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According to Physics-net.com, "gears are a series of continuous levers," which is an accurate assessment of how they work. Drive gear sets are used to multiply torque or to increase shaft speed, and they do so with these two changes being inversely proportional to each other. As output force in a gear transmission increases, speed decreases, and vice versa, and the result is that calculating gear speeds is a straightforward endeavour.

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Define the gear application, which must have at least two engaged or meshed gears. You can find an example of designated gear sets in any motor vehicle. Suppose a rear-wheel drive car has a four-speed manual transmission and overdrive, with a 3:1 final drive ratio in its differential rear axle. The transmission speeds have ratios of 2.5:1, 2:1, 1.5:1 and 1:1, respectively. Overdrive provides a final ratio of 0.7:1. Knowing these ratios, you can calculate all the gear speeds, and how fast the rear wheels will turn---and thus vehicle speed---with respect to engine speed.

Set the input gear speed to 2500 revolutions per minute (rpm) as a basis for calculating output gear speeds. You will apply each of the transmission gear ratios against this 2500-rpm input value.

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Calculate the output speed for the first-gear 2.5:1 transmission gear ratio. If the 2.5:1 ratio is produced by a 12-tooth gear driving a 30-tooth gear: 30 / 12 = 2.5, then transmission output speed is: 2500rpm / 2.5 = 1000rpm in first gear.

Since the transmission output is further reduced in the rear-drive axle by a 3:1 ratio---specifically a 10-tooth pinion gear driving a 30-tooth ring gear, the rear wheels will turn at: 1000rpm / 3.0 = 333.33rpm in first-gear, with the engine turning 2500rpm.

Calculate the other transmission speeds based on the calculation used for first-gear.

The 2:1 second-gear ratio computes as 2500rpm / 2.0 = 1250-rpm transmission output speed, and 1250rpm / 3 = 416.67-rpm rear axle speed.

Third gear is 2500rpm / 1.5 = 1666.67-rpm transmission output speed, and 555.55-rpm rear axle speed.

Fourth gear calculates to 2500rpm / 1.0 = 2500rpm, or no reduction; this is termed 1:1 or direct drive. Axle speed is 2500rpm / 3 = 833.33rpm.

Overdrive is a speed-increaser function, which means it increases the transmission output speed over the input engine speed. To realise a 0.7:1 overdrive ratio, a larger gear would drive a smaller gear, such as a 20-tooth gear driving a 14-tooth gear. The input speed is 2500rpm / 0.7, yielding a transmission output speed of 3571.43rpm, resulting in a rear axle speed of 1190.47rpm.

Perform a reasonability calculation to validate the transmission gear speed computations. This is how an automotive engineer would design a vehicle. If the vehicle has tires with a 72-inch outer circumference, then each axle-rpm will yield 6 feet of forward travel.

First gear will therefore provide 333.33rpm * 6 feet/rpm / 88 = 22.72 miles per hour (mph)

Similarly, second-gear would yield 416.67rpm * 6 feet/rpm / 88 = 28.4mph.

Third-gear is 37.87mph.

Fourth gear is 56.81mph.

Overdrive divides the 56.81-mph fourth-gear speed by 0.7, for a cruise speed of 81.16mph. This is entirely reasonable for an overdrive transmission at 2500rpm engine speed, and about right for Germany's Autobahn.

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