Chapter 3. Transmission Components

Chapter 3. Transmission Components The difference between machine design and structure design An important design problem in a mechanical system is how to transmit and convert power to achieve required motion and function. This task is usually achieved by transmission components and mechanism. This chapter will introduce common transmission components, including Spur gear Helical gear Bevel gear Worm gear Belt and chain Bearings Copyright 00. All rights reserved.

Involute The profile of a gear tooth is a special designed curve called involute. An involute can be conceptually regarded as an imaginary trajectory generated by a cord unwrapped about a cylinder. cord base circle involute Using involute as the tooth profile, there will be rolling and no sliding between teeth when two spur gears mesh each other (imagine two cans connected by a string). Copyright 00. All rights reserved. 3

Spur Gear Terminologies (I) The circle on which the involute is generated is called base circle. When two spur gears mesh each other, there will be two theoretical circles that are tangent to each other. These two circles are called pitch circles, and the diameter of a pitch circle is called pitch diameter. The tangent line between two base circles is called pressure line, and the angle between the tangent line and the horizontal line is called pressure angle. Pressure angle is usually 4.5, 0 and 5 degree, and 0 degree pressure angle is mostly adopted. Copyright 00. All rights reserved. 4

Spur Gear Terminologies (II) The distance measured from a point on one tooth to the same corresponding point on the adjacent tooth is called pitch. If the distance is measured on the pitch circle, this distance is called circular pitch (p), which can be expressed as: p The ratio of the pitch diameter to the number of teeth is called module (unit: mm). Module is the index of the size of gears, and ranges from 0.3mm to 5mm. m p m Copyright 00. All rights reserved. 5

Spur Gear Terminologies (III) The radial distance between the top of the tooth and the pitch circle is called addendum (a). a m The radial distance from the bottom of the tooth to the pitch circle is called dedendum (d). d. 5m The whole depth is the sum of addendum and dedendum. h r a d. 5m Copyright 00. All rights reserved. 6

A Gear Train A gear train is often used to reduce output rotational speed and magnify torque. power source 動力源 coupler 連軸器連軸器 pinion 齒輪 A When a gear and a pinion mesh each other, the tangential velocities at the contact points of the gear and pinion are equal if there is no sliding between the teeth. R G G v t R P P 齒輪 B gear 連軸器 coupler 被驅動機器 driven machine G G P P pinion 小齒輪 gear 大齒輪 Velocity Ratio VR P G R R G P G P G P Copyright 00. All rights reserved. 7

Example. Calculate the Train Value 750 rpm VR AB B A 30 VR E E 66 5 idle gear -4.5 rpm VR BC C B 68 30 idle gear TV VR AB 30 VR BC VR 68 30 E 66 5 8.6 output 750 8.6 4.5 rpm Copyright 00. All rights reserved. 8

Automotive Manual Transmission The gear ratio is high for the st and nd gear in order to produce enough torque for acceleration. The gear ratio is lower for the 3rd and 4th gear. The gear ratio of the 4th gear is usually :, and is called direct drive. The gear ratio for the 5th (or 6th) gear is less than, and is called over drive (O). The automatic transmission utilizes planetary gear train, which is completely different from manual transmissions. Copyright 00. All rights reserved. 9

Example. Automotive Steering System Rack and pinion are used in traditional automotive steering systems. A rack can be seen as a gear with infinite number of teeth. Copyright 00. All rights reserved. 0

Planetary Gear System in Automatic Transmission (I) In a planetary gear system, there is a sun gear, a planet pinion, and a ring gear. Among the three axes, one axis is fixed, one is the driving axis, and the other one is the output axis. So there are 6 combinations: combination 3 4 5 6 sun gear fixed output fixed driving output driving planet pinion driving driving output output fixed fixed ring gear output fixed driving fixed driving output gear ratio < <0.5 > > <, R >, R Copyright 00. All rights reserved.

Planetary Gear System in Automatic Transmission (II) When the sun gear is fixed (combination and 3): ring planet v v R R ring planet R R planet ring When the ring gear is fixed (combination, 4): sun planet v v R R sun planet R R planet sun stationary 靜止 v stationary st gear Combination 4; nd gear Combination 3; 3 rd gear direct drive; 4 th gear Combination (O); Reverse gear Combination 6. Copyright 00. All rights reserved.

Spur Gear Force Analysis The contact force along the pressure line is called normal force W n. The component in the tangential direction is called tangential force W t. The component in the radial direction is called radial force W r. W t is important when calculating the power transmitted, and W r is the force component applying on the shaft of the gear. The relations between W n, W r, and W t are:: Wt W n cos Wr W n sin Wr W t tan Copyright 00. All rights reserved. 3

Example 4. Calculate the Loads on Spur Gears (I) Given: Input power 0kW @750 rpm. pressure angle ψ=0 and the module m=.5. numbers of teeth of gear A and B are 0 and 60, respectively. Find the normal, radial and tangential forces of gear A and B. Copyright 00. All rights reserved. 4

Example 4. Calculate the Loads on Spur Gears (II) m m A A B B d A.5 d A 30mm 0 db.5 db 90mm 60 P(power) T(torque) (rotational speed) (rpm rad 60 sec) 0,000 T A 750 TA 54,567mm 60 B 30 750 583.33(rpm) 90 0,000 T B 583.33 TB 63,703mm 60 Copyright 00. All rights reserved. 5

Example 4. Calculate the Loads on Spur Gears (III) For Gear A: TA 54,567 W t 3637.8 d A 30 Wt 3638 Wn 387 cos cos 0 W r 387sin 0 34 () For Gear B: W t TB db 63703 90 3638 All loads of Gear B are the same as those of Gear A Copyright 00. All rights reserved. 6

Spur Gear Stress Calculation (I) A tooth of a spur gear is often analogous to a cantilever beam when calculating stress. The maximum stress at the root of the gear is: t Wtl Mc 6W tl 3 I Ft Ft The Lewis equation : W t P FY d P d : diametral pitch, the reciprocal of module. Y is called Lewis form factor Copyright 00. All rights reserved. 7

Lewis Form Factor umber of tooth Y umber of tooth Y 0.45 8 0.353 3 0.6 30 0.359 4 0.77 34 0.37 5 0.90 38 0.384 6 0.96 43 0.397 7 0.303 50 0.409 8 0.309 60 0.4 9 0.34 75 0.435 0 0.3 00 0.447 0.38 50 0.460 0.33 300 0.47 4 0.337 400 0.480 6 0.346 rack 0.485 Copyright 00. All rights reserved. 8

Spur Gear Stress Calculation (II) Five factors are involved when calculating stress for spur gears W P t FY d K t K a K K v s K m K t : stress concentration factor. K a : applied factor, between ~3, depending on the smoothness of the applied load. K s : size factor, this factor must be considered when the module is more than 5. K m : load distribution factor, between ~3, must be considered for large width of tooth. K v : dynamic factor, between 0.5~, depending on the precision of the gears and tangential velocities. Copyright 00. All rights reserved. 9

Procedure for Selecting Spur Gears. Select module m of the gear according to the power transmitted and rotational speed.. ecide the number of teeth of the pinion. To avoid interference, 8 is a good number. ecide the number of teeth of the gear according to the required gear ratio. umber of teeth in pinion 7 6 5 4 3 Max. no. of teeth in gear w/o interference 309 0 45 6 6 Compute the tangential load W t. 4. Select a proper set of gears from manufacturers catalog. ecide the face width of the gear F. 5. Compute the stress of the gear. Is the stress too high? If so, reselect gears with larger m or F. Copyright 00. All rights reserved. 0

Helical gears In transmission applications, helical gears are more popular than spur gears. When transmitting power, there are two or more teeth in contact at the same time because the teeth make an helix angle with the axis. The transmission is more gradual, transmission noise is reduced, and the teeth are worn slower. In transmission, the helical gear will generate an axial force, which is its major drawback. Copyright 00. All rights reserved.

Helix Angle and Pressure Angles in Helical Gears The angle between the teeth and the axis of gear is called helix angle, which normally ranges form 5 to 45 degree. There are two pressure angles, transverse plane pressure angle and normal plane pressure angle. The module and circular pitch of helical gears are defined on the transverse plane. transverse plane pressure angle helix angle normal plane pressure angle y Copyright 00. All rights reserved.

Helical Gear Force Analysis n W n is the contact force along the pressure line. Its components are three-dimensional: W t is the tangential force, and W r is the radial force, and there is a new axial component W x. W r W n sin n W r W t tan t t W x W t tany W t W n cos n cosy z y x y n, t and y are not independent. tan tan cosy n t Copyright 00. All rights reserved. 3

Example 5. Calculate the Loads on Helical Gears For gear A: W t d T 54567 30 For gear B: 3638 () W t T 63703 90 3638 () W n cos n Wt cosy 3638 cos 0 cos30 4470 () W r W x 4470sin 0 59 () 3637.8 tan30 00 () Copyright 00. All rights reserved. 4

Bevel Gears Bevel gears are used to transmit power between two shafts that are perpendicular to each other. The sum of the cone angles of two matching bevel gears has to be 90 degrees. Copyright 00. All rights reserved. 6

Example 6. Automotive ifferential When the car is on a straight road, tires on both sides have equal traction, and there is no differential action. The gears and the differential case all turn as a unit. When the car turns, the inner tire has a shorter distance to travel and a higher resistance. outer shaft (00+0) rpm 00 rpm differential case inner shaft (00-0) rpm Copyright 00. All rights reserved. 8

Worm Gears worm Worm gears transmit power between two perpendicular shafts. Worm gears provide high reduction ratio and transmit high torque, but friction is high because there is sliding between the contact surfaces. There are 3 types of worm gears: the spur gear, the single-enveloping type and the double-enveloping type. worm gear Copyright 00. All rights reserved. 9

Worm Gear Terminologies The number of teeth of a worm ( w ) is defined as the number of teeth advanced in the worm gear with one revolution of the worm. Usually w =,, or 4. Axial pitch (p x ) is the distance along the axial direction of the worm between each thread. Lead (L) is the distance advanced along the axial direction with one revolution of the worm. lead angle axial pitch, p x lead, L L tan w L p w x w Copyright 00. All rights reserved. 3

Worm Gear Force Analysis Tangential force of the worm (W tw ) is equal to the axial force of the worm gear (W xg ). Axial force of the worm (W xw ) is equal to the tangential force of the worm gear (W tg ). Copyright 00. All rights reserved. 3

Belts and Chains Belts and chains are flexible transmission components. They can be used to transmit power between two distant shafts, and the center distance is basically adjustable. Belts are used in high rotational speed and low torque applications. Chains are used in low rotational speed and high torque applications. center distance angle of wrap pitch diameter S span Copyright 00. All rights reserved. 33

Procedure for Selecting Belts (I) Basic concerns when selecting belts and pulleys: the size and length of the belt, the pitch diameters and center distance of pulleys. Typical catalogue for belt: type Top width thickness angle 3V 3/8 (9.5mm) 5/6 (8.0mm) 5V 5/8 (5.9mm) 7/3 (3.5mm) 40 8V (5.4mm) 7/8 (.mm) Procedure for belt selection:.select proper belt size from the catalogue according to power transmission and applications. Copyright 00. All rights reserved. 34

Procedure for Selecting Belts (II).Select proper diameters of pulleys. Large bending force might occur if the pulley is too small. Stress on a belt is caused by tension and bending from wrapping the pulley. Proper linear speed for belt transmission ranges from 5-5 m/sec. Ideal linear speed is 0 m/sec. ecide the diameter of the small pulley from the linear speed and working rotational speed. ecide the diameter of the large pulley from the speed ratio. Pulleys should be of standard specifications and can be found in catalogues. 3.ecide the center distance between pulleys. For proper center distance: Angle of wrap (at least 0 ): C 3 sin C sin C Copyright 00. All rights reserved. 35

Procedure for Selecting Chains (I).Select proper chain size (ASI number 5, 30, 40,, 40) from the catalogue according to power transmission and applications..ecide the number of teeth of the sprocket ormally, the number of teeth of the small sprocket should be at least 7 and the large one at most 0, unless the speed is less than 00 rpm. The maximum speed ratio for chain transmission is 7.0. Multi-stage speed reduction should be used if higher speed ratio is desired. Copyright 00. All rights reserved. 37

Copyright 00. All rights reserved. 38 3. ecide the center distance of two sprockets. Center distance should be 30 to 50 times of the pitch. The angle of wrap of the small sprocket should not be less than 0 degree. 4. Calculate the length of the chain. The unit of chain length is pitch, and is an even number. With the calculated length L, determine the center distance C again. 5. ecide the number of chains we need. Lubrication, such as oil-dripping, oil-bathing or oil-spraying is important. C C L 4 4 8 4 L L C Procedure for Selecting Chains (II)

Bearings (I) Bearings are used in rotational shafts to sustain axial and radial loads and permits relative motion between shafts and housing. There are two major types of bearings, rolling contact bearing and journal bearing. A rolling contact bearing usually consist of an inner race, an outer race, a retainer and rollers or balls. Three important concerns for selecting bearings: the capability to sustain radial load, axial load, and misalignment. Single-row, deep-groove ball bearing animation inner race Angular contact ball bearing outer race rollers retainer Copyright 00. All rights reserved. 39

Bearings Capability to Sustain Loads Type Capability to sustain loads radial load axial load Misalignment Single-row, deep-groove ball bearing Good Acceptable Acceptable Angular contact ball bearing Good Excellent Bad Self-aligning bearing Good Acceptable Excellent Cylindrical roller bearing Excellent Bad Acceptable eedle bearing Excellent Acceptable Acceptable Spherical roller bearing Excellent Good Excellent Taper roller bearing Excellent Excellent Bad Copyright 00. All rights reserved. 4

Selecting Rolling Contact Bearings (I) Basic static load rating and basic dynamic load rating are the two most important specifications for rolling contact bearings. Basic static load rating is the maximum static load a bearing can sustain without permanent deformation. Basic dynamic load rating means that, under this dynamic load, the L 0 life of a bearing is,000,000 revolutions. Relation between loads and life of bearings: k=3.00 for ball bearing and k=3.33 for roller bearings Rated life, L 0 life: under certain rated load, 0% of the bearings will fail after L 0 life. L L P P k Copyright 00. All rights reserved. 44

Selecting Rolling Contact Bearings (II) When there are both radial and axial loads on the bearing, we need to calculate the equivalent load. P VXR YT R: radial load T: axial load V: rotation factor (V=.0 when the outer ring fixed; V=. when the inner ring fixed) Copyright 00. All rights reserved. 45

Selecting Rolling Contact Bearings (III) T/C 0 e T/Re T/Re X Y X Y 0.04 0.9.00 0 0.56.30 0.0 0..00 0 0.56.5 0.08 0..00 0 0.56.99 0.04 0.4.00 0 0.56.85 0.056 0.6.00 0 0.56.7 0.070 0.7.00 0 0.56.63 0.084 0.8.00 0 0.56.55 0.0 0.30.00 0 0.56.45 0.7 0.34.00 0 0.56.3 0.8 0.38.00 0 0.56.5 0.4 0.4.00 0 0.56.04 0.56 0.44.00 0 0.56.00 T/C 0 0.04 should be at least 04; C 0 is basic static load rating Copyright 00. All rights reserved. 46

Journal Bearings In journal bearings, there are no rolling contact elements. Rotation of the shaft is enable by lubrication. The friction of journal bearings depends on the coefficient of viscosity (μ), rotational speed (n) and pressure of the lubricant inside the journal bearing (p). n These three factors are usually combined into a bearing parameter. p Copyright 00. All rights reserved. 47

Bearing parameter vs. Coefficient of Friction The lubrication of journal bearings can be classified into boundary lubrication, mixed-film lubrication, and hydrodynamic lubrication. We hope to operate in hydrodynamic lubrication. Copyright 00. All rights reserved. 48