Virtual displacements, principle of

principle of virtual velocities, principle of virtual work

One of the differential variational principles of classical mechanics, expressing the most general conditions for the equilibrium of mechanical systems interconnected by ideal constraints.

According to this principle, a system is in equilibrium in a certain position if and only if the work by the active forces, during all virtual displacements from this position, is zero or less than zero,

$$ \tag{* } \sum _ \nu \mathbf F _ \nu \cdot \delta \mathbf r _ \nu \leq 0, $$

at any moment of time. A virtual displacement is an infinitesimal displacement $ \delta \mathbf r _ \nu $ consistent with the constraints imposed on the system.

If the constraints are of retaining type (bilateral), the virtual displacements are reversible, and condition (*) must be taken with the equality sign; if the constraints are non-retaining (unilateral), there are virtual displacements that are irreversible. If the system is displaced by active forces, its points are acted upon by the constraints through forces of reaction $ \mathbf R _ \nu $( passive forces); these are defined on the assumption that the mechanical effect of the constraints on the system has been totally taken into account, in the sense that the constraints may be replaced by the reactions they produce (the liberation axiom). The constraints are called ideal if the sum of the work elements of their reactions $ \sum _ \nu \mathbf R _ \nu \cdot \delta \mathbf r _ \nu \geq 0 $, the equality sign applying to the virtual reversible displacements, while the equality sign and the "greater-than-zero" sign apply to irreversible displacements. The states of equilibrium of the system are the states $ \mathbf r _ \nu = \mathbf r _ \nu ( t _ {0} ) $ in which the system will remain if it has been placed in them with zero initial velocities $ \mathbf v _ \nu ( t _ {0} ) = 0 $; here it is assumed that for any $ t $ the equations of the constraints are satisfied by the values $ \mathbf r _ \nu = \mathbf r _ \nu ( t _ {0} ) $ and $ \mathbf v _ \nu = 0 $.

Condition (*) contains all the equations and equilibrium laws of systems with ideal constraints; it is therefore justifiable to say that the entire science of statics is expressed by the single general formula (*).

The law of equilibrium, expressed by the principle of virtual displacements, was first stated by G. Ubaldi for the case of a lever and of a floating pulley and block and tackle. G. Galilei established the law for the case of the inclined plane and as the characteristic feature of the equilibrium of simple machines. J. Wallis considered it as a fundamental law of statics and deduced the theory of equilibrium of engines from it. R. Descartes reduced the entire science of statics to a single principle, which was practically identical with the principle of Galilei. J. Bernoulli was the first to grasp the general nature of the principle and its usefulness in solving problems in statics. J.L. Lagrange [1] expressed the principle in its general form, thus reducing statics to a single general formula, and gave a (not entirely rigorous) proof of it for systems restricted by retaining (bilateral) constraints. Lagrange made systematic use of the general formula of statics for the equilibrium of an arbitrary system of forces, and of his own method of applying the formula, to prove general properties of systems of bodies, and to solve various problems in statics, including problems involving equilibria of non-compressible, compressible and elastic liquids. Lagrange considered this principle as the basic principle of mechanics as a whole. A rigorous proof of the principle of virtual work and its extension to unilateral (non-retaining) constraints is due to J. Fourier [2] and M.V. Ostrogradski [3].

References

[1]	J.L. Lagrange, "Mécanique analytique" , 1–2 , Blanchard, reprint , Paris (1965) ((Also: Oeuvres, Vol. 11.))
[2]	J. Fourier, J. École Polytechnique , II (1798) pp. 20
[3]	M.V. Ostrogradskii, "Lectures on analytical mechanics" , Complete works , 1(2) , Moscow-Leningrad (1946) (In Russian)

Comments

If the virtual displacements are not homogeneously distributed over the system, (*) must be replaced by

$$ \int\limits _ { v } \mathbf F \cdot \delta \mathbf r dV \leq 0 , $$

where the integration is over the system.

References

[a1]	J. Szabo, "Geschichte der mechanischen Prinzipien" , Birkhäuser (1977)
[a2]	J. Szabo, "Höhere technische Mechanik" , Springer (1958)
[a3]	R.M. Rosenberg, "Analytical dynamics of discrete systems" , Plenum (1977)