# Introduction: Fundamentals of Gear Transmission
Transmitting power and motion between shafts is a fundamental need in machine design. When the shafts are close together, several solutions can be used, but gears stand out for guaranteeing a **constant and reliable transmission ratio**, even under heavy loads.
## From friction to shape: why teeth
The simplest method for transmitting rotation between parallel shafts is by means of **friction wheels** (cylinders in contact):
- **Advantage:** simplicity and low cost.
- **Drawback:** if the torque exceeds the maximum friction force ($F_t > \mu N$), **slipping** occurs (loss of synchronism and wear).
To overcome this limitation, transmission moves from **force** to **shape**: teeth prevent slipping and keep synchronism. For the transmission ratio to be constant, the tooth profiles must be **conjugate**.
> In this lesson we start from friction cylinders and arrive at the fundamental law of gearing and at the **involute** profile, the industry standard.
## Fundamental Law of Gearing
### From the kinematic problem to the geometric solution
We have seen that we need teeth to transmit high torques without slipping. But it is not enough to add just any shape: the profiles must guarantee that the **transmission ratio remains rigorously constant** throughout the rotation, just as it did with ideal friction cylinders.
A transmission like the one shown in the figure would give a constant *average* velocity ratio, but not an *instantaneous* one. This kinematic requirement translates into a precise geometric condition, first rigorously formulated in the 18th century and known as the **Fundamental Law of Gearing** (or *Willis' theorem*):
> For the transmission ratio between two profiles in contact to be constant, the **common normal** to both profiles at the point of contact must **always** pass through a fixed point on the line of centers, called the **pitch point (P)**.
Mathematically, this point P divides the center distance into segments inversely proportional to the angular velocities:
$i=\frac{\omega_3}{\omega_2}=\frac{O_2P}{O_3P}=\text{constant}$
Any pair of profiles that satisfies this law is called a pair of **conjugate profiles**. In theory, infinitely many pairs of conjugate profiles are possible (cycloidal, trochoidal, etc.). The rolling levers in the following image are an example that, although not very practical, satisfies this condition. However, a fundamental practical problem arises: **how can these gears be manufactured and standardized so that they are interchangeable?**
## The Involute Profile: The Universal Solution
### Why the involute became the standard
Throughout history, several conjugate profiles have been proposed: **cycloids** (used in early clocks), **trochoids**, and others.
```{image} ./figs/imagenes_tema_07/Imagen06.gif
:alt: imagen6
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```
However, all of them shared the same problem: each pair of wheels required specific profiles designed for that particular center distance. A small assembly error or thermal expansion would break the conjugation.
In the mid-18th century, the Swiss mathematician **Leonhard Euler** proposed a revolutionary profile: the **involute of a circle**. This profile has a unique property that solved the interchangeability problem: **it remains conjugate even if the center distance varies**. This feature, together with its ease of manufacture, made it the universal industry standard.
```{image} ./figs/imagenes_tema_07/Imagen04.gif
:alt: imagen4
:width: 320px
:class: no-lightbox no-scaled-link
```
### Geometric definition and generation
The **involute** is the curve traced by a point on a **straight line** that **rolls without slipping** on a fixed circle, called the **base circle**.
An intuitive way to visualize it is to imagine a spool of thread: if we fix the spool and pull the end of the thread while keeping it taut, the end traces out exactly an involute.
**Parametric equations.** Looking at the figure, we can see that the arc $\overset{\frown}{AB}$ coincides with the segment AT, since they represent the wound and unwound states of the thread.
Calling $\rho$ the radius of the base circle, we have
$\rho \tan{\phi} = \rho (\theta+\phi)$
Solving for $\theta$:
$\theta = \tan{\phi} - \phi = Ev(\phi)$
where $Ev(\phi)$ is called the involute function of $\phi$. The angle $\phi$ can be
obtained from the previous equation using the inverse function:
$\phi = Ev^{-1}(\theta)$
Going back to the figure, we can write
$r = \frac{\rho}{\cos{\phi}}=\frac{\rho}{\cos{(Ev^{-1}(\theta))}}$
which is the analytical equation of the involute in polar coordinates.
### Properties that justify its universal use
The advantages of the involute profile are decisive for industrial manufacturing:
1. **It satisfies the law of gearing:** The **normal** to the involute at any point is always **tangent to the base circle**. When two involutes mesh, their common normal is the **common tangent to both base circles**, which is a fixed line called the **line of action** or **path of contact**. This line always passes through the pitch point P, guaranteeing a constant transmission ratio.
2. **Invariance under changes in center distance:** This is the most remarkable property. If the center distance $a$ varies slightly (due to manufacturing tolerances, assembly, or thermal expansion), involute profiles **remain perfectly conjugate**. The only thing that changes is the operating pressure angle $\alpha'$ (it increases if $a$ increases), but the transmission ratio stays constant. This allows **interchangeability**: wheels manufactured by different suppliers can mesh with each other.
3. **Simple manufacture by generation:** Unlike other profiles that require complex form cutters, the involute can be generated with a tool having **straight-sided flanks** (a rack). The generating-cutting process (which we will see later) produces the involute profile automatically, precisely, and economically.
4. **Favorable wear:** The relative sliding between flanks is moderate near the pitch point, which reduces wear.
These advantages explain why virtually all modern industrial gears (except for very specific applications such as precision clockwork) use involute profiles.
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## Standard Nomenclature and Parameters
### The need for standardization
For the involute profile to deliver on its promise of interchangeability, it is essential to precisely define all the geometric parameters of the tooth. Industry has developed international standards (ISO, DIN, AGMA) that establish these parameters.
Standardization allows:
- Two gears manufactured in different places to mesh if they have the same **module** and **pressure angle**.
- Universal cutting tools (cutters, generating racks) to exist for each module.
- Design and verification calculations to be uniform.
The module is therefore one of the most important parameters when selecting a gear. As will be seen later, it is closely related to the **diametral pitch** (teeth per unit of diameter, the inverse quantity to tooth size — it is the inverse of the module expressing the radius in inches), which is why standardized diametral-pitch gauges can be found, such as the one shown in the figure:
### Fundamental terms (isolated wheel)
To fully define a gear tooth we need several concentric circles and geometric parameters:
- **Reference pitch circle** ($d_p$): The theoretical circle on which all calculations are based. It is equivalent to the diameter of the friction wheel that would give the same transmission ratio. It is the fundamental reference. We will often use the pitch radius, $r_p=d_p/2$. When there is no subscript, the symbol $r$ refers to the pitch radius.
- **Module** ($m$): The most important parameter. It defines the **tooth size**. Two gears **can only mesh if they have the same module**. It is defined as:
$m = \frac{2r_p}{Z} = \frac{p}{\pi}$
It is expressed in millimeters (metric system) or in diametral pitch (imperial system). Module values are **standardized**: 1, 1.25, 1.5, 2, 2.5, 3, 4, 5, 6, 8, 10, etc.
- **Circular pitch** ($p$): Distance measured along the pitch circle between two corresponding points of consecutive teeth. $\overset{\frown}{AC}$ in the figure. Related to the module by: $p = \pi m$.
- **Number of teeth** ($Z$): Total number of teeth on the wheel.
- **Outside (addendum) circle** ($d_a$): Outer diameter of the gear (tips of the teeth).
- **Root (dedendum) circle** ($d_f$): Diameter of the bottom of the tooth space. It corresponds to the circle that bounds the space between teeth from below. This space must be deep enough to clear the tips of the mating wheel's teeth.
- **Base circle** ($d_b$): The circle from which the involute "is born". It is fundamental: $d_b = d_p \cos\alpha$.
- **Addendum** ($h_a$): Height of the tooth above the pitch circle. Standard value: $h_a = m$.
- **Dedendum** ($h_f$): Depth of the tooth below the pitch circle. Standard value: $h_f = 1.25m$.
- **Whole depth** ($h$): Sum of the addendum height and the dedendum depth.
- **Backlash**: gap left between a meshing pair of teeth. Needed to allow for tooth deflection, lubricant flow, and thermal expansion.
- **Bottom (tip) clearance** ($c$): gap left between the tip of one tooth and the bottom of the tooth space of the mating wheel when meshing.
- **Reference pressure angle** ($\alpha$): Angle formed by the line of action with the tangent to the pitch circle. It defines the "inclination" of the thrust between teeth. Standardized values: **14.5°** (obsolete), **20°** (the most common), **25°** (for heavy loads). A larger $\alpha$ gives sturdier teeth but slightly reduces smoothness.
- **Tooth thickness** ($s$): Thickness of the tooth, measured on the pitch circle. $\overset{\frown}{AB}$ in the figure.
- **Space width** ($e$): Gap between teeth, measured on the pitch circle.
The pitch equals the tooth thickness plus the space width. $\overset{\frown}{BC}$ in the figure.
- **Face**: Part of the tooth surface located between the pitch circle and the addendum circle.
- **Flank**: Part of the tooth surface located between the pitch circle and the dedendum circle.
- **Face width** ($b$): Width of the tooth measured in the direction parallel to the axis.
### Standardized parameters (20°)
| Quantity | Symbol | Relation |
|---|---:|---|
| Pitch diameter | $d_p$ | $d_p=m\,Z$ |
| Circular pitch | $p$ | $p=\pi\,m$ |
| Module | $m$ | $m=d_p/Z=p/\pi$ |
| Base diameter | $d_b$ | $d_b=d_p\cos\alpha$ |
| Addendum | $h_a$ | $h_a=m$ |
| Dedendum | $h_f$ | $h_f=1.25\,m$ |
| Bottom clearance | $c$ | $c=0.25\,m$ |
| Tooth thickness at pitch circle | $s$ | $s=p/2$ |
### Assembly (mesh) parameters
- **Center distance (at standard/zero mesh):** $\displaystyle a=\frac{d_{p1}+d_{p2}}{2}=\frac{m}{2}(Z_1+Z_2)$.
- **Backlash:** clearance between non-contacting flanks, needed for lubrication and to avoid jamming ($c=0.5m$).
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## Line of Action and Contact Ratio
- **Line of action:** fixed straight line, common tangent to both base circles, along which the meshing force acts.
- **Path of contact:** segment of the line of action between the points where a tooth pair enters and leaves contact, passing through P.
- **Contact ratio** ($\varepsilon_\alpha$): average number of tooth pairs simultaneously in contact. Requirement: $\varepsilon_\alpha>1$ (typical: 1.4-1.8 for spur gears) for continuous and smooth transmission. As will be seen later, helical gears often exceed a value of 2 thanks to their overlap.