Generalized forces

Generalized forces find use in Lagrangian mechanics, where they play a role conjugate to generalized coordinates. They are obtained from the applied forces, Fi, 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.[1]:265

The virtual work of the forces, Fi, acting on the particles Pi, i=1,..., n, is given by

\( \delta W=\sum _{{i=1}}^{n}{\mathbf {F}}_{{i}}\cdot \delta {\mathbf r}_{i} \) \)

where δri is the virtual displacement of the particle Pi.
Generalized coordinates

Let the position vectors of each of the particles, ri, be a function of the generalized coordinates, qj, j=1,...,m. Then the virtual displacements δri are given by

\( \delta {\mathbf {r}}_{i}=\sum _{{j=1}}^{m}{\frac {\partial {\mathbf {r}}_{i}}{\partial q_{j}}}\delta q_{j},\quad i=1,\ldots ,n, \)

where δqj is the virtual displacement of the generalized coordinate qj.

The virtual work for the system of particles becomes

\( \delta W={\mathbf {F}}_{{1}}\cdot \sum _{{j=1}}^{m}{\frac {\partial {\mathbf {r}}_{1}}{\partial q_{j}}}\delta q_{j}+\ldots +{\mathbf {F}}_{{n}}\cdot \sum _{{j=1}}^{m}{\frac {\partial {\mathbf {r}}_{n}}{\partial q_{j}}}\delta q_{j}. \)

Collect the coefficients of δqj so that

\( \delta W=\sum _{{i=1}}^{n}{\mathbf {F}}_{{i}}\cdot {\frac {\partial {\mathbf {r}}_{i}}{\partial q_{1}}}\delta q_{1}+\ldots +\sum _{{i=1}}^{n}{\mathbf {F}}_{{i}}\cdot {\frac {\partial {\mathbf {r}}_{i}}{\partial q_{m}}}\delta q_{m}. \)

Generalized forces

The virtual work of a system of particles can be written in the form

\( \delta W=Q_{1}\delta q_{1}+\ldots +Q_{m}\delta q_{m}, \)

where

\( Q_{j}=\sum _{{i=1}}^{n}{\mathbf {F}}_{{i}}\cdot {\frac {\partial {\mathbf {r}}_{i}}{\partial q_{j}}},\quad j=1,\ldots ,m, \)

are called the generalized forces associated with the generalized coordinates qj, 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 Pi be Vi, then the virtual displacement δri can also be written in the form[2]

\( \delta {\mathbf {r}}_{i}=\sum _{{j=1}}^{m}{\frac {\partial {\mathbf {V}}_{i}}{\partial {\dot {q}}_{j}}}\delta q_{j},\quad i=1,\ldots ,n. \)

This means that the generalized force, Qj, can also be determined as

\( Q_{j}=\sum _{{i=1}}^{n}{\mathbf {F}}_{{i}}\cdot {\frac {\partial {\mathbf {V}}_{i}}{\partial {\dot {q}}_{j}}},\quad j=1,\ldots ,m. \)

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, Pi, of mass mi is

\( {\mathbf {F}}_{i}^{*}=-m_{i}{\mathbf {A}}_{i},\quad i=1,\ldots ,n, \)

where Ai is the acceleration of the particle.

If the configuration of the particle system depends on the generalized coordinates qj, j=1,...,m, then the generalized inertia force is given by

\( Q_{j}^{*}=\sum _{{i=1}}^{n}{\mathbf {F}}_{{i}}^{*}\cdot {\frac {\partial {\mathbf {V}}_{i}}{\partial {\dot {q}}_{j}}},\quad j=1,\ldots ,m.

D'Alembert's form of the principle of virtual work yields

\( \delta W=(Q_{1}+Q_{1}^{*})\delta q_{1}+\ldots +(Q_{m}+Q_{m}^{*})\delta q_{m}. \)

References

Torby, Bruce (1984). "Energy Methods". Advanced Dynamics for Engineers. HRW Series in Mechanical Engineering. United States of America: CBS College Publishing. ISBN 0-03-063366-4.

T. R. Kane and D. A. Levinson, Dynamics, Theory and Applications, McGraw-Hill, NY, 2005.