Generalized forces

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Generalized forces are defined via coordinate transformation of applied forces, \mathbf{F}_i, on a system of n particles, i. The concept finds use in Lagrangian mechanics, where it plays a conjugate role to generalized coordinates.

A convenient equation from which to derive the expression for generalized forces is that of the virtual work, δWa, caused by applied forces, as seen in D'Alembert's principle in accelerating systems and the principle of virtual work for applied forces in static systems. The subscript a is used here to indicate that this virtual work only accounts for the applied forces, a distinction which is important in dynamic systems.[1]:265

\delta W_a = \sum_{i=1}^n \mathbf {F}_{i} \cdot \delta \mathbf r_i
\delta \mathbf r_i is the virtual displacement of the system, which does not have to be consistent with the constraints (in this development)

Substitute the definition for the virtual displacement (differential):[1]:265

\delta \mathbf{r}_i = \sum_{j=1}^m \frac {\partial \mathbf {r}_i} {\partial q_j} \delta q_j
\delta W_a = \sum_{i=1}^n \mathbf {F}_{i} \cdot \sum_{j=1}^m \frac {\partial \mathbf {r}_i} {\partial q_j} \delta q_j

Using the distributive property of multiplication over addition and the associative property of addition, we have[1]:265

\delta W_a = \sum_{j=1}^m \sum_{i=1}^n \mathbf {F}_{i} \cdot \frac {\partial \mathbf {r}_i} {\partial q_j} \delta q_j.

From this form, we can see that the generalized applied forces are then defined by[1]:265

Q_j = \sum_{i=1}^n \mathbf {F}_{i} \cdot \frac {\partial \mathbf {r}_i} {\partial q_j}.

Thus, the virtual work due to the applied forces is[1]:265

\delta W_a = \sum_{j=1}^m Q_j \delta q_j.

[edit] References

  1. ^ a b c d e Torby, Bruce (1984). "Energy Methods", Advanced Dynamics for Engineers, HRW Series in Mechanical Engineering (in English). United States of America: CBS College Publishing. ISBN 0-03-063366-4.