Transcript Spur Gears - School of Engineering | Penn State

Chapter 8 – Kinematics of Gears
Gears!
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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
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Teeth are parallel to the
axis of the gear
Advantages
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Cost
Ease of manufacture
Availability
Disadvantages
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Only works with mating
gear
Axis of each gear must
be parallel
Standard Spur Gears
(Boston Gear Catalog)
Helical Gears
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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
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Mating gear axis can be
parallel or crossed
Can withstand the
largest capacity at
30,000 hp
Worm Gears
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Gears that are 90° to each
other
Advantages
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Quiet / smooth drive
Can transmit torque at right
worm gear
angles
No back driving
Good for positioning
systems
Disadvantage
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Most inefficient due to
excessive friction (sliding)
Needs maintenance
Slower speed applications
worm
Bevel Gears
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Gear axis at 90°, based
on rolling cones
Advantages
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Right angle drives
Disadvantages
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Get axial loading which
complicates bearings and
housings
Spiral Bevel Gears
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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.
2.
3.
4.
5.
Reduce speed
Increase torque
Move power from one point to another
Change direction of power
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
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Splitter – One input with several outputs
Right Angle – Transfers torque thru right angles, can
be as simple as mating bevel gears
www.gamweb.com/ power_series.htm
Types of Gear Boxes: http://en.wikipedia.org/wiki/Gear_box
Other Drives
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Differentials
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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.o
rg/wiki/Differential_(
mechanical_device)
John Deere 3350 tractor cut in Technikmuseum Speyer Museum
Gears vs Belts and Chains
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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
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Recall the main purpose of mating/meshing gears is
to provide speed reduction or torque increase.
Pitch line speed  vt  R  (D / 2)
nP N G N driven
Velocity Ratio  VR 
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nG N P N driver
Pinion
Gear
nP NP
nG NG
vt ( ft / min)  (Dn / 12)
Example:
Want a 3:1 reduction
 NP=22 teeth
 What is NG?
 Solution:
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VR = 3 = NG/NP
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NG = 3*22 = 66 teeth
Figure 8-15, pg. 322
n1, N1
n4, N4
Engine
Pump
Given:
n2, N2
n1 = 500 rpm, N1 = 20t
N2 = 70t, N3 = 18t, N4 = 54t
n3, N3
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!
Line drawn perpendicular at
point of contact always
crosses centerline at same
place then VR = np/nG =
constant
Law of Kinematics
Holds true if teeth
have conjugate
profile!!
DEMO!
Fig 8-7
Pinion
POWER
np
Spur Gear Nomenclature
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Pitch Circle(s)
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The circles remain
tangent throughout entire
engagement
Pitch Diameter
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Diameter of pitch circle
DP – Pitch f of pinion
DG – Pitch f of gear
(power gear or driving gear)
(Driven gear)
Gear Nomenclature
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N = Number of teeth
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Use subscript for specific gear
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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
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Gear Train Rule – Pitch of two gears in mesh
must be identical
DG
P=
NG
DP
NP
Gear Nomenclature
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Diametral Pitch, (Pd) – Number of teeth per inch of
pitch diameter
N
Pd =
D
*Two gears in mesh must have equal Pd:
NG
NP
=
Pd =
DG DP
*Standard diametral pitches can be found in Table 8-1
and 8-2
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
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Spur Gears
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Tooth Profile – Conjugate
shape
Conjugate Profile
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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
Fillet Radius
Pressure Angle
Force
perpendicular at f
Φ = 14.5˚
Φ = 20˚
Φ = 25˚
Figure 8-11
Gear Nomenclature Example
8-1) Gear has 44 teeth, 20, full depth
involute form diametral pitch Pd = 12
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Pitch Diameter
NG
44 teeth
=
DG =
12 t/in
Pd
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= 3.667 inch
Circular Pitch
 DG
() 3.667in
= .2617 in/t
=
Pc =
NG
44 t
Gear Nomenclature Example
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Addendum
a =
1
=
Pd
1
= .0833 in
12 t/in
Dedendum
1.25
1.25
b=
=
= .1042 in
Pd
12 t/in
Gear Nomenclature Example
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Clearance
.25
.25
c =
=
= .0208 in
Pd
12 t/in
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Whole Depth
ht = a+b = .1875 in
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Working Depth
hk = 2*a = .16667 in
Gear Nomenclature Example
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Tooth Thickness
PC
t =
=
2
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.2617in
2
= .1309 in
Outside Diameter
O.D. = DO =
N+2
Pd
= 2.833 in
Gear Nomenclature Notes
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Clearance maybe a problem for small pinions driving
large gears, therefore they won’t mesh and will lock
up (See Table 8-6)
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As NP decreases so does max NG
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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 gears
p = circular pitch = D/N = constant for meshing gears
nP = 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!!
Torque(lb  in) 
63,000 hp
rpm
Conclusion:
•Total speed reduction =
1750/68 = 25.7
•Torque increase = 25.7
•Power = constant!!
Gear Trains
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Train Value = TV = Product of the values for each
gear pair in the train
nin
TV =
= (VR1)(VR2). . . .
nout
Gear Train Alternate Solution
TV = (VR1)(VR2)(VR3)
30 68 68
= 8.4
TV =
*
*
22 30 25
ni
TV = n
out
nout =
ni
TV
=
1750 rpm
8.4
= 208 rpm ccw
Tout = 8.4 Tin !! Lots of Torque
YouTube Gear Animations:
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Speed Reducers:
http://www.youtube.com/watch?v=7LReoWPg_pM&feature=related
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http://www.youtube.com/watch?v=1_jbZVBXjWc&feature=related
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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