Machine Design Databook 2010 Part 14 docx

24-57 F2 the component of force acting on the driven Geneva wheel shaft due to the torque M2t, kN lbf Fig.. 24-57 F2ðmaxÞ maximum force pressure at the point of contact between the roll

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

24.9 GENEVA MECHANISM

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)

F2ðmaxÞ maximum force (pressure) at the point of contact between the

roller pin and slotted Geneva wheel, kN (lbf )

FðmaxÞ the component of maximum friction force at the point of

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

FiðmaxÞ the component of maximum inertia force at the point of contact

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

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

!2ðmaxÞ¼

ddt

max

1
!1

¼

sinz



1
sinz

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!

2ð24-284bÞRefer to Fig 24-56

cos ðmaxÞ¼
 þpffiffiffiffiffiffiffiffiffiffiffiffiffi2þ 2 ð24-284cÞwhere

 ¼14

0 0

10

20

30

2 4 6

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

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

Pav¼MtðavÞn

where Pavin hp, MtðavÞin lbf in, and n in rpm

M2tðmaxÞ¼ M2tþ M2tiðmaxÞ ð24-309Þwhere M2tis constant

M2tiðmaxÞ¼ J 2aðmaxÞ¼J 2aðmaxÞ

!2



2n60

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

¼r1



ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1
2 cos þ 2q

Refer to Table 24-25A

ti¼zþ 2z

602n



ð24-314Þ

tr¼z
2z

602n

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

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

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

!2

!1

max¼ cos

ð24-326Þwhen sin¼ 0, i.e.,
¼ 0, , 2, etc

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

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

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

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)

ða2þ b2Þ

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

b

FIGURE 24-67

TABLE 24-26

Location of shear center for various cross sections

vy

t f

t f e

x t1 t2

L = Length of Dotted line s

e

x tb th

vy y

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

e 1 vx

d

C t h

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 ,

Equation for torque for a cross-section composed of

several narrow rectangles (Table 24-27)

Angle of twist for a cross section composed of several

narrow rectangles

Maximum shear stress

For approximate formulas for torsional shearing

stress and angle of twist for various cross sections

For variation of stress-concentration factor K with

ratio r=c for structural angle (Fig 24-69)

HOLLOW THIN-WALLED TUBES

(Fig 24-70)

The equation for the twisting moment

The angle of twist

By membrane analogy the value ofÞ

 ds

The equation for the shear stress

The difference in level between DC and AB of

membrane

The equation for twisting moment of thin webbed

tubes (or box beams) (Fig 24-71)

The equations for shear stress

x

S 0

FIGURE 24-70

BENDING STRESSES CAUSED BY

TORSION

Torsion of I-beam having one section

restrained from warping

The lateral bending moment in the flanges of an

I-beam subjected to twisting moment at one end, the

other end being fixed, Fig 24-72

The maximum bending moment for long beam

Twisting moment at any section, distance x from the

fixed end

The angle of twist per unit length

The total angle of twist at the free end

Maximum bending moment

C B

The angle of twist at free end if l=k > 2:5

The maximum bending moment if l=k > 2:5

The bending stress

For beams subjected to torsion

TRANSVERSE LOAD ON BEAM OF

CHANNEL SECTION NOT THROUGH

SHEAR CENTER (Fig 24-73)

t

F e

s Shear centre b

FIGURE 24-73

The direct stress

The bending stress

The maximum longitudinal stress (Fig 24-73b)

For geometrical properties, weight, and nominal

dimensions of beams, channels, T-bars, and equal

and unequal angles

t¼MI

Refer to Tables 24-31 and 24-32 and Figs 24-74 to24-79

TABLE 24-30

Formulas for maximum lateral bending moment and angle of twist of beams subjected to torsiona

Maximum lateral bending moment in Type of loading and support flange, lbf in (N m) Angle of twist of beam of length L,  rad

L

MbðmaxÞ¼Mt k

2h

 cot hL2k
2kL



 ¼Mt 2JG

 L

e w

k
Lk



 ¼Mt JG

 L

 ¼12Mt JG

 L

2
k tan h2kL

 ðapprox:Þ

a

Formulas given in Table 24-30 can also be used for Z and channel sections.

4 Lingaiah, K., Machine Design Data Handbook, McGraw-Hill Publishing Company, New York, 1994.

5 Maleev, V L., and J B Hartman, Machine Design, International Text book Company, Scranton,Pennsylvania, 1954

6 Black, P H., and O E Adams, Jr., Machine Design, McGraw-Hill Book Company Inc., New York, 1968

7 Neale, M J., Tribology Handbook, Newnes-Butterworth, London, 1973

8 Bach, Maschinenelemente, 12th ed., P-43

9 Heldt, P M Torque Converters for Transmissions, Chiltan Company, Philadelphia, 1955

10 Newton, K., and W Steeds, The Motor Vehicle, Iliffe and Sons Ltd, London, 1950

11 Steeds, W., Mechanics of Road Vehicle, Iliffe and Sons, Ltd., London, 1960

12 Arkhangelsky, V., et al., Motor Vehicles Engines, Mir Publisher, Moscow, 1971

13 Heldt, P M., High Speed Combustion Engines, 6th ed., Chilton Company, Philadelphia, 1955

14 Thimoshenko, S., and J N Goodier, Theory of Elasticity, McGraw-Hill Book Company, Inc., andKogakkusha Company Ltd., Tokyo, 1951

15 Timoshenko, S., and S Woinowsky-Krieger, Theory of Plates and Shells, McGraw-Hill Book Company, Inc.,New York, 1959

16 Seely, F B., and J O Smith, Advanced Mechanics of Materials, John Wiley and Sons, Inc., 2nd ed., 1959

17 Timoshenko, S., and J M Gere, Mechanics of Materials, Von Nostrand Reinhold Company, New York,1972

18 Goetze, A G Piston Ring Manual, 3rd ed., Burscheid, Germany, 1987

19 Bureau of Indian Standards, New Delhi, India

B width of V or U die, m (in)

c chisel edge length, m (in)

d diameter of the hole, m (in)

diameter of the drill, m (in)

depth of cut, m (in)

diameter of milling cutter, m (in)

shell diameter, m (in)

Dm diameter of machined surface, m (in)

Dw diameter of workpiece or job, m (in)

E Young’s modulus, GPa (kpsi)

work done in punching or shearing of material, N m (lbf ft)

work done per unit volume of removed, J/mm3

Fa¼ Fx axial component of cutting force, kN (lbf )

Fc cutting force, kN (lbf )

Fhc normal cutting force or the resultant force on a

single point metal cutting tool in the horizontal plane,

kN (lbf )

Fmax maximum force at the punch, kN (lbf )

Fn force normal to F
, kN (lbf )

Fn force normal to shear force, kN (lbf )

Fr reaction of the cutting force, kN (lbf )

radial component of the cutting force, kN(lbf )

FR resultant cutting force, kN (lbf )

Fs stripping pressure, kN (lbf )

Ff ¼ Fx feed force, kN (lbf )

Ft¼ Fz¼ Fc tangential component of the cutting force, kN(lbf)

Fy¼ Fr radial component of the cutting force, kN (lbf )

F
frictional force on tool face, kN (lbf )

F¼
Fr frictional force of the saddle on lathe bed, kN (lbf )

F shear force kN (lbf )

shell height, m (in)

hc height of the lathe center, m (in)

hsw swing over the bed of the lathe, m (in)

K constant of proportionality in Eq (25-31)

coefficient, also with subscripts

L length of cut or perimeter of the cut, m (in)

Lm length of job, m (in)

Pc power at cutting tool, kW (hp)

Pg gross or motor power, kW (hp)

Pf tare power, that is the power required to run the machine at no

load, kW (hp)

Pu¼ Ps unit power or specific power, kW (hp)

Q metal removal rate, cm3/min (in3/min)

rc corner radius, m (in)

rc¼ ðt1=t2Þ cutting ratio

rn nose radius, m (in)

R roughness height,m m (m in)

Ri inside radius of bend, m (in)

s feed rate, mm/rev (in/rev)

sz feed per tooth of milling cutter, mm/tooth

t thickness of material to be punched or sheared, m (in)

thickness of chip, m (in)

depth of cut, m (in)

t1 initial thickness of the chip, m (in)

t2 final thickness of the chip, m (in)

uc number of chamfered threads

v velocity, m/s (ft/min)

vc cutting speed, m/min (ft/min)

vf feed rate, mm/min (in/min)

V cutting velocity, m/min (ft/min)

W work done per unit volume, N m/m3(lbf ft/ft3)

x, v, z machine reference co-ordinate axes

z number of teeth on milling cutter

 relief or clearance angle, deg

f side relief or clearance angle, deg

n normal relief or clearance angle, deg

o orthogonal clearance angle, deg

p front or end relief or clearance angle, deg

tr true relief or clearance angle, deg

 helix angle, deg

 rake angle, also with subscripts, deg

f side rake angle, deg

n normal rake angle, deg

o orthogonal rake angle, deg

p back rake angle, deg

gullet angle, deg

" peripheral pitch angle, deg

wedge angle, also with subscripts, deg

chip flow angle, deg

approach angle in Eq (25-104b), deg

c corner angle in Eq (25-104a)

s inclination angle, deg

coefficient of friction

stress, also with suffices, MPa (kpsi)

c compressive stress, MPa (kpsi)

sut ultimate tensile strength, MPa (kpsi)

y yield stress, MPa (kpsi)

shear stress, MPa (kpsi)

nð¼ suÞ resistance of the material to shearing or the ultimate shearing

strength, MPa (kpsi)cutting edge angle, also with subscripts, deg

shear angle, deg

o end cutting edge angle, deg

p principal cutting edge angle, deg

s side cutting edge angle, deg

engagement angle for milling depth, deg

Note: and  with subscript s designates strength properties of material used in

the design which will be used and observed throughout this Machine Design Data

Handbook

Other factors in performance or in special aspects are included from time to

time in this chapter and, being applicable, only in their immediate context, are

not given at this stage

25.1 METAL CUTTING TOOL DESIGN

25.1.1 Forces on a single-point metal cutting

tool

Nomenclature of metal cutting tool

The normal cutting force or resultant force on a

single-point metal cutting tool in horizontal plane

q

ð25-1Þ

FR¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiF2

t þ F2 hc

Face Major flank

Major cutting edge

FIGURE 25-1 Nomenclature of metal cutting tool.

F r (=F y ) = radial force

F t or F c (=F x ) = tangential cutting force

F R = resultant cutting force on the cutting tool

D w

D m x

Note: and  with subscript s designates strength properties of material used in

the design which will be used and observed throughout this Machine Design Data

Handbook

Other... class="text_page_counter">Trang 40

25.1 METAL CUTTING TOOL DESIGN< /h3>

25.1.1 Forces on a single-point metal cutting

tool

Nomenclature