Generalized forces

In analytical mechanics (particularly Lagrangian mechanics), generalized forces are conjugate to generalized coordinates. They are obtained from the applied forces $F i, i = 1, \dots, n$ , 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 $δ r i$ 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 $δ r i$ 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 $δq j$ 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 $q j, j = 1, ..., m$ .

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 $δ r i$ 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 $A i$ 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}^{*}=\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.

Generalized forces

Virtual work

Generalized coordinates

Generalized forces

Velocity formulation

D'Alembert's principle

References

See also