Demonstrate how rapidly Rayleighs method converges for the

A shaft is loaded in bending and torsion such that Ma 5 70 N ? m, Ta 5 45 N ? m, Mm 5 55 N ? m, and Tm 5 35 N ? m. For the shaft, Su 5 700 MPa and Sy 5 560 MPa, and a fully corrected endurance limit of Se 5 210 MPa is assumed. Let Kf 5 2.2 and Kfs 5 1.8. With a design factor of 2.0 determine the minimum acceptable diameter of the shaft using the

The section of shaft shown in the figure is to be designed to approximate relative sizes of d 5 0.75D and r 5 Dy20 with diameter d conforming to that of standard rolling-bearing bore sizes. The shaft is to be made of SAE 2340 steel, heat-treated to obtain minimum strengths in the shoulder area of 175 kpsi ultimate tensile strength and 160 kpsi yield strength with a Brinell hardness not less than 370. At the shoulder the shaft is subjected to a completely reversed bending moment of 600 lbf ? in, accompanied by a steady torsion of 400 lbf ? in. Use a design factor of 2.5 and size the shaft for an infinite life using the DE-ASME Elliptic criterion

The rotating solid steel shaft is simply supported by bearings at points B and C and is driven by a gear (not shown) which meshes with the spur gear at D, which has a 150-mm pitch diameter. The force F from the drive gear acts at a pressure angle of 20. The shaft transmits a torque to point A of TA 5 340 N ? m. The shaft is machined from steel with Sy 5 420 MPa and Sut 5 560 MPa. Using a factor of safety of 2.5, determine the minimum allowable diameter of the 250-mm section of the shaft based on (a) a static yield analysis using the distortion energy theory and (b) a fatigue-failure analysis. Assume sharp fillet radii at the bearing shoulders for estimating stress-concentration factors.

A geared industrial roll shown in the figure is driven at 300 revymin by a force F acting on a 3-in-diameter pitch circle as shown. The roll exerts a normal force of 30 lbfyin of roll length on the material being pulled through. The material passes under the roll. The coefficient of friction is 0.40. Develop the moment and shear diagrams for the shaft modeling the roll force as (a) a concentrated force at the center of the roll, and (b) a uniformly distributed force along the roll. These diagrams will appear on two orthogonal planes.

Design a shaft for the situation of the industrial roll of Prob. 74 with a design factor of 2 and a reliability goal of 0.999 against fatigue failure. Plan for a ball bearing on the left and a cylindrical roller on the right. For deformation use a factor of safety of 2.

The figure shows a proposed design for the industrial roll shaft of Prob. 74. Hydrodynamic film bearings are to be used. All surfaces are machined except the journals, which are ground and polished. The material is 1035 HR steel. Perform a design assessment. Is the design satisfactory?

For the problem specified in the table, build upon the results of the original problem to obtain a preliminary design of the shaft by performing the following tasks. (a) Sketch a general shaft layout, including means to locate the components and to transmit the torque. Estimates for the component widths are acceptable at this point. (b) Specify a suitable material for the shaft. (c) Determine critical diameters of the shaft based on infinite fatigue life with a design factor of 1.5. Check for yielding. (d) Make any other dimensional decisions necessary to specify all diameters and axial dimensions. Sketch the shaft to scale, showing all proposed dimensions. (e) Check the deflections at the gears, and the slopes at the gears and the bearings for satisfaction of the recommended limits in Table 72. Assume the deflections for any pulleys are not likely to be critical. If any of the deflections exceed the recommended limits, make appropriate changes to bring them all within the limits.

For the problem specified in the table, build upon the results of the original problem to obtain a preliminary design of the shaft by performing the following tasks. (a) Sketch a general shaft layout, including means to locate the components and to transmit the torque. Estimates for the component widths are acceptable at this point. (b) Specify a suitable material for the shaft. (c) Determine critical diameters of the shaft based on infinite fatigue life with a design factor of 1.5. Check for yielding. (d) Make any other dimensional decisions necessary to specify all diameters and axial dimensions. Sketch the shaft to scale, showing all proposed dimensions. (e) Check the deflections at the gears, and the slopes at the gears and the bearings for satisfaction of the recommended limits in Table 72. Assume the deflections for any pulleys are not likely to be critical. If any of the deflections exceed the recommended limits, make appropriate changes to bring them all within the limits.

For the problem specified in the table, build upon the results of the original problem to obtain a preliminary design of the shaft by performing the following tasks. (a) Sketch a general shaft layout, including means to locate the components and to transmit the torque. Estimates for the component widths are acceptable at this point. (b) Specify a suitable material for the shaft. (c) Determine critical diameters of the shaft based on infinite fatigue life with a design factor of 1.5. Check for yielding. (d) Make any other dimensional decisions necessary to specify all diameters and axial dimensions. Sketch the shaft to scale, showing all proposed dimensions. (e) Check the deflections at the gears, and the slopes at the gears and the bearings for satisfaction of the recommended limits in Table 72. Assume the deflections for any pulleys are not likely to be critical. If any of the deflections exceed the recommended limits, make appropriate changes to bring them all within the limits.

For the problem specified in the table, build upon the results of the original problem to obtain a preliminary design of the shaft by performing the following tasks. (a) Sketch a general shaft layout, including means to locate the components and to transmit the torque. Estimates for the component widths are acceptable at this point. (b) Specify a suitable material for the shaft. (c) Determine critical diameters of the shaft based on infinite fatigue life with a design factor of 1.5. Check for yielding. (d) Make any other dimensional decisions necessary to specify all diameters and axial dimensions. Sketch the shaft to scale, showing all proposed dimensions. (e) Check the deflections at the gears, and the slopes at the gears and the bearings for satisfaction of the recommended limits in Table 72. Assume the deflections for any pulleys are not likely to be critical. If any of the deflections exceed the recommended limits, make appropriate changes to bring them all within the limits.

For the problem specified in the table, build upon the results of the original problem to obtain a preliminary design of the shaft by performing the following tasks. (a) Sketch a general shaft layout, including means to locate the components and to transmit the torque. Estimates for the component widths are acceptable at this point. (b) Specify a suitable material for the shaft. (c) Determine critical diameters of the shaft based on infinite fatigue life with a design factor of 1.5. Check for yielding. (d) Make any other dimensional decisions necessary to specify all diameters and axial dimensions. Sketch the shaft to scale, showing all proposed dimensions. (e) Check the deflections at the gears, and the slopes at the gears and the bearings for satisfaction of the recommended limits in Table 72. Assume the deflections for any pulleys are not likely to be critical. If any of the deflections exceed the recommended limits, make appropriate changes to bring them all within the limits.

For the problem specified in the table, build upon the results of the original problem to obtain a preliminary design of the shaft by performing the following tasks. (a) Sketch a general shaft layout, including means to locate the components and to transmit the torque. Estimates for the component widths are acceptable at this point. (b) Specify a suitable material for the shaft. (c) Determine critical diameters of the shaft based on infinite fatigue life with a design factor of 1.5. Check for yielding. (d) Make any other dimensional decisions necessary to specify all diameters and axial dimensions. Sketch the shaft to scale, showing all proposed dimensions. (e) Check the deflections at the gears, and the slopes at the gears and the bearings for satisfaction of the recommended limits in Table 72. Assume the deflections for any pulleys are not likely to be critical. If any of the deflections exceed the recommended limits, make appropriate changes to bring them all within the limits.

For the problem specified in the table, build upon the results of the original problem to obtain a preliminary design of the shaft by performing the following tasks. (a) Sketch a general shaft layout, including means to locate the components and to transmit the torque. Estimates for the component widths are acceptable at this point. (b) Specify a suitable material for the shaft. (c) Determine critical diameters of the shaft based on infinite fatigue life with a design factor of 1.5. Check for yielding. (d) Make any other dimensional decisions necessary to specify all diameters and axial dimensions. Sketch the shaft to scale, showing all proposed dimensions. (e) Check the deflections at the gears, and the slopes at the gears and the bearings for satisfaction of the recommended limits in Table 72. Assume the deflections for any pulleys are not likely to be critical. If any of the deflections exceed the recommended limits, make appropriate changes to bring them all within the limits.

For the problem specified in the table, build upon the results of the original problem to obtain a preliminary design of the shaft by performing the following tasks. (a) Sketch a general shaft layout, including means to locate the components and to transmit the torque. Estimates for the component widths are acceptable at this point. (b) Specify a suitable material for the shaft. (c) Determine critical diameters of the shaft based on infinite fatigue life with a design factor of 1.5. Check for yielding. (d) Make any other dimensional decisions necessary to specify all diameters and axial dimensions. Sketch the shaft to scale, showing all proposed dimensions. (e) Check the deflections at the gears, and the slopes at the gears and the bearings for satisfaction of the recommended limits in Table 72. Assume the deflections for any pulleys are not likely to be critical. If any of the deflections exceed the recommended limits, make appropriate changes to bring them all within the limits.

For the problem specified in the table, build upon the results of the original problem to obtain a preliminary design of the shaft by performing the following tasks. (a) Sketch a general shaft layout, including means to locate the components and to transmit the torque. Estimates for the component widths are acceptable at this point. (b) Specify a suitable material for the shaft. (c) Determine critical diameters of the shaft based on infinite fatigue life with a design factor of 1.5. Check for yielding. (d) Make any other dimensional decisions necessary to specify all diameters and axial dimensions. Sketch the shaft to scale, showing all proposed dimensions. (e) Check the deflections at the gears, and the slopes at the gears and the bearings for satisfaction of the recommended limits in Table 72. Assume the deflections for any pulleys are not likely to be critical. If any of the deflections exceed the recommended limits, make appropriate changes to bring them all within the limits.

For the problem specified in the table, build upon the results of the original problem to obtain a preliminary design of the shaft by performing the following tasks. (a) Sketch a general shaft layout, including means to locate the components and to transmit the torque. Estimates for the component widths are acceptable at this point. (b) Specify a suitable material for the shaft. (c) Determine critical diameters of the shaft based on infinite fatigue life with a design factor of 1.5. Check for yielding. (d) Make any other dimensional decisions necessary to specify all diameters and axial dimensions. Sketch the shaft to scale, showing all proposed dimensions. (e) Check the deflections at the gears, and the slopes at the gears and the bearings for satisfaction of the recommended limits in Table 72. Assume the deflections for any pulleys are not likely to be critical. If any of the deflections exceed the recommended limits, make appropriate changes to bring them all within the limits.

In the double-reduction gear train shown, shaft a is driven by a motor attached by a flexible coupling attached to the overhang. The motor provides a torque of 2500 lbf ? in at a speed of 1200 rpm. The gears have 20 pressure angles, with diameters shown in the figure. Use an AISI 1020 cold-drawn steel. Design one of the shafts (as specified by the instructor) with a design factor of 1.5 by performing the following tasks. (a) Sketch a general shaft layout, including means to locate the gears and bearings, and to transmit the torque. (b) Perform a force analysis to find the bearing reaction forces, and generate shear and bending moment diagrams. (c) Determine potential critical locations for stress design. (d) Determine critical diameters of the shaft based on fatigue and static stresses at the critical locations. (e) Make any other dimensional decisions necessary to specify all diameters and axial dimensions. Sketch the shaft to scale, showing all proposed dimensions. (f ) Check the deflection at the gear, and the slopes at the gear and the bearings for satisfaction of the recommended limits in Table 72. (g) If any of the deflections exceed the recommended limits, make appropriate changes to bring them all within the limits.

In the figure is a proposed shaft design to be used for the input shaft a in Prob. 717. A ball bearing is planned for the left bearing, and a cylindrical roller bearing for the right. (a) Determine the minimum fatigue factor of safety by evaluating at any critical locations. Use the DE-ASME Elliptic fatigue criterion. (b) Check the design for adequacy with respect to deformation, according to the recommendations in Table 72.

The shaft shown in the figure is proposed for the application defined in Prob. 372, p. 152. The material is AISI 1018 cold-drawn steel. The gears seat against the shoulders, and have hubs with setscrews to lock them in place. The effective centers of the gears for force transmission are shown. The keyseats are cut with standard endmills. The bearings are press-fit against the shoulders. Determine the minimum fatigue factor of safety using the DE-Gerber fatigue criterion.

Continue Prob. 719 by checking that the deflections satisfy the suggested minimums for bearings and gears in Table 72. If any of the deflections exceed the recommended limits, make appropriate changes to bring them all within the limits.

The shaft shown in the figure is proposed for the application defined in Prob. 373, p. 152. The material is AISI 1018 cold-drawn steel. The gears seat against the shoulders, and have hubs with setscrews to lock them in place. The effective centers of the gears for force transmission are shown. The keyseats are cut with standard endmills. The bearings are press-fit against the shoulders. Determine the minimum fatigue factor of safety using the DE-Gerber failure criterion.

Continue Prob. 721 by checking that the deflections satisfy the suggested minimums for bearings and gears in Table 72. If any of the deflections exceed the recommended limits, make appropriate changes to bring them all within the limits.

The shaft shown in the figure is driven by a gear at the right keyway, drives a fan at the left keyway, and is supported by two deep-groove ball bearings. The shaft is made from AISI 1020

An AISI 1020 cold-drawn steel shaft with the geometry shown in the figure carries a transverse load of 7 kN and a torque of 107 N ? m. Examine the shaft for strength and deflection. If the largest allowable slope at the bearings is 0.001 rad and at the gear mesh is 0.0005 rad, what is the factor of safety guarding against damaging distortion? Using the DE-ASME Elliptic criterion, what is the factor of safety guarding against a fatigue failure? If the shaft turns out to be unsatisfactory, what would you recommend to correct the problem?

A shaft is to be designed to support the spur pinion and helical gear shown in the figure on two bearings spaced 700 mm center-to-center. Bearing A is a cylindrical roller and is to take only radial load; bearing B is to take the thrust load of 900 N produced by the helical gear and its share of the radial load. The bearing at B can be a ball bearing. The radial loads of both gears are in the same plane, and are 2.7 kN for the pinion and 900 N for the gear. The shaft speed is 1200 rev/min. Design the shaft. Make a sketch to scale of the shaft showing all fillet sizes, keyways, shoulders, and diameters. Specify the material and its heat treatment.

A heat-treated steel shaft is to be designed to support the spur gear and the overhanging worm shown in the figure. A bearing at A takes pure radial load. The bearing at B takes the worm-thrust load for either direction of rotation. The dimensions and the loading are shown in the figure; note that the radial loads are in the same plane. Make a complete design of the shaft, including a sketch of the shaft showing all dimensions. Identify the material and its heat treatment (if necessary). Provide an assessment of your final design. The shaft speed is 310 rev/min.

A bevel-gear shaft mounted on two 40-mm 02-series ball bearings is driven at 1720 rev/min by a motor connected through a flexible coupling. The figure shows the shaft, the gear, and the bearings. The shaft has been giving troublein fact, two of them have already failedand the down time on the machine is so expensive that you have decided to redesign the shaft yourself rather than order replacements. A hardness check of the two shafts in the vicinity of the fracture of the two shafts showed an average of 198 Bhn for one and 204 Bhn of the other. As closely as you can estimate the two shafts failed at a life measure between 600 000 and 1 200 000 cycles of operation. The surfaces of the shaft were machined, but not ground. The fillet sizes were not measured, but they correspond with the recommendations for the ball bearings used. You know that the load is a pulsating or shock-type load, but you have no idea of the magnitude, because the shaft drives an indexing mechanism, and the forces are inertial. The keyways are 3 8 in wide by 3 16 in deep. The straight-toothed bevel pinion drives a 48-tooth bevel gear. Specify a new shaft in sufficient detail to ensure a long and trouble-free life.

A 25-mm-diameter uniform steel shaft is 600 mm long between bearings. (a) Find the lowest critical speed of the shaft. (b) If the goal is to double the critical speed, find the new diameter. (c) A half-size model of the original shaft has what critical speed?

Demonstrate how rapidly Rayleighs method converges for the uniform-diameter solid shaft of Prob. 728, by partitioning the shaft into first one, then two, and finally three elements.

Compare Eq. (727) for the angular frequency of a two-disk shaft with Eq. (728), and note that the constants in the two equations are equal. (a) Develop an expression for the second critical speed. (b) Estimate the second critical speed of the shaft addressed in Ex. 75, parts a and b.

For a uniform-diameter shaft, does hollowing the shaft increase or decrease the critical speed? Determine the ratio of the critical speeds for a solid shaft of diameter d to a hollow shaft of inner diameter dy2 and outer diameter d?

The steel shaft shown in the figure carries a 18-lbf gear on the left and a 32-lbf gear on the right. Estimate the first critical speed due to the loads, the shafts critical speed without the loads, and the critical speed of the combination.

A transverse drilled and reamed hole can be used in a solid shaft to hold a pin that locates and holds a mechanical element, such as the hub of a gear, in axial position, and allows for the transmission of torque. Since a small-diameter hole introduces high stress concentration, and a larger diameter hole erodes the area resisting bending and torsion, investigate the existence of a pin diameter with minimum adverse affect on the shaft. Specifically, determine the pin diameter, as a percentage of the shaft diameter, that minimizes the peak stress in the shaft. (Hint: Use Table A16.)

The shaft shown in Prob. 719 is proposed for the application defined in Prob. 372, p. 152. Specify a square key for gear B, using a factor of safety of 1.1.

The shaft shown in Prob. 721 is proposed for the application defined in Prob. 373, p. 152. Specify a square key for gear B, using a factor of safety of 1.1.

A guide pin is required to align the assembly of a two-part fixture. The nominal size of the pin is 15 mm. Make the dimensional decisions for a 15-mm basic size locational clearance fit.

An interference fit of a cast-iron hub of a gear on a steel shaft is required. Make the dimensional decisions for a 1.75-in basic size medium drive fit

A pin is required for forming a linkage pivot. Find the dimensions required for a 45-mm basic size pin and clevis with a sliding fit

A journal bearing and bushing need to be described. The nominal size is 1.25 in. What dimensions are needed for a 1.25-in basic size with a close running fit if this is a lightly loaded journal and bushing assembly?

A ball bearing has been selected with the bore size specified in the catalog as 35.000 mm to 35.020 mm. Specify appropriate minimum and maximum shaft diameters to provide a locational interference fit

A shaft diameter is carefully measured to be 1.5020 in. A bearing is selected with a catalog specification of the bore diameter range from 1.500 in to 1.501 in. Determine if this is an acceptable selection if a locational interference fit is desired.

A gear and shaft with nominal diameter of 35 mm are to be assembled with a medium drive fit, as specified in Table 79. The gear has a hub, with an outside diameter of 60 mm, and an overall length of 50 mm. The shaft is made from AISI 1020 CD steel, and the gear is made from steel that has been through hardened to provide Su 5 700 MPa and Sy 5 600 MPa. (a) Specify dimensions with tolerances for the shaft and gear bore to achieve the desired fit. (b) Determine the minimum and maximum pressures that could be experienced at the interface with the specified tolerances. (c) Determine the worst-case static factors of safety guarding against yielding at assembly for the shaft and the gear based on the distortion energy failure theory. (d) Determine the maximum torque that the joint should be expected to transmit without slipping, i.e., when the interference pressure is at a minimum for the specified tolerances.

A shaft is loaded in bending and torsion such that \(M_a\) = 70 N \(\cdot\) m, \(T_a\) = 45 N \(\cdot\) m, \(M_m\) = 55 N \(\cdot\) m, and \(T_m\) = 35 N \(\cdot\) m. For the shaft, \(S_u\) = 700 MPa and \(S_y\) = 560 MPa, and a fully corrected endurance limit of \(S_e\) = 210 MPa is assumed. Let \(K_f\) = 2.2 and \(K_{fs}\) = 1.8. With a design factor of 2.0 determine the minimum acceptable diameter of the shaft using the (a) DE-Gerber criterion. (b) DE-ASME Elliptic criterion. (c) DE-Soderberg criterion. (d) DE-Goodman criterion. Discuss and compare the results.

The section of shaft shown in the figure is to be designed to approximate relative sizes of d = 0.75D and r = D/20 with diameter d conforming to that of standard rolling-bearing bore sizes. The shaft is to be made of SAE 2340 steel, heat-treated to obtain minimum strengths in the shoulder area of 175 kpsi ultimate tensile strength and 160 kpsi yield strength with a Brinell hardness not less than 370. At the shoulder the shaft is subjected to a completely reversed bending moment of 600 lbf \(\cdot\) in, accompanied by a steady torsion of 400 lbf \(\cdot\) in. Use a design factor of 2.5 and size the shaft for an infinite life using the DE-ASME Elliptic criterion.

The rotating solid steel shaft is simply supported by bearings at points B and C and is driven by a gear (not shown) which meshes with the spur gear at D, which has a 150-mm pitch diameter. The force F from the drive gear acts at a pressure angle of \(20^{\circ}\). The shaft transmits a torque to point A of \(T_A\) = 340 N \(\cdot\) m. The shaft is machined from steel with \(S_y\) = 420 MPa and \(S_{ut}\) = 560 MPa. Using a factor of safety of 2.5, determine the minimum allowable diameter of the 250-mm section of the shaft based on (a) a static yield analysis using the distortion energy theory and (b) a fatigue-failure analysis. Assume sharp fillet radii at the bearing shoulders for estimating stress-concentration factors.

A geared industrial roll shown in the figure is driven at 300 rev/min by a force F acting on a 3-in-diameter pitch circle as shown. The roll exerts a normal force of 30 lbf/in of roll length on the material being pulled through. The material passes under the roll. The coefficient of friction is 0.40. Develop the moment and shear diagrams for the shaft modeling the roll force as (a) a concentrated force at the center of the roll, and (b) a uniformly distributed force along the roll. These diagrams will appear on two orthogonal planes.

Design a shaft for the situation of the industrial roll of Prob. 7–4 with a design factor of 2 and a reliability goal of 0.999 against fatigue failure. Plan for a ball bearing on the left and a cylindrical roller on the right. For deformation use a factor of safety of 2.

The figure shows a proposed design for the industrial roll shaft of Prob. 7–4. Hydrodynamic film bearings are to be used. All surfaces are machined except the journals, which are ground and polished. The material is 1035 HR steel. Perform a design assessment. Is the design satisfactory?

For the problem specified in the table, build upon the results of the original problem to obtain a preliminary design of the shaft by performing the following tasks. (a) Sketch a general shaft layout, including means to locate the components and to transmit the torque. Estimates for the component widths are acceptable at this point. (b) Specify a suitable material for the shaft. (c) Determine critical diameters of the shaft based on infinite fatigue life with a design factor of 1.5. Check for yielding. (d) Make any other dimensional decisions necessary to specify all diameters and axial dimensions. Sketch the shaft to scale, showing all proposed dimensions. (e) Check the deflections at the gears, and the slopes at the gears and the bearings for satisfaction of the recommended limits in Table 7–2. Assume the deflections for any pulleys are not likely to be critical. If any of the deflections exceed the recommended limits, make appropriate changes to bring them all within the limits. Problem Number Original Problem, Page Problem 7-7* 3-68, 151

For the problem specified in the table, build upon the results of the original problem to obtain a preliminary design of the shaft by performing the following tasks. (a) Sketch a general shaft layout, including means to locate the components and to transmit the torque. Estimates for the component widths are acceptable at this point. (b) Specify a suitable material for the shaft. (c) Determine critical diameters of the shaft based on infinite fatigue life with a design factor of 1.5. Check for yielding. (d) Make any other dimensional decisions necessary to specify all diameters and axial dimensions. Sketch the shaft to scale, showing all proposed dimensions. (e) Check the deflections at the gears, and the slopes at the gears and the bearings for satisfaction of the recommended limits in Table 7–2. Assume the deflections for any pulleys are not likely to be critical. If any of the deflections exceed the recommended limits, make appropriate changes to bring them all within the limits. Problem Number Original Problem, Page Problem 7-8* 3-69, 151

For the problem specified in the table, build upon the results of the original problem to obtain a preliminary design of the shaft by performing the following tasks. (a) Sketch a general shaft layout, including means to locate the components and to transmit the torque. Estimates for the component widths are acceptable at this point. (b) Specify a suitable material for the shaft. (c) Determine critical diameters of the shaft based on infinite fatigue life with a design factor of 1.5. Check for yielding. (d) Make any other dimensional decisions necessary to specify all diameters and axial dimensions. Sketch the shaft to scale, showing all proposed dimensions. (e) Check the deflections at the gears, and the slopes at the gears and the bearings for satisfaction of the recommended limits in Table 7–2. Assume the deflections for any pulleys are not likely to be critical. If any of the deflections exceed the recommended limits, make appropriate changes to bring them all within the limits. Problem Number Original Problem, Page Problem 7-9* 3-70, 151

For the problem specified in the table, build upon the results of the original problem to obtain a preliminary design of the shaft by performing the following tasks. (a) Sketch a general shaft layout, including means to locate the components and to transmit the torque. Estimates for the component widths are acceptable at this point. (b) Specify a suitable material for the shaft. (c) Determine critical diameters of the shaft based on infinite fatigue life with a design factor of 1.5. Check for yielding. (d) Make any other dimensional decisions necessary to specify all diameters and axial dimensions. Sketch the shaft to scale, showing all proposed dimensions. (e) Check the deflections at the gears, and the slopes at the gears and the bearings for satisfaction of the recommended limits in Table 7–2. Assume the deflections for any pulleys are not likely to be critical. If any of the deflections exceed the recommended limits, make appropriate changes to bring them all within the limits. Problem Number Original Problem, Page Problem 7-10* 3-71, 151

For the problem specified in the table, build upon the results of the original problem to obtain a preliminary design of the shaft by performing the following tasks. (a) Sketch a general shaft layout, including means to locate the components and to transmit the torque. Estimates for the component widths are acceptable at this point. (b) Specify a suitable material for the shaft. (c) Determine critical diameters of the shaft based on infinite fatigue life with a design factor of 1.5. Check for yielding. (d) Make any other dimensional decisions necessary to specify all diameters and axial dimensions. Sketch the shaft to scale, showing all proposed dimensions. (e) Check the deflections at the gears, and the slopes at the gears and the bearings for satisfaction of the recommended limits in Table 7–2. Assume the deflections for any pulleys are not likely to be critical. If any of the deflections exceed the recommended limits, make appropriate changes to bring them all within the limits. Problem Number Original Problem, Page Problem 7-11* 3-72, 152

For the problem specified in the table, build upon the results of the original problem to obtain a preliminary design of the shaft by performing the following tasks. (a) Sketch a general shaft layout, including means to locate the components and to transmit the torque. Estimates for the component widths are acceptable at this point. (b) Specify a suitable material for the shaft. (c) Determine critical diameters of the shaft based on infinite fatigue life with a design factor of 1.5. Check for yielding. (d) Make any other dimensional decisions necessary to specify all diameters and axial dimensions. Sketch the shaft to scale, showing all proposed dimensions. (e) Check the deflections at the gears, and the slopes at the gears and the bearings for satisfaction of the recommended limits in Table 7–2. Assume the deflections for any pulleys are not likely to be critical. If any of the deflections exceed the recommended limits, make appropriate changes to bring them all within the limits. Problem Number Original Problem, Page Problem 7-12* 3-73, 152

For the problem specified in the table, build upon the results of the original problem to obtain a preliminary design of the shaft by performing the following tasks. (a) Sketch a general shaft layout, including means to locate the components and to transmit the torque. Estimates for the component widths are acceptable at this point. (b) Specify a suitable material for the shaft. (c) Determine critical diameters of the shaft based on infinite fatigue life with a design factor of 1.5. Check for yielding. (d) Make any other dimensional decisions necessary to specify all diameters and axial dimensions. Sketch the shaft to scale, showing all proposed dimensions. (e) Check the deflections at the gears, and the slopes at the gears and the bearings for satisfaction of the recommended limits in Table 7–2. Assume the deflections for any pulleys are not likely to be critical. If any of the deflections exceed the recommended limits, make appropriate changes to bring them all within the limits. Problem Number Original Problem, Page Problem 7-13* 3-74, 152

For the problem specified in the table, build upon the results of the original problem to obtain a preliminary design of the shaft by performing the following tasks. (a) Sketch a general shaft layout, including means to locate the components and to transmit the torque. Estimates for the component widths are acceptable at this point. (b) Specify a suitable material for the shaft. (c) Determine critical diameters of the shaft based on infinite fatigue life with a design factor of 1.5. Check for yielding. (d) Make any other dimensional decisions necessary to specify all diameters and axial dimensions. Sketch the shaft to scale, showing all proposed dimensions. (e) Check the deflections at the gears, and the slopes at the gears and the bearings for satisfaction of the recommended limits in Table 7–2. Assume the deflections for any pulleys are not likely to be critical. If any of the deflections exceed the recommended limits, make appropriate changes to bring them all within the limits. Problem Number Original Problem, Page Problem 7-14* 3-76, 153

For the problem specified in the table, build upon the results of the original problem to obtain a preliminary design of the shaft by performing the following tasks. (a) Sketch a general shaft layout, including means to locate the components and to transmit the torque. Estimates for the component widths are acceptable at this point. (b) Specify a suitable material for the shaft. (c) Determine critical diameters of the shaft based on infinite fatigue life with a design factor of 1.5. Check for yielding. (d) Make any other dimensional decisions necessary to specify all diameters and axial dimensions. Sketch the shaft to scale, showing all proposed dimensions. (e) Check the deflections at the gears, and the slopes at the gears and the bearings for satisfaction of the recommended limits in Table 7–2. Assume the deflections for any pulleys are not likely to be critical. If any of the deflections exceed the recommended limits, make appropriate changes to bring them all within the limits. Problem Number Original Problem, Page Problem 7-15* 3-77, 153

For the problem specified in the table, build upon the results of the original problem to obtain a preliminary design of the shaft by performing the following tasks. (a) Sketch a general shaft layout, including means to locate the components and to transmit the torque. Estimates for the component widths are acceptable at this point. (b) Specify a suitable material for the shaft. (c) Determine critical diameters of the shaft based on infinite fatigue life with a design factor of 1.5. Check for yielding. (d) Make any other dimensional decisions necessary to specify all diameters and axial dimensions. Sketch the shaft to scale, showing all proposed dimensions. (e) Check the deflections at the gears, and the slopes at the gears and the bearings for satisfaction of the recommended limits in Table 7–2. Assume the deflections for any pulleys are not likely to be critical. If any of the deflections exceed the recommended limits, make appropriate changes to bring them all within the limits. Problem Number Original Problem, Page Problem 7-16* 3-79, 153

In the double-reduction gear train shown, shaft a is driven by a motor attached by a flexible coupling attached to the overhang. The motor provides a torque of 2500 lbf \(\cdot\) in at a speed of 1200 rpm. The gears have \(20^{\circ}\) pressure angles, with diameters shown in the figure. Use an AISI 1020 cold-drawn steel. Design one of the shafts (as specified by the instructor) with a design factor of 1.5 by performing the following tasks. (a) Sketch a general shaft layout, including means to locate the gears and bearings, and to transmit the torque. (b) Perform a force analysis to find the bearing reaction forces, and generate shear and bending moment diagrams. (c) Determine potential critical locations for stress design. (d) Determine critical diameters of the shaft based on fatigue and static stresses at the critical locations. (e) Make any other dimensional decisions necessary to specify all diameters and axial dimensions. Sketch the shaft to scale, showing all proposed dimensions. (f ) Check the deflection at the gear, and the slopes at the gear and the bearings for satisfaction of the recommended limits in Table 7–2. (g) If any of the deflections exceed the recommended limits, make appropriate changes to bring them all within the limits.

In the figure is a proposed shaft design to be used for the input shaft a in Prob. 7–17. A ball bearing is planned for the left bearing, and a cylindrical roller bearing for the right. (a) Determine the minimum fatigue factor of safety by evaluating at any critical locations. Use the DE-ASME Elliptic fatigue criterion. (b) Check the design for adequacy with respect to deformation, according to the recommendations in Table 7–2.

The shaft shown in the figure is proposed for the application defined in Prob. 3–72, p. 152. The material is AISI 1018 cold-drawn steel. The gears seat against the shoulders, and have hubs with setscrews to lock them in place. The effective centers of the gears for force transmission are shown. The keyseats are cut with standard endmills. The bearings are press-fit against the shoulders. Determine the minimum fatigue factor of safety using the DE-Gerber fatigue criterion.

Continue Prob. 7–19 by checking that the deflections satisfy the suggested minimums for bearings and gears in Table 7–2. If any of the deflections exceed the recommended limits, make appropriate changes to bring them all within the limits.

The shaft shown in the figure is proposed for the application defined in Prob. 3–73, p. 152. The material is AISI 1018 cold-drawn steel. The gears seat against the shoulders, and have hubs with setscrews to lock them in place. The effective centers of the gears for force transmission are shown. The keyseats are cut with standard endmills. The bearings are press-fit against the shoulders. Determine the minimum fatigue factor of safety using the DE-Gerber failure criterion.

Continue Prob. 7–21 by checking that the deflections satisfy the suggested minimums for bearings and gears in Table 7–2. If any of the deflections exceed the recommended limits, make appropriate changes to bring them all within the limits.

The shaft shown in the figure is driven by a gear at the right keyway, drives a fan at the left keyway, and is supported by two deep-groove ball bearings. The shaft is made from AISI 1020 cold-drawn steel. At steady-state speed, the gear transmits a radial load of 230 lbf and a tangential load of 633 lbf at a pitch diameter of 8 in. (a) Determine fatigue factors of safety at any potentially critical locations using the DE-Gerber failure criterion. (b) Check that deflections satisfy the suggested minimums for bearings and gears.

An AISI 1020 cold-drawn steel shaft with the geometry shown in the figure carries a transverse load of 7 kN and a torque of 107 N \(\cdot\) m. Examine the shaft for strength and deflection. If the largest allowable slope at the bearings is 0.001 rad and at the gear mesh is 0.0005 rad, what is the factor of safety guarding against damaging distortion? Using the DE-ASME Elliptic criterion, what is the factor of safety guarding against a fatigue failure? If the shaft turns out to be unsatisfactory, what would you recommend to correct the problem?

A shaft is to be designed to support the spur pinion and helical gear shown in the figure on two bearings spaced 700 mm center-to-center. Bearing A is a cylindrical roller and is to take only radial load; bearing B is to take the thrust load of 900 N produced by the helical gear and its share of the radial load. The bearing at B can be a ball bearing. The radial loads of both gears are in the same plane, and are 2.7 kN for the pinion and 900 N for the gear. The shaft speed is 1200 rev/min. Design the shaft. Make a sketch to scale of the shaft showing all fillet sizes, keyways, shoulders, and diameters. Specify the material and its heat treatment.

A heat-treated steel shaft is to be designed to support the spur gear and the overhanging worm shown in the figure. A bearing at A takes pure radial load. The bearing at B takes the worm-thrust load for either direction of rotation. The dimensions and the loading are shown in the figure; note that the radial loads are in the same plane. Make a complete design of the shaft, including a sketch of the shaft showing all dimensions. Identify the material and its heat treatment (if necessary). Provide an assessment of your final design. The shaft speed is 310 rev/min.

A bevel-gear shaft mounted on two 40-mm 02-series ball bearings is driven at 1720 rev/min by a motor connected through a flexible coupling. The figure shows the shaft, the gear, and the bearings. The shaft has been giving trouble—in fact, two of them have already failed—and the down time on the machine is so expensive that you have decided to redesign the shaft yourself rather than order replacements. A hardness check of the two shafts in the vicinity of the fracture of the two shafts showed an average of 198 Bhn for one and 204 Bhn of the other. As closely as you can estimate the two shafts failed at a life measure between 600 000 and 1 200 000 cycles of operation. The surfaces of the shaft were machined, but not ground. The fillet sizes were not measured, but they correspond with the recommendations for the ball bearings used. You know that the load is a pulsating or shock-type load, but you have no idea of the magnitude, because the shaft drives an indexing mechanism, and the forces are inertial. The keyways are \(\frac {3}{8}\) in wide by \(\frac {3}{16}\) in deep. The straight-toothed bevel pinion drives a 48-tooth bevel gear. Specify a new shaft in sufficient detail to ensure a long and trouble-free life.

A 25-mm-diameter uniform steel shaft is 600 mm long between bearings. (a) Find the lowest critical speed of the shaft. (b) If the goal is to double the critical speed, find the new diameter. (c) A half-size model of the original shaft has what critical speed?

Demonstrate how rapidly Rayleigh’s method converges for the uniform-diameter solid shaft of Prob. 7–28, by partitioning the shaft into first one, then two, and finally three elements.

Compare Eq. (7–27) for the angular frequency of a two-disk shaft with Eq. (7–28), and note that the constants in the two equations are equal. (a) Develop an expression for the second critical speed. (b) Estimate the second critical speed of the shaft addressed in Ex. 7–5, parts a and b.

For a uniform-diameter shaft, does hollowing the shaft increase or decrease the critical speed? Determine the ratio of the critical speeds for a solid shaft of diameter d to a hollow shaft of inner diameter d/2 and outer diameter d?

The steel shaft shown in the figure carries a 18-lbf gear on the left and a 32-lbf gear on the right. Estimate the first critical speed due to the loads, the shaft’s critical speed without the loads, and the critical speed of the combination.

A transverse drilled and reamed hole can be used in a solid shaft to hold a pin that locates and holds a mechanical element, such as the hub of a gear, in axial position, and allows for the transmission of torque. Since a small-diameter hole introduces high stress concentration, and a larger diameter hole erodes the area resisting bending and torsion, investigate the existence of a pin diameter with minimum adverse affect on the shaft. Specifically, determine the pin diameter, as a percentage of the shaft diameter, that minimizes the peak stress in the shaft. (Hint: Use Table A–16.)

The shaft shown in Prob. 7–19 is proposed for the application defined in Prob. 3–72, p. 152. Specify a square key for gear B, using a factor of safety of 1.1.

The shaft shown in Prob. 7–21 is proposed for the application defined in Prob. 3–73, p. 152. Specify a square key for gear B, using a factor of safety of 1.1.

A guide pin is required to align the assembly of a two-part fixture. The nominal size of the pin is 15 mm. Make the dimensional decisions for a 15-mm basic size locational clearance fit.

An interference fit of a cast-iron hub of a gear on a steel shaft is required. Make the dimensional decisions for a 1.75-in basic size medium drive fit.

A pin is required for forming a linkage pivot. Find the dimensions required for a 45-mm basic size pin and clevis with a sliding fit.

A journal bearing and bushing need to be described. The nominal size is 1.25 in. What dimensions are needed for a 1.25-in basic size with a close running fit if this is a lightly loaded journal and bushing assembly?

A ball bearing has been selected with the bore size specified in the catalog as 35.000 mm to 35.020 mm. Specify appropriate minimum and maximum shaft diameters to provide a locational interference fit.

A shaft diameter is carefully measured to be 1.5020 in. A bearing is selected with a catalog specification of the bore diameter range from 1.500 in to 1.501 in. Determine if this is an acceptable selection if a locational interference fit is desired.

A gear and shaft with nominal diameter of \(35 \mathrm{~mm}\) are to be assembled with a medium drive fit, as specified in Table 7-9. The gear has a hub, with an outside diameter of \(60 \mathrm{~mm}\), and an overall length of \(50 \mathrm{~mm}\). The shaft is made from AISI \(1020 \mathrm{CD}\) steel, and the gear is made from steel that has been through hardened to provide \(S_u=700 \mathrm{MPa}\) and \(S_{\mathrm{y}}=600 \mathrm{MPa}\). (a) Specify dimensions with tolerances for the shaft and gear bore to achieve the desired fit. (b) Determine the minimum and maximum pressures that could be experienced at the interface with the specified tolerances. (c) Determine the worst-case static factors of safety guarding against yielding at assembly for the shaft and the gear based on the distortion energy failure theory. (d) Determine the maximum torque that the joint should be expected to transmit without slipping, i.e., when the interference pressure is at a minimum for the specified tolerances.