Chain Drives

Notations:

P: pitch of chain (m)D: pitch circle diameter (m)D1: Pitch circle of smallest sprocket D2: Pitch circle of larger sprocketD0: Sprocket outside diameterD1:diameter of chain rollerK: Number of chain linksKs: service factorFt: tangential driving forceFc: Centrifugal tension in the chainFs: Tension in the chain due to saggingL: Length of chainm:mass of chain in kg per metren: factor of safetyN1: speed of rotation of smaller sprocket (rpm )N2:speed of rotation of larger sprocketP: Power transmitted by chainT: Number of teeth on the sprocketWb : Breaking strength of chain

θ : angle subtended by one pitch length at the centre of sprocketv : Average velocity of chain (m/s) x : Centre distance between sprockets(m)σ b: Allowable bearing stress in MPa of N/mm²A : Projected bearing area (mm²)re : Tooth flank radiusri : Roller seating radiusα : Roller seating angleha: Tooth height above the pitch polygonDa: Top diameter

Df : Root Diameter

b f 1: tooth widthrx: Tooth side radius ba: Tooth side reliefb f 1∧b f 2:width over teethT1:Number of teeth on smaller sprocketT2:Number of teeth on larger sprocketW: Total load on the driving side of chain

Formula’s for chain drives

1. θ=3600

T

2. p=D sin θ2=D sin 180

T3. D0=D+0.8d1

4. Angleof articulation=θ2

5. Velocity ratio ,V .R .=N 1

N 2

=T 2T 1

6. v=πDN60

=TpN60

7.

L=K . p ;Where K=T1+T 22

+ 2xp

+[ T1+T 22π ]2px

8.

x= p4 [K−

T 1+T22

+√{K−T 1+T22 }

2

−8[{T 2−T 12π }2]]

For velocity transmission ratio of 3,

xmin=D1+D22

+30¿50mm(For best results ,

Rim Profile of sprocket

min distance centre distance should be 50 times the pitch.

Factor of safety

9. n = W b

W10.

W b=106 p2N for roller chains;W b=106 p N permmwidth of chain for silent chain ; p∈mm

11.FT=

Power transmitted (watts)

Speed of chain(ms)

=Pv Newtons

12. F c=mv2newtons

13. F s=kmgx Newtons ;

K :constant which takes intoaccount the arrangementof chain drive .k=2¿6whencentre line of chainisinclined ¿the horizontal at anangleless than400

k=1¿1.5whenthe centre lineof the chaini s inclined¿ the horizontal at anangle greater than400

Tooth Profile Of sprocket

Power transmitted by chains

14. P= WB . v

n . K s

Where K s=service factor=K1K2 K3K1=Load Factor=1 , for constant load = 1.25, for variable load with mild shock = 1.5, for heavy shock loadsK2=Lubrication Factor¿0.8 , for continuous lubrication= 1, for drop lubrication= 1.5 for periodic lubricationK3= Rating factor

15. P= σb Av

K s

Number of teeth on smaller or driving sprocket

16. V max = π D1 N

60 (m/s) Where D1 is the pitch circle diameter of smaller sprocket

17. V min = π D1 N cos (θ/2 )

60 (m/s)

Principle Dimensions of tooth profile

18. re= 0.008d1(T2+180) [max] = 0.12 d1(T+2) [min]

19. r1 = 0.505 d1+0.069(d1)1/3 [max] = 0.505 d1 [min]

20. ∝=1400−900

T [max]

¿1200−90T

[min]

21. ha=0.625 p−0.5 d1+0.8 pT

[max]

= 0.5 (p-d1)

22. D= psin ¿¿¿

23. Da=D+1.25 p−d1[max] ¿D+ p¿

24. Dr=D−2 r125. bn=0.93b1where p≤12.7mm

¿0.95b1where p>12.7mm26. rx = p27. b a= 0.1p to 0.15p28. bf2 and bf3 = (Number of strands – 1)pt +

bf129. Design Power = Rated Power X Service

factor

30. Load (W) = Rated Power

Pitch line velocity

Factor of safety (n) for bush roller and silent chain: Table 14.38 Pg.287 Characteristics of roller chains according IS :2403 – 1991 : Pg 287 B ,table 14.40 a

Power Rating (in kW) of simple roller chain

Speed of smaller sprocket or pinion (rpm)

Power ( kW)06 B 08B 10B 12B 16B

100 0.25 0.64 1.18 2.01 4.83200 0.47 1.18 2.19 3.75 8.94300 0.61 1.70 3.15 5.43 13.06500 1.09 2.72 5.01 8.53 20.57700 1.48 3.66 6.71 11.63 27.731000 2.03 5.09 8.97 15.65 34.891400 2.73 6.81 11.67 18.15 38.471800 3.44 8.10 13.03 19.85 --2000 3.80 8.67 13.49 20.57 --

Permissible speed of smaller sprocket or pinion in r.p.m

Type of chain

Number of teeth on sprocket pinion

Pitch of chain (p) in mm12 15 20 25 30

Bush Roller chain

15 2300 1900 1350 1150 100019 2400 2000 1450 1200 105023 2500 2100 1500 1250 110027 2550 2150 1550 1300 110030 2600 2200 1550 1300 1100

Silent Chain 17-35 3300 2650 2200 1650 1300

Number of teeth on the smaller sprocket

Type of chain Number of teeth at velocity ratio1 2 3 4 5 6

Roller 31 27 25 23 21 17Silent 40 35 31 27 23 19

Maximum allowable speed for chains in r.p.m.

Type of chain Number of teeth

on the smaller sprocket (T1)

Chain pitch (p) in mm12 15 20 25 30

Roller Chain 15 2300 1900 1350 1150 110019 2300 2000 1450 1200 1050

23 2400 2100 1500 1250 110027 2550 2150 1550 1300 110030 2600 2200 1550 1300 1100

Silent Chain 17-35 3300 2650 2200 1650 1300

Thick Cylinders – Principal Stresses

Variation of principal stresses (cylinders with internal pressure)

σ r=−PiDi

2

(Do2−Di2 ) [ D o2

4 r2−1] ; σ t= +PiDi

2

(D o2−Di2 ) [ D o2

4 r2+1]

At the inner surface of the cylinder: r=Di2

, the stresses

are

σ r= - Pi ; σ t=+Pi (Do2+Di2 )

(Do2−D i2)

At the outer surface of the cylinder: r=Do2

, the stresses

are

σ r= 0 ; σ t=+2PiDi

2

(D o2−Di2 )

Variation of stresses (Cylinders with external Pressure)

σ r=−PoDo

2

(Do2−Di2 ) [1− Di2

4 r2 ] ;σ t=

−PoDo2

(D o2−Di2 ) [1+ D o2

4 r2 ]

At the inner surface of the cylinder , r = Di2

, stresses are

σ r=0 ; σ t=+2PoDi

2

(D o2−Di2 )

At the outer surface, r = D02

, stresses areσ r=−Po;σ t=

−Po (D o2+Di2 )(D o2−Di2 )

Compound Cylinders

δ j=D2 PE [ (D32+D22 )

(D32−D 22)+μ ]

δ c=−D2PE [ (D22+D12 )

(D22−D12 )−μ]

δ=δ j+δ cδ=PD 2

E [ 2D22 (D32−D12)

(D32−D22 ) (D 22−D1

2 ) ]Spur Gear

P=F t . v ; P is the Power ( kW) , v is the pitch line velocity ( m/s) , Ft is the tangential force (kN)Ft = FaCosα , Where α is the pressure angle, and Fn is the

normal force.Beam Strength: F beam= bmσ dY ; where 9.5 ≤ b ≤ 12.5m

Fmax=C sF tC v

; Fbeam ≥Fmax

Values of Y

For a 20o involute full depth tooth Y=π (0.154−0.912Z )For a 20o stub tooth Y=π (0.175−0.95Z )

For a 14.5o tooth Y=π (0.124−0.684Z )

Values of Service factor ( Cs) : refer data book page 187 ,table 12.20

Values of Velocity Factor ( Cv): refer data book page 164

Dynamic Load: refer data book page 166 and 167

F en=Fdynamic .FoS=bmσ enY ; Where σ en is the endurance limit ( = 17.5 BHN)

Gear Construction:

1. If the pitch diameter d≤ 14.8m + 60.0 mm , the pinion / gear should be of solid disc type.

2. If the pitch diameter d≥23.5m + 85.0 mm ,the gear to be constructed should be of

3. The thickness of the rim , h= (2 to 4) m . The rim should be tapered 1 : 5 towards the centre

4. The thickness of the stiffening rib, q= (1.0+1.25)h5. Hub diameter , dh= (1.6 to 2.0 ) ds

6. Hub length, lh = 2ds or at least equal to face width b.7. Number of arms (j)

j=4 If d ≤ 500 mmj=6 If 500 mm ≤ d ≤1500 mmj=8 If d ≥ 1500mm

The section modulus of the arm section: Z = Fo (d−dh )2 jσ d

Stress Concentration

The static stress concentration factor :

Fatigue stress concentration factor, Kf : Fbeam = bmσ dY

K f