Generalized forces explained

In analytical mechanics (particularly Lagrangian mechanics), generalized forces are conjugate to generalized coordinates. They are obtained from the applied forces, 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,, of the applied forces.^[1]

The virtual work of the forces,, acting on the particles, is given by$$\backslash delta\; W\; =\; \backslash sum\_^n\; \backslash mathbf\; F\_i\; \backslash cdot\; \backslash delta\; \backslash mathbf\; r\_i$$where is the virtual displacement of the particle .

Generalized coordinates

Let the position vectors of each of the particles,, be a function of the generalized coordinates, . Then the virtual displacements 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 is the virtual displacement of the generalized coordinate .

The virtual work for the system of particles becomes$$\backslash delta\; W\; =\; \backslash mathbf\; F\_1\; \backslash cdot\; \backslash sum\_^m\; \backslash frac\; \backslash delta\; q\_j\; +\; \backslash dots\; +\; \backslash mathbf\; F\_n\; \backslash cdot\; \backslash sum\_^m\; \backslash frac\; \backslash delta\; q\_j.$$Collect the coefficients of so that$$\backslash delta\; W\; =\; \backslash sum\_^n\; \backslash mathbf\; F\_i\; \backslash cdot\; \backslash frac\; \backslash delta\; q\_1\; +\; \backslash dots\; +\; \backslash sum\_^n\; \backslash mathbf\; F\_i\; \backslash cdot\; \backslash frac\; \backslash delta\; q\_m.$$

Generalized forces

The virtual work of a system of particles can be written in the form$$\backslash delta\; W\; =\; Q\_1\backslash delta\; q\_1\; +\; \backslash dots\; +\; Q\_m\backslash delta\; q\_m,$$where$$Q\_j\; =\; \backslash sum\_^n\; \backslash mathbf\; F\_i\; \backslash cdot\; \backslash frac,\backslash quad\; j=1,\backslash ldots,\; m,$$are called the generalized forces associated with the generalized coordinates .

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, then the virtual displacement can also be written in the form^[2] $$\backslash delta\; \backslash mathbf\; r\_i\; =\; \backslash sum\_^m\; \backslash frac\; \backslash delta\; q\_j,\backslash quad\; i=1,\backslash ldots,\; n.$$

This means that the generalized force,, can also be determined as$$Q\_j\; =\; \backslash sum\_^n\; \backslash mathbf\; F\_i\; \backslash cdot\; \backslash frac,\; \backslash quad\; j=1,\backslash ldots,\; m.$$

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,, of mass is$$\backslash mathbf\; F\_i^*=-m\_i\backslash mathbf\; A\_i,\backslash quad\; i=1,\backslash ldots,\; n,$$where is the acceleration of the particle.

If the configuration of the particle system depends on the generalized coordinates, then the generalized inertia force is given by$$Q^*\_j\; =\; \backslash sum\_^n\; \backslash mathbf\; F^*\_\; \backslash cdot\; \backslash frac,\backslash quad\; j=1,\backslash ldots,\; m.$$

D'Alembert's form of the principle of virtual work yields$$\backslash delta\; W\; =\; (Q\_1\; +\; Q^*\_1)\backslash delta\; q\_1\; +\; \backslash dots\; +\; (Q\_m\; +\; Q^*\_m)\backslash delta\; q\_m.$$

See also

• Lagrangian mechanics
• Generalized coordinates
• Degrees of freedom (physics and chemistry)
• Virtual work

Notes and References

1. Book: Torby, Bruce . Advanced Dynamics for Engineers . HRW Series in Mechanical Engineering . 1984 . CBS College Publishing . United States of America . 0-03-063366-4 . Energy Methods.
2. T. R. Kane and D. A. Levinson, Dynamics, Theory and Applications, McGraw-Hill, NY, 2005.