Generalized forces

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, $delta\; W\_a$, 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.cite book |last=Torby |first=Bruce |title=Advanced Dynamics for Engineers |series=HRW Series in Mechanical Engineering |year=1984 |publisher=CBS College Publishing |location=United States of America |language=English |isbn=0-03-063366-4 |chapter=Energy Methods] rp|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):rp|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 haverp|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 byrp|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 isrp|265

:$delta\; W\_a\; =\; sum\_\{j=1\}^m\; Q\_j\; delta\; q\_j$.

References