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 :$\backslash delta\; W\; =\; \backslash sum\_^n\; \backslash mathbf\; \_\; \backslash cdot\; \backslash delta\; \backslash mathbf\; r\_i$ 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 :$\backslash delta\; \backslash mathbf\_i\; =\; \backslash sum\_^m\; \backslash frac\; \backslash delta\; q\_j,\backslash quad\; i=1,\backslash ldots,\; n,$ where δq_j is the virtual displacement of the generalized coordinate q_j. The virtual work for the system of particles becomes :$\backslash delta\; W\; =\; \backslash mathbf\; \_\; \backslash cdot\; \backslash sum\_^m\; \backslash frac\; \backslash delta\; q\_j\; +\backslash ldots+\; \backslash mathbf\; \_\; \backslash cdot\; \backslash sum\_^m\; \backslash frac\; \backslash delta\; q\_j.$ Collect the coefficients of δq_j so that :$\backslash delta\; W\; =\; \backslash sum\_^n\; \backslash mathbf\; \_\; \backslash cdot\; \backslash frac\; \backslash delta\; q\_1\; +\backslash ldots+\; \backslash sum\_^n\; \backslash mathbf\; \_\; \backslash cdot\; \backslash frac\; \backslash delta\; q\_m.$

Generalized forces

The virtual work of a system of particles can be written in the form :$\backslash delta\; W\; =\; Q\_1\backslash delta\; q\_1\; +\; \backslash ldots\; +\; Q\_m\backslash delta\; q\_m,$ where :$Q\_j\; =\; \backslash sum\_^n\; \backslash mathbf\; \_\; \backslash cdot\; \backslash frac\; ,\backslash quad\; j=1,\backslash ldots,\; m,$ 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 formT. R. Kane and D. A. Levinson
Dynamics, Theory and Applications
McGraw-Hill, NY, 2005. :$\backslash delta\; \backslash mathbf\_i\; =\; \backslash sum\_^m\; \backslash frac\; \backslash delta\; q\_j,\backslash quad\; i=1,\backslash ldots,\; n.$ This means that the generalized force, Q_j, can also be determined as :$Q\_j\; =\; \backslash sum\_^n\; \backslash mathbf\; \_\; \backslash cdot\; \backslash frac\; ,\; \backslash quad\; j=1,\backslash ldots,\; m.$

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 :$\backslash mathbf\_i^*=-m\_i\backslash mathbf\_i,\backslash quad\; i=1,\backslash ldots,\; n,$ 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 :$Q^*\_j\; =\; \backslash sum\_^n\; \backslash mathbf\; ^*\_\; \backslash cdot\; \backslash frac\; ,\backslash quad\; j=1,\backslash ldots,\; m.$ D'Alembert's form of the principle of virtual work yields :$\backslash delta\; W\; =\; (Q\_1+Q^*\_1)\backslash delta\; q\_1\; +\; \backslash ldots\; +\; (Q\_m+Q^*\_m)\backslash delta\; q\_m.$

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