# Gear Train - Analysis Using Virtual Work

Analysis Using Virtual Work

For this analysis, we consider a gear train that has one degree-of-freedom, which means the angular rotation of all the gears in the gear train are defined by the angle of the input gear.

The size of the gears and the sequence in which they engage define the ratio of the angular velocity ωA of the input gear to the angular velocity ωB of the output gear, known as the speed ratio, or gear ratio, of the gear train. Let R be the speed ratio, then

The input torque TA acting on the input gear GA is transformed by the gear train into the output torque TB exerted by the output gear GB. If we assume, that the gears are rigid and that there are no losses in the engagement of the gear teeth, then the principle of virtual work can be used to analyze the static equilibrium of the gear train.

Let the angle θ of the input gear be the generalized coordinate of the gear train, then the speed ratio R of the gear train defines the angular velocity of the output gear in terms of the input gear, that is

The formula for the generalized force obtained from the principle of virtual work with applied torques yields

The mechanical advantage of the gear train is the ratio of the output torque TB to the input torque TA, and the above equation yields

Thus, the speed ratio of a gear train also defines its mechanical advantage. This shows that if the input gear rotates faster than the output gear, then the gear train amplifies the input torque. And, if the input gear rotates slower than the output gear, then the gear train reduces the input torque.

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