Machine Design Databook Episode 2 part 7 pdf

The actual factor of safety or reliability factorThe wire diameter for static loading The wire diameter where there is no space limitation ðD ¼ cdÞ ds¼ e naksz¼1:89e nad0:25 Metric ð20-4

The general expression for size factor

Wire diameter

SELECTION OF MATERIALS AND

STRESSES FOR SPRINGS

For materials for springs7

The torsional yield strength

The maximum allowable torsional stress for static

applications according to Joerres8;9;11

The maximum allowable torsional stress according to

Shigley and Mischke9

The shear endurance limit according to Zimmerli10

The torsional modulus of rupture

0:45sut cold-drawn carbon steel

0:50sut hardened and tempered

carbon and low-alloy steel

0:35sut austenitic stainless steel

and nonferrous alloys

sf ¼ 310 MPa ð45 kpsiÞ ð20-47dÞfor unpeened springs

sf ¼ 465 MPa ð67:5 kpsiÞ ð20-47eÞfor peened springs

The weight of the active coil of a helical spring

For free-length tolerances, coil diameter tolerances,

and load tolerances of helical compression springs

DESIGN OF HELICAL COMPRESSION SPRINGS

Design stress

The size factor

The design stress

whereein psi and d in in

Tolerances: mm/mm (in/in) of free length

Spring index (D=d) Number of active

Tolerance: % of load, start with tolerance from Table 20-11 multiplied by LF

Deflection from free length to load, mm (in) Length

First load test at not less than 15% of available deflection; final load test at not more than 85% of available deflection.

Source: Associated Spring, Barnes Group Inc., Bristol, Connecticut.

Source: K Lingaiah and B R Narayana Iyengar, Machine Design Data Handbook, Vol I, Suma Publishers, Bangalore, India, 1986, and K Lingaiah, Machine Design Data Handbook, Vol 11, Suma Publishers, Bangalore, India, 1986.

The actual factor of safety or reliability factor

The wire diameter for static loading

The wire diameter where there is no space limitation

ðD ¼ cdÞ

ds¼ e

naksz¼1:89e

nad0:25 Metric ð20-49cÞwhereein kgf/mm2and d in mm

where na¼ actual factor of safety or reliabilityfactor

na¼FðcompressedÞ

na¼ free length
fully compressed length

free length
working length

¼y þ a

where y is deflection under working load, m (mm),

a is the clearance which is to be added whendetermining the free length of the spring and

is made equal to 25% of the workingdeflection

Generally nais chosen at 1.25

d ¼ 1:445

6naF

d ¼

6naF

Final dimensions (Fig 20-7d)

The number of active coils

The minimum free length of the spring

Outside diameter of cod of helical spring

Solid length (or height) of helical spring

Pitch of spring

Free length of helical spring lf or lo

Maximum working length of helical spring

Minimum working length of helical spring

Springs with different types of ends1;2;3

STABILITY OF HELICAL SPRINGS

The critical axial load that can cause buckling

i ¼yd4G8FD3¼ydG8Fc3¼kydG

where

a ¼ clearance, m (mm)

n ¼ 2 if ends are bent before grinding

¼ 1 if ends are either ground or bent

¼ 0 if ends are neither ground nor bent

where Klis factor taken from Fig 20-8

springs (V L Maleev and J B Hartman, Machine Design, International Textbook Company, Scranton, Pennsylvania, 1954.)

The equivalent stiffness of springs

The critical load on the spring

The critical deflection is explicitly given by

REPEATED LOADING (Fig 20-9)

The variable shear stress amplitude

The mean shear stress

Design equations for repeated loadings1;2;3

1þ v

2þ v

D

Narayana Iyengar, Machine Design Data Handbook, Engineering College Cooperative Society, Bangalore, India, 1962; K Lingaiah and B R Narayana Iyengar, Machine Design Data Handbook, Vol I, Suma Publishers, 1986; K Lingaiah, Machine Design Data Handbook, Vol II, Suma Publishers, Bangalore, India, 1986.)

The Goodman straight-line relation

The Soderberg straight-line relation

Method 2

The static equivalent of cyclic load Fm Fa

The relation betweeneandf for brittle material

The static equivalent of cyclic load for brittle material

The relation between Fm0, Fmaxand Fmin

The diameter of wire for static equivalent load

The wire diameter when there is no space limitation

e

0:4

D0:3 SI ð20-67aÞwhere F in N, ein MPa, D in m, and d in m

d ¼

3nað3Fmax
FminÞ

e

0:4

D0:3 USCS ð20-67bÞwhere F in lbf, ein psi, D in in, and d in in

d ¼ 0:724

3nað3Fmax
FminÞ

e

0:4

D0:3Metric ð20-67cÞwhere F in kgf, ein kgf/mm2, D in mm, and d inmm

d ¼ 1:67

3nað3Fmax
FminÞ

e

0:57

c0:43 SI ð20-68aÞwhere F in N, ein MPa, and d in m

d ¼

3nað3Fmax
FminÞ

e

0:57

c0:43 USCS ð20-68bÞwhere F in lbf, ein psi, and d in in

d ¼ 0:64

3nað3Fmax
FminÞ

e

0:57

c0:43Metric ð20-68cÞwhere F in kgf, ein kgf/mm2, and d in mm

CONCENTRIC SPRINGS (Fig 20-10)

The relation between the respective loads shared by

each spring, when both the springs are of the same

length

The relation between the respective loads shared by

each spring, when both are stressed to the same value

The approximate relation between the sizes of two

concentric springs wound from round wire of the

same material

Total load on concentric springs

The total maximum load on the spring

The load on the inner spring

The load on the outer spring

VIBRATION OF HELICAL SPRINGS

The natural frequency of a spring when one end of the

r

¼ 0:705

ffiffiffiffiffiffi

k0W

r

SI ð20-75Þwhere

fn¼ natural frequency, Hz

W ¼ weight of vibrating system, N

k0¼ scale of spring, N/m

g ¼ 9:8066 m=s2

The natural frequency of a spring when both ends are

fixed

The natural frequency for a helical compression

spring one end against a flat plate and free at the

other end according to Wolford and Smith7

Another form of equation for natural frequency of

compression helical spring with both ends fixed

with-out damping effect

fn¼ 22:3

k0W

1=2

SI ð20-75aÞwhere k0in N/mm, W in N, fnin Hz,

g ¼ 9086:6 mm=s2

fn¼ 4:42

k0W

1=2

USCS ð20-75bÞwhere k0in lbf/in, W in lbf, fnin Hz, g ¼ 32:2 ft=s2

fn¼ 1:28

k0W

r

¼ 1:41

ffiffiffiffiffiffi

k0W

r

SI ð20-76Þwhere k0in N/m, W in N, fnin Hz,

g ¼ 9:0866 mm=s2

fn¼ 44:6

k0W

1=2

SI ð20-76aÞwhere k0in N/mm, W in N, fnin Hz,

g ¼ 9086:6 mm=s2

fn¼ 2:56

k0W

1=2

USCS ð20-76bÞwhere k0in lb/ft, W in lbf, fnin Hz, g ¼ 32:2 ft=s2

fn¼ 8:84

k0W

1=2

USCS ð20-76cÞwhere k0in lbf/in, W in lbf, fnin Hz,

g ¼ 386:4 in=s2

fn¼ 0:25

k0gW



1=2

SI ð20-76eÞwhere

G ¼ shear modulus, MPa

g ¼ 9:8006 m=s2

d and D in mm, fnin Hz, in g/cm3

fn¼3:5ð105Þd

STRESS WAVE PROPAGATION IN

CYLINDRICAL SPRINGS UNDER IMPACT

LOAD

The velocity of torsional stress wave in helical

com-pression springs

The velocity of surge wave (Vs)

The impact velocity (Vimp)

The frequency of vibration of valve spring per minute

fn¼0:11d

D2i

Gg



1=2

USCS ð20-76gÞwhere

G ¼ modulus of rigidity, psi



1=2

SI ð20-76iÞwhere Vin m/s, G in MPa, g ¼ 9:8066 m=s2, ing/cm3

V¼

Gg



1=2

USCS ð20-76jÞwhere Vin in/s, G in psi, g ¼ 386:4 in=s2, inlbf=in3

(It varies from 50 to 500 m/s.)

Vimp¼ 10:1

g

r

SI ð20-77aÞwhere k0in N/m, W in N

fn¼ 2676:12

ffiffiffiffiffiffi

k0W

r

Metric ð20-77bÞwhere k0in kgf/mm, W in kgf

fn¼ 530

ffiffiffiffiffiffi

k0W

r

USCS ð20-77cÞwhere k0in lbf/in, W in lbf

HELICAL EXTENSION SPRINGS (Fig 20-11

to 20-13)

For typical ends of extension helical springs

The maximum stress in bending at point A (Fig

20-12)

The constant K1in Eq (20-78a)

The constant C1in Eq (20-78b)

Cross center loop

or hook

I.D.

required by design

to inside of end ID is inside diameter of adjacent coil in spring body (Associated Spring, Barnes Group, Inc.)

in twist loops (Associated Spring, Barnes Group, Inc.)

The maximum stress in torsion at point B (Fig 20-12)

The constant C2in Eq (20-78d)

For extension helical spring dimensions

For design equations of extension helical springs

The spring rate

The stress

CONICAL SPRINGS [Fig 20-14(a)]

The axial deflection y for i coils of round stock may be

computed by the relation [Fig 20-14(a)]

The axial deflection of a conical spring made of

rectangular stock with radial thickness b and an

axial dimension h [Fig 20-14(c)]

For R1, refer to Fig 20-12

B¼8DF d3 4C2
1

C2¼2R2

For R2, refer to Fig 10-12

In practice C2may be taken greater than 4

NONMETALLIC SPRINGS

Rectangular rubber spring (Fig 20-15)

Approximate overall dimension of the shock absorber

can be obtained by (Fig 20-15)

Spring constant K of an absorber

Dimensions of sleeve and core are found by empirical

relations

TORSION SPRINGS (Fig 20-16)7

The maximum stress in torsion spring

The stress in torsion spring taking into consideration

the correction factor k0

The deflection

The stress in round wire spring

L

D2¼
E2F2

U

The stress is also given by Eq (20-90) without taking

into consideration the direct stress (F/A)

The expressions for k for use in Eq (20-90)

Equation (20-90) for stress becomes

The angular deflection in radians

The spring rate of torsion spring

The spring rate can also be expressed by Eq (20-95),

which gives good results

The allowable tensile stress for torsion springs

The endurance limit for torsion springs

Torsion spring of rectangular cross section

The stress in rectangular wire spring

Axial dimension b after keystoning

Another expression for stress for rectangular

cross-sectional wire torsion spring without taking into

consideration the direct stress ( ¼ F=A)

The spring rate

Torsion bar springs

For allowable working stresses for rubber

compres-sion springs

sy¼ a¼

0:78sut cold-drawn carbon steel

0:87sut hardened and tempered

carbon and low-alloysteels

0:61sut stainless steel

and nonferrous alloys

Cross section of

bar

Angular

Suggested allowable working stresses for rubber compression springs

Limits of allowable stress

1 Lingaiah, K and B R Narayana Iyengar, Machine Design Data Handbook, Engineering College operative Society, Bangalore, India, 1962.

Co-2 Lingaiah, K., and B R Narayana Iyengar, Machine Design Data Handbook, Vol I (SI and Customary MetricUnits), Suma Publishers, Bangalore, India, 1986

3 Lingaiah, K., Machine Design Data Handbook, Vol II (SI and Customary Metric Units), Suma Publishers,Bangalore, India, 1986

4 SAE Handbook, Springs, Vol I, 1981

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

6 Wahl, A M., Mechanical Springs, McGraw-Hill Book Company, New York, 1963

7 Associated Spring, Barnes Group Inc., Bristol, CT, USA

8 Jorres, R E., Springs; Chap 24 in J E Shigley and C R Mischke, eds., Standard Handbook of MachineDesign, McGraw-Hill Book Company, New York, 1986

9 Shigley, J E., and C R Mischke, Mechanical Engineering Design, 5th ed McGraw-Hill Company, New York,1989

10 Zimmerli, F P., Human Failures in Springs Applications, The Mainspring, No 17, Associated SpringCorporation, Bristol, Connecticut, Aug.-Sept 1957

11 Shigley, J E., and C R Mischke, Standard Handbook of Machine Design, McGraw-Hill Book Company, NewYork, 1986

12 Phelan, R M., Fundamentals of Mechanical Design, Tata-McGraw-Hill Publishing Company Ltd, New Delhi,1975

13 Lingaiah, K., Machine Design Data Handbook of Machine Design, 2nd edition, McGraw-Hill PublishingCompany, New York, 1996)

14 Shigley, J E., and C R Mischke, Standard Handbook of Machine Design, 2nd edition, McGraw-HillPublishing Company, New York, 1996

Chironis, N P., Spring Design and Application, McGraw-Hill Book Company, 1961

Norman, C A., E S Ault, and I F Zarobsky, Fundamentals of Machine Design, The Macmillan Company, NewYork, 1951

Shigley, J E., Machine Design, McGraw-Hill Book Company, 1962

21

FLEXIBLE MACHINE ELEMENTS

SYMBOLS11;12;13

a width of pulley face, m (in)

pivot arm length in Rockwood drive, m (in)

a1 width of belt, m (in)

A ¼ 0:4ð
d2=4Þ useful area of cross-section of the wire rope, m2(in2)

dimension in Rockwood drive (Fig 21-5), m (in)

c dimension in Rockwood drive (Fig 21-5), m (in)

C center distance between sprockets (also with suffixes), m (in)

center distance between pulleys, m (in)capacity of conveyor, m3(ft3)

constant depends on the rope diameter, sheave diameter, chain,the bearing, and coefficient of friction [Eqs (21-59) to (21-62)and (21-86) to (21-103)] (also with suffixes)

C1 tooth width in precision roller and bush chains, m (in)

diameter of shaft, m (in)diameter of idler bearing, m (in)diameter of smaller pulley, m (in)diameter of rope, m (in)

pitch diameter of sprocket, m (in)

d1 diameter of small sprocket, m (in)

hub diameter of pulley, m (in)

d2 diameter of large sprocket, m (in)

da tip diameter of sprocket, m (in)

da1 tip diameter of small sprocket, m (in)

da2 tip diameter of large sprocket, m (in)

dc¼ fpFb equivalent pitch diameter, m (in)

df root diameter of sprocket, m (in)

dp pitch diameter of the V-belt small pulley, m (in)

dr diameter of roller pin, m (in)

D pitch diameter of sheave, m (in)

diameter of large pulley, m (in)wire rope drum diameter, m (in) (Fig 21-4)

Dr diameter of reel barrel, m (in) Eq (21-76)

Dd diameter of the drum in mm as measured over the outermost

layer filling the reel drum

E0 corrected elasticity modulus of steel ropes

(78.5 GPa¼ 11.4 Mpsi), GPa (psi)

tension in belt, kN (lbf )minimum tooth side radius, m (in)

Fa correction factor for instructional belt service from Table 21-27

Fc correction factor for belt length from Table 21-26

Fct centrifugal tension, kN (lbf )

Fd correction factor for arc of contact of belt from Table 21-25

F tangential force in the belt, required chain pull, kN (lbf )

Fs tension due to sagging of chain, kN (lbf )

F1 tension in belt on tight side, kN (lbf )

F2 tension in belt on slack side, kN (lbf )

Fc centrifugal force, kN (lbf )

values of coefficient for manila rope, Table 21-32

FR1 the minimum value of tooth flank radius in roller and bush

chains, m (in)

FR2 the maximum value of tooth flank radius in roller and bush

chains, m (in)

g acceleration due to gravity, 9.8066 m/s2(32.2 ft/s2)

G tooth side relief in bush and roller chain, m (in)

h the thickness of wall of rope drum, m (in)

crown height, m (in)

h1 depth of groove in rope drum, m (in)

H ¼ ðDd
DrÞ=2 depth of rope layer in reel drum, m (in)

i number of arms in the pulley,

number of V-belts,number of strands in a chain,transmission ratio

k ¼ ðe
1Þ=e variable in Eqs (21-2d), (21-4a), (21-6), and (21-123), which

ksg coefficient for sag from Table 21-55

l width of chain or length of roller, m (in)

minimum length of boss of pulley, m (in)minimum length of bore of pulley, m (in)length of conveyor belt, m (in)

length of cast-iron wire rope drum, m (in)outside length of coil link chain, m (in)

K1 tooth correction factor for use in Eq (21-116a)

K2 multistrand factor for use in Eq (21-116a)

L length of flat belt, m (in)

pitch length of V-belt, m (in)rope capacity of wire rope reel, m (in)

Lp length of chain in pitches

n number of times a rope passes over a sheave,

number of turns on the drum for one rope memberspeed, rpm

factor of safety

n2 speed of larger pulley, rpm or rps

speed of larger sprocket, rpm or rps

n0¼ nkd stress factor

PT power required by tripper, kW (hp)

pitch of the grooves on the wire rope drum, m (in)

p1 distance between the grooves of two-rope pulley, m (in)

s the amount of shift of the line of action of the load from the

center line on the raising load side of sheave, m (in)

s the average shift of the center line in the load on the effort side

of the sheave, m (in)

S the distance through which the load is raised, m (in)

SA1 the minimum value of roller or bush seating angle, deg

SA2 the maximum value of roller or bush seating angle, deg

SR1 the minimum value of roller or bush seating radius, m (in)

SR2 the maximum value of roller or bush seating radius, m (in)

t nominal belt thickness, m (in)

thickness of rim, m (in)

T tension in ropes, chains, kN (lbf )

TDmin minimum limit of the tooth top diameter, m (in)

TDmax maximum limit of the tooth top diameter, m (in)

v velocity of belt chain, m/s (ft/min)

w specific weight of belt, kN/m3(lbf/in3)

W width between reel drum flanges, m (in)

WB weight of belt, kN/m (lbf/in)

wc weight of chain, kN/m (lbf/in)

WI weight of revolving idler, kN/m (lbf/in) belt

z1 number of teeth on the small sprocket

z2 number of teeth on the large sprocket

1 unit tension in belt on tight side, MPa (psi)

2 unit tension in belt on slack side, MPa (psi)

c centrifugal force coefficient for leather belt, MPa (psi)

br breaking stress for hemp rope, MPa (psi)

 angle between tangent to the sprocket pitch circle and the

center line, deg

 coefficient of friction between belt and pulley

coefficient of journal friction

c coefficient of chain friction