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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, 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. The virtual work of the forces, Fi, acting on the particles Pi, i=1,..., n, is given by :$\delta W = \sum_^n \mathbf _ \cdot \delta \mathbf r_i$ 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 :$\delta \mathbf_i = \sum_^m \frac \delta q_j,\quad i=1,\ldots, n,$ where δqj is the virtual displacement of the generalized coordinate qj. The virtual work for the system of particles becomes :$\delta W = \mathbf _ \cdot \sum_^m \frac \delta q_j +\ldots+ \mathbf _ \cdot \sum_^m \frac \delta q_j.$ Collect the coefficients of δqj so that :$\delta W = \sum_^n \mathbf _ \cdot \frac \delta q_1 +\ldots+ \sum_^n \mathbf _ \cdot \frac \delta q_m.$

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 :$Q_j = \sum_^n \mathbf _ \cdot \frac ,\quad j=1,\ldots, m,$ 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 formT. R. Kane and D. A. Levinson
Dynamics, Theory and Applications
McGraw-Hill, NY, 2005.
:$\delta \mathbf_i = \sum_^m \frac \delta q_j,\quad i=1,\ldots, n.$ This means that the generalized force, Qj, can also be determined as :$Q_j = \sum_^n \mathbf _ \cdot \frac , \quad j=1,\ldots, m.$

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 :$\mathbf_i^*=-m_i\mathbf_i,\quad i=1,\ldots, n,$ 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 :$Q^*_j = \sum_^n \mathbf ^*_ \cdot \frac ,\quad j=1,\ldots, m.$ D'Alembert's form of the principle of virtual work yields :$\delta W = \left(Q_1+Q^*_1\right)\delta q_1 + \ldots + \left(Q_m+Q^*_m\right)\delta q_m.$

References