Spur gear problems

• gears with any tooth number up to 120 are procurable ( constraints are more severe in practice )
• all gears are of steel, to the 20^o full depth system unless otherwise indicated
• mid-range profile shifts apply, where relevant.

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.
   [ 524 rev/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.
   [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. Derive equation ( 10) from which the contact ratio may be calculated (20^o full depth system).
    
8. 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 ]
    
9. 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 ]
    
10. 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.
     
11. 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	allowable stresses, MPa		speed,
    	of teeth	contact	bending	rev/min
    pinion	25	1100	290	150
    wheel	55	1000	280	-

    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	allowable stresses, MPa		speed,
    	of teeth	contact	bending	rev/min
    pinion	20	1300	180	90
    wheel	37	1250	175	-

    Select a suitable module and facewidth for a life of 15 khr.     [ 16, 144 mm ]
     
14. Details of a pair of commercial gears having a transmission accuracy level of 8 are as follows :

    	number	allowable stresses, MPa		speed,
    	of teeth	contact	bending	rev/min
    pinion	10	1320	380	200
    wheel	36	1100	360	-

    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 ]
     
15. 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.
     
16. Estimate the life of a gear whose allowable contact stress is 1.2 GPa and which undergoes the stress spectrum :

    contact stress	σc	(GPa)	1.0	1.1	0.9
    speed	N	(rev/min)	500	400	300
    duration	t	(hours)	2	1	3

    [ 7.9 khr]
     
17. 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	N1	(rev/min)	200	150	100
    duration	t	(min)	10	20	30

    [ 13 khr ]