Virtual Displacements. Principle of Virtual Work

A virtual displacement is an infinitesimal imagined displacement compatible with the constraints at a given instant. The principle of virtual work provides equilibrium conditions without explicitly determining reactions of ideal constraints.

Virtual displacements

Virtual displacements of system particles are denoted $\delta\vec r_i$. They must satisfy the geometric restrictions imposed by the constraints but are not actual displacements occurring during a time interval $dt$.

Virtual work

The elementary virtual work of forces is:

$$\delta A=\sum_i\vec F_i\cdot\delta\vec r_i.$$

For applied couples, corresponding terms $M\,\delta\varphi$ are added.

Ideal constraints

Constraints are ideal if the total virtual work of their reactions is zero for every admissible virtual displacement. Their reactions can then be omitted from the virtual-work equation.

Principle of virtual work

For equilibrium of a system with ideal constraints, under the usual conditions of stationary constraints, it is necessary and sufficient that:

$$\sum_i\vec F_i^{a}\cdot\delta\vec r_i=0$$

for every admissible virtual displacement, where $\vec F_i^{a}$ are the applied active forces.

Generalized coordinates

If the configuration is described by independent coordinates $q_j$, then $\delta\vec r_i=\sum_j(\partial\vec r_i/\partial q_j)\delta q_j$. Virtual work can be written $\delta A=\sum_jQ_j\delta q_j$, where $Q_j$ are generalized forces.

Advantage of the method

The method is particularly effective for mechanisms with many constraint reactions: for ideal constraints, these reactions are automatically eliminated from the equation.

Example

For a lever that can rotate about a fixed axis, an admissible virtual displacement is described by a small rotation $\delta\varphi$. The condition $\delta A=0$ reduces to zero algebraic sum of moments of the active forces about the axis.

Common mistakes

• confusing virtual displacement with actual displacement during $dt$;
• choosing a displacement incompatible with the constraints;
• discarding work of reactions of nonideal constraints;
• treating $\delta$ as an ordinary time differential.