shaft_problems - AD E 51 o N 18-] Shohs and Axles 951 A...

Info iconThis preview shows pages 1–8. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 6
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 8
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: AD E 51 o N 18-] Shohs and Axles 951 A shaft embodies a large number of decisions. To view all these decisions as "open" is to create a design space that cannot be easily visualized. As a result. a designer's most valuable tool—geometric thinking—is severely hampered. An approach for the designer is to come closer to the position of the engineer who has to modify an existing design and has an approximate picture of the result at the out- set. Since the active constraint is likely to be from deformation. consider deformation first. Find the diameter of the uniform—diameter shaft that meets the deflection and slopes at the bearings and at the power transmission elements. Equations {l8»—l) and (184} will be helpful. Finding that diameter will give you an average size for the shaft. Bearing seats and shoulders will create geometric features at the bearing locations. If the bearing locations are "outboard" of the power transmission features. the bearing journal diameter will have little inflttence on deflection and slope: thus either it can be ignored. or the central diameter can be increased slightly. Consider the power transmission features. shoulders, and hub bores. and make some tentative decisions on step geometry (diameters and length). Now the geometry is becoming less opaque. Take your approximate idea of the shaft geometry and perform a deflection and slope analysis. Use Eqs. (18—3) and (IS—4) to find the largest (tum/do” ratio. then multiply all diameters by this ratio. See Ex. [8—2. You will have a stepped shaft that meets all the deflection and slope constraints. Check the. new shoulders (especially with rolling-contact hearings) to see if shoulder height still is within manufacturers' recommended range. If adjustments are necessary. construct a new table. Begin a Strength analysis using DE-Gerber or DE-elliptic theory. Use a desirable shaft material that will not need heat treatment. Heat treatment may increase strength. but the cost of heat treatment will increase the cost of the shaft severalfold. Examine. feature by feature. left bearing shoulder. left gear shoulder. gear keyway, right gear shoulder. right gear keyway. right bearing shoulder, shaft collar locations, snap-ring locations. for example. Examine these features for adequate diameter to see if mater- ial strength or diameter needs improvement. This feature-by-feature examination will tell the designer where the critical location is. If the critical feature is of sufficient strength. the designer has a clearer picture of the final design. The engineer may now have a satisfactory shaft. If some material can be pared off the shaft at other features (still keeping relevant constraints loose), consider this. In smaller production runs where volume—conservative initial forming methods are not used. the cost of addi- tional turning to remove material (make chisz may mitigate against any such size reduction. PROBLEMS The Iii-tooth pinion in the figure drives a double~reduction gear train as shown. All gears have 25" pressure angle. The pinion rotates counterclockwise at 1200 rev/min and transmits 50 hp to the gear train. Our focus is on the pinion shaft. None of the shafts have been designed. (a) For the pinion shaft develop the moment diagram. (b) For cylindrical roller bearings the shaft slope at the joumal should be less than 0.001 rad. For a design factor of nut = 2 estimate the uniform shaft diameter that would meet the deflection constraints. 952 Mechanical Engineering Design Problem lfldl Dimensions In inches. ADESIGN AGES-EN ADESIGN 1 8-2 '| 8-3 18-4 Problem l8—4 Shoulder fillets oi booting seol 0.030in rodlus. others gin radius, except righthond bearing soot transition. 3'; in. The moleriol is 1030 HR. Keywoys g in we by 3 ADESIGN Antonio “5 in deep Dimensions in inches, 18-5 18-6 1 8-7 Investigate the result of Prob. 18—1 for fatigue snength in a preliminary way, whether sue fiection controls. Use a 1030 hot-rolled steel for this purpose. The likely reliability goal for r‘: is 0.999. Assume a keyway in shaft with K, and K” given on page 444 and a fillet radius (a) Use the DE—elliptic fatigue-failure criterion for your factor of safety estimate. (b) Use the DE-Gerber fatigue-failure criterion in your estimate. Having found the pinion shaft of Probs. 18—1 and 18~2 has a tight deflection constraint, much of the shaft as you can. The overhang to mount a coupling upon can presume a 6- r . sion of the shaft beyond the bearing. Draw your shaft. showing all dimensional decisions; _ All the designs completed in response to Prob. 18—3 will differ. For the sake of discussi' -- - sider the design shown in the figure. The designer specified 02—40mm ball bearing on the an 03—40-mrn cylindrical roller beari mg on the right. Check this design for adequacy with to deformation. 1030 HR Bearing shoulder radii 0.030. Kcyway fillet radii radii 0.10. s 7% r 0.354 Lens Check the design in Prob. 13—4 for adequacy in fatigue strength using the DE-elliptic failure criterion. ' Now you have the opportunity to see how your design (Prob. 18—3) compares. Using your " completed in response to Prob. 18-3. perform the adequacy checks suggested in Probs. 5..- 18—5. The experience you have gained with Probs. 18—1 through 18—6 will make the design of 7' Prob. 18-1 easier. Design shaft r: of Prob. [8—1 using a design factor of 2 with a rel iability 1a . 0.995. Amritsar 18-8 ANALYSlS 18-9 DESIGN 18—10 Problem 18— l 0 Material moves under the toll Dimensions tn inches. Answers 18-" ADESEGN 18-12 AGES-ION 13—13 ADESKBN 18—14 fiasmtrsrs 18-15 Shohs ond Axles 953 The design of the intermediate shaft [3 of the speed reducer of Prob. 18ul is more difficult. The moment diagram does not lie in a plane. It is necessary to use moment diagrams in orthogonal planes. Design shaft (7 of Prob. 18—1 for R = 0.995 and a design factor of 2. Whether you have had the time to accomplish the design of the shafts a. b. and r- of Prob. 18—! or not, it is useful to consider the interrelationship to the geometry of the shafts, which the rolling- contact hearings will have. Avoid engaging in an iterative design situation unnecessarily. There will be a bearing reliability goal for the six bearings of the reducer. Plan a strategy for identifying the individual bearing reliabilities. The design work you will save will be your own. A geared industrial roll shown in the figure is driven at 300 revlmin by a force F acting on a 3-in- diameter pitch circle as shown. The roll exerts a normal force of 30 [bf/in of roll length on the ma- terial 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. 3 din. Using a 1035 hot-rolled steel. estimate the necessary diameters at the locations of peak bending moment using a design factor of 2. There are likely to be fillets at both ends of the right-hand bearing seat. where the bending moment is slightly less than the local extreme. Estimating the fa- tigue stress~concentration factor as 2. and using a design factor of 2. what is the approximate nec- essary diameter of the bearing seat using the DE-elliptic fatigue-failure criterion in Prob. Ill—l0? For the situation in Prob. lS—lO. what diameter uniform shaft will meet a deflection bearing slope of (a) 0.001 rad and (b) 0.0005 rad? For Prob. l8—10. find the diameter of a unifonn shaft to meet the slope limitation of 0.0005 rad at the gear mesh. What is the diameter to meet the slope with a factor of safety of 2‘? Design a shaft for the situation of the industrial roll of Prob. 18—] 0 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. l8-10. Hydrodynamic film bearings are to be used. All surfaces are machined except the journals. which are ground and polished. The material is l035 HR steel. Perform a design assessment. 15 the design satisfactory? 954 Mechanical Engineering Design I l I I % kcyway ' I Problem l8—l5 0 Heating shoulder lillets O 030 in. “ll— ”‘ 4 others T's in leEd'lUl'il'lél‘i lieywoy is 3i, in long. Dimensions ll'l inches. ? ‘ _ t 1 ' to 1'3 4 t f flANAW S i 5 1 8-16 As shown in the figure, the axle of a railroad freight car is tapered. with the least di w the rails. This problem will give some insight as to why this is so. A freight-car axle tit-f to its wheels with thejournals outboard of the wheels, and with journal centers about The track gauge is 56% in between the rails, and so the span between the rail cen 7 59% in. Freight-car wheels are usually of diameter 33 in for cars up to Til-ton capaci the worst-case location of the center of mass of the car and load to be 7?. in above The worst-case vertical load is to be 42 750 lhf per axle. The worst-case horizontal a crosswinds and track curvature is l? 100 ibf per axle. through the center of mass. - I bending-moment diagram for the axle. Center of grand ty of loaded car Problem lS—lb A Ioilroori l'reight-cor mile. (iofjoul Qofrail q: Top of rail flit-its! ALYSIS 18-17 in a class C American Association of Railroads standard axle, the diameter of the wheel“ " 7 in. and the diameter of the axle at midspan is 5% in. Using the results of Prob. 18-16, v the bending stress level at the center of the wheel seat and at the axle midspan. is this I It to be expected? ADESIGN 18-18 The section of shaft shown in the figure is to be designed to approximate relative : d = 0.750 and r = D l 20 with diameter d conforming to that of standard metric rolling- bore sizes. The shaft is to be made of SAE 2340 steel. heat-treated to obtain minimum s in the shoulder area of 1226—MPa ultimate tensile strength and llBO—MPa yield strength _ Brinell hardness not less than 368. At the shoulder the shaft is Subjected to a completely bending moment of 70 N - m. accompanied by a steady torsion of45 N - m. Use a design f ' 2.5 and size the shaft for an infinite life. The results should be based on the DE-elliptic f I _ failure criterion. Problem l 8-— l' 8 Section ol 0 shall containing o grinding-relief groove. Unless otherwise speciliecl. the diameter at the loci of the groove of, = d — 2L and though Ihe section ol diameter d is ground. the root ol the groove is still a machined surface. ' 'I -1 flsmmvsm 8 9 flitNA-WSIS 18-20 Problem l8~20 martlAiYSS 18-21 flatworm 18-22 1 8-23 Antwan Shohs and Axles 955 Repeat Prob. Iii—18 using the MSS-Soderberg 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 6-in 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 T,‘ = 3000 Ibf— in. The shaft is machined from steel with S. = 60 kpsi and S... = 80 kpsi. Using a factor of safety of 2.5. determine the minimum allowable diameter of the shaft based on (a) a static yield analysis using the distortion energy theory and (b) a fatigue-failure analysis using the four criteria given in Secs. l8-—3 and 18—4. Use fatigue stress concentration factors of K, = 1.8 and K“ = L3. A shaft is loaded in bending and torsion such that M., = 600 lbf- in. T” = 400 Ibf- in. M... = 5001hf- in. and T... = 300 lhf‘ in. For the shaft. 5.. = 100 kpsi and S. = 80 kpsi. and a fully cor- rected endurance limit of 5F c: 30 kpsi is assumed. Let Kr = 2.2 and Kr. 2 LS. With a design factor of 2.0 determine the minimum acceptable diameter of the shaft using the (a) DIE-Gerber criterion. (h) DIE—elliptic criterion. lc') MSS-Soderberg criterion. in!) DE-Goodman criterion. Discuss and compare the results. A transverse drilled and reamed hole can he 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 trans- mission 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 existance of a pin diameter with minimum adverse affect on the shaft. Then formulate a design rule. (Him: Use Table A—1 6.) A slow-speed spur gear with a 1.75-in bore and a l%-in-long hub has a pitch diameter of 8 in and involute teeth of 20° pressure angle. and it transmits 4.5 hp at “2 rev/min. The shaft is to have a bearing span of 10 in. with gear placed 3 in from the right—hand bearing. Deep-groove ball bear- ings are to he used. A 2-in overhang for a coupling having a l-in diameter seat is to he provided at the left of the left-hand bearing. For a design factor of 2. the bearing life is to be If) kh at a reliability of 0.995 for the bearing pair. Some a priori decisions are shown in the figure. Select appropriate bearings. then dimension the shaft in the neighborhood of the bearings. 956 Mechanical Engineering Design Problem I 8—23 Dimensions "1 inches. Am SIGN 1 8-24 The design of Prob. 18—23 has been completed. resulting in a shaft that is 12;i in long is important to have assurance that the deflection is within limits necessary to the the gear mesh and hearing life. A testing machine that can accurately apply a be - available, but the throat area is such that a specimen and fixture only up to 8 in long . died. For a scale model made half-size, what load should be applied? How should a‘ deflection or slope be interpreted if the model is made of the same material as the prot Problem i344 Dimensions in inches 1 ixlgkeyway _1_R : a _I_R_ ’2 ‘ e“ X T; kcyway fl DE SI G N A 5% F S 5 ME M 18-25 An A15] 1020 cold-drawn steel shaft with the geometry shown in the figure carries a I load of 7 kN and a torque of 107 N - m. Examine the shaft for strength and deflection. lithe allowable slope at the bearings is 0.001 rad and at the gear mesh is 0.0005 rad. what is the- of safety guarding against damaging distortion? What is the factor of safety guarding : y. fatigue failure? If the shaft turns out to be unsatisfactory. what would you recommend to ‘ the problem? 'ikN Problem i8—25 Dimensions in millimelers‘ elsfl tr All fillets 2 mm 375 ANALYSlS E ANALYSIS E ANALYSlS E ANAIXSIS :E§% 18-26 18-27 18-28 18—29 18-30 Problem l3—30 Drmensrons in inches. /fi\ptwem 18—31 Problem l8—3 l [Dimensions Ir: Inches /fi\peaew 18-32 Shells and Axles 957 A l-in-diameter uniform steel shaft is 24 in long between bearings. (a) Find the lowest critical speed of the shaft. (.5) 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. [8-26. by partitioning the shaft into first one. then two. and finally three elements. Compare Eq. (18—40) for the angular frequency of a two-disk shaft with Eq. (IS—4]). and note that the constants in the two equations are equal. to) Develop an expression for the second critical speed. tb) Estimate the second critical speed of the shaft addressed in Ex. 18—5, parts a and b. For a uniform-diameter shaft. does hollowing the shaft increase or decrease the critical speed? The shaft shown in the figure carries a 20~lbf gear on the left and a 35-lbf gear on the right. Esti- mate the first critical speed due to the loads. the shaft’s critical speed without the loads. and the critical speed of the combination. 20 [hr A shaft is to be designed to support the spur pinion and helical gear shown in the figure on two hearings spaced 28 in cettter—to-center. Bearing A is a cylindrical roller and is to take only radial load; bearing B is to take the thrust load of 220 lbf produced by the helical gear and its share of the radial load. The hearing at B can be a ball bearing. The radial loads of both gears are in the same plane. and are 660 lbf for the pinion and 220 lbf for the gear. The hearings are to have a life of 2000 h at a combined reliability of 0.995. The shaft speed is 1150 rev/min. Select the bearings and design the shaft. Make a sketch to scale of the shaft showing all fillet sizes. keyways. shoul- ders. 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 beating at A takes pure radial load. The hearing at 3 takes the worm-thrust load for either direction of rotation. The dimensions and the loading are shown in the figure; note 958 Mechanical Engineering Design Problem 13-32 Dimensmns in inches. Animators 18-33 Problem 18-33 i Dimensions in Inches. that the radial loads are in the same plane. Make a complete design of the shaft, including a- "i of the shaft showing all dimensions. identify the material and its heat treatment (if 11 Provide an assessment of your final design. The shaft speed 310 revfmin. |-+4 4— I4 —+ 3 i‘ . and lbf t 950w RB (-4]r 5600 lbf r = 4300 Ibi-in Ra Rn A bevel-gear shaft mounted on two 40-mm 02-series ball bearings is driven at 1720 revltnin motor connected through a flexible coupling. The figure shows the shaft. the gear. and the - ings. The shaft has been giving trouble—in fact. We of them have already failed—and the. time on the machine is so expensive that you have decided to redesign the shaft yourself ' than order replacements, which would. it is more than likely, also fail in a few months. You --1 not been abie to assemble much information. A hardness check of the two shafts in the via" the fracture of the two shafts showed an average of 198 Elm for one and 204 Bhn of the other closely as you can estimate the two shafts failed at a life measure between 600 000 and l 200- cycles of operation. The surfaces of the shaft were machined. but not ground. The fillet sizes - not measured. but they correspond with the recommendations for the ball bearings used. know that the load is a pulsating or shock-type load. but you have no idea of the magnitude cause the shaft drives an indexing mechanism. and the forces are inertial. The keyways wide by in deep. The straight-toothed bevel pinion drives a 48-tooth bevel gear. Specify a. shaft in sufficient detail to ensure a long and trouble—free life. ‘ ...
View Full Document

Page1 / 8

shaft_problems - AD E 51 o N 18-] Shohs and Axles 951 A...

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online