18.1. Transmission per adhesion¶

(O,i,j) orthonormal reference frame

wheel_1 rotation speed: u (radian/s)
wheel_2 rotation speed: v (radian/s)

speed of M, point of wheel_1:

V(M) = u*R1*j

speed of N, point of wheel_2:

V(N) = v*R2*j

Because of the adhesion of the wheel_1 and wheel_2 in M:

V(M).j = V(N).j
u*R1 = v*R2
v = u*R1/R2

18.2. Transmission with teeth¶

angular_pitch = 2*pi/tooth_nb
circular_pitch = angular_pitch * primitive radius
addendum_radius = primitive_radius + addendum_height
dedendum_radius = primitive_radius + dedendum_height
tooth_height = addendum_height + dedendum_height

circular_pitch_1 = circular_pitch_2
addendum_height_1 < dedendum_height_2
addendum_height_2 < dedendum_height_1
transmission ratio = primitive_radius_1 / primitive_radius_2 = tooth_nb_1 / tooth_nb_2

18.3. Tooth profile¶

Cartesian equation:

Mx(a) = R*cos(a)+a*R*cos(a-pi/2)
My(a) = R*sin(a)+a*R*sin(a-pi/2)

Trigonometry formula remind:

cos(-x) = cos(x)
sin(-x) = -sin(x)
cos(pi/2-x) = sin(x)
sin(pi/2-x) = cos(x)
cos(a-pi/2)=cos(pi/2-a)=sin(a)
sin(a-pi/2)=-sin(pi/2-a)=-cos(a)

Tangent vector:

Mx'(a) = -R*sin(a)+R*cos(a-pi/2)-a*R*sin(a-pi/2) = -a*R*sin(a-pi/2) = a*R*cos(a)
My'(a) =  R*cos(a)+R*sin(a-pi/2)+a*R*cos(a-pi/2) =  a*R*cos(a-pi/2) = a*R*sin(a)

u: rotation speed of the wheel
v: linear speed of tha bar
u(t) = d/dt(a(t))

OM = sqrt(R² + (a*R)²) = R*sqrt(1+a²)

S = OM*u
Sn = S*cos(b)
St = S*sin(b)

Sn = u*R*sqrt(1+a²)*cos(b)
relation between a(t) and b(t)?
tan(b) = (a*R)/R = a
Sn = u*R*sqrt(1+tan²(b))*cos(b)

Trigonometry formula remind:

1+tan²(x) = (cos²(x)+sin²(x))/cos²(x) = 1/cos²(x)

So,:

v = Sn = u*R

v does not depend on the angle a!

St = u*R*sqrt(1+a²)*sin(b) = u*R*tan(b) = u*R*a

u: rotation speed of the wheel
v: linear speed of tha bar
u(t) = d/dt(a(t))

OM = sqrt(R² + (a*R)²) = R*sqrt(1+a²)

S = OM*u
Sn = S*cos(b)
St = S*sin(b)

v = Sn = u*R*sqrt(1+a²)*cos(b)
= u*R*sqrt(1+tan²(b))*cos(b) = u*R

v does not depend on the angle a!

St = u*R*sqrt(1+a²)*sin(b) = u*R*tan(b) = u*R*a

v = u1*R1 = u2*R2
So, u2 = u1*R1/R2

Sn1 = Sn2 because of the contact

Friction between the two wheels:

Sf = St2 - St1 = u2*R2*a2 - u1*R1*a1
= u1*R1*(a2-a1)
But,
a1 = k1-u1*t
a2 = k2+u2*t
Sf = u1*R1*(k1-k2+(u1+u2)*t)

18.4. Gear profile construction¶

18.5. Gear rules¶

• The base diameter of the two directions can be different

• The top-land and bottom-land are not critical part of the profile

  The top-land can be a straight line. The bottom-land is usually a hollow to help the manufacturing.

• The rotation ratio implies by the involutes-of-circles is:

  base_radius_1 / base_radius_2
  

  The rotation ratio implies by the teeth is:

  tooth_nb_1 / tooth_nb_2
  

  In order to get a continuous transmission without cough, we must ensure that:

  base_radius_1 / base_radius_2 = tooth_nb_1 / tooth_nb_2
  

  If you use two base circles for the positive rotation and the negative rotation, then:

  base_radius_positive_1 / base_radius_positive_2 = tooth_nb_1 / tooth_nb_2
  base_radius_negative_1 / base_radius_negative_2 = tooth_nb_1 / tooth_nb_2
  

• The position of the positive involute of circle compare to the negative involute of circle is arbitrary and it is usually defined by the addendum-dedendum-ration on the primitive circle. Just make sure the top-land and bottom-land still exist (positive length). The addendum-dedendum-ration of the second wheel must be the complementary.

Do not mix-up the primitive circle and the base circle. The primitive circle helps defining the addendum and dedendum circles. The base circle defines the involutes of circle.We have the relation:

base_radius < primitive_radius

18.6. Torque transmission¶

F = T1/R1 = T2/R2
T2 = T1*R2/R1

The transmitted torque T2 does not depend on the angle a!

18.7. Gearwheel position¶

The rotation ration depends only on the two base circle diameters. It does not depend on the inter-axis length. The inter-axis length can be set arbitrary within a reasonable range (addendum and dedendum height constraints).