Generalized forces

In analytical mechanics (particularly Lagrangian mechanics), generalized forces are 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 $${\displaystyle \delta W=\sum _{i=1}^{n}\mathbf {F} _{i}\cdot \delta \mathbf {r} _{i}}$$ 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 $${\displaystyle \delta \mathbf {r} _{i}=\sum _{j=1}^{m}{\frac {\partial \mathbf {r} _{i}}{\partial q_{j}}}\delta q_{j},\quad i=1,\ldots ,n,}$$ where δq_j is the virtual displacement of the generalized coordinate q_j.

The virtual work for the system of particles becomes $${\displaystyle \delta W=\mathbf {F} _{1}\cdot \sum _{j=1}^{m}{\frac {\partial \mathbf {r} _{1}}{\partial q_{j}}}\delta q_{j}+\dots +\mathbf {F} _{n}\cdot \sum _{j=1}^{m}{\frac {\partial \mathbf {r} _{n}}{\partial q_{j}}}\delta q_{j}.}$$ Collect the coefficients of δq_j so that $${\displaystyle \delta W=\sum _{i=1}^{n}\mathbf {F} _{i}\cdot {\frac {\partial \mathbf {r} _{i}}{\partial q_{1}}}\delta q_{1}+\dots +\sum _{i=1}^{n}\mathbf {F} _{i}\cdot {\frac {\partial \mathbf {r} _{i}}{\partial q_{m}}}\delta q_{m}.}$$

Generalized forces

The virtual work of a system of particles can be written in the form $${\displaystyle \delta W=Q_{1}\delta q_{1}+\dots +Q_{m}\delta q_{m},}$$ where $${\displaystyle Q_{j}=\sum _{i=1}^{n}\mathbf {F} _{i}\cdot {\frac {\partial \mathbf {r} _{i}}{\partial q_{j}}},\quad j=1,\ldots ,m,}$$ are called the generalized forces associated with the generalized coordinates 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 form^[2] $${\displaystyle \delta \mathbf {r} _{i}=\sum _{j=1}^{m}{\frac {\partial \mathbf {V} _{i}}{\partial {\dot {q}}_{j}}}\delta q_{j},\quad i=1,\ldots ,n.}$$

This means that the generalized force, Q_j, can also be determined as $${\displaystyle Q_{j}=\sum _{i=1}^{n}\mathbf {F} _{i}\cdot {\frac {\partial \mathbf {V} _{i}}{\partial {\dot {q}}_{j}}},\quad j=1,\ldots ,m.}$$

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

D'Alembert's form of the principle of virtual work yields $${\displaystyle \delta W=(Q_{1}+Q_{1}^{*})\delta q_{1}+\dots +(Q_{m}+Q_{m}^{*})\delta q_{m}.}$$

See also

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

References

1. 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.
2. T. R. Kane and D. A. Levinson, Dynamics, Theory and Applications, McGraw-Hill, NY, 2005.