Computational Dynamics, 3rd Edition

Chapter 5. VIRTUAL WORK AND LAGRANGIAN DYNAMICS

The principle of virtual work represents a powerful tool for deriving the static and dynamic equations of multibody systems. Unlike Newtonian mechanics, the principle of virtual work does not require considering the constraint forces, and it requires only scalar work quantities to define the static and dynamic equations. This principle can be used to systematically derive a minimum set of equations of motion of the multibody systems by eliminating the constraint forces. To use the principle of virtual work, the important concepts of the virtual displacements and generalized forces are first introduced and used to formulate the generalized forces of several force elements, such as springs and dampers and friction forces. It is shown in this chapter that the principle of virtual work can be used to obtain a number of equations equal to the number of the system degrees of freedom, thereby providing a systematic procedure for obtaining the embedding form of the equations of motion of the mechanical system. Use of the principle of virtual work in statics and dynamics is demonstrated using several applications. The principle of virtual work is also used in this chapter to derive the well-known Lagrange's equation, in which the generalized inertia force is expressed in terms of the scalar kinetic energy. Several other forms of the generalized inertia forces are also presented, including the form that appears in the Gibbs-Appel equation, in ...