Various forms of this principle have been credited to Johann (Jean) Bernoulli (1667–1748) and Daniel Bernoulli (1700–1782). The expression compatible displacements means that the particles remain in contact and displace together so that the work done by pairs of action/reaction inter-particle forces cancel out. 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. If a system is not in static equilibrium, D'Alembert showed that by introducing the acceleration terms of Newton's laws as inertia forces, this approach is generalized to define dynamic equilibrium. If the principle of virtual work for applied forces is used on individual particles of a rigid body, the principle can be generalized for a rigid body: When a rigid body that is in equilibrium is subject to virtual compatible displacements, the total virtual work of all external forces is zero and conversely, if the total virtual work of all external forces acting on a rigid body is zero then the body is in equilibrium. And, if the input gear rotates slower than the output gear, then the gear train reduces the input torque.ĭynamic equilibrium for rigid bodies This shows that if the input gear rotates faster than the output gear, then the gear train amplifies the input torque. Thus, the speed ratio of a gear train also defines its mechanical advantage. If a force acts on a particle as it moves from point A 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. 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. 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. His idea was to convert a dynamical problem into static problem by introducing inertial force. In 1743 D'Alembert published his Traité de Dynamique where he applied the principle of virtual work, based on Bernoulli's work, to solve various problems in dynamics. This formulation of the principle is today known as the principle of virtual velocities and is commonly considered as the prototype of the contemporary virtual work principles. ![]() Bernoulli's version of virtual work law appeared in his letter to Pierre Varignon in 1715, which was later published in Varignon's second volume of Nouvelle mécanique ou Statique in 1725. He was able to solve problems for both rigid bodies as well as fluids. Working with Leibnizian concepts, Johann Bernoulli systematized the virtual work principle and made explicit the concept of infinitesimal displacement. ![]() 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. It was used by the Greeks, medieval Arabs and Latins, and Renaissance Italians as "the law of lever". The principle of virtual work had always been used in some form since antiquity in the study of statics. Historically, virtual work and the associated calculus of variations were formulated to analyze systems of rigid bodies, but they have also been developed for the study of the mechanics of deformable bodies. The work of a force on a particle along a virtual displacement is known as the virtual work. This displacement is therefore the displacement followed by the particle according to the principle of least action. Among all the possible displacements that a particle may follow, called virtual displacements, one will minimize the action. The work of a force acting on a particle as it moves along a displacement is different for different displacements. In mechanics, virtual work arises in the application of the principle of least action to the study of forces and movement of a mechanical system.
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