**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 ω_{A} of the input gear to the angular velocity ω_{B} 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 *T*_{A} acting on the input gear *G*_{A} is transformed by the gear train into the output torque *T*_{B} exerted by the output gear *G*_{B}. 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 *T*_{B} to the input torque *T*_{A}, 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.

Read more about this topic: Gear Train

### Famous quotes containing the words work, analysis and/or virtual:

“You are made

Rather to wonder at the things you hear

Than to *work* any.”

—William Shakespeare (1564–1616)

“Whatever else American thinkers do, they psychologize, often brilliantly. The trouble is that psychology only takes us so far. The new interest in families has its merits, but it will have done us all a disservice if it turns us away from public issues to private matters. A vision of things that has no room for the inner life is bankrupt, but a psychology without social *analysis* or politics is both powerless and very lonely.”

—Joseph Featherstone (20th century)

“Tragedy dramatizes human life as potentiality and fulfillment. Its *virtual* future, or Destiny, is therefore quite different from that created in comedy. Comic Destiny is Fortune—what the world will bring, and the man will take or miss, encounter or escape; tragic Destiny is what the man brings, and the world will demand of him. That is his Fate.”

—Susanne K. Langer (1895–1985)