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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
rollingcontact 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 DEGerber or DEelliptic 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, snapring
locations. for example. Examine these features for adequate diameter to see if mater
ial strength or diameter needs improvement. This featurebyfeature 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 Iiitooth 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 AGESEN
ADESIGN 1 82 ' 83 184 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, 185 186 1 87 Investigate the result of Prob. 18—1 for fatigue snength in a preliminary way, whether sue ﬁection controls. Use a 1030 hotrolled 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 fatiguefailure criterion for your factor of safety estimate. (b) Use the DEGerber fatiguefailure 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—40mrn 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 DEelliptic failure criterion. ' Now you have the opportunity to see how your design (Prob. 18—3) compares. Using your "
completed in response to Prob. 183. 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. 181 easier. Design shaft r: of Prob. [8—1 using a design factor of 2 with a rel iability 1a .
0.995. Amritsar 188
ANALYSlS 189
DESIGN 18—10 Problem 18— l 0 Material moves under the toll
Dimensions tn inches. Answers 18"
ADESEGN 1812
AGESION 13—13
ADESKBN 18—14
ﬁasmtrsrs 1815 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 3in
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 hotrolled 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 righthand
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 DEelliptic fatiguefailure 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. l810. 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 1 816 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 freightcar axle titf 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. Freightcar wheels are usually of diameter 33 in for cars up to Tilton capaci
the worstcase location of the center of mass of the car and load to be 7?. in above
The worstcase vertical load is to be 42 750 lhf per axle. The worstcase horizontal a
crosswinds and track curvature is l? 100 ibf per axle. through the center of mass.  I bendingmoment diagram for the axle. Center of grand ty
of loaded car Problem lS—lb
A Ioilroori l'reightcor mile. (iofjoul Qofrail q: Top of rail ﬂitits! ALYSIS 1817 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. 1816, 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 1818 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. heattreated 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 DEelliptic f I _ failure criterion. Problem l 8— l' 8 Section ol 0 shall containing o
grindingrelief 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
ﬂitNAWSIS 1820 Problem l8~20 martlAiYSS 1821 ﬂatworm 1822 1 823 Antwan Shohs and Axles 955 Repeat Prob. Iii—18 using the MSSSoderberg 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 6in 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 fatiguefailure 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) DIEGerber criterion. (h) DIE—elliptic criterion. lc') MSSSoderberg criterion. in!) DEGoodman 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 slowspeed spur gear with a 1.75in bore and a l%inlong 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. Deepgroove ball bear
ings are to he used. A 2in overhang for a coupling having a lin diameter seat is to he provided
at the left of the lefthand 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 1 824 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 halfsize, 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 1825 An A15] 1020 colddrawn 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§% 1826 1827 1828 18—29
1830 Problem l3—30 Drmensrons in inches. /ﬁ\ptwem 18—31 Problem l8—3 l [Dimensions Ir: Inches /ﬁ\peaew 1832 Shells and Axles 957 A lindiameter 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 uniformdiameter solid shaft of
Prob. [826. by partitioning the shaft into ﬁrst one. then two. and ﬁnally three elements. Compare Eq. (18—40) for the angular frequency of a twodisk 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 uniformdiameter 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 35lbf 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—tocenter. 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 heattreated 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 wormthrust
load for either direction of rotation. The dimensions and the loading are shown in the ﬁgure; note 958 Mechanical Engineering Design Problem 1332
Dimensmns in inches. Animators 1833 Problem 1833
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 Ibiin
Ra Rn A bevelgear shaft mounted on two 40mm 02series 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 shocktype 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 straighttoothed bevel pinion drives a 48tooth bevel gear. Specify a.
shaft in sufﬁcient detail to ensure a long and trouble—free life. ‘ ...
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 Fall '09
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