Generalized forces find use in
Lagrangian mechanics, where they play a role conjugate to
generalized coordinates. They are obtained from the applied forces, F
i, i=1,..., n, acting on a
system that has its configuration defined in terms of
generalized coordinates. In the formulation of
virtual work, each generalized force is the coefficient of the variation of a generalized coordinate.
Virtual work
Generalized forces can be obtained from the computation of the
virtual work, δW, of the applied forces.
The virtual work of the forces, F
i, acting on the particles P
i, i=1,..., n, is given by
:
where δr
i is the
virtual displacement of the particle P
i.
Generalized coordinates
Let the position vectors of each of the particles, r
i, be a function of the generalized coordinates, q
j, j=1,...,m. Then the virtual displacements δr
i are given by
:
where δq
j is the virtual displacement of the generalized coordinate q
j.
The virtual work for the system of particles becomes
:
Collect the coefficients of δq
j so that
:
Generalized forces
The virtual work of a system of particles can be written in the form
:
where
:
are called the generalized forces associated with the generalized coordinates q
j, j=1,...,m.
Velocity formulation
In the application of the principle of virtual work it is often convenient to obtain virtual displacements from the velocities of the system. For the n particle system, let the velocity of each particle P
i be V
i, then the virtual displacement δr
i can also be written in the form
[T. R. Kane and D. A. Levinson]
Dynamics, Theory and Applications
McGraw-Hill, NY, 2005.
:
This means that the generalized force, Q
j, can also be determined as
:
D'Alembert's principle
D'Alembert formulated the dynamics of a particle as the equilibrium of the applied forces with an inertia force (
apparent force), called
D'Alembert's principle. The inertia force of a particle, P
i, of mass m
i is
:
where A
i is the acceleration of the particle.
If the configuration of the particle system depends on the generalized coordinates q
j, j=1,...,m, then the generalized inertia force is given by
:
D'Alembert's form of the principle of virtual work yields
:
References
See also
*
Lagrangian mechanics
*
Generalized coordinates
*
Degrees of freedom (physics and chemistry)
*
Virtual work
{{DEFAULTSORT:Generalized Forces
Category:Mechanics
Category:Classical mechanics
Category:Lagrangian mechanics