Machine Components: Power
Transmission Elements
ME 72 Engineering Design Laboratory
Power Transmission Design
Issues
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Power source (motor, engine, etc.)
Power required (P = torque × velocity)
Continuous or Intermittent Motion
Operating conditions (start, load duration)
Magnitude of speed (input or output)
Speed modification (output to input,
constant or variable, linear or rotary)
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Constant Speed Mechanical
Transmission Elements
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Gears
Friction wheels
Power Screws
Belts and Chains
Wire Rope
Couplings
Clutches and Brakes
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Relative Shaft Position
Relative Size
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Gear Design Issues
• Types of Gears: spur, helical, herringbone,
internal, rack, bevel, and worm
• Angular Velocity Ratio (gear ratio)
• Power Requirements (speed and torque)
• Gear Tooth Loads
• Gear Tooth Stresses
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Gear Terminology
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Pitch Circle (radius or diameter)
Circular Pitch
Number of Teeth
Pressure Angle
Face Width
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Spur Gears: Force Analysis
• W is the resultant force
• Wt and Wr are tangential and
radial components
• d is the pitch diameter
• φ is the pressure angle
• τ is the applied torque
2τ
d
Wt
W=
cos φ
Wt =
Wr = Wt tan φ
Helical Gears: Force Analysis
• Wt, Wr and Wa are tangential
and radial and axial components
• φn is the normal pressure angle
• φt is the tangent pressure angle
• ψ is the helix angle
2τ
d
Wr = W sin φ n = Wt tan φt
Wt =
Wt = W cos φ n cos ψ
Wa = W cos φ n sin ψ = Wt tan ψ
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Bevel Gears: Force Analysis
• Wt, Wr and Wa are tangential
and radial and axial components
• φ is the pressure angle
• γ is the bevel angle
• rav is the average pitch radius
Wt =
τ
rav
Wr = Wt tan φ cos γ
Wa = Wt tan φ sin γ
Worm Gears: Force Analysis
• Wt, Wr and Wa are tangential
and radial and axial components
• φn is the normal pressure angle
• λ is the worm lead angle
• dworm is the pitch diameter of
the worm
Wt =
2τ
dworm
W = Wt = W cos φ n sin λ
x
W y = Wr = W sin φ n
W z = Wa = W cos φ n cos λ
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Stresses in Gears: Lewis Formula
σb =
Mc Wt l( 2t ) 6Wt l
= b t3 =
w
I
bw t 2
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t2
4x
Wp
3Wt
3Wt pd
σb =
=
= t d
2bw x 2bw pd x bw Y
l=
Y=
2 pd x
= Lewis Form Factor
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Stresses in Gears: AGMA Formula
• Assumptions
– Contact ratio is 1-2
– No interference or
undercutting
– No pointed teeth
– Nonzero backlash
– Standard root fillets
– Friction neglected
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σb=bending stress
Wt=tangential force
pd = pitch diameter
bw=face width
J=geometry factor
Ka=application factor
Km= load distribution factor
Ks=size factor
Kv=dynamic factor
σb =
Wt pd K a K m K s
bw J
Kv
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Power Screws
• Consists of a threaded
shaft and nut
• Used to convert rotary
to linear motion
• Can be designed for
high precision
• Often have multiple
threads
• Non backdrivable
Power Screw Force Balance
• Basic force balance
∑F
= F − µN cos λ − N sin λ = 0
∑F
= N cos λ − µN sin λ = 0
x
– Rewrite in terms of L using:
tan λ =
F = N ( µ cos λ + sin λ )
y
N=
P
(cos λ − µ sin λ )
F=P
T=
L
πd p
( µ cos λ + sin λ )
(cos λ − µ sin λ )
Pd p ( µ cos λ + sin λ )
2 (cos λ − µ sin λ )
Tup =
(
)
−
d
L
π
µ
( p )
Pd p µπd p + L
2
Tdown =
(
)
(πd p + µL)
Pd p µπd p − L
2
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Belt Drives
• Belt drives are used when large distances
between shafts make gears impractical or
when designated speed is too high for chain
drives.
• Belts are used with pulleys or sheaves to
transmit power.
• Belts require tensioning, and are prone to
slip under high loads.
Types of Belts
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Chain Drives
• Chains transmit power
through interlocking links
wrapping on a sprocket.
• Chain drives have a high
load capacity.
• They can be used to
transmit power or impart
timed linear motion.
Types of Chains
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Chain Attachments
Belt/Chain Length
AB = cd 2 −
2
AB =
1
2
(
cd 2 − ( D2 − D1 )
φ1 = π − 2α
φ2 = π + 2α
α = sin −1
)
D2 − D1 2
2
(
D2 − D1
2 cd
2
)
L = 2 AB +
D1
2
φ1 +
D2
2
φ2
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Wire Rope
• Wire rope can also be
used to transmit power.
• It is characterized by the
number of bundles and
wires/bundle.
• The critical design
parameter is the pulley
to rope diameter ratio.
Summary
• Gears transmit power between a variety of
intersecting and non-intersecting shafts.
• Gear forces and tooth stresses are key design
parameters for gear trains.
• Power screws convert rotary to linear motion and
produce high forces from small torques.
• Belts and Chains transmit power through larger
distances than gears.
• Wire rope (or cable) can provide backlash-free
power transmission for a limited range of motion.
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References
• Hindhede, U., Zimmerman, J., Hopkins, B., et al.,
Machine Design Fundamentals: A Practical Approach,
Englewood Cliffs: Prentice Hall, 1983.
• Shigley, J., and Mischke, C., Mechanical Engineering
Design, 5th Ed., San Francisco: McGraw-Hill Inc., 1989.
• Norton, R., Machine Design: An Integrated Approach,
Upper Saddle River: Prentice-Hall, 1998.
• Hamrock, B., Jacobson, B., and Schmid, S., Fundamentals
of Machine Elements, San Francisco, WCB McGraw-Hill,
1999.
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