Chapter 8 – Kinematics of Gears

Gears! Gears are most often used in transmissions to convert an electric motor’s high speed

and low torque to a shaft’s requirements for low speed high torque: Speed is easy to generate, because voltage is easy to generate Torque is difficult to generate because it requires large amounts of current Gears essentially allow positive engagement between teeth so high forces can be

transmitted while still undergoing essentially rolling contact Gears do not depend on friction and do best when friction is minimized Basic Law of Gearing:

–A common normal (the line of action) to the tooth profiles at their point of contact must, in all positions of the contacting teeth, pass through a fixed point on the line-of-centers called the pitch point

–Any two curves or profiles engaging each other and satisfying the law of gearing are conjugate curves, and the relative rotation speed of the gears will be constant

Spur Gears

Teeth are parallel to the axis of the gear

Advantages Cost Ease of manufacture Availability

Disadvantages Only works with mating

gear Axis of each gear must

be parallel

Standard Spur Gears

(Boston Gear Catalog)

Helical Gears Teeth are at an angle to the gear

axis (usually 10° to 45°) – called helix angle

Advantages Smooth and quiet due to gradual

tooth engagements (spur gears whine at high speed due to impact). Helical gears good up to speeds in excess of 5,000 ft/min

More tooth engagement allows for greater power transmission for given gear size.

Parallel to perpendicular shaft arrangement – Fig 8.2

Disadvantage More expensive Resulting axial thrust component

Helical Gears

Mating gear axis can be parallel or crossed

Can withstand the largest capacity at 30,000 hp

Worm Gears

Gears that are 90° to each other

Advantages Quiet / smooth drive Can transmit torque at right

angles No back driving Good for positioning

systems Disadvantage

Most inefficient due to excessive friction (sliding)

Needs maintenance Slower speed applications

worm

worm gear

Bevel Gears Gear axis at 90°, based

on rolling cones Advantages

Right angle drives Disadvantages

Get axial loading which complicates bearings and housings

Spiral Bevel Gears

Same advantage over bevel gears as helical gears have over spur gears!!

Teeth at helix angle Very Strong Used in rear end

applications (see differentials)

Why Use Gears?1. Reduce speed

2. Increase torque

3. Move power from one point to another

4. Change direction of power

5. Split power

Generally this functionality is accomplished by many gears mounted in a gear box!

Boston Gear

Examples of “off the shelf” gearing

Other Drives

Splitter – One input with several outputs Right Angle – Transfers torque thru right angles, can

be as simple as mating bevel gearswww.gamweb.com/ power_series.htm

Types of Gear Boxes: http://en.wikipedia.org/wiki/Gear_box

Other Drives

Differentials Engines typically operate over a

range of 600 to about 7000 revolutions per minute (though this varies, and is typically less for diesel engines), while the car's wheels rotate between 0 rpm and around 1800 rpm. Engine: higher speed, lower torque versus wheels.

www.torsen.com/products/ T-1.htm

How a manual transmission works: http://en.wikipedia.org/wiki/Manual_transmission

How a differential works: http://en.wikipedia.org/wiki/Differential_(mechanical_device)

John Deere 3350 tractor cut in Technikmuseum Speyer Museum

Gears vs Belts and Chains

Gears are much more capable in terms of power rating (helical gear drives capable of > 30,000 hp)

With planetary gear sets large gear ratio’s can be achieved (100:1)

Gear applications include high torque and high speeds

Can have multiple speed reductions by pairing different gears or gear trains (several gears in series)

Gears used for Speed Reducer Recall the main purpose of mating/meshing gears is

to provide speed reduction or torque increase.

driver

driven

P

G

G

P

N

N

N

N

n

nVRRatioVelocity

Pinion

nP NP

Gear

nG NG

)2/(DRvspeedlinePitch t

)12/(min)/( Dnftvt

Example:

Want a 3:1 reduction NP=22 teeth

What is NG? Solution:

VR = 3 = NG/NP

NG = 3*22 = 66 teeth

Figure 8-15, pg. 322

Engine

Pump

n1, N1

n2, N2

n3, N3

n4, N4

Given:

n1 = 500 rpm, N1 = 20tN2 = 70t, N3 = 18t, N4 = 54t

Find: n4

Example: Double Speed Reducer

Solution:

1. n2 = 500 rpm*(20/70) = 142.8 rpm

2. n3 = n2

3. n4 = 142.8 rpm*(18/54) = 47.6 rpm

4. Total reduction = 500/47.6 = 10.5 (0r 10.5:1)

Torque?? Increases by 10.5!!Power?? Stays the same throughout!

Pinion

POWER

np

Law of Kinematics

Holds true if teeth have conjugate profile!!

Fig 8-7

Line drawn perpendicular at point of contact always crosses centerline at same place then VR = np/nG = constant

DEMO!

Spur Gear Nomenclature

Pitch Circle(s) The circles remain

tangent throughout entire engagement

Pitch Diameter Diameter of pitch circle

DP – Pitch of pinion

DG – Pitch of gear (power gear or driving gear)

(Driven gear)

Gear Nomenclature

N = Number of teeth Use subscript for specific gear

NP=Number of teeth on pinion (driver) NG=Number of teeth on gear (driven) NP < NG (for speed reducer) NA=Number of teeth on gear A

Circular Pitch, P is the radial distance from a point on a tooth at the pitch circle to corresponding point on the next adjacent tooth P=(D)/N

Gear Nomenclature Gear Train Rule – Pitch of two gears in mesh

must be identical

DG

NG

=PDP

NP

Gear Nomenclature

Diametral Pitch, (Pd) – Number of teeth per inch of pitch diameter

*Two gears in mesh must have equal Pd:

*Standard diametral pitches can be found in Table 8-1 and 8-2

DN

=Pd

DG

NG ==Pd DP

NP

Gear Nomenclature

Figure 8-8

More Gear Nomenclature: http://en.wikipedia.org/wiki/List_of_gear_nomenclature

Gear Formulas Courtesy of Boston Gear

Gear Formulas Courtesy of Boston Gear (cont’d)

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Go to http://www.bostongear.com/pdf/gear_theory.pdf for the complete 18 page PDF on gearing Engineering Information

Gear Geometry

Spur Gears Tooth Profile – Conjugate

shape Conjugate Profile

Tooth is thicker at base, maximum moment

σ = M/s Pressure Angle (φ) - angle

between tangent and perpendicular line to gear tooth surface

Allows constant velocity ratio between mating gears and smooth power transmission

Conjugate profile

Fillet Radius

Force perpendicular at

Φ = 14.5˚ Φ = 20˚ Φ = 25˚

Pressure Angle

Figure 8-11

Gear Nomenclature Example

8-1) Gear has 44 teeth, full depth involute form diametral pitch Pd = 12 Pitch Diameter

Circular PitchPd

NG 3.667 inch==DG 12 t/in

44 teeth=

NG

DG.2617 in/t==Pc

44 t3.667in

=

Gear Nomenclature Example

Addendum

Dedendum

Pd1

.0833 in==a12 t/in

1=

Pd1.25

.1042 in==b12 t/in

1.25=

Gear Nomenclature Example

Clearance

Whole Depth

ht = a+b = .1875 in

Working Depth

hk = 2*a = .16667 in

Pd.25

.0208 in==c12 t/in

.25=

Gear Nomenclature Example

Tooth Thickness

Outside Diameter

2

PC .1309 in==t2

.2617in=

Pd

N+22.833 in==O.D. DO =

Gear Nomenclature Notes

Clearance maybe a problem for small pinions driving large gears, therefore they won’t mesh and will lock up (See Table 8-6)

As NP decreases so does max NG

If design necessatates small pinion, maybe able to increase clearance by undercutting gear tooth (See Figure 8-14)

Summary of Gear Nomenclature:

DP = Pitch diameter of pinion

DG = Pitch diameter of gear

NP = No. teeth (t) for pinion

NG = No. teeth (t) or gear

Pd = diametral pitch = N/D = constant for meshing gearsp = circular pitch = D/N = constant for meshing gearsnP = speed of pinion (rpm)

nG = speed of gear (rpm)

VR = velocity ratio = nP/nG = NG/NP

Power = constant across mating gears or series system:Pin = Pout

Power in branched system is conserved:Pin = PA + PB + …..

Torque will change!!

rpm

hpinlbTorque

000,63)(

Conclusion:

•Total speed reduction = 1750/68 = 25.7

•Torque increase = 25.7

•Power = constant!!

Gear Trains

Train Value = TV = Product of the values for each gear pair in the train

ninnout

==TV (VR1)(VR2). . . .

Gear Train Alternate Solution

=TV (VR1)(VR2)(VR3)

3022

=TV 8.4=*6830 *

6825

ninout

=TV

=ninout =TV

208 rpm ccw=1750 rpm

8.4

Tout = 8.4 Tin !! Lots of Torque

YouTube Gear Animations:

Speed Reducers: http://www.youtube.com/watch?v=7LReoWPg_pM&feature=related   http://www.youtube.com/watch?v=1_jbZVBXjWc&feature=related   Automotive Differential:

http://www.youtube.com/watch?v=iBLE0_Sjqw4&feature=related Manual Transmission:

http://www.youtube.com/watch?v=MBmLJCeGu7o&feature=related Gear Cutting: http://www.youtube.com/watch?v=fps0OR1eF_s&feature=related http://www.youtube.com/watch?v=xF9CjluRFJ4&feature=related