Editing Virtual work (section)

=== Constraint forces ===
An important benefit of the principle of virtual work is that only forces that do work as the system moves through a [[virtual displacement]] are needed to determine the mechanics of the system.  There are many forces in a mechanical system that do no work during a [[virtual displacement]], which means that they need not be considered in this analysis.  The two important examples are (i) the internal forces in a [[rigid body]], and (ii) the constraint forces at an ideal [[Kinematic pair|joint]].

Lanczos<ref name=Lanczos/> presents this as the postulate:  "The virtual work of the forces of reaction is always zero for any [[virtual displacement]] which is in harmony with the given kinematic constraints."  The argument is as follows.  The principle of virtual work states that in [[Mechanical equilibrium|equilibrium]] the virtual work of the forces applied to a system is zero.  [[Newton's laws of motion|Newton's laws]] state that at [[Mechanical equilibrium|equilibrium]] the applied forces are equal and opposite to the reaction, or constraint forces.  This means the virtual work of the constraint forces must be zero as well.
<!-- removal of unnecessary part, lever and gear train in the next two sections is sufficient
== One degree-of-freedom mechanisms ==
In this section, the principle of virtual work is used for the static analysis of one degree-of-freedom mechanical devices.  Specifically, we analyze  the lever, a pulley system, a gear train, and a four-bar linkage.  Each of these devices moves in the plane, therefore a force {{math|1='''F''' = (''f<sub>x</sub>'', ''f<sub>y</sub>'')}} has two components and acts on a point with coordinates {{math|1='''r''' = (''r<sub>x</sub>'', ''r<sub>y</sub>'')}} and velocity {{math|1='''v''' = (''v<sub>x</sub>'', ''v<sub>y</sub>'').  A moment, also called a [[torque]], ''T'' acting on a body that moves in the plane has one component as does the angular velocity ''ω'' of the body.

Assume the bodies in the mechanism are rigid and the joints are ideal so that the only change in virtual work is associated with the movement of the input and output forces and torques.

=== Applied Forces ===
Consider a mechanism such as a lever that operates so that an input force generates an output force.   Let ''A'' be the point where the input force '''F'''<sub>''A''</sub> is applied, and let ''B'' be the point where the output force '''F'''<sub>''B''</sub> is exerted.  Define the position and velocity of ''A'' and ''B'' by the vectors '''r'''<sub>''A''</sub>, '''v'''<sub>''A''</sub> and '''r'''<sub>''B''</sub>, '''v'''<sub>''B''</sub>, respectively.

Because the mechanism has one degree-of-freedom, there is a single generalized coordinate ''q'' that defines the position vectors '''r'''<sub>''A''</sub>(''q'') and '''r'''<sub>''B''</sub>(''q'') of the input and output points in the system.  The principle of virtual work requires that the generalized force associated with this coordinate be zero, thus
<math display="block"> Q = \mathbf{F}_A \cdot \frac{\partial\mathbf{v}_A}{\partial\dot{q}} - \mathbf{F}_B \cdot \frac{\partial\mathbf{v}_B}{\partial\dot{q}} = 0.</math>

The negative sign on the output force '''F'''<sub>''B''</sub> arises because the convention of virtual work assumes the forces are applied to the device.

=== Applied Torque ===
Consider a mechanism such as a gear train that operates so that an input torque generates an output torque.  Let body ''E''<sub>''A''</sub> have the input moment ''T''<sub>''A''</sub> applied to it, and let body ''E''<sub>''B''</sub> exert the output torque ''T''<sub>''B''</sub>.  Define the angular position and velocity of ''E''<sub>''A''</sub> and ''E''<sub>''B''</sub> by ''θ''<sub>''A''</sub>, ''ω''<sub>''A''</sub> and ''θ''<sub>''B''</sub>, ''ω''<sub>''B''</sub>, respectively.

Because the mechanism has one degree-of-freedom, there is a single generalized coordinate ''q'' that defines the angles ''θ<sub>A</sub>''(''q'') and ''θ<sub>B</sub>''(''q'') of the input and output of the system.  The principle of virtual work requires that the generalized force associated with this coordinate be zero, thus
<math display="block"> Q = T_A \frac{\partial\boldsymbol{\omega}_A}{\partial\dot{q}} - T_B \frac{\partial\boldsymbol{\omega}_B}{\partial\dot{q}}=0.</math>

The negative sign on the output torque ''T''<sub>''B''</sub> arises because the convention of virtual work assumes the torques are applied to the device.
-->