## Gear Typology

### Geometric diversity according to shaft configuration

Although the involute profile is common to almost all gears, the **tooth geometry** and the wheel geometry vary according to the **relative orientation of the shafts** to be connected and the **load, speed, and noise requirements**.

The basic classification follows the spatial relationship between the shafts:

### 1. Parallel shafts: spur and helical gears

- **Spur gears:** The teeth are parallel to the axis of rotation. They are the **simplest to manufacture** and generate no axial forces. High efficiency (98-99%). However, contact between teeth happens **all at once** (the whole line of contact engages simultaneously), which produces **impacts and noise** at high speeds. **Typical use:** household appliances, toys, low/medium-speed gearboxes, simple machine tools.

<img src="./figs/imagenes_tema_07/Imagen11.png" alt="imagen11" width="420px">

- **Helical gears:** The teeth form a **helix** around the axis. Contact is **progressive** (it starts at one end of the tooth and gradually advances), which makes them much **quieter and smoother**. They have a higher contact ratio ($\varepsilon_\alpha$ increases) and can transmit more power. **Drawback:** they generate an **axial force component** that must be absorbed by the bearings. **Typical use:** automotive gearboxes, high-speed industrial transmissions, compressors.

- **Double helical (herringbone):** Two helices of opposite hand on the same wheel. They **cancel out** the axial thrust, keeping all the advantages of helical gears. They are very rugged and capable of transmitting large amounts of power. **Drawback:** complex and expensive to manufacture (requires special cutters or very precise generating machining). **Typical use:** large power gearboxes (turbines, rolling mills, ships), where the cost is justified.

<img src="./figs/imagenes_tema_07/Imagen12.png" alt="imagen12" width="520px">

### 2. Intersecting shafts: bevel gears

When the input and output shafts **intersect** (typically at 90°, but any angle is possible), **bevel gears** are used. Instead of cylinders, the pitch surfaces are **cones**.

- **Straight bevel gears:** The teeth are straight and run from the apex of the cone to its base. They are the bevel version of spur gears. **Pros:** relatively simple, economical manufacturing. **Cons:** noisy at high speed, contact is abrupt. **Typical use:** 90° changes of direction in low/medium-speed applications (pillar drills, simple right-angle gearboxes).

- **Spiral bevel gears (Gleason system):** The teeth are **curved**, similar to helical gears but on a conical surface. Contact is progressive, which makes them **much smoother, quieter, and stronger**. **Cons:** complex manufacturing (requires specialized Gleason or Klingelnberg machines), more expensive. **Typical use:** automotive differentials (front/rear axle), helicopter transmissions, high-speed, high-precision applications.

<img src="./figs/imagenes_tema_07/Imagen13.png" alt="imagen13" width="520px">

### 3. Crossed (skew) shafts, non-intersecting

When the shafts are **neither parallel nor intersecting** (they cross in space), special geometries with significant relative sliding are used:

- **Hypoid gears:** Similar to spiral bevel gears, but with the shafts **offset vertically** (they do not intersect). This geometry lets an automobile lower the driveshaft position (improving cabin space) and use sturdier pinions. They have **a lot of relative sliding**, which generates heat and requires **extreme-pressure (EP) lubricants**. **Typical use:** differentials in modern automobiles.

- **Worm and worm gear:** The pinion is a **worm** (with one or more threads/starts) and the wheel is a helical gear (worm wheel) that wraps around it. It allows **enormous reductions in a single stage** (ratios from 20:1 to 100:1 or more). It is **very quiet** because contact is a "rubbing" type over a large surface.
  - **Irreversibility:** If the lead angle of the worm's helix is small ($\tan\gamma < \mu$, where $\mu$ is the friction coefficient), the worm wheel **cannot drive** the worm. This is useful as a **safety brake** (hoists, lifting tables, adjustment screws).
  - **Cons:** Low efficiency (50-80%) due to high sliding. Generates a lot of **heat**; requires good cooling and lubrication.
  - **Typical use:** Elevators, conveyor belts, positioning mechanisms, compact high-reduction gearboxes, tuning pegs of musical instruments.

<img src="./figs/imagenes_tema_07/Imagen14.png" alt="imagen14" width="620px">

### 4. Motion transformation: rotation ↔ translation

- **Rack and pinion:** The rack is a gear of infinite radius (straight teeth on a bar). As the pinion rotates, it makes the rack move linearly (or vice versa). Relation: $v = r \cdot \omega$ (linear velocity = pitch radius × angular velocity). It is the most direct and precise way to convert rotation into translation. **Watch out for backlash:** in precision applications (CNC) preloaded or double-rack systems are used. **Typical use:** automotive steering, machine-tool tables (lathes, CNC milling machines), automatic doors, elevators (vertical rack).