Helical Gears
advantages ...
greater load capacity
smoother operation
less sensitivity to tooth errors
ref: Gear Drive Systems; Design and
Application, Peter Lynwander
teeth are at angle to rotation, contact is a series of oblique
lines with several lines in contact simultaneously. total
length of contact varies as teeth mesh.
offset adjacent "strings" in involute generator concept on base cylinder by angle ψ
L=
ψB
2*π*RB
2⋅π⋅RB
L = lead
tan ψ B
RB = base_radius
( )
L
( )
tan ψ B =
2⋅π⋅RB
ψ B = base_helix_angle
L
develop normal at any radius on tooth by considering transverse and normal planes intersecting tooth at that point
geometry development
B
E
D
A
0, 0
( Xn , Yn , Zn) , ( Xn1 , Yn1 , Zn1) , ( X_line , Y_line , Z_line) , ( Xg , Yg , Zg) , ( X , Y , Z)
11/22/2004
1
point B ... point on gear for normal with helix (shown off gear)
point A ... point on radial line 0,0 to B perpendicular joining tangent
point D ... tangent point
point E ... point on plane perpendicular to tooth at B, connecting with (transverse) tangent point along RB
( )
AB
tan φN =
AE
( )
AB
tan φT =
AD
( )
cos( ψ ) =
( )
AD AB
AB
⋅
=
= cos( ψ ) ⋅ tan φT = tan φn
AE AD AE
AD
AE
( )
=>
tan φT =
( )
tan φn
 tan( φn) 
or ...
cos( ψ )
φT = atan 
 cos( ψ ) 
other parameters; as in general gear ...
π⋅ D π⋅ d
Pt = circular_pitch_transverse =
=
Ng
Np
D = diameter_gear
Ng = number_of_teeth_gear
d = diameter_pinion
Np = number_of_teeth_pinion
considering an expanded view at any radius ...
Pt
magnified ....
Pn
cos( ψ) =
ψ
Px
ψ
Pt
Pn
Px
Pn
Pt
Pn = circular_pitch_normal = Pt ⋅ cos( ψ )
ψ
if radius were RB ...
ψ
Pb
tan ψB =
Px
( )
Pb = base_pitch_transverse
 Pb 
gear geometry at pitch radius (diameter)
3. Pitch
Pt = circular_pitch_transverse =
pitch_circumference
number_of_teeth
Pn = circular_pitch_normal = Pt ⋅ cos( ψ )
N.B.
P = diametral_pitch_transverse =
above geometry =>
11/22/2004
ψ B = atan
and ...
π⋅ D
Ng
=
 Px 
g = gear
π⋅ d
p = pinion
Np
from geometry above
number_of_teeth
pitch_diameter
tan( ψ ) =
=
 Px 
 Pb 
= atan 
Pt
=
Ng
D
=
=>
Px
2
Np
so ...
d
Px =
Pt
tan( ψ )
π
and ...
Pt =
P
π
Pn =
PN
Px = axial_pitch = Pt ⋅ cot( ψ )
Pb = base_pitch_transverse =
base_circumference
number_of_teeth
π⋅ 2⋅ RB
=
N
=
π⋅ 2⋅ RG RB
⋅
RG
N
RG = pitch_radius =
from geometry way above ... 0.0, A, B, D
RB
( )
=>
cos φt =
RG
D
2
B
φt
π⋅ D
Pb =
⋅ cos φt = Pt ⋅ cos φt
Ng
( )
( )
A
D
RG
RB
φt
PbN = base_pitch_normal
0,0
Pb
PbN
consider geometry at left which is above brought down to base radius
Px is common, not dependent on radius ... PbN forms altitude of
triangle with sides Px and Pb and base sqrt(...) calculate area
ψΒ
area =
Px
ψΒ
1
2
⋅ base⋅ altitude =
and ...
=>
area =
2
1
1
⋅P ⋅P
2 x b
2
2
2
⋅ Pb + Px ⋅ PbN
2
Pb + Px ⋅ PbN = Px⋅ Pb
as ... is right triangle ...
and ...
Px⋅ Pb
PbN =
2
2
Pb + Px
also ...
Px
PbN =
2
2
Pb + Px
( )
( )
from figure and above ..
⋅ Pb = cos ψ B ⋅ Pt ⋅ cos φt
( )
not shown here ...
PbN = Pt ⋅ cos( ψ ) ⋅ cos φn
Ng
shown above ...
P = diametral_pitch_transverse =
D
also ...
π
π
P
PN = diametral_pitch_normal =
=
=
Pn
cos( ψ )
Pt ⋅ cos( ψ )
additional note on tooth loading ...
hp
 2⋅π  ⋅ 1⋅ in ⋅ 1⋅ in

 min  2
= 126051
lbf
in
π
and ...
Pt =
P
HP
HP
Wt = tangential_tooth_load = 126050⋅
= 126050⋅
RPMp⋅ d
RPMg⋅ D
Wt
WT = total_tooth_load_transverse_plane =
cos φt
( )
Wn = tangential_tooth_load_normal_plane =
WN = total_tooth_load_normal_plane =
11/22/2004
π
Pn =
PN
3
Wt
cos( ψ
)
Wt
( )
cos φn ⋅ cos( ψ )