The motion ratio of a mechanism is the ratio of the displacement of the point of interest to that of another point.
The most common example is in a vehicle's suspension, where it is used to describe the displacement and forces in the springs and shock absorbers. The force in the spring is (roughly) the vertical force at the contact patch divided by the motion ratio, and the spring rate is the wheel rate divided by the motion ratio squared.
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This is described as the Installation Ratio in the reference. Motion ratio is the more common term in the industry, but sometimes is used to mean the inverse of the above definition.
Motion ratio in suspension of a vehicle describes the amount of shock travel for a given amount of wheel travel. Mathematically, it is the ratio of shock travel and wheel travel. The amount of force transmitted to the vehicle chassis reduces with increase in motion ratio. A motion ratio close to one is desired in the vehicle for better ride and comfort. One should know the desired wheel travel of the vehicle before calculating motion ratio, which depends much on the type of track the vehicle will run upon.
Selecting the appropriate ratio depends on multiple factors: