Gear Forces and Gear Train

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Power transmission Components used to transmit power: gears, belt, clutch and brakes.

Gear Objective: Student must be able to do force analysis, stress analysis using basic formula (Lewis) and AGMA (bending stress and surface stress)

Type of gear: Spur gear only

a)

Spur Gear

 

b)

Helical gear

   

c)

Teeth is parallel to axis of rotation Can transmit power from one shaft to another parallel shaft

Bevel gear

Teeth is inclined to the axis of rotation Smoother than spur Develop thrust load (helix angle) Can transmit power from one shaft to a parallel and non-parallel shaft

2

 

d)

Teeth on conical surfaces Transmit power between two intersecting shafts

Worm gear 

Transmit power between intersecting shafts

two

3

Formation of Spur Gear i. Friction drive Represented by two pitch circles. Smaller gear: pinion (denoted by letter p) Larger gear: gear (denoted by letter g)

ii. Belt Drive:  Another base circle is introduces represent the base circle of the pulley.

iii. Involute gear tooth drive:

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Terminologies  A pair of gears can be represented as 2 circles Metric Pinion Gear

d2 = N2m d3 = N3m 2

where: N: number of teeth m: module in mm note: mating gear must have same m English Unit Pinion

d 2

Gear

d 3 



 N 2 P

 N 3 P

Where P is diametral pitch (in -1) Relationship between diametral pitch and m m

25.4 P

3

5

Module: is the ratio if diametral pitch and number of teeth m = d/N [mm] Face Widt h (F) : width of the tooth  Addendum [a]  : distance between top face of the tooth to pitch circle Dedendum [b]  : distance between pitch diameter to bottom of the gear In the following : we only concentrates on full depth gear 

Full depth tooth

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When the offset occurs between pitch.  not full-depth tooth, which is called stub

Undercut : resulted from number of tooth is less than the minimum number of tooth suggested. Results: higher stresses at the root of the tooth. (Refer to section 13-7 interference)

Backlash  : gap between mating tooth. The gap can be used for lubrication

Contact Ratio : the number of tooth in contact during meshing. Roughly  spur gear (1.4 to 1.8 )

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Pinion and Gear  : pinion is the smaller gear and gear is the bigger gear

Gear Parameters Metric Unit Module m = d/N English Unit Diametral Pitch P= N/d (inverse to module) One pair of gear must have the same module Pressure Angle: 200, 22.5o, 250

English Unit  Available Size

SI

8 12 pitch gear

 

12 teeth per inch of pitch

6 mm module

diameter Means 12 pitch gear  1

in pitch diameter will have 12T

1.5

in pitch diameter will have 18T

Fillet at the root of the tooth = 0.35/P Range of face width

= m/3

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The addendum circle

r a  r   a

Where r a: addendum circle radius r: pitch circle radius a: addendum

The addendum is

a = 1/P or a = m

The maximum addendum circle without interference is r a (max)  r b2  c 2 sin(  )

If r a > r a(max)  interference

Undercut happens when pinion is less than 18T for 20 o pressure angle, less than 12T for 25o pressure angle.

Contact ratio id

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Example An 18T pinion having 20o  pressure angle with a 36T gear. If the maximum center to center is set 6 in. The pinion has stub teeth and full depth. a. Determine the suitable dimetral pitch and its center to center distance  b. Check whether interference will occur c. Determine the contact ratio

Gear Train

2

V2 = V3

3

Known that

V



 dn 60

where : n : revolution / min it

 d 2 n2   d 3 n3

d 2 n2  d 3 n3

Equation 1

For a pair of gear m 2 = m3 d d2  3 N2 N3

Equation 2

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From Eq 1 and 2

d 2 N2 n3   d 3 N3 n2 Significance:  d increases N increase  d increases n reduces  to reduce rpm requires small pinion and larger gear and vice versa.

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Gear Train (continued)

3

2

4

V2 = V3 and V3 = V4 Therefore

V2 = V3 = V4 N2 From previous formula: N3



N2 n2

n3 n2 

N3 n3

Gear 2 and 3 n3



N2 N3

n2

… (1)

Gear 3 and 4 n4



N3 N4

n3

…(2)

Eq (1) in eq (2)

n4 

nL 

N3 N2 n2 N4 N3

product of  driving tooth numbers nF product of  driven tooth numbers where:

Train value

e

nL : rotational speed of last gear (output) nF : rotational speed of first gear (input)

product of  driving tooth numbers product of  driven tooth numbers

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Planetary Gear

5 4 2 3

Gear 2: Sun gear Gear 3: Arm Gear 4: Planet Gear Gear 5: Ring Gear 

 Assumption Arm Fixed:

4 2 3

Train value

e  

N 2 N 4 N5 N 4 N5 N6

5

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3 MAGIC FORMULAE FOR FBD ANALYSIS ON GEAR Metric Unit Torque

 d    F t    2 

T   

[ Nm]

Ft: tangential force Speed

V  

 dn

60

[m / s ]

d: pitch diameter in [m] n: rotational speed [rpm] Power

 H   F t V 

[ watts]

Englis h Unit Torque

 d   W t    2 

T   

[ Nm]

Ft: tangential force (Ib) Speed

V  

 dn

60

[ ft  / m]

d: pitch diameter in [ft] n: rotational speed [rpm] Power

W   F t V  / 33000

[hp]

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Force Analysis (Free Body Diagram)

Input rpm direction: cw

To transfer power, T must exist. When the pinion rotates, tooth from gear against the movement  direction W t32 must against the direction of rotation Wt32 = H / V T2 = Wt32. d 2/2 Due to pressure angle, W r32 (radial force) is generated Wr32  = Wt32 tan

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On Gear 3,

W t23 and W r23 must in the opposite direction. To be statically analytical, T 3 is against W t23

T3 = Wt23. d 3/2 Note: Wt23 can be calculated using Wt32= H/ V, please remember that all the parameters must be based on gear 3.

Example 1 (Q1 in the exercise handout)

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