Generalized forces

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Generalized forces find use in Lagrangian mechanics, where they play a role conjugate to generalized coordinates. They are obtained from the applied forces, Fi, 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, Fi, acting on the particles Pi, i=1,..., n, is given by

\delta W = \sum_{i=1}^n \mathbf {F}_{i} \cdot \delta \mathbf r_i

where δri is the virtual displacement of the particle Pi.

Generalized coordinates

Let the position vectors of each of the particles, ri, be a function of the generalized coordinates, qj, j=1,...,m. Then the virtual displacements δri 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 δqj is the virtual displacement of the generalized coordinate qj.

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 δqj 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 qj, 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 Pi be Vi, then the virtual displacement δri can also be written in the form2

\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, Qj, 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, Pi, of mass mi is

\mathbf{F}_i^*=-m_i\mathbf{A}_i,\quad i=1,\ldots, n,

where Ai is the acceleration of the particle.

If the configuration of the particle system depends on the generalized coordinates qj, 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

  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.

See also








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