Chapter 8 – Kinematics of Gears
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)
Double ClickOn Image to Print
PDF(will not work in presentation mode)
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