It is performed by a Vernier or a Micrometer with special anvils called a Span Micrometer, see diagram.
The method makes use of one of the basic properties of the involute and means that the dimension is constant irrespective of the point of tangency between the flank of the gear and the anvil of the micrometer.
The calculation will form two parts:
1. The number of teeth over which the measurement is to be made (spanned).
2. The base tangent size (sometimes referred to as the span size).
Span Micrometer
From Gear Data
z = no. of teeth in gear (61)
Mn = normal module (8)
an = normal pressure angle (20)
ß = helix angle (15)
x = profile shift coefficient (0)
Calculated
k = number of teeth spanned (8)
at = transverse pressure angle (0.360356324radians 20.64689649°)
invat = involute of the transverse pressure angle (0.01645339)
avt = transverse pressure angle at a point (0.360356324radians 20.64689649°)
ßb = base helix angle (0.245674211radians 14.07609542º)
Wk = nominal base tangent length (184.6729mm)
Aw = change factor of base tangent length (0.9396926)
Formula: Number of teeth in span
Calculation procedure:
1. Transverse Pressure
Angle: at
= atan(tan(an)/cos(ß))
= atan(tan(20)/cos(15))
= atan(0.3639702/0.9659258)
= 0.360356324 radians (20.64689649º
)
2. Involute of the Transverse Pressure
Angle: invat
= tan(at)-(at)
= tan(20.64689649)-20.64689649
= 0.376809714-0.360356324
= 0.01645339
3. Transverse
Pressure Angle at a Point: avt
= acos((z/(z+2*x*cos(ß)))*cos(at))
= acos((61/(61+2*0*cos(15)))*cos(20.64689649))
= acos((61/(61+2*0*0.9659258))*0.9357712)
= acos((61/61)*0.9357712)
= acos(1*0.9357712)
= acos(0.9357712)
= 0.360356324 radians
(20.64689649°)
4. Base Helix Angle:
ßb
= asin(sin(ß)*cos(an)
= asin(sin(15)*cos(20)
= asin(0.258819*0.9396926)
= asin(0.2432103)
= 14.076074º
= 0.245674211 radians (14.07609542º)
5. Optimum number
of teeth in span: k
= (z/Pi)*((tan(avt)/cos(ßb)2)-2*(x/z)*tan(an)-invat)+0.5
= (61/Pi)*((tan(0.360356324)/cos(0.245674211)2)-2*(0/61)*tan(20)-0.01645339)+0.5
= 19.416903*((0.376810/0.940848)-2*0*0.3639702-0.01645339)+0.5
= 19.416903*(0.400500-2*0*0.363969-0.016453)+0.5
= 19.416903*0.400500+0.5
= int(19.416903*0.400500+0.5)
= 8 teeth
Formula: Base Tangent Length - External gears
1. Actual number of teeth in span: k
(see calculation above)
= 8 teeth
2. Transverse Pressure
Angle: at
= atan(tan(an)/cos(ß)
= atan(tan(20)/cos(15)
= atan(0.3639702/0.9659258)
= 0.360356324 radians (20.64689649º
)
3. Involute of the Transverse Pressure Angle: invat
= tan(at)-(at)
= tan(20.64689649)-20.64689649
= 0.376809714-0.360356324
= 0.01645339
4. Nominal Base Tangent Length: Wk
= (Mn*cos(an))*(((k-0.5)*Pi)+(z*invat))+(2*x*Mn*sin(an))
= (8*cos(20))*(((8-0.5)*Pi)+(61*0.01645339))+(2*0*8*sin(20))
= (8*0.939692)*(((8-0.5)*Pi)+(61*0.01645339))+(2*0*8*0.3420201))
= 7.517541*(23.561945+1.003633)+0
= 7.517541*24.5655779
= 184.6729mm
5. Change Factor of Base
Tangent Length: Aw ......
see Change Factor
= cos(an)
= cos(20)
= 0.9396926