FOUNDATION of MECHANICS 1FOUNDATION of MECHANICS 1
Presentation08: Gears
Outline
• Introduction: power transmission with constant transmission ratio; parallel axes; incident axes; skew axes.
• Elements of geometry and kinematics of spur gears: conjugate profiles; primitive curves; fundamental law of gearings; generation of conjugate profiles by envelop; involute curve; main definitions and properties of gears; spur p; ; p p g ; pgears; rack; gear manufacturing (mention).
• Helical gears: geometry and kinematics; main definitions and properties.Helical gears: geometry and kinematics; main definitions and properties.
• Incident axes transmission: spherical motion; bevel gear geometry & kinematics.
• Skew axes transmission: worm-gear geometry & kinematics; other solutions.
INTRODUCTION
Transmitting power between twoTransmitting power between twoshafts with a constant transmissionratio is one of the most commonproblems in designing machines,that can be solved with a numberof different solutions, i.e. using:
• Linkages
Fl ibl d i• Flexible devices
• Wheels
Friction wheelsDriven wheel
Driving wheel
o Friction wheels
o Gears
INTRODUCTION
Gears and Gearings
Bevel gears Worm-gear
Spur gears Helical gears
INTRODUCTION
Gears and Gearings
Simple gearings
Shaft 3
Simple gearings(gear trains)
Shaft 2
Shaft 1
Planetary (or epyciclic)gearings
(car differential gearing)
ELEMENTS OF KINEMATICS
(21)Conjugate profiles
1 tn(21)
(21) (21)M
M Mt
t
v
v v2
1
M2 (21)
Mn v 0
(21)(2)
(1)Mv
(21) (1)(2)
M
C12
3O1 O2
(21)Mv
(2)Mv
(21)
2 2
(1)(2)
( )M MM
M O
vω
vv
3
12 12 12
(21) (2) (1)0C C C v v v( )C O
1 1( )M O ω
12
(2)2
(1)
( )
( )C 12 23C C
C C
v ω
v ω
2 12 1
1 12 2
( )( )C OC O
12
( )1 ( )C 12 13C C v ω
ELEMENTS OF KINEMATICS
Fundamental Law of Gearing
1 n
M
1 2t
2M
(21)Mv
12 1( )( )C OC O
pitch point
C12
3O1 O2
Mv12 2( )C Opitch point
line of centres C12
3
line of action (line of contact)
The transmission ratio of two mating gears is constant if, and only if, the locationof their relative centre of rotation is stationary with respect to the absolute centres.y p
ELEMENTS OF KINEMATICS
Primitive curves
2
1
3O O
M ≡ C12 12
(21) (21)
(2) (1)
0M C v vO1 O2 (2) (1)
M Mv v
2 12 1( )C O 2 12 1
1 12 2
( ) .( )C O constC O
Primitive curves of the motion are the geometrical loci of the relative centre of rotation (seen by the two links respectively). The relative motion between the links can be described by the rolling motion of the primitive curves.In the case of a gear pair, the primitive curves are two circles (pitch circles).
ELEMENTS OF GEOMETRY&KINEMATICS
Generation of conjugate profiles *
Envelope of a family of curves in the plane is a curve that is a curve tangent to each member of gthe family at least in one point.
Generation of conjugate profiles by Cs2
s
σ1
ε
μ
enveloping proper curves.
The envelope of line μ for a rolling
s1
σ2
ρ2
of line ε over the two primitive curves σ1 and σ2 generates two conjugate profiles. The conjugate
R2
profiles are involute curves.
Being circles the two primitive curve the transmission ratio is
* see also:- Paul B., Kinematics and dynamics of planar machinery, Prentice-Hall, New Jersey, 1979.
curve, the transmission ratio is constant.
- Doughty S., Mechanics of Machines, John-Wiley & Sons, New York, 1988.
ELEMENTS OF GEOMETRY&KINEMATICS
Involute curve *
ρ1
K1
C
Involutecurve
M
ρ1ρ2
K2a
KL
K1
ρBase circle
K2K1
* see also:- Paul B., Kinematics and dynamics of planar machinery, Prentice-Hall, New Jersey, 1979.
K- Doughty S., Mechanics of Machines, John-Wiley & Sons, New York, 1988.
K2
ELEMENTS OF GEOMETRY&KINEMATICS
Involute as a gear tooth *
R
αCs2ρ1
σ1μ
R12
ρ R R′
α
C
s1α
ρ2α
ε
α
σ2
R2
* see also:Doughty S Mechanics of Machines- Doughty S., Mechanics of Machines,
John-Wiley & Sons, New York, 1988.
ELEMENTS OF GEOMETRY&KINEMATICS
Involute as a gear toothO2
1 1 1
' '
K M K L
K M K L
2 2 2
' '
K M K L
K M K L
2
2 1 1 1
1 1' '
K M K L
MM L L
2 2 2
2 2' '
K M K L
MM L L
K2L'2 L2
M'
1 1 1 1 2 2 2 2' 'L L L L
C121O C R
K1
ML1L'1
M
1
C12
α2
2 12 2 ( )O C R
1 12 1 cos( )O C R
O
1
1
2 2 1
α
1R
2 12 2 cos( )
O11 1 2
2R
ELEMENTS OF GEOMETRY&KINEMATICS
Involute as a gear tooth
HH'
'LL 'MM
LL' 'HH R
'HH R 1'HH '
HH RLL
1
cos
'HHMM
R cos( )R
O
ELEMENTS OF GEOMETRY&KINEMATICS
Variation of the centre distance
α α′
ρ
C′
ρ2
ρ1
C
a Δa
a′
'2 1 1 1
'
R RR R
1 2 2 2R R
ELEMENTS OF GEOMETRY&KINEMATICS
Involute as a gear tooth
Pitch circles are the primitive curves of motion and are defined for a gear pair
Base circle is the evolute curve of the involute profile. It is a geometrical characteristic of a certain geardefined for a gear pair. characteristic of a certain gear.
ELEMENTS OF GEOMETRY&KINEMATICS
ODefinitions and properties
O2
2
2
K2L'2 L2
MM'
α
K1M
L1L'1
1
αα
Pressure angle
O1
1
Line of action (or contact)O1
ELEMENTS OF GEOMETRY&KINEMATICS
Definitions and properties
Doughty S., Mechanics of Machines, John-Wiley & Sons, New York, 1988.
ELEMENTS OF GEOMETRY&KINEMATICS
Definitions and properties
11 12 21 22'MM L L L L
O2
1 1 2 2 2
1 22 2Z Z K2
L22 L21
M'
2
1 2Z Z
2K1
ML11L12
M
1
Z
pb2
Base Pitch
O
1
1
O1
ELEMENTS OF GEOMETRY&KINEMATICS
O2Definitions and properties
11 12 21 22H H H HR R
1 1 2 2R R
2 2R R K2
2
2
1 21 2
R RZ Z
2H22
H21
HH
K2
RZ
p 2 Circular Pitch
H11H12
K11
cosppp
p m m:= Module
O
1
cosppR
pb O1
ELEMENTS OF GEOMETRY&KINEMATICS
Definitions and properties
Re
R eiRi
R
R R
Ri
R Re
i
R R eR R i
e
i
R R eR R i
2.25h e i m
ELEMENTS OF GEOMETRY&KINEMATICS
The rack
R
σ1
μ
p0 = m0Cs1
ε
s2
n
= =
0
2
= 2.
5 m
0
== linea di riferimentoPitch (or reference) line
h = =
ELEMENTS OF GEOMETRY&KINEMATICS
Gear manufacturing *
Gears are generally manufactured by means of tool machines (gear cutting machines) that create the tooth profiles through the generation method, by exploiting the relative motion of pure rolling between the two primitive surfaces (cutting primitives) for enveloping involute tooth profiles.
* see also:Paul B., Kinematics and dynamics of planar machinery, Prentice-Hall, Newplanar machinery, Prentice Hall, New Jersey, 1979.
ELEMENTS OF GEOMETRY&KINEMATICS
Gear manufacturing
Cutting gears through the generation method can be easily understood by thinking the gear to be cut as made of mouldable material (e.g. Plasticine) and the cutter gear or rack as rigid (e.g. made of steel).
In gear cutting machines there are three important motions apart from the rolling of theimportant motions, apart from the rolling of the cutting primitive surfaces:
• the cutting motion (e.g. in the case of a cutting rack it is a reciprocating translational motion along the direction of the teeth);
• the feeding motion (to make the cutter• the feeding motion (to make the cutter remove material by little steps)
• the positioning motion (e.g. to take the gear back to the initial position of the rack stroke).
ELEMENTS OF GEOMETRY&KINEMATICS
Gear manufacturing
Cutting machines can be divided into:
• Cutting-slotting machines• Cutting-slotting machines, where the tools is provided with a reciprocating translational motion (e.g. cutting rack, cutting gears, F ll h )Fellows gear-shaper);
• Cutting-hobbing machines with a milling cutter (hob)with a milling cutter (hob) provided with a cutting motion which is rotational and continuous.
ELEMENTS OF GEOMETRY&KINEMATICS
Gear manufacturing
Cutting machines can be divided into:
• Cutting-slotting machines• Cutting-slotting machines, where the tools is provided with a reciprocating translational motion (e.g. cutting rack, cutting gears, F ll h )Fellows gear shaper);
• Cutting-hobbing machines with a milling cutter (hob)with a milling cutter (hob) provided with a cutting motion which is rotational and continuous.
ELEMENTS OF GEOMETRY&KINEMATICS
Gear manufacturing
Other methods (less significant):Other methods (less significant):
• machining the tooth profiles using modular milling cutters;using modular milling cutters;
• generation of internal gears by means of broaching machines.
ELEMENTS OF GEOMETRY&KINEMATICS
Properties
'2 1 1 1
'1 2 2 2
R RR R
α α′
ρ1
C′
O O ′
1 2' ' '
1 2
R R a
R R a
Primitive radii of cuttingρ2
ρ1
CO2
O1 O1
1 2
' '1 1
R a
Primitive radii
aa′
Δa
''
2 1aR
Primitive radii of motion
' ' ' ' '1 2 1 2 1 2
'
( ) cos( ) ( ) cos( ) cos( ) cos( )
( ) ( )
R R R R a aa
P l f ti'cos( ) cos( )a
Pressure angle of motion
ELEMENTS OF GEOMETRY&KINEMATICS
Propertiesp = m
α α′
ρ1
p0 = m0
= =
OO1 O1′
C′
ρ2
h =
2.5
m0
== linea di riferimentoPitch line
O2C
0 02p R mZ Cutting
module/circular pitch
a
a′
Δa
0cos( ) cos( )2
m ZR '0 '
cos( )cos( )
m m
Motion module
'' ' '
2
cos( ) cos( )2
m ZR ' '
cos( )p m
Motion circular pitch
ELEMENTS OF GEOMETRY&KINEMATICS
Contact RR2
K2
2
N2
N2 C
K2
A2B
N2
C
N1B1
CA1
2B2
α
α
1
N1
1
R1
K1
1
N N
O1
1
1 2
1 1 2 2
N N
A B A B
Length of contact
Arc of contact 1 2
1 1 cos( )N NA B
1
ELEMENTS OF GEOMETRY&KINEMATICS
Contact
For a uniform transmission of motion, it is necessary that when a tooth pair is loosing contact another tooth pair is already meshing. This means that the
cos( )A B p N N p p
arc of action must be greater that the circular pitch:
1 1 1 2 cos( ) bA B p N N p p
The ratio between the arc of action and the circular pitch is called contact
A B
ratio and must be greater than 1
1 1 1A Bp
generally 1 2
ELEMENTS OF GEOMETRY&KINEMATICS
Interference
The length of contact is determined by the intersection of the line of action with the two addendum circles. If the
t t t f thcontact occurs out of the segment K1K2 the action is non-conjugate. This situationnon conjugate. This situation is referred to as interference, and must be avoided.
ELEMENTS OF GEOMETRY&KINEMATICS
Interference
Avoiding interference means setting an upper bound to the value of the two gear addendum radii (Re1,2):
2 2 2 21 1 2 2sin ( ) sin ( )e b e bR R a R R a
K1
Re2lim
Rb1
O1 O2
C
N1
Re1lim
O1 O2
Rb1
K
N2
K2
ELEMENTS OF GEOMETRY&KINEMATICS
Interference
2 21 1 1 sin ( )e bR R e R a
2 22 2 2 sin ( )e bR R e R a
For standard gears:
1 2 1 2( 1.25 )e e e m i i i m
1 2
1 2
2 2R RmZ Z
An upper bound for e (i.e. for m) entails a lower bound for Z1 and Z2:
1 2Z Z
Z Z Z Z 1 1min 2 2minZ Z Z Z
ELEMENTS OF GEOMETRY&KINEMATICS
Interference
R2The condition is more critical (implying a tighter limitation) for2
KR2 + e2 lim
(implying a tighter limitation) for “big” driven gears (having a high pitch radius R2).
N2
K2p 2)
C e2 lim
K11
ELEMENTS OF GEOMETRY&KINEMATICS
InterferenceR R The condition is more critical
(implying a tighter limitation) for ll i i (h i l
R2
K2
R2 + e2 lim
small pinions (having a low number of teeth Z2).
N2
C
2
e2 lim
B1
CA1
K
2 lim
The most critical condition is therefore the pair rack-pinion.
R
K1O'1t e e o e t e pa ac p o
R1
O1
ELEMENTS OF GEOMETRY&KINEMATICS
Interference
12 2R
It can be demonstrated that for a pinion-gear pair:
1
1 1min 2 2 22 2 1 1 2
2 22 sin 1 1 2 sin
Rz zR R R R R
For a rack pinion pair:
Zmin
= 20° = 1/2
Zmin
For a rack-pinion pair:
120 ( )R R
R
2
min 0 22
2 2limsin1 1 2 i
R
z
2 sin1 1 2 sin
ELEMENTS OF GEOMETRY&KINEMATICS
Interference
Undercutting is the consequence of interference wheninterference when manufacturing gears with the generation method g(envelop). If the gear to be cut has a number of teeth smaller than Zmin, the tool removes material from the blank where it should notblank where it should not.
min
Standard gear:
20 17Zpinion with Z = 8
min20 17Z