gear system analysis

8/10/2019 Gear System Analysis

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Part VII: Gear Systems: Analysis

This section will review standard gear systems and will provide the basic tools to perform

analysis on these systems. The areas covered in this section are:

1) Gears 101: The geometric details about standard gears (involute)

2) alient features of involute gears

!) Gear"tooth geometry e#uations$) Gear train systems: fi%ed"a%is and &lanetary

') Types of Gears

Gears 101: Details about the involute gear profile:

Gears were created to transmit constant"velocity rotating motion between shafts relying oninematic contact (not friction) to transmit forces. ecall that in order to have constant velocity*

the line of action and line of centers must intersect at a constant location:

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Therefore* the point of contact between the two gears must lie along this line of action. 3nynumber of arbitrary* con4ugate shapes could be defined to complete this tas* however two

profiles of significance wor: involute and cycloidal profiles. The involute profile is the

standard for gear teeth* and is uni#ue in that the involute is con4ugate to itself (at any point alongits profile) to maintain a constant intersection of the line of action and line of centers. The

involute is easy to manufacture and does not depend on distance between gear centers.

5ased on this involute geometry of gear teeth* the geometry of a gear can be standardi6ed andnamed* as in the following figures. The nature of tooth contact is described as well on these

figures.



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5ased on this* the velocity ratio between gears is given as:

onditions of nterchangeability (7or tandard Gears)

1.

2.

!.

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Salient Features of Involute Gears:1. ..

2.

!.

$.

'.

-.

8.

9.

.

10. alient: (sa; li"ent) adj.tanding out from the rest< noticeable< conspicuous< prominent.

(=ebster;s* ollege ,d.)

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Details of Involute Gears

1. ,#uations not found in /orton:

(efer to 7ig. 1)

5ase pitch (distance between one tooth set measured along base circle):

(1)

>ength of action:

(2)

ontact ratio (average number of teeth in contact):

(!)

?iametral &itch (number of teeth per inch):

($)

+odule (mm per tooth):

(')

+inimum number of teeth to avoid interference: (k@1 for full depth teeth)

a) for a rac:

(-)

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b) for two gears in mesh:

(8)

enter distance:

(9)

AoperatingA center distance and pressure angle:

()

5aclash resulting from an increased operating center distance:

(10)

Tooth thicness: (re#uires the tooth thicness at some radius to be nown* generally at the pitch

circle):

(11)

adius and angle at various points along the involute:

(12)

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Gear-rain Systems:

Gears are used in combinations to create a desired tor#ueBvelocity ratio. ombinations of gears

can be divided into two classes: 7i%ed"a%is gear trains* and planetary gear trains.

7i%ed"a%is gear trains:

The sign change occurs for e%ternal gears. The diameters listed are the pitch diameters. /ow

consider a series of gears in mesh:

n this arrangement* the intermediate gears do not affect the overall velocity ratio* and therefore

should be replaced with a more cost effective means of power transmission. nly the outer twogears are useful in achieving the desired velocity ratio. ince the velocity ratio of a single gear

set is practically limited to 10:1 or less (actually* more lie ':1 greater)* compound gears (two

gears constrained to have the same angular velocity) are used in gears trains to achieve largervelocity ratios:



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Planetary Gear rains:

3 &lanetary gear train (see 7ig. below) results when certain gears in the train (called the planet

gears) have moving a%es. The arm* while not a gear* is an essential part of the planetary becauseit defines the motion of the moving planet gear a%es. The planetary is also uni#ue to a standard

gear train in that it re#uires two inputs to define one output (verify this using mobility). 3 good

e%ample is your car;s differential* which has two inputs: one the drive"shaft* and the second aconstraint between the two driven wheels provided by whatever you are driving on (e.g. dry

pavement* one wheel on ice* etc.)

The planetary gear train consists of three parts:

1.

2.

!.



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Planetary Gear !"uation:

The planetary gear train e#uation must be used to solve the angular velocities of elements in the

planetary. The e#uation is:

where:

f* and lidentify two gears in the planetary (call them first and last)*

arepresents the arm*

wla*

wfa*

wla / wfa

wl*

wf

wa.

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#hoosingf, l$ an% a:

hoosing elements for the first* last* and arm is the first step in solving a planetary. olving will

fall into one of the three following scenarios (remember that you must now two pieces of

information to solve the planetary e#uation).

&ase iCou want to find the arm velocity* (wa is not nown) nowing the velocity of two gears:hoose f and l as the two nown gears* and the arm as a* an unnown. olve for wa.

&ase ii Cou want to find the velocity of a gear* and you now the velocity of the arm and one

other gear:hoose l as the desired unnown gear* choose f as the nown gear and a as the nown arm. olve

for wl.

&ase iii Cou want to find the velocity of a gear* and you now the velocity of two gears but not

the arm.7irst* choose f and l as the nown gears and solve the arm velocity* wa. Then go to case ii.

'i(e% Gear trains:

3 general gear train can include both fi%ed a%is and planetary gear trains* or multiple planetaries.olving systems lie these re#uires using the procedures outlined above and looing for elements

that share the same angular velocity between the mi%ed gear trains.



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Gear ypes:

3 gear train consists of one or more gear sets intended to give a specific velocity ratio* or changedirection of motion. Gear and gear train types can be grouped based on their application and

tooth geometry.

Table I: Gear Types Grouped According to Shaft Arrangement

Parallel A(es Interse&ting A(es)on-Interse&ting

*)on-parallel+ A(es

,otary to

ranslation

Spur gears (7ig. 1): pur gears connect parallel shafts* have involute teeth that are parallel tothe shafts* and can have either internal or e%ternal teeth. /otes:

1.

2. .

!.



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eli&al gears (7ig. 2): Delical gears also connect parallel shafts* but the involute teeth are cutat an angle (called the heli% angle) to the a%is of rotation. /ote that two mating helical gears

must have e#ual heli% angle but opposite hand. These are found in automotive transmissions* andany application re#uiring high speed rotation and good performance. /otes:

1.

2.!.

$.

erringbone gears (7ig. !): To avoid a%ial thrust* two helical gears of opposite hand can bemounted side by side* to cancel resulting thrust forces. These are called double helical orherringbone gears



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.evel gears (7ig. $): 5evel gears connect intersecting a%es* and come in several types (listedbelow). 7or bevel gears* the pitch surface is a cone* (it was a cylinder in spur and helical gears)

and mating spiral gears can be modeled as two cones in rolling contact. Types of bevel gears:

1. traight bevel: These are lie spur gears* the teeth have no heli% angle. traight bevelgears can be

a. +iter gears* e#ual si6e gears with a 0 degree shaft angle*b. 3ngular bevel gears* shaft angle other than 0 degrees* orc. rown gears* one gear is flat* has a pitch angle of 0 degree.

2. piral bevel gears(7ig. $a): Teeth have a spiral angle which gives performance

improvements much lie helical gears!. Eerol bevel gears (7ig. $b): Teeth are crowned* so that tooth contact taes place first at

the tooth center.

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ypoi% gears (7ig. '): imilar to spiral bevel gears* but connect non"parallel shafts that do notintersect. The pitch surface of a hypoid gear is a hyperboloid of revolution (rather than a cone*

the pitch surface in bevel gears)* hence the name.

#rosse% heli&al gears (7ig. -): Delical gears that connect sew shafts. The teeth have slidingmotion and therefore lower efficiency. ne application is connecting distributer to cam shaft inpre"electronic ignition vehicles.

/orm Gears (7ig. 8): The driving gear is called a worm* and typically has 1* 2* or four teeth.The low number of teeth on the worm can result in a very large velocity ratio. These can also be

designed to be non"bacdriveable* and can carry high loads. 5ecause of sliding action* efficiency

is low.

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,a& an% Pinion (7ig. 9): These transmit rotary motion (from the pinion) to translationalmotion (of the rac). The rac is a gear with infinite radius< its teeth* although flat sided* are

involute. The rac and pinion is commonly used in steering units and 4acs.

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gear system analysis

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