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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.^[1]^:265

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

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

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

\( \delta \mathbf{r}_i = \sum_{j=1}^m \frac {\partial \mathbf {r}_i} {\partial q_j} \delta q_j,\quad i=1,\ldots, n, \)

where δq_j is the virtual displacement of the generalized coordinate q_j.

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 P_i be V_i, then the virtual displacement δr_i 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, Q_j, 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, P_i, of mass m_i 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 q_j, 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.