Generalized forces

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.^[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 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

\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 δr_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