By Prof S S Chauhan, IEC College of Engineering & Technology, Greater Noida |

Unit-1 Spur Gears & Helical gears Objective: It gives the basic knowledge about the principles of Spur & cross helical Gear design on the basis of static load, dynamic load, wear etc. Objective type Questions 1. The gears are termed as medium velocity gears, if their peripheral velocity is (a) 1-3 m/s (b) 3-15 m/s (c) 15-30 m/s (d) 30-50 m/s 2. The size of a gear is usually specified by (a) pressure angle (b) pitch circle diameter (c) circular pitch (d) diametral pitch (c) tooth thickness (f) none of the above 3. The radial distance from the ……..to the clearance circle is called working depth. (a) addendum circle (b) dedendum circle 4. The product of the diametral pitch and circular pitch is equal to (a) 1 (b) 1/ (c) (d) x no. of teeth |

**Unit-1 Spur Gears & Helical gears** *Objective: It gives the basic knowledge about the principles of Spur & cross helical Gear design on the basis of static load, dynamic load, wear etc.* **Theoretical questions **

1. How are the gears classified and what are the various terms used in spur gear terminology?

2. Mention four important types of gears and discuss their applications, the materials used for them and their construction.

3. What condition must be satisfied in order that a pair o spur gears may have a constant velocity ratio?

4. State the two most important reasons for adopting involute curves for a gear tooth profile.

5. What is interference in gears? How can you remove it?

6. Write the expressions for static, limiting wear-load and dynamic load for spur gears and explain various terms used therein.

7. Discuss the design procedure for spur gears

8. How are the shaft and arms for spur gears designed?

9. Explain the following terms used in helical gears:

(a) Helix angle (b) normal pitch (c) axial pitch

**Unit-1 Spur Gears & Helical gears** *Objective: It gives the basic knowledge about the principles of Spur & cross helical Gear design on the basis of static load, dynamic load, wear etc.* **Numerical Questions**

1. A spur steel pinion (s0 =200 MN/m^{2}) is to drive a spur steel gear (s0=140MN/ m^{2}) . The diameter of the pinion is to be 100mm and the center distance 200mm. The pinion is to transmit 5Kw at 900RPM. The teeth are to be 20^{o} full depths. Determine the necessary module and face width to give the greatest no of teeth.

2. Two spur gears are to be used for a rock crusher and are to be minimum size. The gears are to be design for following requirements: power to be transmitted 18Kw, speed of pinion 1200RPM angular velocity ratio 3.5:1, tooth profile 20^{o} stub, s0 value of pinion 100 MN/m^{2}, s0 value for gear 70 MN/ m^{2}. Design the gear.

3. A pair of spur gears transmitting power from motor to pump impeller shaft is to be designed with as small center distance as possible. The forged steel pinion s0 =160MN/ m^{2}) is to transmit 4Kw at 600 RPM to a cast steel gear (s0=100MN/ m^{2}) with a transmission ratio 9/2 to 1, and 20^{o} full depth involute teeth are to be used. Determine the necessary face width and module.

4. A cast steel 24-tooth spur pinion operating at 1150 RPM transmits 3Kw to a cast steel 56 Tooth spur gear. The gears have the following specifications: module 3 , s0 =160MN/ m^{2}, face width 35mm 14.5^{o} tooth profile design the gear.

5. A cast steel pinion (s0=103MN/ m^{2}) rotating at 900 RPM is to drive a cast iron gear (s0=55MN/ m^{2}) at 144RPM. The teeth are to have standard 20^{o} stub involute profiles and maximum power to be transmitted is 25Kw. determine the proper module, no. of teeth and face width for these gear for stand point of strength , dynamic load and wear. Pinion is surface hardened to BHN 250.

6. A 80mm diameter steel pinion (s0=140MN/ m^{2}) drives a gray iron 240 mm diameter gear (s0=100MN/ m^{2}). The pinion operates at 1200 RPM and transmits 5Kw. The teeth are 20^{o} stub. Determine the greatest no. of teeth and necessary face width

7. A Bronze spur gear (s0=83MN/ m^{2}) is to drive a mild steel pinion (s0=103MN/ m^{2}). The angular velocity ratio is to be 3.5 to 1 . The pressure angle is to be 14.5^{o}. Determine the smallest diameters gears that can be used and the necessary face width to transmit 5Kw at 1800RPM of the pinion no less than 15 teeth are to be used on either gear. Design the gear.

8. A compressor running at 300RPM is driven by a 15Kw 1200RPM. Motor through a 20-degree full depth involute gear. The center distance is 375. Choosing the suitable material for pinion and gear, Design the drive completely. Assume medium shock condition.

9. The counter shaft of lathe runs at 250RPM. The back gear (reverted train) of the lathe required to reduce the speed to 25RPM approximately. Keeping the center distance between the axis of the lathe axis of the gear shaft 150mm, Determine module and no. of teeth of gear wheels of the train. A set of gears having teeth from 20 to 120 rising by 4 available for the purpose.

10. A 45 Kw motor running at 100RPM is required to drive rotatary mill at 120RPM through a pair of spur gears. The distance between axis of the motor and mill shaft is to be about 1m. Design the spur gear to go with the mill shaft. Assume a shaft diameter of 1000mm for the mill. Gear material may be assumed to be cast steel.

11. A reciprocating compressor is to be driven by electric motor running at 900RPM. The compressor speed is 300RPM and it requires a steady torque of 200N-m. Assuming a starting torque 25% greater than mean torque, design a suitable pair of spur gear for the drives. Assume the motor shaft and compressor diameters as 30mm and 37mm respectively.

12. Design a pair of helical gear to transmit 40 Kw from a shaft running at 1500RPM .to another shaft with a speed reduction of 6. Select the suitable materials for pinion and gear. Assume medium shock condition.

13. A 10 Kw, 750RPM induction motor is to drive a pump at 150 RPM through a pair of helical gear. The starting torque of the motor is taken as 2.5times rated torque. Design the gear pair completely and check for dynamic and wear loads

14. A pair of straight spur gears are required to reduce speed from 500 to 100 RPM for 12 hours running time per day continuously. The pinion is made of 40C8 steel normalized and has 20 teeth. The wheel is of cast iron of grade FG200 and has 100teeth. The gear is of 8mm module; 100mm face width and 20-degree pressure angle. Calculate the power rating.

15. A compressor running at 360RPM is driven by a 150 Kw, 1440-RPM motor through a pair of 20^{o} full depth helical gears having helix angle of 25 degree. The center distance is approximately 400mm. the motor pinion is to be forged steel and the driven gear is to be cast steel. Assume medium shock conditions. Design the gear pair.

16. Two helical gears having 14.5-degree normal pressure angle and a helix angle of 23^{o} connect two parallel shaft. The pinion has 48 teeth, the gear has 240 teeth and the module is 3. The pinion has an BHN 250 and the gear has the BHN 200. The face width is 250mm. Determine the wear load

17. Two parallel shafts are connected by a pair of steel helical gear. The pinion transmits 10Kw at 400RPM of the pinion. Both gear made of same material, hardened steel with allowable stress (s0=100MN/ m^{2}). If the velocity ratio is 4.5:1, determine the smallest diameter gears that may be used having sufficient strength. No less than 30 teeth are to be used for either gear, the teeth are of 20 degree stub in diametral plane and the helix angle is 45^{o}.

18. For a helical gear drive an expression for the virtual no of teeth N_{f} in terms of helix angle φ and actual no. of teeth N.

19. A pair of helical gears are used to transmit 15Kw. The teeth are 20^{o} stub in diametral plane and have a helix angle of 45 degree. The pinion has an 80mm pitch diameter and operates at 10000RPM. The gear has a 320mm pitch diameter. If the gears are made of cast steel (s_{0}=100MN/ m^{2}) , determine a suitable module and face width . The pinion is heat treated to BHN of 3000 and the gear has a BHN of 200.

20. A pair of helical gears with a 23^{o} helix angle is to transmit 2.5 Kw at 10000RPM of the pinion. The velocity ratio of 4:1. Both the gear are made of hardened steel with an allowable stress (s_{0}=100MN/ m^{2}) fro each gear. The gears are 20^{o} stub and the pinion is to have 24 teeth. Determine the minimum diameter gear that may be used, and the required BHN.

**Unit-1 Spur Gears & Helical gears** **Mini Project Problems**

Spur gear problems

In the following problems, assume that :

- gears with any tooth number up to 120 are procurable ( constraints are more severe in practice)

- all gears are of steel, to the 20 degree full depth system unless otherwise indicated

- mid-range profile shifts apply, where relevant.

The program Steel Spur Gears should be used to assist solution of asterisked problems, and may be used to check longhand solution of other fatigue problems.

1. Tooth numbers of certain gears in the epicyclic train are indicated; all gears are of the same module. Gear A rotates at 1000 rev/min clockwise while E rotates anticlockwise at 500 rev/min. Determine the speed and direction of rotation of the ring-gear D and of the arm shaft F. If the power output through each of D and F is 1 kW, what are the power transfers through A and E? [ 371 rev/min anticlockwise; 40 rev/min clockwise; 8.77 kW input; 6.77 kW output.

2. The arm of the epicyclic train is driven clockwise at 1450 rev/min by a 5 kW motor. What torque is necessary to lock the 33 teeth gear? What is the speed of the 31 tooth gear? Note the reduction[ 16.3 kNm clockwise, 2.92 rev/min clockwise ]

3. The sun wheels A and D are integral with the input shaft of the compound epicyclic gear illustrated, and the annular wheel C is fixed. The planet wheel B rotates freely on an axle carried by the annular wheel F, and the planet E on an axle mounted on the output shaft's arm. Given the tooth numbers indicated, find the speed of the output shaft when the input shaft rotates at 1000 rev/min.

Ans=524rev/min

4. In the epicyclic train illustrated, the gear C is fixed and the compound planet BD revolves freely on a spindle which is coaxial with the input and output shafts.

(a) Show that if z_{b} z_{e} > z_{c} z_{d} then input and output shafts rotate in the same direction.

(b) 7.5 kW is fed into the input shaft at 500 rev/min, losses are negligible, and tooth numbers are sketched. Determine the torque on the output shaft.

Ans=15.5kNm

5. Select spur gears suitable for speed ratios of (i) 1/√2, and (ii) π , to four significant figures.

6. Determine the practical limits of profile shift on a 6 mm module gear with 19 teeth. If a profile shift of 0.4 is implemented, what are the dedendum, base, pitch, extended pitch and addendum circle diameters of the gear ?

Evaluate the base pitch and the angle γ of Fig A.

[ 103.8, 107.1, 114, 118.8 and 130.8 mm. 17.7 mm, 6.47^{o}]

7. What is the practical range of centre distance for a pair of 4 mm module spur gears with 19 and 35 teeth ? If they are manufactured with profile shifts of 1.5 mm and 2 mm respectively, evaluate the extended pressure angle and the contact ratio.

[ 108.6 ≤ C ≤ 112.8 mm, 24.47^{o}, 1.42 ]

8. Use the design procedure outlined in the Notes to determine gears suitable for a speed ratio of √ 2 ± 0.5 % and a centre distance of 200 ± 1 mm.

[ 6 mm module, with 27 and 38 teeth, and profile shifts of 0.45 say, for pinion and correspondingly 0.38 for wheel ]

9. Evaluate the contact ratio and the fatigue geometric factors I and J for each of the following :

(a) the pairs 13:35, 23:62 and 36:97 (which approximate the ratio 0.3711 to within 0.1%);

(b)* 23:62 teeth, assuming the minimum practical profile shifts for both gears;

(c)* repeat (b) but use the maximum practical profile shifts. Comment upon the trends suggested by these results.

10. The transmission accuracy level number of a pair of open gears is 6. Further particulars of the 25 mm module 300 mm facewidth gears are as follows :

number of teeth | allowable stresses, MPa | speed, | ||

contact | bending | rev/min | ||

pinion | 25 | 1100 | 290 | 150 |

wheel | 55 | 1000 | 280 | - |

11. What life may be expected of the gears whilst transmitting 1 MW uniformly? [ 39 khr ]

Shock loading of the foregoing drive results from unsuspected torsional vibration. If the effective application factor is in fact 1.25, what life may now be expected? [ 5.4 khr ]

12. Two mating gears of commercial quality are to hand with 18 and 56 teeth. Their common facewidth is measured as 50 mm and their addendum diameters as 83.2 and 233.6 mm. Metallurgical analysis reveals that the expected contact and bending stresses of the gears' common material are 1100 and 300 MPa respectively. Estimate the pair's capacity (kW) for a 10 khr life in a shock-free application in which the pinion speed is 300 rev/min. The transmission accuracy level number is 6. [ 9.1 kW ]

13. A gear pair transmits 75 kW with an application factor of 1.5 and reliability of 99%. Particulars of the commercial 6-accuracy level gears are :

number of teeth | allowable stresses, MPa | speed, | ||

contact | bending | rev/min | ||

pinion | 20 | 1300 | 180 | 90 |

wheel | 37 | 1250 | 175 | - |

14. Select a suitable module and facewidth for a life of 15 khr. [ 16, 144 mm ]

15. Details of a pair of commercial gears having a transmission accuracy level of 8 are as follows :

number of teeth | allowable stresses, MPa | speed, | ||

contact | bending | rev/min | ||

pinion | 10 | 1320 | 380 | 200 |

wheel | 36 | 1100 | 360 | - |

16. Select a suitable module and facewidth for a design life of 16 khr whilst transmitting a uniform 125 kW with a reliability of 99%. [ 16, 187 mm ]

17. A commercial gear pair having a transmission accuracy level of 8 is required to transmit 100 kW in a shockfree application with 99% reliability. The speeds of pinion and wheel are 1450 and approximately 470 rev/min. Allowable stresses for contact and for bending of the pinion are 1450 and 400 MPa respectively; for the wheel 1300 and 350 MPa. Select suitable tooth numbers and profile shifts, along with a corresponding module and facewidth for a compact pair with a design life of 20 khr.

18. Estimate the life of a gear whose allowable contact stress is 1.2 GPa and which undergoes the stress spectrum:

contact stress | σ_{c} |
(GPa) | 1.0 | 1.1 | 0.9 |

speed | N | (rev/min) | 500 | 400 | 300 |

duration | t | (hours) | 2 | 1 | 3 |

19. A pair of 8 mm module, 100 mm facewidth commercial gears is manufactured to a transmission accuracy level of 7 and employed in a periodic duty of 1.25 application factor. The 23 tooth pinion's allowable contact stress is 1.2 GPa at 99% reliability, the 47 tooth wheel's is 1.1 GPa.

If power is transmitted to the following cycle, what life may be expected of the pair?

power | P | (kW) | 60 | 45 | 35 |

pinion speed | N_{1} |
(rev/min) | 200 | 150 | 100 |

duration | t | (min) | 10 | 20 | 30 |