virtual work

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Constraint forces

A (holonomic) constraint force \mathbf{F}^{(c)}\, acts perpendicularly to an allowable displacement \delta\mathbf{r}_i\,. For if f(\mathbf{r}_i) = C\, then along some path \mathbf{r}_i = \mathbf{r}_i(s)\,,

\frac{df}{ds} = \frac{\partial f}{\partial \mathbf{r}_i} \cdot \frac{d\mathbf{r}_i}{d s}=0\,,

and since \mathbf{F}^{(c)}_i = \lambda_i \frac{\partial f}{\partial \mathbf{r}_i}\, is exactly the force of constraint, it follows that \mathbf{F}^{(c)}_i \cdot \delta\mathbf{r}_i = 0\,. Therefore we have the following important result:

\delta W^{(c)}_i = \mathbf{F}^{(c)}_i \cdot \delta\mathbf{r}_i = 0\,,

i.e.,

holonomic constraint forces do no virtual work.

Principle of virtual work

For a system in static equilibrium with one or more holonomic constraints,

\sum_i \mathbf{F}_i = 0\,.

We can ignore internal forces, as these occur in pairs, and decompose the force \mathbf{F}_i\, into an applied force and a constraint force:

\sum_i \left( \mathbf{F}^{(a)}_i + \mathbf{F}^{(c)}_i\right) = 0\,.

Consider the virtual work when the system is in equilibrium:

\delta W = \left(\sum_i \mathbf{F}^{(a)}_i + \mathbf{F}^{(c)}_i\right)\cdot\delta\mathbf{r}_i = 0\,.

Since holonomic constraint forces do no virtual work, we see that

\delta W = \sum_i \mathbf{F}^{(a)}_i = 0\,,

i.e.,

applied forces in static equilibrium do no virtual work.

See also


back to virtual displacement
back to Constraints
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