Design of Spur Gears

Design of Spur GearsConcept of Friction WheelsFrictional force P No slipping,When P Frictional force , slipping will occur, to avoid slipping tooth are provided over the wheel, known as gear .

Design of Spur Gears

Advantages1. Exact velocity ratio.2. Transmit large power.3. Small centre distances of shafts.4. High efficiency.5. Reliable service.6. Compact layout.

Advantages and Disadvantages of Gear Drives

Disadvantages1. Manufacturing Cost- Costlier ( Special Machines- Hobbing,

milling etc.)2. Error in gear cutting- Vibration 3. Requirement of lubricating oil for reliable operation

1. According to the position of axes of the shafts

(a) Parallel, (b) Intersecting, and (c) Non-intersecting and non-parallel.

Classification of Gears

herringbone gears.

2. According to the peripheral velocity of the gears.

(a)Low velocity- less than 3 m/s (b) Medium velocity-3 to 15 m /

s (c)High velocity- more than 15

m / s

3. According to the type of gearing

(a) External gearing, (b) Internal gearing, and(c) Rack and pinion

4. According to the position of teeth on the gearsurface. (a)Straight, (b)Inclined, and(c)Curved.

Rack & Pinion

Design of Spur GearsTerms used in Gears• Circular pitch, p = π d/z equation 12.2 , P.No.162• Diametral pitch P =z/ d equation 12.2 , P.No.162• Module m = d/ z , equation 12.4 , P.No.162

Design of Spur Gears1. Pitch circle: imaginary circle that gives same motion similar to actual gear2. Pitch circle diameter: diameter of the pitch circle3. Pitch point: common point of contact between two pitch circles4. Pitch surface: surface of the rolling discs5. Pressure angle or angle of obliquity: angle between the common normal to two gear teeth at the point of contact and the common tangent at the pitch point 14 ½ ° and 20°.6. Addendum: radial distance of a tooth from the pitch circle to the top of the tooth7. Dedendum: radial distance of a tooth from the pitch circle to the bottom of the tooth.8. Addendum circle: circle drawn through the top of the teeth9. Dedendum circle :circle drawn through the bottom of the teeth Root circle diameter = Pitch circle diameter × cos α, where α is the pressure angle.10. Circular pitch 11. Diametral pitch12. Module: T12.2 P.No. 182,DDHB13. Clearance: radial distance from the top of the tooth to the bottom of the tooth in meshing of gears14. Total depth:distance between the addendum and the dedendum circle15. Working depth: radial distance from the addendum circle to the clearance circle16. Tooth thickness:width of the tooth measured along the pitch circle17. Tooth space:width of space between the two adjacent teeth measured along thepitch circle.18. Backlash: difference between tooth space and the tooth thickness

Design of Spur GearsArc of contact. It is the path traced by a point on the pitch

circle from the beginning to the end of engagement of a given pair of teeth. The arc of contact consists of two parts, i.e.

(a) Arc of approach. It is the portion of the path of contact from the beginning of the engagement to the pitch point.

(b) Arc of recess. It is the portion of the path of contact from the pitch point to the end of the engagement of a pair of teeth.

Design of Spur GearsCondition for Constant Velocity Ratio of Gears–Law of Gearing

The common normal to the tooth profile at point of contact should always pass Through a fixed point called the pitch point, in order to obtain a constant velocityRatio.

Design of Spur GearsForms of Teeth1.Cycloidal teeth

Design of Spur Gears2. Involute teeth

Design of Spur GearsSystems of Gear Teeth1 . 4 ½ °Composite system, 2. 14 ½ °Full depth involute system, 3. 20° Full depth involute system, and 4. 20° Sub involute system.T-12.1, 12.4 P.No. 182/183 DDHB

Interference

Design of Spur GearsMinimum Number of Teeth on the Pinion in Order to Avoid Interference

i = Gear ratio or velocity ratio = z2/ z1 = d2 / d1,α = Pressure angle or angle of obliquity

Equation 12.10, P.No.162

Equation 12.11, P.No.162

Design of Spur Gears

Gear Materials

Table: 12.7 P.No.186,DDHB

Design of Spur GearsDesign Considerations for a Gear Drive1. The power to be transmitted.2. The speed of the driving gear,3. The speed of the driven gear or the velocity ratio, and4. The centre distance.The following requirements must be met in the design of

a gear drive :(a)static /Dynamic loading- Sufficient strength of gear

teeth(b)wear characteristics for satisfactory life(c) The use of space and material should be economical.(d) The alignment of the gears and deflections of the

shafts must be(e) The lubrication of the gears must be satisfactory.

Design of Spur GearsBeam Strength of Gear Teeth – Lewis Equation

Design of Spur GearsTooth is assumed as cantilever beam that is subjected to moment M = Ft × h

Ft = 1000 P Cs/ v………… 12.20 a ,P.No. 164,DDHB

Take Cs from T12.8, P.No. 187, as per load on tooth

Ft = Tangential load acting at the tooth,

h= height of the tooth,Half the thickness of the tooth (t) at critical section BC = t/2,I = Moment of inertia about the centre line of the tooth = b.t3/12,b = Width of gear face.From bending equation σ / Distance of outer(or inner surface from NA)= M/Iσ= (Ft × h,.t/2)/(b.t3/12)

or Ft= (σ × b × t2) / 6 h

In this expression, t and h are variables depending upon the size of the tooth (i.e. the circular pitch) and its profile.Let t = X × p , and h = k × p ; where X and k are constants.

WT= (σ × b × X2 pc 2) /( 6 k pc) = (σ × b × y× pc)………12.15,P.No. 163,DDHB

where y= x2/ 6 k or y= t2/ (6 π h m)= Lewis form factor (T-12.5, P.No. 184)Also form factor Y= π y

Design of Spur GearsThe value of y in terms of the number of teeth may be expressed as follows : y =0.684 - 0.124/ z ………………. ….. for 14 ½ composite and full depth involute

system.y =0.0.154 - 0.912/ z ……………………..for 20 full depth involute systemy =0.0.175 - 0.841/ z ……………………..for 20 sub involute systemEquation 12.17 a to 12.17 c , P.No. 163 DDHBσ = σd × Cvwhere σd = Allowable static stress, and (T-12.7,P.No. 186)Cv = Velocity factor ( also see T-12.9 ,P.No. 188) Cv =3.05/ (3.05 + v) for ordinary cut gears , velocities upto 8 m / s.=4.58/(4.58 + v) for carefully cut gears operating at velocities upto 13 m/s.=6.1/(6.1 + v) for very accurately cut and ground metallic gears

operating at velocities upto 20 m/ s.=0.75/(0.75 + √v) for precision gears cut with high accuracy and operating at

velocities upto 20 m/s, Equation 12.19a to 12.19 e, P.No. 164 DDHB

Design of Spur Gears

Design of Spur GearsDynamic Tooth LoadDynamic loads are due to following reasons-

1. Inaccuracies of tooth spacing2. Inaccuracies in tooth profiles3. Deflection of tooth under loads

Dynamic load may be obtained as –Fd= Ft+ Fi where Fd = total dynamic load, Ft = Static load due to transmitted torque, and Fi = Increment load due to dynamic action.The increment load depend upon the pitch line velocity v, , face width,

b material of gears, the accuracy of cut and tangential load. For average conditions, dynamic load is determined by using the

following Buckingham equation as-

C may also be obtained from the T-12.12, P. N0.190 as per error eerror e depending upon pitch line velocity may be taken from T-12.14,P.No. 191, also diagram 12.4, p.no. 205

Design of Spur GearsT12.13, P.No. 191, DDHB

Design of Spur GearsEndurance Strength of tooth may be obtained by putting value of σen in Lewis formulaFen = σen.b.pc.y = σen.b.π m.y equation 12.34 ,P.No.190σen may be obtained from T 12.15, p.no. 192Buckingham suggests the following relationship between Fen and Fd.For steady loads, Fen ≥ 1.25 FD

For pulsating loads, Fen ≥ 1.35 FD

For shock loads, Fen ≥ 1.5 FD

Note :σen = 1.75 × B.H.N. (in MPa) ( This is empirical formula taken from book)

See T 12.15, P.No. 192

Design of Spur GearsWear Tooth LoadFw = d1b.Q.K

where Fw = Maximum or limiting load for wear in newtons,

d1 = Pitch circle diameter of the pinion in mm,

b = Face width of the pinion in mm,Q = Ratio factor=(2×V.R)/(1+V.R.)=(2 Tg)/(Tg+Tp)…for external gears= (2×V.R)/(V.R.-1)=(2 Tg)/(Tg-Tp) for internal gears.V.R. = Velocity ratio = TG / TP,According to Buckingham, the load stress factor K is given by the following relation :

σes may be taken from T-12.16, P.No. 193

Design of Spur Gears

where σes = Surface endurance limit in MPa or N/mm2,

α = Pressure angle,EP = Young's modulus for the material of the pinion in N/mm2, and

Eg = Young's modulus for the material of the gear in N/mm2.σes may be taken from T-12.16, P.No. 193

Design of Spur Gears

Design Procedure for Spur Gears.docx