Module 2- GEARS. Lecture 9 - SPUR GEAR DESIGN

Transcription

1 Module 2- GEARS Lecture 9 - SPUR GEAR DESIGN Contents 9.1 Problem 1 Analysis 9.2 Problem 2 Spur gear 9.1 PROBLEM 1 SPUR GEAR DESIGN In a conveyor system a step-down gear drive is used. The input pinion is made of 18 teeth, 2.5 mm module, 20 o full depth teeth of hardness 330Bhn and runs at 1720 rpm. The driven gear is of hardness 280Bhn and runs with moderate shock at 860 rpm. Face width of wheels is 35 mm. The gears are supported on less rigid mountings, less accurate gears and contact across full face may be assumed. The ultimate tensile strength of pinion and gear materials is 420 and 385MPa respectively. The gears are made by hobbing process. Find the tooth bending strength of both wheels and the maximum power that can be transmitted by the drive with a factor of safety 1.5. The layout diagram is shown in the Fig 9.1. Fig 9.1 Conveyor drive layout

2 Solution: The bending fatigue stress is found from AGMA equation as, Ft K v Ko K (9.1) m bmj We know that, Z 2 = Z 1 x (N 1 /N 2 ) Substituting values from table 1, Z 2 = 18 X (1720/860) = 36 Table 9.1 Data given for gear and pinion N Z m d b Pinion 1720rpm mm 45 mm 35 mm Gear 860 rpm mm 90 mm 35 mm Using the values from Table 9.1, V = π dn/60000 = π x 45 x 1720/60000 = 4.051m/s We know that K v (200V) (9.2) 50 Table 9.2 J values for pinion and gear Z J (sharing) K v K o K m Pinion Gear The J value is obtained from Fig. 9.2 for sharing teeth as in practice. K o and K m values are obtained from Tables 9.3 and 9.4 for the given conditions.

3 Fig Geometric Factor J SPUR GEAR TOOTH BENDING STRESS (AGMA) Table Overload factor K o Driven Machinery Source of power Uniform Moderate Shock Heavy Shock Uniform Light shock Medium shock

4 Table Load distribution factor K m Face width ( mm) Characteristics of Support up Accurate mountings, small bearing clearances, minimum deflection, precision gears Less rigid mountings, less accurate gears, contact across the full face Accuracy and mounting such that less than Over Over Over Over full-face contact exists For pinion: Ft K bmj t K K v o m Ft x1.569x1.25x1.6 35x 2.5x = F (9. 3) And for Gear: Ft σ = Kv Ko Km bmj Ft = x1.569x1.25x1.6 35x 2.5x0.385 = F t (9.4) Fatigue strength of the material is given by, σ e = σ e k L k v k s k r k T k f k m (9.5) Table 9.5 Properties of pinion and gear Prop. σ ut MPa σ e =0.5σ ut MPa k L K v k s Pinion Gear

5 SPUR GEAR PERMISSIBLE TOOTH BENDING STRESS (AGMA) Endurance limit of the material is given by: σ e = σ e k L k v k s k r k T k f k m (9.6) Where, σ e is the endurance limit of rotating-beam specimen From table 9.5, k L = load factor = 1.0 for bending loads k v = size factor = 1.0 for m < 5 mm and = 0.85 for m > 5 mm k s = surface factor, is taken from Fig.9.3 based on the ultimate tensile strength of the material for cut, shaved, and ground gears. k r = reliability factor given in Table 9.5. k T = temperature factor = 1 for T 350 o C = 0.5 for 350 < T 500 o C Fig.9.3 Surface factor K s Reliability of 90%, working temperature <150 o C and reversible is assumed. k f = 1.0 since it is taken in J factor.

6 k m = 1.0 for reverse bending assumed here Table 9.6 K terms of pinion and gear Prop. k r k T k f k m Pinion Gear Table 9.7 Reliability factor R Reliability factor R Factor K r Permissible bending stress e [ ] n ( (9.7) Hence the design equation from bending consideration is, σ [ σ ] (9.8) Factor of safety required = 1.5 ```````````````````````Table 9.8 Strength values of pinion and gear Prop. σ e MPa [σ]= σ e / s MPa σ MPa F T N Pinion F t 947 Gear F t 962 Table 9.8 shows that the pinion is weaker than gear. And maximum tangential force that can be transmitted is: F t = 947 N So, the maximum power that can be transmitted is: W = F t v / 1000 = 947 x /1000 = 3.84 kw

7 9.2 PROBLEM 2 SPUR GEAR DESIGN In a conveyor system a step-down gear drive is used. The input pinion is made of 18 teeth, 2.5 mm module, 20 o full depth teeth of hardness 340Bhn and runs at 1720rpm. The driven gear is of hardness 280Bhn and runs with moderate shock at 860 rpm. Face width of wheels is 35mm. The gears are supported on less rigid mountings, less accurate gears and contact across full face may be assumed. The ultimate tensile strength of pinion and gear materials is 420 and 385MPa respectively. The gears are made by hobbing process. From surface durability consideration, find the maximum power that can be transmitted by the drive with a factor of safety 1.2 for a life of 10 8 cycles. Drive layout is shown in the Fig 9.4. Fig. 9.4 Conveyor drive Layout diagram Data given: i = n 1 /n 2 = 1720/860 = 2 Z 2 = Z 1 x i = 18 X 2 = 36

8 Table 9.9 Data given for pinion and gear n Z m d = mz b Pinion 1720rpm mm 45 mm 35 mm Gear 860 rpm mm 90 mm 35 mm Table 9.10 Properties of gear and pinion Bhn Ø Reliability Life Temp Pinion o 99 % 10 8 <120 o C Gear o 99 % - <120 o C Solution: The induced dynamic contact stress is given by equation below, Ft H Cp KVKoK (9.9) m bd I 1 When both pinion and gear material are made up of steel, from Table 9.11, C p = 191 (9.10) MPa SPUR GEAR CONTACT STRESS Table 9.11 Elastic coefficient Cp for spur gears in MPa Pinion Material(μ=0.3 in all cases) Gear material Steel Cast iron Al Bronze Tin Bronze Steel, E=207Gpa Cast iron, E=131Gpa Al Bronze, E=121Gpa Tin Bronze, E=110Gpa

9 sin cos i I (9.11) 2 i 1 Substituting the values from table 10, o o sin 20 cos20 2 I SPUR GEAR SURFACE DURABILITY From table 3 and 4, V = π dn/60000 = π x 45 x 1720/60000 = 4.051m/s For hobbed gear, 50 (200V) K v (9.12) Table 9.14 K Values of pinion and gear Z K v K o K m Pinion Gear Substituting values from Table 14, we have, C F t H p V o m bd1 I t x1.25x1.6 t K K K F 35x45x F MPa Surface fatigue strength of the material is given by, σ sf = σ sf K L K r K T (9.13) From table 10, for steel life is 10 7 cycles & reliability 99% and from Table 9.15,

10 σ sf = 28(Bhn) 69 = 2.8x = 954MPa K L = 0.9 for 10 8 cycles from Fig.9.2 K R = 1.0. for 99% reliability from Table 9.10 SPUR GEAR SURFACE FATIGUE STRENGTH Table 9.15 Surafce fatigue strength σ sf for metallic spur gears (10 7 cycle life 99% reliability and temperature <120 0 C) Material Steel Nodular iron σ sf (MPa) 2.8 (Bhn)-69MPa 0.95 (2.8(Bhn)-69MPa) Cast iron, grade Cast iron, grade Cast iron, grade Tin Bronze, AGMA 2C (11% Sn) 207 Aluminium Bronze (ASTM ) 448 (Alloy 9C H.T.) Fig. 9.5 Life factor K l

11 SPUR GEAR ENDURANCE LIMIT Table 9.16 Reliability factor K R Reliability (%) K R SPUR GEAR ALLOWABLE SURFACE FATIGUE STRESS (AGMA) We know that, [ σ H ] = σ Sf / f s = 954/1.2 = 795MPa For factor of safety f s = 1.2 Design equation is, σ H [ σ H ] F t = 795 F t = 931 N Maximum Power that can be transmitted is, W = F t V/1000 = 931x4.051/1000 = 3.51kW