Generalized forces find use in
Lagrangian mechanics
In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph- ...
, where they play a role conjugate to
generalized coordinates
In analytical mechanics, generalized coordinates are a set of parameters used to represent the state of a system in a configuration space. These parameters must uniquely define the configuration of the system relative to a reference state.,p. 39 ...
. 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 analytical mechanics, generalized coordinates are a set of parameters used to represent the state of a system in a configuration space. These parameters must uniquely define the configuration of the system relative to a reference state.,p. 39 ...
. 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
A fictitious force is a force that appears to act on a mass whose motion is described using a non-inertial frame of reference, such as a linearly accelerating or rotating reference frame.
It is related to Newton's second law of motion, which trea ...
), called
D'Alembert's principle
D'Alembert's principle, also known as the Lagrange–d'Alembert principle, is a statement of the fundamental classical laws of motion. It is named after its discoverer, the French physicist and mathematician Jean le Rond d'Alembert. D'Alembert ...
. 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
In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph- ...
*
Generalized coordinates
In analytical mechanics, generalized coordinates are a set of parameters used to represent the state of a system in a configuration space. These parameters must uniquely define the configuration of the system relative to a reference state.,p. 39 ...
*
Degrees of freedom (physics and chemistry)
In physics and chemistry, a degree of freedom is an independent physical parameter in the formal description of the state of a physical system. The set of all states of a system is known as the system's phase space, and the degrees of freedo ...
*
Virtual work
{{DEFAULTSORT:Generalized Forces
Mechanics
Classical mechanics
Lagrangian mechanics