but they can come in many different shapes including square and octagonal. Keys and notches can also result in some unique shapes. Hollow Versus Solid Shafts Hollow shafts are lighter than solid shafts of comparable strength but are more expensive to manufacture. Thusly hollow shafts are primarily only used when weight is critical. For example the propeller shafts on rear wheel drive cars require lightweight shafts in order to handle speeds within the operating range of the vehicle. Shigley’s Mechanical Engineering Design

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Shaft Design Shigley’s Mechanical Engineering Design

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Shaft Design Shigley’s Mechanical Engineering Design

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Shaft Design Material Selection Geometric Layout Stress and strength Static strength Fatigue strength Deflection and rigidity Bending deflection Torsional deflection Slope at bearings and shaft-supported elements Shear deflection due to transverse loading of short shafts Vibration due to natural frequency

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Shaft Materials Deflection primarily controlled by geometry, not material Stress controlled by geometry, not material Strength controlled by material property

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Shaft Materials Shafts are commonly made from low carbon, CD or HR steel, such as ANSI 1020–1050 steels. Fatigue properties don’t usually benefit much from high alloy content and heat treatment. Surface hardening usually only used when the shaft is being used as a bearing surface.

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Shaft Materials Shigley’s Mechanical Engineering Design

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Shaft Materials Shigley’s Mechanical Engineering Design

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**Shaft Materials Hot Rolled**

Hot rolling is a mill process which involves rolling the steel at a high temperature (typically at a temperature over 1700° F), which is above the steel’s recrystallization temperature. When steel is above the recrystallization temperature, it can be shaped and formed easily, and the steel can be made in much larger sizes. Hot rolled steel is typically cheaper than cold rolled steel due to the fact that it is often manufactured without any delays in the process, and therefore the reheating of the steel is not required (as it is with cold rolled). When the steel cools off it will shrink slightly thus giving less control on the size and shape of the finished product when compared to cold rolled. Uses: Hot rolled products like hot rolled steel bars are used in the welding and construction trades to make railroad tracks and I-beams, for example. Hot rolled steel is used in situations where precise shapes and tolerances are not required.

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**Shaft Materials Cold Rolled**

Cold rolled steel is essentially hot rolled steel that has had further processing. The steel is processed further in cold reduction mills, where the material is cooled (at room temperature) followed by annealing and/or tempers rolling. This process will produce steel with closer dimensional tolerances and a wider range of surface finishes. The term Cold Rolled is mistakenly used on all products, when actually the product name refers to the rolling of flat rolled sheet and coil products. When referring to bar products, the term used is “cold finishing”, which usually consists of cold drawing and/or turning, grinding and polishing. This process results in higher yield points and has four main advantages:

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**Shaft Materials Cold Rolled**

Cold drawing increases the yield and tensile strengths, often eliminating further costly thermal treatments. Turning gets rid of surface imperfections. Grinding narrows the original size tolerance range. Polishing improves surface finish. All cold products provide a superior surface finish, and are superior in tolerance, concentricity, and straightness when compared to hot rolled. Cold finished bars are typically harder to work with than hot rolled due to the increased carbon content. However, this cannot be said about cold rolled sheet and hot rolled sheet. With these two products, the cold rolled product has low carbon content and it is typically annealed, making it softer than hot rolled sheet. Uses: Any project where tolerances, surface condition, concentricity, and straightness are the major factors

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Shaft Materials Cold drawn steel typical for d < 3 in. HR steel common for larger sizes. Should be machined all over. Low production quantities Lathe machining is typical Minimum material removal may be design goal High production quantities Forming or casting is common Minimum material may be design goal

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**Issues to consider for shaft layout Axial layout of components **

Supporting axial loads Providing for torque transmission Assembly and Disassembly Fig. 7-1

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**Axial Layout of Components**

Fig. 7-2 Choose a shaft configuration to support and locate the two gears and two bearings. (b) Solution uses an integral pinion, three shaft shoulders, key and keyway, and sleeve. The housing locates the bearings on their outer rings and receives the thrust loads. (c) Choose fanshaft configuration. (d) Solution uses sleeve bearings, a straight-through shaft, locating collars, and setscrews for collars, fan pulley, and fan itself. The fan housing supports the sleeve bearings.

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**Supporting Axial Loads**

Axial loads must be supported through a bearing to the frame. Generally best for only one bearing to carry axial load to shoulder Allows greater tolerances and prevents binding Fig. 7-3 Fig. 7-4

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**Providing for Torque Transmission**

Common means of transferring torque to shaft Keys Splines Setscrews Pins Press or shrink fits Tapered fits Keys are one of the most effective Slip fit of component onto shaft for easy assembly Positive angular orientation of component Can design key to be weakest link to fail in case of overload

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**Assembly and Disassembly**

Arrangement showing bearing inner rings press-fitted to shaft while outer rings float in the housing. The axial clearance should be sufficient only to allow for machinery vibrations. Note the labyrinth seal on the right. Fig. 7-5 Similar to the arrangement of Fig. 7–5 except that the outer bearing rings are preloaded. Fig. 7-6

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**Assembly and Disassembly**

In this arrangement the inner ring of the left-hand bearing is locked to the shaft between a nut and a shaft shoulder. The locknut and washer are AFBMA standard. The snap ring in the outer race is used to positively locate the shaft assembly in the axial direction. Note the floating right-hand bearing and the grinding runout grooves in the shaft. Fig. 7-7 This arrangement is similar to Fig. 7–7 in that the left-hand bearing positions the entire shaft assembly. In this case the inner ring is secured to the shaft using a snap ring. Note the use of a shield to prevent dirt generated from within the machine from entering the bearing Fig. 7-8 Shigley’s Mechanical Engineering Design

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**SHAFTS: General Design Rules**

Keep shafts short, with bearings close to the applied loads. This will reduce deflections and bending moments, and increases critical speeds. Place necessary stress raisers away from highly stressed shaft regions if possible. If unavoidable, use generous radii and good surface finishes. Consider local surface-strengthening processes (shot-peening or cold-rolling). Use inexpensive steels for deflection-critical shafts because all steels have essentially the same modulus of elasticity. Early in the design of any given shaft, an estimate is usually made of whether strength or deflection will be the critical factor. A preliminary design is based on that criterion; then, the remaining factor is checked. Shigley’s Mechanical Engineering Design

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**SHAFTS: Rules of Deflection**

Deflections should not cause mating gear teeth to separate more than about .005 in. They should also not cause the relative slope of the gear axes to change more than .03 deg. The shaft deflection across a plain bearing must be small compared to the oil film thickness. The shaft angular deflection at a ball or roller bearing should generally not exceed .04 deg. unless the bearing is self-aligning. Rule Of Thumb: Restrict the torsional deflection to 1° for every 20 diameters of length, sometimes less. Rule Of Thumb: In bending the deflection should be limited to .01 in. per foot of length between supports. Shigley’s Mechanical Engineering Design

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**Shaft design procedure for a shaft experiencing constant loading**

Determine the shaft rotational speed. Determine the power or torque to be transmitted by the shaft. Determine the dimensions of the power transmitting devices and other components mounted on the shaft and specify locations for each device. Specify the locations of the bearings to support the shaft. Propose a general form or scheme for the shaft geometry, considering how each component will be located axially and how power transmission will take place. Determine the magnitude of the torques throughout the shaft. Determine the forces exerted on the shaft. Produce shearing force and bending moment diagrams so that the distribution of bending moments in the shaft can be determined. Select a material for the shaft and specify any heat treatments etc. Determine an appropriate design stress, taking into account the type of loading (whether smooth, shock, repeated, reversed). Analyse all the critical points on the shaft and determine the minimum acceptable diameter at each point to ensure safe design. Specify the final dimensions of the shaft. This is best achieved using a detailed manufacturing drawing, and the drawing should include all the information required to ensure the desired quality. Typically this will include material specifications, dimensions and tolerances (bilateral, runout, datums etc.;), surface finishes, material treatments and inspection procedures. Shigley’s Mechanical Engineering Design

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**Shaft design procedure (Simple Steps)**

Shigley’s Mechanical Engineering Design

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Static Loading Shigley’s Mechanical Engineering Design

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Static Loading Shigley’s Mechanical Engineering Design

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**ASME Design for transmission shaft**

Shigley’s Mechanical Engineering Design

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**Shigley’s Mechanical Engineering Design**

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**General principles that should be observed in shaft design**

Keep shafts as short as possible with the bearings close to applied loads. This will reduce shaft deflection and bending moments and increase critical speeds. If possible, locate stress raisers away from highly stressed regions of the shaft. Use generous fillet radii and smooth surface finishes, and consider using local surface strengthening processes such cold rolling. If weight is critical use hollow shafts. Shigley’s Mechanical Engineering Design

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**General principles that should be observed in shaft design**

Shigley’s Mechanical Engineering Design

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**Shaft Design for Stress**

Stresses are only evaluated at critical locations Critical locations are usually On the outer surface Where the bending moment is large Where the torque is present Where stress concentrations exist

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Shaft Stresses Standard stress equations can be customized for shafts for convenience Axial loads are generally small and constant, so will be ignored in this section Standard alternating and midrange stresses

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Shaft Stresses Customized for round shafts

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Shaft Stresses Combine stresses into von Mises stresses

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Shaft Stresses Substitute von Mises stresses into failure criteria equation. For example, using modified Goodman line, Solving for d is convenient for design purposes

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Shaft Stresses Similar approach can be taken with any of the fatigue failure criteria Equations are referred to by referencing both the Distortion Energy method of combining stresses and the fatigue failure locus name. For example, DE-Goodman, DE-Gerber, etc. In analysis situation, can either use these customized equations for factor of safety, or can use standard approach from Ch. 6. In design situation, customized equations for d are much more convenient.

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Shaft Stresses DE-Gerber where Shigley’s Mechanical Engineering Design

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Shaft Stresses DE-ASME Elliptic DE-Soderberg

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**Shaft Stresses for Rotating Shaft**

For rotating shaft with steady bending and torsion Bending stress is completely reversed, since a stress element on the surface cycles from equal tension to compression during each rotation Torsional stress is steady Previous equations simplify with Mm and Ta equal to 0

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**Checking for Yielding in Shafts**

Always necessary to consider static failure, even in fatigue situation Soderberg criteria inherently guards against yielding ASME-Elliptic criteria takes yielding into account, but is not entirely conservative Gerber and modified Goodman criteria require specific check for yielding

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**Checking for Yielding in Shafts**

Use von Mises maximum stress to check for yielding, Alternate simple check is to obtain conservative estimate of s'max by summing s'a and s'm

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Example 7-1

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Example 7-1

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Example 7-1

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Example 7-1

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Example 7-1 Shigley’s Mechanical Engineering Design

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**Estimating Stress Concentrations**

Stress analysis for shafts is highly dependent on stress concentrations. Stress concentrations depend on size specifications, which are not known the first time through a design process. Standard shaft elements such as shoulders and keys have standard proportions, making it possible to estimate stress concentrations factors before determining actual sizes.

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**Estimating Stress Concentrations**

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**Reducing Stress Concentration at Shoulder Fillet**

Bearings often require relatively sharp fillet radius at shoulder If such a shoulder is the location of the critical stress, some manufacturing techniques are available to reduce the stress concentration Large radius undercut into shoulder Large radius relief groove into back of shoulder Large radius relief groove into small diameter of shaft Fig. 7-9

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Example 7-2

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Example 7-2 Fig. 7-10

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Example 7-2

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Example 7-2

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Example 7-2

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Example 7-2

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Example 7-2

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Example 7-2

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Example 7-2

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Example 7-2

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Example 7-2 Shigley’s Mechanical Engineering Design

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Example 7-2

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Example 7-2

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Example 7-2

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**Deflection Considerations**

Deflection analysis at a single point of interest requires complete geometry information for the entire shaft. For this reason, a common approach is to size critical locations for stress, then fill in reasonable size estimates for other locations, then perform deflection analysis. Deflection of the shaft, both linear and angular, should be checked at gears and bearings.

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**Deflection Considerations**

Allowable deflections at components will depend on the component manufacturer’s specifications. Typical ranges are given in Table 7–2

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**Deflection Considerations**

Deflection analysis is straightforward, but lengthy and tedious to carry out manually. Each point of interest requires entirely new deflection analysis. Consequently, shaft deflection analysis is almost always done with the assistance of software. Options include specialized shaft software, general beam deflection software, and finite element analysis software.

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Example 7-3 Fig. 7-10

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Example 7-3

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Example 7-3 Fig. 7-11

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Example 7-3

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Example 7-3

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**Adjusting Diameters for Allowable Deflections**

If any deflection is larger than allowed, since I is proportional to d4, a new diameter can be found from Similarly, for slopes, Determine the largest dnew/dold ratio, then multiply all diameters by this ratio. The tight constraint will be just right, and the others will be loose.

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Example 7-4

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**Angular Deflection of Shafts**

For stepped shaft with individual cylinder length li and torque Ti, the angular deflection can be estimated from For constant torque throughout homogeneous material Experimental evidence shows that these equations slightly underestimate the angular deflection. Torsional stiffness of a stepped shaft is

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**Critical Speeds for Shafts**

A shaft with mass has a critical speed at which its deflections become unstable. Components attached to the shaft have an even lower critical speed than the shaft. Designers should ensure that the lowest critical speed is at least twice the operating speed.

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**Critical Speeds for Shafts**

For a simply supported shaft of uniform diameter, the first critical speed is For an ensemble of attachments, Rayleigh’s method for lumped masses gives

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**Critical Speeds for Shafts**

Eq. (7–23) can be applied to the shaft itself by partitioning the shaft into segments. Fig. 7–12

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**Critical Speeds for Shafts**

An influence coefficient is the transverse deflection at location I due to a unit load at location j. From Table A–9–6 for a simply supported beam with a single unit load Fig. 7–13

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**Critical Speeds for Shafts**

Taking for example a simply supported shaft with three loads, the deflections corresponding to the location of each load is If the forces are due only to centrifugal force due to the shaft mass, Rearranging,

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**Critical Speeds for Shafts**

Non-trivial solutions to this set of simultaneous equations will exist when its determinant equals zero. Expanding the determinant, Eq. (7–27) can be written in terms of its three roots as

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**Critical Speeds for Shafts**

Comparing Eqs. (7–27) and (7–28), Define wii as the critical speed if mi is acting alone. From Eq. (7–29), Thus, Eq. (7–29) can be rewritten as

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**Critical Speeds for Shafts**

Note that The first critical speed can be approximated from Eq. (7–30) as Extending this idea to an n-body shaft, we obtain Dunkerley’s equation,

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**Critical Speeds for Shafts**

Since Dunkerley’s equation has loads appearing in the equation, it follows that if each load could be placed at some convenient location transformed into an equivalent load, then the critical speed of an array of loads could be found by summing the equivalent loads, all placed at a single convenient location. For the load at station 1, placed at the center of the span, the equivalent load is found from

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Example 7-5 Fig. 7–14

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Example 7-5

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Example 7-5 Shigley’s Mechanical Engineering Design

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Example 7-5

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Example 7-5 Fig. 7–14

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Example 7-5

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Example 7-5

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**Setscrews resist axial and rotational motion **

They apply a compressive force to create friction The tip of the set screw may also provide a slight penetration Various tips are available Fig. 7–15

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Setscrews Resistance to axial motion of collar or hub relative to shaft is called holding power Typical values listed in Table 7–4 apply to axial and torsional resistance Typical factors of safety are 1.5 to 2.0 for static, and 4 to 8 for dynamic loads Length should be about half the shaft diameter

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**Used to secure rotating elements and to transmit torque**

Keys and Pins Used to secure rotating elements and to transmit torque Fig. 7–16

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**Taper pins are sized by diameter at large end Small end diameter is **

Tapered Pins Taper pins are sized by diameter at large end Small end diameter is Table 7–5 shows some standard sizes in inches Table 7–5

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**Keys come in standard square and rectangular sizes**

Shaft diameter determines key size Table 7–6

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Keys Failure of keys is by either direct shear or bearing stress Key length is designed to provide desired factor of safety Factor of safety should not be excessive, so the inexpensive key is the weak link Key length is limited to hub length Key length should not exceed 1.5 times shaft diameter to avoid problems from twisting Multiple keys may be used to carry greater torque, typically oriented 90º from one another Stock key material is typically low carbon cold-rolled steel, with dimensions slightly under the nominal dimensions to easily fit end-milled keyway A setscrew is sometimes used with a key for axial positioning, and to minimize rotational backlash

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**Head makes removal easy Projection of head may be hazardous**

Gib-head Key Gib-head key is tapered so that when firmly driven it prevents axial motion Head makes removal easy Projection of head may be hazardous Fig. 7–17 Shigley’s Mechanical Engineering Design

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**Woodruff keys have deeper penetration **

Useful for smaller shafts to prevent key from rolling When used near a shoulder, the keyway stress concentration interferes less with shoulder than square keyway Fig. 7–17

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Woodruff Key Shigley’s Mechanical Engineering Design

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Woodruff Key

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**Stress Concentration Factors for Keys**

For keyseats cut by standard end-mill cutters, with a ratio of r/d = 0.02, Peterson’s charts give Kt = 2.14 for bending Kt = 2.62 for torsion without the key in place Kt = 3.0 for torsion with the key in place Keeping the end of the keyseat at least a distance of d/10 from the shoulder fillet will prevent the two stress concentrations from combining.

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Example 7-6 Fig. 7–19

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Example 7-6

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Example 7-6

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Retaining Rings Retaining rings are often used instead of a shoulder to provide axial positioning Fig. 7–18

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**This requires sharp radius in bottom of groove. **

Retaining Rings Retaining ring must seat well in bottom of groove to support axial loads against the sides of the groove. This requires sharp radius in bottom of groove. Stress concentrations for flat-bottomed grooves are available in Table A–15–16 and A–15–17. Typical stress concentration factors are high, around 5 for bending and axial, and 3 for torsion Shigley’s Mechanical Engineering Design

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Limits and Fits Shaft diameters need to be sized to “fit” the shaft components (e.g. gears, bearings, etc.) Need ease of assembly Need minimum slop May need to transmit torque through press fit Shaft design requires only nominal shaft diameters Precise dimensions, including tolerances, are necessary to finalize design

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Tolerances Close tolerances generally increase cost Require additional processing steps Require additional inspection Require machines with lower production rates Fig. 1–2

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**Standards for Limits and Fits**

Two standards for limits and fits in use in United States U.S. Customary (1967) Metric (1978) Metric version will be presented, with a set of inch conversions

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**Nomenclature for Cylindrical Fit**

Upper case letters refer to hole Lower case letters refer to shaft Basic size is the nominal diameter and is same for both parts, D=d Tolerance is the difference between maximum and minimum size Deviation is the difference between a size and the basic size Fig. 7–20

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**Tolerance Grade Number**

Tolerance is the difference between maximum and minimum size International tolerance grade numbers 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 IT grades range from IT0 to IT16, but only IT6 to IT11 are generally needed Specifications for IT grades are listed in Table A–11 for metric series and A–13 for inch series

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**Tolerance Grades – Metric Series**

Table A–11

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**Tolerance Grades – Inch Series**

Table A–13

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Deviations Deviation is the algebraic difference between a size and the basic size Upper deviation is the algebraic difference between the maximum limit and the basic size Lower deviation is the algebraic difference between the minimum limit and the basic size Fundamental deviation is either the upper or lower deviation that is closer to the basic size Letter codes are used to designate a similar level of clearance or interference for different basic sizes

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**Fundamental Deviation Letter Codes**

Shafts with clearance fits Letter codes c, d, f, g, and h Upper deviation = fundamental deviation Lower deviation = upper deviation – tolerance grade Shafts with transition or interference fits Letter codes k, n, p, s, and u Lower deviation = fundamental deviation Upper deviation = lower deviation + tolerance grade Hole The standard is a hole based standard, so letter code H is always used for the hole Lower deviation = 0 (Therefore Dmin = 0) Upper deviation = tolerance grade Fundamental deviations for letter codes are shown in Table A–12 for metric series and A–14 for inch series

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**Fundamental Deviations – Metric series**

Table A–12

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**Fundamental Deviations – Inch series**

Table A–14

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Specification of Fit A particular fit is specified by giving the basic size followed by letter code and IT grades for hole and shaft. For example, a sliding fit of a nominally 32 mm diameter shaft and hub would be specified as 32H7/g6 This indicates 32 mm basic size hole with IT grade of 7 (look up tolerance DD in Table A–11) shaft with fundamental deviation specified by letter code g (look up fundamental deviation dF in Table A–12) shaft with IT grade of 6 (look up tolerance Dd in Table A–11) Appropriate letter codes and IT grades for common fits are given in Table 7–9

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**Description of Preferred Fits (Clearance)**

Table 7–9

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**Description of Preferred Fits (Transition & Interference)**

Table 7–9

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**Procedure to Size for Specified Fit**

Select description of desired fit from Table 7–9. Obtain letter codes and IT grades from symbol for desired fit from Table 7–9 Use Table A–11 (metric) or A–13 (inch) with IT grade numbers to obtain DD for hole and Dd for shaft Use Table A–12 (metric) or A–14 (inch) with shaft letter code to obtain dF for shaft For hole For shafts with clearance fits c, d, f, g, and h For shafts with interference fits k, n, p, s, and u

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Example 7-7

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Example 7-8

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**Stress in Interference Fits**

Interference fit generates pressure at interface Need to ensure stresses are acceptable Treat shaft as cylinder with uniform external pressure Treat hub as hollow cylinder with uniform internal pressure These situations were developed in Ch. 3 and will be adapted here

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**Stress in Interference Fits**

The pressure at the interface, from Eq. (3–56) converted into terms of diameters, If both members are of the same material, d is diametral interference Taking into account the tolerances,

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**Stress in Interference Fits**

From Eqs. (3–58) and (3–59), with radii converted to diameters, the tangential stresses at the interface are The radial stresses at the interface are simply The tangential and radial stresses are orthogonal and can be combined using a failure theory

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**Torque Transmission from Interference Fit**

Estimate the torque that can be transmitted through interference fit by friction analysis at interface Use the minimum interference to determine the minimum pressure to find the maximum torque that the joint should be expected to transmit.

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