Machine Design Databook Episode 3 part 7 potx

24-57 roller pin and slotted Geneva wheel, kN lbf contact due to the friction torque M2t, on the driven Genevawheel shaft, kN lbf due to the inertia torque on the driven Geneva wheel s

For external ratchet

For internal ratchet

The ratio of a=d (internal ratchet)

The module

The bending moment on pawl

The bending stress

The diameter of pawl pin

2þ th

s

ð24-272Þwhere th¼ thickness of hub on pawl

MISCELLANEOUS MACHINE ELEMENTS 24.81

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24.9 GENEVA MECHANISM

SYMBOLS2,3

a¼ r1

sin center distance, m (in)

F1 the component of force acting on the crank or the driving shaft

due to the torque, M1t, kN (lbf ) (Fig 24-57)

F2 the component of force acting on the driven Geneva wheel shaft

due to the torque M2t, kN (lbf ) (Fig 24-57)

roller pin and slotted Geneva wheel, kN (lbf )

contact due to the friction torque M2t, on the driven Genevawheel shaft, kN (lbf )

due to the inertia torque on the driven Geneva wheel shaft,

kN (lbf )

i¼z
2

J polar moment of inertia of all the masses of parts attached to

Geneva wheel shaft, m4(in4)

k the working time coefficient of the Geneva wheel

M1t total torque on the driver or crank, N m (lbf in)

M2t total torque on the driven or Geneva wheel, N m (lbf in)

M2ti inertia torque on the Geneva wheel, N m (lbf in)

M2t friction or resistance torque on Geneva wheel, N m (lbf in)

r1 radius to center of driving pin, m (in)

r2 radius of Geneva wheel, m (in)

r02 distance of center of semicircular end of slot from the center of

Geneva wheel, m (in)

ra outside radius of Geneva wheel, which includes correction for

finite pin diameter, m (in)

rp pin radius, m (in)

Rr¼r2

r1 radius ratio

t total time required for a full revolution of the driver or crank, s

ti time required for indexing Geneva wheel, s

tr time during which Geneva wheel is at rest, s

z number of slots on the Geneva wheel

crank angle or angle of driver at any instant, deg (Fig 24-54)

2a angular acceleration, m/s2(ft/s2)

angular acceleration of Geneva wheel, m/s2(ft/s2)

m angular position of the crank or driver radius at which the

product! 2a is maximum, degangle of the driven wheel or Geneva wheel at any instant, deg

(Fig 24-54)

 ¼r1

a the ratio of the driver radius to center distance

 efficiency of Geneva mechanism

 locking angle of driver or crank, rad or deg

 ratio of time of motion of Geneva wheel to time for one

revolution of driver or crank

 ¼360

2z semi-indexing or Geneva wheel angle, or half the anglesubtended by an adjacent slot, deg (Fig 24-54)

crank or driver angle, deg (Fig 24-54)

! ¼2n

60 angular velocity of driver or crank (assumed constant), rad/s

!1,!2 angular velocities of driver or crank and Geneva wheel,

FIGURE 24-54 Design of Geneva mechanism.

The angular velocity (constant) of driver or crank

Gear ratio

The semi-indexing angle or Geneva wheel angle or

half the angle subtended by two adjacent slots

The angle through which the Geneva wheel rotates

EXTERNAL GENEVA WHEEL

The angle of rotation of driver through which the

Geneva wheel is at rest or angle of locking action

MISCELLANEOUS MACHINE ELEMENTS 24.83

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The center distance (Fig 24-55)

The radius ratio

The ratio of crank radius to center distance

The relation between crank angle and Geneva wheel

angle

VELOCITY

The angular velocity of the Geneva wheel

The maximum angular velocity of Geneva wheel at

!2¼ sinð=zÞðcos
sin =zÞ

1
2 sinð=zÞ cos þ sin2=z!1 ð24-283bÞ

ddt



1
sinz

ω1

ω2ψ

φ

FIGURE 24-55 External Geneva mechanism.

The angular acceleration,a 2a, of Geneva wheel

For angular velocity and angular acceleration curves

for three-slot external Geneva wheel with driver

velocity,!1¼ 1 rad/s

The maximum angular acceleration of Geneva wheel

which occurs at ¼ ðmaxÞ

The angular acceleration of Geneva wheel at start and

a 2a¼  sinð=zÞ cos2ð=zÞ sin

1
2 sinð=zÞ cos þ sin2ð=zÞ!

2

ð24-284bÞRefer to Fig 24-56

where

 ¼14

0 0

10

20

30

2 4 6

a 2a is the symbol used for angular acceleration of Geneva wheel; is the crank or driver angle at any given instant.

MISCELLANEOUS MACHINE ELEMENTS 24.85

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The ratio of tito t

The ratio of trto t

The sum of angles of ( þ )

The time required for indexing Geneva wheel, in

seconds

The time during which Geneva wheel is at rest, in

seconds

The working time coefficient of Geneva wheel

Ratio of time of motion of Geneva wheel to time for

one revolution of crank or driver

The required speed of the driver shaft or crankshaft

SHOCK OR JERK

The jerk or shock, J2, on Geneva wheel

The jerk or shock at ¼ 0

The jerk or shock at start, i.e.,

The length of the slot (Fig 24-54)

The condition to be satisfied by diameter on which the

driver or crank is mounted

The condition to be satisfied by the diameter on which

Geneva wheel is mounted

z

602n



ð24-290Þ

tr¼zþ 2z

602n



<12



ð24-293Þ

n¼zþ 2z

602tr



ð24-294Þwhere n in rpm

zþ cos

z
1

ð24-298Þ

d1< 2a3¼ 2ða
r2Þ ¼ 2a



1
cosz

ð24-299Þor

d1

a < 2



1
cosz

ð24-301Þ

TORQUE ACTING ON SHAFTS OF

GENEVA WHEEL AND DRIVER

The total torque acting on Geneva wheel shaft

The torque on the shaft of crank or driver

The efficiency of Geneva mechanism

INSTANTANEOUS POWER

The instantaneous power on the crank or driving

shaft

Calculation of average power

The average torque MtiðavÞfor complete cycle

The average torque for first half-cycle

 ¼ 0.80 to 0.90 when Geneva wheel shaft

is mounted on journal bearings (24-304a)

 ¼ 0.95 when drive shaft is mounted onrolling contact bearings (24-304b)

 ¼ 0.75 when the diameter of bearingsurface is larger than the outsidediameter of Geneva wheel (24-304c)

P¼ Mt!

75 103 Customary Metric ð24-305cÞwhere P in hpm, Mtin kgf mm, and! in rad/s

MISCELLANEOUS MACHINE ELEMENTS 24.87

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The average power required on the crank or driving

shaft

Calculation of maximum power

The maximum torque on the driven shaft of Geneva

FIGURE 24-57 Forces acting on Geneva wheel.

The maximum power required on the shaft of the

FORCES AT THE POINT OF CONTACT

(Fig 24-57)

The maximum force at the point of contact between

the roller pin and slotted Geneva wheel

The component of maximum friction force at the

point of contact due to the friction torque M2t on

the driven Geneva wheel shaft

For maximum values of F2i

For design data for external Geneva mechanism

INTERNAL GENEVA WHEEL

The time required for indexing Geneva wheel, s

The time during which Geneva wheel is at rest, s

The ti=t ratio

The tr=t ratio

The working time coefficient of Geneva wheel

The relationship between crank or driver angle and

Geneva wheel angle

Refer to Table 24-25A

ti¼zþ 2z

602n



ð24-314Þ

tr¼z
2z

602n

MISCELLANEOUS MACHINE ELEMENTS 24.89

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The angular velocity of Geneva wheel

The maximum angular velocity of Geneva wheel

The angular acceleration, 2a, of Geneva wheel

For values of 2aat start and finish of indexing

For curves of angular velocity and angular

accelera-tion of internal Geneva wheel

The contact forces between the slotted wheel and the

pin on the driving crank of the internal Geneva wheel

are calculated in a manner similar to that for the

external Geneva wheel

Materials

Chromium steel 15 Cr65case-hardened to Rc58 to 65

is used for the roller pin on the driver or crank

Chromium steel 40 Cr 1 hardened and tempered to Rc

45 to 55 is used for the sides of slotted Geneva wheel

0 0.2 0.4 0.6 0.8

FIGURE 24-58 Angular velocity and angular acceleration for four-slot internal Geneva wheel.

24.10 UNIVERSAL JOINT

SYMBOLS2,3

Kct correction factor to be applied to torque to be transmitted

Kct correction factor to be applied to power to be transmitted

l length (also with subscripts), m (in)

Mt torque to be transmitted by universal joint, N m (lbf in)

Mtd design torque, N m (lbf in)

P power to be transmitted by universal joint, kW (hp)

Pd design power, kW (hp)

angle between two intersecting shafts 1 and 2, deg

angle of rotation of the driver shaft 1, deg

 angle of rotation of the driven shaft 2, deg

!1,!2 angular velocities of driver and driven shafts respectively, rad/s

SINGLE UNIVERSAL JOINT (Figs 24-59 and

24-61a)

The relation between

The relation between the angular velocities of driving

shaft 1 or driver (!1) to the driven shaft 2 or the

ω2

r β

FIGURE 24-59 A single universal joint.

MISCELLANEOUS MACHINE ELEMENTS 24.91

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The maximum value of!2=!1

The minimum value of!2=!1

The angular acceleration of the driven shaft 2, if!1is

constant

The value of
for which the angular acceleration of

the driven shaft is maximum

The power transmitted by universal joint

The design torque of universal joint

The design power of universal joint

For calculation of torque and power transmitted by

universal joint for various angles of inclination

For design data of universal joint

DOUBLE UNIVERSAL JOINT (Figs 24-60

and 24-61b)

The angular velocities ratio for a double universal

joint which will produce a uniform velocity ratio at

all times between the input and output ends

d2

dt2 ¼d!2

dt ¼ cos 2ð1
sin2
sin2 2!2 ð24-327Þcos 2
ðmaxÞ¼ 
pffiffiffiffiffiffiffiffiffiffiffiffiffi2þ 2 ð24-328Þwhere ¼ ð2
sin2 2

The angular acceleration of driven shaft is maximumwhen
is approximately equal to 458, 1358, etc., whenthe arms of cross are inclined at 458 to the planecontaining the axes of the two shafts

(b) Double universal joint

di = 10 to 50 (a) Single universal joint di = 6 to 50

FIGURE 24-62 Angle between two intersecting

shafts vs correction factor (Kct).

Angle of inclination (β), deg 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

d i×d o

325 rpm

FIGURE 24-65(a) Design curves for single universal joint with plain bearings for

MISCELLANEOUS MACHINE ELEMENTS 24.95

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USE OF CURVES IN FIGS 24-62 TO 24-65

Worked example 1

A single universal joint has to transmit a torque of

10 kgf m at 1500 rpm The angle between intersectingshafts is 258 The joint is subjected to a minorshock The shock factorðKsÞ is 1.5 Design a universaljoint with needle bearings for a life of 800 h.SOLUTION From Fig 24-62 correction factor for

is Kct¼ 1:2 Design torque¼ Mtd

¼ MtKsKct¼ 10  1:5  1:2 ¼ 18 kgf m (176.5 N m).Speed life ¼ nL ¼ 1500  800 ¼ 120  104rpm h.From Fig 24-64 for Mtd¼ 18 kgf m (176.5 N m) and

nL¼ 120  104rpm h, the size of a single universaljoint isðdi doÞ 40  75 mm

Worked example 2

Design a single universal joint with plain bearings totransmit 2 kW power at 325 rpm The angle betweentwo intersecting shafts is 27.58

SOLUTION From Fig 24-63 correction factorfor CN¼ 0:35 Design power ¼ Pd

¼ ðP=KCNÞ ¼ ð2=0:35Þ ¼ 5:7 kW From Fig 24-65athe size of a single universal joint for Pd¼ 5:7 kWand speed¼ n ¼ 325 rpm is ðdi doÞ 40  75 mm.The permissible torque for this size of joint(Fig 24-65a) is 17.5 kgf m (171.5 N m)

Maximum allowable rotational play Test torque Angular rotational play Tolerance on

FIGURE 24-65(b) Taper pin joint The length of the taper

pin should conform to diameter doin Table 24-25B.

24.11 UNSYMMETRICAL BENDING AND

TORSION OF NONCIRCULAR CROSS-SECTION

MACHINE ELEMENTS

SYMBOLS2,3

a semimajor axis of elliptical section, m (in)

width of rectangular section, m (in) (in2)

A area of cross section, m2(in)

b semi-minor axis of elliptical section, m (in)

height of rectangular section, m (in)

c distance of the plane from neutral axis, m (in)

thickness of narrow rectangular cross section (Fig 24-68)

e the distance from a point in the shear center S (Table 24-26)

G modulus of rigidity, GPa (MPsi)

I moment of inertia, area (also with suffixes), m4(cm4) (in4)

Iu, Iv moment of inertia of cross-sectional area, respectively, m4

(cm4) (in4)

Jk polar moment of inertia, m4(cm4) (in4)

k1, k2 constants from Table 24-28 for use in Eqs 343) and

(24-344)

Mb bending moment, N m (lbf ft)

Mt twisting moment, N m (lbf ft)

Mbu¼ Mbcos
bending moment about the U principal centroidal axis or any

axis parallel thereto

Mbv¼ Mbsin
bending moment about the V principal centroidal axis or any

axis parallel thereto

Q the first moment of the section, m4(cm4) (in4)

S the length of the center of the ring section of the thin tube, m

(in)

t width of cross section at the plane in which it is desired to find

the shear stress, m (in)thickness of the wall of the thin-walled section, m

u, v coordinates of any point in the section with reference to

principal centroidal axes

V shear force on the cross section, kN (lbf)

Vy resultant shear force acting at the shear center, kN (lbf)

x the distance of the section considered from the fixed end (Fig

24-73)

x, y coordinates in x and y directions

b bending stress (also with suffixes), MPa (psi)

 shear stress (also with suffixes), MPa (psi)

variable thickness of thin tube wall (Fig 24-70), m (in)

angle measured from the V principal centroidal axis, deg

MISCELLANEOUS MACHINE ELEMENTS 24.97

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For the equations for locating the shear centers of

various thin open sections

ELLIPTICAL CROSS SECTION

Shear stress acting in the x direction on the xz plane

(Fig 24-66)

Shear stress acting in the y direction on the yz plane

(Fig 24-66)

Maximum shear stress on the periphery at the

extremities of the minor axis (Fig 24-66 and Table

24-27)

Minimum shear stress on the periphery at the

extremities of the major axis

Angle of twist (Fig 24-66)

lRECTANGULAR CROSS SECTION

The maximum shear stress at point A on the

boundary, close to the center (Fig 24-67 and Table

24-27)

Angle of twist (Table 24-27)

NARROW RECTANGULAR CROSS

SECTIONS (Fig 24-68)

Equation for twisting moment (Fig 24-68)

Equation for angle of twist

The maximum shear stress

x y

τmaxb

FIGURE 24-67 MISCELLANEOUS MACHINE ELEMENTS 24.99

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TABLE 24-26

Location of shear center for various cross sections

Section Location of shear center Section Location of shear center

vy

t f

t f e

e ¼ðtw=tfÞh þ 6ðb13ðb2
b2Þþ b2Þfor b2< b1

b1

h1
43

eb

x t1 t2

b1

h1
43

h1

h2

 2

1 þ16

y

s e b

2

4h1 a

 3 b a

n ¼ 3 þ 12



b þ h1 a



þ 4



h1a

2

3 þh1 a



c h

L = Length of Dotted line s

e

x

tb th

vy y

e ¼hþ Lðth=tbÞ2Awhere A ¼ area

C is at the centroid of triangle

e ¼ 0:47a for narrow triangle

ð > 128Þ approx.

TABLE 24-26 (Cont.)

Location of shear center for various cross sections

Section Location of shear center Section Location of shear center

b2 e2

d

C t h

ht 3

w þ tf b 3



c ¼2hh2ðb1þ 2b1þ b2Þh ; d ¼2hbðb12þ bþ b2Þ2

I x; Iy ¼ moment of inertia of section about

x and y axes, respectively

Ixy¼ product of moment of inertia

t 3 dx

MISCELLANEOUS MACHINE ELEMENTS 24.101

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TABLE 24-27

Approximate formulas for torsional shearing stress and angle of twist for various cross sections

Shearing stress, lbf/in2 Angle of twist per unit length ,