Gear Trains PDF

Kinematics of Machines Sub Code : 18ME36 CIE Marks : 100 Hrs/week : 3L + 2T + 0P SEE Marks : 100 Credits :3 Exam : 3 Hrs Unit 4 : 06 Hrs ...

Kinematics of Machines Sub Code : 18ME36 CIE Marks : 100 Hrs/week : 3L + 2T + 0P SEE Marks : 100 Credits :3 Exam : 3 Hrs Unit 4 : 06 Hrs Gear Trains Dept. Of Mechanical Engineering, RVCE, Bangalore  Gear Trains A gear train is an arrangement of two or more successively meshing gears through which power can be transmitted between the driving & driven shafts. Train Value Train value is the ratio of speed of the driven gear to that of the driving gear. It is the reciprocal of the velocity ratio. Direction of rotation Velocity ratioWhen in Gear gears Drives :mesh externally they rotate in n1 d 2 z2 the opposite direction and when they mesh   , where n1 Speed internally, of driving they rotate pulley, in n2 Speed the same ofdirection. driven pulley n2 d 1 z1 d1 Pitch circle diameter (PCD) of driver gear, d 2 PCD of of driven gear z1 No of teeth on driver gear, z2 No of teeth on driven gear P R Venkatesh, Mech Dept,RVCE,B'lore  Types of Gear Trains A gear train may be broadly classified into the following. 1.Simple Gear Train 2.Compound Gear Train 3.Reverted Gear Train 4.Epicyclic Gear Train P R Venkatesh, Mech Dept,RVCE,B'lore  SIMPLE GEAR A simple gear train is one in which TRAIN each shaft carries only one gear. From the fig, gear A is the driving gear and gear D is the driven gear. A B C B & C are the intermediate gears or Idler gears. The idler gears do not affect the N  z  velocity ratio but simply bridge the Velocity ratio  A   C   NC   z A  gap between the driver & the driven A gears. Also if odd number of intermediate B gears are used, the driver & the driven gears rotate in the same direction. C If even number of intermediate gears are used, the driver & the driven gears rotate in the opposite D directions. N  z  Velocity ratio  A   D   ND   zA   SIMPLE GEAR TRAIN ANIMATION  N A   zB   N B   zC  From the fig,     and also      NB   zA   NC   zB   N A   N B   z B   zC   N A   zC             =  N N z  B  C  A  B z N  C   A z  zA   30  The speed of gear C, NC   ×N A  NC   ×90 =180 rpm.  zP CR Venkatesh,  Mech Dept,RVCE,B'lore  15   COMPOUND GEAR In a compound gear train the TRAIN intermediate shaft carries two or more gears which are keyed C to it. A B D Compound gears are used when a high velocity ratio is required in a limited space. The intermediate gears will have an effect on the overall velocity ratio. N  z  N  z  From the fig,  A   B  and also  C   D   NB   zA   N D   zC  N  N  z  z    A   C   B   D   N B   N D   z A   zC  As gears B & C are on same shaft, N B  N C  N A   z B z D     =  N  D  A C z  z Speed of first driver Product of no of teeth on driven gears i.e.  SpeedPofR Venkatesh, the last driven Mech Product of no of teeth on driving gears Dept,RVCE,B'lore REVERTED GEAR A reverted gear train is a TRAIN compound gear train in which the first & the last D gears are on the same axis. A Hence, in a reverted gear train, the center distances for the two gear pairs must be same. B C Reverted gear trains are used in automotive transmissions, lathe back gears, and in clocks.  d A  d B   dC  d D  As    ,  2   2  But d=mz, and the module 'm' is same for all gears,  z A  z B  zC  z D P R Venkatesh, Mech Dept,RVCE,B'lore EPICYCLIC GEAR TRAIN An epicyclic gear train is one in which the axis of one or more gears moves relative to the frame. Large speed reductions are obtained with an epicyclic train. They are compact in size and Used in automobile differential. P R Venkatesh, Mech Dept,RVCE,B'lore  Problem 1 A simple train of wheels consists of successively engaging three wheels having number of teeth 40, 50 & 70 respectively. Find its velocity ratio. If the driving wheel having 40 teeth runs at 210 rpm clockwise, find the speed of the driven wheel and its direction of rotation. P R Venkatesh, Mech Dept,RVCE,B'lore  210 Rpm A B C 40 T 50 T 70 T  NB   zA   NC   zB  From the fig,     and also      N A   zB   N B   zC   N B   NC   z A   zB   NC   z A             =   N A   N B   z B   zC   N A   zC   zA   40  The speed of gear C, NC   ×N A  NC   ×210 =120 rpm.  zC   70  As there are odd number of idler gears, the driven gear rotates at 120 P R Venkatesh, Mech rpm clockwise. (i.e. same as that of driving gear) Dept,RVCE,B'lore  Problem 2 In a simple gear train consists of four wheels having number of teeth 30, 40, 50 & 60 teeth respectively. Determine the speed and the direction of rotation of the last gear if the first makes 600 rpm, clockwise. P R Venkatesh, Mech Dept,RVCE,B'lore  600 Rpm A B C D 30 T 40 T 50 T 60 T  N D   z A   z B   zC  From the fig,          N A   z B   zC   z D   ND   z A     =   NA   zD   zA   30  The speed of gear D, ND   ×N A  NC   ×600 =300 rpm.  zD   60  As there are even number of idler gears, the driven gear rotates at 300 P R Venkatesh, Mech rpm counter clockwise. (i.e. opposite to that of driving gear) Dept,RVCE,B'lore  Problem 3 A compound gear train consists of 4 gears, A, B, C & D and they have 20, 30, 40 & 60 teeth respectively. A is keyed to the driving shaft, and D is keyed to the driven shaft, B & C are compound gears. B meshes with A & C meshes with D. If A rotates at 180 rpm, find the rpm of D. P R Venkatesh, Mech Dept,RVCE,B'lore  40T 20T 60T 30T A B D C  NB   zA   N D   zC  From the fig,     and also      N A   zB   NC   zD   N B   N D   z A   zC           N A   NC   zB   zD  As gears B & C are on same shaft, N B  N C  ND   z A zC     =  N  A  B z  z D   20 40  ND   180 =80 RPM  30 60  P R Venkatesh, Mech Dept,RVCE,B'lore  Problem 4 Fig shows a train of gears from the spindle of a lathe to the lead screw used for cutting a screw thread of a certain pitch. If the spindle speed is 150 rpm, what is the lead screw speed? Gears 2 & 3 form a compound gear. 75 T Lead screw 20T 50 T Spindle 25 T 1 2 3 4 P R Venkatesh, Mech Dept,RVCE,B'lore  75 T Lead screw 20T 50 T Spindle 25 T 1 2 3 4 From the fig, velocity ratio  Speed of the driven shaft   Product of the no of teeth on driver       Speed of the driving shaft   Product of the no of teeth on driven   N4   z1 z3    =  As N1 =150rpm,  N1   z 2 z 4   20 25  N4   150 = 20 RPM  75 50  P R Venkatesh, Mech Dept,RVCE,B'lore  Problem 5 Fig shows a reverted gear train used in a lathe headstock. If the motor runs at 1200 rpm, find the speed of the spindle. 100 T 60 T 2 3 Motor Shaft Spindle 1 4 50 T P R Venkatesh, Mech Dept,RVCE,B'lore  100 T 60 T 2 3 Motor Shaft Spindle 1 4 50 T As the center distance between the shafts is same,  d1  d 2   d 3  d 4       d1  d 2  = d 3  d 4   2   2   d  The circular pitch =   = m  d mz where 'm' is known as module.  z  For two gears in mesh, circular pitch and hence the module is same. P R Venkatesh, Mech Dept,RVCE,B'lore  100 T 60 T 2 3 Motor Shaft Spindle 1 4 50 T As z1  z2  z3  z4 50  100 60  z4 No of teeth on gear 4=90 teeth.  z1 z3   Speed of the spindle N 4   N1  z2 z4   50×60   N4 =  ×1200=400 rpm  100×90  P RDept,RVCE,B'lore Venkatesh, Mech  Problem 6 In an epicyclic gear train, an arm carries two gears A & B having 36 and 45 teeth respectively. If the arm rotates at 150 rpm in the anticlockwise direction, about center of gear A which is fixed, determine the speed of speed of B. If the gear A instead of being fixed, makes 300 rpm in the clockwise direction, what will be the speed of gear B? B A Arm C P R Venkatesh, Mech Dept,RVCE,B'lore  Sl.No Condition of motion Arm C Gear A Gear B Fix the arm & give  ZA  1 0 1   1 rev to gear A  ZB   ZA  2 Multiply by x 0 x  x  ZB   ZA  3 Add y y yx y  x  ZB  Speed of gear B when A is fixed : (Taking ccw direction as + ve) As the arm makes 150 rpm ccw, y= 150 Gear A is fixed and hence y  x 0  x  150  ZA   36  Speed of gear B=y    x 150    ( 150) 270  ZB   45  i.e. Speed of gear B=270 rpm counter clockwise  Sl.No Condition of motion Arm C Gear A Gear B Fix the arm & give  ZA  1 0 1   1 rev to gear A Z  B  ZA  2 Multiply by x 0 x  x  ZB  Z  3 Add y y yx y  A x  ZB  Speed of gear B when A makes 300 rpm clockwise : (Taking ccw direction as + ve) As the arm makes 150 rpm ccw, y= 150 Gear A makes 300 rpm cw, hence y  x  300  150  x  300  x  450  ZA   36  Speed of gear B=y    x 150    ( 450) 510  ZB   45  i.e. Speed of gear B=510 rpm counter clockwise  Problem 7 In the reverted gear train shown in fig, the number of teeth on gears B, C & D are 75, 30 and 90 respectively. Find the speed and direction of gear C when the gear B is fixed and the arm makes 100 rpm clockwise. B D C E A P R Venkatesh, Mech Dept,RVCE,B'lore  Compound gear Sl.No Condition of motion Arm A Gear B Gear C D E Fix the arm & give Z  Z  1 +1 rev to compound gear 0 1  E  D D-E  ZB   ZC  Z  Z  2 Multiply by x 0 x  E x   D x  ZB   ZC  Z  Z  3 Add y y yx y  E x y  D x  ZB   ZC  From fig, (Z B  Z E ) ( Z C  Z D )  (75  Z E ) (30  90)  Z E 45 teeth Wheel B is fixed and arm A makes 100 rpm clockwise. i.e. y=100  ZE   45  Hence y    x 0  100    x 0  ZB   75  Z   90   x 166.67  Speed of gear C=y   D  x 100    166.67  400  ZC   30  i.e. Speed of gear C=400 rpm counterclockwise P R Venkatesh, Mech Dept,RVCE,B'lore  Problem 8 An epicyclic gear train consists of a sun wheel S, a stationary internal gear E, and three identical planet wheels P carried on a star- shaped planet carrier C. The size of different toothed wheels are such that the planet carrier C rotates at 1/5th the speed of the sun wheel. The minimum number of teeth on any wheel is 16. The driving torque on the sun wheel is 100 Nm. Determine; Number of teeth on different wheels of the train. Torque necessary to keep the internal gear stationary. P R Venkatesh, Mech Dept,RVCE,B'lore  E P S C P P Schematic arrangement of gears P R Venkatesh, Mech Dept,RVCE,B'lore  E P S C P P Condition of Carrier sunwheel Planet Internal Gear Sl.No motion C S P E Fix the carrier C Z   Z  Z   Zs  1 & give +1 rev to 0 1  s   s   P     sunwheel S  ZP   ZP   ZE  Z  E Z  Z  2 Multiply by x 0 x  s x  s x  ZP   ZE  Z  Z  3 Add y y yx y  s x y  s x  ZP   ZE   E P S C P P  Problem 9 An internal wheel B with 80 teeth is keyed to shaft F. A fixed internal wheel C with 82 teeth is concentric with B. A compound wheel D-E gears with the two internal wheels; D has 28 teeth and gears with C while E gears with B. The compound wheels revolve freely on a pin which projects from a disc of a shaft A coaxial with F. If all the wheels have the same pitch and shaft A makes 800 rpm, what is the speed of shaft F? Sketch the arrangement. P R Venkatesh, Mech Dept,RVCE,B'lore  E D C  B F Schematic arrangement of gears P R Venkatesh, Mech Dept,RVCE,B'lore  E D C  B F Condition Compound gear Sl.No Arm Gear B Gear C of motion E&D Fix the arm & give Z  Z   ZD  1 0 1  B   B    +1 rev to gear B  ZE   ZE Z   C  ZB   ZB   ZD  2 Muliply by x 0 x  x    x  ZE  Z  E  C Z Z  Z   ZD  3 Add y y yx y  B x y  B   x  ZE   ZE   ZC  P R Venkatesh, Mech Dept,RVCE,B'lore  E D C  B F From fig, (rC  rB ) ( rD  rE ) As r=mz where m=module which is same for all gears; (ZC  Z B ) ( Z D  Z E )  (82  80) (28  Z E )  Z E 26 teeth Since the disc is keyed to the shaft and the shaft makes 800 rpm, y=800 Z  Z   80   28  Wheel C is fixed. Hence y   B   D  x 0  800      x 0  Z E   ZC   26   82   x  761.43 Speed of gear B=800  (  761.43) 38.57 i.e. Speed of gear B=speed of shaft F=38.57 rpm (same as direction of shaft A) P R Venkatesh, Mech Dept,RVCE,B'lore