Derivation For Special Cases
To date nobody has shown that D'Alembert's principle is equivalent to Newton's Second Law. This is true only for some very special cases e.g. rigid body constraints. However, an approximate solution to this problem does exist.
Consider Newton's law for a system of particles, i. The total force on each particle is
where
-
are the total forces acting on the system's particles, are the inertial forces that result from the total forces.
Moving the inertial forces to the left gives an expression that can be considered to represent quasi-static equilibrium, but which is really just a small algebraic manipulation of Newton's law:
Considering the virtual work, done by the total and inertial forces together through an arbitrary virtual displacement, of the system leads to a zero identity, since the forces involved sum to zero for each particle.
The original vector equation could be recovered by recognizing that the work expression must hold for arbitrary displacements. Separating the total forces into applied forces, and constraint forces, yields
If arbitrary virtual displacements are assumed to be in directions that are orthogonal to the constraint forces (which is not usually the case, so this derivation works only for special cases), the constraint forces do no work. Such displacements are said to be consistent with the constraints. This leads to the formulation of d'Alembert's principle, which states that the difference of applied forces and inertial forces for a dynamic system does no virtual work:.
There is also a corresponding principle for static systems called the principle of virtual work for applied forces.
Read more about this topic: D'Alembert's Principle
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