Virtual work

[2] The principle of virtual work had always been used in some form since antiquity in the study of statics.

It was used by the Greeks, medieval Arabs and Latins, and Renaissance Italians as "the law of lever".

[3] The idea of virtual work was invoked by many notable physicists of the 17th century, such as Galileo, Descartes, Torricelli, Wallis, and Huygens, in varying degrees of generality, when solving problems in statics.

[4] In 1768, Lagrange presented the virtual work principle in a more efficient form by introducing generalized coordinates and presented it as an alternative principle of mechanics by which all problems of equilibrium could be solved.

A systematic exposition of Lagrange's program of applying this approach to all of mechanics, both static and dynamic, essentially D'Alembert's principle, was given in his Mécanique Analytique of 1788.

[3] Although Lagrange had presented his version of least action principle prior to this work, he recognized the virtual work principle to be more fundamental mainly because it could be assumed alone as the foundation for all mechanics, unlike the modern understanding that least action does not account for non-conservative forces.

, then, for each possible trajectory that the particle may take, it is possible to compute the total work done by the force along the path.

Now consider particle P that moves from point A to point B again, but this time it moves along the nearby trajectory that differs from r(t) by the variation δr(t) = εh(t), where ε is a scaling constant that can be made as small as desired and h(t) is an arbitrary function that satisfies h(t0) = h(t1) = 0.

The terms Qi are called the generalized forces associated with the virtual displacement δr.

Kane[5] shows that these generalized forces can also be formulated in terms of the ratio of time derivatives.

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.

A lever is modeled as a rigid bar connected to a ground frame by a hinged joint called a fulcrum.

The lever is operated by applying an input force FA at a point A located by the coordinate vector rA on the bar.

The lever then exerts an output force FB at the point B located by rB.

Let the coordinate vector of the point P that defines the fulcrum be rP, and introduce the lengths

Now introduce the unit vectors eA and eB from the fulcrum to the point A and B, so

Now, denote as FA and FB the components of the forces that are perpendicular to the radial segments PA and PB.

This notation and the principle of virtual work yield the formula for the generalized force as

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

The result is D'Alembert's form of the principle of virtual work, which is used to derive the equations of motion for a mechanical system of rigid bodies.

Consider a single rigid body which moves under the action of a resultant force F and torque T, with one degree of freedom defined by the generalized coordinate q.

This inertia force can be computed from the kinetic energy of the rigid body,

Thus, dynamic equilibrium of a system of n rigid bodies with m generalized coordinates requires that

If the generalized forces Qj are derivable from a potential energy V(q1,...,qm), then these equations of motion take the form

We start by looking at the total work done by surface traction on the body going through the specified deformation:

The physical interpretation of the above equation is, the External virtual work is equal to internal virtual work when equilibrated forces and stresses undergo unrelated but consistent displacements and strains.

For practical applications: These two general scenarios give rise to two often stated variational principles.

For example, to derive the principle of virtual displacements in variational notations for supported bodies, we specify: The virtual work equation then becomes the principle of virtual displacements: This relation is equivalent to the set of equilibrium equations written for a differential element in the deformable body as well as of the stress boundary conditions on the part

Conversely, (f) can be reached, albeit in a non-trivial manner, by starting with the differential equilibrium equations and the stress boundary conditions on

The virtual work principle is also valid for large real displacements; however, Eq.