Gear train

s showing tangent contact between their pitch circles, each illustrated with broken blue lines; the gear on the left has 10 teeth and the gear on the right has 15 teeth. Dimensions and terms The pitch circle of a given gear is determined by the tangent point contact between two meshing gears; for example, two spur gears mesh together when their pitch circles are tangent, as illustrated. The number of teeth per gear is an integer determined by the pitch circle and circular pitch. Relationships The circular pitch of a gear can be defined as the circumference of the pitch circle using its pitch radius divided by the number of teeth : : F_\theta = T_A \frac{\partial\omega_A}{\partial\omega} - T_B \frac{\partial \omega_B}{\partial\omega}= T_A - \frac{T_B}{R_{AB}} = 0. This can be rearranged to: : R_{AB} = \frac{T_B}{T_A} Since is the gear ratio of the gear train, the input torque applied to the input gear A and the output torque on the output gear B are related by the same gear or speed ratio. Mechanical advantage The torque ratio of a gear train is also known as its mechanical advantage; as demonstrated, the gear ratio and speed ratio of a gear train also give its mechanical advantage. : \mathrm{MA} \equiv \frac{T_B}{T_A} = R_{AB} The mechanical advantage of a pair of meshing gears for which the input gear A has teeth and the output gear B has teeth is given by : \mathrm{MA} = R_{AB} = \left| \frac{\omega_A}{\omega_B} \right| = \frac{N_B}{N_A} This shows that if the output gear B has more teeth than the input gear A, then the gear train amplifies the input torque. In this case, the gear train is called a speed reducer and since the output gear must have more teeth than the input gear, the speed reducer amplifies the input torque. ==Implementations==

Gear trains with two gears The simplest example of a gear train has two gears. The input gear (also known as the drive gear or driver) transmits power to the output gear (also known as the driven gear). The input gear will typically be connected to a power source, such as a motor or engine. In such an example, the output of torque and rotational speed from the output (driven) gear depend on the ratio of the dimensions of the two gears or the ratio of the tooth counts. Idler gears in the middle which does not affect the overall gear ratio but reverses the direction of rotation of the gear on the right In a sequence of gears chained together, the ratio depends only on the number of teeth on the first and last gear. The intermediate gears, regardless of their size, do not alter the overall gear ratio of the chain. However, the addition of each intermediate gear reverses the direction of rotation of the final gear. An intermediate gear which does not drive a shaft to perform any work is called an idler gear. Sometimes, a single idler gear is used to reverse the direction, in which case it may be referred to as a reverse idler. For instance, the typical automobile manual transmission engages reverse gear by means of inserting a reverse idler between two gears. Idler gears can also transmit rotation among distant shafts in situations where it would be impractical to simply make the distant gears larger to bring them together. Not only do larger gears occupy more space, the mass and rotational inertia (moment of inertia) of a gear is proportional to the square of its radius. Instead of idler gears, a toothed belt or chain can be used to transmit torque over distance. Formula If a simple gear train has three gears, such that the input gear A meshes with an intermediate gear I which in turn meshes with the output gear B, then the pitch circle of the intermediate gear rolls without slipping on both the pitch circles of the input and output gears. This yields the two relations : \frac = \frac{N_I}{N_A}, \quad \frac = \frac{N_B}{N_I}. The speed ratio of the overall gear train is obtained by multiplying these two equations for each pair (A-I and I-B) to obtain : R = \frac = \frac{N_B}{N_A}. This is because the number of idler gear teeth cancels out when the gear ratios of the two subsets are multiplied: :\begin{align} R_{final} &= R_{AI} \cdot R_{IB} \\ &= \left( \frac{N_I}{N_A} \right) \cdot \left( \frac{N_B}{N_I} \right) = \left( \frac{N_B}{N_A} \right) \end{align} Notice that this gear ratio is exactly the same as for the case when the gears A and B engage directly. The intermediate gear provides spacing but does not affect the gear ratio. For this reason it is called an idler gear. The same gear ratio is obtained for a sequence of idler gears and hence an idler gear is used to provide the same direction to rotate the driver and driven gear. If the driver gear moves in the clockwise direction, then the driven gear also moves in the clockwise direction with the help of the idler gear. Example on a piece of farm equipment, with a ratio of 42:13 = 3.23:1 In the photo, assume the smallest gear (gear A, in the lower right corner) is connected to the motor, which makes it the drive gear or input gear. The somewhat larger gear in the middle (gear I) is called an idler gear. It is not connected directly to either the motor or the output shaft and only transmits power between the input and output gears. There is a third gear (gear B) partially shown in the upper-right corner of the photo. Assuming that gear is connected to the machine's output shaft, it is the output or driven gear. Considering only gears A and I, the gear ratio between the idler and the input gear can be calculated as if the idler gear was the output gear. The input gear A in this two-gear subset has 13 teeth () and the idler gear I has 21 teeth (). Therefore, the gear ratio for this subset is :R_{AI} = \frac{N_I}{N_A} = \frac{21}{13} This is approximately 1.62 or 1.62:1. At this ratio, it means the drive gear (A) must make 1.62 revolutions to turn the output gear (I) once. It also means that for every one revolution of the driver (A), the output gear (I) has made = , or 0.62, revolutions. The larger gear (I) turns slower. The third gear in the picture (B) has = 42 teeth. Now consider the gear ratio for the subset consisting of gears I and B, with the idler gear I serving as the input and third gear B serving as the output. The gear ratio between the idler (I) and third gear (B) is thus :R_{IB} = \frac{N_B}{N_I} = \frac{42}{21} or 2:1. The final gear ratio of the compound system is . For every 3.23 revolutions of the smallest gear A, the largest gear B turns one revolution, or for every one revolution of the smallest gear A, the largest gear B turns 0.31 () revolution, a total reduction of about 1:3.23 (Gear Reduction Ratio (GRR) is the inverse of Gear Ratio (GR)). Since the idler gear I contacts directly both the smaller gear A and the larger gear B, it can be removed from the calculation, also giving a ratio of . The idler gear serves to make both the drive gear and the driven gear rotate in the same direction, but confers no mechanical advantage. Double reduction gear A double reduction gear set comprises two pairs of gears, each individually single reductions, in series. In the diagram, the red and blue gears give the first stage of reduction and the orange and green gears give the second stage of reduction. The total reduction is the product of the first stage of reduction and the second stage of reduction. It is essential to have two coupled gears, of different sizes, on the intermediate layshaft. If a single intermediate gear was used, the overall ratio would be simply that between the first and final gears, the intermediate gear would only act as an idler gear: it would reverse the direction of rotation, but not change the ratio. Belt and chain drives Special gears called sprockets can be coupled together with chains, as on bicycles and some motorcycles. Alternatively, belts can have teeth in them also and be coupled to gear-like pulleys. Again, exact accounting of teeth and revolutions can be applied with these machines. For example, a belt with teeth, called the timing belt, is used in some internal combustion engines to synchronize the movement of the camshaft with that of the crankshaft, so that the valves open and close at the top of each cylinder at exactly the right time relative to the movement of each piston. A chain, called a timing chain, is used on some automobiles for this purpose, while in others, the camshaft and crankshaft are coupled directly together through meshed gears. Regardless of which form of drive is employed, the crankshaft-to-camshaft gear ratio is always 2:1 on four-stroke engines, which means that for every two revolutions of the crankshaft the camshaft will rotate once. ==Automotive applications==

Automobile powertrains generally have two or more major areas where gear sets are used. For internal combustion engine (ICE) vehicles, gearing is typically employed in the transmission, which contains a number of different sets of gears that can be changed to allow a wide range of vehicle speeds while operating the ICE within a narrower range of speeds, optimizing efficiency, power, and torque. Because electric vehicles instead use one or more electric traction motor(s) which generally have a broader range of operating speeds, they are typically equipped with a single-ratio reduction gear set instead. The second common gear set in almost all motor vehicles is the differential, which contains the final drive to and often provides additional speed reduction at the wheels. Moreover, the differential contains gearing that splits torque equally between the two wheels while permitting them to have different speeds when traveling in a curved path. The transmission and final drive might be separate and connected by a driveshaft, or they might be combined into one unit called a transaxle. The gear ratios in transmission and final drive are important because different gear ratios will change the characteristics of a vehicle's performance. timing gears on a Ford Taunus V4 engine — the small gear is on the crankshaft, the larger gear is on the camshaft. The crankshaft gear has 34 teeth, the camshaft gear has 68 teeth and runs at half the crankshaft RPM.(The small gear in the lower left is on the balance shaft.) As noted, the ICE itself is often equipped with a gear train to synchronize valve operation with crankshaft speed. Typically, the camshafts are driven by gearing, chain, or toothed belt. Example : In first gear, the engine makes 2.97 revolutions for every revolution of the transmission's output. In fourth gear, the gear ratio of 1:1 means that the engine and the transmission's output rotate at the same speed, referred to as the "direct drive" ratio. Fifth and sixth gears are known as overdrive gears, in which the output of the transmission revolves faster than the engine's output. The Corvette above is equipped with a differential that has a final drive ratio (or axle ratio) of 3.42:1, meaning that for every 3.42 revolutions of the transmission's output, the wheels make one revolution. The differential ratio multiplies with the transmission ratio, so in 1st gear, the engine makes revolutions for every revolution of the wheels. The car's tires can almost be thought of as a third type of gearing. This car is equipped with 295/35-18 tires, which have a circumference of  inches. This means that for every complete revolution of the wheel, the car travels . If the Corvette had larger tires, it would travel farther with each revolution of the wheel, which would be like a higher gear. If the car had smaller tires, it would be like a lower gear. With the gear ratios of the transmission and differential and the size of the tires, it becomes possible to calculate the speed of the car for a particular gear at a particular engine RPM. For example, it is possible to determine the distance the car will travel for one revolution of the engine by dividing the circumference of the tire by the combined gear ratio of the transmission and differential. : d = \frac{c_t}{gr_t \times gr_d} It is also possible to determine a car's speed from the engine speed by multiplying the circumference of the tire by the engine speed and dividing by the combined gear ratio. : v_c = \frac{c_t \times v_e}{gr_t \times gr_d} Note that the answer is in inches per minute, which can be converted to mph by dividing by 1056. Close-ratio transmissions are generally offered in sports cars, sport bikes, and especially in race vehicles, where the engine is tuned for maximum power in a narrow range of operating speeds, and the driver or rider can be expected to shift often to keep the engine in its power band. Factory four- or five-speed transmission ratios generally have a greater difference between gear ratios and tend to be effective for ordinary driving and moderate performance use. Wider gaps between ratios allow a higher 1st gear ratio for better manners in traffic, but cause engine speed to decrease more when shifting. Narrowing the gaps will increase acceleration at speed, and potentially improve top speed under certain conditions, but acceleration from a stopped position and operation in daily driving will suffer. Range is the torque multiplication difference between 1st and 4th gears; wider-ratio gear-sets have more, typically between 2.8 and 3.2. This is the single most important determinant of low-speed acceleration from stopped. Progression is the reduction or decay in the percentage drop in engine speed in the next gear, for example after shifting from first to second gear. Most transmissions have some degree of progression in that the RPM drop on the first–second shift is larger than the RPM drop on the second–third shift, which is in turn larger than the RPM drop on the third–fourth shift. The progression may not be linear (continuously reduced) or done in proportionate stages for various reasons, including a special need for a gear to reach a specific speed or RPM for passing, racing and so on, or simply economic necessity that the parts were available. Range and progression are not mutually exclusive, but each limits the number of options for the other. A wide range, which gives a strong torque multiplication in first gear for excellent manners in low-speed traffic, especially with a smaller motor, heavy vehicle, or numerically low axle ratio such as 2.50, means the progression percentages must be high. The amount of engine speed, and therefore power, lost on each up-shift is greater than would be the case in a transmission with less range, but less power in first gear. A numerically low first gear, such as 2:1, reduces available torque in 1st gear, but allows more choices of progression. There is no optimal choice of transmission gear ratios or a final drive ratio for best performance at all speeds, as gear ratios are compromises, and not necessarily better than the original ratios for certain purposes. ==See also==