# 18. Gear Profile Theory¶

## 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
```

### 18.1.1. Issue¶

The maximal torque transmission is limited by the adhesion capacity.

### 18.1.2. Idea¶

Create hollows and bums around the wheel to get a contact point force transmission.

## 18.2. Transmission with teeth¶

### 18.2.1. One wheel description¶

```
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
```

### 18.2.2. Conditions for working gear¶

```
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
```

Problematic: How to design the tooth-profile?

## 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).