1 gears
-
Upload
solo-hermelin -
Category
Science
-
view
702 -
download
2
Transcript of 1 gears
1
Gears
SOLO HERMELIN
Updated: 26.11.06Run the Power Point
http://www.solohermelin.com
2
Antikythera Mechanism (cc. 87 B.C.)
Antikythera mechanism is an ancient mechanism,made from bronze in a wooden frame, is believed tobe an analog computer designed to track the movementsof heavenly objects. The device the motion of sun, moonAnd Mercury, Venus, Mars, Jupiter and Saturn theCelestial body known to Ancient Greeks.
http://www.voyager.in/Antikythera_mechanism
Discovered in 1900in a shipwreck of theGreek island ofAntikythera.
SOLO
3
Antikythera (87 B.C.(SOLO
Original Reconstruction 1
Reconstruction 2Front view
Reconstruction 2Back vew
4
Antikythera (87 B.C.(SOLO
36842.1319
254
32
127
24
48
38
64 ==××
Moon to Sun ratio In 19 years, the moon goes through 235 phase cycles
from fool moon to crescent moon and back to fool moonand passes through the zodiac 254 times.
7
Pitch Circles
P
O 1
O 2
R 1
R 2
ϕ
Pressure
Line RB1
RB2
BaseCircles
Spur Gear Nomenclature
Pitch circles – Theoretical circles upon which all calculations are based with centers O1 and O2 and diameters D1 and D2. The two pitch circles are tangent at
the pitch point P.
Pressure Line – The line passing through P and making an angle φ with the tangent to the pitch
circles along which the tooth of one gear presses the tooth of the second gear.
Base circles – Circles tangent to pressure line with centers O1 and O2 and diameters
DB1 and DB2.
ϕϕ cos&cos 2211 DDDD BB ==
Circular pitch p – The distance measured along the pitch circle from a point on one tooth to the
corresponding point on the adjacent point. Since we assume that the two
gears always maintain contact during rotation the circular pitches are equal.
N1, N2 – Number of teeth on the two gears.
ω1, ω2 – Angular rates of the two gears.
2211 ωω NN =
2
2
1
1
N
D
N
Dp
ππ ==
1
2
2
1
2
1
ωω==
N
N
D
D
SOLO
8
Spur Gear Teeth
Spur Gear Tooth shape must be such that contact between the tooth of one gear to thecorresponding tooth of the second gear iscontinuously maintained until the contact occurs with the adjacent tooth.
If the shapes of the teeth on the two gearshave this property they are called conjugates.
From the figure we can see that during therotation of the driver the contact begins atpoint A and continues until it ends at point B.
The segment AB is on the pressure line.
Among infinite possibilities of conjugateshapes the one that is almost exclusively usedin the gear design is the involute.
SOLO
9
Spur Gear Teeth
Among infinite possibilities of conjugateshapes the one that is almost exclusively usedin the gear design is the involute.
The involute of a circle (base circle) is obtainedby an imaginary string IA (see Figures) wound on
the circle and then unwounded (IC, IB(, while holding it taut.
Base circleO
β
I
θ ψA
β1BR
β1BR
r
x
y
Cβ
Cθ Cr
( )θ,rB
( )CCrC θ,
( )( )βββ
βββcossin
sincos
1
1
−=+=
B
B
Ry
Rx
The involute equation
The involute has the following properties:
1. All lines normal to the involute are tangent to the base circle
2. The radius of curvature of the involute at a point P (r,θ) is given by ρ=RB1β. The base circle is the locus of the center of curvature
of the involute.
SOLO
10
Spur Gear Teeth
In the same way
Point A on the involute has the radius RA and thethickness tA along the circle RA.
SOLO
AB
AR
BR
bRAφ
Bφ
2/Bt2/At
O
E
D
FG
toothinvolute
base circle
B
involute
OG
BG
OG
DGDOG φtan===∠
∩
BBBB invDOGDOB φφφφ =−=−∠=∠ :tan
AAA invDOA φφφ =−=∠ :tan
Point B on the involute has the radius RB and thethickness tB along the circle RB.
A
A
AA
A
R
tinv
R
tDOADOE 2
121
+=+∠=∠ φ
B
B
BB
B
R
tinv
R
tDOBDOE 2
121
+=+∠=∠ φ
−+= BA
B
ABB invinv
R
tRt φφ
22
11
Width ofspace
Face width
Top land
Addendumcircle
Tooth
thicknessAddendum
Dedendum
Dedendum
circle
Clearencecircle
Clearence
Pitch
circle
Face
Flank
Bottom la
nd
Spur Gear Teeth SOLO
Addendum – the radial distance between the top land of the tooth and the pitch circle
Dedendum – the radial distance between the bottom land of the tooth and the pitch circle
Addendum circle – the circle passing through the top land of the tooth
Dedendum circle – the circle passing through the bottom land of the tooth
Clearance circle – the circle tangent to the addendum of the mating gear
12
Spur Gear Force Equation SOLO
If no backlash R1θ1 = R2θ2
21
21 θθ
=
R
R
2
2
1
1
N
D
N
Dp
ππ ==
Mating condition of the teeth on the two gears is
Circular path of gear 1 = Circular path of gear 2
1
2
1
2
1
2
N
N
D
D
R
R ==2
1
22
1
21 θθθ
=
=
N
N
R
R
The force applied by gear 1 on gear 2 is Fn . Since the tooth surface shape is an involute and the force at the point of contact is normal to the surface, it will be tangent to the base circle that defines the involute. Therefore Fn will always be on the pressure line and tangent to both base circles.
ϕϕ cos
2
23'
cos
1
1
21 R
b
rLawsNewton
n
R
b
r
R
TF
R
Tth
==
22
12
2
11 rrr T
N
NT
R
RT ==
Pressure line
Base circle
Pitch circle
Pitch circleBase circle
P B
Pinion(driver)
Gear(driven )
C
O1
ϕBegin
contact
EndcontactA
nF
nF
1bR1R
2bR
2R
ϕ
ϕ
Tr1, Tr2 are the reaction moments onthe two gears
13
Spur Gear Equations of Motion SOLO
Rotor
Stator
Gimbal
Gears
Motor
Body
1θ
1R
2R
2θ2T
1rTBω
( )Bbn JRFT ωθ −=− 1111
( )Bbn JTRF ωθ +=− 2222
1
2
2
1
θθ
=
N
N
we obtain
multiply the second equation by and add both equations using 2
1
2
1
2
1
b
b
R
R
R
R
N
N ==
BJN
NJJ
N
NJT
N
NT ωθ
−−
+=
− 2
2
1112
2
2
112
2
11
BJN
NJJ
N
NJTT
N
N ωθ
−+
+=−
1
1
2221
2
1
2221
1
2
multiply the second equation by and subtract from first2
1
2
1
2
1
b
b
R
R
R
R
N
N ==
Bbn JN
NJJ
N
NJT
N
NTRF ωθ
+−
−−
+= 2
2
1112
2
2
112
2
1112
Bbn JN
NJJ
N
NJT
N
NTRF ωθ
+−
−+
+= 1
1
2221
2
1
221
1
2222
1
2
N
N×
1
2
N
N×
14
Dedendum circle
Pressure line
Base circle
Pitch circle
Addendum circle
Addendum circlePitch circleBase circle
Dedendum circle
Angle ofapproach
Angle ofrecess
Angle ofapproach
Angle ofrecess
PA
B
Pinion(driver)
Gear(driven )
ϕ
O2
O1
ϕ
ϕ
EndcontactBegin
contact
Spur Gear Teeth SOLO
16
Angular Bevel GearsSOLO
Γ - Pitch angle
Σ - Shaft angle
α - Addendum angle
δ - Dedendum angle
ΓO - Face angle
ΓR - Root angle
21 Γ+Γ=Σ
( ) 2221 sincoscossinsinsin ΓΣ−ΓΣ=Γ−Σ=Γ
ΣΣ−
ΓΓ=
ΓΣΓ
sin
cos
sin
cos
sinsin
sin
2
2
2
1
22
1
tan
1cos
sin
sin
sin
1
Γ=
Σ+
ΓΓ
Σ
AO – Cone distance
D – Pitch diameter
0
11 2
sinA
D=Γ0
22 2
sinA
D=Γ
2
2
1
1
N
D
N
Dp
ππ ==
Circular path of gear 1 = Circular path of gear 2
1
2
1
2
1
2
N
N
D
D
R
R ==2
1
22
1
21 θθθ
=
=
N
N
R
R
2
1
2
1
sin
sin
D
D=ΓΓ
22
1
tan
1cos
sin
1
Γ=
Σ+
Σ D
D
2
12
cos
sintan
NN+Σ
Σ=Γ
17
SOLO
Γ - Pitch angle
Perpendicular Shafts Straight Bevel Gears ) Σ = 90° - Shaft angle (
α - Addendum angle
δ - Dedendum angle
ΓO - Face angle
ΓR - Root angle
AO – Cone distance
D – Pitch diameter
2
2
1
1
N
D
N
Dp
ππ ==
Circular path of gear 1 = Circular path of gear 2
1
2
1
2
1
2
N
N
D
D
R
R ==2
1
22
1
21 θθθ
=
=
N
N
R
R
2
12
cos
sintan
NN+Σ
Σ=Γ1
22tan
N
N=Γ2
11tan
N
N=Γ90=Σ
20
Crossed Shaft Helical GearSOLO
Helical Gears are used to transmit rotary motion between parallel and nonparallel shafts.
22
Normal circular path of gear 1 = Normal circular path of gear 2
2
22
1
11 coscos
N
D
N
Dpn
ψψ ==
2211 ωω NN = 11
22
1
2
2
1
cos
cos
ψψ
ωω
D
D
N
N ==
21 ψψ ±=Σ
iψ - Gear i helix angle
iN - Gear i teeth number
iD - Gear i diameter
iω - Gear i angular velocity
Crossed Shaft Helical GearSOLO
23
Worm and Worm Gear
If a thread on a helical gear makes a complete revolutionon the pitch cylinder, the resulting gear is called a worm.
1
tanD
L
πλ =
λ - lead angle
ψ - helix angle
D1 – worm pitch diameter
N1 – number of threads
p1 – worm circular pitch
1
11 N
Dp
π=
px – worm linear pitch
λλ cossin1 xpp =
L - lead of the worm
λπλ tantan 1111 DpNpNL x ===
Worm
SOLO
λ
1ψ1D
λ
1ψ1D
xp
Different threadsN1 = 3
λ
111 pND =π
1p
xpNL 1=xp
Thread 1
Thread 2 Thread 3
1
11 N
Dp
π=
Unwrapped completerevolution of the worm
24
Worm and Worm Gear
The mating gear for a worm is called a worm gear orworm wheel.
The mating condition of the worm gear with the worm is thatworm axial pitch = worm gear circular pitch
2ppx =
2
22 N
Dp
π=1N
Lpx =
1
tanD
L
πλ =
λπ
tan2
1
22
1
D
D
D
L
N
N ==
SOLO
iψ - Gear i helix angle
iN - Gear i teeth number
iD - Gear i diameter
iω - Gear i angular velocity
25
Worm and Worm GearSOLO
Worm gears are used to transmit rotational motion between nonparallel and nonintersecting shafts.
26
Worm and Worm Gear
1
tanD
L
πλ =
SOLO
xp
x
p
R =1
11θ
λθθ tan111
11 Rp
pRx x == 111 2
tan θπ
λθ L
Rx ==
Rotation of the worm by an angle θ1 will move the pointon the worm that made contact with the gear in the wormaxis direction by x.
2
22
p
R
p
x
x
θ= 22θRx =
22θ Rx =
λθ tan11 Rx = 2
1
2
1
1
2 tanN
N
R
R == λθθ
This movement by x is equivalent to a rotation θ2 of the worm gear, such that:
λπ
tan2
1
22
1
D
D
D
L
N
N ==
λ
1ψ1D
λ
1ψ1D
xp
Different threadsN1 = 3
λ
111 pND =π
1p
xpNL 1=xp
Thread 1
Thread 2 Thread 3
1
11 N
Dp
π=
Unwrapped completerevolution of the worm
27
Worm and Worm GearSOLO
2
22
1
11 coscos
N
D
N
Dpn
ψψ ==
22
1
22
11
2
1
1
2 tancos
cos
D
L
D
D
D
D
N
N
πλ
ψψ
θθ ====
λπψλψ −==212
Normal circular path of worm=Normal circular path of worm gear
1
tanD
L
πλ =
λ - lead angle ψ - helix angle
λcosxn pp =Normal circular path of worm
28
Worm and Worm Gear
Static Equations
1DnF
λ
11,θTworm
2pnF
worm gear
2RFn - contact force
- lead angleλ - pressure anglein normal plane
nφ
nφ
λ
nnF φcos
nF
λφ coscos nnF
nnF φsinλφ sincos nnF
nφ
λφ coscos nnF
nnF φsin
λnnF φcos
nF
λφ sincos nnF
x
yz
nFµλµ cosnF
λµ sinnF
nFµλµ sinnF
λµ cosnF
( )λµλφ cossincos11 −= nnr FRT
( )λµλφ sincoscos22 += nnr FRT
22 ,θT
0&1 11 >θθ x
SOLO
Worm efficiency calculation
( )λµλφ sincoscos22 += nnr FRT
( )λµλφ cossincos11 −= nnr FRT
λµλφλµλφ
cossincos
sincoscos
1
22
1
−+=
n
n
r
r
T
TRR
Without friction
λλ
µ
sin
cos
0
1
22
1
=
=
r
r
T
TRR
λµλφλµλφ
λλ
η µ
cossincossincoscos
sincos
1
22
1
0
1
22
1
−+== =
→
n
n
r
r
r
r
WGW
T
TRR
T
TRR
0tancos
cotcos >+−=→ θ
λµφλµφη
n
nWGW
( )0>θ
or
29
Worm and Worm Gear
Static Equations
1DnF
λ
11,θTworm
2pnF
worm gear
2RFn - contact force
- lead angleλ - pressure anglein normal plane
nφ
nφ
λ
nnF φcos
nF
λφ coscos nnF
nnF φsinλφ sincos nnF
nφ
λφ coscos nnF
nnF φsin
λnnF φcos
nF
λφ sincos nnF
x
yz
( )λµλφ cossincos11 += nnr FRT
( )λµλφ sincoscos22 −= nnr FRT
22 ,θT
nFµ
λµ sinnF
λµ cosnF
nFµ
λµ cosnF
λµ sinnF
0&1 11 <θθ x
SOLO
or
Worm Gear efficiency calculation
( )λµλφ sincoscos22 −= nnr FRT
( )λµλφ cossincos11 += nnr FRT
λµλφλµλφ
sincoscos
cossincos
2
11
2
−+=
n
n
r
r
T
TRR
Without friction
λλ
µ
cos
sin
0
2
11
2
=
=
r
r
T
TRR
λµλφλµλφ
λλ
η µ
sincoscoscossincos
cossin
2
11
2
0
2
11
2
−+== =
→
n
n
r
r
r
r
WWG
T
TRR
T
TRR
0cotcos
tancos <+−=→ θ
λµφλµφη
n
nWWG
( )0<θ
30
Worm and Worm Gear
Equations of Motion (no Friction(
1DnF
λ
1Tworm
2pnF
worm gear
2RFn - contact force
- lead angleλ - pressure anglein normal plane
nφ
nφ nnF φsin
λ
nnF φcos
nF
λφ sincos nnF
λφ coscos nnF
nφ
λφ coscos nnF
nnF φsin
λnnF φcos
nF
λφ sincos nnF
x
yz
SOLO
( ) λφθω sincos1111 nnBx RFTJ −=+
( ) 2222 coscos TRFJ nnBy −=+ λφθω
1
2
2
1
2
1 tanθθλ
==
R
R
N
N
( ) ( ) λθωλθω tantan2
1212
2
1211 R
RTT
R
RJJ ByBx −=+++
22
11
2
1211
2
2
121 T
N
NT
N
NJJ
N
NJJ ByBx
−=
++
+ ωωθ
or
we obtain
multiply the first equation by and add both equations
BxBynn JN
NJJ
N
NJT
N
NTRF ωωθλφ
11
2221
2
1
221
1
222 coscos2
−+
−+
+=
multiply the second equation by and add both equations using
λtan2
1
R
R
1
2
1
2
tan N
N
R
R −=−λ
BxBynn JJN
NJ
N
NJT
N
NTRF ωωθλφ
122
112
2
2
112
2
111 sincos2 −
+
−−
+=
2
1
N
N×
211
221
1
22
2
1
212 TT
N
NJJ
N
N
N
NJJ ByBx −
=+
+
+ ωωθ
1
2
N
N×
31
Worm and Worm Gear
1T
λψ =1
xp 2p
np
xnnF φcos
nnF φcos
λφ coscos nnF λφ coscos nnF
λφ sincos nnF
worm
worm geertooth
worm geeraxis
worm geer
λφ sincos nnF
22 ,θT
1,θ
λφ sincos nnF
λφ coscos nnF
nnF φcos
nnF φsin
λ1θµ signFn
nnF φsin
λφ sincos nnFnnF φcos
λφ coscos nnF
1θµ signFn
λ
λλ
zy
SOLO
Equations of Motion (including Friction)
( ) ( )θλµλφθω signRFTJ nnBx cossincos1111 +−=+
( ) ( ) 2222 sincoscos TsignRFJ nnBy −−=+ θλµλφθω
( ) ( )λ
θµλθωλθωcos
tantan 1
2
1212
2
1211
signRF
R
RTT
R
RJJ n
ByBx −−=+++
λθµωωθ
cos1
2
121
2
1211
2
2
121
signFR
N
NTT
N
NJJ
N
NJJ n
ByBx −
−=
++
+
or
we obtain1
2
2
1
2
1 tanθθλ
==
R
R
N
N
multiply the second equation by and add both equations using
λtan2
1
R
R
λθµωωθ
sin2
211
22
1
212
2
1
222
signFR
TTN
NJ
N
NJ
N
NJJ n
ByBx −−
=+
+
+
1
2
N
N×
36
SOLO Gears
References
1. Wolfram Stadler, “Analytical Robotics and Mechatronics”, McGraw-Hill, 1995
2. Hamilton H. Mabie and Fred W. Ocvirk, “Mechanism and Dynamics of Machinery”, SI Version, Third Edition, John Wiley & Sons, 1978
3. Joseph E. Shigley, Charles R. Mischke, Richard G. Budynas, “Mechanical Engineering Design”, Seventh Edition, McGraw-Hill, 2004
Antikythera References
1. Ivars Peterson, “Newton’s Clock, Chaos in the Solar System”, Freeman, 1993
http://en.wikipedia.org/wiki/Antikythera mechanism2.
January 4, 2015 37
SOLO
TechnionIsraeli Institute of Technology
1964 – 1968 BSc EE1968 – 1971 MSc EE
Israeli Air Force1970 – 1974
RAFAELIsraeli Armament Development Authority
1974 – 2013
Stanford University1983 – 1986 PhD AA